logical problem solving skills

• Onsite training

3,000,000+ delegates

15,000+ clients

1,000+ locations

• KnowledgePass
• Log a ticket

01344203999 Available 24/7

What is Logical Thinking? A Beginner's Guide 

What is Logical Thinking? A Beginner's Guide: Discover the essence of Logical Thinking in this detailed guide. Unveil its importance in problem-solving, decision-making, and analytical reasoning. Learn techniques to develop this crucial skill, understand common logical fallacies, and explore how Logical Thinking can be applied effectively in various aspects of life and work.

Exclusive 40% OFF

Training Outcomes Within Your Budget!

We ensure quality, budget-alignment, and timely delivery by our expert instructors.

Share this Resource

• Strategic Planning and Thinking Course
• Leadership Skills Training
• Practical Thinking Course
• Attention Management Training
• Introduction to Supervising a Team

Whether you're solving a complex problem, engaging in critical discussions, or just navigating your daily routines, Logical Thinking plays a pivotal role in ensuring that your thoughts and actions are rational and coherent. In this blog, we will discuss What is Logical Thinking in detail, its importance, and its components. You'll also learn about the various ways that make up Logical Thinking and how to develop this essential skill.    

Table of contents  

1)  Understanding Logical Thinking 

2)  Components of Logical Thinking 

3)  Why is Logical Thinking important? 

4)  What are Logical Thinking skills?   

5)  Developing Logical Thinking skills 

6)  Exercises to improve Logical Thinking 

7)  Conclusion 

Understanding Logical Thinking  

Logical Thinking is the capacity to employ reason and systematic processes to analyse information, establish connections, and reach well-founded conclusions. It entails a structured and rational approach to problem-solving and decision-making. 

For example, consider a scenario where you're presented with a puzzle. To logically think through it, you would assess the provided clues, break down the problem into smaller elements, and systematically find potential solutions. You'd avoid hasty or emotion-driven judgments and rely on evidence and sound reasoning to arrive at the correct answer, showcasing the essence of Logical Thinking in problem-solving.

C omponents of Logical Thinking  

After knowing What is Logic al Thinking, let’s move on to the key components of Logical Thinking. Logical Thinking comprises several key components that work together to facilitate reasoned analysis and problem-solving. Here are the following key components of Logical Thinking.  

1)  Deductive reasoning : Deductive reasoning involves drawing specific conclusions from general premises or facts. It's like moving from a broad idea to a more specific conclusion. For example, if all humans are mortal, and Socrates is a human, then you can logically conclude that Socrates is mortal. 

2)   I nductive reasoning : Inductive reasoning is the procedure of forming general conclusions based on specific observations or evidence. It's the opposite of deductive reasoning. For instance, if you observe that the sun has risen every day, you might inductively reason that the sun will rise again tomorrow.  

3)  Causal inference : Causal inference is the ability to identify cause-and-effect relationships between events, actions, or variables. It involves understanding that one event or action can lead to another event as a consequence . In essence, it's the recognition that a specific cause produces a particular effect.  

4)  Analogy : Analogical reasoning or analogy involves drawing similarities and making comparisons between two or more situations, objects, or concepts. It's a way of applying knowledge or understanding from one context to another by recognising shared features or characteristics. Analogical reasoning is powerful because it allows you to transfer what you know in one domain to another, making it easier to comprehend and solve new problems. 

Why is Logical Thinking Important?  

1)  Effective problem-solving : Logical Thinking equips individuals with the ability to dissect complex problems, identify patterns, and devise systematic solutions. Whether it's troubleshooting a technical issue or resolving personal dilemmas, Logical Thinking ensures that problems are approached with a structured and efficient methodology. 

2)  Enhanced decision-making : Making sound decisions is a cornerstone of success in both personal and professional life. Logical Thinking allows individuals to evaluate options, consider consequences, and choose the most rational course of action. This is particularly critical in high-stakes situations. 

3)   Critical thinking : Logical Thinking is at the core of critical thinking. It encourages individuals to question assumptions, seek evidence, and challenge existing beliefs. This capacity for critical analysis fosters a deeper understanding of complex issues and prevents the acceptance of unfounded or biased information. 

4)  Effective communication : In discussions and debates, Logical Thinking helps individuals express their ideas and viewpoints clearly and persuasively. It enables individuals to construct well-structured arguments, provide evidence, and counter opposing views, fostering productive and respectful communication. 

5)  Academic and professional success : Logical Thinking is highly valued in educational settings and the workplace. It allows students to excel academically by tackling challenging coursework and assignments. In the professional world, it's a key attribute for problem-solving, innovation, and career advancement. 

6)  Avoiding Logical fallacies : Logical Thinking equips individuals with the ability to recognise and avoid common logical fallacies such as circular reasoning, straw man arguments, and ad hominem attacks. This safeguards them from being deceived or manipulated by flawed or deceptive arguments. 

Supercharge Your Sales with our Sales Negotiation Skills Course . Check out now and close deals like a pro!  

What are Logical Thinking skills ?  

Logical Thinking skills are cognitive abilities that allow individuals to process information, analyse it systematically, and draw reasonable conclusions. These skills enable people to approach problems, decisions, and challenges with a structured and rational mindset .  

Developing Logical Thinking skills  

Developing strong Logical Thinking skills is essential for improved problem-solving, decision-making, and critical analysis. Here are some key strategies to help you enhance your Logical Thinking abilities.   

1)  Practice critical thinking : Engage in activities that require critical thinking, such as analysing articles, solving puzzles, or evaluating arguments. Regular practice sharpens your analytical skills.  

2)  L earn formal logic : Study the principles of formal logic, which provide a structured approach to reasoning. This can include topics like syllogisms, propositional logic, and predicate logic. 

3)  I dentify assumptions : When faced with a problem or argument, be aware of underlying assumptions. Question these assumptions and consider how they impact the overall reasoning. 

4)  B reak down problems : When tackling complex problems, break them down into smaller, more manageable components. Analyse each component individually before looking at the problem as a whole . 

5)   Seek diverse perspectives : Engage in discussions and debates with people who hold different viewpoints. This helps you consider a range of perspectives and strengthens your ability to construct and counter -arguments. 

6)  Read widely : Reading a variety of materials, from academic articles to literature, exposes you to different modes of reasoning and argumentation. This broadens your thinking and enhances your ability to connect ideas.  

7)  Solve puzzles and brain teasers : Engaging in puzzles, riddles, and brain teasers challenges your mind and encourages creative problem-solving. It's an enjoyable way to exercise your Logical Thinking. 

8)  Develop mathematical skills : Mathematics is a discipline that heavily relies on Logical Thinking. Learning and practising mathematical concepts and problem-solving techniques can significantly boost your logical reasoning skills. 

Unlock Your sales potential with our Soft Skills Training For Sales Professionals . Elevate your sales game today and boost your success – register now!  

Exercises to improve Logical Thinking  

Enhancing your Logical Thinking skills is achievable through various exercises and activities. Here are some practical exercises to help you strengthen your Logical Thinking abilities:  

1)   Sudoku puzzles : Solve Sudoku puzzles, as they require logical deduction to fill in the missing numbers.  

2)   Crossword puzzles : Crosswords challenge your vocabulary and logical word placement.  

3)  Brain teasers : Engage in brain teasers and riddles that encourage creative problem-solving.  

4)  Chess and board games : Play strategic board games like chess, checkers, or strategic video games that require forward thinking and planning.  

5)  Logical argumentation : Engage in debates or discussions where you must construct reasoned arguments and counter opposing viewpoints.  

6)  Coding and programming : Learn coding and programming languages which promote structured and Logical Thinking in problem-solving. 

7)  Mathematical challenges : Solve mathematical problems and equations, as mathematics is inherently logical.  

8)   Mensa puzzles : Work on Mensa puzzles, which are designed to test and strengthen Logical Thinking skills. 

9)  Logic games : Play logic-based games like Minesweeper or Mastermind.  

10)   Logical analogy exercises : Practice solving analogy exercises, which test your ability to find relationships between words or concepts.  

11)  Visual logic puzzles : Tackle visual logic puzzles like nonograms or logic grid puzzles. 

12)  Critical reading : Read books, articles, or academic papers and critically analyse the arguments and evidence presented. 

13)  Coding challenges : Participate in online coding challenges and competitions that require logical problem-solving in coding. 

14)  Scientific method : Conduct simple science experiments or projects, applying the scientific method to develop hypotheses and draw logical conclusions.  

15)   Poker or card games : Play card games like poker, where you must strategi se and make logical decisions based on probabilities and information. 

16)  Analyse real-world situations : Analyse real-world situations or news stories, evaluating the information, causes, and potential consequences. 

These exercises will help you practice and enhance your Logical Thinking skills in a fun and engaging way, making them an integral part of your problem-solving and decision-making toolkit. 

Take your sales to the next level! Elevate your skills with our Sales Training . Join us today!  

Concluson  

In this blog, we have discussed What is Logical Thinking, its importance, its components and ways to improve this skill. When you learn how to think logically, you start gathering each and every information as much as possible, analyse the facts, and methodically choose the best way to go forward with your decision. Logical Thinking is considered the most important tool in brainstorming ideas, assessing issues and finding solutions. 

Master the art of effective meetings. Elevate your Meeting Skills for success. Get started now!  

Frequently Asked Questions

Upcoming business skills resources batches & dates.

Fri 26th Apr 2024

Fri 24th May 2024

Fri 28th Jun 2024

Fri 26th Jul 2024

Fri 23rd Aug 2024

Fri 27th Sep 2024

Fri 25th Oct 2024

Fri 22nd Nov 2024

Fri 27th Dec 2024

Get A Quote

WHO WILL BE FUNDING THE COURSE?

My employer

By submitting your details you agree to be contacted in order to respond to your enquiry

• Business Analysis
• Lean Six Sigma Certification

Share this course

Our biggest spring sale.

We cannot process your enquiry without contacting you, please tick to confirm your consent to us for contacting you about your enquiry.

By submitting your details you agree to be contacted in order to respond to your enquiry.

We may not have the course you’re looking for. If you enquire or give us a call on 01344203999 and speak to our training experts, we may still be able to help with your training requirements.

Or select from our popular topics

• ITIL® Certification
• Scrum Certification
• Change Management Certification
• Business Analysis Courses
• Microsoft Azure Certification
• Microsoft Excel Courses
• Microsoft Project
• Explore more courses

Press esc to close

Fill out your  contact details  below and our training experts will be in touch.

Fill out your   contact details   below

Thank you for your enquiry!

One of our training experts will be in touch shortly to go over your training requirements.

Back to Course Information

Fill out your contact details below so we can get in touch with you regarding your training requirements.

* WHO WILL BE FUNDING THE COURSE?

Preferred Contact Method

No preference

Back to course information

Fill out your  training details  below

Fill out your training details below so we have a better idea of what your training requirements are.

HOW MANY DELEGATES NEED TRAINING?

HOW DO YOU WANT THE COURSE DELIVERED?

Online Instructor-led

Online Self-paced

WHEN WOULD YOU LIKE TO TAKE THIS COURSE?

Next 2 - 4 months

WHAT IS YOUR REASON FOR ENQUIRING?

Looking for some information

Looking for a discount

I want to book but have questions

One of our training experts will be in touch shortly to go overy your training requirements.

Your privacy & cookies!

Like many websites we use cookies. We care about your data and experience, so to give you the best possible experience using our site, we store a very limited amount of your data. Continuing to use this site or clicking “Accept & close” means that you agree to our use of cookies. Learn more about our privacy policy and cookie policy cookie policy .

We use cookies that are essential for our site to work. Please visit our cookie policy for more information. To accept all cookies click 'Accept & close'.

Explore Jobs

• Jobs Near Me
• Remote Jobs
• Full Time Jobs
• Part Time Jobs
• Entry Level Jobs
• Work From Home Jobs

Find Specific Jobs

• $15 Per Hour Jobs
• $20 Per Hour Jobs
• Hiring Immediately Jobs
• High School Jobs
• H1b Visa Jobs

Explore Careers

• Business And Financial
• Architecture And Engineering
• Computer And Mathematical

Explore Professions

• What They Do
• Certifications
• Demographics

Best Companies

• Health Care
• Fortune 500

Explore Companies

• CEO And Executies
• Resume Builder
• Career Advice
• Explore Majors
• Questions And Answers
• Interview Questions

The Most Important Logical Thinking Skills (With Examples)

• Logical Skills
• Promotional Skills
• Bookkeeping Skills
• Typing Skills
• Sales Skills
• Science And Math Skills
• Physical Strength And Dexterity Skills
• Customer Service Skills
• Instructional Skills
• Logical-Thinking Skills
• Mechanical Skills
• Memorization Skills
• Motivational Skills
• Artistic Talent

Find a Job You Really Want In

Logical thinking skills like critical-thinking, research, and creative thinking are valuable assets in the workplace. These skills are sought after by many employers, who want employees that take into account facts and data before deciding on an important course of action. This is because such solutions will ensure the organization’s processes can continue to operate efficiently.

So, if you’re a job seeker or employee looking to explore and brush up on your logical thinking skills, you’re in luck. This article will cover examples of logical thinking skills in the workplace, as well as what you can do to showcase those skills on your resume and in interviews.

Key Takeaways:

Logical thinking is problem solving based on reasoning that follows a strictly structured progression of analysis.

Critical thinking, research, creativity, mathematics, reading, active listening, and organization are all important logical thinking skills in the workplace.

Logical thinking provides objectivity for decision making that multiple people can accept.

Deduction follows valid premises to reach a logical conclusion.

It can be very helpful to demonstrate logical thinking skills at a job interview.

What is logical thinking?

10 examples of logical thinking skills, examples of logical thinking in the workplace, what is deductive reasoning, logical thinking in a job interview, logical thinking skills faq, final thoughts.

• Sign Up For More Advice and Jobs

Logical thinking is the ability to reason out an issue after observing and analyzing it from all angles . You can then form a conclusion that makes the most sense. It also includes the ability to take note of reactions and feedback to aid in the formation of the conclusion.

Logical thinking skills enable you to present your justification for the actions you take, the strategies you use, and the decisions you make. You can easily stand in front of your clients, peers, and supervisors and defend your product, service, and course of action if the necessity arises.

Logical thinking is an excellent way of solving complex problems. You can break the problem into smaller parts; solve them individually in a sequence, then present the complete solution. However, it is not infallible.

So, when a problem in the workplace feels overwhelming, you may want to think about it logically first.

Logical thinking skills are a skill set that enables you to reason logically when solving problems. They enable you to provide well-reasoned answers to any issues that arise. They also empower you to make decisions that most people will consider rational.

Critical-thinking skills. If you are a critical thinker, then you can analyze and evaluate a problem before making judgments. You need to improve your critical thinking process to become a logical thinker.

Your critical thinking skills will improve your ability to solve problems. You will be the go-to employee concerning crises. People can rely on you to be reasonable whenever an issue arises instead of letting biases rule you.

Research skills. If you are a good researcher , then you can search and locate data that can be useful when presenting information on your preferred subject.

The more relevant information you have about a particular subject, the more accurate your conclusions are likely to be. The sources you use must be reputable and relevant.

For this reason, your ability to ferret out information will affect how well you can reason logically.

Creative thinking skills. If you are a creative thinker , then you can find innovative solutions to problems.

You are the kind of person that can think outside the box when brainstorming ideas and potential solutions. Your thinking is not rigid. Instead, you tend to look at issues in ways other people have not thought of before.

While logical thinking is based on data and facts, that doesn’t mean it is rigid. You can creatively find ways of sourcing that data or experimenting so that you can form logical conclusions. Your strategic thinking skills will also help enable you to analyze reactions or collect feedback .

Mathematical skills. If you are skilled in mathematics , then you can work well with numbers and represent mathematical ideas using visual symbols. Your brain must be able to compute information.

Business is a numbers game. That means you must have some knowledge of mathematics. You must be able to perform basic mathematical tasks involving addition, subtractions, divisions, multiplications, etc.

So, to become a logical thinker, you must be comfortable working with numbers. You will encounter them in many business-related complex problems. And your ability to understand them will determine whether you can reach an accurate logical conclusion that helps your organization.

Reading skills. If you are a good reader , then you can make sense of the letters and symbols that you see. Your ability to read will determine your competency concerning your logical thinking and reasoning skills.

And that skill set will come in handy when you are presented with different sets of work-related statements from which you are meant to conclude. Such statements may be part of your company policy, technical manual, etc.

Active listening skills. Active listening is an important communication skill to have. If you are an active listener, then you can hear, understand what is being said, remember it, and respond to it if necessary.

Not all instructions are written. You may need to listen to someone to get the information you need to solve problems before you write it down. In that case, your active listening skills will determine how well you can remember the information so that you can use it to reason things out logically.

Information ordering skills. If you have information ordering skills, then you can arrange things based on a specified order following the set rules or conditions. These things may include mathematical operations, words, pictures, etc.

Different organizations have different business processes. The workflow in one organization will be not similar to that of another organization even if both belong to the same industry.

Your ability to order information will depend on an organization’s culture . And it will have a major impact on how you can think and reason concerning solutions to your company problems.

If you follow the wrong order, then no matter how good your problem-solving techniques are your conclusions may be wrong for your organization.

Persuasion skills. Logical thinking can be useful when persuading others, especially in the workplace.

For example, lets say one of your co-workers wants to take a project in an impulsive direction, which will increase the budget. However, after you do your research, you realize a budget increase would be impossible.

You can then use your logical thinking skills to explain the situation to your co-worker , including details facts and numbers, which will help dissuade them from making an uninformed decision.

Decision making skills. Decision making skills go hand and hand with logical thinking, as being able to think logically about solutions and research topics will make it far easier to make informed decisions.

After all, no one likes making a decision that feels like a shot in the dark, so knowing crucial information about the options aviable to you, and thinking about them logically, can improve your confidence around decision making.

Confidence skills. Confidence that stems from an emotional and irrational place will always be fragile, but when you have more knowledge available to you through logical thinking, you can be more confident in your confidence skills.

For instance, if an employee asked you to answer an important question, you will have a lot more confidence in your answer if you can think logically about it, as opposed to having an air of uncertainty.

To improve your logic skills, it would be wise to practice how to solve problems based on facts and data. Below are examples of logical thinking in the workplace that will help you understand this kind of reasoning so that you can improve your thinking:

The human resource department in your organization has determined that leadership skills are important for anyone looking to go into a senior management position. So, it decides that it needs proof of leadership before hiring anyone internally. To find the right person for the senior management position , every candidate must undertake a project that involves a team of five. Whoever leads the winning team will get the senior managerial position.

This example shows a logical conclusion that is reached by your organization’s human resource department. In this case, your HR department has utilized logical thinking to determine the best internal candidate for the senior manager position.

It could be summarized as follows:

Statement 1: People with excellent leadership skills that produce winning teams make great senior managers. Statement 2: Candidate A is an excellent leader that has produced a winning team. Conclusion: Candidate A will make an excellent senior manager .

A marketing company researches working women on behalf of one of their clients – a robotics company. They find out that these women feel overwhelmed with responsibilities at home and in the workplace. As a result, they do not have enough time to clean, take care of their children, and stay productive in the workplace. A robotics company uses this research to create a robot cleaner that can be operated remotely . Then they advertise this cleaner specifically to working women with the tag line, “Working women can do it all with a little bit of help.” As a result of this marketing campaign, their revenues double within a year.

This example shows a logical conclusion reached by a robotics company after receiving the results of marketing research on working women. In this case, logical thinking has enabled the company to come up with a new marketing strategy for their cleaning product.

Statement 1: Working women struggle to keep their homes clean. Statement 2: Robot cleaners can take over cleaning duties for women who struggle to keep their homes clean. Conclusion: Robot cleaner can help working women keep their homes clean.

CalcX. Inc. has created a customer survey concerning its new finance software. The goal of the survey is to determine what customers like best about the software. After reading through over 100 customer reviews and ratings, it emerges that 60% of customers love the new user interface because it’s easy to navigate. CalcX. Inc. then decides to improve its marketing strategy. It decides to train every salesperson to talk about the easy navigation feature and how superior it is to the competition. So, every time a client objects to the price, the sales rep could admit that it is expensive, but the excellent user interface makes up for the price. At the end of the year, it emerges that this strategy has improved sales revenues by 10%.

The above example shows how logical thinking has helped CalcX. Sell more software and improve its bottom line.

Statement 1: If the majority of customers like a particular software feature, then sales reps should use it to overcome objections and increase revenues. Statement 2: 60% of the surveyed customers like the user interface of the new software, and; they think it makes navigation easier. Conclusion: The sales reps should market the new software’s user interface and the fact that it is easy to navigate to improve the company’s bottom line.

A political candidate hires a focus group to discuss hot-button issues they feel strongly about. It emerges that the group is torn on sexual reproductive health issues, but most support the issue of internal security . However, nearly everyone is opposed to the lower wages being paid due to the current economic crisis. Based on the results of this research, the candidate decides to focus on improving the economy and security mechanisms in the country. He also decides to let go of the sexual productive health issues because it would potentially cause him to lose some support.

In this case, the political candidate has made logical conclusions on what topics he should use to campaign for his seat with minimal controversies so that he doesn’t lose many votes.

This situation could be summarized as follows:

Statement 1: Most people find sexual reproductive health issues controversial and cannot agree. Statement 2: Most people feel that the internal security of the country is in jeopardy and something should be done about it. Statement 3: Most people want higher wages and an improved economy. Statement 4: Political candidates who want to win must avoid controversy and speak up on things that matter to people. Conclusion: To win, political candidates must focus on higher wages, an improved economy, and the internal security of the country while avoiding sexual reproductive health matters.

Deductive reasoning is an aspect of logical reasoning. It is a top-down reasoning approach that enables you to form a specific logical conclusion based on generalities. Therefore, you can use one or more statements, usually referred to as premises, to conclude something.

For example:

Statement 1: All mothers are women Statement 2: Daisy is a mother. Conclusion: Daisy is a woman.

Based on the above examples, all mothers are classified as women, and since Daisy is a mother, then it’s logical to deduce that she is a woman too.

It’s worth noting though, that deductive reasoning does not always produce an accurate conclusion based on reality.

Statement 1: All caregivers in this room are nurses. Statement 2: This dog, Tom, is a caregiver . Conclusion: This dog, Tom, is a nurse .

From the above example, we have deduced that Tom, the dog, is a nurse simply because the first statement stated that all caregivers are nurses. And yet, in reality, we know that dogs cannot be nurses. They do not have the mental capacity to become engaged in the profession.

For this reason, you must bear in mind that an argument can be validly based on the conditions but it can also be unsound if some statements are based on a fallacy.

Since logical thinking is so important in the workplace, most job interviewers will want to see you demonstrate this skill at the job interview. It is very important to keep in mind your logical thinking skills when you talk about yourself at the interview.

There are many ways in which an interviewer may ask you to demonstrate your logical thinking skills. For example:

You may have to solve an example problem. If the interviewer provides you a problem similar to one you might find at your job, make sure to critically analyze the problem to deduce a solution.

You may be asked about a previous problem or conflict you had to solve. This classic question provides you the opportunity to show your skills in action, so make sure to highlight the objectivity and logic of your problem solving.

Show your logic when talking about yourself. When given the opportunity to talk about yourself, highlight how logic comes into play in your decision making. This could be in how you picked the job position, why you choose your career or education, or what it is about yourself that makes you a great candidate.

Why is it important to think logically?

It’s important to think logically because it allows you to analyze a situation and come up with a logical solution. It allows for you to reason through the important decisions and solve problems with a better understanding of what needs to be done. This is necessary for developing a strong career.

Why is logic important?

Logic is important because it helps develop critical thinking skills. Critical thinking skills are important because they help you analyze and evaluate a problem before you make a decision. It also helps you improve your problem-solving skills to allow you to make better decisions.

How do you improve your logical thinking skills?

When improving your logical thinking skills make sure you spend time on a creative hobby and practice questioning. Creative hobbies can help reduce stress levels, and lower stress leads to having an easier time focusing on tasks and making logical thinking. Creative hobbies can include things like drawing, painting, and writing.

Another way to improve your logical thinking is to start asking questions about things. Asking questions allows for you to discover new things and learn about new topics you may not have thought about before.

What are logical thinking skills you need to succeed at work?

There are many logical thinking skills you need to succeed in the workplace. Our top four picks include:

Observation

Active Listening

Problem-solving

Logical thinking skills are valuable skills to have. You need to develop them so that you can become an asset to any organization that hires you. Be sure to include them in your resume and cover letter .

And if you make it to the interview, also ensure that you highlight these skills. You can do all this by highlighting the career accomplishments that required you to use logical thinking in the workplace.

It’s Your Yale – Consider Critical Thinking Skills to Articulate Your Work Quality

How useful was this post?

Click on a star to rate it!

Average rating / 5. Vote count:

No votes so far! Be the first to rate this post.

Roger Raber has been a content writer at Zippia for over a year and has authored several hundred articles. Having retired after 28 years of teaching writing and research at both the high school and college levels, Roger enjoys providing career details that help inform people who are curious about a new job or career. Roger holds a BA in English from Cleveland State University and a MA from Marygrove college.

Recent Job Searches

• Registered Nurse Jobs Resume Location
• Truck Driver Jobs Resume Location
• Call Center Representative Jobs Resume Location
• Customer Service Representative Jobs Resume
• Delivery Driver Jobs Resume Location
• Warehouse Worker Jobs Resume Location
• Account Executive Jobs Resume Location
• Sales Associate Jobs Resume Location
• Licensed Practical Nurse Jobs Resume Location
• Company Driver Jobs Resume

Related posts

What Are Compassion Skills? (With Examples)

What Is Work-Life Balance? (And How To Improve It)

The Most Important Science And Math Skills (With Examples)

Mechanical Engineer Skills For A Resume

• Career Advice >
• Resume Skills >
• Defined Skills >
• Logical Thinking Skills

• Search the site

How to Think Logically (And Permanently Solve Serious Problems)

Anthony Metivier | March 5, 2024 | Podcast , Thinking

Podcast: Download

Subscribe: Apple Podcasts | RSS

Yes, but not so fast.

You want to make sure you’re using the right kinds of logic for the problems at hand.

For example, you might need a non-classical logic instead of classical logic to approach a particular problem.

You see, logical thinkers do what I’m doing now:

They put the brakes on when they encounter problems and start to spin those problems around.

Why? Because logic itself often involves digging deeper and analyzing different perspectives.

For example, one of the forms of logical thinking you’re about to discover would have you instantly ask… 

Is there more than one kind of logic for solving life’s problems quickly? Or can I explore alternatives outside of logic? 

A logical thinker might do the same thing to the very idea of a “problem” itself.

This is done by “mentally rotating” the topic at hand and seeing how it might in fact not be a problem at all.

It might be a path to a solution. 

How to Think Logically: 9 Ways to Improve Your Logical Thinking Skills

At the end of the day, using the right form of logic is more about the best possible solution than the problem, but we do need to make sure we understand the problem first.

If you’ve listened to Elon Musk talk about first principles thinking, that’s a form of logic he’s using to help humans thrive on distant planets after earth dies. And communicate better here on our precious planet while we still can.

Those are real problems, and the right forms of logic are needed.

The best part?

There are a whole lot more ways to think logically to solve global and personal problems alike, so let’s get started 

One: Take A Deep Dive Into Logical Thinking

Improving logical reasoning begins by knowing the types of logic at your disposal.

Exploring the history of logic is well worth your time because it will help you see how humans discovered these principles and refined them over time through practice . 

As you’ll soon discover, many cultures have identified and used logical forms such as:

• Philosophical logic
• Informal logic
• Formal logic
• Modal logic
• Mathematical logic
• Paraconsistent logic
• Semantic logic
• Inferential logic
• Systematic logic

Related to this, you have the difference between what philosopher Elijah Millgram calls theoretical reasoning vs. practical reasoning. The first involves figuring out the facts, the second is the process of determining what courses of action to take based on what is ideally a set of accurate facts.

Now, usually what people who want to think more logically are actually after is the first category, or philosophical logic . This is also called “reasoning” and includes the skills of:

• Causal inference

Deductive reasoning is what we think of when we think of Sherlock Holmes , who builds his cases by arguing from general principles. He uses these to describe a specific series of events and solve various mysteries. 

Inductive reasoning is essentially the reverse of this process. Instead of using general principles to arrive at specifics, you use specific details to generalize. For example, you might notice that I post on this blog almost every week, and use inductive reasoning to logically determine that I am a consistent blogger. 

Causal inference helps you understand the scientific reason why and how things change. For example, why are you reading this article? I can logically infer that it is because you want to experience change and become a better thinker.

(Or maybe you want to experience more, such as all of these 11 benefits of critical thinking .)

Analogy or analogical reasoning involves making comparisons based on established examples or models. 

For example, we know that nearly every memory champion openly admits that they have normal memory that doesn’t work especially well without using mnemonic devices . By analogy, we can infer that any person with average memory abilities can become a memory champion. 

How long should you study logic? I’d suggest at least 90 days so you can get the bird’s eye overview and enough of the granular details.

Logical thinkers always make sure they have a bird’s eye view and the granular details at the same time.

Plus, as you’ll soon discover on this page, there are other fields you can read from to improve your logical thinking.

