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What Students Are Saying About the Value of Math

We asked teenagers: Do you see the point in learning math? The answer from many was “yes.”

By The Learning Network

“Mathematics, I now see, is important because it expands the world,” Alec Wilkinson writes in a recent guest essay . “It is a point of entry into larger concerns. It teaches reverence. It insists one be receptive to wonder. It requires that a person pay close attention.”

In our writing prompt “ Do You See the Point in Learning Math? ” we wanted to know if students agreed. Basic arithmetic, sure, but is there value in learning higher-level math, such as algebra, geometry and calculus? Do we appreciate math enough?

The answer from many students — those who love and those who “detest” the subject alike — was yes. Of course math helps us balance checkbooks and work up budgets, they said, but it also helps us learn how to follow a formula, appreciate music, draw, shoot three-pointers and even skateboard. It gives us different perspectives, helps us organize our chaotic thoughts, makes us more creative, and shows us how to think rationally.

Not all were convinced that young people should have to take higher-level math classes all through high school, but, as one student said, “I can see myself understanding even more how important it is and appreciating it more as I get older.”

Thank you to all the teenagers who joined the conversation on our writing prompts this week, including students from Bentonville West High School in Centerton, Ark, ; Harvard-Westlake School in Los Angeles ; and North High School in North St. Paul, Minn.

Please note: Student comments have been lightly edited for length, but otherwise appear as they were originally submitted.

“Math is a valuable tool and function of the world.”

As a musician, math is intrinsically related to my passion. As a sailor, math is intertwined with the workings of my boat. As a human, math is the building block for all that functions. When I was a child, I could very much relate to wanting a reason behind math. I soon learned that math IS the reason behind all of the world’s workings. Besides the benefits that math provides to one’s intellect, it becomes obvious later in life that math is a valuable tool and function of the world. In music for example, “adolescent mathematics” are used to portray functions of audio engineering. For example, phase shifting a sine wave to better project sound or understanding waves emitted by electricity and how they affect audio signals. To better understand music, math is a recurring pattern of intervals between generating pitches that are all mathematically related. The frets on a guitar are measured precisely to provide intervals based on a tuning system surrounding 440Hz, which is the mathematically calculated middle of the pitches humans can perceive and a string can effectively generate. The difference between intervals in making a chord are not all uniform, so guitar frets are placed in a way where all chords can sound equally consonant and not favor any chord. The power of mathematics! I am fascinated by the way that math creeps its way into all that I do, despite my plentiful efforts to keep it at a safe distance …

— Renan, Miami Country Day School

“Math isn’t about taking derivatives or solving for x, it’s about having the skills to do so and putting them to use elsewhere in life.”

I believe learning mathematics is both crucial to the learning and development of 21st century students and yet also not to be imposed upon learners too heavily. Aside from the rise in career opportunity in fields centered around mathematics, the skills gained while learning math are able to be translated to many facets of life after a student’s education. Learning mathematics develops problem solving skills which combine logic and reasoning in students as they grow. The average calculus student may complain of learning how to take derivatives, arguing that they will never have to use this after high school, and in that, they may be right. Many students in these math classes will become writers, musicians, or historians and may never take a derivative in their life after high school, and thus deem the skill to do so useless. However, learning mathematics isn’t about taking derivatives or solving for x, it’s about having the skills to do so and putting them to use elsewhere in life. A student who excels at calculus may never use it again, but with the skills of creativity and rational thinking presented by this course, learning mathematics will have had a profound effect on their life.

— Cam, Glenbard West

“Just stop and consider your hobbies and pastimes … all of it needs math.”

Math is timing, it’s logic, it’s precision, it’s structure, and it’s the way most of the physical world works. I love math — especially algebra and geometry — as it all follows a formula, and if you set it up just right, you can create almost anything you want in at least two different ways. Just stop and consider your hobbies and pastimes. You could be into skateboarding, basketball, or skiing. You could be like me, and sit at home for hours on end grinding out solves on a Rubik’s cube. Or you could be into sketching. Did you know that a proper drawing of the human face places the eyes exactly halfway down from the top of the head? All of it needs math. Author Alec Wilkinson, when sharing his high school doubting view on mathematics, laments “If I had understood how deeply mathematics is embedded in the world …” You can’t draw a face without proportions. You can’t stop with your skis at just any angle. You can’t get three points without shooting at least 22 feet away from the basket, and get this: you can’t even ride a skateboard if you can’t create four congruent wheels to put on it.

— Marshall, Union High School, Vancouver, WA

“Math gives us a different perspective on everyday activities.”

Even though the question “why do we even do math?” is asked all the time, there is a deeper meaning to the values it shares. Math gives us a different perspective on everyday activities, even if those activities in our routine have absolutely nothing to do with mathematical concepts itself. Geometry, for instance, allows us to think on a different level than simply achieving accuracy maintains. It trains our mind to look at something from various viewpoints as well as teaching us to think before acting and organizing chaotic thoughts. The build up of learning math can allow someone to mature beyond the point where if they didn’t learn math and thought through everything. It paves a way where we develop certain characteristics and traits that are favorable when assisting someone with difficult tasks in the future.

— Linden, Harvard-Westlake High School, CA

“Math teaches us how to think.”

As explained in the article, math is all around us. Shapes, numbers, statistics, you can find math in almost anything and everything. But is it important for all students to learn? I would say so. Math in elementary school years is very important because it teaches how to do simple calculations that can be used in your everyday life; however middle and high school math isn’t used as directly. Math teaches us how to think. It’s far different from any other subject in school, and truly understanding it can be very rewarding. There are also many career paths that are based around math, such as engineering, statistics, or computer programming, for example. These careers are all crucial for society to function, and many pay well. Without a solid background in math, these careers wouldn’t be possible. While math is a very important subject, I also feel it should become optional at some point, perhaps part way through high school. Upper level math classes often lose their educational value if the student isn’t genuinely interested in learning it. I would encourage all students to learn math, but not require it.

— Grey, Cary High School

“Math is a valuable tool for everyone to learn, but students need better influences to show them why it’s useful.”

Although I loved math as a kid, as I got older it felt more like a chore; all the kids would say “when am I ever going to use this in real life?” and even I, who had loved math, couldn’t figure out how it benefits me either. This was until I started asking my dad for help with my homework. He would go on and on about how he used the math I was learning everyday at work and even started giving me examples of when and where I could use it, which changed my perspective completely. Ultimately, I believe that math is a valuable tool for everyone to learn, but students need better influences to show them why it’s useful and where they can use it outside of class.

— Lilly, Union High School

“At the roots of math, it teaches people how to follow a process.”

I do believe that the math outside of arithmetic, percentages, and fractions are the only math skills truly needed for everyone, with all other concepts being only used for certain careers. However, at the same time, I can’t help but want to still learn it. I believe that at the roots of math, it teaches people how to follow a process. All mathematics is about following a formula and then getting the result of it as accurately as possible. It teaches us that in order to get the results needed, all the work must be put and no shortcuts or guesses can be made. Every equation, number, and symbol in math all interconnect with each other, to create formulas that if followed correctly gives us the answer needed. Everything is essential to getting the results needed, and skipping a step will lead to a wrong answer. Although I do understand why many would see no reason to learn math outside of arithmetic, I also see lessons of work ethics and understanding the process that can be applied to many real world scenarios.

— Takuma, Irvine High School

“I see now that math not only works through logic but also creativity.”

A story that will never finish resembling the universe constantly expanding, this is what math is. I detest math, but I love a never-ending tale of mystery and suspense. If we were to see math as an adventure it would make it more enjoyable. I have often had a closed mindset on math, however, viewing it from this perspective, I find it much more appealing. Teachers urge students to try on math and though it seems daunting and useless, once you get to higher math it is still important. I see now that math not only works through logic but also creativity and as the author emphasizes, it is “a fundamental part of the world’s design.” This view on math will help students succeed and have a more open mindset toward math. How is this never-ending story of suspense going to affect YOU?

— Audrey, Vancouver, WA union high school

“In some word problems, I encounter problems that thoroughly interest me.”

I believe math is a crucial thing to learn as you grow up. Math is easily my favorite subject and I wish more people would share my enthusiasm. As Alec Wilkinson writes, “Mathematics, I now see, is important because it expands the world.” I have always enjoyed math, but until the past year, I have not seen a point in higher-level math. In some of the word problems I deal with in these classes, I encounter problems that thoroughly interest me. The problems that I am working on in math involve the speed of a plane being affected by wind. I know this is not riveting to everyone, but I thoroughly wonder about things like this on a daily basis. The type of math used in the plane problems is similar to what Alec is learning — trigonometry. It may not serve the most use to me now, but I believe a thorough understanding of the world is a big part of living a meaningful life.

— Rehan, Cary High School

“Without high school classes, fewer people get that spark of wonder about math.”

I think that math should be required through high school because math is a use-it-or-lose-it subject. If we stop teaching math in high school and just teach it up to middle school, not only will many people lose their ability to do basic math, but we will have fewer and fewer people get that spark of wonder about math that the author had when taking math for a second time; after having that spark myself, I realized that people start getting the spark once they are in harder math classes. At first, I thought that if math stopped being required in high school, and was offered as an elective, then only people with the spark would continue with it, and everything would be okay. After thinking about the consequences of the idea, I realized that technology requires knowing the seemingly unneeded math. There is already a shortage of IT professionals, and stopping math earlier will only worsen that shortage. Math is tricky. If you try your best to understand it, it isn’t too hard. However, the problem is people had bad math teachers when they were younger, which made them hate math. I have learned that the key to learning math is to have an open mind.

— Andrew, Cary High School

“I think math is a waste of my time because I don’t think I will ever get it.”

In the article Mr. Wilkinson writes, “When I thought about mathematics at all as a boy it was to speculate about why I was being made to learn it, since it seemed plainly obvious that there was no need for it in adult life.” His experience as a boy resonates with my experience now. I feel like math is extremely difficult at some points and it is not my strongest subject. Whenever I am having a hard time with something I get a little upset with myself because I feel like I need to get everything perfect. So therefore, I think it is a waste of my time because I don’t think I will ever get it. At the age of 65 Mr. Wilkinson decided to see if he could learn more/relearn algebra, geometry and calculus and I can’t imagine myself doing this but I can see myself understanding even more how important it is and appreciating it more as I get older. When my dad was young he hated history but, as he got older he learned to appreciate it and see how we can learn from our past mistakes and he now loves learning new things about history.

— Kate, Cary High School

“Not all children need to learn higher level math.”

The higher levels of math like calculus, algebra, and geometry have shaped the world we live in today. Just designing a house relates to math. To be in many professions you have to know algebra, geometry, and calculus such as being an economist, engineer, and architect. Although higher-level math isn’t useful to some people. If you want to do something that pertains to math, you should be able to do so and learn those high levels of math. Many things children learn in math they will never use again, so learning those skills isn’t very helpful … Children went through so much stress and anxiety to learn these skills that they will never see again in their lives. In school, children are using their time learning calculus when they could be learning something more meaningful that can prepare them for life.

— Julyssa, Hanover Horton High School

“Once you understand the basics, more math classes should be a choice.”

I believe that once you get to the point where you have a great understanding of the basics of math, you should be able to take more useful classes that will prepare you for the future better, rather than memorizing equations after equations about weird shapes that will be irrelevant to anything in my future. Yes, all math levels can be useful to others’ futures depending on what career path they choose, but for the ones like me who know they are not planning on encountering extremely high level math equations on the daily, we should not have to take math after a certain point.

— Tessa, Glenbard West High School

“Math could shape the world if it were taught differently.”

If we learned how to balance checkbooks and learn about actual life situations, math could be more helpful. Instead of learning about rare situations that probably won’t come up in our lives, we should be learning how to live on a budget and succeed money-wise. Since it is a required class, learning this would save more people from going into debt and overspending. In schools today, we have to take a specific class that doesn’t sound appealing to the average teenager to learn how to save and spend money responsibly. If it was required in math to learn about that instead of how far Sally has to walk then we would be a more successful nation as a whole. Math could shape the world differently but the way it is taught in schools does not have much impact on everyday life.

— Becca, Bentonville West High School

“To be honest, I don’t see the point in learning all of the complicated math.”

In a realistic point of view, I need to know how to cut a cake or a piece of pie or know how to divide 25,000 dollars into 10 paychecks. On the other hand, I don’t need to know the arc and angle. I need to throw a piece of paper into a trash can. I say this because, in all reality and I know a lot of people say this but it’s true, when are we actually going to need this in our real world lives? Learning complicated math is a waste of precious learning time unless you desire to have a career that requires these studies like becoming an engineer, or a math professor. I think that the fact that schools are still requiring us to learn these types of mathematics is just ignorance from the past generations. I believe that if we have the technology to complete these problems in a few seconds then we should use this technology, but the past generations are salty because they didn’t have these resources so they want to do the same thing they did when they were learning math. So to be honest, I don’t see the point in learning all of the complicated math but I do think it’s necessary to know the basic math.

— Shai, Julia R Masterman, Philadelphia, PA

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Why Is Math Important? 9 Reasons Why Math Skills Improve Quality of Life

Written by Ashley Crowe

• Parent Resources

Why is math so important in life?

• 9 Benefits of a great math education

Why students struggle to master certain math concepts

• Set your child’s math skills up for life with Prodigy Math

Math isn't just an important subject in school — it’s essential for many of your daily tasks. You likely use it every day to perform real-life skills, like grocery shopping, cooking and tracking your finances. 

What makes math special is that it’s a universal language — a powerful tool with the same meaning across the globe. Though languages divide our world, numbers unite us. Math allows us to work together towards new innovations and ideas. 

In this post, learn why math is important for kids and adults. Plus, find out why learning even the most basic math can significantly improve your family’s quality of life.

You simply can’t make it through a day without using some sort of basic math. Here’s why.

A person needs an understanding of math, measurements and fractions to cook and bake. Many people may also use math to count calories or nutrients as part of their diet or exercise routine. 

You also need math to calculate when you should leave your house to arrive on time, or how much paint you need to redo your bedroom walls. 

And then the big one, money. Financial literacy is an incredibly important skill for adults to master. It can help you budget, save and even help you make big decisions like changing careers or buying a home. 

Mathematical knowledge may even be connected to many other not-so-obvious benefits. A strong foundation in math can translate into increased understanding and regulation of your emotions, improved memory and better problem-solving skills.

The importance of math: 9 benefits of a great math education

Math offers more opportunities beyond grade school, middle school and high school. Its applications to real-life scenarios are vast. 

Though many students sit in math class wondering when they’ll ever use these things they’re learning, we know there are many times their math skills will be needed in adulthood. 

The importance of mathematics to your child’s success can’t be overstated. Basic math is a necessity, but even abstract math can help hone critical thinking skills — even if your child chooses not to pursue a STEM-style career. Math can help them succeed professionally, emotionally and cognitively. Here’s why.

1. Math promotes healthy brain function

“Use it or lose it.” We hear this said about many skills, and math is no exception. 

Solving math problems and improving our math skills gives our brain a good workout. And it improves our cognitive skills over time. Many studies have shown that routinely practicing math keeps our brain healthy and functioning well.

2. Math improves problem-solving skills

At first, classic math problems like Johnny bringing home 42 watermelons and returning 13 of them can just seem a silly exercise. But all those math word problems our children solve really do improve their problem solving skills. Word problems teach kids how to pull out the important information and then manipulate it to find a solution.

Later on, complex life problems take the place of workbooks, but problem-solving still happens the same way. When students understand algorithms and problems more deeply, they can decode the facts and more easily solve the issue. Real-life solutions are found with math and logic.

3. Math supports logical reasoning and analytical thinking

A strong understanding of math concepts means more than just number sense. It helps us see the pathways to a solution. Equations and word problems need to be examined before determining the best method for solving them. And in many cases, there’s more than one way to get to the right answer. 

It’s no surprise that logical reasoning and analytical thinking improve alongside math skills. Logic skills are necessary at all levels of mathematical education.

4. Math develops flexible thinking and creativity

Practicing math has been shown to improve investigative skills, resourcefulness and creativity.

This is because math problems often require us to bend our thinking and approach problems in more than one way. The first process we try might not work. We need flexibility and creativity to think of new pathways to the solution. And just like anything else, this way of thinking is strengthened with practice.

5. Math opens up many different career paths

There are many careers that use a large number of math concepts. These include architects, accountants, and scientists. 

But many other professionals use math skills every day to complete their jobs. CEOs use math to analyze financials. Mailmen use it to calculate how long it will take them to walk their new route. Graphic designers use math to figure out the appropriate scale and proportions in their designs. 

No matter what career path your child chooses, math skills will be beneficial.

Math skills might become even more important for today's kids!

Math can certainly open up a lot of opportunities for many of us. But did you know that careers which heavily use math are going to be among the fastest-growing jobs by the time kids today start their careers? These jobs include:

• Statisticians
• Data scientists
• Software developers
• Cybersecurity analysts

It's not just STEM jobs that will require math either. Other popular, high-growth careers like nursing and teaching now ask for a minimum knowledge of college-level math.

