drill vs problem solving practice

Do "drill and practice" instructional strategies work in education?

"Drill & practice" does not have to be a dirty phrase in education. It's up to educators to properly balance constructivism vs. behaviorism.

Over the past several years, the world’s thought leaders in education have increasingly clamored for more collaborative, constructivist, project-based activities in the classroom, at the expense of drill and practice instructional strategy .

Schools should focus less on “memorization” and more on “skills” , the theory goes, to better prepare students for the 21st-century workplace. We agree ... but with some limitations. Why? Because the drill and practice method of teaching  works.

The drill and practice method of teaching is still useful in education

This virtuous shift toward competency-based learning is arguably one of the most important trends in the advancement of education this decade. Yet, there is a risk of taking it too far.

Repetitive study—otherwise known as "drill and practice" instructional strategy—and knowledge-based assessments can still be very helpful in a lot of cases! The ubiquity of Google and Wikipedia is no substitute for having your own mind full of immediately accessible information.

For example, do we want to produce a generation of:

• Physicists who don’t know their basic multiplication tables by heart?
• Cultured citizens who can’t identify France on a map?
• Doctors who can’t make a simple diagnosis without consulting WebMD?
• Peace Corps volunteers in Peru who can’t conjugate Spanish verbs without consulting an app?

The list goes on.

Knowledge is, and will always be, an important objective of the education process itself, and the best way to fully acquire it is often through good old-fashioned repetition and through the drill and practice method of teaching.

Indeed, research shows that those students who can remember more about a topic are also able to think more critically about it . They're actually better at analysis . It's just a matter of gracefully integrating drill and practice instructional strategy into a more holistic learning experience.

Check out our video 'Why rote memorization still matters & how to do it right'...

Brainscape automates the drill and practice method of teaching via confidence-based repetition

Over the past several years of building and managing Brainscape, an adaptive flashcard learning platform , we've collected valuable feedback from tons of educators —ranging from primary school teachers to college professors—about what really works in the classroom. And the role of drill and practice instructional strategy, especially optimized by the use of flashcards, remains important for reasons we'll dive into in just a bit.

(If you want to learn more about what works to help your students retain knowledge , check out our in-depth guide on optimizing student performance in the classroom .)

Constructivism vs. behaviorism

Of course, many progressive educators will argue that real-life simulations, on-the-job training, constructivist activities, and project-based learning are significantly more effective at “teaching” such new concepts than rote memorization and other drill and practice instructional strategies. They are only half-right.

The problem is that no single cost-effective constructivist activity will guarantee that you will be exposed to all the concepts you need to know—or that you will fully remember the concepts that you are exposed to.

While it may be preferable to first expose students to knowledge in a more constructivist manner, concepts still need to be systematically reviewed to be internalized for the long term. This is particularly important in higher education, where the targeted accumulation of knowledge can be critical for success or certification in a particular field.

So we think the debate should be shifted from "either-or" to "both-and." In other words, to combine the two learning philosophies, where appropriate. Educators should encourage their students to do any studying or drilling on their own time, using the most personalized study tools possible, to leave more physical class time for collaborative, skill-building activities.

By introducing anchoring repetitive learning to classroom topics, you can solve the age-old argument that certain methods, like " flashcards result in learning out of context ."

Fortunately, educators are now better positioned than ever to “outsource” this drill and practice instructional strategy outside of the classroom, thanks largely to adaptive web and mobile study technologies like Brainscape .

Teachers and students can now easily find, create, and share online study material s tied to almost any curriculum on the planet. Students of all ages can benefit from scientifically optimized study algorithms that help them learn more in less time , thereby leaving more class time for those richer activities that schools are best at.

A final word on drill and practice instructional strategy

We are now entering the golden age of the Flipped Classroom model we just described. As long as curriculum designers don’t insist that kids waste time memorizing the wrong things (e.g. trivial historical dates, every sub-species of mollusk, excessive numbers of unimportant historical figures, etc.), educators should be able to design lesson plans that appropriately segment the constructivist and behaviorist components of the learning process.

And any little bits of memorization might just end up being good for students’ brains in the first place!

Flashcards for serious learners .

• Quick links Applicants & Students Important Apps & Links Alumni Faculty and Staff Community Admissions How to Apply Cost & Aid Tuition Calculator Registrar Orientation Visit Campus Academics Register for Class Programs of Study Online Degrees & Programs Graduate Education International Student Services Study Away Student Support Bookstore UIS Life Dining Diversity & Inclusion Get Involved Health & Wellness COVID-19 United in Safety Residence Life Student Life Programs UIS Connection Important Apps UIS Mobile App Advise U Canvas myUIS i-card Balance Pay My Bill - UIS Bursar Self-Service Email Resources Bookstore Box Information Technology Services Library Orbit Policies Webtools Get Connected Area Information Calendar Campus Recreation Departments & Programs (A-Z) Parking UIS Newsroom Connect & Get Involved Update your Info Alumni Events Alumni Networks & Groups Volunteer Opportunities Alumni Board News & Publications Featured Alumni Alumni News UIS Alumni Magazine Resources Order your Transcripts Give Back Alumni Programs Career Development Services & Support Accessibility Services Campus Services Campus Police Facilities & Services Registrar Faculty & Staff Resources Website Project Request Web Services Training & Tools Academic Impressions Career Connect CSA Reporting Cybersecurity Training Faculty Research FERPA Training Website Login Campus Resources Newsroom Campus Calendar Campus Maps i-Card Human Resources Public Relations Webtools Arts & Events UIS Performing Arts Center Visual Arts Gallery Event Calendar Sangamon Experience Center for Lincoln Studies ECCE Speaker Series Community Engagement Center for State Policy and Leadership Illinois Innocence Project Innovate Springfield Central IL Nonprofit Resource Center NPR Illinois Community Resources Child Protection Training Academy Office of Electronic Media University Archives/IRAD Institute for Illinois Public Finance
• Request Info Request info for....     Undergraduate/Graduate     Online     Study Away     Continuing & Professional Education     International Student Services     General Inquiries

Drill and Practice

Drill and practice is a behaviorist aligned technique in which students are given the same materials repeatedly until mastery is achieved. In each iteration, students are given similar questions to answer or activities to perform, with a certain percentage of correct responses or actions moving the student to the next level of difficulty.

Appropriate Content Areas

Most common in Kinesthetics, Coaching, Music, Mathematics, Language, Typing, and Biological Sciences. Not as common in adult education.

Anatomy Drill and Practice  at the Student Companion site to Tortora & Grabowski,  Introduction to the Human Body , 5th edition.

Goals and Objectives

Several goals can be attributed to drill and practice exercises. They can be used to build confidence as more answers are correctly provided. They also help to reinforce important materials. Learners are also provided and opportunity to practice critical skills and knowledge sets. Sample objectives are shown next.

During and after performing the activity, students will…

• increase skill at performing the given task…
• increase speed at performing the given task…
• internalize the given information until it is an automatic assumption…

…as determined by successfully attending to 80% of rubric items.

Prerequisites

In general, there are no prerequisites for drill and practice. It is commonly used as a core set activity such as learning an alphabet. Prerequisites may exist if more advanced knowledge sets are to be practiced. For example, you would not use drill and practice in multiplication if the student does not already know addition.

Materials and Resources

The instructor must provide instruction in what is to be practiced. Depending on the curriculum, worksheets may be produced on which the students work.

Guiding Questions for this Lesson

There is not usually a guiding question in drill and practice exercises, perhaps adding to the opposition of its use in more advanced thinking skills. It is basically a technique used in process and declarative knowledge building activities.

Lesson Outline and Procedure

Before beginning a drill and practice exercise, the students must be informed and taught the underlying principles. Once the principle has been demonstrated or instructed, the students are given an activity, procedure, or worksheet to complete. A set time limit is usually employed. Assessment should quickly follow performance with opportunities for additional reinforcement and skill building. Effective use of drill and practice will depend on linking the activity to the actual skill that the instructor intends to develop.

Drill and practice activities can also be provided to students to do on their own time and at their own pace, with assessment provided after they have been completed. When appropriate, computer-based assessment can be used to provide immediate feedback without instructor overseeing the activity, such as in many mathematics drills.

Teaching Strategies

• Kinesthetic drills should be in a real life situation.
• Avoid criticisms of student ability. It may be that the prerequisites have not been met and the activity is not appropriate for the given student.
• Finish every drill.
• Studies have shown that children learn better individually rather than in pairs when performing drill and practice exercises.