Two: Understand the Problems You’re Trying to Solve Deeply

Ever taken a quiz and realized you answered before thinking about the question? You could have gotten it correctly, but your impulses took over and you lost precious points. 

It’s not that you were being illogical. You just didn’t take the time to fully understand the question, and the reason why you failed to do so might have been logical. For example, from one perspective, in some contexts it might be perfectly logical to rush through an exam if you’re running out of time. 

But generally, we want to be sure that we deeply understand the problems we face. That is why Abraham Lincoln famously said:

“Give me six hours to chop down a tree and I will spend the first four sharpening the axe.” 

Lincoln is using an analogy here, one in which the “axe” stands in as an analogy. It speaks to spending the time needed to make sure you’re using the right tools for the job. Moreover, you make sure they are in top shape before you use them.

All the more reason to learn more about the different forms of logic. It will put more tools in your tool box and enable you to keep them sharp.

Here are 9 more critical thinking strategies to help you keep your axe sharp.

Three: Learn More About Language

A lot of people struggle to think logically because they don’t understand enough about what words mean.

Logical thinking involves nuance, so the more you know about words and their meanings, the greater mental precision in decision-making you’ll enjoy. 

To improve, here’s how to memorize vocabulary . It will help you add more meanings to words and add more definitions to those you already know. Learning word origins and how prefixes and suffixes work will help you too.

On top of learning more about words and their meanings, learning about language and logic will help, such as studying syllogisms and logical fallacies .

Go deep and learn as much as you can about fallacies so you really know your stuff. It’s easy to fall into thinking traps if you don’t.

For example, some people like to accuse others of slippery slope fallacy, without realizing that there are actually six kinds of this fallacy. 

If you want to think logically, it pays to be thorough. That’s why we’ll focus on thoroughness next.

Four: Read Quickly Without Sacrificing Thoroughness

Improving vocabulary is huge for improving logical thinking, and it will help you read faster .

But to improve your logical skills over time, you need to read thoroughly. 

I suggest you read bigger books and more of them, starting with the key textbooks in your field of interest. 

By going for the biggest and most authoritative books, you’ll be reading more logically .

Establishing foundations in your mind by reading authoritative textbooks will help you develop pattern recognition. This skill leads to faster use of the logical forms of inference we discussed in the first part of this article.

Five: Listen To Long Form Content

Short form content is causing people to make snap judgments and interrupt people before they’ve heard the full story. Logical thinkers protect themselves by practicing listening for long periods of time.

Not only is it helpful to read longer books, but you’ll learn to think much more logically when you listen to logical people think out loud.

Debates are a great way to do this and the Internet makes it possible to find many of them. 

It’s important to pay attention to both sides of the argument, however.

As you listen, practice thinking yourself by mentally rehearsing the evidence you would provide in support of your views. Also think about how you would respond.

Another tip:

Notice the holes in the arguments proposed by the debaters and list out the ways you would fill in the gaps. 

And if you want to remember more of what goes on during debates, Memory Palace Mastery is here to help.

Six: Expand Your Competence Using Multiple Media

I’ve just suggested that you experience “thinking out loud” and model it yourself.

But you’ll want to go beyond completing logical exercises in your mind. You should also:

To practice speaking logically, engage in as many discussions as you can about real problems. Sure, there’s a place for talking about movies and sports. But if you want to know how to think more logically, you’ve got to practice it yourself in real time.

Writing is always key for developing logical thinking, so I suggest you keep a journal. This simple practice will help you see your own thinking process and improve it over time. 

Combined, you will have many opportunities for self-analysis. If you can record your conversations and look at transcripts of them, all the better. 

Seven: Ask Better Questions

A lot of us ask the typical W5 questions and let it rest at that:

But to practice thinking logically, you want to go beyond these questions. Ask in addition to these questions: 

• According to whom?
• According to what precedent?
• Where isn’t this true?
• When hasn’t this been the case?

There are many variations on these questions you can ask, and I cover more along these lines in our community’s post on how to think faster .

Eight: Learn Game Theory

One of the lesser known ways to learn logical thinking is to study games and metagames. 

In brief, game theory studies areas of competition where people regularly make decisions. These decisions are influenced by other people in the area and in turn influence others. 

By modeling the ways people interact in competitive contexts, you can learn to think more logically and avoid cognitive biases that harm your performance in life. 

You’ll enjoy avoiding many problems because game theory helps train your mind to anticipate the possible outcomes of various decisions. By thinking through consequences in advance, you save yourself a lot of trouble.

Note: You can perform game theory on the past as well by thinking through what would have happened had people acted differently. This philosophical approach is called working through the counterfactuals of a historical situation and can be used on your personal life and large groups.

Some people think that game theory has limited value for everyday life, but I don’t think they’re being… logical about that. We all find ourselves in situations where we are influenced to act in certain ways and understanding these pressures will help you respond in much better ways. 

A key example is by using the Monty Hall Problem or Three Door Problem to make decisions . 

Logical exercises like The Monty Hall Problem help you think through what to do when you face choices in life.

Some people squabble over whether it is in fact logical to use this problem in life, but I can attest to its value.

For example, when I see an opportunity to do something different and feel like I want to default to my previous choices, I bring this game theoretical example to mind and remind myself to travel the “path less travelled.” 

Is the math on my side?

I think so, because I’ve gone on many adventures that logic dictates could not have happened had I chosen to stick with the same thing.

To learn more about these situations, check out the stories I share in The Victorious Mind: How to Master Memory, Meditation and Mental Well-Being .

Nine: Use Rules And Embrace Limitations

 I didn’t use to like rules. In some ways I still don’t.

But one day I was enjoying dinner with Tony Buzan, memory expert, mind map innovator and co-founder of the World Memory Championships. 

I told him about how I sometimes would switch memory systems while under time trials for numbers and playing cards. 

He said, “The rules will set you free.” 

Tony Buzan with Anthony Metivier and Phil Chambers

This is important because life, as in memory training, often gives us the opportunity to use multiple techniques.

For example, when remembering numbers, we could choose the Dominic System or the Major System , though as I discovered, it doesn’t pay off to switch from one to the other during a time trial.

But by willing to limit ourselves and stick to the “rules” of just one system, we can improve our performance.

This is true in life too, where you can learn certain rules of thumb and stick to them.

To take another example, learning the logic of Chip and Dan Heath’s W.R.A.P. technique and practicing it over time has been a tremendously helpful problem solving model for me. In fact, it’s probably the approach that has improved my critical thinking the fastest .

In fact, it’s so helpful, it is “illogical” to forget not to use it when making decisions. That’s why I memorized it using a special memory technique called ars combinatoria , something that was very important in the history of how logical thinking developed.

What rules of thumb that help you “limit” yourself to a productive form of thinking and decision making can you adopt? 

Thinking Logically Is A Rewarding Process To Enjoy For Life

Have you enjoyed learning these nine ways to improve your logical thinking? 

I hope so and hope you will make practicing some of these approaches a personal hobby.

You can easily practice logical thinking while meditating or working with an alternative to logic like Zen.

As a final tip, it would only be logical for me to recommend the opposite of logic. 

You see, there are practices like Zen which evolved to help us see and experience the limits of logic. Zen turns language against itself to help us experience mental relief from the problems we think so hard about. 

One of the best critical thinking books that situates the topic in the larger realm of computational thinking for both humans and machines is Gödel Escher Bach . For a collection of koans to explore, The Gateless Gate by Mumon is an interesting source.

I mention the opposite of logic not only because it is logical to do so. To fully experience the rewards of logical thinking, you need to be able to step outside of thinking altogether. 

Questioning deeply is not enough. We need to question the process of questioning itself as a lifelong learning habit.

So on that note, let the questioning begin. Let me know which of these ways to improve your thinking you’re going to try out and what questions about logic do you still have?

Related Posts

Developing independent thinking doesn't have to be a hard task. These tips reveal how easily…

Real life critical thinking examples are hard to come by. This post gives you 7…

Most critical thinking quotes have nothing to do with the critical part. These 5 quotes…

Most barriers to critical to critical thinking are easily removed. Read this post to boost…

Last modified: March 5, 2024

About the Author / Anthony Metivier

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Save my name, email, and website in this browser for the next time I comment.

I accept the Privacy Policy

POPULAR POSTS

• How to Remember Things: 18 Proven Memory Techniques
• How to Build a Memory Palace: Proven Memory Palace Technique Approach
• How to Learn a New Language Fast: A Step-by-Step Guide
• Digital Amnesia: 5 Ways To Stop The Internet From Ruining Your Memory
• 15 Brain Exercises & Memory Exercises For Rapid Remembering
• The Memory Palace of Matteo Ricci
• Effortless Conference Interpreting: 7 Career-Making Memory Tips
• How To Teach Your Kids Memory Techniques
• 5 Unconventional, But Proven Note Taking Techniques

Recent Posts

• Do Brain Games Work? Here’s A Better Way To Fix Your Memory
• From Life Crushing Anxiety To PERFECT Calm: USA Memory Champ John Graham’s BEST Stress-Busting Tips
• How To Prepare For A Debate: 8 Ways To Win & Build Reputation
• 3 Polymath Personality Traits Masterful People Nurture & Amplify
• Lifelong Learning: The Benefits in Life and in Today’s Job Market

WANT TO LEARN SIMPLE EVERYDAY THINGS WITHOUT FORGETTING?

Enter your email below to get instant access to my FREE course that gives you a proven step-by-step process for remembering anything you want . You'll discover how to:

• Speak any language fluently
• Recall complicated formulas, math equations, or numbers.
• Master the technical terms for your field of work or study.
• Recite poetry, jokes, and even long speeches word-for-word
• Quickly absorb the most important ideas from books, textbooks, or lectures...

Unlock your natural ability to learn and remember anything 3x faster now!

Pay with Confidence

Memory Courses

• The Magnetic Memory Method Masterclass
• The Magnetic Memory Method Masterplan
• How To Learn And Memorize The Vocabulary Of Any Language
• How To Learn And Memorize Poetry
• How To Memorize Names and Faces
• How to Memorize Math, Numbers, Simple Arithmetic and Equations
• How to Remember Your Dreams

Quick Links

• Testimonials
• Privacy Policy
• Privacy Tools
• Terms of Service

P.O. Box 933 Mooloolaba, QLD 4557 Australia

Memory Improvement Blog

• How to Build A Memory Palace
• Eidetic Memory
• Episodic Memory
• Photographic Memory
• Improve Memory for Studying
• Memorization Techniques
• How to Memorize Things Fast
• Brain Exercises

• The Magnetic Memory Method Podcast
• Memory Improvement Resources for Learning And Remembering

GET YOUR FREE 4-PART VIDEO MINI-COURSE

Improve your memory in record time!  Just enter your email below now to subscribe.

• How to learn anything at lightning fast speeds
• Improve your short and long term memory almost overnight
• Learn any language with and recall any information with ease

How to Train Your Problem-Solving Skills

From the hiccups that disrupt your morning routines to the hurdles that define your professional paths, there is always a problem to be solved. 

The good news is that every obstacle is an opportunity to develop problem-solving skills and become the best version of yourself. That’s right: It turns out you can get better at problem-solving, which will help you increase success in daily life and long-term goals.  

Read on to learn how to improve your problem-solving abilities through scientific research and practical strategies.

Understanding Problem-Solving Skills

You may be surprised to learn that your problem-solving skills go beyond just trying to find a solution. Problem-solving skills involve cognitive abilities such as analytical thinking, creativity, decision-making, logical reasoning, and memory. 

Strong problem-solving skills boost critical thinking, spark creativity, and hone decision-making abilities. For you or anyone looking to improve their mental fitness , these skills are necessary for career advancement, personal growth, and positive interpersonal relationships. 

Core Components of Problem-Solving Skills Training

To effectively train your problem-solving skills, it’s important to practice all of the steps required to solve the problem. Think of it this way: Before attempting to solve a problem, your brain has already been hard at work evaluating the situation and picking the best action plan. After you’ve worked hard preparing, you’ll need to implement your plan and assess the outcome by following these steps:  

• Identify and define problems: Recognizing and clearly articulating issues is the foundational step in solving them.
• Generate solutions: Employing brainstorming techniques helps you develop multiple potential solutions.
• Evaluate and select solutions: Using specific criteria to assess solutions helps you choose the most effective one.
• Implement solutions: Developing and executing action plans, including preparing for potential obstacles, guides you to positive outcomes.
• Review and learn from outcomes: Assessing the success of solutions and learning from the results for future improvement facilitates future success. 

Strategies for Developing Problem-Solving Skills

There are many practical exercises and activities that can improve problem-solving abilities.

Cultivate a Problem-Solving Mindset

• Adopt a growth mindset: A growth mindset involves transforming phrases like “I can’t” into “I can’t yet.” Believing in the capacity to improve your skills through effort and perseverance can lead to greater success in problem-solving.
• Practice mindfulness: Mindfulness can enhance cognitive flexibility , allowing you to view problems from multiple perspectives and find creative solutions.

Enhance Core Cognitive Skills 

• Strengthen your memory: Engage in activities that challenge your memory since accurately recalling information is crucial in problem-solving. Techniques such as mnemonic devices or memory palaces can be particularly effective.
• Build your critical thinking: Regularly question assumptions, evaluate arguments, and engage in activities that require reasoning, such as strategy games or debates.

Apply Structured Problem-Solving Techniques

• Use the STOP method: This stands for Stop , Think , Observe , and Plan . It's a simple yet effective way to approach any problem methodically, ensuring you consider all aspects before taking action.
• Try reverse engineering: Start with the desired outcome and work backward to understand the steps needed to achieve that result. This approach can be particularly useful for complex problems with unclear starting points.

Incorporate Technology into Your Training

• Engage with online courses and workshops: Many platforms offer courses specifically designed to enhance problem-solving skills, ranging from critical thinking to creative problem-solving techniques.
• Use cognitive training apps: Apps like Elevate provide targeted, research-backed games and workouts to improve cognitive skills including attention, processing speed, and more. 

Practice with Real-World Applications and Learn from Experience

• Tackle daily challenges: Use everyday issues as opportunities to practice problem-solving. Whether figuring out a new recipe or managing a tight budget, applying your skills in real-world situations can reinforce learning.
• Keep a problem-solving journal: Record the challenges you face, the strategies you employ, and the outcomes you achieve. Reflecting on your problem-solving process over time can provide insights into your strengths and areas for improvement.

Embracing Problem-Solving as a Lifelong Journey

Since problems arise daily, it’s important to feel confident in solving them. 

And you can do just that by downloading the Elevate brain training app. Elevate offers 40+ games and activities designed to improve problem-solving, communication, and other cognitive skills in a personalized way that’s backed by science. Pretty cool, right? 

Consider downloading the Elevate app on Android or iOS now—it’ll be the easiest problem you solve all day. 

Related Articles

How Problem-Solving Games Can Boost Your Brain

• Discover why problem-solving games are fun and effective ways to train your brain. 

Improving Your Problem-Solving Skills

• Discover how to improve your problem-solving skills and make logical, informed decisions.  

Best Ways to Boost Your Mental Fitness

• Mental fitness refers to your ability to sustain your overall well-being. Learn tips to improve yours.  

Advisory boards aren’t only for executives. Join the LogRocket Content Advisory Board today →

• Product Management
• Solve User-Reported Issues
• Find Issues Faster
• Optimize Conversion and Adoption

A guide to problem-solving techniques, steps, and skills

You might associate problem-solving with the math exercises that a seven-year-old would do at school. But problem-solving isn’t just about math — it’s a crucial skill that helps everyone make better decisions in everyday life or work.

Problem-solving involves finding effective solutions to address complex challenges, in any context they may arise.

Unfortunately, structured and systematic problem-solving methods aren’t commonly taught. Instead, when solving a problem, PMs tend to rely heavily on intuition. While for simple issues this might work well, solving a complex problem with a straightforward solution is often ineffective and can even create more problems.

In this article, you’ll learn a framework for approaching problem-solving, alongside how you can improve your problem-solving skills.

The 7 steps to problem-solving

When it comes to problem-solving there are seven key steps that you should follow: define the problem, disaggregate, prioritize problem branches, create an analysis plan, conduct analysis, synthesis, and communication.

1. Define the problem

Problem-solving begins with a clear understanding of the issue at hand. Without a well-defined problem statement, confusion and misunderstandings can hinder progress. It’s crucial to ensure that the problem statement is outcome-focused, specific, measurable whenever possible, and time-bound.

Additionally, aligning the problem definition with relevant stakeholders and decision-makers is essential to ensure efforts are directed towards addressing the actual problem rather than side issues.

2. Disaggregate

Complex issues often require deeper analysis. Instead of tackling the entire problem at once, the next step is to break it down into smaller, more manageable components.

Various types of logic trees (also known as issue trees or decision trees) can be used to break down the problem. At each stage where new branches are created, it’s important for them to be “MECE” – mutually exclusive and collectively exhaustive. This process of breaking down continues until manageable components are identified, allowing for individual examination.

The decomposition of the problem demands looking at the problem from various perspectives. That is why collaboration within a team often yields more valuable results, as diverse viewpoints lead to a richer pool of ideas and solutions.

3. Prioritize problem branches

The next step involves prioritization. Not all branches of the problem tree have the same impact, so it’s important to understand the significance of each and focus attention on the most impactful areas. Prioritizing helps streamline efforts and minimize the time required to solve the problem.

Over 200k developers and product managers use LogRocket to create better digital experiences

4. Create an analysis plan

For prioritized components, you may need to conduct in-depth analysis. Before proceeding, a work plan is created for data gathering and analysis. If work is conducted within a team, having a plan provides guidance on what needs to be achieved, who is responsible for which tasks, and the timelines involved.

5. Conduct analysis

Data gathering and analysis are central to the problem-solving process. It’s a good practice to set time limits for this phase to prevent excessive time spent on perfecting details. You can employ heuristics and rule-of-thumb reasoning to improve efficiency and direct efforts towards the most impactful work.

6. Synthesis

After each individual branch component has been researched, the problem isn’t solved yet. The next step is synthesizing the data logically to address the initial question. The synthesis process and the logical relationship between the individual branch results depend on the logic tree used.

7. Communication

The last step is communicating the story and the solution of the problem to the stakeholders and decision-makers. Clear effective communication is necessary to build trust in the solution and facilitates understanding among all parties involved. It ensures that stakeholders grasp the intricacies of the problem and the proposed solution, leading to informed decision-making.

Exploring problem-solving in various contexts

While problem-solving has traditionally been associated with fields like engineering and science, today it has become a fundamental skill for individuals across all professions. In fact, problem-solving consistently ranks as one of the top skills required by employers.

Problem-solving techniques can be applied in diverse contexts:

• Individuals — What career path should I choose? Where should I live? These are examples of simple and common personal challenges that require effective problem-solving skills
• Organizations — Businesses also face many decisions that are not trivial to answer. Should we expand into new markets this year? How can we enhance the quality of our product development? Will our office accommodate the upcoming year’s growth in terms of capacity?
• Societal issues — The biggest world challenges are also complex problems that can be addressed with the same technique. How can we minimize the impact of climate change? How do we fight cancer?

Despite the variation in domains and contexts, the fundamental approach to solving these questions remains the same. It starts with gaining a clear understanding of the problem, followed by decomposition, conducting analysis of the decomposed branches, and synthesizing it into a result that answers the initial problem.

Real-world examples of problem-solving

Let’s now explore some examples where we can apply the problem solving framework.

Problem: In the production of electronic devices, you observe an increasing number of defects. How can you reduce the error rate and improve the quality?

Before delving into analysis, you can deprioritize branches that you already have information for or ones you deem less important. For instance, while transportation delays may occur, the resulting material degradation is likely negligible. For other branches, additional research and data gathering may be necessary.

Once results are obtained, synthesis is crucial to address the core question: How can you decrease the defect rate?

While all factors listed may play a role, their significance varies. Your task is to prioritize effectively. Through data analysis, you may discover that altering the equipment would bring the most substantial positive outcome. However, executing a solution isn’t always straightforward. In prioritizing, you should consider both the potential impact and the level of effort needed for implementation.

By evaluating impact and effort, you can systematically prioritize areas for improvement, focusing on those with high impact and requiring minimal effort to address. This approach ensures efficient allocation of resources towards improvements that offer the greatest return on investment.

Problem : What should be my next job role?

When breaking down this problem, you need to consider various factors that are important for your future happiness in the role. This includes aspects like the company culture, our interest in the work itself, and the lifestyle that you can afford with the role.

However, not all factors carry the same weight for us. To make sense of the results, we can assign a weight factor to each branch. For instance, passion for the job role may have a weight factor of 1, while interest in the industry may have a weight factor of 0.5, because that is less important for you.

By applying these weights to a specific role and summing the values, you can have an estimate of how suitable that role is for you. Moreover, you can compare two roles and make an informed decision based on these weighted indicators.

Key problem-solving skills

This framework provides the foundation and guidance needed to effectively solve problems. However, successfully applying this framework requires the following:

• Creativity — During the decomposition phase, it’s essential to approach the problem from various perspectives and think outside the box to generate innovative ideas for breaking down the problem tree
• Decision-making — Throughout the process, decisions must be made, even when full confidence is lacking. Employing rules of thumb to simplify analysis or selecting one tree cut over another requires decisiveness and comfort with choices made
• Analytical skills — Analytical and research skills are necessary for the phase following decomposition, involving data gathering and analysis on selected tree branches
• Teamwork — Collaboration and teamwork are crucial when working within a team setting. Solving problems effectively often requires collective effort and shared responsibility
• Communication — Clear and structured communication is essential to convey the problem solution to stakeholders and decision-makers and build trust

How to enhance your problem-solving skills

Problem-solving requires practice and a certain mindset. The more you practice, the easier it becomes. Here are some strategies to enhance your skills:

• Practice structured thinking in your daily life — Break down problems or questions into manageable parts. You don’t need to go through the entire problem-solving process and conduct detailed analysis. When conveying a message, simplify the conversation by breaking the message into smaller, more understandable segments
• Regularly challenging yourself with games and puzzles — Solving puzzles, riddles, or strategy games can boost your problem-solving skills and cognitive agility.
• Engage with individuals from diverse backgrounds and viewpoints — Conversing with people who offer different perspectives provides fresh insights and alternative solutions to problems. This boosts creativity and helps in approaching challenges from new angles

Final thoughts

Problem-solving extends far beyond mathematics or scientific fields; it’s a critical skill for making informed decisions in every area of life and work. The seven-step framework presented here provides a systematic approach to problem-solving, relevant across various domains.

Now, consider this: What’s one question currently on your mind? Grab a piece of paper and try to apply the problem-solving framework. You might uncover fresh insights you hadn’t considered before.

Featured image source: IconScout

LogRocket generates product insights that lead to meaningful action

Get your teams on the same page — try LogRocket today.

Share this:

• Click to share on Twitter (Opens in new window)
• Click to share on Reddit (Opens in new window)
• Click to share on LinkedIn (Opens in new window)
• Click to share on Facebook (Opens in new window)
• #career development
• #tools and resources

Stop guessing about your digital experience with LogRocket

Recent posts:.

Leader Spotlight: Empowering analytics and business intelligence teams, with Akash Gupta

Akash Gupta discusses the importance of empowering analytics and business intelligence teams to find “golden nuggets” of insights.

What are product lines? Types, examples, and strategies

Product lines are more than just a collection of products. They are a reflection of a company’s strategic vision and market positioning.

Leader Spotlight: The impact of macroeconomic trends on product roles, with Lori Edwards

Lori Edwards, Director of Product at Niche, discusses challenges with the transition from an individual contributor to a people manager.

Techniques for building rapport in professional settings

Effective rapport fosters trust, facilitates communication, and creates a foundation for successful collaboration and conflict resolution.

Leave a Reply Cancel reply

For enquiries call:

+1-469-442-0620

What Is Logical Thinking – Significance, Components, And Examples

Home Blog Career What Is Logical Thinking – Significance, Components, And Examples

Logical thinking skills play a significant role in developing careers because they help you reason through vital decisions, generate creative ideas, set goals, and solve problems. You may encounter multiple challenges in your life when you enter the job industry or advance your career. Therefore, need strong logical reasoning skills to solve your problems.

But you must know ‘what is logical thinking’ before you move forward or come up with solutions.

What Is Logical Thinking?

Logical thinking is your ability to think in a disciplined manner or base significant thoughts on evidence and facts. The process involves incorporating logic into an individual’s thinking abilities when analyzing a problem to devise a solution. Logical thinking may require Soft Skills Courses because it involves progressive analysis systems.

Now that you know the logical thinking meaning, you can undertake Knowledgehut Training to become probable, reasonable, and actionable with your thoughts. Many fields, such as project management , can benefit from logical thinking skills. Also, consider obtaining some accredited PMP certification programs as well.

Importance Of Logical Thinking

According to a global report , problem-solving, a critical and logical thinking aspect, is one of the top skills employers look for in job candidates. So, it explains the demand for logical thinking or reasoning abilities.

You have already gone through the logical reasoning meaning earlier. Now, it is time to understand its importance through the following points.

1. It Encourages Independent Abilities

You may require multiple demonstrations and examples in your life to learn and comprehend processes. However, prolonged and frequent demonstration systems do not work because problem-solving requires reasoning and analysis. So, you must acquire independent reasoning abilities that define logical thinking.

2. It Promotes Creativity and Innovation

Think out of the box to devise creative solutions to your problems. Here is where logical thinking comes in handy because it allows you to innovate better ideas and give a controlled sense to the events happening in your life.

3. It Helps Enhance Analytical Thinking

You weigh down all possible results and evaluate different options to ensure a favorable outcome for your decisions. Logical reasoning enables you to master multiple choice questions in various ways to get the desired answer by thinking better about the solution.

4. It Helps Strengthen the Brain

If you think about logical reasoning meaning, it involves diverse tasks that help activate various parts of your brain - memory, visual-shape memory, verbal-logic memory, etc. The process helps strengthen your brain and enables you to distinguish significant facets of life.

5. It Helps Enhance Focus

Logical thinking is one of the best ways to increase your concentration. The reasoning ability tests require your focus on problem-solving and include multiple methods and strategies to keep you hooked and develop positive self-esteem.

Ways To Improve Your Logical Thinking

Logical thinking ability definition helps you understand that you must possess this significant skill to move forward in life. So, you must improve and develop your logical thinking through proper activities and exercises. Here is a breakdown of tips to help improve your logical thinking abilities.

• Learn from your life’s mistakes.  
• Anticipate what lies ahead of you and other future happenings.  
• Take complex mental tests.  
• Stimulate your brain through multiple activities.  
• Differentiate between observation and inferences.  
• Try to recognize repetitive patterns like a sequence of numbers.  
• Indulge in analytical values like critical thinking, interpreting, deciding, and concluding facts.

Logical Thinking Skills

The best way to define logical reasoning skills is the ability to focus on tasks and activities by following a chain of thought processes and relating statements to one another. The process allows you to find a logical solution to your problem.

How To Build Logical Thinking Skills?

Work on your logical thinking development to enhance your problem-solving abilities. Here is a breakdown of the techniques to help you overcome your thinking obstacles and understand what the concept of logical thinking is.

• Do not view things from your perspective and understand other people’s opinions.  
• Think before you start doing things by devising efficient strategies.  
• Analyze the meaning of words and sentences carefully.  
• Enhance your thinking skills through games and mystery books.

How To Think Logically in Five Steps?

Logical reasoning means rationalizing your thoughts and creating positive outcomes. The process combines situational awareness and the ability to regulate emotions to enable efficient decision-making. Here is how you can think logically before making decisions.

1. Take Part in Creative Activities

Creative activities like painting, writing, drawing, music, etc., help stimulate your brain and promote logical thinking. Creative thinking also helps develop problem-solving abilities to make you a better performer.

2. Practice Asking Meaningful Questions

Try asking questions regularly to gain a comprehensive perspective of the facts. It will enable you to approach problems creatively and logically and devise solutions strategically.

3. Spend Time with Other People

Try developing meaningful relationships with other people to help broaden your views and perspectives. Socializing with them will help you think logically and provide alternative viewpoints to solutions.

4. Learn New Skills

You must learn new skills frequently to sharpen your logical reasoning abilities. Take opportunities to learn as often as possible and practice your skills daily to help thoughtfully approach situations.

5. Visualize the Outcome of Your Decisions

You must consider your decisions and their impact on your future to help assess positive outcomes. Visualizing the outcome of your choices and decisions will help you strengthen your logical thinking skills.

Components Of Logical Thinking

When someone asks you what the meaning of logical thinking is, your answer should be emotional reasoning and intelligence. It means you possess self-awareness of your feelings and prevent them from affecting your decision-making process.

You must know four significant components after understanding  what  the logical thinking concept is.   

1. Deductive Reasoning

Deductive Reasoning or Deduction is a significant component of logical thinking that seeks to reach specific conclusions. The process makes it easier for you to gain a simplified understanding and indulge in rational and logical thought processes.

2. Inductive Reasoning

Inductive reasoning or induction enables you to think more logically and rely on generalizations. Your general notions depend on anecdotal experiences, facts, and personal observations of your life that are either true or false.

3. Causal Inference

Causal inference involves recognizing the change and evolvement in reasoning things to help you think logically. The process enables taking specific actions and making a logical or causal inference to reason your activities.

