Children′s underachievement in mathematics is a significant problem with almost one in four leaving elementary school without reaching the mathematics level expected of them, and some failing even to achieve the numeracy skills expected of a 7-year-old. In some cases, these problems endure into adulthood, and a fifth of adults have numeracy skills below the basic level needed for everyday situations. Today, numeracy skills have a bigger impact on life chances than poor literacy. Numeracy is the new literacy.
While it is true that some people are better at mathematics than others, it is also true that the vast majority of people are fully capable of learning K–12 mathematics. Virtually everyone is capable of learning the numeracy content and skills required for good citizenship: an understanding of arithmetic procedures, algebraic thinking, basic concepts of geometry, and use of probability deep enough to apply it to problems in our daily lives.
Learning mathematics does not come as naturally as learning to speak, but our brains do have the necessary equipment. So, learning math is somewhat like learning to read: we can do it, but it takes time and effort and requires mastering increasingly complex skills and content. Just about everyone will get to the point where they can read a serious newspaper, and just about everyone will get to the point where they can do high school–level algebra and geometry. At the same time, not everyone can or wants to reach the point of writing a novel or solving a complex calculus problem.
Many factors contribute to differences in mathematics achievement, personal—past experiences, attitude, motivation, language and intellectual ability, and environmental—social, educational. It is clear that domain-specific numerical skills (number concept, numbersense, and numeracy; spatial sense and geometry) and knowledge are important for success with mathematics, but other cognitive factors also play an equally important role. In particular, the domain-general skills of holding and manipulating information in the mind (working memory) and other such executive functions (EF) have been found to be critical. Working memory, for example, helps children keep information in mind as they are doing a mathematics word problem or a long procedure. The use of executive functions in mathematics is often one of the key determinants of a student’s success in complex mathematics.
Understanding the impact of executive function skills on mathematics is important both for parents and teachers. Rather than viewing children’s difficulties with math as being the result of not understanding particular math operations, it may be that issues such as poor working memory, organization and planning skills are having an impact on the child’s mathematical abilities. As a result, it is helpful to consider the role of these executive skills in teaching, learning, and acquiring mathematical competencies.
Components of Executive Function
The executive functions—the set of higher-order mental skills that allow one to plan and organize, make considered decisions, see and make connections, evaluate and apportion intellectual resources on tasks, monitor the progress, manage time and focus attention, are important to all learning but more important in mathematics. Research suggests that executive function skills, more specifically—updating (monitoring and manipulating) information in the mind (working memory), focusing on relevant information and suppressing distracting information and unwanted responses (inhibition) and flexible thinking (shifting)—play a critical role in the development of learning skills such as metacognition for mathematics learning.
The most prominent EF skills called upon in different components of mathematics learning are: inhibition, updating our working memory, and shifting. These skills play a key role in learning and using complex cognitive, affective, and psycho-motoric skills. EF skills use these resources in planning and coordinating with the other slave systems, the episodic buffer (a place between working and long-term memories to bring and hold information about to be transferred to or retrieved from the long-term memory) and the long-term memory. It plans the sequencing and monitoring of cognitive operations; therefore, all its components and functions collectively are known as executive function.
Executive functions are seen as predictors of individual differences in mathematical abilities. For example, executive functions differ between low achieving and typically achieving children, and the absence of executive functions can be, in some cases, the cause of math learning disabilities. Working memory ability, for example, compared to preparatory mathematical abilities predicts math learning disabilities even over and above the predictive value of preparatory and basic mathematical abilities.
Executive functions form the basis of abilities such as problem solving and flexible thinking and are the foundations for the skill-set needed for mathematical way of thinking. Lack of executive function skills is evident when a student cannot attempt and solve problems without extensive external cues and guidance. This is particularly so when this behavior is prevalent in novel situations.
Executive function skills begin to emerge in infancy but are among the last cognitive abilities to mature and continue to develop into late adolescence. Teachers and parents can make judgments about children’s executive functions by observing their behaviors during math learning, but these skills can also formally be measured using formal assessment tools such as the WCST (Wisconsin Card Sorting Test). This test challenges people to adapt to changing rules and situations, and WCST scores can be used as the primary outcome measure of executive functions. Still, what is even more important is to know what to do when an individual displays the lack of these skills.