Accommodations

What accommodations may be needed for students with disabilities or other special needs? Clearly, some physical disabilities will prohibit some kinesthetic activities from being performed. Time may also become a factor for some students, such as those with dyslexia, in performing some tasks since speed is often part of the assessment in drill and practice exercises.

Drill and practice exercises are short term activities that are usually completed in under 10 minutes. There should be follow-up reinforcement at later times.

Ideas for Lesson Evaluation and Teacher Reflection

How did the students like the lesson? End of semester evaluations should ask about the usefulness and learning accomplished through such activities.

How was student learning verified? Participation can be assessed in discussion sessions. In general, student work is assessed by the number of correct responses or actions in a given time frame with process providing partial credit towards the final grade.

Additional Resources

• Audioblox, (n.d.)  Teaching myth: Repetition and drill dull creativity . Retrieved January 9, 2007, from  http://www.audiblox2000.com/repetition.htm
• Handal, B., & Herrington, A. (2003). Re-examining categories of computer-based learning in mathematics education. Contemporary Issues in Technology and Teacher Education [Online serial],  3 (3). Retrieved January 9, 2007, from  http://www.citejournal.org/vol3/iss3/mathematics/article1.cfm
• Norman, D. A., & Spohrer, J. C. (1996). Learner-centered education. IT Forum, 12 . Retrieved January 9, 2007, from http://itech1.coe.uga.edu/itforum/paper12/paper12.html  – “Rote learning and drill-and-practice are still essential to transform understanding into automated skill, making the information and procedures available to the mind without conscious effort.

• Get IGI Global News

• All Products
• Book Chapters
• Journal Articles
• Video Lessons
• Teaching Cases
• Recommend to Librarian
• Recommend to Colleague
• Fair Use Policy

• Access on Platform

Export Reference

• e-Journal Collection
• e-Book Collection Select
• Social Sciences Knowledge Solutions e-Journal Collection
• Computer Science and IT Knowledge Solutions e-Journal Collection

The Differences between Problem-Based and Drill and Practice Games on Motivations to Learn

Problem-Based education has been put forward as the most fruitful approach when it comes to serious game design (Aldrich, 2009; Gee, 2005). In Problem-Based learning, students start with a problem. This problem is rather loosely defined as something ‘for which an individual lacks a ready response’ (Hallinger, 1992, p. 27). Problem-Based education distinguishes between well- and ill-defined problems. Ill-defined problems are those ‘in which one or several aspects of the situation is not well specified, the goals are unclear, and there is insufficient information to solve them’ (Ge & Land, p5 in Ertmer et al., 2008). Shaffer’s (Shaffer, Squire, Halverson, & Gee, 2005; Shaffer, 2008) suggestion for epistemic games, in which players adopt the perspective of a professional to confront complex problems in simulation-like game, aligns with Problem-Based learning approach.

Drill & Practice learning teaches the ‘what’ and the ‘when’, but not the ‘why’ and the ‘how’. Ke (2008) suggests that students in Drill & Practice Learning merely memorize facts. As a result, this kind of learning may not facilitate creative thought or stimulate problem-solving skills. Or, as Reeve et al. (2004) state, it may not present students with the opportunity to experiment, explore and struggle with the learning content to find the truth for themselves. Games such as Math Gran Prix (Atari Inc., 1982), Math Blaster (Davidson & Associates, 1994), and Dr. Kawashima’s Brain Training (Nintendo SDD, 2005) align with the Drill & Practice learning. In these games, there is only one solution to a mathematical challenge, and players are prompted to input the correct one.

Complete Article List

2. Teaching and Learning

2.2 instructional strategies, direct instruction.

In general usage, the term direct instruction refers to (1) instructional approaches that are structured, sequenced, and led by teachers, and/or (2) the presentation of academic content to students by teachers, such as in a lecture or demonstration. In other words, teachers are “directing” the instructional process or instruction is being “directed” at students.

The basic techniques of direct instruction not only extend beyond lecturing, presenting, or demonstrating, but many are considered to be foundational to effective teaching. For example:

• Establishing learning objectives for lessons, activities, and projects, and then making sure that students understand the goals.
• Purposefully organizing and sequencing a series of lessons, projects, and assignments that move students toward understanding and the achievement of specific academic goals.
• Reviewing instructions for an activity or modeling a process—such as a scientific experiment—so that students know what they are expected to do.
• Providing students with clear explanations, descriptions, and illustrations of the knowledge and skills being taught.
• Asking questions to make sure of student understanding after a lesson.

As seen in Figure Two, teachers rarely use either direct instruction or some other teaching approach—in practice, diverse strategies are frequently blended together. For these reasons, negative perceptions of direct instruction likely result more from a widespread overreliance on the approach, and from the tendency to view it as an either/or option, rather than from its inherent value to the instructional process (Carnine, Silbert, Kameenui, & Tarver, 1997).

Drill and Practice

Megan Schreder talks to her fifth-grade classroom.

Lecture is a convenient instructional strategy. Material can be delivered efficiently since there are no interruptions from students. Lecture still allows the teacher to relate new material to other topics in the course, define and explain key terms, and relate material to students’ interests.

Lecture is an instructional strategy that places students in a passive role. Essentially the lecturer is the expert and the students are having knowledge poured into their brains. The material and presentation are solely the intellectual product of the teacher. Students sit silently at desks that face the lecturer.

Often lecture topics are not remembered well because retrieval pathways to memory have not been established by students actively participating in the instruction. Students have not taken the presented material and created their own interpreted meaning. The lecturer usually does not know if students understand the topic because there is no feedback from students (Lujan, H. & DiCarlo, S, 2006).

Question and Answer

Megan Schreder asks her students questions during a Q&A session with her class.

The technique of question and answer allows the application of knowledge by students and offers a more reflective response.  By asking questions, teachers are inviting brief responses from students, which incorporate their prior knowledge and some interpretation of that knowledge. This allows indications of whether students were listening and understand the material being presented.  Questions serve both to motivate students to listen and to assess how much and how well they know the material. Incorporating this instructional approach allows both the teacher to ask students questions and students to ask the teacher questions, fostering a better understanding of th e  lesson  ( Paul & Elder, 2007).   

In this instructional strategy, the role of the teacher shifts to leading an exchange of ideas about a specific topic. The teacher is no longer the sole provider of the content as students gain a voice for their ideas and the research they have conducted. At times, the teacher may assign students individual concepts that they have to speak about during the discussion. Some control of what course the discussion takes devolves to students. All of the content planned for the lesson might not be discussed. In fact, after reflecting on the day’s discussion a teacher might have to begin the next day’s discussion on important content that had been overlooked or squeezed out of the lesson.

Teachers need to develop strategies so that the voices of all students are heard. In addition, for effective class discussions students need to listen to what their classmates are saying so the points made during the dialogue allow students to make sense of the new ideas. As the discussion takes place, time should be taken for the teacher or better yet, a student to summarize the important points (Brookfield & Preskill, 2012).

Mental modeling

When a person perceives how something works in the real world and then formalizes that thought process a mental model is created. Mental modeling is a student-centered pedagogical strategy that helps students to solve problems or make decisions. For example, a mathematics teacher verbally modeling the thought process she is using while solving a problem in front of the class is using mental modeling. When teachers model the process of thinking or doing, the strategy of mental modeling becomes clearer to students. Students may then explain their own mental models to learn the strategy and improve their use of it.

Mental modeling often starts with a question, for example: why does lake effect snow occur? “What if” questions are also good starting points, for example: What if gravity ceased entirely? Strategies used by teachers and students engaged in mental modeling include observation, asking questions, as well as location and analysis of information. The level of cognitive load in mental modeling is high making it a strategy that should be employed often.

Teachers are encouraged to help students select the right mental model and help students select relevant information to develop their model. Teachers should create or find problems, case studies, lab activities, and projects at the appropriate grade level for their students. For students to have success they need to possess the appropriate background knowledge and supports to develop an accurate mental model. Often students encounter more success when they focus on the process instead of the outcome (Hestenes, D, 2010).

When students investigate to answer a question about a particular topic, they are using inquiry or inquiry-based learning. When teachers use inquiry-based learning, students or teachers may identify questions, however in any case questions posed should be open ended. Inquiry learning may be experienced individually; but it is beneficial when students work with other students. Differing perspectives and varied resources are important to inquiry-based projects.

Providing responses to questions such as “Why is the sky blue?” demands high-order thinking skills from both the student and the teacher. Allowing students to explore a broad topic, and to choose questions in which they are invested creates the best environment for successful inquiry-based projects. Students benefit from learning and negotiating through group investigation in order to answer a question.