Analogical reasoning or analogy enables you to find the things between two different perspectives. Analogy helps you know and understand every situation to help you think logically and make rational decisions.

Example s Of Thinking Logically on Different Occasions  

What is a logical thinking example? I f you are asking yourself this question, look at the following situations for reference.  

1. Logical Thinking When You Are in Disagreement

You and your friend discuss the upcoming cricket match, and both disagree on who will be the opening batsman. You try logically reasoning out the facts and back out by stating that your friend’s prediction is correct.

2. Logical Thinking to Complete Your Work

You had planned a day out with friends for the weekend, but you got caught up with some pending work. The logical way to sort the situation would be to complete your work beforehand and head out for your getaway.

3. Logical Thinking When Making a Tough Decision

You get a good job opportunity in another city, but it makes you emotional thinking you have to leave your hometown. The logical way is to think of the opportunities awaiting you in the other place and decide to take the job.   

4. Logical Thinking When You Do Not Know the Answer

If you do not know the answer to a few questions about your recent assignment, the logical way of solving them is by approaching your teacher and asking for clarification.   

5. Other Logical Thinking Examples

Logical thinking involves reasoning skills to study problems and find rational conclusions or solutions. One of the best examples is the following situation.

You are facing some problems in the office. So, you use the available facts using your logical reasoning skills to address them.

Here is another example of logical reasoning.

You develop a fever ahead of an important meeting that you cannot miss at any cost. The logical way to solve the problem is to attend the meeting virtually instead of remaining physically present.

In Conclusion

Logical thinking is an act of analyzing situations and using reasoning abilities to study the problem and make a rational conclusion.  When you become a logical thinker, you gather all the information you can, assess the facts, and methodically decide the best way to move forward with your decision. Most people consider logical thinking an essential tool to brainstorm ideas, analyze problems, and find answers at home, workplace, or in educational institutions.

Secure your future with the highest paying jobs in the world. Stay ahead in a rapidly evolving job market and unlock a world of financial possibilities!  

Frequently Asked Questions (FAQs)

You can consider yourself a logical thinker if you are attentive, get your facts straight, and have clear ideas about situations.

Yes, logical thinking is a soft skill that is tangible, easy to practice, and improves your reasoning abilities.

Economists, software developers, accountants, chemical engineers, technical writers, criminologists, and other related careers use logical thinking.

Logical thinkers are good at observing and analyzing situations, feedback, and reactions to draw rational conclusions.

Mounika Narang

Mounika Narang is a project manager having a specialisation in IT project management and Instructional Design. She has an experience of 10 years  working with Fortune 500 companies to solve their most important development challenges. She lives in Bangalore with her family.

Avail your free 1:1 mentorship session.

Something went wrong

Invest in Your Future: Upcoming High-Paying Career Courses to Consider

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

7 Module 7: Thinking, Reasoning, and Problem-Solving

This module is about how a solid working knowledge of psychological principles can help you to think more effectively, so you can succeed in school and life. You might be inclined to believe that—because you have been thinking for as long as you can remember, because you are able to figure out the solution to many problems, because you feel capable of using logic to argue a point, because you can evaluate whether the things you read and hear make sense—you do not need any special training in thinking. But this, of course, is one of the key barriers to helping people think better. If you do not believe that there is anything wrong, why try to fix it?

The human brain is indeed a remarkable thinking machine, capable of amazing, complex, creative, logical thoughts. Why, then, are we telling you that you need to learn how to think? Mainly because one major lesson from cognitive psychology is that these capabilities of the human brain are relatively infrequently realized. Many psychologists believe that people are essentially “cognitive misers.” It is not that we are lazy, but that we have a tendency to expend the least amount of mental effort necessary. Although you may not realize it, it actually takes a great deal of energy to think. Careful, deliberative reasoning and critical thinking are very difficult. Because we seem to be successful without going to the trouble of using these skills well, it feels unnecessary to develop them. As you shall see, however, there are many pitfalls in the cognitive processes described in this module. When people do not devote extra effort to learning and improving reasoning, problem solving, and critical thinking skills, they make many errors.

As is true for memory, if you develop the cognitive skills presented in this module, you will be more successful in school. It is important that you realize, however, that these skills will help you far beyond school, even more so than a good memory will. Although it is somewhat useful to have a good memory, ten years from now no potential employer will care how many questions you got right on multiple choice exams during college. All of them will, however, recognize whether you are a logical, analytical, critical thinker. With these thinking skills, you will be an effective, persuasive communicator and an excellent problem solver.

The module begins by describing different kinds of thought and knowledge, especially conceptual knowledge and critical thinking. An understanding of these differences will be valuable as you progress through school and encounter different assignments that require you to tap into different kinds of knowledge. The second section covers deductive and inductive reasoning, which are processes we use to construct and evaluate strong arguments. They are essential skills to have whenever you are trying to persuade someone (including yourself) of some point, or to respond to someone’s efforts to persuade you. The module ends with a section about problem solving. A solid understanding of the key processes involved in problem solving will help you to handle many daily challenges.

7.1. Different kinds of thought

7.2. Reasoning and Judgment

7.3. Problem Solving

READING WITH PURPOSE

Remember and understand.

By reading and studying Module 7, you should be able to remember and describe:

• Concepts and inferences (7.1)
• Procedural knowledge (7.1)
• Metacognition (7.1)
• Characteristics of critical thinking:  skepticism; identify biases, distortions, omissions, and assumptions; reasoning and problem solving skills  (7.1)
• Reasoning:  deductive reasoning, deductively valid argument, inductive reasoning, inductively strong argument, availability heuristic, representativeness heuristic  (7.2)
• Fixation:  functional fixedness, mental set  (7.3)
• Algorithms, heuristics, and the role of confirmation bias (7.3)
• Effective problem solving sequence (7.3)

By reading and thinking about how the concepts in Module 6 apply to real life, you should be able to:

• Identify which type of knowledge a piece of information is (7.1)
• Recognize examples of deductive and inductive reasoning (7.2)
• Recognize judgments that have probably been influenced by the availability heuristic (7.2)
• Recognize examples of problem solving heuristics and algorithms (7.3)

Analyze, Evaluate, and Create

By reading and thinking about Module 6, participating in classroom activities, and completing out-of-class assignments, you should be able to:

• Use the principles of critical thinking to evaluate information (7.1)
• Explain whether examples of reasoning arguments are deductively valid or inductively strong (7.2)
• Outline how you could try to solve a problem from your life using the effective problem solving sequence (7.3)

7.1. Different kinds of thought and knowledge

• Take a few minutes to write down everything that you know about dogs.
• Do you believe that:
• Psychic ability exists?
• Hypnosis is an altered state of consciousness?
• Magnet therapy is effective for relieving pain?
• Aerobic exercise is an effective treatment for depression?
• UFO’s from outer space have visited earth?

On what do you base your belief or disbelief for the questions above?

Of course, we all know what is meant by the words  think  and  knowledge . You probably also realize that they are not unitary concepts; there are different kinds of thought and knowledge. In this section, let us look at some of these differences. If you are familiar with these different kinds of thought and pay attention to them in your classes, it will help you to focus on the right goals, learn more effectively, and succeed in school. Different assignments and requirements in school call on you to use different kinds of knowledge or thought, so it will be very helpful for you to learn to recognize them (Anderson, et al. 2001).

Factual and conceptual knowledge

Module 5 introduced the idea of declarative memory, which is composed of facts and episodes. If you have ever played a trivia game or watched Jeopardy on TV, you realize that the human brain is able to hold an extraordinary number of facts. Likewise, you realize that each of us has an enormous store of episodes, essentially facts about events that happened in our own lives. It may be difficult to keep that in mind when we are struggling to retrieve one of those facts while taking an exam, however. Part of the problem is that, in contradiction to the advice from Module 5, many students continue to try to memorize course material as a series of unrelated facts (picture a history student simply trying to memorize history as a set of unrelated dates without any coherent story tying them together). Facts in the real world are not random and unorganized, however. It is the way that they are organized that constitutes a second key kind of knowledge, conceptual.

Concepts are nothing more than our mental representations of categories of things in the world. For example, think about dogs. When you do this, you might remember specific facts about dogs, such as they have fur and they bark. You may also recall dogs that you have encountered and picture them in your mind. All of this information (and more) makes up your concept of dog. You can have concepts of simple categories (e.g., triangle), complex categories (e.g., small dogs that sleep all day, eat out of the garbage, and bark at leaves), kinds of people (e.g., psychology professors), events (e.g., birthday parties), and abstract ideas (e.g., justice). Gregory Murphy (2002) refers to concepts as the “glue that holds our mental life together” (p. 1). Very simply, summarizing the world by using concepts is one of the most important cognitive tasks that we do. Our conceptual knowledge  is  our knowledge about the world. Individual concepts are related to each other to form a rich interconnected network of knowledge. For example, think about how the following concepts might be related to each other: dog, pet, play, Frisbee, chew toy, shoe. Or, of more obvious use to you now, how these concepts are related: working memory, long-term memory, declarative memory, procedural memory, and rehearsal? Because our minds have a natural tendency to organize information conceptually, when students try to remember course material as isolated facts, they are working against their strengths.

One last important point about concepts is that they allow you to instantly know a great deal of information about something. For example, if someone hands you a small red object and says, “here is an apple,” they do not have to tell you, “it is something you can eat.” You already know that you can eat it because it is true by virtue of the fact that the object is an apple; this is called drawing an  inference , assuming that something is true on the basis of your previous knowledge (for example, of category membership or of how the world works) or logical reasoning.

Procedural knowledge

Physical skills, such as tying your shoes, doing a cartwheel, and driving a car (or doing all three at the same time, but don’t try this at home) are certainly a kind of knowledge. They are procedural knowledge, the same idea as procedural memory that you saw in Module 5. Mental skills, such as reading, debating, and planning a psychology experiment, are procedural knowledge, as well. In short, procedural knowledge is the knowledge how to do something (Cohen & Eichenbaum, 1993).

Metacognitive knowledge

Floyd used to think that he had a great memory. Now, he has a better memory. Why? Because he finally realized that his memory was not as great as he once thought it was. Because Floyd eventually learned that he often forgets where he put things, he finally developed the habit of putting things in the same place. (Unfortunately, he did not learn this lesson before losing at least 5 watches and a wedding ring.) Because he finally realized that he often forgets to do things, he finally started using the To Do list app on his phone. And so on. Floyd’s insights about the real limitations of his memory have allowed him to remember things that he used to forget.

All of us have knowledge about the way our own minds work. You may know that you have a good memory for people’s names and a poor memory for math formulas. Someone else might realize that they have difficulty remembering to do things, like stopping at the store on the way home. Others still know that they tend to overlook details. This knowledge about our own thinking is actually quite important; it is called metacognitive knowledge, or  metacognition . Like other kinds of thinking skills, it is subject to error. For example, in unpublished research, one of the authors surveyed about 120 General Psychology students on the first day of the term. Among other questions, the students were asked them to predict their grade in the class and report their current Grade Point Average. Two-thirds of the students predicted that their grade in the course would be higher than their GPA. (The reality is that at our college, students tend to earn lower grades in psychology than their overall GPA.) Another example: Students routinely report that they thought they had done well on an exam, only to discover, to their dismay, that they were wrong (more on that important problem in a moment). Both errors reveal a breakdown in metacognition.

The Dunning-Kruger Effect

In general, most college students probably do not study enough. For example, using data from the National Survey of Student Engagement, Fosnacht, McCormack, and Lerma (2018) reported that first-year students at 4-year colleges in the U.S. averaged less than 14 hours per week preparing for classes. The typical suggestion is that you should spend two hours outside of class for every hour in class, or 24 – 30 hours per week for a full-time student. Clearly, students in general are nowhere near that recommended mark. Many observers, including some faculty, believe that this shortfall is a result of students being too busy or lazy. Now, it may be true that many students are too busy, with work and family obligations, for example. Others, are not particularly motivated in school, and therefore might correctly be labeled lazy. A third possible explanation, however, is that some students might not think they need to spend this much time. And this is a matter of metacognition. Consider the scenario that we mentioned above, students thinking they had done well on an exam only to discover that they did not. Justin Kruger and David Dunning examined scenarios very much like this in 1999. Kruger and Dunning gave research participants tests measuring humor, logic, and grammar. Then, they asked the participants to assess their own abilities and test performance in these areas. They found that participants in general tended to overestimate their abilities, already a problem with metacognition. Importantly, the participants who scored the lowest overestimated their abilities the most. Specifically, students who scored in the bottom quarter (averaging in the 12th percentile) thought they had scored in the 62nd percentile. This has become known as the  Dunning-Kruger effect . Many individual faculty members have replicated these results with their own student on their course exams, including the authors of this book. Think about it. Some students who just took an exam and performed poorly believe that they did well before seeing their score. It seems very likely that these are the very same students who stopped studying the night before because they thought they were “done.” Quite simply, it is not just that they did not know the material. They did not know that they did not know the material. That is poor metacognition.

In order to develop good metacognitive skills, you should continually monitor your thinking and seek frequent feedback on the accuracy of your thinking (Medina, Castleberry, & Persky 2017). For example, in classes get in the habit of predicting your exam grades. As soon as possible after taking an exam, try to find out which questions you missed and try to figure out why. If you do this soon enough, you may be able to recall the way it felt when you originally answered the question. Did you feel confident that you had answered the question correctly? Then you have just discovered an opportunity to improve your metacognition. Be on the lookout for that feeling and respond with caution.

concept :  a mental representation of a category of things in the world

Dunning-Kruger effect : individuals who are less competent tend to overestimate their abilities more than individuals who are more competent do

inference : an assumption about the truth of something that is not stated. Inferences come from our prior knowledge and experience, and from logical reasoning

metacognition :  knowledge about one’s own cognitive processes; thinking about your thinking

Critical thinking

One particular kind of knowledge or thinking skill that is related to metacognition is  critical thinking (Chew, 2020). You may have noticed that critical thinking is an objective in many college courses, and thus it could be a legitimate topic to cover in nearly any college course. It is particularly appropriate in psychology, however. As the science of (behavior and) mental processes, psychology is obviously well suited to be the discipline through which you should be introduced to this important way of thinking.

More importantly, there is a particular need to use critical thinking in psychology. We are all, in a way, experts in human behavior and mental processes, having engaged in them literally since birth. Thus, perhaps more than in any other class, students typically approach psychology with very clear ideas and opinions about its subject matter. That is, students already “know” a lot about psychology. The problem is, “it ain’t so much the things we don’t know that get us into trouble. It’s the things we know that just ain’t so” (Ward, quoted in Gilovich 1991). Indeed, many of students’ preconceptions about psychology are just plain wrong. Randolph Smith (2002) wrote a book about critical thinking in psychology called  Challenging Your Preconceptions,  highlighting this fact. On the other hand, many of students’ preconceptions about psychology are just plain right! But wait, how do you know which of your preconceptions are right and which are wrong? And when you come across a research finding or theory in this class that contradicts your preconceptions, what will you do? Will you stick to your original idea, discounting the information from the class? Will you immediately change your mind? Critical thinking can help us sort through this confusing mess.

But what is critical thinking? The goal of critical thinking is simple to state (but extraordinarily difficult to achieve): it is to be right, to draw the correct conclusions, to believe in things that are true and to disbelieve things that are false. We will provide two definitions of critical thinking (or, if you like, one large definition with two distinct parts). First, a more conceptual one: Critical thinking is thinking like a scientist in your everyday life (Schmaltz, Jansen, & Wenckowski, 2017).  Our second definition is more operational; it is simply a list of skills that are essential to be a critical thinker. Critical thinking entails solid reasoning and problem solving skills; skepticism; and an ability to identify biases, distortions, omissions, and assumptions. Excellent deductive and inductive reasoning, and problem solving skills contribute to critical thinking. So, you can consider the subject matter of sections 7.2 and 7.3 to be part of critical thinking. Because we will be devoting considerable time to these concepts in the rest of the module, let us begin with a discussion about the other aspects of critical thinking.

Let’s address that first part of the definition. Scientists form hypotheses, or predictions about some possible future observations. Then, they collect data, or information (think of this as making those future observations). They do their best to make unbiased observations using reliable techniques that have been verified by others. Then, and only then, they draw a conclusion about what those observations mean. Oh, and do not forget the most important part. “Conclusion” is probably not the most appropriate word because this conclusion is only tentative. A scientist is always prepared that someone else might come along and produce new observations that would require a new conclusion be drawn. Wow! If you like to be right, you could do a lot worse than using a process like this.

A Critical Thinker’s Toolkit 

Now for the second part of the definition. Good critical thinkers (and scientists) rely on a variety of tools to evaluate information. Perhaps the most recognizable tool for critical thinking is  skepticism (and this term provides the clearest link to the thinking like a scientist definition, as you are about to see). Some people intend it as an insult when they call someone a skeptic. But if someone calls you a skeptic, if they are using the term correctly, you should consider it a great compliment. Simply put, skepticism is a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided. People from Missouri should recognize this principle, as Missouri is known as the Show-Me State. As a skeptic, you are not inclined to believe something just because someone said so, because someone else believes it, or because it sounds reasonable. You must be persuaded by high quality evidence.

Of course, if that evidence is produced, you have a responsibility as a skeptic to change your belief. Failure to change a belief in the face of good evidence is not skepticism; skepticism has open mindedness at its core. M. Neil Browne and Stuart Keeley (2018) use the term weak sense critical thinking to describe critical thinking behaviors that are used only to strengthen a prior belief. Strong sense critical thinking, on the other hand, has as its goal reaching the best conclusion. Sometimes that means strengthening your prior belief, but sometimes it means changing your belief to accommodate the better evidence.

Many times, a failure to think critically or weak sense critical thinking is related to a  bias , an inclination, tendency, leaning, or prejudice. Everybody has biases, but many people are unaware of them. Awareness of your own biases gives you the opportunity to control or counteract them. Unfortunately, however, many people are happy to let their biases creep into their attempts to persuade others; indeed, it is a key part of their persuasive strategy. To see how these biases influence messages, just look at the different descriptions and explanations of the same events given by people of different ages or income brackets, or conservative versus liberal commentators, or by commentators from different parts of the world. Of course, to be successful, these people who are consciously using their biases must disguise them. Even undisguised biases can be difficult to identify, so disguised ones can be nearly impossible.

Here are some common sources of biases:

• Personal values and beliefs.  Some people believe that human beings are basically driven to seek power and that they are typically in competition with one another over scarce resources. These beliefs are similar to the world-view that political scientists call “realism.” Other people believe that human beings prefer to cooperate and that, given the chance, they will do so. These beliefs are similar to the world-view known as “idealism.” For many people, these deeply held beliefs can influence, or bias, their interpretations of such wide ranging situations as the behavior of nations and their leaders or the behavior of the driver in the car ahead of you. For example, if your worldview is that people are typically in competition and someone cuts you off on the highway, you may assume that the driver did it purposely to get ahead of you. Other types of beliefs about the way the world is or the way the world should be, for example, political beliefs, can similarly become a significant source of bias.
• Racism, sexism, ageism and other forms of prejudice and bigotry.  These are, sadly, a common source of bias in many people. They are essentially a special kind of “belief about the way the world is.” These beliefs—for example, that women do not make effective leaders—lead people to ignore contradictory evidence (examples of effective women leaders, or research that disputes the belief) and to interpret ambiguous evidence in a way consistent with the belief.
• Self-interest.  When particular people benefit from things turning out a certain way, they can sometimes be very susceptible to letting that interest bias them. For example, a company that will earn a profit if they sell their product may have a bias in the way that they give information about their product. A union that will benefit if its members get a generous contract might have a bias in the way it presents information about salaries at competing organizations. (Note that our inclusion of examples describing both companies and unions is an explicit attempt to control for our own personal biases). Home buyers are often dismayed to discover that they purchased their dream house from someone whose self-interest led them to lie about flooding problems in the basement or back yard. This principle, the biasing power of self-interest, is likely what led to the famous phrase  Caveat Emptor  (let the buyer beware) .  

Knowing that these types of biases exist will help you evaluate evidence more critically. Do not forget, though, that people are not always keen to let you discover the sources of biases in their arguments. For example, companies or political organizations can sometimes disguise their support of a research study by contracting with a university professor, who comes complete with a seemingly unbiased institutional affiliation, to conduct the study.

People’s biases, conscious or unconscious, can lead them to make omissions, distortions, and assumptions that undermine our ability to correctly evaluate evidence. It is essential that you look for these elements. Always ask, what is missing, what is not as it appears, and what is being assumed here? For example, consider this (fictional) chart from an ad reporting customer satisfaction at 4 local health clubs.

Clearly, from the results of the chart, one would be tempted to give Club C a try, as customer satisfaction is much higher than for the other 3 clubs.

There are so many distortions and omissions in this chart, however, that it is actually quite meaningless. First, how was satisfaction measured? Do the bars represent responses to a survey? If so, how were the questions asked? Most importantly, where is the missing scale for the chart? Although the differences look quite large, are they really?

Well, here is the same chart, with a different scale, this time labeled:

Club C is not so impressive any more, is it? In fact, all of the health clubs have customer satisfaction ratings (whatever that means) between 85% and 88%. In the first chart, the entire scale of the graph included only the percentages between 83 and 89. This “judicious” choice of scale—some would call it a distortion—and omission of that scale from the chart make the tiny differences among the clubs seem important, however.

Also, in order to be a critical thinker, you need to learn to pay attention to the assumptions that underlie a message. Let us briefly illustrate the role of assumptions by touching on some people’s beliefs about the criminal justice system in the US. Some believe that a major problem with our judicial system is that many criminals go free because of legal technicalities. Others believe that a major problem is that many innocent people are convicted of crimes. The simple fact is, both types of errors occur. A person’s conclusion about which flaw in our judicial system is the greater tragedy is based on an assumption about which of these is the more serious error (letting the guilty go free or convicting the innocent). This type of assumption is called a value assumption (Browne and Keeley, 2018). It reflects the differences in values that people develop, differences that may lead us to disregard valid evidence that does not fit in with our particular values.

Oh, by the way, some students probably noticed this, but the seven tips for evaluating information that we shared in Module 1 are related to this. Actually, they are part of this section. The tips are, to a very large degree, set of ideas you can use to help you identify biases, distortions, omissions, and assumptions. If you do not remember this section, we strongly recommend you take a few minutes to review it.

skepticism :  a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided

bias : an inclination, tendency, leaning, or prejudice

• Which of your beliefs (or disbeliefs) from the Activate exercise for this section were derived from a process of critical thinking? If some of your beliefs were not based on critical thinking, are you willing to reassess these beliefs? If the answer is no, why do you think that is? If the answer is yes, what concrete steps will you take?

7.2 Reasoning and Judgment

• What percentage of kidnappings are committed by strangers?
• Which area of the house is riskiest: kitchen, bathroom, or stairs?
• What is the most common cancer in the US?
• What percentage of workplace homicides are committed by co-workers?

An essential set of procedural thinking skills is  reasoning , the ability to generate and evaluate solid conclusions from a set of statements or evidence. You should note that these conclusions (when they are generated instead of being evaluated) are one key type of inference that we described in Section 7.1. There are two main types of reasoning, deductive and inductive.

Deductive reasoning

Suppose your teacher tells you that if you get an A on the final exam in a course, you will get an A for the whole course. Then, you get an A on the final exam. What will your final course grade be? Most people can see instantly that you can conclude with certainty that you will get an A for the course. This is a type of reasoning called  deductive reasoning , which is defined as reasoning in which a conclusion is guaranteed to be true as long as the statements leading to it are true. The three statements can be listed as an  argument , with two beginning statements and a conclusion:

Statement 1: If you get an A on the final exam, you will get an A for the course

Statement 2: You get an A on the final exam

Conclusion: You will get an A for the course

This particular arrangement, in which true beginning statements lead to a guaranteed true conclusion, is known as a  deductively valid argument . Although deductive reasoning is often the subject of abstract, brain-teasing, puzzle-like word problems, it is actually an extremely important type of everyday reasoning. It is just hard to recognize sometimes. For example, imagine that you are looking for your car keys and you realize that they are either in the kitchen drawer or in your book bag. After looking in the kitchen drawer, you instantly know that they must be in your book bag. That conclusion results from a simple deductive reasoning argument. In addition, solid deductive reasoning skills are necessary for you to succeed in the sciences, philosophy, math, computer programming, and any endeavor involving the use of logic to persuade others to your point of view or to evaluate others’ arguments.

Cognitive psychologists, and before them philosophers, have been quite interested in deductive reasoning, not so much for its practical applications, but for the insights it can offer them about the ways that human beings think. One of the early ideas to emerge from the examination of deductive reasoning is that people learn (or develop) mental versions of rules that allow them to solve these types of reasoning problems (Braine, 1978; Braine, Reiser, & Rumain, 1984). The best way to see this point of view is to realize that there are different possible rules, and some of them are very simple. For example, consider this rule of logic:

therefore q

Logical rules are often presented abstractly, as letters, in order to imply that they can be used in very many specific situations. Here is a concrete version of the of the same rule:

I’ll either have pizza or a hamburger for dinner tonight (p or q)

I won’t have pizza (not p)

Therefore, I’ll have a hamburger (therefore q)

This kind of reasoning seems so natural, so easy, that it is quite plausible that we would use a version of this rule in our daily lives. At least, it seems more plausible than some of the alternative possibilities—for example, that we need to have experience with the specific situation (pizza or hamburger, in this case) in order to solve this type of problem easily. So perhaps there is a form of natural logic (Rips, 1990) that contains very simple versions of logical rules. When we are faced with a reasoning problem that maps onto one of these rules, we use the rule.

But be very careful; things are not always as easy as they seem. Even these simple rules are not so simple. For example, consider the following rule. Many people fail to realize that this rule is just as valid as the pizza or hamburger rule above.

if p, then q

therefore, not p

Concrete version:

If I eat dinner, then I will have dessert

I did not have dessert

Therefore, I did not eat dinner

The simple fact is, it can be very difficult for people to apply rules of deductive logic correctly; as a result, they make many errors when trying to do so. Is this a deductively valid argument or not?

Students who like school study a lot

Students who study a lot get good grades

Jane does not like school

Therefore, Jane does not get good grades

Many people are surprised to discover that this is not a logically valid argument; the conclusion is not guaranteed to be true from the beginning statements. Although the first statement says that students who like school study a lot, it does NOT say that students who do not like school do not study a lot. In other words, it may very well be possible to study a lot without liking school. Even people who sometimes get problems like this right might not be using the rules of deductive reasoning. Instead, they might just be making judgments for examples they know, in this case, remembering instances of people who get good grades despite not liking school.

Making deductive reasoning even more difficult is the fact that there are two important properties that an argument may have. One, it can be valid or invalid (meaning that the conclusion does or does not follow logically from the statements leading up to it). Two, an argument (or more correctly, its conclusion) can be true or false. Here is an example of an argument that is logically valid, but has a false conclusion (at least we think it is false).

Either you are eleven feet tall or the Grand Canyon was created by a spaceship crashing into the earth.

You are not eleven feet tall

Therefore the Grand Canyon was created by a spaceship crashing into the earth

This argument has the exact same form as the pizza or hamburger argument above, making it is deductively valid. The conclusion is so false, however, that it is absurd (of course, the reason the conclusion is false is that the first statement is false). When people are judging arguments, they tend to not observe the difference between deductive validity and the empirical truth of statements or conclusions. If the elements of an argument happen to be true, people are likely to judge the argument logically valid; if the elements are false, they will very likely judge it invalid (Markovits & Bouffard-Bouchard, 1992; Moshman & Franks, 1986). Thus, it seems a stretch to say that people are using these logical rules to judge the validity of arguments. Many psychologists believe that most people actually have very limited deductive reasoning skills (Johnson-Laird, 1999). They argue that when faced with a problem for which deductive logic is required, people resort to some simpler technique, such as matching terms that appear in the statements and the conclusion (Evans, 1982). This might not seem like a problem, but what if reasoners believe that the elements are true and they happen to be wrong; they will would believe that they are using a form of reasoning that guarantees they are correct and yet be wrong.

deductive reasoning :  a type of reasoning in which the conclusion is guaranteed to be true any time the statements leading up to it are true

argument :  a set of statements in which the beginning statements lead to a conclusion

deductively valid argument :  an argument for which true beginning statements guarantee that the conclusion is true

Inductive reasoning and judgment

Every day, you make many judgments about the likelihood of one thing or another. Whether you realize it or not, you are practicing  inductive reasoning   on a daily basis. In inductive reasoning arguments, a conclusion is likely whenever the statements preceding it are true. The first thing to notice about inductive reasoning is that, by definition, you can never be sure about your conclusion; you can only estimate how likely the conclusion is. Inductive reasoning may lead you to focus on Memory Encoding and Recoding when you study for the exam, but it is possible the instructor will ask more questions about Memory Retrieval instead. Unlike deductive reasoning, the conclusions you reach through inductive reasoning are only probable, not certain. That is why scientists consider inductive reasoning weaker than deductive reasoning. But imagine how hard it would be for us to function if we could not act unless we were certain about the outcome.

