In addition to the *executive functions* discussed in Executive Function Part I and II (cognitive inhibition, switching retrieval strategies* , *and identifying, activating, manipulating relevant information), another

*executive function*is the

*capacity to coordinate performance*on two or more separate tasks and shift from one task to another. Each arithmetic fact and procedure is a compendium of multiple tasks involving subtasks. For example, keeping track of the component tasks (multiplication facts and partial products, place value, addition) in computing 23×7 need to be organized mentally and performed. To succeed in this process is the task of the executive function. However, given that the sub-tasks are parts of an integrated skill, the requirement for coordination is presumably low relative to performing multiple independent tasks.

Shifting is the flexibility to switch between different tasks, making decisions, and choosing strategies in multi-step and multi-operational problems and procedures. Solving complex mathematics problems requires prioritization because operations must be solved in a specific order. Impulse control is essential to stick with these problems long enough to completely solve them. Many children lose points in math not because they got the answer wrong but simply because they gave up too soon. Not enough storage space in their working memory prevents them from connecting the logic strings that many math problems require; organization skills are required to know which formula to apply and where to look to find the right ones; flexible thinking is necessary to help the math student forget about the previous problem and cleanly move on to the next. By focusing efforts on building these executive function skills, math proficiency is sure to improve.

Shifting ability predicts performance in mathematics. Shifting is required to switch between different procedures (e.g. adding or subtracting) when solving complex mathematical problems. For factual knowledge, working memory is likely to play a role in acquiring new facts as both sum and answer need to be held in mind together in order to strengthen the relationship between them. Shifting is an essential skill in multi-step and multi-concept operations, for example, simplifying an expression using the order of operations: **G**rouping—transparent and hidden, **E**xponents, **M**ultiplication and **D**ivision in order of appearance, and **A**ddition and **S**ubtraction in order of appearance (**GEMDAS**), long-division, operations on fractions (adding fractions with different denominators—even finding the least common denominator requires shifting), solving a system of linear equations, etc. Competence in shifting can be achieved with mnemonic devices, graphic organizers, and organized sets of task sequence.

Solving problems requires understanding the task. This means analyzing tasks and setting goals and sub-goals. Doing task-analysis improves prioritization while fixed routines and mnemonic devices inhibit distractions that strengthen impulse control. Exercises that emphasize time management can also help children stay focused. These improve both organizational skills and flexible thinking in moving from one task to the next. Training in those areas can accompany mathematics lessons for better performance overall.

**Organization Skills and Their Role in Mathematics Learning
**Organization skills help a child take a systematic approach to problem solving by creating order out of disorder and requiring a step-by-step series of calculations, or executing a standard procedure. These executive function skills are crucial to becoming proficient in mathematics.

Organization skills range from learning how to collect all materials –physical objects/equipment/instruments necessary to understanding and completing a task, collecting and classifying the information (content) from the problem and stepping back and examining the complexity of the situation to organizing one’s thinking. For example, children use organizational skills when they take time to gather all of their notes before starting to study for a test or identifying what definitions, axioms, theorems, and postulates are needed in writing a proof.

Organization skills deal with:

(a) *Organization of physical resources*: Even the physical environment and workspace are key elements of this type of organization. Many students do not know how to use the space on writing paper—where to begin and what direction. There is no organization in the way they record information on paper and pursue calculations. There is no clear path to their work. This material-spatial disorganization – tendency to lose or misplace things; writing problems in disorganized fashion on the paper; difficulty bringing home or returning assignments in a timely way comes in the way of learning, particularly in mathematics.

(b) *Organization of cognitive resources*: Many intelligent students have adequate to higher cognitive abilities, but they do not have efficient strategies to organize their thoughts, systems, strategies, and approaches to solving problems. This ranges from note taking to summarizing. This includes (i) transitional disorganization – difficulty shifting gears smoothly, often resulting in rushing from one activity to the next or the opposite not being able to shift from one task to other; difficulty settling down to work or preparing to leave for school, and (ii) prospective retrieval disorganization – difficulty remembering to do something that was planned in advance, such as forgetting the deadline of a project until the night before it is due.

(c) *Organization of emotional resources*: Because of their lack of organization, many students feel overwhelmed by mathematics assignments. This includes temporal-sequential disorganization – confusion about time and sequencing of tasks; procrastination; difficulty estimating how long a task will take to complete.

These disorganizations result in frustrations and then math anxiety among students.