6. Math may boost emotional health

While this research is still in its early days, what we have seen is promising. 

The parts of the brain used to solve math problems seem to work together with the parts of the brain that regulate emotions. This suggests that math practice can actually help us cope with difficult situations. In these studies, the better someone was with numerical calculations, the better they were at regulating fear and anger. Strong math skills may even be able to help treat anxiety and depression.

7. Math improves financial literacy

Though kids may not be managing their finances now, there's going to be plenty of times where math skills are going to make a massive difference in their life as an adult.

Budgeting and saving is a big one. Where can they cut back on their spending? How will budgeting help them reach their financial goals? Can they afford this new purchase now? 

As they age into adulthood, It will benefit your child to understand how loans and interest work before purchasing a house or car. They should fully grasp profits and losses before investing in the stock market. And they will likely need to evaluate job salaries and benefits before choosing their first job.

8. Math sharpens your memory

Learning mental math starts in elementary school. Students learn addition tables, then subtraction, multiplication and division tables. As they master those skills, they’ll begin to memorize more tips and tricks, like adding a zero to the end when multiplying by 10. Students will memorize algorithms and processes throughout their education. 

Using your memory often keeps it sharp. As your child grows and continues to use math skills in adulthood, their memory will remain in tip top shape.

9. Math teaches perseverance

“I can do it!’ 

These are words heard often from our toddlers. This phrase is a marker of growth, and a point of pride. But as your child moves into elementary school, you may not hear these words as often or with as much confidence as before. 

Learning math is great for teaching perseverance. With the right math instruction, your child can see their progress and once again feel that “I can do it” attitude. The rush of excitement a child experiences when they master a new concept sticks in their memory. And they can reflect back on it when they’re struggling with a new, harder skill. 

Even when things get tough, they’ll know they can keep trying and eventually overcome it — because they’ve done it before.

Tip: Set goals to inspire and motivate your child to learn math

If your child has a  Prodigy Math Membership , you can use your parent account to set learning goals for them to achieve as they play our online math game.

The best bit? Every time they complete a goal, they'll also get a special in-game reward!

Many students experience roadblocks and hurdles throughout their math education. You might recognize some of these math struggles below in your child. But don’t worry! Any struggle is manageable with the right support and help. Together, you and your child can tackle anything. 

Here are some of the most common math struggles. 

• Increasing complexity

Sometimes the pace of class moves a bit faster than your child can keep up with. Or the concepts are just too abstract and difficult for them to wrap their mind around in one lesson. Some math ideas simply take more time to learn. 

• Wrong teaching style

A good teaching style with plenty of practice is essential to a high-quality math education. If the teacher’s style doesn’t mesh well with how your child learns, math class can be challenging. 

• Fear of failure  

Even as adults, we can feel scared to fail. It’s no surprise that our children experience this same same fear, especially with the many other pressures school can bring. 

• Lack of practice  

Sometimes, all your child needs is a little more practice. But this can be easier said than done. You can help by providing them with plenty of support and encouragement to help them get that practice time in.

• Math anxiety

Algorithms and complex problems can strike anxiety in the heart of any child (and many adults). Math anxiety is a common phenomenon. But with the right coping strategies it can be managed. 

Set your child’s math skills up for success with Prodigy Math

Now we've discovered just how important math is in both our everyday and life decisions, let's set the next generation up for success with the right tools that'll help them learn math.

Prodigy Math is a game-based, online learning platform that makes learning math fun for kids. As kids play and explore a safe, virtual world filled with fun characters and pets to collect, they'll answer math questions. These questions are curriculum-aligned and powered by an adaptive algorithm that can help them master math skills more quickly.

Plus, with a free parent account , you'll also get to be a big part of their math education without needing to be a math genius. You'll get to:

• Easily keep up with their math learning with a monthly Report Card
• See how they're doing in math class when their teacher uses Prodigy Math
• Send them motivational messages to encourage their perseverance in math

Want to play an even bigger role in helping your child master math? Try our optional Math Memberships for extra in-game content for your child to enjoy and get amazing parent tools like the ability to set in-game goals and rewards for them to achieve.

See why Prodigy can make math fun below!

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• School Education /

Essay on Importance of Mathematics in our Daily Life in 100, 200, and 350 words.

• Updated on  
• Dec 22, 2023

Mathematics is one of the core aspects of education. Without mathematics, several subjects would cease to exist. It’s applied in the science fields of physics, chemistry, and even biology as well. In commerce accountancy, business statistics and analytics all revolve around mathematics. But what we fail to see is that not only in the field of education but our lives also revolve around it. There is a major role that mathematics plays in our lives. Regardless of where we are, or what we are doing, mathematics is forever persistent. Let’s see how maths is there in our lives via our blog essay on importance of mathematics in our daily life. 

Table of Contents

• 1 Essay on Importance of Mathematics in our Daily life in 100 words 
• 2 Essay on Importance of Mathematics in our Daily life in 200 words
• 3 Essay on Importance of Mathematics in our Daily Life in 350 words

Essay on Importance of Mathematics in our Daily life in 100 words 

Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Also Read:- Importance of Internet

Essay on Importance of Mathematics in our Daily life in 200 words

Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same. 

From making instalments to dialling basic phone numbers it all revolves around mathematics. 

Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations. 

Without mathematics and numbers, none of this would be possible.

Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler. 

Also Read:-   My Aim in Life

Essay on Importance of Mathematics in our Daily Life in 350 words

Mathematics is what we call a backbone, a backbone of science. Without it, human life would be extremely difficult to imagine. We cannot live even a single day without making use of mathematics in our daily lives. Without mathematics, human progress would come to a halt. 

Maths helps us with our finances. It helps us calculate our daily, monthly as well as yearly expenses. It teaches us how to divide and prioritise our expenses. Its knowledge is essential for investing money too. We can only invest money in property, bank schemes, the stock market, mutual funds, etc. only when we calculate the figures. Let’s take an example from the basic routine of a day. Let’s assume we have to make tea for ourselves. Without mathematics, we wouldn’t be able to calculate how many teaspoons of sugar we need, how many cups of milk and water we have to put in, etc. and if these mentioned calculations aren’t made, how would one be able to prepare tea? 

In such a way, mathematics is used to decide the portions of food, ingredients, etc. Mathematics teaches us logical reasoning and helps us develop problem-solving skills. It also improves our analytical thinking and reasoning ability. To stay in shape, mathematics helps by calculating the number of calories and keeping the account of the same. It helps us in deciding the portion of our meals. It will be impossible to think of sports without mathematics. For instance, in cricket, run economy, run rate, strike rate, overs bowled, overs left, number of wickets, bowling average, etc. are calculated. It also helps in predicting the result of the match. When we are on the road and driving, mathetics help us keep account of our speeds, the distance we have travelled, the amount of fuel left, when should we refuel our vehicles, etc. 

We can go on and on about how mathematics is involved in our daily lives. In conclusion, we can say that the universe revolves around mathematics. It encompasses everything and without it, we cannot imagine our lives. 

Also Read:- Essay on Pollution

Ans: Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Ans: Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same.  From making instalments to dialling basic phone numbers it all revolves around mathematics. Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations. Without mathematics and numbers, none of this would be possible. Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler.  

Ans: Archimedes is considered the father of mathematics.

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September 2, 2017

Why Is It Important to Study Math?

What’s the point of learning math? Why is it so important that kids are exposed to mathematical thinking? And what do parents and teachers need to know about learning  real  math? Keep on reading to find out.

By Math Dude Jason Marshall

PeopleImages  Getty Images

Today is a very special episode of the Math Dude. To begin with, it’s episode 300. And because we humans have 10 fingers, we love to give special meaning to multiples of 10. But while that’s fun, it’s not the big news of the day or what makes this episode special to me. The big news is that this 300th episode is my last. Between my day job as a physics and astronomy professor and my day-and-night job of being “Dad” to an awesome and bustling 3-year-old, my free time for Math Dude duties has dwindled. And although I will surely miss all of you math fans, after seven years on the job, it's time to say goodbye.

But before I go, I have one more thing to say—and I think it’s the most important thing I’ve ever said on the show. It’s not something that I would (or even could) have said when I wrote the first episode seven years ago, because I wasn’t yet a father and so I wasn’t yet watching somebody discover the world for the first time. So please take a few minutes and listen, because I think this is something that everybody who has kids or might have kids or works with kids or might work with kids should know.

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Here it is:  Math is a playground … so play!  Allow me to explain.

»Continue reading on QuickAndDirtyTips.com

Teaching Math: The Best Learning Practice Essay

Learning math is not an easy task for many students, especially if they are of a young age. It is not always interesting to deal with numbers and develop various calculating activities to meet educational standards and get high grades. However, students have to learn mathematics as an obligatory part of the elementary school system. Therefore, to facilitate a learning process and raise students’ interest, teachers offer various practices and improve their understanding of children’s needs, abilities, and expectations. The examples introduced on video lessons show that teachers may cooperate with children in a variety of ways. Still, the main task is to involve students in a learning process by giving vivid examples, clear instructions, and options to choose from. In this paper, the attention to three teaching strategies, which are explicit teaching, developmental activities to support a diversity of levels, and reflection, will be discussed and explained through the prism of mathematics classes and teachers’ intentions to improve their classroom work.

Motivation and interest are the two crucial aspects of student education. It does not matter what subject has to be learned or what activities must be developed, teachers have to make sure that their students are ready and eager to participate in classroom activities. In both videos, teachers make use of explicit teaching as the main learning practice for their students to be offered. This type of learning aims at directing student attention towards certain subjects and topics in the already established learning environment. In math classes, it is not enough for teachers to provide students with an opportunity to learn new material and follow the given instructions (Selling, 2016). Explicit learning is used to make mathematical practices interesting to students through real-life representations, generalizations, problem-solving, and justifying (Selling, 2016). When students can use their own examples and explanations of the theoretical material given, it is easy for them to realize why all these tasks cannot be ignored, and what the essence of all these tasks is.

Another significant step in classroom learning is the attention to developmental activities in terms of which a diversity of levels and group work can be taken into consideration. Student learning and achievements depend directly on the teachers’ level of professionalism (Bayar, 2014). On the one hand, following the instructions and recognizing standards cannot be ignored in the classroom. On the other hand, students may easily get bored with all those requirements being set. In both videos, teachers underline the necessity to promote development activities and observe what students can do, want to do, and try to avoid. Sometimes, it is better to observe the work of students in groups or make them work individually and investigate their strengths and weaknesses. In schools, children of different backgrounds should cooperate, demonstrating their ability to use knowledge and their tolerance for other students’ mistakes. Teachers have to underline the importance of development and create the required environment to support but never offense or reproach a child.

Finally, the teachers from both videos agree that reflection has to be one of the main learning activities in the classroom. It helps not only to clarify what students learn but also focuses on the gaps that still exist. The possibility to reflect on the already gained experience and personal progress is the skill that has to be developed through teachers’ and students’ cooperation (Kiemer, Grőschner, Pehmer, & Seidel, 2015). In addition, reflection as a learning activity has a number of crucial benefits for students. First, they learn how to use the classroom material in real life. Second, sharing their thoughts, doubts, and ideas, students are able to comprehend the true importance of mathematics. Finally, students who are able to reflect on their activities demonstrate intentions to learn new information and participate in classroom activities. Therefore, the choice of this practice is not only a sign of a high-level professionalism of an educator, but an example of how teachers should respect and support their students.

To conclude, it is necessary to say that the offered videos help to realize how to improve mathematics classes and how to choose appropriate learning activities. The justification of this choice is based not only on the benefits students may gain in the classroom. The recognition of explicit learning, developmental activities, and reflection as the best practices to study math is explained by the possibility to focus on the merits and demerits of the subject and underline the importance of student participation. Though rules and plans cannot be ignored in teaching, educators should never forget that student motivation, interest, and achievement are the core values. Students may vary in their backgrounds, personal experiences, and skills. The task of teachers is to promote development, organize group projects, and reflect on the achievement made. Math can be one of the greatest subjects for students to deal with in elementary school, and explicit learning alone with group activities and reflections can help to achieve the best results in teaching practice.

Bayar, A. (2014). The components of effective professional development activities in terms of teachers’ perspective. International Online Journal of Educational Sciences, 6 (2), 319-327.

Kiemer, K., Grőschner, A., Pehmer, A. K., & Seidel, T. (2015). Effects of a classroom discourse intervention on teachers’ practice and students’ motivation to learn mathematics and science. Learning and Instruction, 35 , 94-103.

Selling, S. K. (2016). Making mathematical practices explicit in urban middle and high school mathematics classrooms. Journal for Research in Mathematics Education, 47 (5), 505-551.

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IvyPanda. (2021, June 26). Teaching Math: The Best Learning Practice. https://ivypanda.com/essays/teaching-math-the-best-learning-practice/

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Why Math Is Important: A Student’s View (Updated for 2024)

A couple of weeks ago, I asked my son to write an extra essay for a project we were working on for the Classical Conversations Practicum . I allowed him to work on that essay instead of his math lesson for the day.

Suddenly, my daughter, Ada, did not want to do her math lesson for the day. I explained that my son was writing an essay instead and she asked to be allowed to do the same.

I thought, “Hmm… This is a good time for my daughter to think about why math is important and come up with her own reasons for studying this subject.” She wrote the following essay and we thought it might help parents and students to read Ada’s thoughts.

Pure mathematics is, in its way, the poetry of logical ideas.” – Albert Einstein

“Why Is Math Important?” by Ada Bianco

“Everyone agrees that learning math can be difficult, but some people believe math is important and some people believe math is not important. Math is important for three reasons:

• Math is everywhere.
• Children need math.
• God created math.

Math Is Everywhere

The first reason math is important is because it is everywhere. It is used in everyday life. It is useful, but it is more than just useful. Math is there to help us, to keep us well ordered, to help us learn new things and to help teach us new things. Students will become adults who will use math in their jobs. All kinds of careers use math. For example, musicians, accountants, fashion designers, and mothers use math. However, math is not only used for things you do. It also brings order to everything around you. The world is organized essentially because it was made with math.

Children Need Math

The second reason math is important is children need math. Now, as we all know, children are as chaotic as a volcanic eruption. But as they grow, children need to learn patience. Patience is precisely what math teaches us. It also teaches us curiosity. For example, why is this rule used here? Why would that number be negative? Why is that equation set up like that? These are questions they will learn to ask if they are taught math. The parents’ job is to help their children grow up to become good people who are patient and wise, who want to learn even more about anything and everything. Their future depends on what they have learned and if they have learned mathematics, then they will be able to do many different things—maybe even anything—when they are adults

God Created Math

The third reason math is important is God created it. This is a reason most adults use to convince their children that math is not boring and unimportant, so it may seem unoriginal. I believe, however, it is something that needs to be stated. God created the universe as well as math. The universe is full of math and it is orderly because of math. The sun is a certain distance from the earth; everything is organized in such a way that no matter what has happened we have always been safe. We need math. From this, you should be able to see how much we really do use and need math. We would not be able to process or even do everyday things without it. Math, in addition to these things, helps us to know God. God gave us math to live well and to serve Him. With everything we learn using math in science, we learn more about the world, which can help draw us closer to God.

Some people say math is unimportant because you don’t need math other than basic math principles—you can live without more complicated math. They say, if you need it, then simply use a calculator and leave the more complicated math to people who like math, the mathematicians. This, however, is not correct. You need math and could not live well without math, even including more complicated math concepts. God made us with a sense of curiosity so we can learn, do, and think about all sorts of things. Math is that thing that connects everything together, everything people love to do: music, cooking, painting, and everything else. Math is important.

Math is important because math is everywhere, children need math, and God created math. This matters to me and other children because math determines our future and how we choose to live.”

Understanding Why We Learn Subjects

Often, it’s not just our students who struggle to understand why math is important. Some of us can use a reminder as well. Hopefully Ada’s essay inspired you with reasons why we and our students should learn math. What students learn from math, just as with studying any other subject, can be applied to all areas of life. Having your student write a persuasive essay like the one above can be a great way to help them understand how a subject they are studying is useful in everyday life.

Of course, your student’s topic doesn’t have to be math. It can be history, geography, English, literature, Latin, science, or any other subject. Whatever subject they write about, this exercise is undeniably useful for helping them to understand the point of studying that subject. Perhaps they’ll even surprise you with reasons you haven’t thought of.

-Ada Bianco

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Math Essay Ideas for Students: Exploring Mathematical Concepts

Are you a student who's been tasked with writing a math essay? Don't fret! While math may seem like an abstract and daunting subject, it's actually full of fascinating concepts waiting to be explored. In this article, we'll delve into some exciting math essay ideas that will not only pique your interest but also impress your teachers. So grab your pens and calculators, and let's dive into the world of mathematics!

• The Beauty of Fibonacci Sequence

Have you ever wondered why sunflowers, pinecones, and even galaxies exhibit a mesmerizing spiral pattern? It's all thanks to the Fibonacci sequence! Explore the origin, properties, and real-world applications of this remarkable mathematical sequence. Discuss how it manifests in nature, art, and even financial markets. Unveil the hidden beauty behind these numbers and show how they shape the world around us.