Inductive reasoning can be represented as logical arguments consisting of statements and a conclusion, just as deductive reasoning can be. In an inductive argument, you are given some statements and a conclusion (or you are given some statements and must draw a conclusion). An argument is  inductively strong   if the conclusion would be very probable whenever the statements are true. So, for example, here is an inductively strong argument:

• Statement #1: The forecaster on Channel 2 said it is going to rain today.
• Statement #2: The forecaster on Channel 5 said it is going to rain today.
• Statement #3: It is very cloudy and humid.
• Statement #4: You just heard thunder.
• Conclusion (or judgment): It is going to rain today.

Think of the statements as evidence, on the basis of which you will draw a conclusion. So, based on the evidence presented in the four statements, it is very likely that it will rain today. Will it definitely rain today? Certainly not. We can all think of times that the weather forecaster was wrong.

A true story: Some years ago psychology student was watching a baseball playoff game between the St. Louis Cardinals and the Los Angeles Dodgers. A graphic on the screen had just informed the audience that the Cardinal at bat, (Hall of Fame shortstop) Ozzie Smith, a switch hitter batting left-handed for this plate appearance, had never, in nearly 3000 career at-bats, hit a home run left-handed. The student, who had just learned about inductive reasoning in his psychology class, turned to his companion (a Cardinals fan) and smugly said, “It is an inductively strong argument that Ozzie Smith will not hit a home run.” He turned back to face the television just in time to watch the ball sail over the right field fence for a home run. Although the student felt foolish at the time, he was not wrong. It was an inductively strong argument; 3000 at-bats is an awful lot of evidence suggesting that the Wizard of Ozz (as he was known) would not be hitting one out of the park (think of each at-bat without a home run as a statement in an inductive argument). Sadly (for the die-hard Cubs fan and Cardinals-hating student), despite the strength of the argument, the conclusion was wrong.

Given the possibility that we might draw an incorrect conclusion even with an inductively strong argument, we really want to be sure that we do, in fact, make inductively strong arguments. If we judge something probable, it had better be probable. If we judge something nearly impossible, it had better not happen. Think of inductive reasoning, then, as making reasonably accurate judgments of the probability of some conclusion given a set of evidence.

We base many decisions in our lives on inductive reasoning. For example:

Statement #1: Psychology is not my best subject

Statement #2: My psychology instructor has a reputation for giving difficult exams

Statement #3: My first psychology exam was much harder than I expected

Judgment: The next exam will probably be very difficult.

Decision: I will study tonight instead of watching Netflix.

Some other examples of judgments that people commonly make in a school context include judgments of the likelihood that:

• A particular class will be interesting/useful/difficult
• You will be able to finish writing a paper by next week if you go out tonight
• Your laptop’s battery will last through the next trip to the library
• You will not miss anything important if you skip class tomorrow
• Your instructor will not notice if you skip class tomorrow
• You will be able to find a book that you will need for a paper
• There will be an essay question about Memory Encoding on the next exam

Tversky and Kahneman (1983) recognized that there are two general ways that we might make these judgments; they termed them extensional (i.e., following the laws of probability) and intuitive (i.e., using shortcuts or heuristics, see below). We will use a similar distinction between Type 1 and Type 2 thinking, as described by Keith Stanovich and his colleagues (Evans and Stanovich, 2013; Stanovich and West, 2000). Type 1 thinking is fast, automatic, effortful, and emotional. In fact, it is hardly fair to call it reasoning at all, as judgments just seem to pop into one’s head. Type 2 thinking , on the other hand, is slow, effortful, and logical. So obviously, it is more likely to lead to a correct judgment, or an optimal decision. The problem is, we tend to over-rely on Type 1. Now, we are not saying that Type 2 is the right way to go for every decision or judgment we make. It seems a bit much, for example, to engage in a step-by-step logical reasoning procedure to decide whether we will have chicken or fish for dinner tonight.

Many bad decisions in some very important contexts, however, can be traced back to poor judgments of the likelihood of certain risks or outcomes that result from the use of Type 1 when a more logical reasoning process would have been more appropriate. For example:

Statement #1: It is late at night.

Statement #2: Albert has been drinking beer for the past five hours at a party.

Statement #3: Albert is not exactly sure where he is or how far away home is.

Judgment: Albert will have no difficulty walking home.

Decision: He walks home alone.

As you can see in this example, the three statements backing up the judgment do not really support it. In other words, this argument is not inductively strong because it is based on judgments that ignore the laws of probability. What are the chances that someone facing these conditions will be able to walk home alone easily? And one need not be drunk to make poor decisions based on judgments that just pop into our heads.

The truth is that many of our probability judgments do not come very close to what the laws of probability say they should be. Think about it. In order for us to reason in accordance with these laws, we would need to know the laws of probability, which would allow us to calculate the relationship between particular pieces of evidence and the probability of some outcome (i.e., how much likelihood should change given a piece of evidence), and we would have to do these heavy math calculations in our heads. After all, that is what Type 2 requires. Needless to say, even if we were motivated, we often do not even know how to apply Type 2 reasoning in many cases.

So what do we do when we don’t have the knowledge, skills, or time required to make the correct mathematical judgment? Do we hold off and wait until we can get better evidence? Do we read up on probability and fire up our calculator app so we can compute the correct probability? Of course not. We rely on Type 1 thinking. We “wing it.” That is, we come up with a likelihood estimate using some means at our disposal. Psychologists use the term heuristic to describe the type of “winging it” we are talking about. A  heuristic   is a shortcut strategy that we use to make some judgment or solve some problem (see Section 7.3). Heuristics are easy and quick, think of them as the basic procedures that are characteristic of Type 1.  They can absolutely lead to reasonably good judgments and decisions in some situations (like choosing between chicken and fish for dinner). They are, however, far from foolproof. There are, in fact, quite a lot of situations in which heuristics can lead us to make incorrect judgments, and in many cases the decisions based on those judgments can have serious consequences.

Let us return to the activity that begins this section. You were asked to judge the likelihood (or frequency) of certain events and risks. You were free to come up with your own evidence (or statements) to make these judgments. This is where a heuristic crops up. As a judgment shortcut, we tend to generate specific examples of those very events to help us decide their likelihood or frequency. For example, if we are asked to judge how common, frequent, or likely a particular type of cancer is, many of our statements would be examples of specific cancer cases:

Statement #1: Andy Kaufman (comedian) had lung cancer.

Statement #2: Colin Powell (US Secretary of State) had prostate cancer.

Statement #3: Bob Marley (musician) had skin and brain cancer

Statement #4: Sandra Day O’Connor (Supreme Court Justice) had breast cancer.

Statement #5: Fred Rogers (children’s entertainer) had stomach cancer.

Statement #6: Robin Roberts (news anchor) had breast cancer.

Statement #7: Bette Davis (actress) had breast cancer.

Judgment: Breast cancer is the most common type.

Your own experience or memory may also tell you that breast cancer is the most common type. But it is not (although it is common). Actually, skin cancer is the most common type in the US. We make the same types of misjudgments all the time because we do not generate the examples or evidence according to their actual frequencies or probabilities. Instead, we have a tendency (or bias) to search for the examples in memory; if they are easy to retrieve, we assume that they are common. To rephrase this in the language of the heuristic, events seem more likely to the extent that they are available to memory. This bias has been termed the  availability heuristic   (Kahneman and Tversky, 1974).

The fact that we use the availability heuristic does not automatically mean that our judgment is wrong. The reason we use heuristics in the first place is that they work fairly well in many cases (and, of course that they are easy to use). So, the easiest examples to think of sometimes are the most common ones. Is it more likely that a member of the U.S. Senate is a man or a woman? Most people have a much easier time generating examples of male senators. And as it turns out, the U.S. Senate has many more men than women (74 to 26 in 2020). In this case, then, the availability heuristic would lead you to make the correct judgment; it is far more likely that a senator would be a man.

In many other cases, however, the availability heuristic will lead us astray. This is because events can be memorable for many reasons other than their frequency. Section 5.2, Encoding Meaning, suggested that one good way to encode the meaning of some information is to form a mental image of it. Thus, information that has been pictured mentally will be more available to memory. Indeed, an event that is vivid and easily pictured will trick many people into supposing that type of event is more common than it actually is. Repetition of information will also make it more memorable. So, if the same event is described to you in a magazine, on the evening news, on a podcast that you listen to, and in your Facebook feed; it will be very available to memory. Again, the availability heuristic will cause you to misperceive the frequency of these types of events.

Most interestingly, information that is unusual is more memorable. Suppose we give you the following list of words to remember: box, flower, letter, platypus, oven, boat, newspaper, purse, drum, car. Very likely, the easiest word to remember would be platypus, the unusual one. The same thing occurs with memories of events. An event may be available to memory because it is unusual, yet the availability heuristic leads us to judge that the event is common. Did you catch that? In these cases, the availability heuristic makes us think the exact opposite of the true frequency. We end up thinking something is common because it is unusual (and therefore memorable). Yikes.

The misapplication of the availability heuristic sometimes has unfortunate results. For example, if you went to K-12 school in the US over the past 10 years, it is extremely likely that you have participated in lockdown and active shooter drills. Of course, everyone is trying to prevent the tragedy of another school shooting. And believe us, we are not trying to minimize how terrible the tragedy is. But the truth of the matter is, school shootings are extremely rare. Because the federal government does not keep a database of school shootings, the Washington Post has maintained their own running tally. Between 1999 and January 2020 (the date of the most recent school shooting with a death in the US at of the time this paragraph was written), the Post reported a total of 254 people died in school shootings in the US. Not 254 per year, 254 total. That is an average of 12 per year. Of course, that is 254 people who should not have died (particularly because many were children), but in a country with approximately 60,000,000 students and teachers, this is a very small risk.

But many students and teachers are terrified that they will be victims of school shootings because of the availability heuristic. It is so easy to think of examples (they are very available to memory) that people believe the event is very common. It is not. And there is a downside to this. We happen to believe that there is an enormous gun violence problem in the United States. According the the Centers for Disease Control and Prevention, there were 39,773 firearm deaths in the US in 2017. Fifteen of those deaths were in school shootings, according to the Post. 60% of those deaths were suicides. When people pay attention to the school shooting risk (low), they often fail to notice the much larger risk.

And examples like this are by no means unique. The authors of this book have been teaching psychology since the 1990’s. We have been able to make the exact same arguments about the misapplication of the availability heuristics and keep them current by simply swapping out for the “fear of the day.” In the 1990’s it was children being kidnapped by strangers (it was known as “stranger danger”) despite the facts that kidnappings accounted for only 2% of the violent crimes committed against children, and only 24% of kidnappings are committed by strangers (US Department of Justice, 2007). This fear overlapped with the fear of terrorism that gripped the country after the 2001 terrorist attacks on the World Trade Center and US Pentagon and still plagues the population of the US somewhat in 2020. After a well-publicized, sensational act of violence, people are extremely likely to increase their estimates of the chances that they, too, will be victims of terror. Think about the reality, however. In October of 2001, a terrorist mailed anthrax spores to members of the US government and a number of media companies. A total of five people died as a result of this attack. The nation was nearly paralyzed by the fear of dying from the attack; in reality the probability of an individual person dying was 0.00000002.

The availability heuristic can lead you to make incorrect judgments in a school setting as well. For example, suppose you are trying to decide if you should take a class from a particular math professor. You might try to make a judgment of how good a teacher she is by recalling instances of friends and acquaintances making comments about her teaching skill. You may have some examples that suggest that she is a poor teacher very available to memory, so on the basis of the availability heuristic you judge her a poor teacher and decide to take the class from someone else. What if, however, the instances you recalled were all from the same person, and this person happens to be a very colorful storyteller? The subsequent ease of remembering the instances might not indicate that the professor is a poor teacher after all.

Although the availability heuristic is obviously important, it is not the only judgment heuristic we use. Amos Tversky and Daniel Kahneman examined the role of heuristics in inductive reasoning in a long series of studies. Kahneman received a Nobel Prize in Economics for this research in 2002, and Tversky would have certainly received one as well if he had not died of melanoma at age 59 in 1996 (Nobel Prizes are not awarded posthumously). Kahneman and Tversky demonstrated repeatedly that people do not reason in ways that are consistent with the laws of probability. They identified several heuristic strategies that people use instead to make judgments about likelihood. The importance of this work for economics (and the reason that Kahneman was awarded the Nobel Prize) is that earlier economic theories had assumed that people do make judgments rationally, that is, in agreement with the laws of probability.

Another common heuristic that people use for making judgments is the  representativeness heuristic (Kahneman & Tversky 1973). Suppose we describe a person to you. He is quiet and shy, has an unassuming personality, and likes to work with numbers. Is this person more likely to be an accountant or an attorney? If you said accountant, you were probably using the representativeness heuristic. Our imaginary person is judged likely to be an accountant because he resembles, or is representative of the concept of, an accountant. When research participants are asked to make judgments such as these, the only thing that seems to matter is the representativeness of the description. For example, if told that the person described is in a room that contains 70 attorneys and 30 accountants, participants will still assume that he is an accountant.

inductive reasoning :  a type of reasoning in which we make judgments about likelihood from sets of evidence

inductively strong argument :  an inductive argument in which the beginning statements lead to a conclusion that is probably true

heuristic :  a shortcut strategy that we use to make judgments and solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

availability heuristic :  judging the frequency or likelihood of some event type according to how easily examples of the event can be called to mind (i.e., how available they are to memory)

representativeness heuristic:   judging the likelihood that something is a member of a category on the basis of how much it resembles a typical category member (i.e., how representative it is of the category)

Type 1 thinking : fast, automatic, and emotional thinking.

Type 2 thinking : slow, effortful, and logical thinking.

• What percentage of workplace homicides are co-worker violence?

Many people get these questions wrong. The answers are 10%; stairs; skin; 6%. How close were your answers? Explain how the availability heuristic might have led you to make the incorrect judgments.

• Can you think of some other judgments that you have made (or beliefs that you have) that might have been influenced by the availability heuristic?

7.3 Problem Solving

• Please take a few minutes to list a number of problems that you are facing right now.
• Now write about a problem that you recently solved.
• What is your definition of a problem?

Mary has a problem. Her daughter, ordinarily quite eager to please, appears to delight in being the last person to do anything. Whether getting ready for school, going to piano lessons or karate class, or even going out with her friends, she seems unwilling or unable to get ready on time. Other people have different kinds of problems. For example, many students work at jobs, have numerous family commitments, and are facing a course schedule full of difficult exams, assignments, papers, and speeches. How can they find enough time to devote to their studies and still fulfill their other obligations? Speaking of students and their problems: Show that a ball thrown vertically upward with initial velocity v0 takes twice as much time to return as to reach the highest point (from Spiegel, 1981).

These are three very different situations, but we have called them all problems. What makes them all the same, despite the differences? A psychologist might define a  problem   as a situation with an initial state, a goal state, and a set of possible intermediate states. Somewhat more meaningfully, we might consider a problem a situation in which you are in here one state (e.g., daughter is always late), you want to be there in another state (e.g., daughter is not always late), and with no obvious way to get from here to there. Defined this way, each of the three situations we outlined can now be seen as an example of the same general concept, a problem. At this point, you might begin to wonder what is not a problem, given such a general definition. It seems that nearly every non-routine task we engage in could qualify as a problem. As long as you realize that problems are not necessarily bad (it can be quite fun and satisfying to rise to the challenge and solve a problem), this may be a useful way to think about it.

Can we identify a set of problem-solving skills that would apply to these very different kinds of situations? That task, in a nutshell, is a major goal of this section. Let us try to begin to make sense of the wide variety of ways that problems can be solved with an important observation: the process of solving problems can be divided into two key parts. First, people have to notice, comprehend, and represent the problem properly in their minds (called  problem representation ). Second, they have to apply some kind of solution strategy to the problem. Psychologists have studied both of these key parts of the process in detail.

When you first think about the problem-solving process, you might guess that most of our difficulties would occur because we are failing in the second step, the application of strategies. Although this can be a significant difficulty much of the time, the more important source of difficulty is probably problem representation. In short, we often fail to solve a problem because we are looking at it, or thinking about it, the wrong way.

problem :  a situation in which we are in an initial state, have a desired goal state, and there is a number of possible intermediate states (i.e., there is no obvious way to get from the initial to the goal state)

problem representation :  noticing, comprehending and forming a mental conception of a problem

Defining and Mentally Representing Problems in Order to Solve Them

So, the main obstacle to solving a problem is that we do not clearly understand exactly what the problem is. Recall the problem with Mary’s daughter always being late. One way to represent, or to think about, this problem is that she is being defiant. She refuses to get ready in time. This type of representation or definition suggests a particular type of solution. Another way to think about the problem, however, is to consider the possibility that she is simply being sidetracked by interesting diversions. This different conception of what the problem is (i.e., different representation) suggests a very different solution strategy. For example, if Mary defines the problem as defiance, she may be tempted to solve the problem using some kind of coercive tactics, that is, to assert her authority as her mother and force her to listen. On the other hand, if Mary defines the problem as distraction, she may try to solve it by simply removing the distracting objects.

As you might guess, when a problem is represented one way, the solution may seem very difficult, or even impossible. Seen another way, the solution might be very easy. For example, consider the following problem (from Nasar, 1998):

Two bicyclists start 20 miles apart and head toward each other, each going at a steady rate of 10 miles per hour. At the same time, a fly that travels at a steady 15 miles per hour starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner until he is crushed between the two front wheels. Question: what total distance did the fly cover?

Please take a few minutes to try to solve this problem.

Most people represent this problem as a question about a fly because, well, that is how the question is asked. The solution, using this representation, is to figure out how far the fly travels on the first leg of its journey, then add this total to how far it travels on the second leg of its journey (when it turns around and returns to the first bicycle), then continue to add the smaller distance from each leg of the journey until you converge on the correct answer. You would have to be quite skilled at math to solve this problem, and you would probably need some time and pencil and paper to do it.

If you consider a different representation, however, you can solve this problem in your head. Instead of thinking about it as a question about a fly, think about it as a question about the bicycles. They are 20 miles apart, and each is traveling 10 miles per hour. How long will it take for the bicycles to reach each other? Right, one hour. The fly is traveling 15 miles per hour; therefore, it will travel a total of 15 miles back and forth in the hour before the bicycles meet. Represented one way (as a problem about a fly), the problem is quite difficult. Represented another way (as a problem about two bicycles), it is easy. Changing your representation of a problem is sometimes the best—sometimes the only—way to solve it.

Unfortunately, however, changing a problem’s representation is not the easiest thing in the world to do. Often, problem solvers get stuck looking at a problem one way. This is called  fixation . Most people who represent the preceding problem as a problem about a fly probably do not pause to reconsider, and consequently change, their representation. A parent who thinks her daughter is being defiant is unlikely to consider the possibility that her behavior is far less purposeful.

Problem-solving fixation was examined by a group of German psychologists called Gestalt psychologists during the 1930’s and 1940’s. Karl Dunker, for example, discovered an important type of failure to take a different perspective called  functional fixedness . Imagine being a participant in one of his experiments. You are asked to figure out how to mount two candles on a door and are given an assortment of odds and ends, including a small empty cardboard box and some thumbtacks. Perhaps you have already figured out a solution: tack the box to the door so it forms a platform, then put the candles on top of the box. Most people are able to arrive at this solution. Imagine a slight variation of the procedure, however. What if, instead of being empty, the box had matches in it? Most people given this version of the problem do not arrive at the solution given above. Why? Because it seems to people that when the box contains matches, it already has a function; it is a matchbox. People are unlikely to consider a new function for an object that already has a function. This is functional fixedness.

Mental set is a type of fixation in which the problem solver gets stuck using the same solution strategy that has been successful in the past, even though the solution may no longer be useful. It is commonly seen when students do math problems for homework. Often, several problems in a row require the reapplication of the same solution strategy. Then, without warning, the next problem in the set requires a new strategy. Many students attempt to apply the formerly successful strategy on the new problem and therefore cannot come up with a correct answer.

The thing to remember is that you cannot solve a problem unless you correctly identify what it is to begin with (initial state) and what you want the end result to be (goal state). That may mean looking at the problem from a different angle and representing it in a new way. The correct representation does not guarantee a successful solution, but it certainly puts you on the right track.

A bit more optimistically, the Gestalt psychologists discovered what may be considered the opposite of fixation, namely  insight . Sometimes the solution to a problem just seems to pop into your head. Wolfgang Kohler examined insight by posing many different problems to chimpanzees, principally problems pertaining to their acquisition of out-of-reach food. In one version, a banana was placed outside of a chimpanzee’s cage and a short stick inside the cage. The stick was too short to retrieve the banana, but was long enough to retrieve a longer stick also located outside of the cage. This second stick was long enough to retrieve the banana. After trying, and failing, to reach the banana with the shorter stick, the chimpanzee would try a couple of random-seeming attempts, react with some apparent frustration or anger, then suddenly rush to the longer stick, the correct solution fully realized at this point. This sudden appearance of the solution, observed many times with many different problems, was termed insight by Kohler.

Lest you think it pertains to chimpanzees only, Karl Dunker demonstrated that children also solve problems through insight in the 1930s. More importantly, you have probably experienced insight yourself. Think back to a time when you were trying to solve a difficult problem. After struggling for a while, you gave up. Hours later, the solution just popped into your head, perhaps when you were taking a walk, eating dinner, or lying in bed.

fixation :  when a problem solver gets stuck looking at a problem a particular way and cannot change his or her representation of it (or his or her intended solution strategy)

functional fixedness :  a specific type of fixation in which a problem solver cannot think of a new use for an object that already has a function

mental set :  a specific type of fixation in which a problem solver gets stuck using the same solution strategy that has been successful in the past

insight :  a sudden realization of a solution to a problem

Solving Problems by Trial and Error

Correctly identifying the problem and your goal for a solution is a good start, but recall the psychologist’s definition of a problem: it includes a set of possible intermediate states. Viewed this way, a problem can be solved satisfactorily only if one can find a path through some of these intermediate states to the goal. Imagine a fairly routine problem, finding a new route to school when your ordinary route is blocked (by road construction, for example). At each intersection, you may turn left, turn right, or go straight. A satisfactory solution to the problem (of getting to school) is a sequence of selections at each intersection that allows you to wind up at school.

If you had all the time in the world to get to school, you might try choosing intermediate states randomly. At one corner you turn left, the next you go straight, then you go left again, then right, then right, then straight. Unfortunately, trial and error will not necessarily get you where you want to go, and even if it does, it is not the fastest way to get there. For example, when a friend of ours was in college, he got lost on the way to a concert and attempted to find the venue by choosing streets to turn onto randomly (this was long before the use of GPS). Amazingly enough, the strategy worked, although he did end up missing two out of the three bands who played that night.

Trial and error is not all bad, however. B.F. Skinner, a prominent behaviorist psychologist, suggested that people often behave randomly in order to see what effect the behavior has on the environment and what subsequent effect this environmental change has on them. This seems particularly true for the very young person. Picture a child filling a household’s fish tank with toilet paper, for example. To a child trying to develop a repertoire of creative problem-solving strategies, an odd and random behavior might be just the ticket. Eventually, the exasperated parent hopes, the child will discover that many of these random behaviors do not successfully solve problems; in fact, in many cases they create problems. Thus, one would expect a decrease in this random behavior as a child matures. You should realize, however, that the opposite extreme is equally counterproductive. If the children become too rigid, never trying something unexpected and new, their problem solving skills can become too limited.

Effective problem solving seems to call for a happy medium that strikes a balance between using well-founded old strategies and trying new ground and territory. The individual who recognizes a situation in which an old problem-solving strategy would work best, and who can also recognize a situation in which a new untested strategy is necessary is halfway to success.

Solving Problems with Algorithms and Heuristics

For many problems there is a possible strategy available that will guarantee a correct solution. For example, think about math problems. Math lessons often consist of step-by-step procedures that can be used to solve the problems. If you apply the strategy without error, you are guaranteed to arrive at the correct solution to the problem. This approach is called using an  algorithm , a term that denotes the step-by-step procedure that guarantees a correct solution. Because algorithms are sometimes available and come with a guarantee, you might think that most people use them frequently. Unfortunately, however, they do not. As the experience of many students who have struggled through math classes can attest, algorithms can be extremely difficult to use, even when the problem solver knows which algorithm is supposed to work in solving the problem. In problems outside of math class, we often do not even know if an algorithm is available. It is probably fair to say, then, that algorithms are rarely used when people try to solve problems.

Because algorithms are so difficult to use, people often pass up the opportunity to guarantee a correct solution in favor of a strategy that is much easier to use and yields a reasonable chance of coming up with a correct solution. These strategies are called  problem solving heuristics . Similar to what you saw in section 6.2 with reasoning heuristics, a problem solving heuristic is a shortcut strategy that people use when trying to solve problems. It usually works pretty well, but does not guarantee a correct solution to the problem. For example, one problem solving heuristic might be “always move toward the goal” (so when trying to get to school when your regular route is blocked, you would always turn in the direction you think the school is). A heuristic that people might use when doing math homework is “use the same solution strategy that you just used for the previous problem.”

By the way, we hope these last two paragraphs feel familiar to you. They seem to parallel a distinction that you recently learned. Indeed, algorithms and problem-solving heuristics are another example of the distinction between Type 1 thinking and Type 2 thinking.

Although it is probably not worth describing a large number of specific heuristics, two observations about heuristics are worth mentioning. First, heuristics can be very general or they can be very specific, pertaining to a particular type of problem only. For example, “always move toward the goal” is a general strategy that you can apply to countless problem situations. On the other hand, “when you are lost without a functioning gps, pick the most expensive car you can see and follow it” is specific to the problem of being lost. Second, all heuristics are not equally useful. One heuristic that many students know is “when in doubt, choose c for a question on a multiple-choice exam.” This is a dreadful strategy because many instructors intentionally randomize the order of answer choices. Another test-taking heuristic, somewhat more useful, is “look for the answer to one question somewhere else on the exam.”

You really should pay attention to the application of heuristics to test taking. Imagine that while reviewing your answers for a multiple-choice exam before turning it in, you come across a question for which you originally thought the answer was c. Upon reflection, you now think that the answer might be b. Should you change the answer to b, or should you stick with your first impression? Most people will apply the heuristic strategy to “stick with your first impression.” What they do not realize, of course, is that this is a very poor strategy (Lilienfeld et al, 2009). Most of the errors on exams come on questions that were answered wrong originally and were not changed (so they remain wrong). There are many fewer errors where we change a correct answer to an incorrect answer. And, of course, sometimes we change an incorrect answer to a correct answer. In fact, research has shown that it is more common to change a wrong answer to a right answer than vice versa (Bruno, 2001).

The belief in this poor test-taking strategy (stick with your first impression) is based on the  confirmation bias   (Nickerson, 1998; Wason, 1960). You first saw the confirmation bias in Module 1, but because it is so important, we will repeat the information here. People have a bias, or tendency, to notice information that confirms what they already believe. Somebody at one time told you to stick with your first impression, so when you look at the results of an exam you have taken, you will tend to notice the cases that are consistent with that belief. That is, you will notice the cases in which you originally had an answer correct and changed it to the wrong answer. You tend not to notice the other two important (and more common) cases, changing an answer from wrong to right, and leaving a wrong answer unchanged.

Because heuristics by definition do not guarantee a correct solution to a problem, mistakes are bound to occur when we employ them. A poor choice of a specific heuristic will lead to an even higher likelihood of making an error.

algorithm :  a step-by-step procedure that guarantees a correct solution to a problem

problem solving heuristic :  a shortcut strategy that we use to solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

confirmation bias :  people’s tendency to notice information that confirms what they already believe

An Effective Problem-Solving Sequence

You may be left with a big question: If algorithms are hard to use and heuristics often don’t work, how am I supposed to solve problems? Robert Sternberg (1996), as part of his theory of what makes people successfully intelligent (Module 8) described a problem-solving sequence that has been shown to work rather well:

• Identify the existence of a problem.  In school, problem identification is often easy; problems that you encounter in math classes, for example, are conveniently labeled as problems for you. Outside of school, however, realizing that you have a problem is a key difficulty that you must get past in order to begin solving it. You must be very sensitive to the symptoms that indicate a problem.
• Define the problem.  Suppose you realize that you have been having many headaches recently. Very likely, you would identify this as a problem. If you define the problem as “headaches,” the solution would probably be to take aspirin or ibuprofen or some other anti-inflammatory medication. If the headaches keep returning, however, you have not really solved the problem—likely because you have mistaken a symptom for the problem itself. Instead, you must find the root cause of the headaches. Stress might be the real problem. For you to successfully solve many problems it may be necessary for you to overcome your fixations and represent the problems differently. One specific strategy that you might find useful is to try to define the problem from someone else’s perspective. How would your parents, spouse, significant other, doctor, etc. define the problem? Somewhere in these different perspectives may lurk the key definition that will allow you to find an easier and permanent solution.
• Formulate strategy.  Now it is time to begin planning exactly how the problem will be solved. Is there an algorithm or heuristic available for you to use? Remember, heuristics by their very nature guarantee that occasionally you will not be able to solve the problem. One point to keep in mind is that you should look for long-range solutions, which are more likely to address the root cause of a problem than short-range solutions.
• Represent and organize information.  Similar to the way that the problem itself can be defined, or represented in multiple ways, information within the problem is open to different interpretations. Suppose you are studying for a big exam. You have chapters from a textbook and from a supplemental reader, along with lecture notes that all need to be studied. How should you (represent and) organize these materials? Should you separate them by type of material (text versus reader versus lecture notes), or should you separate them by topic? To solve problems effectively, you must learn to find the most useful representation and organization of information.
• Allocate resources.  This is perhaps the simplest principle of the problem solving sequence, but it is extremely difficult for many people. First, you must decide whether time, money, skills, effort, goodwill, or some other resource would help to solve the problem Then, you must make the hard choice of deciding which resources to use, realizing that you cannot devote maximum resources to every problem. Very often, the solution to problem is simply to change how resources are allocated (for example, spending more time studying in order to improve grades).
• Monitor and evaluate solutions.  Pay attention to the solution strategy while you are applying it. If it is not working, you may be able to select another strategy. Another fact you should realize about problem solving is that it never does end. Solving one problem frequently brings up new ones. Good monitoring and evaluation of your problem solutions can help you to anticipate and get a jump on solving the inevitable new problems that will arise.