Teachers who wish to engage in inquiry-based learning set the stage for this process in three ways:

• Assess students to determine their knowledge of the topic, and lay groundwork when that knowledge does not exist.
• Match the scope of the inquiry question to the learning level of students.
• Provide resources and/or provide internet search strategies for locating credible resources that will inform the inquiry.

The teacher’s role in inquiry-based learning is one of mentor and advisor. Students may struggle through problems; however, if the struggle occurs at a level that students may be successful, this struggle is worthwhile. The teacher’s most difficult role, in this case, is to resist answering questions that would inform the inquiry and therefore negate the process for the student!

Inquiry based learning requires time and patience; however this teaching strategy lays groundwork for real-world learning in which students will engage throughout their lives (Sharples, Collins, Feißt, Gaved, Mulholland, Paxton, & Wright, 2011).

Discovery Learning

“Discovery learning is a type of learning where learners construct their own knowledge by experimenting with a domain and inferring rules from the results of these experiments” (Van Joolingen, 2000, p.385).

In today’s educational realm, discovery learning is also called problem-based learning or experiential learning. Students participate through a hands-on approach and learning is interactive. Through discovery learning students are encouraged to explore with little guidance from the instructor. Discovery learning is based on the beliefs of Piaget (Ültanır, 2012), in which students are provided with a topic, and from that point students choose how they are going to learn, discover new information, synthesize the information and do so without correction from the teacher. The teacher does feed back to the student, as do the other members of the class, once the project is complete.

It is important that teachers create specific goals and guide students through discovery learning using pre-determined structures, for example, groupwork, fieldwork, or interaction with others. Unless this is the case, students may have too much freedom resulting in a lack of rigor within the method. However, Mayer (2004) states, “In many ways, guided discovery appears to offer the best method for promoting constructivist learning. The challenge of teaching by guided discovery is to know how much and what kind of guidance to provide and to know how to specify the desired outcome of learning.” (p.14)

In group work, students are assigned one or more partners to collaborate with on ideas in a strategy like think-pair-share or problem solving. Before students begin working, the teacher explains the objectives, expectations, and details of the activity or project. This explanation is meant to ensure all group members understand the goal of the group. As the group works together it is expected that all members teach and learn from each other. At the end of the group activity the teacher may debrief with groups or may provide a grade on a group artifact.

Students often need to be oriented on how to work effectively with their peers. Listening to group members’ ideas and not attaching self-worth to proposed ideas go a long way toward reaching the goals of the activity. Compromise is a skill that requires practice to be effective.  Alignment of group activities with the Social and Emotional Learning (SEL) Benchmarks (New York State, 2018) provides a well-defined way to identify and advance the skills students need to be effective group members.

When engaging students in groupwork, teachers should circulate to monitor the groups’ progress toward accomplishing the objectives of the lesson. Asking groups what they are discussing and why that is important to the topic assists in reinforcing the idea that the group activity is educational. As teachers see group behavior that is not on-task, the teacher should not hesitate to address this with the group. This reinforces to all groups that students are individually accountable for their behavior in the group. They are not “lost in a crowd”. (Blatchford, Kutnick, Baines, & Galton, 2003).

• Foundations of Education. Authored by : SUNY Oneonta Education Department. License : CC BY: Attribution
• Direct Instruction. Authored by : S Abbot (Ed.). Provided by : Great Schools Partnership. Located at : http://edglossary.org/direct-instruction/ . Project : The Glossary of Education Reform. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
• homework routine. Authored by : Woodley Wonderworks. Located at : https://www.flickr.com/photos/wwworks/4195916777/in/photostream/ . License : CC BY: Attribution
• Classroom Pictures. Authored by : Paul Mann. Provided by : Alaska Air National Guard. Located at : https://www.168wg.ang.af.mil/News/Photos/igphoto/2001727086/ . License : Public Domain: No Known Copyright

Privacy Policy

• Memberships

Drill Down technique

Drill Down technique: this article explains the Drill Down technique in a practical way. After reading this article, you’ll understand the basics of this powerful tool for problem solving .

What is the Drill Down technique?

The Drill Down technique is a method for gaining insight into the root causes of a problem within a department or area. After the root causes are known, a larger plan can be devised to address the problem.

A Drill Down is not the same as a diagnosis, but rather a broad and deep general examination. Furthermore, the technique is not used to find out the causes of all problems, but only the 20 percent of the causes behind 80 percent of the effects. This is a principle from the Pareto-analysis .

Method of the Drill Down technique

The technique starts with a table describing the main problem in the leftmost column. The factors and causes that create this problem are then described right next to it in the second column. The idea is to “drill through” until the real causes of the problem are identified. Solutions are then built based on these causes.

The idea here is that it is easier to deal with poor time management than poor quality customer service in general. In addition, some other causes of poor customer service are also discussed.

Step 1: Note down the most important problem

The aim of the first step is to take inventory of all core problems. Be specific in this, and do not generalise or use plurals such as “we” and “they”. Also mention the names of people who are affected by the problem. This is the only way to work on solutions effectively.

Have every individual connected in any way to the problem at hand participate in this Drill Down. You will benefit from this because they each bring their own insight to the brainstorming table. Don’t focus on a rare event or trivial problems. Don’t focus on the pursuit of unrealistic perfection, either.

Leave the search for solutions to the following steps. In the first step it is especially important that the problems are summarised.

Step 2: Identify the causes of the problems

In the second step, the more deeply rooted reasons causing the problems are identified. Often problems arise in different departments because it is not clear who is responsible, or because someone does not account for his or her responsibilities. Direct causes must be distinguished from underlying causes.

To find out the root cause of a problem, the Five Times Why method can be used, for example. Below is an example:

• Problem: The project team is working overtime too often and is in danger of burning out
• Why? There isn’t enough capacity to meet the team’s demands
• Why? Because new responsibilities have been added without extra resources
• Why? Because the manager did not correctly estimate the amount of work before taking responsibility
• Why? Because the manager is unable to anticipate problems and make plans

Relevant individuals should not be left out while performing this Drill Down technique. At the same time, remember that people tend to respond defensively to criticism. It is the manager’s job to find out the truth and to come up with a good solution. In practice, this can mean that people have to be trained, relocated, or even fired.

After this step, take a short break and then start developing a plan.

Step 3: Make a plan

The third step is to develop a plan that addresses the root causes of a problem. Such an implementation plan works like a script: everything that has to be done and by whom is visualised and recorded. Risk management also plays an important role in this. The likelihood of achieving goals is set against the costs and risks. The plan must consist of at least:

• Specific tasks and responsibilities
• Measuring variables

Step 4: Implement the plan

Execute the established implementation plan and be transparent in documenting progress. Report at least once a month on actual progress and expectations for the coming period.

Drill Down technique in combination with other methods

The Drill Down technique fits seamlessly with other forms and methods of problem solving. The closest method is the 5-Why analysis . Both methods aim to get to the heart of a problem instead of solving all sorts of other problems first.

Neither method provides a quick way to a solution, but that isn’t the solution that should be sought anyway. Instead, it makes much more sense to have a clear understanding of the situational aspects of doing business.

It is very important that everyone in a company is on the same page when it comes to using the Drill Down technique. The method will not be optimally effective within the company if only a small part of the team uses the method.

Take the time to teach everyone how to get to the root of a problem by zooming in with the Drill Down technique. As indicated, the Drill Down technique does not automatically solve problems, but when used properly it can certainly help to move forward.

Drill Down Technique: pitfalls in general problem solving

Problem solving is not achieved by simply employing methods and frameworks and following them blindly. It is a very broad discipline in which various effects occur that can hinder the way to the solution. In general problem solving and research, there are the following pitfalls to watch out for.

Confirmation bias

Confirmation bias is the tendency for people to seek or interpret information in a way that confirms a person’s previous knowledge, values or beliefs. It is an important type of bias that has a significant effect on the effective performance of problem-solving methods such as the Drill Down technique.

People show this bias when they collect or remember information and interpret it in a biased way. For example, a team member may choose information while preparing for a new task that supports their beliefs and ignore what is not supportive. This effect is strongest when people envision desired outcomes, when a problem is emotionally charged, and for deeply held beliefs.

Perceptual expectations

A perceptual expectation in psychology is also called a set. A set is a group of expectations that shape a specific experience by making people sensitive to certain types of information. It is the disposition or habit to perceive things in a certain way.

This was demonstrated in an experiment by Abraham Luchins in the 1940s. In this experiment, participants were asked to fill a pitcher with a specific amount of water with the aid of only three other pitchers of different capacities.