• The Mathematics of Music

Did you know that music and mathematics go hand in hand? Dive into the relationship between these two seemingly unrelated fields and develop your writing skills . Explore the connection between harmonics, frequencies, and mathematical ratios. Analyze how musical scales are constructed and why certain combinations of notes create pleasant melodies while others may sound dissonant. Explore the fascinating world where numbers and melodies intertwine.

• The Geometry of Architecture

Architects have been using mathematical principles for centuries to create awe-inspiring structures. Explore the geometric concepts that underpin iconic architectural designs. From the symmetry of the Parthenon to the intricate tessellations in Islamic art, mathematics plays a crucial role in creating visually stunning buildings. Discuss the mathematical principles architects employ and how they enhance the functionality and aesthetics of their designs.

• Fractals: Nature's Infinite Complexity

Step into the mesmerizing world of fractals, where infinite complexity arises from simple patterns. Did you know that the famous Mandelbrot set , a classic example of a fractal, has been studied extensively and generated using computers? In fact, it is estimated that the Mandelbrot set requires billions of calculations to generate just a single image! This showcases the computational power and mathematical precision involved in exploring the beauty of fractal geometry.

Explore the beauty and intricacy of fractal geometry, from the famous Mandelbrot set to the Sierpinski triangle. Discuss the self-similarity and infinite iteration that define fractals and how they can be found in natural phenomena such as coastlines, clouds, and even in the structure of our lungs. Examine how fractal mathematics is applied in computer graphics, art, and the study of chaotic systems. Let the captivating world of fractals unfold before your eyes.

• The Game Theory Revolution

Game theory isn't just about playing games; it's a powerful tool used in various fields, from economics to biology. Dive into the world of strategic decision-making and explore how game theory helps us understand human behavior and predict outcomes. Discuss in your essay classic games like The Prisoner's Dilemma and examine how mathematical models can shed light on complex social interactions. Explore the cutting-edge applications of game theory in diverse fields, such as cybersecurity and evolutionary biology. If you still have difficulties choosing an idea for a math essay, find a reliable expert online. Ask them to write me an essay or provide any other academic assistance with your math assignments.

• Chaos Theory and the Butterfly Effect

While writing an essay, explore the fascinating world of chaos theory and how small changes can lead to big consequences. Discuss the famous Butterfly Effect and how it exemplifies the sensitive dependence on initial conditions. Delve into the mathematical principles behind chaotic systems and their applications in weather forecasting, population dynamics, and cryptography. Unravel the hidden order within apparent randomness and showcase the far-reaching implications of chaos theory.

• The Mathematics Behind Cryptography

In an increasingly digital world, cryptography plays a vital role in ensuring secure communication and data protection. Did you know that the global cybersecurity market is projected to reach a staggering $248.26 billion by 2023? This statistic emphasizes the growing importance of cryptography in safeguarding sensitive information.

Explore the mathematical foundations of cryptography and how it allows for the creation of unbreakable codes and encryption algorithms. Discuss the concepts of prime numbers, modular arithmetic, and public-key cryptography. Delve into the fascinating history of cryptography, from ancient times to modern-day encryption methods. In your essay, highlight the importance of mathematics in safeguarding sensitive information and the ongoing challenges faced by cryptographers.

Writing a math essay doesn't have to be a daunting task. By choosing a captivating topic and exploring the various mathematical concepts, you can turn your essay into a fascinating journey of discovery. Whether you're uncovering the beauty of the Fibonacci sequence, exploring the mathematical underpinnings of music, or delving into the game theory revolution, there's a world of possibilities waiting to be explored. So embrace the power of mathematics and let your creativity shine through your words!

Remember, these are just a few math essay ideas to get you started. Feel free to explore other mathematical concepts that ignite your curiosity. The world of mathematics is vast, and each concept has its own unique story to tell. So go ahead, unleash your inner mathematician, and embark on an exciting journey through the captivating realm of mathematical ideas!

Tobi Columb, a math expert, is a dedicated educator and explorer. He is deeply fascinated by the infinite possibilities of mathematics. Tobi's mission is to equip his students with the tools needed to excel in the realm of numbers. He also advocates for the benefits of a gluten-free lifestyle for students and people of all ages. Join Tobi on his transformative journey of mathematical mastery and holistic well-being.

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Why is Math Important: Benefits of Learning Math at School

August 25, 2023

11 minutes read

Why is math important is a question worth exploring. Mathematics, a subject beyond mere numbers and formulas, constitutes the core of our existence. Its influence extends far beyond the confines of textbooks, penetrating the very essence of modern life. The topic — why is math the most important subject — also carries weight within the realm of education, which is why kids may be asked to write a why is math important essay in class. 

As we embrace math in education, we enable ourselves to unravel the mysteries of our reality. Through this article, you will discover answers to the question, why is math so important, and understand the many benefits of immersing ourselves in mathematics.

Why Is Math Important for Kids to Learn?

Math plays a significant role in everyone’s educational journey, bringing many benefits beyond just numbers. From the basics like counting and recognizing shapes to more complicated aspects like algebra, geometry, and calculus, studying math grounds students intellectually. At its heart, math teaches discipline and accuracy.

As people study math, they learn to take logical steps, follow the rules, and pay attention to the details. These skills make their studies easier and help them in other areas of life, teaching them how to approach problems systematically. Math also hones critical thinking and analysis.

It’s essential to know the answer to the question — why is math important for kids. When faced with math problems, we learn to spot patterns, make connections, and develop hypotheses. This natural problem-solving pathway helps us understand how things work and resolve complex issues. Besides, math literacy is a must-have in a world full of data and tech. Knowing the ins and outs of math gives kids the ability to interpret numbers and make well-thought-out decisions in terms of finance, health, and science. 

Also, math takes students and even teachers to the apex of creativity! When both parties explore numbers, shapes, and equations, they use their imaginations and develop new ways to solve problems and develop ideas. Finally, math encourages collaboration. Group activities and conversations about math help them communicate better, learn together, and make friends. 

Having understood the overview of math’s relevance in people’s lives, let’s delve deeper into why is math important in everyday life for kids.

Math Hones Complex Problem-Solving Skills

Knowing the answer to the ‘why is math important in life’ question enables kids to break down complex problems into smaller components, identify pertinent variables, and use appropriate formulas or methods to arrive at practical solutions. Math equips kids with a structured approach to problem-solving, empowering them to overcome obstacles and adapt to a dynamic world.

The capacity to methodically resolve issues enables them to approach various challenges with unwavering confidence and creativity, whether resolving complex technical troubleshooting issues, streamlining workflows, or interpersonal conflicts. 

It Promotes Critical Thinking

Knowing 5 reasons why math is important reveals math’s role in fostering critical thinking. The journey of solving mathematical problems is crucible for developing critical thinking. As kids immerse themselves in scrutinizing data, solving maze-like word problems, and developing logical strategies, they develop a robust skill — evaluating information from diverse perspectives. 

This ability to see recurring patterns and coherent conclusions is essential to making informed decisions. In debate and dialogue, kids with sharp critical thinking demonstrate the ability to obtain reliable sources, deconstruct complex arguments, and participate meaningfully in discussions. When faced with unexpected circumstances or a whirlwind of rapidly changing scenarios, this honed analytical skill allows them to objectively weigh new information, seamlessly adjust strategies, and deftly navigate the tides of change.

Math Improves Kids Financial Knowledge

Why math is the most important subject is validated within academics, but we can look beyond that. Math knowledge is a vital component of financial literacy, as it provides kids with the understanding and tools to make informed decisions that shape their financial well-being. Mathematics is central in helping individuals develop the essential skills to decipher complex financial concepts.

From understanding the dynamics of interest rates and the complex effects of investing to evaluating risk and return profiles, mathematics provides the basis for building a solid financial foundation. Using these mathematical insights, kids can create an adequate budget that meets their goals and desires. But financial literacy goes beyond self-interest; it enables them to contribute positively to their communities. 

By making intelligent philanthropic decisions or supporting local businesses, financially savvy kids become agents of change that drive economic growth and community development. Financial literacy provides clarity about a student’s perspective on broader financial issues. 

Using mathematical reasoning, they can engage in informed discussions about public policy, evaluate economic proposals, and make informed choices with far-reaching societies. The combination of mathematics and financial literacy, allowing them to secure their financial future and actively participate in creating a more financially stable and fair society, makes us more confident in answering the question, why is math important?

It Helps Kids Develop Technical Skills

In a digital age where technology permeates every aspect of modern life, the question of ‘why is discrete math important’ is quickly answered. Look at cybersecurity, for example. In 2023, it is among the most sought-after technical skills as companies try to protect their networks and data from breaches and uphold customers’ privacy.

A good understanding of mathematics opens up the ability to understand, analyze and innovate in a complex digital environment. Knowledge of mathematics allows kids to contribute to the development of technology actively. 

As technology evolves and shapes the future, mathematicians are uniquely positioned to drive progress. Using mathematical principles, they confidently explore the digital world, contributing to developing new solutions, advanced applications, and transformative breakthroughs that move society into uncharted territories of technological innovation.

Math Opens The Door to More Career Opportunities

Kids know why math is important and impacts job opportunities because of how many more career paths it offers them.

Beyond the bounds of traditional math-oriented roles like engineering and finance, the need for math skills has permeated many industries. Meanwhile, dynamic marketing has used statistical analysis to discover consumer behavior, improve customer segmentation, and drive strategic campaigns. 

As artificial intelligence and automation redefine industries, kids with a solid foundation in mathematics have the adaptability and innovation to thrive in new areas of employment in the future. From harnessing the power of big data to building data-driven narratives, these math-savvy professionals are at the forefront of shaping the future of work.

Learning Math Improves Analytical Skills

Mathematical analysis is crucial for developing analytical thinking, an invaluable skill in our complex, information-saturated world. So why is it important to learn math to improve analytical skills? In an age where navigating massive data sets and deciphering multifaceted challenges is the norm, the ability to discern complex situations and evaluate evidence becomes valuable. 

In a world where career paths and problem-solving paradigms are evolving at an unprecedented rate, the enrichment provided by mathematical and analytical ability is a cornerstone of success. Whether driving an industry into the future or developing innovative solutions to global problems, kids with these skills are built to make a lasting and transformative impact.

Progressive Scientific Discovery

The question — why is math and science important — is a run-off of the belief that math is often the language of science. Math is an indispensable tool for pushing the boundaries of scientific research and inquiry. It is the hidden force behind the breakthrough discoveries that allow scientists to bridge the gap between theoretical concepts and empirical observations.

Clinical trial design and medical analysis are governed by mathematical principles, which aid researchers in evaluating the efficacy of interventions and treatments. Statistical methods rooted in mathematics can provide insight into the effects of new drugs, the spread of diseases, and the impact of public health initiatives. This quantitative approach that improves medical knowledge and saves lives by guiding evidence-based medical practice shows why learning math is important.

It Helps Kids Develop Mental Stamina and Endurance

Facing complex math problems develops mathematical and endurance skills. It promotes mental strength and a will to overcome difficulties. When kids become aware of this, before you point out 5 reasons why math is not important they can already give you countless reasons why math is important. That is because they have gone through the rigors of solving math and now understand that mastery requires dedication, persistence, and a willingness to face failure. 

They develop an inherent resilience beyond mathematics as they solve complex problems and grapple with confusing concepts. This little thing becomes the foundation of personal and professional success. Those who successfully navigate the difficulties of mathematics are better prepared to face the complexity of the modern world. 

As we ponder why math is important in life, we should know that math provides a compass for navigating complexity in a world of information and rapid developments. This article could have still gone ahead to give an extra 10 reasons why math is important as its importance is countless. But you get the point already!

Mastering mathematics nurtures critical thinking, problem-solving abilities, and analytical reasoning — qualities necessary in a world filled with complex challenges and diverse opportunities. These all make us understand why math is important in our daily lives.

But why is learning math important at Brighterly? Brighterly recognizes the transformative power of mathematics and its role in shaping resilient individuals. They provided a platform that supports math understanding and learning. So register now to embark on a journey of discovery, where interactive lessons, engaging activities, and a supportive community await.

Jessica is a a seasoned math tutor with over a decade of experience in the field. With a BSc and Master’s degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.

Are you seeking to transform math into something captivating and enjoyable for your students? This article is just the thing you need! It contains fascinating and fun math trivia questions and answers designed to fascinate kids. Whether you are a teacher or parent, rest assured the trivia questions math presented in this article will benefit […]

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When studying geometry, one essential skill is the ability to classify quadrilaterals. Quadrilaterals — those shapes with four sides — are as much a part of a child’s world as the games they play and the stories they read. Imagine a world where every tile on the floor, every window frame, and every board screen […]

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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity (2009)

Chapter: 9 conclusions and recommendations.

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9 Conclusions and Recommendations Over the past several decades there has been an increased focus on the importance of the preschool period—between ages 3 and 5—in providing children with the opportunities they need to get off to a successful start in formal schooling. Many policy makers are now intent on implementing universal public preschool because of the mounting evidence that high-qual- ity preschool can help ameliorate inequities in educational opportunity and begin to address achievement gaps. The importance of supporting literacy in these early childhood settings is widely accepted, but little attention is given to mathematics. However, research on children’s capacity to learn mathematics, when combined with evidence that early success in mathemat- ics is linked to later success in both mathematics and reading, makes it clear that basic literacy consists of both reading and mathematics. Improvements in early childhood mathematics education can provide young children with the foundational educational resources that are critical for school success. Furthermore, the increasing importance of science and technology in ev- eryday life and for success in many careers highlights the need for a strong foundation in mathematics. Historically, mathematics has been viewed by many as unimportant to or developmentally inappropriate for young children’s learning experi- ences. However, the research synthesized in this report makes it clear that these beliefs are unfounded. In the course of normal development, young children develop key mathematical ideas and skills that include counting; adding and subtracting; finding which is more (or less); working with shapes by moving, combining, and comparing them to learn some of their properties; experiencing and labeling spatial terms (e.g., above, below); 331 

332 MATHEMATICS LEARNING IN EARLY CHILDHOOD and understanding length measurement as the number of length units that makes the total; as well as representing and communicating mathematics understanding to others. Relying on a comprehensive review of the research, this report lays out the critical areas that should be the focus of young children’s early mathematics education, explores the extent to which they are currently in- corporated into early childhood settings, and identifies the changes needed to improve the quality of mathematics experiences for young children. The committee describes these critical areas of mathematics in terms of teaching- learning paths that can be used to promote optimal learning. Such a path describes the skills and knowledge that are foundational to later learning and lays out a likely sequence of the steps toward greater competence. One can look closely along the path to gauge what children will be able to do next and to design instructional activities that will help them move along the path. The notion of such teaching-learning paths is a framing assump- tion for the conclusions and recommendations of this report. To ensure that all children enter elementary school with the mathemati- cal foundation they need for success, the committee recommends a major national initiative in early childhood mathematics. The success of such an initiative requires that parents, early childhood teachers, policy makers, and communities reconceptualize the way they think about and understand young children’s mathematics. The early childhood education system (e.g., workforce, early childhood programs, and policies) will need to work co- herently together toward this goal. Furthermore, families and communities must also adopt this goal if they are serious about improving children’s mathematics education. In this chapter, the committee summarizes the major conclusions of the report organized around the chapters, articulates the key recommenda- tions that flow from these conclusions, and lays out an agenda for future research. CHILDREN’S COMPETENCE AND POTENTIAL TO LEARN MATHEMATICS The committee’s review of developmental research with infants and toddlers demonstrates that the knowledge and competencies relevant to mathematics are present from early in life. As early as infancy, babies are curious about their world and are able to think about it in mathematical ways. Preverbal number knowledge is shared by humans from diverse cul- tural backgrounds as well as by other species. For example, by 10 months of age, young infants can distinguish a set of two items from a set of three items, and over time they are able to distinguish the number of items in sets with larger numbers. Building on this foundation, young children continue 

CONCLUSIONS AND RECOMMENDATIONS 333 to expand their knowledge and competence and enjoy their early informal experiences with mathematics, such as spontaneously counting toys, excit- edly asking who has more, or pointing out shapes. Conclusion 1: Young children have the capacity and interest to learn meaningful mathematics. Learning such mathematics enriches their current intellectual and social experiences and lays the foundation for later learning. Knowledge and competencies acquired through everyday experiences provide a starting point for mathematics learning. Infants’ and toddlers’ natural curiosity initially sparks their interest in understanding the world from a mathematical perspective, and the adults and communities that educate and care for them also provide experiences that serve as the basis for further mathematics learning. Children’s everyday environments are rich with mathematics learning opportunities, for example, using relational words, such as more than/less than, and counting and sorting objects by shape or size. These foundational, everyday mathematics experiences can be built on to move children further along in their understanding of math- ematical concepts. Conclusion 2: Children learn mathematics, in part, through everyday experiences in the home and the larger environment beginning in the first year of life. Children need rich mathematical interactions and guidance, both at home and school to be well prepared for the challenges they will meet in formal schooling. Parents, other caregivers, and teachers can play a fundamental role in the organization of learning experiences that support mathematics because they can expose children to mathematically rich envi- ronments and engage them in mathematics activities. For example, parents and caregivers can teach children to see and name small quantities, count, and point out shapes in the world, “Here are two crackers. You have one in each hand. These crackers are square.” One important way that young children’s mathematics learning can be enhanced is through adult support and instruction that is connected to and extends their preexisting mathematics knowledge. For example, a situa- tion in which a young child insists on having “more” teddy bears than his playmate provides an opportunity for the adult to engage the child with a mathematical question (e.g., who has more and how can you find out?). In this instance, the adult can use several key mathematical ideas to help the child understand who has more bears, such as using the number word list to count, 1-to-1 counting correspondence, cardinality (i.e., knowing the total 