Please note that this as  an  effective problem-solving sequence, not  the  effective problem solving sequence. Just as you can become fixated and end up representing the problem incorrectly or trying an inefficient solution, you can become stuck applying the problem-solving sequence in an inflexible way. Clearly there are problem situations that can be solved without using these skills in this order.

Additionally, many real-world problems may require that you go back and redefine a problem several times as the situation changes (Sternberg et al. 2000). For example, consider the problem with Mary’s daughter one last time. At first, Mary did represent the problem as one of defiance. When her early strategy of pleading and threatening punishment was unsuccessful, Mary began to observe her daughter more carefully. She noticed that, indeed, her daughter’s attention would be drawn by an irresistible distraction or book. Fresh with a re-representation of the problem, she began a new solution strategy. She began to remind her daughter every few minutes to stay on task and remind her that if she is ready before it is time to leave, she may return to the book or other distracting object at that time. Fortunately, this strategy was successful, so Mary did not have to go back and redefine the problem again.

Pick one or two of the problems that you listed when you first started studying this section and try to work out the steps of Sternberg’s problem solving sequence for each one.

a mental representation of a category of things in the world

an assumption about the truth of something that is not stated. Inferences come from our prior knowledge and experience, and from logical reasoning

knowledge about one’s own cognitive processes; thinking about your thinking

individuals who are less competent tend to overestimate their abilities more than individuals who are more competent do

Thinking like a scientist in your everyday life for the purpose of drawing correct conclusions. It entails skepticism; an ability to identify biases, distortions, omissions, and assumptions; and excellent deductive and inductive reasoning, and problem solving skills.

a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided

an inclination, tendency, leaning, or prejudice

a type of reasoning in which the conclusion is guaranteed to be true any time the statements leading up to it are true

a set of statements in which the beginning statements lead to a conclusion

an argument for which true beginning statements guarantee that the conclusion is true

a type of reasoning in which we make judgments about likelihood from sets of evidence

an inductive argument in which the beginning statements lead to a conclusion that is probably true

fast, automatic, and emotional thinking

slow, effortful, and logical thinking

a shortcut strategy that we use to make judgments and solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

udging the frequency or likelihood of some event type according to how easily examples of the event can be called to mind (i.e., how available they are to memory)

judging the likelihood that something is a member of a category on the basis of how much it resembles a typical category member (i.e., how representative it is of the category)

a situation in which we are in an initial state, have a desired goal state, and there is a number of possible intermediate states (i.e., there is no obvious way to get from the initial to the goal state)

noticing, comprehending and forming a mental conception of a problem

when a problem solver gets stuck looking at a problem a particular way and cannot change his or her representation of it (or his or her intended solution strategy)

a specific type of fixation in which a problem solver cannot think of a new use for an object that already has a function

a specific type of fixation in which a problem solver gets stuck using the same solution strategy that has been successful in the past

a sudden realization of a solution to a problem

a step-by-step procedure that guarantees a correct solution to a problem

The tendency to notice and pay attention to information that confirms your prior beliefs and to ignore information that disconfirms them.

a shortcut strategy that we use to solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

Introduction to Psychology Copyright © 2020 by Ken Gray; Elizabeth Arnott-Hill; and Or'Shaundra Benson is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

Share This Book

Analytical Thinking, Critical Analysis, and Problem Solving Guide

• Post author: Samir Saif
• Post published: September 5, 2023
• Post category: marketing skills
• Post comments: 4 Comments
• Post last modified: November 10, 2023
• Reading time: 9 mins read

Analytical thinking; is a mental process that entails dissecting an issue or situation into its constituent parts, investigating their relationships, and reaching conclusions based on facts and logic.

It is not about trusting instincts or making assumptions; rather, it is about studying details, recognizing patterns, and developing a full understanding. Whether you’re a seasoned professional, an aspiring entrepreneur, or a curious mind, improving analytical thinking can help you solve problems more effectively.

Table of Contents

Analytical Thinking’s Importance in Problem Solving

Certainly! Analytical thinking entails the capacity to gather pertinent information, critically assess evidence, and reach logical conclusions. It enables you to:

• Identify Root Causes: Analytical thinking allows you to delve deeper into a problem to find the underlying causes rather than just addressing surface-level symptoms.
• Reduce Risks: Analytical thinking can help discover potential risks and obstacles connected with various solutions. This kind of thinking encourages constant progress and the generation of new ideas.
• Improve Communication: Analytical thinking enables you to deliver clear and well-structured explanations while giving answers to others.
• Adaptability : Analytical thinking gives you a flexible attitude.
• Learning and Development: Analytical thinking improves your cognitive skills, allowing you to learn from prior experiences and apply those lessons to new situations.
• Problem Prevention: By examining previous difficulties, you can find trends and patterns.
• Analytical thinking is, in essence, the foundation of effective problem-solving. It enables you to approach problems methodically, make well-informed judgments, and eventually get better results.

Key Components of Analytical Thinking

Analytical thinking is a multifaceted process including a beautifully woven tapestry of observation, inquiry, and logic. Engage your curiosity as you approach a complex task and see patterns emerge, similar to stars in the night sky.

These patterns direct your thinking toward greater comprehension. Your understanding grows as you progress, and your analytical thinking becomes a light of clarity, guiding people through the fog of complexity.

Your tapestry is complete as you approach the shores of conclusion, a tribute to the power of analytical thinking. Embrace your curiosity, navigate the waters of observation, and let the stars of logic guide you. Remember that the art of analytical thinking is a magnificent journey that leads to enlightenment.

Using analytical reasoning in real-life situations

Absolutely! Let’s get started with analytical thinking! Consider yourself in a busy city, attempting to discover the shortest route to your goal. Instead than taking the first option that comes to mind, you take a moment to think about your possibilities.

This is the initial stage in analytical thinking: evaluating the situation. As you contemplate, you balance the advantages and disadvantages of each route, taking into account issues such as traffic, distance, and potential bypasses. This information gathering approach assists you in making an informed decision.

Breaking down the problem

Then you go to the second phase, which entails breaking the problem down into smaller portions. You break down the difficult job of navigating the city into manageable components, much like a puzzle.

This technique allows you to identify future difficulties and devise creative solutions. For example, you may observe a construction zone on one route but recall a shortcut that may save you time.

Read Also:  Goal Alignment: Key Strategies for Success

Analyzing the information

You employ critical thinking to assess the material you’ve received as you go. As you consider the significance of each component—time, distance, and traffic—patterns and connections emerge.

You begin to make connections and discover that, while a faster route may appear enticing, heavy traffic at certain times of day might make it a frustrating experience.

Make a decision

Making a decision in the last step necessitates a complete comprehension of the circumstance as well as critical analysis. Analytical thinking entails investigating alternatives, comprehending nuances, and making informed decisions.

This approach can lead to optimal, well-thought-out, and adaptable solutions, whether navigating a city, tackling a complex project, or making life decisions. Analytic thinking allows one to make informed judgments that benefit both the situation and the individual.

Strategies to Enhance Analytical Thinking Skills

Developing strong analytical thinking abilities is a journey that opens up new possibilities for comprehension and issue solving.

Consider yourself on an exciting mental journey where every challenge is an opportunity for improvement. Here’s a step-by-step guide to cultivating and improving your analytical thinking talents.

Accept curiosity

Begin by embracing your curiosity. Allow your thoughts to roam, pondering about the hows and whys of the world around you.

Allow yourself to immerse yourself completely in the complexities of a complex topic, such as climate change. “What are the underlying causes of this phenomenon?” Two decent places to start are “How do different variables interact to shape its outcomes?”.

Improve your observing abilities

Then, put your observation abilities to the test. Pay close attention to details that would otherwise go undetected. Instead of just gazing at the colors and shapes, try to figure out the brushstrokes, the play of light and shadow, and the feelings they create, as if you were studying a painting.

When analyzing data, look underneath the surface figures for trends, anomalies, and patterns that can reveal hidden insights.

Accept critical thinking

Learn to think critically as you progress. Examine your assumptions and look for alternative points of view. Assume you’re looking into a business problem, such as declining sales.

Instead than jumping to conclusions, investigate the matter from all angles. Consider changes in the sector, client preferences, and even internal corporate processes. This broader viewpoint can lead to creative solutions.

Read Also:  Business Development: Strategies and Tips for Success

Experiment with logical reasoning

Also, practice logical reasoning. Improve your ability to connect the dots and build logical chains of reasoning. As if you were assembling a jigsaw puzzle, each piece must fit snugly into the whole.

Consider how numerous variables such as population growth, infrastructure, and transportation systems logically interconnect when dealing with a complex issue such as urban congestion.

Improve your problem-solving skills

Develop your problem-solving abilities as well. For example, if you’re struggling with a personal issue, such as time management, break it down into smaller components. Analyze your daily routine to discover bottlenecks and develop a strategy to overcome them.

Foster continuous learning

Finally, encourage ongoing learning by broadening your knowledge base and investigating new domains. Imagine yourself as a discerning thinker analyzing the world’s intricacies and unraveling secrets.

Remember that progress, not perfection, is the goal. Every task, question, and conundrum you solve puts you one step closer to being an analytical juggernaut. Continue to explore and study to see your critical thinking skills soar to new heights.

Applying analytical reasoning to work

Assume you are a business owner who wants to boost client happiness. An analytical thinker would collect and analyze client input to uncover frequent pain issues.

You can adopt targeted adjustments that address the fundamental causes of unhappiness by detecting patterns in feedback data.

How can you demonstrate analytical skills on a resume?

Analytical skills on your CV can set you apart and leave a lasting impression on potential employers. Make your CV into a canvas, describing specific instances where your analytical skills were put to use.

Share how you methodically dissected a challenging topic or situation, revealing insights that aided your decision-making.

If you were tasked with optimizing a company’s supply chain, for example, dig further into data on inventory levels, production rates, and distribution deadlines.

Explain how your study found a bottleneck in the distribution network, leading to a realignment suggestion that saved the organization time and money.

Storytelling is key. Create a fascinating story about how your analytical abilities helped solve a tough problem, demonstrating your abilities and attracting the reader.

Your CV should read like a motivational trip through your analytical abilities, inspiring companies with your future contributions to their organization.

What is a case study of analytical thinking?

Absolutely! Let me give you an excellent example of analytical thinking that perfectly expresses its essence. Maya, a young scientist in this example, is dedicated to discovering a long-term solution for safe drinking water in rural areas.

She performs extensive research on water supplies, toxins, and local circumstances, looking for patterns and anomalies. She develops the concept that heavy rains increase runoff, resulting in higher levels of water contamination.

Maya designs controlled experiments in a lab setting to test her idea, acquiring quantifiable information through manipulation and observation.

Maya’s investigation continues, and she explores the big picture, imagining a multi-faceted solution that involves rainwater gathering, enhanced filtration systems, and community education.

She anticipates problems and works with engineers, social workers, and community leaders to refine her ideas and ensure their viability.

Her journey exemplifies how analytical thinking can lead to transformational solutions, and it motivates us to tackle complex challenges with curiosity, diligence, and the hope that careful analysis may design a better future.

Final Thoughts

Analytical thinking is more than just a cognitive skill; it’s a mindset that empowers you to unravel complexity, make informed choices, and navigate challenges with confidence.

You will be better able to handle the intricacies of the modern world as your analytical thinking skills increase, whether in business, academics, or daily life. Accept the power of analytical thinking, and your decision-making and problem-solving abilities will soar.

Share this content

• Opens in a new window X
• Opens in a new window Facebook
• Opens in a new window WhatsApp

You Might Also Like

Marketing Skills in Business: Driving Success in the Digital Age

Marketing Skills Mastery: A Comprehensive Success Guide

This post has 4 comments.

Pingback: Performance Appraisal to Improve Employee Engagement

Pingback: Self-Reflection Power: Uncovering Your True Potential

Pingback: Self-Esteem Development: Increasing Confidence and Empowerment

Pingback: Stress-Tolerance Mindset: Effective Stress Management Strategies

Leave a Reply Cancel reply

Logical Thinking

What is logical thinking.

Logical thinking is a fundamental cognitive ability that allows individuals to analyze, reason, and make sound decisions based on objective facts and evidence. It entails the ability to think critically, systematically, and coherentlWithin the context of problem-solving and decision-making, logical thinking enables individuals to identify patterns, recognize relationships, and draw logical conclusions.

Key Features of Logical Thinking

1. Deductive Reasoning: Logical thinking involves deducing specific conclusions from general principles or premises. It follows a top-down approach, using logic and established rules to reach valid conclusions.

2. Inductive Reasoning: This component of logical thinking involves inferring general principles or conclusions from specific observations or instances. Inductive reasoning utilizes patterns, data, and examples to arrive at probable conclusions.

3. Analytical Skills: Logical thinking requires strong analytical skills to break down complex problems or situations into smaller, more manageable components. By breaking down the elements and identifying the relationships between them, individuals can better understand the larger picture and draw logical conclusions.

4. Critical Thinking: Logical thinking relies heavily on critical thinking to evaluate arguments, ideas, and evidence objectively. It involves questioning assumptions, identifying biases, and applying logical principles to assess the validity of statements and arguments.

5. Problem-solving: Individuals with strong logical thinking skills excel in problem-solving. They can approach problems analytically, logically, and systematically, breaking them down into smaller parts, assessing possible solutions, and choosing the most effective course of action.

6. Decision-making: Logical thinking is closely linked to effective decision-making. By evaluating all available information, assessing potential outcomes, and considering the logical consequences of each option, individuals can make informed decisions that are rational and objective.

Why is Logical Thinking Important?

Logical thinking is a crucial skillset in various aspects of life and work. Whether in academic pursuits, professional endeavors, or everyday situations, honing logical thinking abilities can lead to better problem-solving, more effective decision-making, and improved overall cognitive functioning.

In academic settings, logical thinking enables students to excel in subjects such as mathematics, science, and philosophy, where reasoning and analytical skills are paramount. In the workplace, logical thinking is highly valued across disciplines, including business, engineering, law, and technology, as it allows employees to solve complex problems and make sound decisions based on objective analysis.

Furthermore, logical thinking fosters a more rational and critical approach to information, discouraging the acceptance of fallacious arguments or misinformation. It promotes a deeper understanding of complex issues and encourages individuals to challenge assumptions, leading to a more well-informed and intellectually engaged society.

Why Assess a Candidate's Logical Thinking Skill Level?

Assessing a candidate's logical thinking skill level is essential for organizations seeking to hire individuals who can approach complex problems with precision and make informed decisions based on objective analysis. Evaluating logical thinking abilities during the hiring process can provide numerous benefits, including:

Better Problem-Solving: Logical thinking is directly linked to effective problem-solving. By assessing a candidate's logical thinking skills, organizations can identify individuals who possess the ability to analyze and break down complex problems into manageable components, leading to more efficient and innovative solutions.

Enhanced Decision-Making: Logical thinking enables individuals to weigh evidence, consider multiple perspectives, and draw logical conclusions. By assessing a candidate's logical thinking abilities, organizations can identify individuals who can make well-informed and rational decisions based on objective analysis, minimizing the risks associated with subjective or biased decision-making.

Improved Critical Thinking: Critical thinking is closely intertwined with logical thinking. By evaluating a candidate's logical thinking abilities, organizations can gauge their capacity to evaluate arguments, identify logical fallacies, and assess the validity of information. This ensures that organizations hire individuals who can think critically and approach information with a logical and analytical mindset.

Efficient Resource Allocation: Assessing a candidate's logical thinking skill level helps organizations allocate resources more effectively. Individuals with strong logical thinking abilities can analyze situations, identify potential challenges or risks, and develop well-structured action plans. This allows organizations to optimize their resources and mitigate potential setbacks or roadblocks.

Promotion of Innovation: Logical thinking is crucial for fostering innovation and creativity within organizations. Candidates with strong logical thinking skills are more likely to think outside the box, consider alternative solutions, and explore new possibilities. Assessing logical thinking abilities can help organizations identify individuals who can bring fresh perspectives and contribute to innovative problem-solving.

Reduced Errors and Mistakes: Effective logical thinking minimizes errors and mistakes in decision-making and problem-solving processes. By assessing a candidate's logical thinking skills, organizations can ensure that they hire individuals who can critically evaluate information, identify inconsistencies, and avoid errors that may have significant consequences in critical or sensitive situations.

By assessing a candidate's logical thinking skill level, organizations can ensure that they hire individuals who possess the cognitive abilities necessary for success in problem-solving, decision-making, and critical analysis. Alooba's comprehensive assessment platform can help organizations evaluate logical thinking skills effectively and identify top-quality candidates to drive their success.

Assessing a Candidate's Logical Thinking Skill Level with Alooba

Alooba's advanced assessment platform offers a seamless solution for evaluating a candidate's logical thinking skill level. With our comprehensive range of assessment tools and features, organizations can confidently and efficiently assess candidates' logical thinking abilities. Here's how Alooba can help:

Customizable Tests : Alooba allows organizations to create customized logical thinking tests tailored to their specific needs. Whether it's deductive reasoning, inductive reasoning, analytical skills, or critical thinking, organizations can design tests that accurately measure a candidate's logical thinking abilities.

Versatile Test Formats : Alooba offers various test formats designed to assess logical thinking skills effectively. From multiple-choice tests that evaluate a candidate's conceptual understanding to practical assessments where candidates analyze data or write SQL statements, Alooba covers a wide range of logical thinking scenarios.

Objective Evaluation : Alooba's assessment platform utilizes an autograding system for objective evaluation of logical thinking tests. This ensures consistent and fair assessment results, enabling organizations to compare and rank candidates based on their logical thinking abilities accurately.

In-depth Assessments : For a more comprehensive evaluation of logical thinking skills, Alooba provides in-depth assessments. These assessments allow candidates to demonstrate their logical thinking abilities through tasks such as diagramming, coding, or written responses. Expert evaluators manually assess these tasks, providing valuable insights into a candidate's capabilities.

Alooba Interview Product : Alooba's structured interviews with predefined topics and questions offer an additional avenue to assess a candidate's logical thinking skills. Interviewers can use a marking guide for objective evaluation, ensuring consistency in the assessment process.

Alooba's Vision : Alooba's vision is to create a world where everyone can get the job they deserve. By assessing candidates' logical thinking skill level, organizations can make fair and informed hiring decisions, matching top-quality candidates with the opportunities they deserve.

With Alooba's comprehensive assessment platform, organizations can confidently evaluate and measure a candidate's logical thinking skill level. Save time, streamline your hiring process, and discover the candidates with the logical thinking abilities your organization needs to excel. Unleash the power of logical thinking assessments with Alooba.

Components of Logical Thinking

Logical thinking encompasses various subtopics, each contributing to an individual's overall proficiency in this valuable cognitive skill. Understanding the components of logical thinking provides a comprehensive view of the abilities necessary for effective problem-solving and decision-making. Here are some key components to consider:

Deductive Reasoning : Deductive reasoning involves drawing specific conclusions from general principles or premises. It requires individuals to apply logical rules and principles to reach valid and sound conclusions based on the given information.

Inductive Reasoning : Inductive reasoning involves making general conclusions based on specific observations or instances. Individuals utilize patterns, data, and examples to infer broader principles, allowing them to form probable conclusions.

Analytical Skills : Analytical skills are crucial for logical thinking, enabling individuals to break down complex problems or situations into smaller, more manageable components. This component focuses on identifying relationships, patterns, and underlying structures to gain a deeper understanding of the overall problem.

Critical Thinking : Critical thinking is closely associated with logical thinking, requiring individuals to assess, analyze, and evaluate arguments and evidence objectively. It involves questioning assumptions, identifying biases, and employing logical principles to determine the validity and soundness of statements and arguments.

Problem-Solving Strategies : Logical thinking plays a significant role in effective problem-solving strategies. It involves the systematic and coherent approach of breaking down problems, identifying the core issues, exploring possible solutions, and selecting the most appropriate course of action.

Decision-Making Processes : Logical thinking provides the foundation for robust decision-making processes. It allows individuals to evaluate all available information objectively, consider potential outcomes, and analyze the logical consequences of each decision. Logical thinking ensures rational and informed decision-making.

Pattern Recognition : Pattern recognition is vital in logical thinking, as it involves the ability to identify regularities, repetitions, and systematic relationships within data or information. Individuals proficient in pattern recognition can detect underlying structures and use them to draw logical insights.

Logical Communication : Logical thinking also extends to effective communication. It includes the ability to present ideas, arguments, and solutions in a logical and coherent manner, allowing others to understand and follow the train of thought.

By understanding and developing these key components, individuals can enhance their logical thinking skills, enabling them to approach problems and decision-making with precision and clarity. With Alooba's assessment platform, you can measure and evaluate these components to identify candidates who possess strong logical thinking abilities, ensuring you make informed hiring decisions.

Practical Applications of Logical Thinking

Logical thinking is a versatile cognitive skill that finds application in various aspects of life, work, and decision-making processes. By utilizing logical thinking, individuals can approach challenges, solve problems, and navigate complex situations with clarity and sound judgment. Here are some practical applications where logical thinking is commonly employed:

Problem-Solving: Logical thinking is essential for effective problem-solving. Whether it's troubleshooting technical issues, resolving conflicts, or finding innovative solutions, logical thinking enables individuals to break down problems, analyze the components, and apply logical reasoning to identify the most suitable course of action.

Critical Analysis: Logical thinking plays a significant role in critical analysis. It helps individuals evaluate information, arguments, and evidence objectively, enabling them to identify flaws, inconsistencies, or biases. By employing logical thinking, individuals can make informed judgments and arrive at well-supported conclusions.

Decision-Making: Logical thinking provides a foundation for rational decision-making. It allows individuals to consider all available information, assess potential outcomes, and analyze the logical implications of each decision. Logical thinking helps individuals make sound decisions based on objective analysis, minimizing the influence of emotions or biases.

Scientific and Mathematical Reasoning: Logical thinking is fundamental in scientific and mathematical reasoning processes. It involves following logical steps, applying rules and principles, and drawing logical conclusions. In fields such as physics, computer science, and mathematics, logical thinking ensures rigorous and accurate problem-solving.

Data Analysis: Logical thinking is crucial in data analysis. It enables individuals to identify patterns, make connections between variables, and draw logical insights from large datasets. By applying logical thinking, individuals can derive meaningful information from data, leading to informed decisions and insights.

Strategic Planning: Logical thinking is invaluable in strategic planning processes. It assists individuals in assessing the current situation, identifying goals, analyzing potential options, and formulating logical strategies. Logical thinking ensures that plans are coherent, feasible, and aligned with the organization's objectives.

Communication: Logical thinking enhances effective communication. It enables individuals to organize their thoughts in a logical and coherent manner, ensuring clear and concise communication of ideas, arguments, and instructions. Logical thinking helps individuals convey their message effectively and facilitate understanding.

By recognizing the practical applications of logical thinking, individuals and organizations can harness this cognitive skill to their advantage. Alooba's assessment platform allows you to evaluate and identify candidates with strong logical thinking skills, ensuring you have the right talent to tackle complex challenges and make informed decisions.

Roles that Require Strong Logical Thinking Skills

Logical thinking is a valuable skillset that plays a crucial role in numerous job functions. Certain roles particularly benefit from individuals who possess strong logical thinking skills. Here are some of the key roles where logical thinking abilities are highly relevant:

Data Analyst : Data analysts rely on logical thinking to interpret and analyze complex data sets. They apply logical reasoning to identify patterns, draw meaningful insights, and make data-driven recommendations.

Data Scientist : Data scientists leverage logical thinking to apply statistical models, algorithms, and machine learning techniques. They use logical reasoning to process data, test hypotheses, and develop predictive models.

Data Engineer : Data engineers employ logical thinking to design, construct, and maintain data systems. They utilize logical reasoning to develop efficient database architectures and ensure data integrity.

Insights Analyst : Insights analysts rely on logical thinking to interpret market trends, consumer behavior, and business performance. They use logical reasoning to draw meaningful conclusions from data and provide valuable insights.

Marketing Analyst : Marketing analysts utilize logical thinking to evaluate marketing strategies, measure campaign effectiveness, and analyze customer data. They apply logical reasoning to optimize marketing initiatives and drive business growth.

Product Analyst : Product analysts use logical thinking to assess market trends, user feedback, and product performance. They apply logical reasoning to identify opportunities for improvement and make data-informed decisions for product development.

Analytics Engineer : Analytics engineers employ logical thinking to design and develop systems for data analysis. They apply logical reasoning to implement data pipelines, automate data processes, and ensure accurate data reporting.

Artificial Intelligence Engineer : Artificial intelligence engineers rely on logical thinking to develop intelligent systems and algorithms. They apply logical reasoning to design and optimize algorithms for machine learning and decision-making.

Back-End Engineer : Back-end engineers use logical thinking to develop and maintain server-side applications. They apply logical reasoning to ensure seamless data flow, optimize system performance, and resolve technical issues.

Data Architect : Data architects employ logical thinking to design and structure data systems. They use logical reasoning to create data models, define data governance policies, and ensure data accuracy and integrity.

Data Governance Analyst : Data governance analysts rely on logical thinking to establish and enforce data management policies. They apply logical reasoning to ensure compliance, data quality, and secure data access.

Deep Learning Engineer : Deep learning engineers utilize logical thinking to design and implement deep learning models. They use logical reasoning to optimize neural networks, analyze model performance, and improve complex algorithms.

These roles showcase the significance of logical thinking in various job functions. By evaluating candidates' logical thinking skills using Alooba's assessment platform, organizations can identify top talent for these roles and ensure that their teams possess the necessary cognitive abilities to excel in these positions.

Associated Roles

Analytics engineer.

Analytics Engineers are responsible for preparing data for analytical or operational uses. These professionals bridge the gap between data engineering and data analysis, ensuring data is not only available but also accessible, reliable, and well-organized. They typically work with data warehousing tools, ETL (Extract, Transform, Load) processes, and data modeling, often using SQL, Python, and various data visualization tools. Their role is crucial in enabling data-driven decision making across all functions of an organization.

Artificial Intelligence Engineer

Artificial Intelligence Engineers are responsible for designing, developing, and deploying intelligent systems and solutions that leverage AI and machine learning technologies. They work across various domains such as healthcare, finance, and technology, employing algorithms, data modeling, and software engineering skills. Their role involves not only technical prowess but also collaboration with cross-functional teams to align AI solutions with business objectives. Familiarity with programming languages like Python, frameworks like TensorFlow or PyTorch, and cloud platforms is essential.

Back-End Engineer

Back-End Engineers focus on server-side web application logic and integration. They write clean, scalable, and testable code to connect the web application with the underlying services and databases. These professionals work in a variety of environments, including cloud platforms like AWS and Azure, and are proficient in programming languages such as Java, C#, and NodeJS. Their expertise extends to database management, API development, and implementing security and data protection solutions. Collaboration with front-end developers and other team members is key to creating cohesive and efficient applications.

Data Analyst

Data Analysts draw meaningful insights from complex datasets with the goal of making better decisions. Data Analysts work wherever an organization has data - these days that could be in any function, such as product, sales, marketing, HR, operations, and more.

Data Architect

Data Architects are responsible for designing, creating, deploying, and managing an organization's data architecture. They define how data is stored, consumed, integrated, and managed by different data entities and IT systems, as well as any applications using or processing that data. Data Architects ensure data solutions are built for performance and design analytics applications for various platforms. Their role is pivotal in aligning data management and digital transformation initiatives with business objectives.

Data Engineer

Data Engineers are responsible for moving data from A to B, ensuring data is always quickly accessible, correct and in the hands of those who need it. Data Engineers are the data pipeline builders and maintainers.

Data Governance Analyst

Data Governance Analysts play a crucial role in managing and protecting an organization's data assets. They establish and enforce policies and standards that govern data usage, quality, and security. These analysts collaborate with various departments to ensure data compliance and integrity, and they work with data management tools to maintain the organization's data framework. Their goal is to optimize data practices for accuracy, security, and efficiency.

Data Scientist

Data Scientists are experts in statistical analysis and use their skills to interpret and extract meaning from data. They operate across various domains, including finance, healthcare, and technology, developing models to predict future trends, identify patterns, and provide actionable insights. Data Scientists typically have proficiency in programming languages like Python or R and are skilled in using machine learning techniques, statistical modeling, and data visualization tools such as Tableau or PowerBI.

Deep Learning Engineer

Deep Learning Engineers’ role centers on the development and optimization of AI models, leveraging deep learning techniques. They are involved in designing and implementing algorithms, deploying models on various platforms, and contributing to cutting-edge research. This role requires a blend of technical expertise in Python, PyTorch or TensorFlow, and a deep understanding of neural network architectures.