After Luchins gave the participants this problem that could be solved by a simple technique, he gave them new assignments for other pitchers. This new problem could be solved by the same method, or by a newer and simpler method.

Luchins found that many of his participants tended to use the same old technique, despite the possibility for a better method. Thus, the mental set describes a person’s tendency to solve problems in a way that has previously proven successful.

As in Luchins’ experiment, choosing a method that has worked in the past is sometimes no longer sufficient or optimal for the new problem. Therefore, it is necessary for people to transcend their mental set.

Functional fixation

Functional fixation is a cognitive bias that limits a person to using or accessing an object only as it is traditionally used. This fixation also occurs when solving a problem through the Drill Down Technique. The concept of functional fixation stems from the Gestalt psychological movement.

This movement emphasizes holistic processing. Karl Duncker defined functional fixation as a mental block against using an object in a new way that is necessary to solve a problem. This block limits an individual’s ability to complete a task or solve a problem, as it does not look beyond the original purpose of the components of the solution.

Functional fixation is the inability to see, for example, the use of a hammer as anything different than for hitting nails.

Unnecessary limitations

Unnecessary limitations- or constraints, is a barrier that occurs when people subconsciously set limits on the task at hand. A well-known example of this is the point problem. In this assignment nine points are arranged in a square of three by three.

The task is to draw no more than four lines, without removing the pen or pencil from the paper, to connect all the dots. In the minds of the people who have never seen this problem before, the thought probably arises that the line does not come out of the square of the points. Unnecessary restrictions in this case are about literally thinking ‘outside the box’.

The term group mindset is also linked to unnecessary restrictions. Group thinking, or adopting the mentality of the group members, occurs when team members start to think the same. This is common, but also ensures that people take longer to start thinking “outside the box”.

Irrelevant information

Irrelevant information is information presented within the context of a problem but unrelated to the specific problem. Within the context of the problem, irrelevant information has no influence on whether or not the problem is solved. In fact, irrelevant information is often detrimental to the problem-solving process. Irrelevant information is a common problem that people struggle with. This is mainly because people are not aware of the existence of irrelevant information.

One of the reasons that irrelevant information is so effective in keeping people from the solution is how it is presented. The way information is presented can make a big difference for the level of interpreted difficulty of the problem. Below is a well-known example of irrelevant information in the Buddhist monk problem.

A monk starts walking up a mountain at sunrise one day and reaches the temple at the top of the mountain at sunset. After a few days of meditation, he leaves at sunrise to descend from the mountain. He arrives at sunset. There is a spot along the path the monk takes both ways where he will pass at the same time of the day.

Now it is your turn

What do you think? Do you recognise yourself in the explanation of the Drill Down method? Is this tool used in your own working environment? If not, do you think this could be valuable in your work? What other helpful troubleshooting methods and tools do you know? What do you believe are pros and cons of the Drill Down technique? Do you have any tips or solutions?

Share your experience and knowledge in the comments box below.

More information

• de Aguiar Ciferri, C. D., Ciferri, R. R., Forlani, D. T., Traina, A. J. M., & da Fonseca de Souza, F. (2007, March). Horizontal fragmentation as a technique to improve the performance of drill-down and roll-up queries . In Proceedings of the 2007 ACM symposium on Applied computing (pp. 494-499).
• Joglekar, M., Garcia-Molina, H., & Parameswaran, A. (2017). Interactive data exploration with smart drill-down . IEEE Transactions on Knowledge and Data Engineering, 31(1), 46-60.
• McDonald, A., & Leyhane, T. (2005). Drill down with root cause analysis . Nursing management, 36(10), 26-31.

How to cite this article: Janse, B. (2020). Drill Down Technique . Retrieved [insert date] from toolshero: https://www.toolshero.com/problem-solving/drill-down-technique/

Add a link to this page on your website: <a href=”https://www.toolshero.com/problem-solving/drill-down-technique/”>toolshero: Drill Down Technique</a>

Published on: 22/11/2020 | Last update: 04/03/2022

Did you find this article interesting?

Your rating is more than welcome or share this article via Social media!

Average rating 4.5 / 5. Vote count: 13

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

We are sorry that this post was not useful for you!

Let us improve this post!

Tell us how we can improve this post?

Ben Janse is a young professional working at ToolsHero as Content Manager. He is also an International Business student at Rotterdam Business School where he focusses on analyzing and developing management models. Thanks to his theoretical and practical knowledge, he knows how to distinguish main- and side issues and to make the essence of each article clearly visible.

Related ARTICLES

Convergent Thinking: the Definition and Theory

CATWOE Analysis: theory and example

Means End Analysis: the basics and example

Systems Thinking: Theory and Definition

5 Whys Root Cause Analysis (Toyoda)

Positive Deviance (PD) explained

Also interesting.

Crowdsourcing: the meaning, definition and some examples

Systematic Inventive Thinking (SIT)

Linear Thinking by Edward De Bono explained

Leave a reply cancel reply.

You must be logged in to post a comment.

BOOST YOUR SKILLS

Toolshero supports people worldwide ( 10+ million visitors from 100+ countries ) to empower themselves through an easily accessible and high-quality learning platform for personal and professional development.

By making access to scientific knowledge simple and affordable, self-development becomes attainable for everyone, including you! Join our learning platform and boost your skills with Toolshero.

POPULAR TOPICS

• Change Management
• Marketing Theories
• Problem Solving Theories
• Psychology Theories

ABOUT TOOLSHERO

• Free Toolshero e-book
• Memberships & Pricing

Effective Technology Integration for Disabled Children pp 25–33 Cite as

Drill-and-Practice Programs

• Malka Margalit Ph.D. 2  

122 Accesses

Computer-based drill-and-practice is designed to provide immediate corrective instruction, and to reinforce previously learned information, thus developing fluency or automaticity in the skill. In order to advance knowledge within the mastery-learning paradigm that assumes that most students can learn most things to a specific level of competence in varying amounts of time (Ackerman, 1987), the nature of the computer makes it irrefutably the ideal means for providing endless practice in a needed curricular area, until reaching the desired level of fluent performance.

This is a preview of subscription content, log in via an institution .

Buying options

• Available as PDF
• Read on any device
• Instant download
• Own it forever
• Available as EPUB and PDF
• Compact, lightweight edition
• Dispatched in 3 to 5 business days
• Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Unable to display preview.  Download preview PDF.

Author information

Authors and affiliations.

School of Education, Tel-Aviv University, 69978, Ramat-Aviv, Israel

Malka Margalit Ph.D.

You can also search for this author in PubMed   Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1990 Springer-Verlag New York Inc.

About this chapter

Cite this chapter.

Margalit, M. (1990). Drill-and-Practice Programs. In: Effective Technology Integration for Disabled Children. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9006-0_3

Download citation

DOI : https://doi.org/10.1007/978-1-4613-9006-0_3

Publisher Name : Springer, New York, NY

Print ISBN : 978-1-4613-9008-4

Online ISBN : 978-1-4613-9006-0

eBook Packages : Springer Book Archive

Share this chapter

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Publish with us

Policies and ethics

• Find a journal
• Track your research

Should U.S. Students Do More Math Practice and Drilling?

Brief, fast-paced practice with corrective yet supportive feedback should help..

Posted August 18, 2018

Should U.S. students be doing more math practice and drilling in their classrooms? That’s the suggestion from last week’s most emailed New York Times op-ed . The op-ed’s author argued that more practice and drilling could help narrow math achievement gaps. These gaps occur in the U.S. by the primary grades .

Yet others worry that more math practice and drilling will stifle creativity . They argue that routine practice and drill interferes with understanding underlying mathematical concepts. Instead, young students should acquire a good understanding of mathematics concepts before being given practice opportunities to become procedurally fluent. The suggested way to do so is for teachers to use instructional approaches in which mathematical concepts and strategies are explored and constructed through student-centered, discovery activities. Whether U.S. students learn math better through traditional teacher-directed activities emphasizing practice or, instead, through student-centered activities emphasizing reasoning and discovery has been debated for decades .

Yet increasingly there are good reasons to believe that more math practice and drilling would help U.S. students do better in math, particularly those who are already struggling in elementary school.

Why might this be so? Routine practice and drilling following explicit teacher-directed instruction should help students become quick and accurate in solving basic operations, thereby becoming procedurally fluent. Becoming procedurally fluent in turn should help students by freeing up their cognitive capacities to solve more complex tasks. For example, students who practice to quickly and accurately recognize the meaning of the equal sign do better at solving word problems. Practice may be particularly important for elementary school students who are struggling in math. This is because these students often have underlying difficulties in attention , working memory , and language that interfere with their learning during less structured student-centered activities.