334 MATHEMATICS LEARNING IN EARLY CHILDHOOD number of items in the set), and comparing the number of bears in the two sets. These kinds of mathematics learning opportunities help children learn to mathematize or engage in processes that involve focusing on the math- ematical aspects of an everyday situation, learn to represent and elaborate a model of the situation, and use that model to solve problems. Conclusion 3: Children need adult support and instruction to build and extend their early knowledge and learn to focus on and elaborate the mathematical aspects of everyday situations—to mathematize. The committee was keenly aware of the influence that developmental and contextual variations have on children’s learning opportunities and the quality of their educational environments both inside and outside the class- room. Understanding individual differences in children’s ­development—for example, in executive function or in opportunities to learn about math- ematics in their everyday environments—is fundamental to supporting the development of competence in mathematics. Although all children need extensive exposure to mathematics, there is a wide range of individual variation across all domains of learning. This affects the kinds of learning experiences and instruction that individual children need. The need to sup- port early childhood mathematics education in ways that are appropriate for diverse learners and contexts is a theme throughout the committee’s discussion of early childhood mathematics. Conclusion 4: Due to individual variation, which is related to a com- bination of previous experiences, opportunities to learn, and innate ability, some children need more extensive support in mathematics than others. It is important to understand the sources of observed differences in children’s competence and not confuse one source of individual variation for another. For example, low performance might be attributed to a deficit in a child’s ability to learn mathematics, when it actually results from other factors, such as that child’s lack of opportunities to learn mathematics or difficulties stemming from linguistic and cultural barriers between teacher and child. Opportunities to explore the mathematics of everyday life differ de- pending on children’s background, including their socioeconomic status (SES) and cultural group. Mathematics knowledge and skills vary within and between cultural groups due to a variety of factors, including language and relative emphasis placed on mathematics. Cultural, linguistic, and socioeconomic factors interact in complex ways that are difficult to tease apart. 

CONCLUSIONS AND RECOMMENDATIONS 335 The committee was particularly concerned about mathematics teaching and learning for children from low socioeconomic backgrounds because of the particular challenges they face that can have an impact on their knowl- edge and competence in mathematics. For example, they may be more likely to attend schools with fewer resources and have less support for mathemat- ics at home. Thus, although children with very low and high mathematics knowledge and competence are found across all SES groups, those with low SES will need particular attention. Importantly, providing young children with high-quality mathematics instruction can help to ameliorate systematic inequities in educational outcomes and later career opportunities. Conclusion 5: Young children in lower socioeconomic groups enter school, on average, with less mathematics knowledge and skill than their higher socioeconomic status peers. Formal schooling has not been successful in closing this gap for low socioeconomic status children. In addition to needing instructional support in mathematics, evidence indicates that young children also need to be supported in their social- emotional development as an integral part of their education. Specifically, during the early education years, children develop general competencies and approaches to learning that include their capacity to regulate their emotions and behavior, to focus their attention, and to communicate effectively with others. In turn, mathematics learning can help to promote the development of these general competencies. Conclusion 6: All learning, including learning mathematics, is facili- tated when young children also are developing skills to regulate their own learning, which includes regulating emotions and behavior, focus- ing their attention, and communicating effectively with others. FOUNDATIONAL AND ACHIEVABLE MATHEMATICS FOR YOUNG CHILDREN On the basis of research evidence about children’s knowledge and competence during the early childhood years, as well as on the established consensus of the early childhood mathematics community (see, for example, the NCTM Curriculum Focal Points), the committee identified two areas of mathematics on which to focus: (1) number, including whole number, operations, and relations, and (2) geometry, spatial thinking, and mea- surement. In each of these areas, the committee offers guidance about the teaching-learning paths based on what is known from developmental and classroom-based research. Each child’s progression along these mathematics teaching-learning paths is a function of his or her own level of develop- 

336 MATHEMATICS LEARNING IN EARLY CHILDHOOD ment as well as opportunities and experiences, including instruction. The teaching-learning paths can provide the basis for curriculum and can be used by teachers to assess where each child is along the path. Although it is true that young children are more competent in math- ematics than many early childhood teachers, parents, and the general public believe, there are limits to what they can do in mathematics. The committee kept this in mind throughout the study process, and thus the teaching-learning paths presented in this report are both foundational and achievable. The first content area is number, including whole number, operations, and relations. Working with number (e.g., learning to count) is the primary goal of many early childhood programs; however, when given the oppor- tunity, children are capable of demonstrating competence in more sophis- ticated mathematics activities related to whole number, operations, and relations. For example, cardinality—knowing how many are in a set—is a key part of children’s number learning. Relations and operations are extensions of understanding number. The relations core consists of such skills as constructing the relations more than, less than, and equal to. The operations core includes addition and subtraction. The second major content area is geometry, spatial thinking, and mea- surement. Children’s foundational mathematics involves geometry or learn- ing about space and shapes in two and three dimensions (e.g., learning to recognize shapes in many different orientations, sizes, and shapes). A fun- damental understanding of shape begins with experiences in which children are shown varied examples and nonexamples and understand attributes of shapes that are mathematically relevant as well as those (e.g., orientation, size) that are not. As children progress along the teaching-learning path, they need opportunities to discuss and describe shapes, and, on the basis of these experiences, they gain abilities to combine shapes into pictures and eventually learn to take apart and put together shapes to create new shapes. Young children also need instructional activities involving spatial orienta- tion and spatial visualization. For example, they can use mental representa- tions of their environment and, on the basis of the representation, model relationships between objects in their environment. Importantly, children’s knowledge of measurement helps them connect number and geometry be- cause measurement involves covering space and quantifying this coverage. Later, children can compare lengths by measuring objects with manipulable units, such as centimeter cubes. Number is particularly important to later success in school mathemat- ics, as number and related concepts make up the majority of mathemat- ics content covered in later grades. However, it is important to point out that concepts related to number (and relations and operations) can also be explored through geometry and measurement. In addition, geometry 

CONCLUSIONS AND RECOMMENDATIONS 337 and measurement provide rich contexts in which children can deepen their mathematical reasoning abilities. Conclusion 7: Two broad mathematical content areas are particularly important as a focus for mathematics instruction in the early years: (1) number (which includes whole number, operations, and relations) and (2) geometry, spatial thinking, and measurement. In the context of these core content areas, young children should engage in both general and specific thinking processes that underpin all levels of mathematics. These include the general processes of representing, problem solving, reasoning, connecting, and communicating, as well as the more specific processes of unitizing, decomposing and composing, relating and ordering, looking for patterns and structures, and organizing and clas- sifying information. In other words, children should learn to mathematize their world: focusing on the mathematical aspects of an everyday situation, learning to represent and elaborate the quantitative and spatial aspects of a situation to create a mathematical model of the situation, and using that model to solve problems. Conclusion 8: In the context of each of these content areas, young chil- dren should engage in both general and specific mathematical thinking processes as described above and in Chapter 2. THE EARLY CHILDHOOD EDUCATION SYSTEM The early childhood education “delivery system,” which educates and cares for children before kindergarten entry, has a great deal of diversity and is best characterized as a loosely sewn-together patchwork of different kinds of programs and providers that vary widely in the extent to which they articulate and act on their educational missions or are explicitly de- signed to provide education services. Program types range from friends and relatives who care for children in the home through informal arrangements, to large centers staffed by teachers offering a structured curriculum. This diversity in the early childhood education system characterizes the education and care arrangements of young children in the United States today. About 40 percent of young children spend their day in a home-based setting, either with a parent or some other caregiving adult (this percentage includes children in home-based relative and nonrelative care as well as children who do not have any regular early education and care arrange- ments), and about 60 percent are in some kind of center-based care (this includes children in center-based non-Head Start and Head Start settings). Depending on the type of setting, different regulations regarding edu- 

338 MATHEMATICS LEARNING IN EARLY CHILDHOOD cational standards or expectations may be in place, which in turn influence the nature and quality of young children’s learning experiences from setting to setting. Increasingly, policy makers are focused on how to provide high- quality preschool education for more children, especially to those whose families cannot afford to pay for it. A number of states are moving toward state-funded preschool education to provide early education and care for these children. Across all settings, there is a need to increase the amount and qual- ity of time devoted to mathematics. Formal settings with an educational agenda represent the greatest opportunity for implementing a coherent, sequenced set of learning experiences in mathematics. For this reason, the committee focused attention on the kind of curriculum and instruction that can be implemented in centers and preschools. The committee gave more limited attention to how to increase support for mathematics in informal settings. These approaches are discussed in the section “Beyond the Educa- tion System.” Curriculum and Instruction Having laid out a vision for optimal teaching-learning paths in early childhood mathematics, the committee turned to the evidence base re- lated to curriculum and instruction. The committee first examined the extent to which the content and learning experiences embodied in the teaching-learning paths are represented in current curricula and preschool classrooms. Next, the committee explored what is known about effective mathematics instruction for young children and what might need to be done to improve existing practice. The committee looked for evidence to address two sets of questions: What is known about how much mathematics in- struction is available currently to children in preschool settings and of what quality? What is known about the best methods of instruction and effective curriculum to teach mathematics to young children? Although few system- atic data exist, the committee was able to identify some useful sources. We conducted original analyses of the standards documents pertaining to early childhood for 49 states and those pertaining to kindergarten for the 10 states with the largest student populations. On the basis of these analyses, the committee concludes: Conclusion 9: Current state standards for early childhood do not, on average, include much mathematics. When mathematics is included, there is a pattern of wide variation among states in the content that is covered. Although standards represent broad guidance from the states regard- ing appropriate content for early childhood settings, they do not provide a 

CONCLUSIONS AND RECOMMENDATIONS 339 window on what actually occurs in classrooms. For the latter, the commit- tee examined data from a large-scale study of instruction in state-funded preschools drawn from 11 states as well as several, small-scale studies of curriculum. The results show that when mathematics activities are incor- porated into early childhood classrooms, they are often presented as part of an integrated or embedded curriculum, in which the teaching of math- ematics is secondary to other learning goals. This kind of integration occurs when, for example, a storybook has some mathematical content but is not designed to bring mathematics to the forefront, a teacher counts or does simple arithmetic during snack time, or points out the mathematical ideas children might encounter during play with blocks. However, data suggest that heavy reliance on integrated or embedded mathematics activities may contribute to too little time being spent on mathematics in early childhood classrooms. Furthermore, the time that is spent may be on activities in which the integrity and depth of the mathematics is questionable. Few of the existing comprehensive early childhood curriculum approaches provide enough focused mathematics instruction for children to progress along the teaching-learning paths recommended by the committee. Conclusion 10: Most early childhood programs spend little focused time on mathematics, and most of it is of low instructional quality. Many opportunities are therefore missed for learning mathematics over the course of the preschool day. Evidence examined by the committee suggests that instructional time focused on mathematics is potentially more effective than embedded math- ematics. Emerging evidence from a few studies of rigorous mathematics curricula show that children who experience focused mathematics activi- ties in which mathematics teaching is the major goal have higher gains in mathematics and report enjoying mathematics more than those who do not. Furthermore, these studies indicate that a planned, sequenced curriculum can support young children’s mathematical development in a sensitive and responsive manner. Supplemental opportunities to use mathematics during mathematical play, sociodramatic play, and with concrete materials (e.g., blocks, puzzles, manipulatives, interactive computer software) can provide children with the opportunity to “practice” mathematics in a meaningful and engaging context. Conclusion 11: Children’s mathematics learning can be improved if they experience a planned, sequenced curriculum that uses the research- based teaching-learning paths described in this report, as well as inte- grated mathematics experiences (e.g., mathematics in the context of a storybook) that extend mathematical thinking through play, explora- tion, creative activities, and practice. 

340 MATHEMATICS LEARNING IN EARLY CHILDHOOD Effective mathematics curricula use a variety of instructional ap- proaches, such as a combination of individual, small-group, and whole- group activities focused on mathematics that move children along the research-based teaching-learning paths described in this report. Further- more, in all these contexts, intentional teaching enhances the mathematics learning of young children. Intentional teaching varies from teacher-guided activities to responsive feedback that builds on and extends the child’s under­ standing. It is also important to engage children in math talk—­discussion between adults and children that focuses on mathematics concepts, such as how many objects are in a set or how to arrive at an answer—as this facilitates their mathematical development by increasing the connections they make between mathematics concepts, words, and ideas. It should be noted that the committee does not endorse any specific model or curricu- lum; rather we hope to convey that the research-based principles described in this report should guide choices about development of early childhood mathematics curriculum and instruction. Conclusion 12: Effective early mathematics curricula use a variety of instructional approaches and incorporate intentional teaching. Evidence also indicates that instruction is more effective when it can build on information about the child’s current level of understanding. Such responsive instruction can be accomplished when teachers know how to use formative assessment to guide instruction. Formative assessment is an important component of what teachers need to know to effectively guide children along the mathematics teaching-learning paths. Conclusion 13: Formative assessment provides teachers with informa- tion about children’s current knowledge and skills to guide instruction and is an important element of effective mathematics teaching. Evidence from studies of early childhood education indicates that any approach to curriculum and pedagogy is more effective if undertaken in the context of a positive learning environment. Positive relationships between children and their teachers are a key aspect of high-quality early childhood education. In this kind of classroom, children are provided with a safe and nurturing environment that promotes learning and positive interactions between teachers and peers. Conclusion 14: Successful mathematics learning requires a positive learning environment that fully engages children and promotes their enthusiasm for learning. 

CONCLUSIONS AND RECOMMENDATIONS 341 Workforce and Professional Development The early childhood workforce—those who serve both instructional and noninstructional roles in early childhood settings—is central to sup- porting the academic, social, emotional, and physical development of young children. This workforce consists of people who serve in a variety of roles, are located in a variety of settings, and have a wide range of education and training backgrounds. About 24 percent of early childhood workers are in center-based settings, 28 percent are in regulated home-based settings, and about 48 percent work in informal care arrangements outside both of these systems. Although the majority of early childhood professionals work in informal care settings, the majority of children are in center-based settings. Even in a single setting, individuals fill different roles, such as lead teacher, assistant teacher, classroom aide, or program administrator. Level and type of training can vary by both role and setting. For example, family child-care providers may have little or no specific training in early child- hood education, a teachers’ assistant may have some formal coursework, and center-based lead teachers may have a 4-year college degree (or even a graduate degree) with specialization in early childhood. This diversity of roles and educational backgrounds creates challenges for addressing the workforce needs related to supporting early childhood mathematics. Individuals in different roles are likely to need different kinds of knowledge and training to support children’s mathematics. Depending on level of education, there are also likely to be differences in individuals’ knowledge of mathematics, of children’s development in mathematics, and of how to support mathematics learning. In addition, the field of early childhood has historically placed great emphasis on teaching its workforce to support children’s social and emo- tional development, placing less attention on cognitive development and academic domains. Indeed, academic activities, such as mathematics learn- ing, can be a context in which social-emotional development flourishes. In large part, the heavy emphasis on social-emotional development in early childhood is based on misinterpretations of cognitive development theories; that is, the notion of young children engaging in more abstract thinking, such as mathematics, was believed to be at odds with the development and learning of preschool-age children. Research on early childhood mathemat- ics has disproved this notion, but the idea is still pervasive in the field and continues to be a challenge in moving from research to practice. Conclusion 15: Many in the early childhood workforce are not aware of what young children are capable of in mathematics and may not recognize their potential to learn mathematics. 