Different arithmetic strategies tax EF skills differently as mathematics strategies involve different combinations of and emphasis on linguistic, procedural, conceptual and factual mathematical knowledge. Thus, different mathematics situations may create different EF demands in understanding, achieving fluency, and applying.
Inhibition as an Executive Function Skill in Mathematics
The first executive function is cognitive inhibition—attending selectively to different inputs and focus appropriately. Cognitive inhibitory control abilities predict performance in mathematics. Suppressing distracting information and unwanted responses and engaging in focused action (e.g., in word problems and in similar looking concepts and skills) is called inhibition. It is the ability to actively inhibit or delay a response to achieve a pre-determined goal and focus on a particular desired action. Cognitive inhibition might be measured by presenting someone with information or a cognitive task, then introducing a new cognitive task, which either competes with and threatens to interfere or is aligned with the previous task— and seeing if the subject can suppress that interference or not. Students without any executive function deficit are able to make the shift well; however, those with a deficit will struggle. A student with this executive function deficit will need some kind of reminder, graphic organizer, scaffolding or task analysis to help them to engage in the task properly. It is not surprising that children with SLD, ADD, and ADHD show signs of cognitive interference that impedes their learning.
In mathematics, this kind of interference is quite common; it is clearly a feature of multi-digit multi-concept operations problems carried out as a series of subtasks, where attention needs to paid to selected parts of the problem at different times. For example, in the long-division algorithm, at one moment one is thinking of multiplication to find the quotient, the next moment one has to subtract, etc. Similarly, in estimating the product 23 x 7, the initial focus is on place value and then rounding (23 is about 20 and 7 is about 10, so first rough estimate is 30×10 = 300) to get a sense of the outcome.
To achieve in mathematics, it is important to suppress unwanted behavioral and cognitive processes (automatic and/or overlearned). This initially could be done with help (cues, sympathetic and encouraging support, graphic organizers, lists, mnemonic devices, concrete materials, etc.); ultimately, however, it should be done without cues and support. This support should help in suppressing irrelevant automatic responses and one should be helped in actively engaging in strategic processes. The strategies should be aimed towards attaining a short- and/or long-term goal, such as engaging in conscious, reflective problem-solving, searching, selecting, and applying appropriate strategies, making strategic decisions and evaluating their impact on a task. The help should also involve maintaining that delay when encountering interference and resistance. Working memory—a dynamic mechanism with a capacity to store information over short periods of time acts as an aid in this effort of inhibition and other related cognitively demanding activities.
Inhibition is an important factor in applying strategies and problem solving (one has to choose the right strategy out of several possible ones of varying efficiencies). Suppressing unwanted habitual or overlearned responses allows the student to search for, develop and/or implement more efficient actions and strategies.
Inhibition is necessary to suppress answers to related but incorrect number facts (e.g. inhibit 6 when asked 3×3; inhibit 8 when asked for 42, etc.). Part of such interference is due to students’ poor conceptual understanding and partly due to visual perceptual difficulties.
Inhibition, along with shifting, is also needed when a student is learning new concepts as she has to inhibit an automatic procedural approach (an overlearned activity in our classrooms) and she needs to shift attention towards the true conceptual numerical relationships involved in the concept. For example, when students have become too dependent on a procedure, they have difficulty in acquiring mental math capacities. For example, consider the simple problem of 16 – 9 being handled by a middle school student who has not learned the strategies based on decomposition/ recomposition and knows the subtraction procedure of regrouping:
The arithmetic procedures should be introduced to children only after they have acquired efficient strategies of mental math using decomposition/recomposition.
Inhibition is a variable, not a stable, developmental mental process; it is dependent on the task, concept, procedure, or the problem. The context plays a significant role in its application. Inhibition first appears in development around age three or four but continues to develop through adolescence. It is also the primary executive function that precedes and supports the development of other executive functions.