Self-Awareness is an example of organization as an executive skill helpful in learning and achieving in mathematics. Teachers not only require their students to complete math examples correctly but also to explain their rationale and reasoning, which reinforces their achievements in mathematics. Self-Awareness involves the capacity to think about one’s thinking and then share it in a way that others can understand. Self-Awareness skills help kids understand their own strengths and weaknesses and can be helpful in determining areas in which more study is required.

**How Do Executive Functions Work?**

How do the executive functions work—and especially how do these help us to learn? In particular, how do they function in learning mathematics? What is the role of the understanding of the functioning of the executive function in teachers’ instructional decisions? Generally, teachers’ instructional decisions are based on a mix of theories learned in teacher education, trial and error, knowledge of the craft and content, and gut instinct. Such knowledge often serves us well, but is there anything sturdier to rely on? That is where the appropriate knowledge of EF comes to play.

Many teachers are not aware of the importance of EF skills in learning mathematics. While the mechanisms by which EF skills support the acquisition as well as the application of mathematics knowledge are far from clear, a basic understanding about EF is essential to inform classroom practice to help students with and without EF skill deficits.

The executive function skills help us make decisions such as: focus on task(s), classify and organize information, make connections and see patterns, refer tasks from one slave system to other, break the main task into subtasks, sequence the tasks, delegate, allocate and apportion resources to different functions, and maximize the functions of the slave systems.

EF evaluates the outcome of tasks and decisions, monitors the progress, reports the progress to different systems, becomes the communicator of the success and failure of the tasks, and experiences the results of the endeavor, prepares for the next experience, and even arranges for new experiences. For example, in the long-division algorithm, the executive function skills of *inhibition* (when to estimate, multiply, subtract, and bring down), *updating* (decide: “What is the next step?” “How do I use this information? “Where do I place the quotient, if the quotient is not working should I try 2?”), *shifting* (from one operation to another—divide, multiply, then subtract, etc.), and *mental-attentional capacity* (*M*-capacity) contributes to and helps children’s ability to keep the sequence of tasks in this procedure.

When children reach fluency in a procedure, they are ready to acquire the competence in solving word problems such as those involving division. At each juncture of the procedure, different EF functions (inhibition, updating, shifting, and *M*-capacity) are called upon. For example, updating mediates the relationship between multiplication performance (controlling for reading comprehension score) and latent attentional factors *M*-capacity and inhibition. Updating plays a more important role in predicting performance on multiple-step problems than age, whereas age and updating are equally important predictors on one-step problems.

Correlational studies provide evidence of a relationship between EF skills and mathematics which may be stronger than the relationship between EF skills and other areas of academic performance. However, we are not sure of the one-to-one relationship between EF skills (inhibition, shifting, working memory) and the different components of mathematics: *factual* (e.g. 6 + 4 = 10), *conceptual* (e.g. knowing that addition is the inverse of subtraction) and *procedural* (e.g. ′carrying′ when adding above 10 in multi-digit number additions) knowledge.

Individuals differ in their profile of performance across linguistic, conceptual, and procedural components and may have strengths in one component but not in others, suggesting that different mathematics components rely on differential sets of EF skills and/or their mathematics learning personalities. Similarly, the role and contribution of executive function skills differ across these components. For example, while working memory ability is related to fraction computation, it is not a predictor of conceptual understanding of fractions. In contrast, inhibition has been linked to the application of additive concepts. We need to understand how EF skills support different aspects of mathematical competence. The following description and the summary chart show the interrelationships between the mathematics components and the EF skills.

**Concepts and Understanding
**

*Working Memory (*Recalling prior knowledge to relate to new ideas; keeping multiple ideas in mind at once; making connections)*Self Awareness*(Being able to explain and communicate one’s own reasoning in writing or to others; being able to think about and explain the steps one uses to solve different kinds of problems; being able to explain the reasoning behind completing a math problem a certain way)

**Computational Procedures**

*Working Memory*(Keeping different steps involved in solving a problem in mind; recalling which formulas to use to solve which a problem; Keeping parts to a multistep problem in mind, etc.)*Focus*and*inhibition*(Determining the primacy of a task; Sustaining attention to the task; Not getting distracted by the irrelevant information in the middle of completing a problem; Setting goals and working to meet them)*Planning*(What kind of the problem is this; Planning the steps one will use in solving the problem; Thinking ahead about what steps to take and what options one has for solving it;)*Organization*(Organizing the work on the page so that it is clear—where to start, what unit to use in the diagram, does it match the given information, organizing images/notes on page; deciding on the sequence of steps; organizing information in a word problem)