342 MATHEMATICS LEARNING IN EARLY CHILDHOOD Professional development, which typically provides training to those already in the workforce, can be a vital mechanism for providing teachers with new or updated skills and knowledge that they need and for reach- ing those in the workforce who have little or no formal training. Based on studies at the K-12 level, effective approaches to in-service professional development in mathematics are ongoing, grounded in theory, tied to a cur- riculum, job-embedded, and delivered at least partially onsite by a knowl- edgeable trainer who allows teachers time for reflection. The committee reviewed emerging data from studies conducted in early childhood settings that support these findings. These studies indicate that professional devel- opment focused on understanding children’s developmental progression in mathematics in the context of a research-based curricular sequence can improve teachers’ instructional effectiveness. An effort to provide profes- sional development to teachers is one important component of successfully improving instruction, but sustainable change will also require collabora- tion from administrators, teachers, and parents. Conclusion 16: In-service education of teachers and other staff to support mathematics teaching and learning is essential to effective implementation of early childhood mathematics education. Useful pro- fessional development will require a sustained effort that involves help- ing teachers to (a) understand the necessary mathematics, the crucial teaching-learning paths, and principles of intentional teaching and curriculum and (b) learn how to implement a curriculum. Evidence reviewed by the committee about the formal preparation of early childhood educators (courses taken as part of an associate or under- graduate degree) indicates that there are few opportunities to learn about children’s development in mathematics or how to teach early childhood mathematics. To better prepare early childhood educators in mathematics, additional courses and additional materials in existing courses that cover children’s development in mathematics and mathematics pedagogy are needed. Furthermore, licensure and credentialing systems exert a great deal of influence over the content and experience of pre-service education pro- grams in early childhood, and few incorporate mathematics requirements. Conclusion 17: Pre-service preparation of early childhood educators typically includes few opportunities to learn about children’s math- ematical development or how to support it. Licensure and certifica- tion requirements for credentialing teachers and programs are both potential leverage points for increasing the amount of attention given to supporting mathematics. 

CONCLUSIONS AND RECOMMENDATIONS 343 In addition to the challenges already outlined regarding the diverse training and settings of the workforce, attracting and retaining qualified in- dividuals to work in early childhood is difficult due to poor compensation, lack of benefits, and high turnover rates in the field. This situation presents an additional challenge to designing pre-service and in-service experiences that can improve early childhood educators’ knowledge of how to support young children’s learning in mathematics. Conclusion 18: Improving the training and knowledge requirements for early childhood teachers will present significant challenges unless exist- ing issues of recruitment, compensation, benefits, and high turnover are also addressed. BEYOND THE EDUCATION SYSTEM A significant number (about 40 percent) of children do not attend cen- ters but instead are educated and cared for by a parent, relative, or another adult in homes. Parents or other caregivers serve as children’s first teachers; evidence reviewed by the committee indicates that they can play a key role in shaping children’s early mathematics learning through such activities as encouraging play with blocks and other manipulatives, teaching number words, playing counting and board games, sorting, classifying, writing, and viewing educational television programs while talking with children about what they are watching. Math talk has been shown to be a particu- larly effective way for adults to support the development of mathematical ideas. In fact, math talk beginning as early as infancy is related to children’s mathematics knowledge at preschool entry. In addition, informal learning environments, such as libraries, museums, and community centers, have the potential to be resources that parents and caregivers can use to engage children in mathematics activities. Conclusion 19: Families can enhance the development of mathematical knowledge and skills as they set expectations and provide stimulating environments. Evidence indicates, however, that low-SES families are less likely than families from higher socioeconomic groups to engage in the kind of prac- tices that promote language and mathematics competence. Although many types of educational programs have been designed to promote the use of these practices with low-SES parents, there is little evidence about the qualities that make such efforts successful. Educational programs for par- ents based on models that place parents in the traditional role of students 

344 MATHEMATICS LEARNING IN EARLY CHILDHOOD learning from “experts” have difficulty sustaining family participation long enough to be successful. Conclusion 20: Educational programs for parents have the potential to enhance the mathematical experiences provided by parents; however, there is little evidence about how to design such programs to make them effective. The resources available to parents and other caregivers as well as those available through informal educational environments (e.g., libraries, muse- ums, community centers) can also be an effective mechanism for supporting children’s mathematics learning. Educational television programming and software, for example, can teach children about mathematics. The com- mittee reviewed research on software and educational programs, as well as models of community-based programs that promote mathematics, and concludes: Conclusion 21: Given appropriate mathematical content and adult support, the media (e.g., television, computer software) as well as community-based learning opportunities (e.g., museums, libraries, community centers) can engage and educate young children in math- ematics. Such resources can provide additional mathematics learning opportunities for young children, especially those who may not have access to high-quality early education programs. RECOMMENDATIONS As the committee’s conclusions make clear, there is much work to be done to provide young children with the learning opportunities in math- ematics that they need. Thus, the committee thinks it is critically important to begin an intensive national effort to enhance opportunities to learn mathematics in early childhood settings to ensure that all children enter school with the mathematical foundations they need for academic success. The research-based principles and mathematics teaching-learning paths de- scribed in this report can also reduce the disparity in educational outcomes between children from low-SES backgrounds and their higher SES peers. The research to date about how young children learn key concepts in mathematics has clear implications for practice, yet these findings are not widely known or implemented by early childhood educators or even those who teach early childhood educators. This report has focused on synthesiz- ing and translating this evidence base into a usable form that can be used to guide a national effort. Thus the committee recommends: 

CONCLUSIONS AND RECOMMENDATIONS 345 Recommendation 1: A coordinated national early childhood mathemat- ics initiative should be put in place to improve mathematics teaching and learning for all children ages 3 to 6. A number of specific recommendations for action follow from this overarching recommendation. The specific steps and the individuals or or- ganizations that must be involved in enacting them are outlined below. Recommendation 2: Mathematics experiences in early childhood set- tings should concentrate on (1) number (which includes whole num- ber, operations, and relations) and (2) geometry, spatial relations, and measurement, with more mathematics learning time devoted to num- ber than to the other topics. The mathematical process goals should be integrated in these content areas. Children should understand the concepts and learn the skills exemplified in the teaching-learning paths described in this report. In both content areas, sufficient time should be devoted to instruction to allow children to become proficient with the concepts and skills outlined in the teaching-learning paths. In addition, the general and specific math- ematical process goals (see Chapter 2) must be integrated with the content in order to allow children to make connections between mathematical ideas and deepen their mathematical reasoning abilities. This new content focus will require that everyone involved rethink how they view and understand the mathematics that is learned in early childhood. Early childhood learn- ing goals, programs, curricula, and professional development will need to be informed by and adapted to the research-based teaching-learning paths laid out in this report. The committee therefore recommends: Recommendation 3: All early childhood programs should provide high-quality mathematics curricula and instruction as described in this report. Early childhood programs will each need to implement a thoughtfully planned curriculum that includes a sequence of teacher-guided mathemat- ics activities as well as child-focused, teacher-supported experiences. Such curricula must be based on models of instruction that are appropriate for young children and support their emotional and social development as well as their cognitive development. As noted previously, effective mathematics curricula use a variety of instructional approaches and should incorporate opportunities for children to extend their mathematical thinking through play, exploration, creative activities, and practice. Programs will need to review, revise, and align their existing stan- 

346 MATHEMATICS LEARNING IN EARLY CHILDHOOD dards, professional development, curriculum, and materials to achieve the teaching-learning paths for early childhood mathematics education pre- sented in this report. It is especially important that children living in pov- erty receive such high-quality experiences so that they start first grade on a par with children from more advantaged backgrounds. This means that implementation of our recommendations by programs serving economically disadvantaged children, such as Head Start and publicly funded early child- hood programs, is particularly urgent. To make the recommended changes, early childhood programs will need explicit policy directives to do so. To encourage this, the committee recommends: Recommendation 4: States should develop or revise their early child- hood learning standards or guidelines to reflect the teaching-learning paths described in this report. Given the fresh knowledge and perspectives this report affords, it is important that states review their early learning and development stan- dards and guidelines to ensure that they reflect an appropriate emphasis on early mathematics. To that end, we call for all states to examine their early learning and development guidelines, first, to determine that sufficient emphasis is given to the importance of mathematics for young children’s development and, second, to ensure that the mathematics content focuses on (1) number (including whole number, operations, and relations) and (2) geometry, spatial thinking, and measurement. Recommendation 5: Curriculum developers and publishers should base their materials on the principles and teaching-learning paths described in this report. Teachers and early childhood programs need appropriate materials in order to support children’s mathematical development and learning. Cur- riculum developers and publishers who produce materials for curriculum, instruction, and assessment should revise and update them so that they reflect the principles articulated in this report. The success of this overall effort will need to focus on reaching both the existing early childhood workforce and pre-service educators to provide them with skills and knowledge they need to teach mathematics. Thus, we make several recommendations related to teachers and the workforce. Recommendation 6: An essential component of a coordinated national early childhood mathematics initiative is the provision of professional development to early childhood in-service teachers that helps them 

CONCLUSIONS AND RECOMMENDATIONS 347 (a) to understand the necessary mathematics, the crucial teaching- l ­earning paths, and principles of intentional teaching and curriculum and (b) to learn how to implement a curriculum. Applying teachers’ theoretical knowledge to a curriculum with a strong mathematics component provides them with the opportunity to get feed- back and reflect on the instructional practices that they will actually be im- plementing in the classroom. Professional development should also focus on teachers’ beliefs about children’s mathematics, the activities and resources in the classroom that can promote children’s mathematical development, and their knowledge of curriculum-linked assessment practices. All of these important areas should be included in professional development delivered by a highly qualified teacher educator. To implement high-quality mathematics instruction, the committee also recommends that early childhood educators be taught to use a range of effective instructional strategies in a variety of formats, including whole- group, pair/small-group, and individual work; exploration and practice; and play and focused activities. Serious efforts to improve the preparation of early childhood ­ teachers will need to include the state licensure/certification, accreditation and recogni- tion, and credentialing systems that assess teachers’ competence and program quality. The early childhood mathematics described in this report should be reflected in the core components of these systems and programs. Recommendation 7: Coursework and practicum requirements for early childhood educators should be changed to reflect an increased emphasis on children’s mathematics as described in the report. These changes should also be made and enforced by early childhood organizations that oversee credentialing, accreditation, and recognition of teacher professional development programs. The committee also recognizes the need to go beyond the formal early childhood education system to reach families and communities—both of which have a strong impact on young children’s learning. An important component of reaching all children will need to include strategies aimed at children who are in other settings, such as homes or family child care. Recommendation 8: Early childhood education partnerships should be formed between family and community programs so that they are equipped to work together in promoting children’s mathematics. For example, family education and support programs, such as the Head Start Family and Community Partnerships Program, should include infor- 

348 MATHEMATICS LEARNING IN EARLY CHILDHOOD mation that provides guidance to families and communities as to why they should and how they can help children develop key mathematical ideas and skills. Furthermore, professionals working with families should be given training focused on early mathematics knowledge and skills, as well as have access to programs and resources on home-based mathematics activities. To this end, there is a need for development of more resources that can support mathematics in informal settings and through media and technology. Recommendation 9: There is a need for increased informal program- ming, curricular resources, software, and other media that can be used to support young children’s mathematics learning in such settings as homes, community centers, libraries, and museums. FUTURE RESEARCH In its work, the committee conducted a comprehensive review of the existing evidence related to mathematics development and learning in early childhood. As noted, we have determined that the evidence base is robust enough to guide a major national initiative in early mathematics. Yet gaps remain in the knowledge base about children’s mathematics education. We think it is critical that the research base continue to advance in a number of key areas outlined below. Implications for English language learners.  Increasingly, early childhood classrooms serve significant numbers of children whose first language is not English; these children will be held to the same expectations for future achievement as children whose home language is English. To date, little published research has investigated the teaching and learning of mathemat- ics with preschool age children who are simultaneously learning English. The committee recommends research be conducted that can help identify the best methods of enhancing the mathematical learning of young children who speak a first language other than English. Research on the role of teachers in providing effective instruction.  In re- cent years, researchers have made progress in understanding the process of teaching mathematics at the elementary school level. This research stresses the role of teachers’ knowledge and skill including their knowledge of mathematics, their understanding of children’s mathematical thinking and learning and their pedagogical content knowledge (i.e., their knowledge of how to structure the classroom and curriculum and to engage children in activities so that the mathematics is accessible). However, there has been much less attention to similar issues in early childhood settings. Research is needed to determine the extent to which the findings from research in 

CONCLUSIONS AND RECOMMENDATIONS 349 the higher grades apply to mathematics instruction in early childhood and what might be unique to early childhood. Evaluation of curricula.  In the course of our review of early childhood mathematics, it became clear that many of the available curricula have not been rigorously evaluated for effectiveness. High-quality curriculum re- search is needed that tracks the effectiveness of curricula during implemen- tation, using the theories and instructional models that were originally used to guide development of the curriculum. This research must also consider how diversity in children’s backgrounds and across learning environments influences implementation and effectiveness. To achieve these goals, the committee recommends that curriculum research and development move through phases: from early reviews of relevant research to the creation of learning materials to help children along the teaching-learning paths in this report, to cycles of baseline evaluation, and finally to confirmatory evalu- ation using rigorous designs, with all phases integrating quantitative and qualitative methodologies. Research of this type will help ensure that early childhood programs can make informed, evidence-based choices among curricula. Effective teacher preparation.  Much of the recent research on the prepa- ration of early childhood educators has focused on whether the bachelor’s degree is an effective marker for teachers’ competency. While this line of inquiry has been helpful in identifying some of teachers’ skills that are related to positive child learning outcomes, research in the field needs to move beyond the B.A./non-B.A. distinction. The committee recommends that research on the effectiveness of early childhood teachers focus on the content and quality of teacher education programs rather than on whether or not teachers have a bachelor’s degree. Parental involvement.  It is unclear why families from low SES back- grounds often do not participate in educational activities and what can be done to promote their involvement in these programs. The committee therefore recommends the conduct of better descriptive studies that exam- ine what parents understand about supporting their children’s mathematics learning and how to promote parents involvement in these efforts. Further- more, if parents do have knowledge about how to support their children’s mathematical development but are not putting this knowledge into practice, it is important that research examine the impediments that stand in the way of their actively promoting early childhood mathematics. Interventions for children with mathematics learning disabilities.  Explo- ration of learning difficulties or disabilities in mathematics is a nascent area 

350 MATHEMATICS LEARNING IN EARLY CHILDHOOD of research that needs expansion. Further exploration is needed to better understand what early number competencies are predictive of future success in mathematics. Such research can help identify children at risk for learning difficulties or disabilities in mathematics during the preschool years, de- velop targeted interventions for such children, and test their effectiveness. 

Early childhood mathematics is vitally important for young children's present and future educational success. Research demonstrates that virtually all young children have the capability to learn and become competent in mathematics. Furthermore, young children enjoy their early informal experiences with mathematics. Unfortunately, many children's potential in mathematics is not fully realized, especially those children who are economically disadvantaged. This is due, in part, to a lack of opportunities to learn mathematics in early childhood settings or through everyday experiences in the home and in their communities. Improvements in early childhood mathematics education can provide young children with the foundation for school success.

Relying on a comprehensive review of the research, Mathematics Learning in Early Childhood lays out the critical areas that should be the focus of young children's early mathematics education, explores the extent to which they are currently being incorporated in early childhood settings, and identifies the changes needed to improve the quality of mathematics experiences for young children. This book serves as a call to action to improve the state of early childhood mathematics. It will be especially useful for policy makers and practitioners-those who work directly with children and their families in shaping the policies that affect the education of young children.

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• Published: 19 December 2019

Problematizing teaching and learning mathematics as “given” in STEM education

• Yeping Li 1 &
• Alan H. Schoenfeld 2  

International Journal of STEM Education volume  6 , Article number:  44 ( 2019 ) Cite this article

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Mathematics is fundamental for many professions, especially science, technology, and engineering. Yet, mathematics is often perceived as difficult and many students leave disciplines in science, technology, engineering, and mathematics (STEM) as a result, closing doors to scientific, engineering, and technological careers. In this editorial, we argue that how mathematics is traditionally viewed as “given” or “fixed” for students’ expected acquisition alienates many students and needs to be problematized. We propose an alternative approach to changes in mathematics education and show how the alternative also applies to STEM education.

Introduction

Mathematics is commonly perceived to be difficult (e.g., Fritz et al. 2019 ). Moreover, many believe “it is ok—not everyone can be good at math” (Rattan et al. 2012 ). With such perceptions, many students stop studying mathematics soon after it is no longer required of them. Giving up learning mathematics may seem acceptable to those who see mathematics as “optional,” but it is deeply problematic for society as a whole. Mathematics is a gateway to many scientific and technological fields. Leaving it limits students’ opportunities to learn a range of important subjects, thus limiting their future job opportunities and depriving society of a potential pool of quantitatively literate citizens. This situation needs to be changed, especially as we prepare students for the continuously increasing demand for quantitative and computational literacy over the twenty-first century (e.g., Committee on STEM Education 2018 ).

The goal of this editorial is to re-frame issues of change in mathematics education, with connections to science, technology, engineering, and mathematics (STEM) education. We are hardly the first to call for such changes; the history of mathematics and philosophy has seen ongoing changes in conceptualization of the discipline, and there have been numerous changes in the past century alone (Schoenfeld 2001 ). Yet changes in practice of how mathematics is viewed, taught, and learned have fallen far short of espoused aspirations. While there has been an increased focus on the processes and practices of mathematics (e.g., problem solving) over the past half century, the vast majority of the emphasis is still on what content should be presented to students. It is thus not surprising that significant progress has not been made.