Insights Analyst

Insights Analysts play a pivotal role in transforming complex data sets into actionable insights, driving business growth and efficiency. They specialize in analyzing customer behavior, market trends, and operational data, utilizing advanced tools such as SQL, Python, and BI platforms like Tableau and Power BI. Their expertise aids in decision-making across multiple channels, ensuring data-driven strategies align with business objectives.

Marketing Analyst

Marketing Analysts specialize in interpreting data to enhance marketing efforts. They analyze market trends, consumer behavior, and campaign performance to inform marketing strategies. Proficient in data analysis tools and techniques, they bridge the gap between data and marketing decision-making. Their role is crucial in tailoring marketing efforts to target audiences effectively and efficiently.

Product Analyst

Product Analysts utilize data to optimize product strategies and enhance user experiences. They work closely with product teams, leveraging skills in SQL, data visualization (e.g., Tableau), and data analysis to drive product development. Their role includes translating business requirements into technical specifications, conducting A/B testing, and presenting data-driven insights to inform product decisions. Product Analysts are key in understanding customer needs and driving product innovation.

Other names for Logical Thinking include Problem Solving , and Critical Thinking .

Ready to Assess Logical Thinking Skills?

Discover how Alooba can help you evaluate candidates' logical thinking abilities and make informed hiring decisions. Book a discovery call with our experts now.

Over 50,000 Candidates Can't Be Wrong

Our Customers Say

I was at WooliesX (Woolworths) and we used Alooba and it was a highly positive experience. We had a large number of candidates. At WooliesX, previously we were quite dependent on the designed test from the team leads. That was quite a manual process. We realised it would take too much time from us. The time saving is great. Even spending 15 minutes per candidate with a manual test would be huge - hours per week, but with Alooba we just see the numbers immediately.

Shen Liu , Logickube ( Principal at Logickube )

We get a high flow of applicants, which leads to potentially longer lead times, causing delays in the pipelines which can lead to missing out on good candidates. Alooba supports both speed and quality. The speed to return to candidates gives us a competitive advantage. Alooba provides a higher level of confidence in the people coming through the pipeline with less time spent interviewing unqualified candidates.

Scott Crowe , Canva ( Lead Recruiter - Data )

How can you accurately assess somebody's technical skills, like the same way across the board, right? We had devised a Tableau-based assessment. So it wasn't like a past/fail. It was kind of like, hey, what do they send us? Did they understand the data or the values that they're showing accurate? Where we'd say, hey, here's the credentials to access the data set. And it just wasn't really a scalable way to assess technical - just administering it, all of it was manual, but the whole process sucked!

Cole Brickley , Avicado ( Director Data Science & Business Intelligence )

The diversity of our pool has definitely improved so we just have many more candidates from just different backgrounds which I am a huge believer in. It makes the team much better, it makes our output much better and gives us more voices in terms of building the best product and service that we can.

Piers Stobbs , Cazoo ( Chief Data Officer )

I wouldn't dream of hiring somebody in a technical role without doing that technical assessment because the number of times where I've had candidates either on paper on the CV, say, I'm a SQL expert or in an interview, saying, I'm brilliant at Excel, I'm brilliant at this. And you actually put them in front of a computer, say, do this task. And some people really struggle. So you have to have that technical assessment.

Mike Yates , The British Psychological Society ( Head of Data & Analytics )

We were very quickly quite surprised with the quality of candidates we would get from Alooba. We ended up hiring eight different analysts via Alooba in about a year's time, which is quite extraordinary for us because we actually have almost never used a recruitment agency for any role. It has been our best outsourcing solution by far.

Oz Har Adir , Vio.com ( Founder & CEO )

For data engineering & analytics these take-home assignments we were doing ourselves are a bit time consuming so we wanted to automate that and also reduce the time candidates were spending on the assessment.

Sharin Fritz , Personio ( Tech Talent Acquisition )

An official website of the United States government

The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

• Publications
• Account settings

Preview improvements coming to the PMC website in October 2024. Learn More or Try it out now .

• Advanced Search
• Journal List

Does mathematics training lead to better logical thinking and reasoning? A cross-sectional assessment from students to professors

Clio cresswell.

1 School of Mathematics and Statistics, The University of Sydney, Sydney, Australia

Craig P. Speelman

2 School of Arts and Humanities, Edith Cowan University, Joondalup, Australia

Associated Data

All relevant data are within the paper and its Supporting Information files.

Mathematics is often promoted as endowing those who study it with transferable skills such as an ability to think logically and critically or to have improved investigative skills, resourcefulness and creativity in problem solving. However, there is scant evidence to back up such claims. This project tested participants with increasing levels of mathematics training on 11 well-studied rational and logical reasoning tasks aggregated from various psychological studies. These tasks, that included the Cognitive Reflection Test and the Wason Selection Task, are of particular interest as they have typically and reliably eluded participants in all studies, and results have been uncorrelated with general intelligence, education levels and other demographic information. The results in this study revealed that in general the greater the mathematics training of the participant, the more tasks were completed correctly, and that performance on some tasks was also associated with performance on others not traditionally associated. A ceiling effect also emerged. The work is deconstructed from the viewpoint of adding to the platform from which to approach the greater, and more scientifically elusive, question: are any skills associated with mathematics training innate or do they arise from skills transfer?

Introduction

Mathematics is often promoted as endowing those who study it with a number of broad thinking skills such as: an ability to think logically, analytically, critically and abstractly; having capacity to weigh evidence with impartiality. This is a view of mathematics as providing transferable skills which can be found across educational institutions, governments and corporations worldwide. A view material to the place of mathematics in curricula.

Consider the UK government’s commissioned inquiry into mathematics education “Making Mathematics Count” ascertaining the justification that “mathematical training disciplines the mind, develops logical and critical reasoning, and develops analytical and problem-solving skills to a high degree” [ 1 p11]. The Australian Mathematical Sciences Institute very broadly states in its policy document “Vision for a Maths Nation” that “Not only is mathematics the enabling discipline, it has a vital productive role planning and protecting our well-being” (emphasis in original) [ 2 ]. In Canada, British Columbia’s New 2016 curriculum K-9 expressly mentions as part of its “Goals and Rationale”: “The Mathematics program of study is designed to develop deep mathematical understanding and fluency, logical reasoning, analytical thought, and creative thinking.” [ 3 ]. Universities, too, often make such specific claims with respect to their teaching programs. “Mathematics and statistics will help you to think logically and clearly, and apply a range of problem-solving strategies” is claimed by The School of Mathematical Sciences at Monash University, Australia [ 4 ]. The School of Mathematics and Statistics at The University of Sydney, Australia, directly attributes as part of particular course objectives and outcomes skills that include “enhance your problem-solving skills” as part of studies in first year [ 5 ], “develop logical thinking” as part of studies in second year, which was a statement drafted by the lead author in fact [ 6 ], and “be fluent in analysing and constructing logical arguments” as part of studies in third year [ 7 ]. The University of Cambridge’s Faculty of Mathematics, UK, provides a dedicated document “Transferable Skills in the Mathematical Tripos” as part of its undergraduate mathematics course information, which again lists “analytic ability; creativity; initiative; logical and methodical reasoning; persistence” [ 8 ].

In contrast, psychological research, which has been empirically investigating the concept of transferability of skills since the early 1900s, points quite oppositely to reasoning skills as being highly domain specific [ 9 ]. Therefore, support for claims that studying mathematics engenders more than specific mathematics knowledge is highly pertinent. And yet it is largely absent. The 2014 Centre for Curriculum Redesign (CCR) four part paper “Mathematics for the 21st Century: What Should Students Learn?” concludes in its fourth paper titled “Does mathematics education enhance higher-order thinking skills?” with a call to action “… there is not sufficient evidence to conclude that mathematics enhances higher order cognitive functions. The CCR calls for a much stronger cognitive psychology and neuroscience research base to be developed on the effects of studying mathematics” [ 10 ].

Inglis and Simpson [ 11 ], bringing up this very issue, examined the ability of first-year undergraduate students from a high-ranking UK university mathematics department, on the “Four Cards Problem” thinking task, also known as the Wason Selection Task. It is stated as follows.

Each of the following cards have a letter on one side and a number on the other.

Here is a rule: “if a card has a D on one side, then it has a 3 on the other”. Your task is to select all those cards, but only those cards, which you would have to turn over in order to find out whether the rule is true or false. Which cards would you select?

This task involves understanding conditional inference, namely understanding the rule “If P then Q” and with this, deducing the answer as “P and not Q” or “D and 7”. Such logical deduction indeed presents as a good candidate to test for a potential ability of the mathematically trained. This task has also been substantially investigated in the domain of the psychology of reasoning [ 12 p8] revealing across a wide range of publications that only around 10% of the general population reach the correct result. The predominant mistake being to pick “D and 3”; where in the original study by Wason [ 13 ] it is suggested that this was picked by 65% of people. This poor success rate along with a standard mistake has fuelled interest in the task as well as attempts to understand why it occurs. A prevailing theory being the so named matching bias effect; the effect of disproportionately concentrating on items specifically mentioned in the situation, as opposed to reasoning according to logical rules.

Inglis and Simpson’s results isolated mathematically trained individuals with respect to this task. The participants were under time constraint and 13% of the first-year undergraduate mathematics students sampled reached the correct response, compared to 4% of the non-mathematics (arts) students that was included. Of note also was the 24% of mathematics students as opposed to 45% of the non-mathematics students who chose the standard mistake. The study indeed unveiled that mathematically trained individuals were significantly less affected by the matching bias effect with this problem than the individuals without mathematics training. However, the achievement of the mathematically trained group was still far from masterful and the preponderance for a non-standard mistake compared with non-mathematically trained people is suggestive. Mathematical training appears to engender a different thinking style, but it remains unclear what the difference is.

Inglis, Simpson and colleagues proceeded to follow up their results with a number of studies concentrated on conditional inference in general [ 14 , 15 ]. A justification for this single investigatory pathway being that if transfer of knowledge is present, something subtle to test for in the first place, a key consideration should be the generalisation of learning rather than the application of skills learned in one context to another (where experimenter bias in the choice of contexts is more likely to be an issue). For this they typically used sixteen “if P then Q” comprehension tasks, where their samples across a number of studies have included 16-year-old pre-university mathematics students (from England and Cyprus), mathematics honours students in their first year of undergraduate university study, third year university mathematics students, and associated control groups. The studies have encompassed controls for general intelligence and thinking disposition prior to training, as well as follows ups of up to two years to address the issue of causation. The conclusive thinking pattern that has emerged is a tendency of the mathematical groups towards a greater likelihood of rejecting the invalid denial of the antecedent and affirmation of the consequent inferences. But with this, and this was validated by a second separate study, the English mathematics group actually became less likely to endorse the valid modus tollens inference. So again, mathematical training appears to engender a different thinking style, but there are subtleties and it remains unclear what the exact difference is.

This project was designed to broaden the search on the notion that mathematics training leads to increased reasoning skills. We focused on a range of reasoning problems considered in psychological research to be particularly insightful into decision making, critical thinking and logical deduction, with their distinction in that the general population generally struggles with answering them correctly. An Australian sample adds diversity to the current enquiries that have been European focussed. Furthermore, in an effort to identify the impact of mathematics training through a possible gradation effect, different levels of mathematically trained individuals were tested for performance.

Well-studied thinking tasks from a variety of psychological studies were chosen. Their descriptions, associated success rates and other pertinent details follows. They were all chosen as the correct answer is typically eluded for a standard mistake.

The three-item Cognitive Reflection Test (CRT) was used as introduced by Frederick [ 16 ]. This test was devised in line with the theory that there are two general types of cognitive activity: one that operates quickly and without reflection, and another that requires not only conscious thought and effort, but also an ability to reflect on one’s own cognition by including a step of suppression of the first to reach it. The three items in the test involve an incorrect “gut” response and further cognitive skill is deemed required to reach the correct answer (although see [ 17 ] for evidence that correct responses can result from “intuition”, which could be related to intelligence [ 18 ]).

In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?

Bat and ball

A bat and a ball cost $1.10 in total. The bat costs a dollar more than the ball. How much does the ball cost?

The solutions are: 47 days for the Lily Pads problem, 5 minutes for the Widgets problem and 5 cents for the Bat and Ball problem. The considered intuitive, but wrong, answers are 24 days, 100 minutes and 10 cents, respectively. These wrong answers are attributed to participants becoming over focused on the numbers so as to ignore the exponential growth pattern in the Lily Pads problem, merely complete a pattern in numbers in the Widgets problem, and neglect the relationship “more than” in the Bat and Ball problem [ 19 ]. The original study by Frederick [ 16 ] provides a composite measure of the performance on these three items, with only 17% of those studied (n = 3428) reaching the perfect score. The CRT has since been studied extensively [ 19 – 21 ]. Research using the CRT tends not to report performance on the individual items of the test, but rather a composite measure of performance. Attridge and Inglis [ 22 ] used the CRT as a test for thinking disposition of mathematics students as one way to attempt to disentangle the issue of filtering according to prior thinking styles rather than transference of knowledge in successful problem solving. They repeat tested 16-year old pre-university mathematics students and English literature students without mathematics subjects at a one-year interval and found no difference between groups.

Three problems were included that test the ability to reason about probability. All three problems were originally discussed by Kahneman and Tversky [ 23 ], with the typically poor performance on these problems explained by participants relying not on probability knowledge, but a short-cut method of thinking known as the representativeness heuristic. In the late 1980s, Richard Nisbett and colleagues showed that graduate level training in statistics, while not revealing any improvement in logical reasoning, did correlate with higher-quality statistical answers [ 24 ]. Their studies lead in particular to the conclusion that comprehension of, what is known as the law of large numbers, did show improvement with training. The first of our next three problems targeted this law directly.

A certain town is served by two hospitals. In the larger hospital, about 45 babies are born each day, and in the smaller hospital, about 15 babies are born each day. As you know, about 50 percent of all babies are boys. However, the exact percentage varies from day to day. Sometimes it may be higher than 50 percent, sometimes lower. For a period of one year, each hospital recorded the number of days on which more than 60 percent of the babies born were boys. Which hospital do you think recorded more such days? (Circle one letter.)

• (a) the larger hospital
• (b) the smaller hospital
• (c) about the same (that is, within 5 percent of each other)

Kahneman and Tversky [ 23 ] reported that, of 50 participants, 12 chose (a), 10 chose (b), and 28 chose (c). The correct answer is (b), for the reason that small samples are more likely to exhibit extreme events than large samples from the same population. The larger the sample, the more likely it will exhibit characteristics of the parent population, such as the proportion of boys to girls. However, people tend to discount or be unaware of this feature of sampling statistics, which Kahneman and Tversky refer to as the law of large numbers. Instead, according to Kahneman and Tversky, people tend to adhere to a fallacious law of small numbers, where even small samples are expected to exhibit properties of the parent population, as illustrated by the proportion of participants choosing the answer (c) in their 1972 study. Such thinking reflects use of the representativeness heuristic, whereby someone will judge the likelihood of an uncertain event based on how similar it is to characteristics of the parent population of events.

Birth order

All families of six children in a city were surveyed. In 72 families the exact order of births of boys and girls was GBGBBG.

• (a) What is your estimate of the number of families surveyed in which the exact order of births was BGBBBB?
• (b) In the same survey set, which, if any, of the following two sequences would be more likely: BBBGGG or GBBGBG?

All of the events listed in the problem have an equal probability, so the correct answer to (a) is 72, and to (b) is “neither is more likely”. Kahneman and Tversky [ 23 ] reported that 75 of 92 participants judged the sequence in (a) as less likely than the given sequence. A similar number (unspecified by Kahneman and Tversky, but the statistical effect was reported to be of the same order as in (a)) reported that GBBGBG was the more likely sequence. Again, Kahneman and Tversky suggested that these results reflected use of the representativeness heuristic. In the context of this problem, the heuristic would have taken the following form: some birth orders appear less patterned than others, and less patterned is to be associated with the randomness of birth order, making them more likely.

Coin tosses

In a sequence of coin tosses (the coin is fair) which of the following outcomes would be most likely (circle one letter):

• (a) H T H T H T H T
• (b) H H H H T T T T
• (c) T T H H T T H H
• (d) H T T H T H H T
• (e) all of the above are equally likely

The correct answer in this problem is (e). Kahneman and Tversky [ 23 ] reported that participants tend to choose less patterned looking sequences (e.g., H T T H T H H T) as more likely than more systematic looking sequences (e.g., H T H T H T H T). This reasoning again reflects the representativeness heuristic.

Three further questions from the literature were included to test problem solving skill.

Two drivers

Two drivers set out on a 100-mile race that is marked off into two 50-mile sections. Driver A travels at exactly 50 miles per hour during the entire race. Driver B travels at exactly 45 mph during the first half of the race (up to the 50-mile marker) and travels at exactly 55 mph during the last half of the race (up to the finish line). Which of the two drivers would win the race? (Circle one letter.)

• (a) Driver A would win the race
• (b) Driver B would win the race
• (c) the two drivers would arrive at the same time (within a few seconds of one another)

This problem was developed by Pelham and Neter [ 25 ]. The correct answer is (a), which can be determined by calculations of driving times for each Driver, using time = distance/velocity. Pelham and Neter argue, however, that (c) is intuitively appealing, on the basis that both drivers appear to have the same overall average speed. Pelham and Neter reported that 67% of their sample gave this incorrect response to the problem, and a further 13% selected (b).

Petrol station

Imagine that you are driving along the road and you notice that your car is running low on petrol. You see two petrol stations next to each other, both advertising their petrol prices. Station A’s price is 65c/litre; Station B’s price is 60c/litre. Station A’s sign also announces: “5c/litre discount for cash!” Station B’s sign announces “5c/litre surcharge for credit cards.” All other factors being equal (for example, cleanliness of the stations, number of cars waiting at each etc), to which station would you choose to go, and why?

This problem was adapted from one described by Galotti [ 26 ], and is inspired by research reported by Thaler [ 27 ]. According to Thaler’s research, most people prefer Station A, even though both stations are offering the same deal: 60c/litre for cash, and 65c/litre for credit. Tversky and Kahneman [ 28 ] explain this preference by invoking the concept of framing effects. In the context of this problem, such an effect would involve viewing the outcomes as changes from some initial point. The initial point frames the problem, and provides a context for viewing the outcome. Thus, depending on the starting point, outcomes in this problem can be viewed as either a gain (in Station A, you gain a discount if you use cash) or a loss (in Station B, you are charged more (a loss) for using credit). Given that people are apparently more concerned about a loss than a gain [ 29 ], the loss associated with Station B makes it the less attractive option, and hence the preference for Station A. The correct answer, though, is that the stations are offering the same deal and so no station should be preferred.

And finally, a question described by Stanovich [ 30 , 31 ] as testing our predisposition for cognitive operations that require the least computational effort.

Jack looking at Anne

Jack is looking at Anne, but Anne is looking at George. Jack is married, but George is not. Is a married person looking at an unmarried person? (Circle one letter.)

• (c) Cannot be determined

Stanovich reported that over 80% of people choose the “lazy” answer (c). The correct answer is (a).

The above questions survey, in a clear problem solving setting, an ability to engage advanced cognitive processing in order to critically evaluate and possibly override initial gut reasoning, an ability to reason about probability within the framework of the law of large numbers and the relationship between randomness and patterning, an ability to isolate salient features of a problem and, with the last question in particular, an ability to map logical relations. It might be hypothesised that according to degrees of mathematical training, in line with the knowledge base provided and the claims of associated broad and enhanced problem-solving abilities in general, that participants with greater degrees of such training would outperform others on these questions. This hypothesis was investigated in this study. In addition, given that no previous study on this issue has examined the variety of problems used in this study, we also undertook an exploratory analysis to investigate whether there exist any associations between the problems in terms of their likelihood of correct solution. Similarities between problems might indicate which problem solving domains could be susceptible to the effects of mathematics training.

A questionnaire was constructed containing the problems described in the previous sections plus the Four Cards Problem as tested by Inglis and Simpson [ 11 ] for comparison. The order of the problems was as follows: 1) Lily Pads; 2) Hospitals; 3) Widgets; 4) Four Cards; 5) Bat and Ball; 6) Birth Order; 7) Petrol Station; 8) Coin Tosses; 9) Two Drivers; 10) Jack looking at Anne. It was administered to five groups distinctive in mathematics training levels chosen from a high-ranking Australian university, where the teaching year is separated into two teaching semesters and where being a successful university applicant requires having been highly ranked against peers in terms of intellectual achievement:

• Introductory—First year, second semester, university students with weak high school mathematical results, only enrolled in the current unit as a compulsory component for their chosen degree, a unit not enabling any future mathematical pathway, a typical student may be enrolled in a Biology or Geography major;
• Standard—First year, second semester, university students with fair to good high school mathematical results, enrolled in the current mathematics unit as a compulsory component for their chosen degree with the possibility of including some further mathematical units in their degree pathway, a typical student may be enrolled in an IT or Computer Science major;
• Advanced1—First year, second semester, university mathematics students with very strong interest as well as background in mathematics, all higher year mathematical units are included as possible future pathway, a typical student may be enrolled in a Mathematics or Physics major;
• Advanced2—Second year, second semester, university mathematics students with strong interest as well as background in mathematics, typically a direct follow on from the previously mentioned Advanced1 cohort;
• Academic—Research academics in the mathematical sciences.

Participants

123 first year university students volunteered during “help on demand” tutorial times containing up to 30 students. These are course allocated times that are supervised yet self-directed by students. This minimised disruption and discouraged coercion. 44 second year university students completed the questionnaire during a weekly one-hour time slot dedicated to putting the latest mathematical concepts to practice with the lecturer (whereby contrast to what occurs in tutorial times the lecturer does most of the work and all students enrolled are invited). All these university students completed the questionnaire in normal classroom conditions; they were not placed under strict examination conditions. The lead author walked around to prevent discussion and coercion and there was minimum disruption. 30 research academics responded to local advertising and answered the questionnaire in their workplace while supervised.

The questionnaires were voluntary, anonymous and confidential. Participants were free to withdraw from the study at any time and without any penalty. No participant took this option however. The questionnaires gathered demographic information which included age, level of education attained and current qualification pursued, name of last qualification and years since obtaining it, and an option to note current speciality for research academics. Each problem task was placed on a separate page. Participants were not placed under time constraint, but while supervised, were asked to write their start and finish times on the front page of the survey to note approximate completion times. Speed of completion was not incentivised. Participants were not allowed to use calculators. A final “Comments Page” gave the option for feedback including specifically if the participants had previously seen any of the questions. Questionnaires were administered in person and supervised to avoid collusion or consulting of external sources.

The responses were coded four ways: A) correct; B) standard error (the errors discussed above in The Study); C) other error; D) left blank.

The ethical aspects of the study were approved by the Human Research Ethics Committee of the University of Sydney, protocol number [2016/647].

The first analysis examined the total number of correct responses provided by the participants as a function of group. Scores ranged from 1 to 11 out of a total possible of 11 (Problem 6 had 2 parts) ( Fig 1 ). An ANOVA of this data indicated a significant effect of group (F(4, 192) = 20.426, p < .001, partial η 2 = .299). Pairwise comparisons using Tukey’s HSD test indicated that the Introductory group performed significantly worse than the Advanced1, Advanced2 and Academic groups. There were no significant differences between the Advanced1, Advanced2 and Academic groups.

Error bars are one standard error of the mean.

Overall solution time, while recorded manually and approximately, was positively correlated with group, such that the more training someone had received, the longer were these solution times (r(180) = 0.247, p = .001). However, as can be seen in Fig 2 , this relationship is not strong.

A series of chi-squared analyses, and their Bayesian equivalents, were performed on each problem, to determine whether the distribution of response types differed as a function of group. To minimise the number of cells in which expected values in some of these analyses were less than 5, the Standard Error, Other Error and Blank response categories were collapsed into one category (Incorrect Response). For three of the questions, the expected values of some cells did fall below 5, and this was due to most people getting the problem wrong (Four Cards), or most people correctly responding to the problem (Bat and Ball, Coin Tosses). In these cases, the pattern of results was so clear that a statistical analysis was barely required. Significant chi-squared results were examined further with pairwise posthoc comparisons (see Table 1 ).

Superscripts label the groups (e.g., Introductory = a). Within the table, these letters refer to which other group a particular group was significantly different to according to a series of pairwise post hoc chi squared analyses (Bonferroni corrected α = .005) (e.g., ‘d’ in the Introductory column indicates the Introductory and the Advanced2 (d) group were significantly different for a particular problem).

The three groups with the least amount of training in mathematics were far less likely than the other groups to give the correct solution (χ 2 (4) = 31.06, p < .001; BF 10 = 45,045) ( Table 1 ). People in the two most advanced groups (Advanced2 and Academic) were more likely to solve the card problem correctly, although it was still less than half of the people in these groups who did so. Further, these people were less likely to give the standard incorrect solution, so that most who were incorrect suggested some more cognitively elaborate answer, such as turning over all cards. The proportion of people in the Advanced2 and Academic groups (39 and 37%) who solved the problem correctly far exceeded the typical proportion observed with this problem (10%). Of note, also, is the relatively high proportion of those in the higher training groups who, when they made an error, did not make the standard error, a similar result to the one reported by Inglis and Simpson [ 11 ].

The cognitive reflection test

In the Lily Pads problem, although most people in the Standard, Advanced1, Advanced2 and Academic groups were likely to select the correct solution, it was also the case that the less training someone had received in mathematics, the more likely they were to select an incorrect solution (χ 2 (4) = 27.28, p < .001; BF 10 = 15,554), with the standard incorrect answer being the next most prevalent response for the two lower ability mathematics groups ( Table 1 ).

Performance on the Widgets problem was similar to performance on the Lily Pads problem in that most people in the Standard, Advanced1, Advanced2 and Academic groups were likely to select the correct solution, but that the less training someone had received in mathematics, the more likely they were to select an incorrect solution (χ 2 (4) = 23.76, p< .001; BF 10 = 516) ( Table 1 ). As with the Lily Pads and Widget problems, people in the Standard, Advanced1, Advanced2 and Academic groups were highly likely to solve the Bat and Ball problem (χ 2 (4) = 35.37, p < .001; BF 10 = 208,667). Errors were more likely from the least mathematically trained people (Introductory, Standard) than the other groups ( Table 1 ).

To compare performance on the CRT with previously published results, performance on the three problems (Lily Pads, Widgets, Bat and Ball) were combined. The number of people in each condition that solved 0, 1, 2, or 3 problems correctly is presented in Table 2 . The Introductory group were evenly distributed amongst the four categories, with 26% solving all three problems correctly. Around 70% of the rest of the groups solved all 3 problems correctly, which is vastly superior to the 17% reported by Frederick [ 16 ].

Responses to the Hospitals problem were almost universally split between correct and standard errors in the Standard, Advanced1, Advanced2 and Academic groups. Although this pattern of responses was also evident in the Introductory group, this group also exhibited more non-standard errors and non-responses than the other groups. However, the differences between the groups were not significant (χ 2 (4) = 4.93, p = .295; BF 10 = .068) ( Table 1 ). Nonetheless, the performance of all groups exceeds the 20% correct response rate reported by Kahneman and Tversky [ 23 ].

The two versions of the Birth Order problem showed similar results, with correct responses being more likely in the groups with more training (i.e., Advanced1, Advanced2 and Academic), and responses being shared amongst the various categories in the Introductory and Standard groups (χ a 2 (4) = 24.54, p < .001; BF 10 = 1,303; χ b 2 (4) = 25.77, p < .001; BF 10 = 2,970) ( Table 1 ). Nonetheless, performance on both versions of the problem in this study was significantly better than the 82% error rate reported by Kahneman and Tversky [ 23 ].

The Coin Tosses problem was performed well by all groups, with very few people in any condition committing errors. There were no obvious differences between the groups (χ 2 (4) = 3.70, p = .448; BF 10 = .160) ( Table 1 ). Kahneman and Tversky [ 23 ] reported that people tend to make errors on this type of problem by choosing less patterned looking sequences, but they did not report relative proportions of people making errors versus giving correct responses. Clearly the sample in this study did not perform like those in Kahneman and Tversky’s study.

Responses on the Two Drivers problem were clearly distinguished by a high chance of error in the Introductory and Standard groups (over 80%), and a fairly good chance of being correct in the Advanced1, Advanced2 and Academic groups (χ 2 (4) = 46.16, p < .001; BF 10 = 1.32 x 10 8 ) ( Table 1 ). Academics were the standout performers on this problem, although over a quarter of this group produced an incorrect response. Thus, the first two groups performed similarly to the participants in the Pelham and Neter [ 25 ] study, 80% of whom gave an incorrect response.

Responses on the Petrol Station problem were marked by good performance by the Academic group (73% providing a correct response), and just over half of each of the other groups correctly solving the problem. This difference was not significant (χ 2 (4) = 4.68, p = .322: BF 10 = .059) ( Table 1 ). Errors were fairly evenly balanced between standard and other, except for the Academic group, who were more likely to provide a creative answer if they made an error. Thaler [ 27 ] reported that most people get this problem wrong. In this study, however, on average, most people got this problem correct, although this average was boosted by the Academic group.