**Fluency**

*Working Memory*(Keeping all of the different components to a problem in mind while solving it; thinking about previous steps while doing the current one; retrieving previously learned information to apply it to the current problem/task; applying math rules; etc.)*Planning*(Thinking ahead about what kind of fact/procedure/problem this is, and what options one has for solving it; planning the steps one will use to solve the problem; prioritizing strategies to be used)*Self Awareness*(Thinking about one’s own reasoning and whether or not it makes sense as one tries to construct a fact/execute the procedure/solve a problem; thinking about the steps you used to solve previous problems; self-correcting and checking one’s work)

**Flexibility in Thought and Action**

*Shifting*between different representations written in sentences, computation, etc.; being able to switch one’s approach/strategy when it is not working)

The above model describes the relationships between executive function skills and components of mathematical knowledge. The solid lines indicate direct relationships between the mathematical component and the EF skills. Dashed lines represent relationships that change over the course of development and age. When a student has mastered facts, concepts, and procedures using efficient and generalizable skills, it automatically results in flexibility of thought.

*Nuts and Bolts**: Recognizing and Assisting Executive Function
*

**Both mathematics ability and EF skills improve during development and therefore the relationship between the two will also change as children get older. In other words, the executive function skill levels are not fixed. Everyone has the ability to improve executive function skills with practice while improving proficiency in math at the same time.**

*Strategies for Improving Math Skills & Executive Functions*

A series of studies have indicated the importance of developing executive functions in early ages for future academic and math success. For example, visuo-spatial short-term memory is an excellent predictor of math abilities and verbal working memory is crucial in the recall and application of math formulas when doing calculations.

The majority of current theories and practices of numerical cognition and pedagogies for mathematics learning do not incorporate the role and contribution of EF processes into their models (e.g., lesson plans and interventions). The interplay between domain-general and domain-specific skills in the development of mathematics proficiency suggests that it is essential that both are integrated into theoretical and teaching frameworks. Although there has been much recent attention to young children’s development of executive functions and early mathematics, few pedagogical programs have integrated the two.

Developing both executive function processes and mathematical proficiencies is essential for children with and without learning disabilities, and high-quality mathematics education may have the dual benefit of teaching this important content area and developing executive function processes. This can be accomplished by paying special attention to the selection of quantitative and spatial models for teaching (Visual Cluster cards, dominos, dice, Ten Frames, Cuisenaire rods, Invicta Balance) rather than to the counting of random objects, number line, fingers, etc. to early numeracy and mathematical outcomes.

Understanding the nature of executive functions and their role in learning, functioning, and success is an important part of developing the pedagogy for mathematics learning and teaching. A review from cognitive sciences shows that it begins with the parents, for example, certain parental behaviors—meaningful praise, affection, sensitivity to the child’s needs, and meaningful encouragement of effort in initiating and finishing tasks, along with intellectual stimulation, meaningful and high expectations, support for autonomy, and well-structured and consistent rules—can help children develop robust executive function skills.

Playing games[2] both traditional (e.g., card games, Connect Four, Stratego, Battleships, Concentration, Simon, etc.) and computer/Internet assisted (e.g., such as Lumosity) games help develop and challenge the executive functions. For example, the game *Word Bubbles* challenges verbal fluency, the ability to quickly choose words from a mental vocabulary; *Brain Shift* challenges task switching, the process of adapting to circumstances and switching goals; and many other games challenge other cognitive skills involved in executive functioning. Playing games such as Tetris and working on visual spatial skills can develop skills not only in visually-based mathematics such as geometry or trigonometry but also in considering the step-by-step processes in more complex mathematics.

These games can be adapted to the player and task with increasing difficulty as a player improves. Games and tasks should be accessible and moderately challenging. Games and these training exercises aim at improving flexibility of thought. Complex math word problems often require flexibility in thinking and may require more problem-solving and trial-and-error approaches, games are effective means for such a goal.

Although there is empirical evidence to support both domain-general and domain-specific models, but more and specific skills learning is favored in studies that focus on children’s training that emphasize domain-specific perspective. Research, for example, has shown that children’s visual-spatial WM fails to explain variance in their word reading and passage comprehension similarly verbal WM fails to account for difficulty in mathematics achievement. Verbal WM accounts for statistically significant variance in performance on these verbal tasks, even when relevant verbal skills (e.g., word reading) are controlled.