We propose a two-fold reframing. The first shift is to re-emphasize the nature of mathematics—indeed, all of STEM—as a sense-making activity. Mathematics is typically conceptualized and presented as a body of content to be learned and processes to be engaged in, which can be seen in both the NCTM Standards volumes and the Common Core Standards. Alternatively, we believe that all of the mathematics studied in K-12 can be viewed as the codification of experiences of both making sense and sense making through various practices including problem solving, reasoning, communicating, and mathematical modeling, and that students can and should experience it that way. Indeed, much of the inductive part of mathematics has been lost, and the deductive part is often presented as rote procedures rather than a form of sense making. If we arrange for students to have the right experiences, the formal mathematics can serve to organize and systematize those experiences.

The second shift is suggested by the first, with specific attention to classroom instruction. Whether mathematics or STEM, the main focus of most instruction has been on the content and practices of the discipline, and what the teacher should do in order to make it accessible to students. Instead, we urge that the main focus should be on the student’s experience of the discipline – on the affordances the environment provides the student for disciplinary sense making. We will introduce the Teaching for Robust Understanding (TRU) Framework, which can be used to problematize instruction and guide needed reframing. The first dimension of TRU (The Discipline) focuses on the re-framing discussed above: is the content conceptualized as something rich and connected that can be experienced and codified in meaningful ways? The second dimension (Cognitive Demand) examines opportunities students have to do that kind of sense-making and codification. The third (Equitable Access to Content) examines who has such opportunities: is there equitable access to the core ideas? Dimension 4 (Agency, Ownership, and Identity) asks, do students encounter the discipline in ways that enable them to see themselves as sense makers, building both agency and positive disciplinary identities? Finally, dimension 5 (Formative Assessment) asks, does instruction routinely use formative assessment, allowing student thinking to become public so that instruction can be adjusted accordingly?

We begin with a historical background, briefly discussing different views regarding the nature of mathematics. We then problematize traditional approaches to mathematics teaching and learning. Finally, we discuss possible changes in the context of STEM education.

Knowing the background: the development of conceptions about the nature of mathematics

The scholarly understanding of the nature of mathematics has evolved over its long history (e.g., Devlin 2012 ; Dossey 1992 ). Explicit discussions regarding the nature of mathematics took place among Greek mathematicians from 500 BC to 300 AD (see, https://en.wikipedia.org/wiki/Greek_mathematics ). In contrast to the primarily utilitarian approaches that preceded them, the Greeks pioneered the study of mathematics for its own sake and pursued the development and use of generalized mathematical theories and proofs, especially in geometry and measurement (Boyer 1991 ). Different perspectives about the nature of mathematics were gradually developed during that time. Plato perceived the study of mathematics as pursuing the truth that exists in external world beyond people’s mind. Mathematics was treated as a body of knowledge, in the ideal forms, that exists on its own, which human’s mind may or may not sense. Aristotle, Plato’s student, believed that mathematicians constructed mathematical ideas as a result of the idealization of their experience with objects (Dossey 1992 ). In this perspective, Aristotle emphasized logical reasoning and empirical realization of mathematical objects that are accessible to the human senses. The two schools of thought that evolved from Plato’s and Aristotle’s contrasting conceptions of the nature of mathematics have had important implications for the ensuing development of mathematics as a discipline, and for mathematics education.

Several more schools of thought were developed as mathematicians tackled new problems in mathematics (Dossey 1992 ). Davis and Hersh ( 1980 ) provides an entertaining and informative account of these developments. Three major schools of thought in the early 1900s dealt with paradoxes in the real number system and the theory of sets: (1) logicism, as an outgrowth of the Platonic school, accepts the external existence of mathematics and emphasizes the form rather than the interpretation in a specific setting; (2) intuitionism, as influenced by Aristotle’s ideas, only accepts the mathematics to be developed from the natural numbers forward via “valid” patterns of mental reasoning (not empirical realization in Aristotle’s thought); and (3) formalism, also aligned with Aristotle’s ideas, builds mathematics upon the formal axiomatic structures to free mathematics from contradictions. These three schools of thought are similar in that they view the contents of mathematics as products , but they differ in whether products are viewed as pre-existing or created through experience. The development of these three schools of thought illustrates that the view of mathematics as products has its long history in mathematics.

With the gradual development of school mathematics since 1900s (Stanic and Kilpatrick 1992 ), the conception of the nature of mathematics has increasingly received attention from mathematics educators. Which notion of mathematics mathematics education adopts and uses has a direct and strong impact on the way of school mathematics being presented and approached in education. Although the history of school mathematics is relatively short in comparison with mathematics itself, we can find ample examples about the influence of different views of mathematics on curriculum and classroom instruction in the USA and other education systems (e.g., Dossey et al. 2016 ; Li and Lappan 2014 ; Li, Silver, and Li 2014 ; Stanic and Kilpatrick 1992 ). For instance, the “New Math” movement of 1950s and 1960s used the formalism school of thought as the core of reform efforts. The content was presented in a structural format, using the set theoretic language and conceptions. But the result was not a successful progression toward a school mathematics that is best for students and teachers (e.g., Kline 1973 ). Alternatively, Dossey ( 1992 ), in his review of the nature of mathematics, identified and selected scholars’ works and ideas applicable to both professional mathematicians and mathematics educators (e.g., Davis and Hersh 1980 ; Hersh 1986 ; Tymoczko 1986 ). Those scholars' ideas rested on what professional mathematicians do, not what mathematicians think about what mathematics is. Dossey ( 1992 ) specifically cited Hersh ( 1986 ) to emphasize mathematics is about ideas and should be accepted as a human activity, not strictly governed by any one school of thought.

Devlin ( 2000 ) argued that mathematics is not a single entity but has four different faces: (1) computation, formal reasoning, and problem solving; (2) a way of knowing; (3) a creative medium; and (4) applications. Further, he contended school mathematics typically focuses on the first face, makes some reference to the fourth face, but pays almost no attention to the other two faces. His conception of mathematics assembles ideas from the history of mathematics and observes mathematical activities occurring across different settings.

Our brief review shows that the nature of mathematics can be understood as having different faces, rather than being governed by any single school of thought. At the same time, the ideas of Plato and Aristotle continue to influence the ways that mathematicians, mathematics educators, and the general public perceive mathematics. Despite nearly a half century of process-oriented research (see below), let alone Pólya’s work on problem solving, mathematics is still perceived of largely as products —a body of knowledge as highlighted in the three schools (logicist, intuitionist, formalist) of thought, rather than ideas that call for active thinking and creation. The evolving conceptions about the nature of mathematics in history suggests there is room for us to decide how mathematics can be perceived, rather than being bounded by a pre-occupied notion of mathematics as “given” or “fixed.” Each and every learner can experience mathematics through different practices and “own” mathematics as a human activity.

Problematizing what is important for students to learn in and through mathematics

The evolving conceptions about the nature of mathematics suggest that choices exist when deciding what and how to teach and learn mathematics but they do not specify what and how to make the choice. Decisions require articulating options for conceptions of what is important for students to learn in and through mathematics and evaluating the advantages and drawbacks for the students for each option.

According to Stanic and Kilpatrick ( 1992 ), the history of school mathematics curricula presents two important and real changes over the years: one is at the turn of the twentieth century when school mathematics was reformed as a unified and applied curriculum to accommodate dramatically increased student populations from diverse backgrounds, and the other is the “New Math” movement of the 1950s and 1960s, intended to integrate modern mathematics into school curriculum. The perceived failure of the “New Math” movement led to the “Back to Basics” movement in the 1970s, followed by “Problem Solving” in the 1980s, and then the Curriculum Standards movement in the 1990s and after. The history shows school mathematics curricula have emphasized teaching and learning mathematical knowledge and skills, together with problem solving and some applications of mathematics, a picture that is consistent with what Devlin ( 2000 ) refers to as the 1st face and some reference to the 4th face of mathematics.

Therefore, although there have been reforms in mathematics curriculum and instruction, there are hardly real changes in how mathematics is conceptualized and presented in school education in the USA (Stanic and Kilpatrick 1992 ) and other education systems (e.g., Leung and Li 2010 ; Li and Lappan 2014 ). The dominant conception remains mathematics as products , frequently referring to a body of static knowledge and skills that need to be learned and acquired (Fisher 1990 ). This continues to be largely the case in practice, despite advances in conceptualization (see below).

It should be noted that conceptualizing mathematics as “a body of knowledge and skills” is not wrong, especially with such a long history of knowledge creation and accumulation in mathematics, but it is not adequate for school mathematics nowadays. The set of concepts and procedures, after years of development, exceeds what could be covered in any school curricula. Moreover, this body of knowledge and skills keeps growing, as the product of human intelligence and scholarship in mathematics. Devlin ( 2012 ) pointed out that school mathematics mainly covers what was developed in the Greek mathematics, plus just two further advances from the seventh century: calculus and probability theory. It is no wonder if someone questions the value of learning such a small set of knowledge and skills developed more than a thousand years ago. Meanwhile, this body of knowledge and skills are often abstract, static, and “foreign” to many students and teachers who learned to perceive mathematics as an external entity in existence (Plato’s notion) rather than Aristotelian emphasis on experimentation (Cooney 1987 ). It is thus not surprising for so many students and teachers to claim that mathematics is difficult (e.g., Fritz et al. 2019 ) and “it is ok—not everyone can be good at math” (Rattan et al. 2012 ).

What can be made meaningful should be critically important to those who want to (or need to) learn and teach mathematics. In fact, there is significant evidence that students often try to make sense of mathematics that is “presented” or “given” to them, although they made numerous errors that can be decoded to study their thinking (e.g., Ashlock 2010 ). Indeed, misconceptions are best thought of not as errors that need to be “fixed,” but as plausible abstractions on the basis of what students have learned—i.e., attempts at sense-making (Smith et al. 1993 ). Conceiving mathematics as about “ideas,” we can help students to play, own, experience, and think about some key ideas just like what they do in many other activities, such as game play (Gee 2005 ). Definitions of concepts and formal languages and procedures can be postponed until students are ready to consider why and how they are needed. Mathematics should be taken and accepted as a human activity (Dossey 1992 ), and developing students’ mathematical thinking (about ideas) should be emphasized in learning mathematics itself (Devlin 2012 ) and in STEM (Li et al. 2019a ).

Along with the shift from products to ideas in mathematics, scholars have already focused on how people work with ideas in mathematics. Elaborated in detail by Schoenfeld ( in press ), the revolution began with George Pólya (1887–1985) who had a fundamental interest in having students learn and understand content via problem solving. For Pólya, mathematics was about inquiry, sense making, and understanding how and why mathematical ideas (instead of content as products) fit together the way they do. The call for problem solving in the 1980s in the USA was (at least partially) inspired by Pólya’s ideas after a decade of “back to basics” in the 1970s. It has been recognized since that the practices of mathematics (including problem solving) are every bit as important as the content itself, and the two shouldn’t be separated. In the follow-up standards movement, the content and practices have been the “warp and weave” of the fabric doing mathematics, as articulated in Principles and Standards for School Standards (NCTM 2000 ). There were five content standards and five process standards (i.e., problem solving, reasoning, connecting, communicating, representing). It is widely acknowledged, also in the Common Core State Standards in the USA (CCSSI 2010 ), that both content and processes/practices are essential and they form the base for next steps.

Problematizing how mathematics is taught and learned, with connections to STEM education

How the ways that mathematics is often taught cause concerns.

Conceiving mathematics as a body of facts and procedures to be “mastered” has been long-standing in mathematics education practice, and it often results in students’ learning by rote memorization. For example, Schoenfeld ( 1988 ) provided a detailed account of the disasters of a “well-taught” mathematics course, documenting a 10th-grade geometry class taught by a confident and experienced teacher. The teacher taught and managed his class well, and his students also did well on standardized examinations, which focused on content and procedures. At the same time, however, Schoenfeld pointed out that the students developed counterproductive views of mathematics. Although the students developed some level of proficiency in content and procedures, they gained (or were reinforced in) the kinds of beliefs about mathematics as being fragmented and disconnected. Schoenfeld argued that the course led students to develop a robust and counterproductive set of beliefs about the nature of geometry.

Seeking possible origins about students’ counterproductive beliefs about mathematics from mathematics instruction motivated Schoenfeld’s study (Schoenfeld 1988 ). Such an intuitive motivation is also evident in other studies. Keitel ( 2006 ) compared the lessons of two teachers (T1 and T2) in Germany who taught their classes very differently. T1 regularly taught the class emphasizing routine individual practice and memorization of specific algebraic rules. T1 emphasized the importance of such practices for test taking, and the students followed his instruction. Even when T1 one day introduced a non-routine problem that connects algebra and geometry, the overwhelming emphasis on mastering routines and algorithms seemed to overshadow in dealing such a non-routine problem. In contrast, T2’s teaching emphasized students’ initiatives and collaboration, although T2 also used formal routine tasks. At the end, students in T2’s class reported positively about their experience, enjoyed working together, and appreciated the opportunities of thinking mathematically. Studies by Schoenfeld ( 1988 ) and Keitel ( 2006 ) indicate how students’ experience in mathematics classes influences their perceptions of mathematics and also imply the importance of learning about teachers’ perceptions of mathematics that likely guide their instructional practice (Cooney 1987 ).

Rattan et al. ( 2012 ) found that teachers with different perceptions of mathematics teach differently. Specifically, Rattan et al. looked at these teachers holding an entity (fixed) theory of mathematics intelligence (G1) versus incremental theory (G2). Through their studies, Rattan and colleagues found that G1 teachers more readily judged students to have low ability, comforted students for low mathematical ability, and used “kind” strategies (e.g., assigning less homework) unlikely to promote their engagement with the field than G2 teachers. Students who received comfort-oriented feedback perceived their teachers’ entity theory and low expectations and reported lowered motivation and expectations for their own performance. The results suggest how teachers’ inadequate perceptions of mathematics and beliefs about the nature of students’ mathematical intelligence contributed to low achievement, diminished self-esteem and viewed mathematics is only a set of static facts and procedures. Further, the results suggest that how mathematics is taught influences more than students’ proficiency with mathematics content in a class. Sun ( 2018 ) made a similar argument after synthesizing existing literature and analyzing classroom observation data.

Based on the 2012 US national survey of science and mathematics education conducted by Horizon Research, Banilower et al. ( 2013 ) reported that a vast majority of mathematics teachers, from 81% at the high school level to 90% at the elementary level, believe that students should be given definitions of new vocabulary at the beginning of instruction on a mathematical idea. Also, many teachers believe that they should explain an idea to students before having them consider evidence for it and that hands-on activities should be used primarily to reinforce ideas students have already learned. The report suggests many teachers emphasized pedagogical practices of “give” and “present,” perhaps influenced by conceptions of mathematics that are more Platonic than Aristotelian, similar to what was reported about teachers’ practices more than two decades ago (Cooney 1987 ).

How to change?

Given that the evidence demonstrates a compelling case for changing how mathematics is taught, we turn our attention to suggesting how to realize this transformation. Changing how mathematics is taught and learned is not a new endeavor for both mathematics educators and mathematicians (e.g., Li, Silver, and Li 2014 ; Schoenfeld in press ). For example, the “Moore Method,” developed and used by Robert Lee Moore (a famous topologist) in the early twentieth century, shifted instruction from teacher-centered lecturing to student-centered mathematical development (Coppin et al. 2009 ). In its purest form, students were presented with mathematical definitions and asked to develop and/or prove theorems from them after class, without reading mathematics books or using other resources. When students returned to the class, they were asked to prove a theorem. As a result, students developed the mathematics themselves, instead of the instructor presenting the proofs and doing the mathematics for students. The method has had its own success in producing many great mathematicians; however, the high-pressure environment also drowned many students who might have been successful otherwise (Schoenfeld in press ).

Although the “Moore Method” was used primarily in advanced mathematics courses at the post-secondary level, it illustrates how a different conception of mathematics led to a different instructional approach in which students developed mathematics. However, it might be the opposite end of a spectrum, in comparison to approaches that present mathematics to students in accommodating and easy-to-digest ways that can be as much easy as possible. Neither extreme is a good option for K-12 students. Again, it becomes important for us to consider options that can support the value of learning mathematics.

Our discussion in the previous section highlights the importance of taking mathematics as a human activity, ensuring it is meaningful to students, and developing students’ mathematical thinking about ideas, rather than simply absorbing a set of static and disconnected knowledge and skills. We call for a shift in teaching mathematics based on Platonic conceptions to approaches based on more of Aristotelian conceptions. In essence, Plato emphasized ideal forms of mathematical objects, perhaps inaccessible through people’s sense making efforts. As a result, learners lack ownership of the ideal forms of mathematical objects, because mathematical objects cannot and should not be created by human reasoning. In contrast, Aristotle emphasized that mathematical objects are developed through logic reasoning and empirical realization. In other words, mathematical objects exist only when they can be sensed and verified by people's efforts. This differs from Plato’s passive perspective, highlights human ownership of mathematical ideas and encourages people to make mathematics make sense, termed as making sense by McCallum ( 2018 ). Aristotelian conceptions view mathematics as objects that learners can actively develop and structure as mathematically meaningful, which is more in line with what research mathematicians do. McCallum ( 2018 ) argued that both sense-making and making-sense stances are needed for a complete view of mathematics and learning, recognizing that not attending to both stances carries risks. “Just as it is a risk of the sense-making stance that the mathematics gets ignored, it is a risk of the making-sense stance that the sense-maker gets ignored.” (McCallum 2018 ).