The idea that procedural fluency acquired through routine practice and drilling somehow limits conceptual understanding of math is a “myth” according to a recent research review. Instead, routine practice and drilling—especially when coupled with corrective feedback and ambitious but attainable goal-setting —should help students learn better. Such distributed practice is “ necessary if not sufficient for acquiring expertise .” Procedural fluency and conceptual understanding influence each other bidirectionally over time. And giving students opportunities to practice is viewed as a key element of effective math instruction .

Students provided with explicit instruction by teachers who provide frequent practice opportunities show achievement gains that are similar in size to the gains needed to narrow achievement gaps. And recent work suggests that routine practice and drilling in math might be an especially important instructional practice for elementary school teachers to use.

For example, my colleagues and I analyzed a sample of over 13,000 U.S. students to examine what types of instructional practices predicted greater math achievement by the end of first grade. The achievement measures were independently administered while first grade teachers self-reported how frequently they used various types of instructional approaches.

We were able to control for many factors that might otherwise explain any observed relations between the types of instructional approaches used by first grade teachers and the math achievement of their students. These factors included the math and reading achievement of the students at the end of kindergarten as well as characteristics of their families, classrooms, and schools.

What did we find? Across four types of instructional approaches, we found that only teacher-directed instruction consistently predicted greater math achievement in first grade. Teacher-directed instruction predicted greater achievement by students who had struggled in math during kindergarten and by students who had not. Less traditional types of instruction, like using movement activities or music to teach math, did not predict greater achievement.

Yet first grade teachers were more likely to use these ineffective instructional approaches when teaching classrooms with greater shares of struggling students. Student-centered instruction predicted greater achievement but, importantly, only by students who had not previously struggled in mathematics. And the most effective teacher-directed instructional practice that we examined? Routine practice and drill.

So, should U.S. students do more math practice and drilling in their classrooms? At least to an extent, and particularly for elementary school students who are already struggling. Doing so in ways that are brief, fast-paced, and that provide corrective yet supportive feedback should result in greater procedural fluency and, over time, the conceptual understanding and higher-order thinking skills that we want all students to have.

Paul L. Morgan, Ph.D., is the Empire Innovation Professor and Social and Health Equity Endowed Professor in the School of Public Health's Department of Health Policy, Management and Behavior at the University at Albany, SUNY, where he is the Director of the Institute for Social and Health Equity.

• Find a Therapist
• Find a Treatment Center
• Find a Psychiatrist
• Find a Support Group
• Find Teletherapy
• United States
• Brooklyn, NY
• Chicago, IL
• Houston, TX
• Los Angeles, CA
• New York, NY
• Portland, OR
• San Diego, CA
• San Francisco, CA
• Seattle, WA
• Washington, DC
• Asperger's
• Bipolar Disorder
• Chronic Pain
• Eating Disorders
• Passive Aggression
• Personality
• Goal Setting
• Positive Psychology
• Stopping Smoking
• Low Sexual Desire
• Relationships
• Child Development
• Therapy Center NEW
• Diagnosis Dictionary
• Types of Therapy

Understanding what emotional intelligence looks like and the steps needed to improve it could light a path to a more emotionally adept world.

• Coronavirus Disease 2019
• Affective Forecasting
• Neuroscience

How Drill And Practice Teaching Method Helps In Learning Math?

Last Updated on October 3, 2023 by Editorial Team

Prepare to embark on an adventure filled with numerical wonders and problem-solving prowess. Imagine stepping into a vibrant math circus, where students become mathematical acrobats, defying the limits of their skills and soaring to new heights of knowledge.

In this realm, the drill and practice method emerges as a magical tool, helping students master mathematical concepts. With each repetition, numbers come alive, formulas dance, and equations find their harmonious balance. It’s a symphony of drills, where practice becomes the key to unlocking mathematical brilliance.

But wait, this method is more than just a routine—it’s a catalyst for growth. It transforms math into an exhilarating adventure, where students build fluency, sharpen problem-solving skills, and unlock the secrets of numerical wizardry. Through targeted practice, mathematical doors swing open, revealing the beauty and logic that lie within.

So, join us under the big top of mathematics, where the drill and practice teaching method reigns supreme. Let’s embark on this extraordinary journey, where numbers become our allies, and mathematical mastery becomes a thrilling feat. Get ready to unleash the magic of drill and practice, as we soar to mathematical greatness, one equation at a time!

Drill and practice teaching method: Navigating through the meaning

The drill and practice teaching method is an instructional approach that focuses on the repetitive practice of specific skills or knowledge. It involves presenting students with targeted exercises, problems, or tasks that require them to repeatedly apply and reinforce what they have learned. This method aims to strengthen and solidify understanding, enhance retention, and promote automaticity in the mastery of concepts.

The purpose of drill and practice is to provide students with ample opportunities to practice and internalize essential skills or knowledge. By engaging in repetitive exercises, students develop fluency, accuracy, and efficiency in applying the learned content. This effectiveness of drill and practice has proven successful in building foundational skills, such as basic math operations, vocabulary acquisition, spelling, grammar rules, and procedural knowledge.

“Unlocking the secrets of math: The power of an effective teaching method”

The drill and practice teaching method finds extensive use in mathematics education due to its ability to reinforce foundational skills and promote mastery of mathematical concepts. The main goal of this method is to help students develop proficiency and fluency with a particular skill or concept. Here are ten ways in which the drill and practice teaching method can help students in math:

1. Improves mastery and fluency:

Repetitive practice can help students develop automaticity and fluency with math skills and concepts, allowing them to apply them more quickly and accurately in real-world situations. When students have to think less about how to perform a math operation and can just do it automatically, they are more likely to make fewer mistakes and solve problems more efficiently.

2. Provides immediate feedback:

With the drill and practice method, students can see their progress as they work through practice problems and quizzes, which can help them identify areas where they need more practice.

3. Allows for individualized instruction:

As students become more proficient and fluent in math skills and concepts, they may feel more confident in their ability to solve problems and tackle new challenges. This can help them approach math with a positive attitude and feel more motivated to engage with the material.

4. Builds confidence:

Drill and practice activities can help students develop a strong understanding of number sense, including basic arithmetic operations (addition, subtraction, multiplication, and division). Regular practice with number facts and calculations improves fluency and efficiency.

5. Encourages self-directed learning:

Drill and practice exercises are effective for memorizing essential math facts, such as multiplication tables, division facts, and number patterns. Repetition aids in automatic recall, allowing students to solve problems more quickly and accurately.

6. Promotes efficient problem-solving:

Drill and practice can reinforce problem-solving strategies, such as using algorithms, applying formulas, or using logical reasoning. Regular practice helps students become more proficient in selecting appropriate strategies and applying them effectively.

7. Facilitates retention:

Drill and practice activities encourage mental math skills by challenging students to perform calculations mentally and make quick estimations. Regular practice enhances mental agility and computational fluency.

8. Enhances problem-solving skills:

Drill and practice provide students with opportunities to practice mathematical procedures, such as long division, fraction operations, decimal conversions, and geometric formulas. Repetitive practice helps students internalize these procedures and become more proficient in their application.

9. Increases motivation:

By engaging in drill and practice exercises, students can improve their accuracy in mathematical calculations, reducing errors and promoting precision in their work.

10. Can be used in a variety of settings:

Drill and practice activities can reinforce mathematical vocabulary and terminology, ensuring students have a solid understanding of mathematical language and can effectively communicate their ideas. 

What does the research state?

Research on drill and practice as a teaching method has generally shown that it can be effective for improving performance on specific skills or concepts. Furthermore, research [ 1 ] has shown that drills and practice can be effective ways to teach basic math facts, spelling words, and other skills that require automaticity or quick recall.

However, the effectiveness of drill and practice may depend on the specific learning goals and objectives, as well as the needs and characteristics of the students. In some cases, drill and practice may be less effective for more complex skills or for students who are struggling to learn new material.

Research [ 2 ] has also shown that drill methods along with other methods such as concept-based learning, prove effective in enhancing students’ comprehension of concepts and fostering high-level thinking, promoting a deeper understanding of the subject matter, and facilitating critical analysis and synthesis of information. Thus, the drill and practice method is a great method when used with other approaches to learning maths. 

The drill and practice teaching method can be helpful for improving math skills because it allows students to repeatedly practice specific math concepts or techniques. This can help students to become more proficient and accurate in their math skills, as they are able to develop muscle memory and automaticity through repetition. 