Inhibition is likely to be especially important at younger ages to suppress less sophisticated strategies, e.g. counting on from the first addend, in order to use more sophisticated strategies, such as counting on from the larger addend or using decomposition/recomposition.
Inhibitory mechanisms are distinguished by the related psychological constructs that they act upon, such as behavior and cognition. Thus, inhibition can occur at the behavioral level (e.g. immediate response control or lack of control to an environmental stimulus or failure— “I knew I could not do fractions” “I hate math.” The child throws tantrums.) and/or at the cognitive level—pre-conceived response or application of a strategy to a task or lack of flexibility in thinking (e.g. attentional inertia—repeating the question vocally or sub-vocally— “8×7” “eight times seven” “eight times seven”) or using counting as a strategy in addition and subtraction for even the simplest of problems even if better strategies are available (e.g., strategies are too difficult for me. Just give me the trick. Why can’t you tell me just how to do it?”) In the case of mathematics learning, cognitive and behavioral dis-inhibition may manifest concurrently as one may trigger the other.
Behavioral inhibition is a tendency to over display some behaviors and actions or a lack of display of appropriate speech and action when the child encounters an unfamiliar or challenging event. Behaviorally, it may lead to resistance, tantrums, oppositional behavior, lack of interest and abandonment of the task, thereby disinterest in the learning activity. Behavioral inhibition can also be of secondary nature, e.g., when a child encounters a task that she has been unsuccessful at previously.
Some effective solutions to this problem is to provide the student:
(a) small but meaningful and measurable successes in tasks relevant to the new concept and procedure (e.g., knowing the 45 sight facts with integers using Visual Cluster cards before the lesson on integers for sixth and seventh grade students is taught), and
(b) intentionally relating this success to the cause of the success, thereby, improving student’s metacognition. For example, the teacher helps a student (or the whole class) to memorize a particular multiplication table, say the table of 4 (using distributive property—decomposition/ recomposition of fact into known facts) and then gives 25 fraction problems where the numbers involved in numerators and denominators are multiples of 4 and asks them to reduce the fractions to the lowest terms. And then the teacher points out that since the children had memorized the table (cause of success), the problem of converting a given fraction into lowest terms was so easy (success). Success is the greatest motivator for participating in the learning process, but it also helps developing inhibition actions.
At the cognitive level, the behavior may be application of an overlearned skill or strategy or lack of inhibition or inflexibility in responding to a task. Cognitive inhibition is defined as the active over play or suppression of previously activated cognitive representations. It may also include the inability to remove incorrect inferences from memory when correct inference is available or overusing an inefficient strategy when efficient and elegant strategies are available. This inflexibility can be improved. This is possible when we provide efficient and effective strategies of learning mathematics and enough practice rather than the strategies that are laborious and cannot be generalized and extrapolated.
Some cognitive and neuropsychological research suggests separation when dealing with cognitive and behavioral inhibition, but I have found that providing progressively successful experiences and developing metacognition skills makes it possible to handle both of these issues at the same time, except in extreme cases.
Lack of inhibition has impact on several curricular components of mathematics, but it particularly affects strategic word problem solving. Because inhibition is an important variable with regard to text comprehension in word problems and also affects memory during word recognition, having poor inhibition skills leads to poor recognition of words, formulas, definitions, and results in mathematics problem solving. This is due to the fact that mathematical terms are compact and abstract—each word is packed with contextual meanings. Each word contains schema for a concept or a procedure. This is particularly so with compound terms in mathematics. For example, when students encounter problems related to greatest common factor (GCF) and least common multiple (LCM), many give wrong answers as they focus on the first word in the term. Similarly, when asked to calculate the perimeter or area of a rectangle (or circumference and area of a circle), many students give wrong answers. This is the result of cognitive interference of co-existing schemas and lack of inhibition on the part of the student. I believe this is partly due to lack of effective linguistic, concrete and visual models. It is also due to lack of proper teaching and partly due to lack of differentiation and inhibition of secondary tasks.