In addition, there is the issue of personal identity: if students come to avoid mathematics because they are uncomfortable with it (in fact, mathematics anxiety has become a widespread problem for all ages across the globe, see Luttenberger et al. 2018 ) then mathematics instruction has failed them, regardless of test scores.

In the following, we discuss sense-making and making-sense stances first with specific examples from mathematics. Then, we discuss connections to STEM education.

Sense making is much more than the acquisition of knowledge and skills

Sense making has long been emphasized in mathematics education community. William A. Brownell is a well-known, early 20 th century scholar who advocated the value of sense making in the learning of mathematics. For example, Brownell ( 1945 ) discussed how arithmetic can and should be taught and learned not only as procedures, but also as a meaningful system of thinking. He shared many examples like the following division,

Brownell suggested to ask questions: what does the 5 of 576 mean? Why must 57 be the first partial dividend? Do you actually divide 8 into 57, or into 57…’s? etc., instead of simply letting students memorize how to carry out the procedure. What Brownell advocated has been commonly accepted and emphasized in mathematics education nowadays as sense making (e.g., Schoenfeld 1992 ).

There can be different ways of sense making of the same computation. As an example, the sense making process for the above long division can come out with mental math as: I am looking to see how close I can get to 570 with multiples of 80; 7 multiples of 80 gives me 560, which is close. Of course, given base 10 notation, that’s the same as 8 multiples of 70, which is why the 7 goes over the 57. And when I subtract 560, there are 16 left over, so that’s another 2 8 s. Such a sense-making process also works, as finding the answer (quotient, k ) of 576 ÷ 8 is the same operation as to find k that satisfies 576 = k × 8. In mathematics, division and multiplication are alternate but equivalent ways of doing the same operation.

To help students build numerical reasoning and make sense of computations, many teachers use number talks in their classrooms for students to practice and share these mental math and computation strategies (e.g., Parrish 2011 ). In fact, new terms are being created and used in mathematics education about sense making, such as number sense (e.g., Sowder 1992 ) and symbol sense (Arcavi, 1994 ). Some instructional programs, such as Cognitively Guided Instruction (see, e.g., Carpenter et al., 1997 , 1998 ), make sense making the core of instructional activities. We argue that such activities should be more widely adopted.

Making sense makes the other side of mathematical practice visible, and values idea development and ownership

The making-sense stance, as termed by McCallum ( 2018 ), is not commonly practiced as it is pertinent to expert mathematician’s practices. Where sense making (as discussed previously) emphasizes the process of making sense of what is being learned, making sense emphasizes the process of making mathematics make sense. Making sense highlights the importance for students to experience mathematics through creating, designing, developing, and connecting mathematical ideas. As an example, for the above division computation, 8 \( \overline{\Big)576\ } \) , students may wonder why the division procedure is performed from left to right, which is different from the other operations (addition, subtraction, and multiplication) that are all performed from right to left. In fact, students can be encouraged to explore if the division can also be performed from right to left (i.e., starting from the one’s place). They may discover, with possible support from the teacher, that the division can be done in this way. However, once the division is moved to the high-value places, it will require the process to go back down to the low-value places for completion. In other words, the division process starting from the low-value place would require repeated processes of returning to the low-value places; as a result, it is inefficient. As mathematical procedure is designed to improve efficiency, the division procedure is thus set to be carried out from the high-value place to low-value place (i.e., from left to right). Students who work this out experience mathematics more deeply than the sense-making described by Brownell ( 1945 ).

There are plenty of making-sense opportunities in classroom instruction. For example, kindergarten children are often given opportunities to play with manipulatives like cube trains and snub cubes, to explore and learn about patterns, numbers, and measurement through various connections. The recording of such activities typically results in numerical expressions or operations of these connections. In addition, such activities can also serve as a context to encourage students to design and create a way of “recording” these connections directly with a drawing line next to the connected train cubes. Such a design activity will help students to develop the concept of a number line that includes the original/starting point, unit, and direction (i.e., making mathematics make sense), instead of introducing the number line to students as a mathematical concept being “given” years later.

Learning how to provide students with opportunities to develop mathematics may occur with experience. Huang et al. ( 2010 ) found that expert and novice teachers in China both valued students’ mastering of mathematical knowledge and skills and their development in mathematical thinking methods and abilities. However, novice teachers were particularly concerned about the effectiveness of their guidance, in contrast to expert teachers who emphasized the development of students’ mathematical thinking and higher-order thinking abilities and properly dealing with important and difficult content points. The results suggest that teachers’ perceptions and pedagogical practices can change and improve over time. However, it may be worth asking if support for teacher development would accelerate the process.

Connecting changes in mathematics and STEM education

Although it is commonly acknowledged that mathematics is foundational to STEM, mathematics is being related to STEM education at a distance in practice and also in scholarship development (English 2016 , see additional notes at the end of this editorial). Holding the conception of mathematics as products does not support integrating mathematics with other STEM disciplines, as mathematics can be perceived simply as a set of tools for these disciplines. At the same time, mathematics and science have often proceeded along parallel tracks, with mathematics focused on “problem solving” while science has focused on “inquiry.” To better connect mathematics and other disciplines in STEM, we should focus on ideas and thinking development in mathematics (Li et al. 2019a ), unifying instruction from the student perspective (the Teaching for Robust Understanding framework, discussed below).

Emphasizing both sense making and making sense in mathematics education opens opportunities for connections with similar practices in other STEM disciplines. For example, sense making is very much emphasized in science education (Hogan 2019 ; Kapon 2017 ; Odden and Russ 2019 ), often combined with reflections in engineering (Kilgore et al. 2013 ; Turns et al. 2014 ), and also in the context of using technology (e.g., Antonietti and Cantoia 2000 ; Dick and Hollebrands 2011 ). Science is fundamentally about discovery and understanding of the natural world. This notion provides a natural link to mathematical modeling (e.g., Burkhardt 1981 ). Beyond that, in science education, sense making places a heavy focus on the construction and evaluation of explanation (Kapon 2017 ), and can even be defined as a process of constructing an explanation to resolve a perceived gap or conflict in knowledge (Odden and Russ 2019 ). Design and making play vital roles in engineering and technology education (Dym et al., 2005 ), as is student reflection on these experiences (e.g., Turns et al. 2014 ). Indeed, STEM disciplines share the same conceptual process of sense making as learners, individually or in a group, actively engage with the natural or man-made world, explore it, and then develop, test, refine, and use ideas together with specific explanation. If mathematics was conceived as an “empirical” discipline, connections with other STEM disciplines would be strengthened. In philosophical terms, Lakatos ( 1976 ) made similar claims Footnote 1 .

Similar to the emphasis on sense making placed in the Mathematics Curriculum Standards (e.g., NCTM, 1989 , 2000 ), Next Generation Science Standards (NGSS Lead States 2013 ) prompted a shift in science education away from simply knowing science content and procedures to practicing and using science, together with engineering, to make sense of the world and create the future. In a review, Fitzgerald and Palincsar ( 2019 ) concluded sense making is a productive lens for investigating and characterizing great teaching across multiple disciplines.

Mathematics has stronger linkages to creation and design than traditionally imagined. Therefore, its connections to engineering and technology could be much stronger. However, the deep-rooted conception of mathematics as products has traditionally discouraged students and teachers from considering and valuing design and design thinking (Li et al. 2019b ). Conceiving mathematics as making sense should help promote conceptual changes in mathematical practice to value idea generation and design activity. Connections generated from such a shift will support teaching and learning not only in individual STEM disciplines, but also in integrated STEM education.

At the same time, although STEM education as a commonly recognized field does not have a long history (Li 2014 , 2018a ), its rapid development can help introduce ideas for exploring how mathematics can be taught and learned. For example, the concept of projects is common in engineering professional practice, and the project-based learning (PjBL) as an instructional approach is a key component in some engineering programs (e.g., Berger 2016 ; de los Ríos et al. 2010 ; Mills and Treagust 2003 ). de los Ríos et al. ( 2010 ) highlighted three main advantages of PjBL: (1) development in technical, personal, and contextual competences; (2) students’ engagement with real problems from professional contexts; and (3) collaborative learning facilitated through the integration of teaching and research. Such advantages are important for students’ learning of mathematics and are aligned well with efforts to develop 21 st century skills, including problem solving, communication, collaboration, and critical thinking.

Design-based learning (DBL) is another instructional approach commonly used in engineering and technology fields. Gómez Puente et al. ( 2013 ) conducted a sampled review and concluded that DBL projects consist of open-ended, hands-on, authentic, and multidisciplinary design tasks. Teachers using DBL facilitate both the process for students to gain domain-specific knowledge and thinking activities to generate innovative solutions. Such features could be adapted for mathematics education, especially integrated STEM education, in concert with design and design thinking. In addition to a few examples discussed above about making sense in mathematics, there is a growing body of publications developed by and for mathematics teachers with specific examples of investigations, design projects, and instructional activities associated with STEM (Li et al. 2019b ).

A framework for helping students to gain important experiences in and through mathematics, as connected to other disciplines in STEM

For observing and evaluating classroom instruction in general and mathematics classroom instruction in specific, there are several widely used frameworks and rubrics available. However, a trial use of selected frameworks with sampled mathematics classroom instruction episodes suggested their disagreements on what counts as high-quality instruction, especially with aspects on disciplinary thinking being valued and relevant classroom practices (Schoenfeld et al. 2018 ). The results suggest the importance of choice making, when we consider a framework in discussing and evaluating teaching practices.

Our discussion above highlights the importance of shifting away from viewing mathematics simply as a set of static knowledge and skills, to focusing on ideas and thinking development in teaching and learning mathematics. Further discussion of several aspects of changes specifies the needs of developing and using practices associated with sense making, making sense, and connecting mathematics and STEM education for changes.

To support effective mathematics instruction, we propose the use of the Teaching for Robust Understanding (TRU) framework to help characterize powerful learning environments. With the conception of mathematics as “empirical” and a focus on students’ experience, then the focus of instruction should also be changed. We argue that shift is from instruction conceived as “what should the teacher do” to instruction conceived as “what mathematical experiences should students have in order for them to develop into powerful thinkers?” It is the shift in the frame of TRU that makes it so powerful and pertinent for all these proposed changes. Moreover, TRU only uses a small number of actionable dimensions after distilling the literature on teaching for robust or powerful understanding. That makes TRU a practical mechanism for problematizing instruction.

Figure 1 presents the TRU Math framework that identifies five key dimensions along which powerful classroom environments can be characterized: the mathematics; cognitive demand; equitable access; agency, ownership, and identity; and formative assessment. These five dimensions were distilled from an extensive literature review, thus capturing what the literature considers to be essential. They were tested against classroom videotapes and data on student performance, and the results indicated that classrooms that did well on the TRU dimensions produced students who did correspondingly well on tests of mathematical knowledge, thinking, and problem solving (e.g., Schoenfeld 2014 , 2019 ). In brief, the argument regarding the importance of the five dimensions of TRU Math is as follows. First, the quality of the mathematics discussed (dimension 1) is critical. What individual students learn is unlikely to be richer than what they experience in the classroom. Whether individual students’ understanding rises to the level of what is discussed/presented in the classroom depends on other factors, which are captured in the remaining four dimensions. For example, you surely have had the experience, at a lecture, of hearing beautiful content presented, and then not being able to do any of the assigned problems! The remaining four dimensions capture aspects needed to support the development of all students with respect to sense making, making sense, ownership, and feedback loop. Dimension 2: Cognitive Demand. Are students engaged in sense making and making sense? Are they engaged in “productive struggle”? Dimension 3: Equitable Access. Are all students fully engaged with the central content and practices of the domain so that every student can profit from it? Dimension 4: Agency, Ownership, and Identity. Do all students have opportunities to develop idea ownership and mathematical agency? Dimension 5: Formative Assessment. Are students encouraged and supported to share their thinking with a meaningful feedback loop for instructional adjustment and improvement?

The TRU Mathematics Framework: The five dimensions of powerful mathematics classrooms

The first key point about TRU is that students learn more in classrooms that are powerful along the five TRU dimensions. Second, the shift of attention from the teacher to the environment is fundamentally important. The key question is not “Is the teacher doing particular things to support learning?”; instead, it is, “Are students experiencing instruction so that it is conducive to their growth as mathematical thinkers and learners?” Third, the framework is not prescriptive; it respects teacher autonomy. There are many ways to be an excellent teacher. The question is, Does the learning environment created by the teacher provide each student rich opportunities along the five dimensions of the framework? Specifically, in describing the dimensions of powerful instruction, the framework serves to problematize instruction. Asking “how am I doing along each dimension; how can I improve?” can lead to richer instruction without prescribing or imposing a particular style or particular norms on teachers.

Extending to STEM education

Now, we suggest the following. If you teach biology, chemistry, physics, engineering, or any other STEM field, replace “mathematics” in Fig. 1 with your discipline. The first dimension is about rich content and practices in your field. And the remaining four dimensions are about necessary aspects of your students’ classroom engagement with the discipline. Practices associated with sense making, making sense, and STEM education are all be reflected in these five dimensions, with central attention on students’ experience in such classroom environments. Although the TRU framework was originally developed for characterizing effective mathematics classroom environments, it has been carefully framed in a way that is applicable to many different disciplines (Schoenfeld 2014 ). Our discussion above already specified why sense making, making sense, and specific instructional approaches like PjBL and DBL are shared across disciplines in STEM education. Thus, the TRU framework is applicable to other STEM disciplines. The natural analogue of the TRU framework for any field is given in Fig. 2 .

The domain-general version of the TRU framework

Both the San Francisco Unified School District and the Chicago Public Schools adopted the TRU Math framework and found results within mathematics sufficiently promising that they expanded their efforts to all subject areas for professional development and instruction, using the domain-general TRU framework. Work is still in its early stages. Current efforts might be best conceptualized as a laboratory for exploration rather than a promissory note for improvement across all different disciplines. It will take time to accumulate data to show effectiveness. For further information about the domain-general TRU framework and tools for professional development are available at the TRU framework website, https://truframework.org/

Finally, as a framework, TRU is not a set of specific tools or guidelines, although it can be used to guide their development. To help lead our discussion to something more practical, we can use the framework to check and identify aspects that are typically under-emphasized and move them to center stage in order to improve classroom instruction. Specifically, the following is a list of sample under-emphasized norms and practices that can be identified (Schoenfeld in press ).

Establishing a climate of inquiry, in which mathematics is experienced as a discipline of exploration and sense making.

Developing students’ ownership of ideas through the process of developing, sharing, refining, and using ideas; concepts and language can come later.

Focusing on big ideas, and not losing the forest for the trees.

Making student thinking central to classroom discourse.

Ensuring that classroom discourse is respectful and inviting.

Where to start? Begin by problematizing teaching and the nature of learning environments

Here we start by stipulating that STEM disciplines as practiced, are living, breathing fields of inquiry. Knowledge is important; ideas are important; practices are important. The list given above applied to all STEM disciplines, not just mathematics.

The issue, then, is developing teacher capacity to craft environments that have the properties described immediately above. Here we share some thoughts, and the topic itself can well be discussed extensively in another paper. To make changes in teaching, it should start with assessing and changing teaching practice itself (Hiebert and Morris 2012 ). Opening up teachers’ perceptions of teaching practices should not be done by telling teachers what to do!—the same rules of learning apply to teachers as they apply to students. Learning environments for teachers should offer teachers the same opportunities for rich engagement, challenge, equitable access, and ownership as we hope students will experience (Schoenfeld 2015 ). Working together with teachers to study and reflect on their teaching practices in light of the TRU framework, we can help teachers to find out what their students are experiencing and why changes are needed. The framework can also help guide teachers to learn what changes would be needed, and to try out changes to learn how their students’ learning may differ. It is this iterative and concrete process that can hopefully help shift participating teachers’ perceptions of mathematics. Many tools for problematizing teaching are available at the TRU web site (see https://truframework.org/ ). If teachers can work together with a focus on selected lessons like what teachers often do in China, the process would help form a school-based learning community that can contribute to not only participating teachers’ practice change but also their expertise improvement (Huang et al. 2011 ; Li and Huang 2013 ).

As reported before (Li 2018b ), publications in the International Journal of STEM Education show a mix of individual-disciplinary and multidisciplinary education in STEM over the past several years. Although one journal’s publications are limited in its scope of providing a picture about the scholarship development related to mathematics and STEM education, it can allow us to get a sense of related development.