Additionally, the drill and practice method can be useful for reinforcing previously learned material, helping students retain important math concepts over time. At the same time, educators and parents can go through certain examples , that can explain the idea better how to employ this teaching method.

• Lehtinen, Erno & Hannula-Sormunen, Minna & McMullen, Jake & Gruber, Hans. (2017). Cultivating mathematical skills: from drill-and-practice to deliberate practice. ZDM. 49. 10.1007/s11858-017-0856-6. 
• Lufri, Fitri, R., & Yogica, R. (2018). Effectiveness of concept-based learning model, drawing and drill methods to improve student’s ability to understand concepts and high-level thinking in animal development course. Journal of Physics: Conference Series , 1116 , 052040. https://doi.org/10.1088/1742-6596/1116/5/052040

An engineer, Maths expert, Online Tutor and animal rights activist. In more than 5+ years of my online teaching experience, I closely worked with many students struggling with dyscalculia and dyslexia. With the years passing, I learned that not much effort being put into the awareness of this learning disorder. Students with dyscalculia often misunderstood for having  just a simple math fear. This is still an underresearched and understudied subject. I am also the founder of  Smartynote -‘The notepad app for dyslexia’, 

Leave a Comment Cancel reply

You must be logged in to post a comment.

• The LD@school Team
• Collaborate with Us
• Terms Of Use
• Executive Functions
• Mathematics
• Mental Health
• Social-Emotional Development
• What are Non-verbal Learning Disabilities?
• Accommodations, Modifications & Alternative Skill Areas
• Glossary of Terms
• Resources |
• Learning Modules |
• Educators’ Institute |
• Document Library |

LDs in Mathematics: Evidence-Based Interventions, Strategies, and Resources

Print Resource

By Hanna A. Kubas and James B. Hale

Mathematics. Some love it, some loathe it, but there are many myths about math achievement and math learning disabilities (LDs). The old belief – boys are naturally better at math than girls – may be more a consequence of teacher differences or societal expectations than individual differences in math skill (Lindberg, Hyde, Petersen, & Linn, 2010).

Similarly, the old belief that reading is a left brain task, and math is a right brain task, is not a useful dichotomy as clearly multiple shared and distinct brain regions explain these academic domains (e.g., Ashkenazi, Black, Abrams, Hoeft, & Menon, 2013).

Math is a language with symbols that represent quantity facts instead of language facts (i.e., vocabulary), so rules (syntax) are important for both (Maruyama, Pallier, Jobert, Sigman, & Dehaene, 2012). You might be surprised to learn that approximately 7% of school-aged children have a LD in mathematics (Geary, Hoard, Nugent, & Bailey 2012).

Let’s first explore the fundamental skills needed for math achievement.

Number Sense / Numerical Knowledge

Children develop knowledge of quantity even before math instruction in schools, and kindergarten number sense is predictive of math computation and problem solving skills in elementary school (Jordan et al., 2010). These basic math skills include understanding of number magnitudes, relations, and operations (e.g., adding). Children link basic number sense to symbolic representations of quantity (numbers); the math “language”. Poor early number sense predicts math LDs in later grades (Mazzocco & Thompson, 2005).

Math Computation vs. Math Fluency

Children often rely on various strategies when solving simple calculation problems, but math computation requires caring out a sequence of steps on paper or in your mind (working memory) to arrive at an answer. Math fluency refers to how quickly and accurately students can answer simple math problems without having to compute an answer (i.e., from memory 6 x 6 = 36), with no “steps”, calculation, or number sense needed.

Children with fluency deficits often use immature counting strategies and often do not shift from computation to storing and retrieving math facts from memory, taking more time to provide an answer. Difficulty with retrieval of math facts is a weakness/deficit associated with math LDs (Geary et al., 2007; Gersten, Jordan, & Flojo, 2005). Without math fact automaticity, working memory may be taxed when doing computation, and the child “loses his place” in the problem while computing each part to arrive at a final answer.

Developmental Sequence of Math Skills

1. Finger Counting Strategies : Students first display both addends/numbers with their fingers; this is the most immature strategy.

2. Verbal Counting Strategies: Next, students begin to develop basic adding skills and typically go through three phases.

• Sum:  counting both addends/numbers starting from 1, this is a beginning math counting skill;
• Max: counting from the smaller number; and finally
• Min: counting from the larger number (most efficient strategy).

3. Decomposition (Splitting) Strategies: Students learn that a whole can be decomposed into parts in different ways, a good problem solving strategy for unknown math facts

4. Automatic Retrieval from Long-Term Memory: Students become faster and more efficient at pairing problems they see with correct answers stored in long-term memory (as is the case with sight word reading), no computation is required

The Role of Visual-Spatial Skills

Basic arithmetic skills are factual, detailed “left hemisphere” functions (similar to basic reading), but Byron Rourke (2001) discovered many students with nonverbal or “right hemisphere” LDs had math calculation problems, suggesting left was verbal and right nonverbal.

Students need “right hemisphere” visual/spatial skills to align numbers when setting up multistep math problems , they need to need to be able to understand and spatially represent relationships and magnitude between numbers, and they need to be able to interpret spatially represented information (Geary, 2013).

Neuropsychology has also taught us that children with visual/spatial problems may neglect the left side of stimuli (the left visual field is contralateral to the right hemisphere) (Hale & Fiorello, 2004; Rourke, 2000).

Math Reasoning and Problem-Solving

Word problems require both receptive and expressive language skills , unlike simple calculation, so students with language-based LDs may struggle even if math skills are good. Students must translate math problem sentences/words into numbers and equations , so they must identify what the sentences are asking them to do in terms of calculation, and then perform the calculation

Students with LDs are typically poor strategic learners and problem solvers , and often manifest strategy deficits that hinder performance, particularly on tasks that require higher level processing (Montague, 2008). So there is a strong relationship between fluid reasoning, executive functioning, and quantitative reasoning (Hale et al., 2008). Students with LDs often benefit from explicit instruction in selecting, applying, monitoring, evaluating use of appropriate strategies to solve word problems.

The Brain, Math, and LDs

Click here to access a printable PDF version of LD@school's diagram of brain areas and math skills .

Strategies for Promoting Math Computation and Fluency

Note :  Your understanding of foundational mathematical concepts and skills is critical for targeted interventions that are developed, implemented, monitored, evaluated, and modified until treatment efficacy is obtained!

Remember: early identification and intervention are key!

Click here to access a printable PDF version of the Strategies for Promoting Math Computation and Fluency explained below .

Strategic Number Counting

Fuchs et al. 2009

Goal : Improve counting strategies (e.g., MIN; decomposition) to efficiently pair problem stems and answers

Skills Targeted:  Explicitly teach math counting strategies when number sense or algorithm adherence is limited

Target Age Group:  Elementary students struggling with basic computation and quantity-number association

Description:

• Direct instruction of efficient counting (g., MIN for addition), followed by guided practice.
• For two-number addition, students start with larger number and count for smaller number/addend.
• For two-number subtraction, students start at ‘minus number’ and count up to ‘starting number,’ tallying numbers
• Flashcards used to math fact encoding, storage, and/or retrieval deficits. Optional number line can enhance method.

E mpirical Support:

• Fuchs et al. (2009) found strategic counting led to better math fact fluency compared to control groups, even better if combined with intensive drill and practice
• Strategic counting with and without deliberate practice better math fluency, with deliberate practice better than controls (Fuchs et al., 2010)

Additional Resources:

• Click here to access a number line and a strategic number counting instruction score sheet available on the Intervention Central .

Drill and Practice

Fuchs et al. 2008

Goal:  Drill and practice interventions help children quickly and accurately recall simple math facts

Skills Targeted:  Practice and repetition of math fact calculations

Target Age Group : Students struggling with basic math facts, especially with limited automaticity

• May be paper-and-pencil and/or computerized drill and practice in either a game or drill format, typically includes modeling, practice, frequent administration, and brief, timed practice, self-management, and reinforcement
• Drill and practice with math problem solving strategies may be more effective
• Software to ensure correct student response; math facts appear for 1-3 seconds, and students reproduce the whole equation and answer from short-term memory
• Students visually encode both the number question and answer for long-term memory storage
• Connection between math fact rehearsal and increased fact retention and generalization (Burns, 2005; Codding et al., 2010; Duhon et al., 2012)
• Promotes efficient paring of problems and the correct answers (Fuchs et al., 2008)
• Computer versions improve math fact retrieval fluency (Burns et al., 2010; Slavin & Lake, 2008)
• Click here to access a website for free math fact flashcards .
• Click here to access free math computation worksheets and answer keys for addition, subtraction, multiplication, and division .