Further support of a domain-specific view comes from scholarly reviews of WM deficits among children with learning difficulties. Children with serious learning problems exhibit WM deficits across verbal and visual-spatial domains, however, verbal WM deficits appear more important to the children with reading difficulties. Visual-spatial deficits, by contrast, seem more relevant for children with mathematics difficulties. Moreover, the researchers of most previous WM training with children that uses visual-spatial WM tasks does not transfer to academic performance related to reading skills. Similarly, WM training that focuses on verbal WM tasks shows little training effects that transfer to visual-spatial WM or related academic performance in arithmetic.

Recent reviews of working memory (WM) training have concluded that, for children between the ages of 8 and 15, WM training involving visual-spatial tasks or a combination of visual-spatial and verbal tasks can improve visual-spatial WM, but with limited effects on the academic performance. Therefore, in our training with children in clinical settings we have found that any training to improve EF that does not include domain-specific numerical content has little or no impact on executive functioning and mathematics achievement – for example, when children use mainly non-computerized games with either numerical or non-numerical content. Visuo-spatial working memory improves in both groups compared to controls, but only the numerical training group shows an improvement in numerical skills, suggesting that training needs to be domain-specific.

There is research to show that specific mathematics tutoring to children′s cognitive skills (including EF skills) improves mathematics achievement. Attention and working memory measures predict performance on mathematics measures at the end of such training, suggesting that children′s EF skills do have an impact on their ability to learn new mathematical material.

Only by exploring the differential role of EF skills in multiple components of mathematical knowledge in different age groups, as well as distinguishing between the acquisition and skilled application of this knowledge, will we understand the subtleties in the relationship between EF skills and mathematics learning and build a structure for an instructional design.

There *is* one surprising but well-supported way to improve executive function in both children and adults: aerobic exercise. A review of research concludes that “ample evidence indicates that regular engagement in aerobic exercise can provide a simple means for healthy people to optimize a range of executive functions.”

Of course, the big question is: How to improve executive function? Which activities, if any, will increase a person’s executive functions—chances of remaining mentally sharp in engaging demanding learning activities? Research shows that to improve executive function, one should work hard at something—cognitive, emotional, or physical. Many labs studying brain functions have observed that the critical brain regions increase in activity when people perform difficult tasks, whether the effort is physical or mental. You can therefore help keep these regions thick and healthy through vigorous exercise and bouts of strenuous mental effort.

Of course, the big question is: How to improve executive function? Which activities, if any, will increase a person’s executive functions—chances of remaining mentally sharp in engaging demanding learning activities? Research shows that to improve executive function, one should work hard at something—cognitive, emotional, or physical. Many labs studying brain functions have observed that the critical brain regions increase in activity when people perform difficult tasks, whether the effort is physical or mental. You can therefore help keep these regions thick and healthy through vigorous exercise and bouts of strenuous mental effort.

Of course, the big question is: How to improve executive function? Which activities, if any, will increase a person’s executive functions—chances of remaining mentally sharp in engaging demanding learning activities? Research shows that to improve executive function, one should work hard at something—cognitive, emotional, or physical. Many labs studying brain functions have observed that the critical brain regions increase in activity when people perform difficult tasks, whether the effort is physical or mental. You can therefore help keep these regions thick and healthy through vigorous exercise and bouts of strenuous mental effort.

**School-aged children.** Studies of children have found that regular aerobic exercise can expand their working memory—the capacity that allows us to mentally manipulate facts and ideas to solve problems—as well as improve their selective attention and their ability to inhibit disruptive impulses. Regular exercise and overall physical fitness have been linked to academic achievement, as well as to success on specific tasks.

**Young adults. **Executive functioning reaches its peak levels in young adults, and yet it can be improved still further with aerobic exercise. Studies on young adults find that those who exercise regularly post quicker reaction times, give more accurate responses, and are more effective at detecting errors when they engage in fast-paced tasks.

**Older adults. **Research on older adults has found that regular aerobic exercise can boost the executive functions that typically deteriorate with age, including the ability to pay focused attention, to switch among tasks, and to hold multiple items in working memory.

[1] More on *working memory* in the ** previous two posts** and more information related to

*executive function*in the next post—Part II of this topic.

[1] See the previous posts on *Working Memory and Mathematics Learning* Part I and Part II.

[2] See *Games and Their Uses in Mathematics Learning (Sharma, 2008). *