If taking a closer look at the journal’s publications over the past three years from 2016 to 2018, we found that the number of articles published with a clear focus on mathematics is relatively small: three (out of 21) in 2016, six (out of 34) in 2017, and five (out of 56) in 2018. At the same time, we should point out that these publications from 2016 to 2018 seem to reflect a trend, over these three years, of moving toward issues that can go beyond mathematics itself, as what was noted before (Li 2018b ). Specifically, for these three articles published in 2016, they are all about mathematics education at either elementary school (Ding 2016 ; Zhao et al. 2016 ) or university levels (Schoenfeld et al. 2016 ). Out of the six published in 2017, three are on mathematics education (Hagman et al. 2017 ; Keller et al. 2017 ; Ulrich and Wilkins 2017 ) and the other three on either teacher professional development (Borko et al. 2017 ; Jacobs et al. 2017 ) or connection with engineering (Jehopio and Wesonga 2017 ). For the five published in 2018, two are on mathematics education (Beumann and Wegner 2018 ; Wilkins and Norton 2018 ) and the other three have close association with other disciplines in STEM (Blotnicky et al. 2018 ; Hayward and Laursen 2018 ; Nye et al. 2018 ). This trend likely reflects a growing interest, with close connection to mathematics, in both mathematics education community and a broader STEM education community of developing and sharing multidisciplinary and interdisciplinary scholarship.

Availability of data and materials

Not applicable

Interestingly, Lakatos was advised by both Popper and Pólya—his ideas being in some ways a unification of Pólya’s emphasis on mathematics as an empirical discipline and Popper’s reflections on the nature of the scientific endeavor.

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Many thanks to Jeffrey E. Froyd for his careful review and detailed comments on an earlier version of this editorial that help improve its readability and clarity. Thanks also go to Marius Jung for his valuable feedback.

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Why early math is just as important as early reading

by: Hank Pellissier | Updated: February 27, 2023

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According to new research, the importance of mathematics in early childhood cannot be underestimated. So what grade do U.S. preschools deserve in math instruction?

Answer: F (At least most of them.)

Math is nearly absent in American preschools and prekindergarten classes. One study calculated that at preschools where kids spend six hours a day, math gets an average of only 58 seconds per day . Not even a full minute.

What’s more, those spare moments of math are often taught incorrectly . Learning to recite numbers from one to 10 doesn’t get kids very far, because often kids are just memorizing , according to Stanford math professor Jo Boaler, which does little to lay the groundwork for future problem solving and logical thought.

From “Talk, Read, Sing!” campaigns to closing the 30 million word gap, American parents, educators, and policymakers have embraced the importance of early literacy, yet we collectively presume it’s fine to tackle math later. Meanwhile, research clearly shows that early math exposure is crucial for later success in math.

Why is mathematics important in early childhood? “Early math skills have the greatest predictive power, followed by reading and then attention skills,” reports a psychology squad led by Greg J. Duncan , in School readiness and later achievement , published in Developmental Psychology in 2007. Follow-up studies continue to confirm the importance of early math skills. The more math-oriented activities kids do before kindergarten, the better they’ll understand math in school. Early math skills foretell higher aptitude in high school math and higher rates of college enrollment. And a 2014 Vanderbilt study determined that for “both males and females, mathematical precocity early in life predicts later creative contributions and leadership in critical occupational roles.”

Why counting doesn’t cut it

To date, the most common approach to teaching early math skills has been to surround young kids with numbers alongside their letters and encourage them to practice counting just as they practice singing the alphabet. But researchers say that this approach is short-changing children. In her essay “ Math Matters, Even for Little Kids ,” Stanford professor Deborah Stipek (co-written with Alan Schoenfeld and Deanna Gomby) explains the parallel to the alphabet: “Learning to count by rote teaches children number words and order, but it does not teach them number sense, any more than singing the letters L-M-N-O-P in the alphabet song teaches phonemic awareness.”

As for the magic of counting to 10, University of Chicago professor Susan Levine explains: “Kids can rattle off their numbers early, often from 1 to 10, and parents are surprised and impressed. But it’s a list with no meaning. When you say ‘give me 3 fish,’ they give you a handful.”

While parents and preschool teachers reinforce literacy lessons daily by reading together, singing, pointing out letters and letter sounds, math exposure often begins and ends with the counting.

Early math skills kids are born with

Evidence points to early addition and subtraction being an innate ability . In a 1992 study at the University of Arizona , for example, 6-month-old babies were shown one baby doll. As the babies watched, a screen was placed in front of the doll and then a second doll was placed behind the screen. When the screen was removed, scientists could tell that, at just 6 months old, babies expected to see two dolls. In instances when there were fewer or more dolls when the screen was removed, the babies stared longer because the results were wrong, a “violation of expectation.”

“ Subitize ” — from the Latin word for “suddenly” — is the ability to quickly identify the number of items in a small group. When Dustin Hoffman’s character in Rainman looked down at the pile of spilled toothpicks and knew without counting that there were 246, that was an example of advanced subitization. Preschoolers can differentiate between one and three items; by age 7, this increases to between four and seven items. It’s more than just a cool party trick: Research indicates that developing the ability to subitize larger numbers can increase math skills in a few ways . One example is “counting on” or being able to start at 5 and continue counting up, which is a math strategy first graders will need as they begin tackling adding and subtracting.

Numbers in new and different contexts

One way to build on kids’ innate math abilities is to focus on helping them count in contexts that are meaningful to them. To practice counting on, start with a number they recognize, like two toy dinosaurs. Add another and say, “three,” then add another and say “four,” helping them to connect the number names with the increase in objects.

Toddlers are natural sorters. By age 2, they start recognizing and making comparisons, such as more, same, and different. They enjoy organizing multiple objects into specific “sets”, i.e., groups or categories. Asking a child to sort their prized dinosaurs into groups, such as big and small, is the basis of a few important early math skills. First, they can compare the groups. (There are more small dinosaurs.) They can count how many are in each group (two big ones, six small ones). They can re-sort in different ways (the ones with spikes here, the ones without spikes there; arranging the dinosaurs in order from shortest to tallest). Numeracy is an intimidating word, but it simply means understanding what each number represents and beginning to understand the implications of number operations: What happens when you take one of the dinosaurs away?

The National Association for Education of Young Children notes that young children are building scientific inquiry skills when they sort, compare, describe, and put things in order in terms of observable characteristics, like the dinosaur’s height.

Amazing measuring

Children are rightfully fascinated by variability in size. They delight in the enormity of elephants, trees, and skyscrapers, and the minuteness of ants and caterpillars. The allure of discovering what’s comparatively bigger or smaller fosters their curiosity about inches, pounds, gallons, miles, and other systems of measurement. And that’s what parents and educators should encourage, according to Douglas H. Clements and Julie Sarama, authors of Learning and Teaching Early Math: The Learning Trajectories Approach . Clements and Sarama suggest that measurement is the best way for young kids to learn about math. They go so far as to say it’s better than counting. “We use length consistently in our everyday lives,” they write. “[Measurement] can help develop other areas… including reasoning and logic. Also, by its very nature, [measuring] connects two critical domains of early mathematics: geometry and number.”

Building blocks and the language of space

The next time you’re cleaning up your child’s blocks or Legos, just remember: they’re building their math brains. Boosting spatial skills via block play has been proven beneficial in many studies. For example, the complexity of a child’s LEGO play during the preschool years is correlated with higher math achievement in high school .

Exposing preschoolers to geometrical shapes including circles, squares, triangles, and rectangles helps them build a skill called visual literacy. Researchers Clements and Sarama discovered in one study that kids who learned shapes and spatial skills also showed pronounced benefits in math and writing readiness and even increased their IQ scores. (Related: How to teach your preschooler shapes and spatial skills .)

Clements urges parents and teachers to teach kids what he calls the “Language of space” – words like front, back, behind, top, bottom, over, under, last, first, next, backward, in, on, deep, shallow, triangle, square, corner, edge, etc. Stipek, former dean at the Stanford Graduate School of Education , suggests “when you’re reading a picture book to your child, point out position and spatial representation. Say, ‘the tree is behind the car’ and ‘the roof is a triangle.’” Helping a preschooler understand these relative terms is more than a math vocabulary lesson, says parent educator Nancy Gnass. It’s hard for a child to follow spatial directions commonly given to kindergartners – stand behind the blue line, put the blocks on bottom shelf, this is a quiet corner – if they don’t know what behind, bottom, or corner mean, she says. Preparing your child for directional language they’ll be expected to comply with regularly once they start school can head off misunderstandings and perceived behavior issues, like not understanding rather than deliberately not following directions, she says.

Patterns aren’t just pretty

Visual art and dance provide excellent ways to teach patterns, defined in A Math Dictionary for Kids  as ”a repeated design or recurring sequence.” According to Zero to Three, recognizing and creating patterns helps children learn to make predictions, understand what comes next, make logical connections, and use reasoning skills. Kids start to put together the “growing pattern” in counting and “relationship pattern” that’s the basis for multiplication.

Movement patterns can also imbue a trip to the park with mathematical benefits. Encourage your kinetic kid to walk-tiptoe-jump-repeat or skip-hop-run-in-a-circle-repeat, or stop, drop, and roll; repeat until they’re exhausted (and educated). (Related: Cool ways to teach your preschooler patterns )

Bring on the math games

The best way for parents to “mathematize” their children is to use math in the routines of daily life, either as games or as entertaining ways to solve problems. “Make math fun!” advises Eric Wilson, Lead Teacher at Pacific Primary School in San Francisco . “Young children work very hard when they’re playing. Play is the perfect learning environment.” Puzzles , building blocks , board games, and card games have all been studied, with researchers concluding that all of these elevate math skills.

Chutes and Ladders, for example, excels at teaching numbers, says Stanford professor Deborah J. Stipek, and playing with dice teaches addition.

Attitude matters

Possibly the most important thing any parent or teacher can do for preschoolers’ early math skills is encourage children to believe that they can succeed . “Self-efficacy, which is an individual’s belief in whether he or she can succeed at a particular activity, plays an integral role in student success,” writes math educator Lynda Colgan of Queen’s University in Ontario, Canada in an article for TVO.org — and that extends to any subject.

“When children are positive about learning and feel able to succeed, they are more likely to be successful.”

Why worry about math now?

“Math is the language of logic,” explains Dr. Jie-Qi (Jackie) Chen , professor of Child Development at the Erikson Institute, a principal investigator of the Early Math Collaborative , and co-author of Big ideas of early mathematics: What teachers of young children need to know .

“Math builds reasoning, which leads to comprehension,” she says. “Developing a mentally organized way of thinking is critical.” To make that happen, Chen says, “We need to provide high-quality math education at an early age.”

She’s right, but are PreK parents and educators listening? Let’s not pretend our children will “catch up” in later grades. In the Program for International Student Assessment, an international assessment that measures 15-year-old students’ reading, mathematics, and science literacy every three years, U.S. test scores in math are embarrassing . In 2012, out of 34 OECD contestants, the USA ranked #27 in math. (We’re #17 in Reading, and #20 in Science.)  In the 2018 PISA math assessment the U.S. ranked #37 out of 76 participating nations. American students are especially weak in “performing mathematics tasks with higher cognitive demand… and interpreting mathematical aspects in real-world problems.” Quite dismally, 26 percent of 15-year-olds in the U.S. fail to reach the PISA baseline Level 2 of mathematics, where they would “begin to demonstrate the skills that will enable them to participate effectively and productively in life.”

So, what can parents do? Without being a total pain-in-the-behind (I’m using a “spatial” term here) you could talk with your child’s preschool director about how they approach math in their daily activities. Talk with them about the value of sorting, measurement, patterns, the language of space, block play, building on innate mental addition skills — and most of all the value of a positive attitude toward math. You might even print out this report: Math Matters: Children’s Mathematical Journeys Start Early   by Deborah J. Stipek and Alan H. Shoenfeld and share it with the preschool staff.

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Essay on Importance of Mathematics for Students

Importance of Mathematics

Mathematics is the queen of science. We cannot imagine our life without mathematics. Mathematics makes human life more Organized. The knowledge of numbers and their basic operation is very important to survive in today’s world.

Essay on Importance of Mathematics in Daily Life for Students

Mathematics has its application in daily life such as Time -management, Finance management, Profit and loss in business, Cooking, Purchasing and selling, Consumption of electricity, and patrol. There is hardly any field of science, which does not use mathematics as a tool to arrive at the desired result.

Engineering , Medical , Pharmacy, Geography, Economics, Accountancy and all field related to science uses mathematical methods to study the subjects.

Definition of Mathematics:

How we can define mathematics “Mathematics is the body of knowledge based on numbers and figures”.

Importance of Mathematics in Daily Life:

Math has multiple advantages in day-to-day life. Let’s discuss the importance of Math in day-to-day life.

The most important thing about Maths is that it helps to develop your brain. We know that we can strengthen our muscles by exercise in the same way we can strengthen neurons in our brain by doing Maths. Our brain is made of neuron’s when we do Math neurons in our brain are stimulated.

Research work done by Dr. Tanya Evans of Stanford University proves that the student who performs well in Math can recruit certain brain regions more reliably and have more gray matter in those regions, as compared to those who perform poorly in math.

If you want to succeed in life then you must learn the art of time management math helps you to read the time and manage it. If you understand fractions in math in a better way you can understand time in a better way. Learning to read the analog clock is the first step of time management and it is possible only because of math. In school children are taught to read both times of watching the 12-hour clock and 24-hour clock.

Math helps you with your finances it is said that you should never spend more than your income. Math helps you to calculate your monthly expenditure and helps you keep your monthly expenses as low as possible. Math helps you to calculate what part of your monthly income you are spending on different services and goods. Math helps you to keep your monthly expenditure below your monthly income.

It also helps to calculate your saving according to your future needs. Knowledge of math is also required in investing money in property, bank schemes Mutual funds, or the stock market. This thing required sound knowledge of math and economics.

Math and knowledge of proportion can make you an excellent cook. Let’s take an example if you are making tea. Then to make one cup of tea you will need half a cup of milk, half a cup of water, 1 spoon of tea powder, and 2 spoons of sugar. Now if you have to make two cups of tea you have two take the above ingredients in twice the quantity i.e. 1 cup of milk, 1 cup of water, 2 spoons of tea, and 4 spoons of sugar.

In this way, math helps you to decide the proportion of food ingredients according to their consumers. All this is possible because of math. It is observed that people who are good at math are also having very good problem-solving skills.

Math teaches us to think logically and creatively. Math improves our thinking and improves our Analytical thinking and reasoning ability.

We all face problems in our life our ability to solve problems depends on the way we think. All of us can think but very few think rationally. math develops rational thinking in man and hence improves his problem-solving skills.

Math also helps you to stay fit and healthy. To keep your body in shape you should know which exercise and the number of repetitions you must perform. The counting repetitions during exercise are based on math. You also need a healthy diet to stay fit. We already discussed that knowledge of math plays a vital role in deciding the proportion of ingredients in the diet.

Math helps you to understand sports in a better way. Take the example of cricket knowledge of math helps us to understand cricket scores in a better way. Batting average, bowling average, run rate, strike rate all these cricket terminologies are based on math. Using math sometimes you can also predict the winning team before the match.

Math also helps us while we drive our 2-wheeler or 4-wheeler. It helps us to keep track of the distance covered and the distance to be covered. How much petrol is left inside the tank? What should be the speed of the vehicle to reach the destination in time? All these calculations are based on math.

Conclusion:

Math is the backbone of all science we cannot imagine human life without science. To survive in this world you will need the math you cannot live for a single day without the use of math. The importance of math is like oxygen in the air without which no human progress is possible.

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Using interpretable machine learning approaches to predict and provide explanations for student completion of remedial mathematics

• Published: 10 May 2024

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The successful completion of remedial mathematics is widely recognized as a crucial factor for college success. However, there is considerable concern and ongoing debate surrounding the low completion rates observed in remedial mathematics courses across various parts of the world. This study applies explainable artificial intelligence (XAI) tools to interpret predictions on whether students will complete mathematics remediation. Various machine learning models are compared, with random forest emerging as superior in predicting non-completion. Global interpretations using correlation analysis, logistic regression, feature importance, permutation importance, and SHapley Additive exPlanations (SHAP) summary plots identify significant predictors such as college grade point average (G.P.A), high school G.P.A, starting point in the remedial sequence, number of failed remedial courses, delay in remediation, Rate My Professor scores, and age. Additionally, local interpretations using Local Interpretable Model-Agnostic Explanations (LIME) and Diverse Counterfactual Explanations (DiCE) analyses were utilized to garner tailored advise for at-risk students. It was observed that instructor attributes cannot be overlooked, especially, when exploring local interpretations. Future research should consider other features such as a students’ socio-economic status (SES), employment status, and placement exam scores. Future studies could also involve data from multiple institutions and examine user experience in implementing these models.

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Mgonja, T. Using interpretable machine learning approaches to predict and provide explanations for student completion of remedial mathematics. Educ Inf Technol (2024). https://doi.org/10.1007/s10639-024-12647-6

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DOI : https://doi.org/10.1007/s10639-024-12647-6

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