Research & Training Blog

By Rick Bryck, Ph.D., Senior Director, Landmark College Institute for Research and Training 

It’s no secret that math can be anxiety provoking for many of us – adults, professionals, and scholars alike. For students with learning disabilities or attention or executive function challenges, math can sometimes feel next to impossible. 

One approach to making math more engaging and fun—yes, fun—involves making it harder—yes, harder (well, in a sense harder) by employing rich, problem-solving questions. These types of math problems often ditch the “skill and drill,” rote memorization needed for remembering a particular formula and plugging in the right number. While there is a time and place for building this type of fluency and skill, “skill and drill” types of pedagogy can be rather, well, uninspiring. 

To get students to a deeper level of thinking, we can give them more challenging but enriching problem-solving questions. These often require outside-the-box, divergent, and creative thinking. They enable students to demonstrate their brilliance and unique perspectives and ways of thinking by allowing multiple paths to finding the solution, or a single solution to the problem. 

 

            Here’s an example:

                        Standard problem

                        “Find the area of a garden that is 5 feet by 7 feet long.”

                        Solution: length x width = 5 x 7 = 35 ft²

             Enriched version:

“You have 35 ft of wire fencing to enclose a rectangular garden. What are the possible lengths and widths of the garden.”

In the enriched version, creative thinking is encouraged and often needed. It has multiple solutions and encourages exploration. Further, it fosters strategy use (trial and error, drawing representations) and understanding of area concepts (how length and width affect total area), and finally, it encourages students to explain their solutions. 

Solving problems in this way can develop more positive beliefs about math, more engagement and motivation in math, and better understanding of math concepts (e.g., Boaler, 2002; Lester, 2013). Plenty of teachers and curricula use question sets requiring this deeper level of thinking and problem-solving skills; however, too often it is assumed or implied that these questions should only be used once students reach a certain level of mathematical proficiency, or worse, are only appropriate for the “highest fliers” in the class, those that score best on standard math assessments.  

All learners are capable of this richer thinking and problem solving, illustrating their unique perspectives, approaches, and thinking. But there is that aforementioned difficulty component: These types of problems often require more steps, more planning, and more monitoring of one’s performance and strategy—the kinds of skills requiring strong executive function (EF) and organizational skills. They also tend to be laden with more complex language than “regular” math problems. In short, there is often a higher cognitive load (often of the extraneous variety) in complex problem tasks, making them particularly troublesome for a wide range of students, including neurodivergent learners.

 

But wait, didn’t we say challenge was a good thing?

It is! What can we do if cognitive load and EF demands are high? We can follow some tried and true approaches to reducing extraneous load and providing EF scaffolds, as we outline below.

Much of the following suggestions and supporting evidence comes from work on an EF+Math funded project which tests the efficacy, and continual improvement, of a math platform, CueThink, which provides scaffolded, stepped processes for problem solving, including peer-to-peer interaction and embedded supports for EF. 

Three Strategies to Support Executive Function in Math

Break the Problem Down

CueThink’s approach rests on a four-stage method for supporting effective student problem-solving: explore, plan, solve, and review, based on classic work on a four-stage process (Polya,1945/2014). It employs, on the surface, a somewhat simple tactic of breaking a problem down into distinct parts or processes. This is a shared tenet of the “Landmark College approach” to supporting students with complex assignments and projects of “micro-uniting” the necessary tasks of a project (and yes, a complex math problem can be considered a mini-task) into manageable, bite-sized chunks. In short, this jives with modern, theoretical accounts of our information processing limits.

The estimate of an average adult’s working memory capacity limits (essentially, our attentional focus for complex thinking) is between 3 and 4 “items” at a time. We can think critically, and “do something” with this information, but only with a few things at a time. This limit tends to be exacerbated for individuals with EF challenges. However, it is critically important to understand this is NOT the same as overall ability or intelligence! 