Responses on the Jack looking at Anne problem generally were standard errors, except for the Advanced2 and Academic groups, which were evenly split between standard errors and correct responses (χ 2 (4) = 18.03, p = .001; BF 10 = 46) ( Table 1 ). Thus, apart from these two groups, the error rate in this study was similar to that reported by Stanovich [ 30 ], where 80% of participants were incorrect.

A series of logistic regression analyses were performed in order to examine whether the likelihood of solving a particular problem correctly could be predicted on the basis of whether other problems were solved correctly. Each analysis involved selecting performance (correct or error) on one problem as the outcome variable, and performance on the other problems as predictor variables. Training (amount of training) was also included as a predictor variable in each analysis. A further logistic regression was performed with training as the outcome variable, and performance on all of the problems as predictor variables. The results of these analyses are summarised in Table 3 . There were three multi-variable relationships observed in these analyses, which can be interpreted as the likelihood of solving one problem in each group being associated with solving the others in the set. These sets were: (1) Lily Pads, Widgets and Petrol Station; (2) Hospitals, Four Cards and Two Drivers; (3) Birth Order and Coin Tosses. Training also featured in each of these sets, moderating the relationships as per the results presented above for each problem.

P = Problem (1 = Four Cards; 2 = Lily Pads; 3 = Widgets; 4 = Bat & Ball; 5 = Hospitals; 6a = Birth Order (a); 6b = Birth Order (b); 7 = Coin Tosses; 8 = Two Drivers; 9 = Petrol Station; 10 = Jack looking at Anne).

training = Amount of training condition.

p = significance level of logistic regression model.

% = percentage of cases correctly classified by the logistic regression model.

✓ = significant predictor, α < .05.

* = logistic regression for the training outcome variable is multinomial, whereas all other logistic regressions are binomial.

The final “Comments Page” revealed the participants as overwhelmingly enjoying the questions. Any analysis of previous exposure to the tasks proved impossible as there was little to no alignment on participant’s degree of recall, if any, and even perceptions of what exposure entailed. For example, some participants confused being exposed to the particular tasks with being habitually exposed to puzzles, or even mathematics problems, more broadly.

In general, the amount of mathematics training a group had received predicted their performance on the overall set of problems. The greater the training, the more problems were answered correctly, and the slower the recorded response times. There was not an obvious difference between the Advanced1, Advanced2 and Academic groups on either of these measures, however there were clear differences between this group and the Introductory and Standard groups, with the former exhibiting clearly superior accuracy. While time records were taken approximately, so as to avoid adding time pressure as a variable, that the Advanced1, Advanced2 and Academic groups recorded more time in their consideration of the problems, may suggest a “pause and consider” approach to such problems is a characteristic of the advanced groups. This is in line with what was suggested by an eye-movement tracking study of mathematically trained students attempting the Four Cards Problem; where participants that had not chosen the standard error had spent longer considering the card linked to the matching bias effect [ 14 ]. It is important to note, however, that longer response times may reflect other cognitive processes than deliberation [ 32 ].

Performance on some problems was associated with performance on other problems. That is, if someone correctly answered a problem in one of these sets, they were also highly likely to correctly answer the other problems in the set. These sets were: (1) Lily Pads, Widgets and Petrol Station; (2) Hospitals, Four Cards and Two Drivers; (3) Birth Order and Coin Tosses. This is different with how these problems have been typically clustered a priori in the research literature: (I) Lily Pads, Widgets and Bat and Ball (CRT); (II) Hospitals and Two Drivers (explained below); (III) Hospitals, Birth Order and Coin Tosses (representativeness heuristic); (IV) Birth Order and Coin Tosses (probability theory). Consideration of these problem groupings follows.

Correctly answering all three problems in (I) entailed not being distracted by particular pieces of information in the problems so as to stay focused on uncovering the real underlying relationships. The Lily Pads and Widget problems can mislead if attention is over focused on the numbers, and conversely, the Petrol Station problem can mislead if there is too much focus on the idea of a discount. While the Lily Pads and Widget problems are traditionally paired with the Bat and Ball problem in the CRT, it may be that performance on the Bat and Ball problem did not appear as part of this set due to an added level of difficulty. With the problems in (I), avoiding being distracted by certain parts of the questions at the expense of others almost leads directly to the correct answer. However, with the Bat and Ball problem, further steps in mathematical reasoning still need to occur in answering which two numbers add together to give a result while also subtracting one from the other for another.

With the problems in (II) it is of interest that the Two Drivers problem was created specifically to be paired with the Hospitals problem to test for motivation in problem solving [ 23 ]. Within this framework further transparent versions of these problems were successfully devised to manipulate for difficulty. The Two Drivers problem was amended to have Driver B travelling at exactly 5 mph during the first half of the race and at exactly 95 mph during the last half of the race. The Hospitals problem was amended so the smaller hospital would have “only 2” babies born each day and where for a period of one year the hospitals recorded the number of days on which all of the babies born were boys. Could the association in (II) be pointing to how participants overcome initial fictitious mathematical rules? Maybe they reframe the question in simpler terms to see the pattern. The Four Cards Problem also elicited a high number of incorrect answers where, associated with mathematical training, the standard incorrect solution was avoided for more cognitively elaborate ones. Indeed, a gradation effect appeared across the groups where the standard error of the “D and 3” cards becomes “D only” ( Table 4 ). Adrian Simpson and Derrick Watson found a comparable result across their two groups [14 p61]. This could again be pointing to having avoided an initial fictitious rule of simply concentrating on items directly found in the question, participants then seek to reframe the question to unearth the logical rule to be deduced. An added level of difficulty with this question may be why participants become trapped in a false answer. The eye-movement tracking study mentioned above supports this theory.

The problems in (III) fit naturally together as part of basic probability theory, a topic participants would have assimilated, or not, as part of various education curricula. While the equal likelihood of all possible outcomes with respect to a coin toss may be culturally assimilated, the same may not be as straightforward for birth gender outcomes where such assumptions could be swayed by biological hypothesis or folk wisdom [ 33 ]. The gradation of the results in terms of mathematical training does not support this possibility.

The effect of training on performance accuracy was more obvious in some problems compared to others, and to some extent, this was related to the type of problem. For instance, most of the problems in which performance was related to training (Four Cards, CRT [Lily Pads, Widgets, Bat and Ball], Two Drivers, Jack looking at Anne) could be classed as relying on logical and/or critical thinking. The one exception was the Birth Order problems, which are probability related.

In contrast, two of the three problems in which training did not appear to have much impact on performance (Hospitals and Coin Tosses) require domain-specific knowledge. The Hospitals problem requires a degree of knowledge about sampling statistics. This is a topic of quite distinct flavour that not all mathematically trained individuals gain familiarity with. On the other hand, all groups having performed well on the Coin Tosses problem is in line with a level of familiarity with basic probability having been originally presented at high school. While the questioning of patterning as negatively correlated with randomness is similar to that appearing in the Birth Order question, in the Birth Order question this aspect is arguably more concealed. These results and problem grouping (III) could be pointing to an area for improvement in teaching where the small gap in knowledge required to go from answering the Coin Tosses problem correctly to achieving similarly with the Birth Order problem could be easily addressed. A more formal introduction to sampling statistics in mathematical training could potentially bridge this gap as well as further be extended towards improvement on the Hospitals problem.

The other problem where performance was unrelated to training, the Petrol Station problem, cannot be characterised similarly. It is more of a logical/critical thinking type problem, where there remains some suggestion that training may have impacted performance, as the Academic group seemed to perform better than the rest of the sample. An alternate interpretation of this result is therefore that this problem should not be isolated but grouped with the other problems where performance is affected by training.

Although several aspects of the data suggest mathematics training improves the chances that someone will solve problems of the sort examined here, differences in the performance of participants in the Advanced1, Advanced2 and Academic groups were not obvious. This is despite the fact that large differences exist in the amount of training in these three groups. The first two groups were undergraduate students and the Academic group all had PhDs and many were experienced academic staff. One interpretation of this result is current mathematics training can only take someone so far in terms of improving their abilities with these problems. There is a point of demarcation to consider in terms of mathematical knowledge between the Advanced1, Advanced2 and Academic groups as compared to the Introductory and Standard groups. In Australia students are able to drop mathematical study at ages 15–16 years, or choose between a number of increasingly involved levels of mathematics. For the university in this study, students are filtered upon entry into mathematics courses according to their current knowledge status. All our groups involved students who had opted for post-compulsory mathematics at high school. And since our testing occurred in second semester, some of the mathematical knowledge shortfalls that were there upon arrival were bridged in first semester. Students must pass a first semester course to be allowed entry into the second semester course. A breakdown of the mathematics background of each group is as follows:

• The Introductory group’s mathematics high school syllabus studied prior to first semester course entry covered: Functions, Trigonometric Functions, Calculus (Introduction to Differentiation, Applications of the Derivative, Antiderivatives, Areas and the Definite Integral), Financial Mathematics, Statistical Analysis. The Introductory group then explored concepts in mathematical modelling with emphasis on the importance of calculus in their first semester of mathematical studies.
• The Standard group’s mathematics high school syllabus studied prior to first semester course entry covered: Functions, Trigonometric Functions, Calculus (Rates of Change, Integration including the method of substitution, trigonometric identities and inverse trigonometric functions, Areas and Volumes of solids of revolution, some differential equations), Combinatorics, Proof (with particular focus on Proof by Mathematical Induction), Vectors (with application to projectile motion), Statistical Analysis. In first semester their mathematical studies then covered a number of topics the Advanced1 group studied prior to gaining entrance at university; further details on this are given below.
• The Advanced1 group’s mathematics high school syllabus studied prior to first semester course entry covered: the same course content the Standard group covered at high school plus extra topics on Proof (develop rigorous mathematical arguments and proofs, specifically in the context of number and algebra and further develop Proof by Mathematical Induction), Vectors (3 dimensional vectors, vector equations of lines), Complex Numbers, Calculus (Further Integration techniques with partial fractions and integration by parts), Mechanics (Application of Calculus to Mechanics with simple harmonic motion, modelling motion without and with resistance, projectiles and resisted motion). The Standard group cover these topics in their first semester university studies in mathematics with the exclusion of further concepts of Proof or Mechanics. In first semester the Advanced1 group have built on their knowledge with an emphasis on both theoretical and foundational aspects, as well as developing the skill of applying mathematical theory to solve practical problems. Theoretical topics include a host of theorems relevant to the study of Calculus.

In summary, at the point of our study, the Advanced1 group had more knowledge and practice on rigorous mathematical arguments and proofs in the context of number and algebra, and more in-depth experience with Proofs by Induction, but the bulk of extra knowledge rests with a much deeper knowledge of Calculus. They have had longer experience with a variety of integration techniques, and have worked with a variety of applications of calculus to solve practical problems, including a large section on mechanics at high school. In first semester at university there has been a greater focus on theoretical topics including a host of theorems and associated proofs relevant to the topics studied. As compared to the Introductory and Standard groups, the Advanced1 group have only widened the mathematics knowledge gap since their choice of post-compulsory mathematics at high school. The Advanced2 group come directly from an Advanced1 cohort. And the Academics group would have reached the Advanced1 group’s proficiency as part of their employment. So, are specific reasoning skills resulting from this level of abstract reasoning? Our findings suggest this should certainly be an area of investigation and links in interestingly with other research work. In studying one of the thinking tasks in particular (the Four Cards Problem) and its context of conditional inference more specifically, Inglis and Simpson [ 15 ] found a clear difference between undergraduates in mathematics and undergraduates in other university disciplines, yet also showed a lack of development over first-year university studies on conditional inference measures. A follow up study by Attridge and Inglis [ 22 ] then zeroed in on post-compulsory high school mathematical training and found that students with such training did develop their conditional reasoning to a greater extent than their control group over the course of a year, despite them having received no explicit tuition in conditional logic. The development though, whilst demonstrated as not being the result of a domain-general change in cognitive capacity or thinking disposition, and most likely associated with the domain-specific study of mathematics, revealed a complex pattern of endorsing more of some inferences and less of others. The study here focused on a much broader problem set associated with logical and critical thinking and it too is suggestive of a more complex picture in how mathematics training may be contributing to problem solving styles. A more intricate pattern to do with the impact of mathematical training on problem solving techniques is appearing as required for consideration.

There is also a final interpretation to consider: that people in the Advanced 1, Advanced2 and Academic groups did not gain anything from their mathematics training in terms of their ability to solve these problems. Instead, with studies denying any correlation of many of these problems with what is currently measured as intelligence [ 30 ], they might still be people of a particular intelligence or thinking disposition to start with, who have been able to use that intelligence to not only solve these problems, but also survive the challenges of their mathematics training.

That the CRT has been traditionally used as a measure of baseline thinking disposition and that performance has been found to be immutable across groups tested is of particular interest since our results show a clear possible training effect on these questions. CRT is tied with a willingness to engage in effortful thinking which presents as a suitable ability for training. It is beyond the scope of this study, but a thorough review of CRT testing is suggestive of a broader appreciation and better framework to understand thinking disposition, ability and potential ability.

Mathematical training appears associated with certain thinking skills, but there are clearly some subtleties that need to be extricated. The thinking tasks here add to the foundational results where the aim is for a firmer platform on which to eventually base more targeted and illustrative inquiry. If thinking skills can be fostered, could first year university mathematics teaching be improved so that all samples from that group reach the Advanced1 group level of reasoning? Do university mathematics courses become purely about domain-specific knowledge from this point on? Intensive training has been shown to impact the brain and cognition across a number of domains from music [ 34 ], to video gaming [ 35 ], to Braille reading [ 36 ]. The hypothesis that mathematics, with its highly specific practice, fits within this list remains legitimate, but simply unchartered. With our current level of understanding it is worth appreciating the careful wording of the NYU Courant Institute on ‘Why Study Math?’ where there is no assumption of causation: “Mathematicians need to have good reasoning ability in order to identify, analyze, and apply basic logical principles to technical problems.” [ 37 ].

Limitations

One possible limitation of the current study is that the problems may have been too easy for the more advanced people, and so we observed a ceiling effect (i.e., some people obtained 100% correct on all problems). This was most obvious in the Advanced1, Advanced2 and Academic groups. It is possible that participants in these groups had developed logical and critical thinking skills throughout their mathematical training that were sufficient to cope with most of the problems used in this study, and so this would support the contention that training in mathematics leads to the development of logical and critical thinking skills useful in a range of domains. Another interpretation is that participants in these groups already possessed the necessary thinking skills for solving the problems in this study, which is why they are able to cope with the material in the advanced units they were enrolled in, or complete a PhD in mathematics and hold down an academic position in a mathematics department. This would then suggest that training in mathematics had no effect on abstract thinking skills—people in this study possessed them to varying extents prior to their studies. This issue might be settled in a future study that used a greater number of problems of varying difficulties to maximise the chances of finding a difference between the three groups with the most amount of training. Alternatively, a longitudinal study that followed people through their mathematics training could determine whether their logical and critical thinking abilities changed throughout their course.

A further limitation of the study may be that several of the reasoning biases examined in this study were measured by only one problem each (i.e., Four Cards Problem, Two Drivers, Petrol Station, Jack looking at Anne). A more reliable measure of these biases could be achieved by including more problems that tap into these biases. This would, however, increase the time required of participants during data collection, and in the context of this study, would mean a different mode of testing would likely be required.

Broad sweeping intuitive claims of the transferable skills endowed by a study of mathematics require evidence. Our study uniquely covers a wide range of participants, from limited mathematics training through to research academics in the mathematical sciences. It furthermore considered performance on 11 well-studied thinking tasks that typically elude participants in psychological studies and on which results have been uncorrelated with general intelligence, education levels and other demographic information [ 15 , 16 , 30 ]. We identified different performances on these tasks with respect to different groups, based on level of mathematical training. This included the CRT which has developed into a method of measuring baseline thinking disposition. We identified different distributions of types of errors for the mathematically trained. We furthermore identified a performance threshold that exists in first year university for those with high level mathematics training. This study then provides insight into possible changes and adjustments to mathematics courses in order for them to fulfil their advertised goal of reaching improved rational and logical reasoning for a higher number of students.

It is central to any education program to have a clear grasp of the nature of what it delivers and how, but arguably especially so for the core discipline that is mathematics. In 2014 the Office of The Chief Scientist of Australia released a report “Australia’s STEM workforce: a survey of employers” where transferable skills attributed to mathematics were also ones that employers deemed as part of the most valuable [ 38 ]. A better understanding of what mathematics delivers in this space is an opportunity to truly capitalise on this historical culture-crossing subject.

Supporting information

Acknowledgments.

The authors would like to thank Jacqui Ramagge for her proof reading and input, as well as support towards data collection.

Funding Statement

The authors received no specific funding for this work.

Data Availability

• PLoS One. 2020; 15(7): e0236153.

Decision Letter 0

17 Mar 2020

PONE-D-20-01159

Does mathematics training lead to better logical thinking and reasoning? A cross-sectional assessment from students to professors

Dear Professor Speelman,

Thank you for submitting your manuscript to PLOS ONE. I have sent it to two expert reviewers and have received their comments back. As you can see at the bottom of this email, both reviewers are positive about your manuscript but raise some issues that you would need to address before the manuscript can be considered for publication. Notably, reviewer #1 points out that the manuscript should include a discussion on the reasons why individuals with math training may have improved reasoning skills (e.g., logical intuitions versus deliberate thinking). The reviewer also rightly mentions that your sample sizes are limited, notably for the most advanced groups. This should be discussed and acknowledged. Reviewer #2 has a number of conceptual and methodological points that you will also have to address. The reviewer provides very thorough comments and I will not reiterate the points here. However, note that both reviewers suggest that you need to improve the figures and I agree with them.   

We would appreciate receiving your revised manuscript by May 01 2020 11:59PM. When you are ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter.

To enhance the reproducibility of your results, we recommend that if applicable you deposit your laboratory protocols in protocols.io, where a protocol can be assigned its own identifier (DOI) such that it can be cited independently in the future. For instructions see: http://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols

Please include the following items when submitting your revised manuscript:

• A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). This letter should be uploaded as separate file and labeled 'Response to Reviewers'.
• A marked-up copy of your manuscript that highlights changes made to the original version. This file should be uploaded as separate file and labeled 'Revised Manuscript with Track Changes'.
• An unmarked version of your revised paper without tracked changes. This file should be uploaded as separate file and labeled 'Manuscript'.

Please note while forming your response, if your article is accepted, you may have the opportunity to make the peer review history publicly available. The record will include editor decision letters (with reviews) and your responses to reviewer comments. If eligible, we will contact you to opt in or out.

We look forward to receiving your revised manuscript.

Kind regards,

Jérôme Prado

Academic Editor

Journal Requirements:

When submitting your revision, we need you to address these additional requirements:

1. Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at http://www.plosone.org/attachments/PLOSOne_formatting_sample_main_body.pdf and http://www.plosone.org/attachments/PLOSOne_formatting_sample_title_authors_affiliations.pdf

2. Please include additional information regarding the survey or questionnaire used in the study and ensure that you have provided sufficient details that others could replicate the analyses. For instance, if you developed a questionnaire as part of this study and it is not under a copyright more restrictive than CC-BY, please include a copy, in both the original language and English, as Supporting Information. Please also let us know if it would be possible to provide the anonymized data points necessary to replicate the statistical analyses, for instance, as shown in fig 1 and 2. If so, please deposit those to a suitable data repository or include them in the Supporting Information files.

3. Thank you for stating the following financial disclosure:

"The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript."

• Please provide an amended Funding Statement that declares *all* the funding or sources of support received during this specific study (whether external or internal to your organization) as detailed online in our guide for authors at http://journals.plos.org/plosone/s/submit-now .  
• Please state what role the funders took in the study.  If any authors received a salary from any of your funders, please state which authors and which funder. If the funders had no role, please state: "The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript."

Please include your amended statements within your cover letter; we will change the online submission form on your behalf.

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer #1: I think this is a very good and interesting manuscript trying to answer an important research question. I propose some changes that I believe should be applied before publication.

1. Each reasoning bias is measured with only one problem. In reasoning research, it is rather common to measure each type of reasoning problem with a series of structurally equivalent reasoning problems, so the results will be independent of contexts effects and will be generalizable to that type of problem. Here, the authors only measured each reasoning bias with one single problem and this might be problematic (see, for example: Fiedler & Hertel, 1994). I think this can be addressed by simply discussing it in the limitation section.

2. This is rather a minor issue, but the discussion on the CRT problems is not up-to-date (page 7). Most recent experiments on dual process theory suggest that people who are able to correctly solve these reasoning problems (including the CRT) do so intuitively, and not because they engaged in careful deliberation (Bago & De Neys, 2019). Intelligence made people have better intuitive responses (Thompson, Pennycook, Trippas & Evans, 2018). Similarly, this problems persists in the discussion about reaction times (page 25). Longer reaction times does not necessarily mean that people engaged in deliberation (see: Evans, Kyle, Dillon & Rand, 2015). Response time might be driven by decision conflict or response rationalization. These issues could be clarified with some changes in the wording or some footnotes on page 7 and 25. Furthermore, it would be interesting to have a discussion on how mathematical education helps people overcome their biases. Is it because it creates better intuition, or helps people engage in deliberation? An interesting question this manuscript does not discuss. It’s on the authors whether or not they discuss this latter point now, but the changes on page 7 and 25 should be made.

3. A more serious problem is the rather small sample size (especially in the more advanced groups). This small sample size makes the appearance of both false negatives and false positives more likely. Perhaps, the authors could compute the Bayes Factors for the chi-square or logistic regression test, so we can actually see how strong the evidence is for or against the null. This is especially important as the authors run a great number of explorative analysis (Table 3), and some of those results might need to be interpreted with great caution (depending on the Bayes Factor).

The graphs are not looking good, they should comply with APA formatting. At the very least, the axis titles should be meaningful and measure units should be written there.

The presentation order of the problems is quite unusual; why isn’t it random? Why did the authors decide on this order?

Reviewer #2: The study reported in this paper compared five groups of participants with varying levels of mathematical expertise on a set of reasoning tasks. The study is interesting and informative. It extends the current literature on this topic (which is reviewed very nicely in the introduction). However, there are some issues with the current analysis and interpretation that should be resolved prior to publication. I have therefore recommended major revisions. My comments are organised in the order in which they came up in the paper and they explain my responses to the questions above.

1. Line 114 – “general population” a bit misleading – they were also students but from other disciplines.

2. Line 124 onwards reads:

“The ultimate question to consider here is: are any skills associated with mathematics training innate or do they arise from skills transfer? Though to investigate how mathematical training affects reasoning skills, randomised sampling and randomised intervention to reveal causal relationships are clearly not viable. With so many possible confounding variables and logistical issues, it is even questionable what conclusions such studies might provide. Furthermore, a firm baseline from which to propose more substantive investigations is still missing.”

I find this paragraph slightly problematic because the current study doesn’t inform us on this ultimate question, so it makes the outline of the current study in the following paragraph feel unsatisfactory. I think the current study is important but prefacing it with this paragraph underplays that importance. And I think a randomised controlled study, although not viable, would give the answers we need because the random allocation to groups would allow us to rule out any confounding variables. Finally, the last sentence in this paragraph is unclear to me.

3. In the descriptions of the five participants groups the authors refer to the group’s level of interest in mathematics, but this seems like an overgeneralisation to me. Surely the introductory group could contain a biology student who also happens to be good at mathematics and very much enjoy it? I would be more comfortable with the descriptions if the parts about interest level were removed.

4. How many of the 123 first year students were in each of the three first year groups?

5. Line 313 – the standard group is referred to as “university mathematics students”, but they are not taking mathematics degreed.

6. Line 331 - what is a practice class?

7. Were the data collection settings quiet? From the description it sounds like groups of participants were completing the study at the same time in the same room, but the authors should make this explicit for the sake of the method being reproducible. E.g. how many students were in the room at the time?

8. Line 355-356 – the authors should not use the term “marginally worse” because this is statistically inappropriate – in a frequentist approach results are either significant or non-significant.

9. Line 340 – “approximate completion times were noted.”

This doesn’t sound rigorous enough to justify analysing them. Their analysis is interesting, but the authors should remind readers clearly whenever the response times are analysed or discussed that their recording was only manual and approximate.

10. I suggest replacing Figure 1 with a bar chart showing standard error of the mean on the error bars. A table with mean score out of 11 and the standard deviation for each group may also be useful. Figure 2 should be a scatterplot rather than a box and whisker plot.

11. Was the 0-11 total correct score approximately normally distributed across the full sample?

12. Chi square analysis requires at least 5 cases in each cell, was this met? It seems not since Table 1 shows lots of cells in the “no response” row having 0% of cases.

13. The chi-square analyses should be followed up with post hoc tests to see exactly where the differences between groups are. The descriptions as they stand aren’t that informative (as readers can just look at Table 1) without being backed up by post hoc tests.

14. For each chi square analysis in the text, I would find it easier to read if the test statistics came at the top of the paragraph, before the description.

15. Line 381-383 – “Of note, also, is the relatively low proportion of those in the higher training groups who, when they made an error, did not make the standard error, a similar result to the one reported by Inglis and Simpson [11]."

I think this is supposed to say that a low proportion did make the standard error or that a high proportion did not make the standard error.

16. Line 403 - p values this small should be reported as p < .001 rather than p = .000 since they aren’t actually 0.

17. Line 476 – “…if a particular outcome variable was predicted significantly by a particular predictor variable, the converse relationship was also observed”

Isn’t that necessarily the case with regression analyses, like with correlations?

18. I don’t think the logistic regression analyses add much to the paper and at the moment they come across as potential p-hacking since they don’t clearly relate to the research question. To me they make the paper feel less focused. Having said that, there is some interesting discussion of them in the Discussion section. I’d recommend adding some justification to the introduction for why it is interesting to look at the relationships among tasks (without pretending to have made any specific hypotheses about the relationships, of course).

19. Line 509 would be clearer if it read “between these groups and the introductory and standard groups”

20. Lines 597 – 620 - This is an interesting discussion, especially the suggestion that advanced calculus may be responsible for the development. No development in reasoning skills from the beginning of a mathematics degree onwards was also found by Inglis and Simpson (2009), who suggested that the initial difference between mathematics and non-mathematics undergraduates could have been due to pre-university study of mathematics. Attridge & Inglis (2013) found evidence that this was the case (they found no difference between mathematics and non-mathematics students at age 16 but a significant difference at the end of the academic year, where the mathematics students had improved and the non-mathematics students had not).

Could the authors add some discussion of whether something similar may have been the case with their Australian sample? E.g. do students in Australia choose whether, or to what extent, to study mathematics towards the end of high school? If not, the description of the groups suggests that there were at least differences in high school mathematics attainment between groups 1-3, even if they studied the same mathematics curriculum. Do the authors think that this difference in attainment could have led to the differences between groups in the current study?

21. Line 617 – “Intensive training has been shown to impact the brain and cognition across a number of domains from music, to video gaming, to Braille reading [31].”

Reference 31 appears to only relate to music. Please add references for video gaming and Braille reading.

22. I recommend editing the figures from SPSS’s default style or re-making them in Excel or DataGraph to look more attractive.

23. I cannot find the associated datafile anywhere in the submission. Apologies if this is my mistake.

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files to be viewed.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/ . PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email us at gro.solp@serugif . Please note that Supporting Information files do not need this step.

Author response to Decision Letter 0

20 Apr 2020

All responses are detailed against the specific reviewers' comments in the Response to Reviewers document

Submitted filename: Response to Reviewers.docx

Decision Letter 1

11 Jun 2020

PONE-D-20-01159R1

Does mathematics training lead to better logical thinking and reasoning? A cross-sectional assessment from students to professors.

Dear Dr. Speelman,

Thank you for submitting your revised manuscript to PLOS ONE. I have sent it to reviewer #2 and have now received the reviewer's comment. As you can see, the reviewer thinks that the manuscript is improved but has some outstanding issues that you would need to address in another round of revision. I notably agree with the reviewer that you should provide the raw data, allowing readers to replicate your analyses. Therefore, I invite you submit a revised version of your manuscript.

Please submit your revised manuscript by Jul 26 2020 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at gro.solp@enosolp . When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

• A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.
• A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.
• An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see:  http://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols

Reviewer #2: The manuscript has improved but there are still a few issues that should be resolved prior to publication.

1. On lines 96, 97, 100 and 102, the references to “general population” should be changed to reflect the fact that these participants were non-mathematics (arts) students.

2. Line 306 – change “mathematics students” to “university students”.

3. The method section doesn’t specify the gender split and mean age of the sample.

4. Table 3 - values the p values listed as .000 should be changed to <.001.

5. Table 3 - I suggest repeating the list of problem numbers and names in the legend. It may make for a long legend but would make it much easier for the reader to interpret the table.

6. I am not sure what the new post hoc tests are comparing. What I expected was to see group 1 compared to groups 2, 3, 4 and 5, and so on. This would tell us which groups are statistically different from each other. At the moment we only know from the overall chi square tests whether there are any differences among the groups or not, we don’t know specifically which groups are statistically different from each other and which ones are not. We only have the authors’ interpretations based on the observed counts.

7. Line 584 - change “performance was correlated with training” to “performance was related to training” to avoid any confusion since a correlation analysis was not performed.