Cover-Copy-Compare

Skinner et al. 1997

Goal:  Improve accuracy and speed in basic math facts

Skills Targeted:  Students taught self-management through modeling, guided practice, and corrective feedback

Target Age Group : Students learning basic math facts, those with executive, sequential, or integration problems

• Students learn 5-step strategy to solve simple math equations and self-evaluating correct responses
• Students look at math problem, cover it, copy it, and evaluate response to compare to original
• For errors, brief error correction procedure undertaken before next item introduced
• Strategy requires little teaching time or student training
• CCC procedures enhance math accuracy and fluency across general education (Codding et al., 2009; Grafman & Cates, 2010) and special education (Poncy et al., 2007; Skinner et al., 1997)
• Meta-analysis of many studies shows CCC improves math performance, especially when coupled with other evidence-based methods (e.g., token economies, goal setting, correct digits, increased response opportunity; Joseph et al., 2012)
• Click here to access a CCC intervention description including worksheet and performance log at Intervention Central .

Detect-Practice-Repair

Poncy, Skinner & O’Mara, 2006

Goal:  Promote efficient basic math fact practice targeting problems not completed accurately and/or fluently

Skills Targeted:  Encoding and retrieval of math facts from long-term memory

Target Age Group : Students developing basic math facts, may be useful for executive memory difficulties

• DPR is a 3-phase test-teach-test procedure for individualizing math fact instruction for basic fact groups (e.g., addition)
• (1)  Detect  phase - metronome determined rate to determine automatic (< 2 seconds) vs. slow (>2 second) math fact responding
• (2)  Practice  phase using Cover-Copy-Compare (CCC; see description above)
• (3)  Repair  phase using 1-minute math sprint with items requiring practice embedded in automatic ones
• DPR validated across grades, skills, and research designs (Poncy et al., 2013)
• Improves subtraction, multiplication, and division fluency (Axtell et al., 2009; Poncy et al., 2006; 2010; Parkhurst et al., 2010)
• Differentiation possible because DPR targets specific difficulties (Poncy et al., 2013)

Reciprocal Peer Tutoring

Fuchs et al., 2008

Goal:  Peer tutoring procedure includes explicit timing, immediate response feedback, and overcorrection

Skills Targeted : Basic math fact retrieval and automaticity through constant engagement in dyads

Target Age Group : All students, but especially useful for students with poor attention or persistence

• Students are paired up and take turns serving as the “tutor”
• Flashcards with problem on one side (e.g., 2 x 3 = ___) and answer on other side (e.g., 6)
• Student tutors shows flashcard, tutee responds verbally
• Tutor states either “correct,” (and puts in correct stack) or “incorrect” (and puts in incorrect stack)
• If incorrect, tutee writes problem and correct answer 3 times on paper
• Roles change after 2 minutes; then students complete 1 minute math probes and grade each other
• Multicomponent approach + other evidence-based efforts improve math fact rates (Rhymer et al., 2000)
• Improves math achievement, engagement, and prosocial interactions (Rohrbeck et al., 2003)
• Improves achievement, self-concept, and attitudes (Bowman-Perrot et al., 2013; Tsuei, 2012)

Strategies for Promoting Math Problem-Solving

Click here to access a printable PDF version of the Strategies for Promoting Math Problem-Solving explained below .

Schema Theory Instruction

Jitendra et al. 2002

Goal : Teaches mathematical problem structures, strategies to solve, and transfer to solve novel problems

Skills Targeted:  Expanding student math problem solving schemas

Target Age Group:  Students in any grade learning math problem solving skills, helps conceptual “gestalt”

• Encourages math problem solving schemas for word problems, identifying new, unfamiliar, or unnecessary information, and grouping novel problem features into broad schema for strategy use
• Explicit instruction in recognizing, understanding, and solving problems based on mathematical structures; can be used with schema-broadening instruction for generalization (e.g., Fuchs et al., 2008)
• Randomized controlled trials show improved math word problem solving (Fuchs et al., 2008, 2009)
• Schema-based approach generalizes into better math word problem solving (Jitendra et al., 2002; 2007; Xin, Jitendra, & Deatline-Buchman, 2005)

Mercer & Miller, 1992

Goal : Self-regulated strategy instruction method for increasing math problem solving skills

Skills Targeted:  Targets self-teaching, self-monitoring, and self-support strategies for identifying salient math words in sentences, determining and completing operation, and checking accuracy

Target Age Group:  Students struggling with executive monitoring and evaluation skills

• Teaches 8-step math word problems strategy and self-regulation
• The mnemonic FAST DRAW cues students, can use as checklist
• Increases math achievement and improves math attitude (Tok & Keskin, 2012)
• Increases math achievement in math LD (Miller & Mercer, 1997; Cassel & Reid, 1996)

Click here to access LD@school’s template for the FAST DRAW mnemonic .

Cognitive Strategy Instruction

Montague & Dietz, 2009

Goal : Teach multiple cognitive strategies to enhance math problem solving skills

Skills Targeted:  Focuses on cognitive processes, including executive functions (self-regulation/metacognition)   

Target Age Group:  Useful for differentiating instruction based on processing weaknesses

• Teaches 7-step cognitive strategy for solving math word problems, with 3-step metacognitive self-coaching routine for each step
• Direct instruction includes structured lesson plans, cognitive modeling, guided practice cues and prompts, distributed practice, frequent teacher-student interaction, immediate corrective feedback, positive reinforcement, overlearning, and mastery
• Read  the problem for understanding
• Paraphrase  the problem in your own words
• Visualize  a picture or a diagram to accompany the written problem
• Hypothesize  a plan to solve the problem
• Estimate /predict the answer
• Compute  the answer
• Check  your answer to make sure everything is right  Say, Ask, Check  metacognitive routine in each of the 7-step cognitive processes
• Say  requires self-talk to identify and direct self when solving problem
• Ask  requires self-questioning, promoting self-talk internal dialogue
• Check  requires self-monitoring strategy for checking understanding and accuracy
• Self-regulation strategies foster math problem solving in meta-analyses (Kroesbergen & van Luit, 2003)
• Cognitive strategy instruction increases math problem solving skills in general education (Mercer & Miller, 1992, Montague et al. 2011) and ADHD and LD (Iseman & Naglieri, 2011)
• Click here to access the Say-Ask-Check handout for student self-coaching .

Related Resources on the LD@school Website

Click here to access the article Math Heuristics .

Click here to access the article Helping Students with LDs Learn to Diagram Math Problems .

Click here to access the answer to the question: There is a lot of information about identifying learning disabilities in mathematics. However, information about strategies and ideas for working with these disabilities is limited. What strategies work? .

Click here to access the video Using Collaborative Teacher Inquiry to Support Students with LDs in Math .

Click here to access the recording of the webinar Understanding Developmental Dyscalculia: A Math Learning Disability .

Ashkenazi, S., Black, J. M., Abrams, D. A., Hoeft, F., & Menon, V. (2013). Neurobiological underpinnings of math and reading learning disabilities.  Journal of learning disabilities ,  46 (6), 549-569.

Axtell, P. K., McCallum, R. S., Mee Bell, S., & Poncy, B. (2009). Developing math automaticity using a classwide fluency building procedure for middle school students: A preliminary study.  Psychology in the Schools ,  46 (6), 526-538.

Bowman-Perrott, L., Davis, H., Vannest, K., Williams, L., Greenwood, C., & Parker, R. (2013). Academic benefits of peer tutoring: A meta-analytic review of single-case research.  School Psychology Review ,  42 (1), 39-55.

Burns, M. K. (2005). Using incremental rehearsal to increase fluency of single-digit multiplication facts with children identified as learning disabled in mathematics computation.  Education and Treatment of Children , 237-249.

Cassel, J., & Reid, R. (1996). Use of a self-regulated strategy intervention to improve word problem-solving skills of students with mild disabilities.  Journal of Behavioral Education ,  6 (2), 153-172.

Codding, R. S., Archer, J., & Connell, J. (2010). A systematic replication and extension of using incremental rehearsal to improve multiplication skills: An investigation of generalization.  Journal of Behavioral Education ,  19 (1), 93-105.

Codding, R. S., Chan-Iannetta, L., Palmer, M., & Lukito, G. (2009). Examining a classwide application of cover-copy-compare with and without goal setting to enhance mathematics fluency.  School Psychology Quarterly ,  24 (3), 173.

Duhon, G. J., House, S. H., & Stinnett, T. A. (2012). Evaluating the generalization of math fact fluency gains across paper and computer performance modalities.  Journal of school psychology ,  50 (3), 335-345.