Thus, this four-stage approach allows dedicated time to think through multiple, often necessary, steps to solving non-rote problems. It also inherently encourages learners to slow down a bit, think through their approach and strategy, and even try different strategies if their first choice is fruitful.
 

Provide Scaffolds for Executive Function

Executive function—the cognitive processes needed for goal-directed behavior—has been shown to be highly related to, and predictive of, math proficiency (e.g., Cragg & Gilmore, 2014). All three of the “core” executive functions are crucial for math proficiency. Working memory is needed for holding onto critical pieces of information while processing other information (think carrying numbers in division), inhibitory control is needed to ignore irrelevant information in a problem, or stopping oneself from falling for problem lures, and cognitive flexibility is a hallmark of all symbolic thinking, including thinking in the codes and language of math, as well as needed for fluid switching between mathematical concepts or multi-step operations. 

Strengthening core EFs is one approach then for supporting math learning and can be embedded within math learning. At the same time, we can consider ways to scaffold problems, or steps with high EF demands by giving needed supports when cognitive and EF demands are high. Scaffolding implies that these supports will ultimately help the learning of these skills independently, and the supports or scaffolds can slowly be lessened or eventually removed. Identifying and then lowering extraneous cognitive load—in this case, components of the problem that are inherent to, but not integral to the learning task or objective at hand—is one such approach. Helping students break down and understand complex language (or simply reducing or simplifying the wordiness of problems), highlight the most critical features, and block out distracting or irrelevant wording or information are just some examples of EF scaffolding. Breaking a problem down into its constituent parts, as described above, can also be considered an EF scaffold. 

A related approach is to employ principles of Universal Design. That is, provide for multiple means of engaging with material, multiple ways of representing concepts, and allowing for multiple means of assessing student understanding of material. In the context of math problem solving, this could be using and encouraging multiple tools or applets for understanding, such as graphs, charts, number lines, tables, words, sketches, or manipulatives, and allowing for screen readers and speech-to-text functionality.

Use Metacognitive Prompting

Difficulty getting started on a problem, or knowing where to start on a problem, especially multipart, intricate problems, is a common EF “breakdown” we see in students. Borrowing from principles of EF coaching, one technique for getting “unstuck” is to allow the individual time and space to generate their own approaches or strategies in a non-directive way, as opposed to conferring a teacher- or textbook-preferred method onto the student. A bit of verbal, or “metacognitive,” prompting can often facilitate this, and can take many forms.

A few examples include:

• “What would your first step be?”
• “What has worked before for you when you got stuck?”
• “What is one step you could take right now?”
• “What are two different strategies you could try?”
• “What information is most likely to be useful in solving the problem?”
• “What would a reasonable estimate for the answer be?”

One of the biggest benefits of metacognitive strategies derives from slowing down thinking. This might sound counterintuitive, but is often what is needed to consider a problem and all its constituent parts, formulate a plan, consider alternate approaches, and carry out multiple steps and calculations. 

Stopping oneself from “first thought” thinking, especially if of the negative self-talk variety, can be every bit as, or even more, crucial as being fast at math. Relatedly, the metacognitive skill of monitoring one’s work and progress—and knowing when to change gears or tactics—can be essential for persevering through a difficult problem.
 

Supporting Evidence

This is perhaps not the first time you have heard some, or all, of these strategies espoused. Indeed, these general classes of instructional techniques and approaches have been explored previously in the context of math pedagogy. However, few, if any, empirical studies have attempted a unified effort of combining metacognitive and EF supports in one package to explore changes in math problem-solving proficiency.

Recent work by the CueThink team and colleagues exploring a combined approach is briefly summarized here:

-- Implicit to our theory and approach is the idea that EF, metacognition, and one’s own belief about math (self-efficacy, math anxiety, etc.) are all crucial factors to success and math problem solving proficiency. Yet few empirical studies have researched this question, at least when looking at these variables at the same time. Pulling from our above-mentioned studies, in short, we did find strong evidence from a series of multiple regression (attempts to explain how multiple variables affect a single measured outcome) analyses that EF skills, metacognitive prediction ability, and individuals’ beliefs about math were critical “predictors” (factors) of both math accuracy (raw scores) and math understanding (giving credit for correct work, even if final answer was inaccurate) on rigorous math problem-solving questions (Rhodes, Bryck, and Gutierrez de Blume, 2023).
 