8. Data file – I had expected the data file to give the raw data rather than summary data, i.e. with each participant in a separate row, and a column indicating their group membership, a column giving their age, a column for sex etc (including all the demographics mentioned in the method), and a column for each reasoning question. This would allow other researchers to replicate the regression analyses and look at other relationships within the dataset. Without being able to replicate all analyses in the paper, the data file does not meet the minimal data set definition for publication in PLOS journals: https://journals.plos.org/plosone/s/data-availability .

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool,  https://pacev2.apexcovantage.com/ . PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at  gro.solp@serugif . Please note that Supporting Information files do not need this step.

Author response to Decision Letter 1

16 Jun 2020

Please see "Response to Reviewers" document

Decision Letter 2

PONE-D-20-01159R2

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

An invoice for payment will follow shortly after the formal acceptance. To ensure an efficient process, please log into Editorial Manager at http://www.editorialmanager.com/pone/ , click the 'Update My Information' link at the top of the page, and double check that your user information is up-to-date. If you have any billing related questions, please contact our Author Billing department directly at gro.solp@gnillibrohtua .

If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact gro.solp@sserpeno .

Additional Editor Comments (optional):

Acceptance letter

Dear Dr. Speelman:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact gro.solp@sserpeno .

If we can help with anything else, please email us at gro.solp@enosolp .

Thank you for submitting your work to PLOS ONE and supporting open access.

PLOS ONE Editorial Office Staff

on behalf of

Dr. Jérôme Prado

The Purpose of Logic Puzzles: Developing Critical Thinking and Problem-Solving Skills (And Having Fun!)

Logic puzzles are a popular class of puzzle that challenges the mind and requires logical thinking. The grid style is the most common variety referred to as a “logic puzzle”, but logic puzzles come in various forms and there are benefits to be enjoyed by people of all ages!

The purpose of logic puzzles is to provide a mental workout and improve cognitive skills in the form of a fun, leisurely activity that can be relaxing and satisfying to complete.

Logic puzzles require the player to use reasoning and deduction to solve a problem. They are designed to test the player’s ability to think logically and critically. They are solved using a variety of techniques, including process of elimination, pattern recognition, and logical inference.

The potential benefits of solving logic puzzles are numerous, including improving memory, enhancing problem-solving skills, and reducing stress.

Types of Logic Puzzles

There are many different types of logic puzzles, each with its own unique set of rules and challenges. Some of the most popular variations include:

• Grid puzzles:  These puzzles require the solver to fill in a grid with information based on a set of clues – see our collection of logic grid puzzles if you would like to play one now!

• Sudoku:  Sudoku is an increasingly popular puzzle that could also be considered a type of grid puzzle. The objective of Sudoku is to fill a 9×9 grid with numbers so that each row, column, and 3×3 sub-grid contains all the digits from 1 to 9. The puzzle begins with some numbers already filled in, and the player must use logic and deduction to fill in the rest of the grid. There are also many variations to this standard format.
• Syllogisms :  Syllogisms usually comprise of 3 statements, and the puzzle player uses logic to deduce whether the third statement is true or false.
• Brain teasers:  Brain teasers are puzzles that require the solver to use logic and critical thinking to arrive at the correct answer. They often involve math or visual puzzles.

The answer to the brain teaser can be found at the end of this article.

The Purpose of Logic Puzzles

Logic puzzles are often used as a form of entertainment, but they can also serve a greater purpose – both in school children who are developing critical thinking and reasoning skills, and adults who like to improve these skills or give their brains a regular workout to ward off cognitive decline.

In this section, we will explore the benefits of solving logic puzzles and how they can help in problem-solving.

Benefits of Solving Logic Puzzles

One of the most significant benefits of solving logic puzzles is that they can help improve cognitive skills . Studies have shown that regularly engaging in activities that require critical thinking and problem-solving can help improve memory, concentration, and overall cognitive function.

Children’s development : By solving puzzles, children learn to think logically, analyze information, and make connections between different pieces of information. This helps them to develop their reasoning skills, which are important for academic success.

Preventing cognitive decline : According to a large study of over 50’s who do puzzles such as crosswords and sudokus, as reported by ScienceDaily , “researchers calculate that people who engage in word puzzles have brain function equivalent to ten years younger than their age, on tests assessing grammatical reasoning and eight years younger than their age on tests measuring short term memory.”

Another benefit of solving logic puzzles is that it can help reduce stress and anxiety . When individuals engage in activities that require their full attention, they can become more focused and present in the moment. This can help reduce stress and anxiety levels by providing a mental break from daily stressors.

Finally, solving logic puzzles can be a fun and engaging way to pass the time. As an added bonus – while some are brief and straight to the point, other logic problems have a creative and engaging back story that can be an entertaining read, immersing the reader in the world created by the writer. If the story is based on real-world facts, it can also be an opportunity to learn about a new topic.

How Logic Puzzles Help in Problem-Solving

When solving a logic puzzle, individuals must use a combination of deductive reasoning, critical thinking, and trial and error to arrive at the correct solution. These skills can be applied to real-life situations, as individuals learn to approach problems in a systematic and logical way.

Logic puzzles help in problem-solving by teaching people how to break down complex problems into smaller, more manageable parts. By analyzing the clues and information provided in a logic puzzle, individuals learn to identify patterns and relationships between different pieces of information. This helps them to develop their critical thinking skills, which are essential for solving problems in all areas of life.

Additionally, solving logic puzzles can help individuals develop their lateral thinking skills. Lateral thinking is the ability to approach problems from a different perspective and to think creatively to arrive at a solution. This type of thinking can be valuable in a variety of situations, as it allows people to come up with innovative solutions to problems.

The more logic puzzles you solve, the better you will get at them. Practice regularly to improve your skills!

Browse our collection of logic grid puzzles or first learn how to solve a logic grid puzzle .

Answer to the brain teaser: Dozens.

• share  
• save  
• email  

To provide the best experiences, we and our partners use technologies like cookies to store and/or access device information. Consenting to these technologies will allow us and our partners to process personal data such as browsing behavior or unique IDs on this site and show (non-) personalized ads. Not consenting or withdrawing consent, may adversely affect certain features and functions.

Click below to consent to the above or make granular choices. Your choices will be applied to this site only. You can change your settings at any time, including withdrawing your consent, by using the toggles on the Cookie Policy, or by clicking on the manage consent button at the bottom of the screen.

• IBPS RRB Exam 2023 - Free Course
• Current Affairs
• General Knowledge
• SSC CGL Pre.Yrs.Papers
• SSC CGL Practice Papers
• SBI Clerk PYQ
• IBPS PO PYQ
• IBPS Clerk PYQ
• SBI PO Practice Paper

Logical Reasoning Questions and Answers

• Number Series Reasoning Questions and Answers
• Alphanumeric Series Reasoning Questions and Answers
• Analogy Reasoning Questions and Answers
• Making Judgements: Reasoning Questions
• Course of Action: Logical Reasoning Questions
• Statement and Conclusion Logical Reasoning
• Cause and Effect: Logical Reasoning Questions and Answers
• Statement and Argument-Analytical Reasoning
• Logical Deduction Questions and Answers (2023)
• HCL Placement Paper | Verbal Reasoning Set - 2
• Reasoning Tricks to Solve Coding -Decoding and Calendar Problems
• Statement and Assumption in Logical Reasoning
• Venn Diagram

Logical Reasoning _ Verbal Reasoning

• Verbal Reasoning: Logical Arrangement Of Words
• Placement | Reasoning | Blood Relationship
• Syllogism: Verbal Reasoning Questions and Answers
• Cubes: Verbal Reasoning Questions and Answers
• Seating Arrangement : Aptitude Questions and Answers
• Direction Sense test
• Data Sufficiency in Logical Reasoning

Logical Reasoning _ Non-Verbal Reasoning

• Mirror Image: Verbal Reasoning
• Picture Analogies Questions - Non Verbal Reasoning

Logical Reasoning involves the ability to use and understand logical connections between facts or ideas.

• In verbal reasoning , questions are expressed in words or statements and require the reader to think critically about the language used in order to choose the correct answer from the given options.
• Non-verbal reasoning meanwhile involves questions presented as images and figures, requiring the reader to comprehend how one element relates to another before selecting the right answer out of a list of potential answers.

Logical Reasoning is a key component of many competitive and reasoning ability-testing exams in India and abroad. Reasoning questions allow organizations to assess a candidate’s problem-solving skills, critical thinking capabilities, and capacity for logical and analytical thinking. 

Aptitude Questions such as Quantitative Aptitude and Logical Reasoning are considered essential skills for success in a wide range of competitive exams worldwide. These two sections often form the backbone of entrance exams, whether it’s for a public sector job in India or a university admission test in the United States.

Go through the following article to learn more about the various types of reasoning ability queries generally included in competitive tests.

Logical Reasoning Topics

Logical Reasoning is a crucial section in various competitive exams, and aspirants must study these topics to improve their problem-solving abilities and score better.

Types of Questions included in logical reasoning:

• Verbal Questions
• Puzzle Questions
• Image-Based Questions
• Sequence Questions

Topic-wise practice questions on logical reasoning:

• Number Series
• Letter and Symbol Series
• Verbal Classification
• Essential Part
• Artificial Language
• Matching Definitions
• Making Judgments
• Logical Problems
• Logical Games
• Analyzing Arguments
• Course of Action
• Statement and Conclusion
• Theme Detection
• Cause and Effect
• Statement and Argument
• Logical Deduction
• Letter Series
• Verification of the Truth of the Statement
• Coding Decoding
• Assertion and Reason
• Statement and Assumptions
• Logical Venn Diagram

Verbal Reasoning

Verbal reasoning is the cognitive ability to understand and interpret information presented in written or spoken language and apply logical reasoning to draw conclusions and solve problems.

It involves analyzing and evaluating information, making inferences and deductions, and identifying relationships between concepts and ideas. Verbal reasoning often tests a candidate’s language comprehension, critical thinking, and analytical skills and is commonly used in aptitude tests, job interviews, and higher education admissions.

A strong grasp of verbal reasoning can help individuals communicate effectively, think critically, and make informed decisions in their personal and professional lives.

Verbal Reasoning Questions and Answers Topics

• Logical Sequence of Words
• Blood Relation Test
• Series Completion
• Cube and Cuboid
• Seating Arrangement
• Character Puzzles
• Direction Sense Test
• Classification
• Data Sufficiency
• Arithmetic Reasoning
• Verification of Truth

Non-Verbal Reasoning

Non-verbal reasoning is the cognitive ability that involves questions presented as images and figures, requiring the reader to comprehend how one element relates to another before selecting the right answer out of a list of potential answers.

Non-verbal reasoning often tests a candidate’s ability to think creatively, solve problems, and make quick decisions, and is commonly used in aptitude tests, job interviews, and higher education admissions.

A strong grasp of non-verbal reasoning can help individuals develop their creativity, spatial awareness, and problem-solving abilities, making them more effective at tackling complex challenges in their personal and professional lives.

If you are a government exam aspirant or a student preparing for college placements, the reasoning is the topic that you need to practice thoroughly. Below are some topics that need to be practiced well for the reasoning section of the exam. So, let’s go through the following article to learn more about the various types of reasoning queries generally included in competitive tests.

Non-Verbal Reasoning Questions and Answers Topics

• Analytical Reasoning
• Mirror Images
• Water Images
• Embedded Images
• Pattern Completion
• Figure Matrix
• Paper Folding
• Paper Cutting
• Rule Detection
• Grouping of Images
• Dot Situation
• Shape Construction
• Image Analysis
• Cubes and Dice
• Picture Analogies

Logical reasoning is an important assessment tool for a wide range of competitive examinations. Questions in this section are designed to judge a candidate’s analytical and logical thinking abilities. Various types of reasoning questions are included in this section to test the student’s capacity for problem-solving, deduction, and inference.

Practicing questions is the only way to prepare for the reasoning test section. This way, even those who may struggle in this section can have an equal chance at success during exams or applications. The article contains concepts, questions, and topics of the reasoning section from the competitive exams and the placement exams’ point of view. 

FAQs – Logical Reasoning

Q1. what is logical reasoning  .

Logical reasoning involves the ability to use and understand logical connections between facts or ideas. The reasoning is a critical component of many tests and interviews. In order to perform well, it can be beneficial to practice doing reasoning questions with solutions available. 

Q2. What are logical reasoning questions? 

Logical reasoning questions can be both verbal and non-verbal: In verbal logical reasoning questions, questions are expressed in words or statements and require the reader to think critically about the language used in order to choose the correct answer from the given options and in non-verbal logical reasoning questions, it involves questions presented as images and figures, requiring the reader to comprehend how one element relates to another before selecting the right answer out of a list of potential answers.

Q3. What is the approach to solving reasoning questions? 

Follow the steps given below for preparation: 1. Practice with a timer and solve questions within the time limit. 2. Read the question carefully and try to understand the logic behind it. 3. Practice as many questions as you can and brush up on your skills.

Q4. Which book is good for the preparation of reasoning question sets? 

Students can practice from the following books: 1. A Modern Approach to Verbal & Non-Verbal Reasoning by R.S. Agarwal 2. Shortcuts in Reasoning (Verbal, Non-Verbal, Analytical & Critical) for Competitive Exams by Disha Experts 3. How to Crack Test of Reasoning by Arihant Experts

Q5. What is the syllabus of the Reasoning Aptitude section for competitive exams? 

Reasoning Aptitude covers a wide range of topics. Those topics are already given in the article. Aspirants must go through the article to learn about those topics and practice them thoroughly.

Please Login to comment...

Similar reads.

• SSC/Banking

Improve your Coding Skills with Practice

What kind of Experience do you want to share?

These skills can help you save your job

“I think, therefore I am” is one of the most profound statements by mathematician and philosopher Descartes. It speaks about finding truth in the midst of doubt and uncertainty — a skill that is getting increasingly more valuable today.

From an interview to the latest job talk, you might often be asked to demonstrate your skills around two terms that help you navigate towards the truth through uncertainty — critical thinking and problem solving.

Why are these skills needed today?

Given the rapid advances in technology and the way the future of work and jobs are unfolding, there is definitely more uncertainty today.

In fact, children studying in schools today may grow up to work in jobs we may not even recognise today. Jobs also are shape-shifting in some cases with newer jobs getting discovered as we adapt to working with machines.

Hence, instead of just preparing for short-term need-based jobs, inculcating the skills of critical thinking and problem solving can stand a person in good stead for newer challenges we might face.

What does critical thinking and problem solving look like in action?

Imagine going through vast amounts of information and being able to synthesise that, make logical and evidence-based conclusions. That’s the essence of critical thinking.

Continuing further to problem solving, it helps us find possible answers to a problem and work on the intended solutions.

Logic plays a key role in critical thinking. Daniel Kahneman in his seminal book “Thinking Fast and Slow” spoke about two kinds of thinking that we as humans do: Immediate, gut-based thinking that is often intuitive; and deep, deliverable, thinking.

Both kinds of thinking are required to make different kinds of decisions and to attack different kinds of problems that we will face in our work life.

From a logical point of view, there are two ways to approach this: Deductive logic and inductive logic

In a deductive logic and reasoning approach, we start from individual data points. We try to stitch the patterns we see from that and then arrive at the conclusion.

In inductive logic, we start from a possible hypothesis about the problem we are addressing. This hypothesis could be the result of our intuitive systems. Based on that, we are able to use data in a more streamlined way to either prove or disprove our hypothesis.

At every step, it is important to be aware of the possibility of bias creeping in.

Let’s look at a couple of real-life situations.

Say the customer satisfaction numbers for a company are reducing over time. How can you find a way to improve that situation? Such a problem might require both critical thinking and problem solving.

Using deductive logic, you might start looking at multiple data points across customer touch points to understand the key causes for concern.

On the other hand, using inductive logic, you might first create a hypothesis, like “this is due to customer service levels dropping in channel x.” Then, you start looking at data to see how the picture unfolds.

While both are valid approaches, the second one can save time in an urgent business situation.

Another example. These situations are often tested during interviews. Imagine you are asked “how do you estimate the market demand for petrol pumps in the city?”

Now that you know the two approaches, you can apply a similar logic and get to the possible approaches. The interviewer is looking at your thinking process, not at the exact answer.

There are tools such as structured thinking that take us through a step-by-step approach to focus on insights and problem solving. And reading is another way in which we can keep building our critical thinking skills.

This is also the reason why aptitude in reading, writing, mathematics and logical reasoning is tested in many competitive examinations.

The only difference is that the need for these skills may not end with clearing the exams. These need to be honed lifelong.

One of Coursera’s most popular courses is “learning how to learn.” That constant learnability can be our best guard against certain uncertainty.

For more news like this visit The Economic Times .

Coding with Kids: Unleashing Creativity and Logic

Lomit Patel

• Coding for Kids: Unlocking the Power of Programming
• STEM Coding for Kids: Fun Kits, Robots & Free Resources

Hey there! As a parent or educator, you know kids are natural explorers, always eager to learn and create. What if I told you that coding with kids is the perfect way to nurture their curiosity and help them develop valuable skills for the future?

Coding with kids is about more than just about turning them into mini-programmers. Imagine this as sparking their creative flames while polishing up on logical reasoning and outfitting them with epic problem-busting gear. And the best part? It’s a ton of fun!

Best Free Coding Resources for Coding With Kids

Being a good parent is all about giving your children pathways where they can both learn plenty and bloom. And in today’s digital age, coding with kids has become an essential skill that can open up a world of possibilities for your child’s future.

But where do you start? With so many coding with kids resources, finding the right ones for your kid can be overwhelming. That’s why I’ve rounded up an awesome selection of coding resources for kids that won’t cost a dime—think interactive websites, user-friendly apps, engaging classes, and exciting camps.

Websites for Coding With Kids

Tynker is a popular website that offers free coding courses and resources for kids. It features interactive coding lessons, tutorials, and activities that teach programming concepts in a fun and engaging way.

The website is designed for students of all ages and skill levels, making it an excellent resource for kids who are just starting to learn coding. With Tynker, your child can learn coding concepts like loops, conditionals, and functions while building their own games and animations.

Apps for Coding With Kids

Tynker Junior is a free coding app designed for young children aged 5-7. It introduces basic programming concepts through a visual, drag-and-drop interface.

With Tynker Junior, kids can create their own interactive stories and games by snapping together graphical programming blocks. The app encourages creativity, logical thinking, and problem-solving skills, making it a great way to introduce your child to the world of coding.

Classes for Coding With Kids

Many libraries and community centers offer free coding classes for kids. These classes are often taught by volunteers or local tech professionals and provide a structured learning experience .

At this spot, kids start their journey into coding by playing around with cool programs like Tynker and Python. Imagine your child cracking coding puzzles, side-by-side with classmates, all while expert guides light their way.

Camps for Coding With Kids

CoderDojo is a global network of free coding clubs for young people aged 7-17. These clubs, or “dojos,” are run by volunteers and provide a fun and social learning environment.

At CoderDojo, kids can learn various programming languages , develop websites, apps, and games, and collaborate with other young coders. We pop up our camps right where everyone can join in—local libraries, schools, and even universities welcome kids from every walk of life.

Top Coding With Kids Apps to Learn Programming

Learning to code is like learning a new language—the earlier you start, the easier it is to pick up. And with the rise of coding apps for kids, it’s never been more fun and accessible to introduce your child to the world of programming.

But with so many apps out there, how do you choose the right one for your kid? Here are my top picks for coding apps that will help your child learn programming in a fun and engaging way.

codeSpark Academy

codeSpark Academy is a coding app designed for kids aged 4-9. Imagine grasping basic coding ideas while navigating through exciting puzzles and artistic assignments – that’s what this method is all about.

With codeSpark Academy, kids learn to sequence commands, use loops, and apply conditional logic to guide their characters through various adventures. Kids will find themselves at home with this app’s colorful look and user-friendly layout that eases them into learning step by step.

Hopscotch is a coding app that allows kids to create their own games, animations, and stories using a visual programming language. It is designed for children aged 8 and above and offers a drag-and-drop interface to make coding accessible and fun.

With Hopscotch as their guide, children can unlock the secrets of coding—from mastering variables to navigating conditionals and tinkering with functions—and all that learning is channeled into creating something amazing from scratch. They can also share their creations with the Hopscotch community and remix projects made by others to learn new coding techniques.

Imagine learning to code in an easy-breezy way—that’s what Mimo offers through its playful lessons on everything from Python to JavaScript, not forgetting the backbone of websites – HTML and CSS. While it is not exclusively designed for kids, it provides a user-friendly interface and bite-sized lessons that make it suitable for older children and teenagers who want to learn real-world coding skills.

With Mimo, kids can learn programming concepts through hands-on coding exercises, quizzes, and projects. The app also offers a mobile code compiler that allows users to write and run code directly on their devices, making it easy to practice coding on the go.

Lightbot is a puzzle-based coding app that teaches programming logic and concepts through a series of challenges. Players guide a robot to light up tiles on a grid by issuing a sequence of commands.

As players progress through increasingly complex levels, the app introduces concepts like procedures, loops, and conditionals. Lightbot is suitable for kids aged 4 and above and helps develop problem-solving and computational thinking skills through fun, game-like challenges.

Essential Concepts for Coding With Kids to Master

Learning to code is not just about memorizing syntax and commands – it’s about understanding the fundamental concepts that underpin all programming languages. By mastering these essential coding concepts, kids can develop a strong foundation for future learning and problem-solving.

Here are the top 5 coding concepts that every kid should learn:

Variables are like containers that hold values, such as numbers, text, or true/false statements. They are used in almost every programming language and are essential for creating dynamic and interactive programs.

Teaching kids about variables helps them understand how to store and manipulate data in their code. They can use variables to keep track of scores in a game, store user input, or create personalized messages in their programs.

Loops are like your personal code DJ, spinning the same beat over and over. They’re perfect for when you’ve got to hammer out a task repeatedly, say sketching a cool design or whipping through a bunch of items on your list.

Teaching kids about loops helps them understand how to write efficient and concise code. They can use loops to create animated stories , generate patterns, or process large amounts of data in their programs.

Conditionals

Conditionals in programming are like the crossroads where you get to decide which path your code takes, based on specific conditions. They are often expressed as “if-then” statements, where a specific action is performed only if a particular condition is met.

Teaching kids about conditionals helps them understand how to create programs that can respond to different situations and make logical decisions based on input or data. They can use conditionals to create interactive stories, games, or quizzes that change based on user input.

Functions are like little boxes of code that do one job really well. Think of them as breaking down big tasks into manageable parts – making both organizing your work and understanding it way simpler.

When children start understanding how functions work, it’s like giving them a magic wand for their coding projects. They can avoid repeating themselves and piece together programs that are simpler to tweak and care for. Imagine having a toolbox where you pull out handy bits of code for drawing things or figuring stuff out – that’s what functions let you do.

Objects are like containers that hold related data (properties) and functions (methods) that operate on that data. They are a fundamental concept in object-oriented programming (OOP), which is a common paradigm used in many programming languages.

Imagine teaching young minds about objects; it’s essentially showing them the ropes on making their code tidy while crafting components they can use multiple times without starting from scratch each time. They can use objects to create interactive games, simulations, or animations that involve multiple elements with different properties and behaviors.

Engaging Activities and Projects for Coding With Kids

Kids will jump at the chance to code when you introduce them to projects brimming with fun and creativity. By building their own games, animations, apps, and websites, kids can see the tangible results of their coding efforts and develop a sense of pride and accomplishment in their work.

Here are some of my favorite coding activities and projects for kids:

Building Games

Crafting games isn’t just fun—it’s a sneakily clever way to get kids comfortable with coding. They’ll pick up everything from the basics of variables and loops to conditionals and functions without even realizing they’re learning. By creating simple games like tic-tac-toe, guess the number, or snake, kids can see how these concepts are used in a practical context.

There are many game development platforms designed specifically for kids, such as Scratch, Tynker, and Roblox Studio. Starting from scratch isn’t necessary on these kid-friendly sites that offer plug-and-play solutions like snap-together interfaces designed for dragging and dropping elements seamlessly; they also boast libraries filled with customizable game blueprints plus comprehensive how-tos every step of the way.

Creating Animations

If you’re learning to code as a kid, imagine turning lines of code into your very own animation – now that’s exciting. Kids tapping into platforms like Scratch or Alice find themselves at the helm of story creation—animating personalities on-screen paired perfectly with chosen soundtracks.

Animation isn’t just about bringing drawings to life; it’s a playground where kids learn core coding principles – think sequence control and interaction cues – all while channeling their inner artist and storyteller. They can create short films , music videos, or interactive storybooks that showcase their creativity and coding abilities.

Developing Apps

Imagine this: young minds crafting mobile apps as a playground for sharpening their coding skills. It’s hands-on learning that ends with seeing their own designs working on something as familiar as a smartphone. Using app development platforms like MIT App Inventor or Thunkable, kids can create simple apps like quizzes, calculators, or drawing tools.

In these projects where kids build apps from Tynker, they’re not only coding but learning by doing as they explore interface aesthetics, tackle command reactions head-on (event handling), and sort massive amounts of info. They can also learn how to integrate features like sensors, cameras, and location services into their apps, making them more interactive and engaging.

Designing Websites

Crafting websites turns out to be a super fun coding project for youngsters. With just a bit of HTML, CSS, and JavaScript under their belts, young creators are setting up web spaces filled with vibrant images and clips tailored exactly how they like them.

Through building websites, children get hands-on with designing from scratch, organizing content beautifully, and bringing elements to life interactively. Dive into creating everything from deeply personal weblogs and showcase portfolios on the internet, right down to fan pages celebrating whatever makes your heart race—all while picking up invaluable programming know-how bound to serve you well down line in any job market.

Preparing Kids for a Future in Programming

As a parent, you want to give your child every advantage in life. And in today’s digital age, learning to code is one of the best ways to set them up for success in the future.

Learning to code goes beyond just picking up a programming language or crafting the next hit app. The goal? To foster skills and ways of thinking in your child that will be assets, regardless of the profession they choose to pursue.

Problem-Solving Skills

Learning to code helps kids develop problem-solving skills, which are essential for success in programming and many other fields. When coding, kids learn how to break down complex problems into smaller, manageable steps, and how to approach challenges systematically and logically.

These skills are transferable to other areas of life and can help kids become better critical thinkers and decision-makers. By learning to approach problems methodically, kids can develop a growth mindset and learn to persevere in the face of challenges.

Logical Thinking

Coding requires logical thinking, which involves analyzing problems, identifying patterns, and making decisions based on facts and evidence. By learning to code, kids develop their logical thinking skills, which are crucial for understanding complex systems, debugging code, and creating efficient algorithms.

These skills are valuable not only in programming but also in fields like mathematics, science, and engineering. By learning to think logically and systematically , kids can develop a deeper understanding of how the world works and how to solve problems in a rational and evidence-based way.

While coding is often associated with logic and structure, it also requires creativity and imagination. When kids learn to code, they have the opportunity to create their own projects, design user interfaces, and develop unique solutions to problems.

Coding encourages kids to think outside the box, experiment with new ideas, and express themselves through technology. By learning to be creative with code, kids can develop a sense of agency and empowerment, knowing that they have the skills and tools to bring their ideas to life.

Persistence

Learning to code can be challenging, and kids may encounter obstacles and frustrations along the way. However, coding also teaches persistence and resilience, as kids learn to troubleshoot errors, iterate on their designs, and keep trying until they succeed.

Ever noticed how picking apart what went wrong – whether you’re debugging code or dealing with everyday hiccups – teaches kids not just patience but perseverance? That’s where real growth happens. By learning to persist through challenges, kids can develop grit and determination that will serve them well in any endeavor they pursue.

The journey from learning code to creating personal projects is one where kids pick up not just skills but also a hefty dose of belief in themselves along with pure satisfaction. It’s pretty cool how bringing one’s thoughts out onto the display not only sparks joy but also invites others in. This vibe boosts morale big time – stirring an eagerness to explore further.

This confidence can spill over into other areas of their lives, helping them feel more capable and empowered to tackle new challenges. By learning to

Start your kid’s coding journey with free resources like Code.org and Scratch Jr. They make learning fun and accessible, setting a solid foundation for future success. Dive into apps, classes, camps, or even game building to keep them engaged. Coding not only boosts problem-solving skills but also sparks creativity and confidence in kids.

Showing young ones how to navigate through codes transcends traditional teaching—it’s like we’re exploring uncharted territories side by side, far beyond mere coding lessons. It’s about fostering a love for learning, encouraging them to think outside the box, and helping them develop skills that will serve them well in any future endeavor.

So, whether your kid is building their first game, animating a story, or designing a website, remember that by coding with kids they’re not just playing around; they’re laying the foundation for a bright future. Yep, that’s you—right at the heart of all this awesomeness.

So, are we diving headfirst into this adventure or what? Let’s code, create, and watch our kids soar to new heights!

About Lomit Patel

Related articles.

A Coding Project to Honor Martin Luther King Jr. Day

Learn Coding: A Practical Guide for Parents and Teachers

Tynker Summer Code Jam Winner: Elizabeth Gets Creative with Code

How to Explain Algorithms to Kids

My coding path.

Take your child from novice to expert. Just follow the path.

Block Coding Courses

Apply your coding skills! Find your passion!

Other Advanced Tynker Courses

Graduate to Python and other text coding languages with Tynker’s advanced elective courses.