Fuchs, L. S., & Fuchs, D. (2006). A framework for building capacity for responsiveness to intervention. School Psychology Review ,  35 (4), 621.

Fuchs, L. S., Powell, S. R., Seethaler, P. M., Cirino, P. T., Fletcher, J. M., Fuchs, D., & Hamlett, C. L. (2010). The effects of strategic counting instruction, with and without deliberate practice, on number combination skill among students with mathematics difficulties.  Learning and individual differences , 20 (2), 89-100.

Fuchs, L. S., Powell, S. R., Seethaler, P. M., Cirino, P. T., Fletcher, J. M., Fuchs, D., ... & Zumeta, R. O. (2009). Remediating number combination and word problem deficits among students with mathematics difficulties: A randomized control trial.  Journal of Educational Psychology ,  101 (3), 561.

Fuchs, L. S., Seethaler, P. M., Powell, S. R., Fuchs, D., Hamlett, C. L., & Fletcher, J. M. (2008). Effects of preventative tutoring on the mathematical problem solving of third-grade students with math and reading difficulties.  Exceptional Children ,  74 (2), 155-173.

Geary, D. C. (2013). Early foundations for mathematics learning and their relations to learning disabilities. Current Directions in Psychological Science ,  22 (1), 23-27.

Geary, D. C., Hoard, M. K., Byrd‐Craven, J., Nugent, L., & Numtee, C. (2007). Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability.  Child development , 78 (4), 1343-1359.

Geary, D. C., Hoard, M. K., Nugent, L., & Bailey, D. H. (2012). Mathematical cognition deficits in children with learning disabilities and persistent low achievement: A five-year prospective study.  Journal of Educational Psychology ,  104 (1), 206-217.

Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties.  Journal of learning disabilities ,  38 (4), 293-304.

Grafman, J. M., & Cates, G. L. (2010). The differential effects of two self‐managed math instruction procedures: Cover, Copy, and Compare versus Copy, Cover, and Compare.  Psychology in the Schools , 47 (2), 153-165.

Hale, J. B., & Fiorello, C. A. (2004).  School neuropsychology: A practitioner’s handbook.  New York, NY: Guilford Press.

Hale, J. B., Fiorello, C. A., Dumont, R., Willis, J. O., Rackley, C., & Elliott, C. (2008). Differential Ability Scales–Second Edition (neuro)psychological Predictors of Math Performance for Typical Children and Children with Math Disabilities.  Psychology in the Schools, 45,  838-858 .

Iseman, J. S., & Naglieri, J. A. (2011). A cognitive strategy instruction to improve math calculation for children with ADHD and LD: A randomized controlled study.  Journal of Learning Disabilities ,  44 (2), 184-195.

Jitendra, A. (2002). Teaching students math problem-solving through graphic representations.  Teaching Exceptional Children ,  34 (4), 34-38.

Jitendra, A. K., DuPaul, G. J., Volpe, R. J., Tresco, K. E., Junod, R. E. V., Lutz, J. G., ... & Mannella, M. C. (2007). Consultation-based academic intervention for children with attention deficit hyperactivity disorder: School functioning outcomes.  School Psychology Review ,  36 (2), 217.

Jordan, N. C., Glutting, J., & Ramineni, C. (2010). The importance of number sense to mathematics achievement in first and third grades.  Learning and individual differences ,  20 (2), 82-88.

Joseph, L. M., Konrad, M., Cates, G., Vajcner, T., Eveleigh, E., & Fishley, K. M. (2012). A meta‐analytic review of the cover‐copy‐compare and variations of this self‐management procedure.  Psychology in the Schools , 49 (2), 122-136.

Kroesbergen, E. H., & Van Luit, J. E. (2003). Mathematics interventions for children with special educational needs a meta-analysis.  Remedial and Special Education ,  24 (2), 97-114.

Lindberg, S. M., Hyde, J. S., Petersen, J. L., & Linn, M. C. (2010). New trends in gender and mathematics performance: a meta-analysis.  Psychological bulletin ,  136 (6), 1123.

Maruyama, M., Pallier, C., Jobert, A., Sigman, M., & Dehaene, S. (2012). The cortical representation of simple mathematical expressions.  NeuroImage ,  61 (4), 1444-1460.

Mayes, S. D., & Calhoun, S. L. (2006). Frequency of reading, math, and writing disabilities in children with clinical disorders.  Learning and individual Differences ,  16 (2), 145-157.

Mazzocco, M. M., & Thompson, R. E. (2005). Kindergarten predictors of math learning disability.  Learning Disabilities Research & Practice ,  20 (3), 142-155.

Mercer, C. D., & Miller, S. P. (1992). Teaching students with learning problems in math to acquire, understand, and apply basic math facts.  Remedial and Special Education ,  13 (3), 19-35.

Montague, M. (2008). Self-regulation strategies to improve mathematical problem solving for students with learning disabilities.  Learning Disability Quarterly ,  31 (1), 37-44.

Montague, M., & Dietz, S. (2009). Evaluating the evidence base for cognitive strategy instruction and mathematical problem solving.  Exceptional Children ,  75 (3), 285-302.

Montague, M., Enders, C., & Dietz, S. (2011). Effects of cognitive strategy instruction on math problem solving of middle school students with learning disabilities.  Learning Disability Quarterly ,  34 (4), 262-272.

Parkhurst, J., Skinner, C. H., Yaw, J., Poncy, B., Adcock, W., & Luna, E. (2010). Efficient class-wide remediation: Using technology to identify idiosyncratic math facts for additional automaticity drills. International Journal of Behavioral Consultation and Therapy ,  6 (2), 111.

Poncy, B. C., Fontenelle IV, S. F., & Skinner, C. H. (2013). Using detect, practice, and repair (DPR) to differentiate and individualize math fact instruction in a class-wide setting.  Journal of Behavioral Education ,  22 (3), 211-228.

Poncy, B. C., Skinner, C. H., & Jaspers, K. E. (2007). Evaluating and comparing interventions designed to enhance math fact accuracy and fluency: Cover, copy, and compare versus taped problems.  Journal of Behavioral Education ,  16 (1), 27-37.

Poncy, B. C., Skinner, C. H., & O'Mara, T. (2006). Detect, Practice, and Repair: The Effects of a Classwide Intervention on Elementary Students' Math-Fact Fluency.  Journal of Evidence-Based Practices for Schools .

Rhymer, K. N., Dittmer, K. I., Skinner, C. H., & Jackson, B. (2000). Effectiveness of a multi-component treatment for improving mathematics fluency.  School Psychology Quarterly ,  15 (1), 40.

Rohrbeck, C. A., Ginsburg-Block, M. D., Fantuzzo, J. W., & Miller, T. R. (2003). Peer-assisted learning interventions with elementary school students: A meta-analytic review.  Journal of Educational Psychology ,  95 (2), 240.

Rourke, B. P. (2000). Neuropsychological and psychosocial subtyping: A review of investigations within the University of Windsor laboratory.  Canadian Psychology/Psychologie Canadienne ,  41 (1), 34.

Skinner, C. H., McLaughlin, T. F., & Logan, P. (1997). Cover, copy, and compare: A self-managed academic intervention effective across skills, students, and settings.  Journal of Behavioral Education ,  7 (3), 295-306.

Slavin, R. E., & Lake, C. (2008). Effective programs in elementary mathematics: A best-evidence synthesis. Review of Educational Research ,  78 (3), 427-515.

Tok, Ş., & Keskin, A. (2012). The Effect of Fast Draw Learning Strategy on the Academic Achievement and Attitudes Towards Mathematics.  International Journal of Innovation in Science and Mathematics Education (formerly CAL-laborate International) ,  20 (4).

Tsuei, M. (2012). Using synchronous peer tutoring system to promote elementary students’ learning in mathematics.  Computers & Education ,  58 (4), 1171-1182.

Xin, Y. P., Jitendra, A. K., & Deatline-Buchman, A. (2005). Effects of mathematical word Problem—Solving instruction on middle school students with learning problems.  The Journal of Special Education ,  39 (3), 181-192.

Share This Story, Choose Your Platform!

Signup for our newsletter.

Get notified when new resources are posted on the website!

Sign up to receive our electronic newsletter!

Recent Posts

• Working with SLPs to Support Communication and Literacy
• New Upcoming Webinar: Leading and Staying on Track – Leveraging Data to Monitor and Implement Informed Programs That Make an Impact
• Webinar Recording: SORing After Primary – Morphing into Meaning