-- Perhaps the most fundamental and relevant research question we asked in this work is, is this combined approach effective at improving math (problem solving) scores? In a quasi-experimental design, we demonstrated moderate to strong improvements in math problem solving in a group of middle school students using CueThink compared to a group receiving instruction as usual. These improvements held for, and helped close the achievement gap, in a sample of traditionally underserved (Black, LatinX, and/or lower SES) students (Rhodes, Bryck, and Sethuraman, 2022). To read more about this study, see our white paper here.
 

-- These gains in math problem-solving scores for students using CueThink seem to be moderated by variations in student EF ability, in particular, individual differences in working memory capacity. Students who scored higher on an initial WM test showed larger gain in math problem-solving (Bryck & Rhodes, 2024). It is critical to note that EF skills can be dynamic and were tested at only one point in this analysis; moreover, students who initially scored lower on WM still showed statistically significant gains on the tested math problem-solving questions. Yet the take home message here is individual differences in EF, and/or WM, should be considered and differentiated and/or Universally Designed instruction should be employed to reach the widest range of learners.

 

So, what can I do?

Well, if you have read this far, the obvious answer is to think about ways to incorporate metacognitive prompting, EF scaffolds, and reductions of cognitive load!

One way to do this is to look for and employ “low floor, high ceiling” problems, which are those that essentially allow multiple entry points to learning but can also foster deep learning (and engagement!) with math.

Educators should also contemplate methods to assist students in maintaining and applying EF and metacognitive aids outside of a particular setting—i.e., help students utilize these strategies across different classroom tasks and evaluations. Teachers should be mindful of the diverse levels of EF ability, both within and between students, and offer scaffolding and assistance promptly as needed.

 

Acknowledgments

See the CueThink website for more information on how it works and information on how it can be used by individual students or school districts, can be gained from their website.

The research reported here was supported by the EF+Math Program of the Advanced Education Research and Development Fund (AERDF) through funds provided to CueThink. The perspectives expressed are those of the authors and do not necessarily represent views of the EF+Math Program or AERDF.

A huge thank you to our collaborators at CueThink: 

• Sheela Sethuraman, Founder and CEO, CueThink
• Dr. Sam Rhodes, Assistant Professor, Elementary and Special Education, Georgia Southern University 
• Joann Wang, Senior Product Specialist, CueThink 

 

References 

Boaler, J. (2002). Experiencing school mathematics: Traditional and reform approaches to teaching and their impact on student learning. Routledge.

Bryck, R.L. & Rhodes, S., (2024). Improvement in Math Problem Solving is Moderated by Working Memory. Proceedings of the annual Research Council on Mathematics Learning (RCML) conference, Columbia, SC.

Cragg, L., & Gilmore, C. (2014). Skills underlying mathematics: The role of executive function in the development of mathematics proficiency. Trends in Neuroscience and Education, 3(2), 63-68. An overview of relationships between executive functions and mathematics ability

Lester, F. K. (2013). Thoughts About Research On Mathematical Problem- Solving Instruction. The Mathematics Enthusiast, 10(1),245-278. https://doi.org/10.54870/1551-3440.1267

Pólya, G. (1945/2014). How to solve it: A new aspect of mathematical method. Princeton, NJ: Princeton University Press.

Rhodes, S., Bryck, R.L., and Gutierrez de Blume, A. (2023). Exploring Factors Influencing Mathematics Problem Solving. Proceedings of the forty-fifth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA).

Rhodes, S., Bryck, R.L., Sethuraman, S. (2022). CueThinkEF+ Significantly Increases Students’ Problem Solving Performance [White Paper].