Working memory (WM) refers to the capacity to store information temporarily when engaging in cognitively demanding activities. Compared to short-term memory, WM plays a more influential role in children’s mathematics performance. This is because many mathematics tasks such as concepts and procedures involve multiple steps with intermediate solutions that must be remembered for a short time to accomplish the task at hand. For example, when reading a word problem, children must remember first the terms and expression for comprehension and relate them to previously learned information while simultaneously integrating incoming information in quantity as they progress through a text. As they proceed with the words of the text, they invoke symbols, formulas, concepts, and procedures they need to hold in the working memory. Several studies have shown that training improves children’s WM and academic skills, like reading comprehension and mathematics reasoning.

Working memory is thus important for the mathematics achievement of children who demonstrate a specific difficulty with mathematics. Children with mathematics disabilities have particular difficulty with the central executive component of working memory, especially when numerical information is involved.

Working memory is the record keeper during learning and problem solving (e.g., monitoring and manipulating information in mind that arises in partial calculations—partial sums, products, quotients, partial simplification in algebraic expressions and equations, etc.). In other words, executive function (EF) is like the executive that leads the learning process in all its aspects. It thinks for us. EF’s functioning is a major determinant in our learning.

Working memory is important at all ages in order to hold interim answers while performing other parts of a sum. In the process of learning, we are constantly updating the status and quality of information at hand by the incoming information. To keep track of the incoming information, seeking the related information from the long-term memory and making connections takes place in the working memory. Thus, the information in the working memory is dynamic, always in flux and change. Keeping track of changes and updating requires constant attention. This updating (in working memory) involves an attentional control system (the *central executive*), supported by two subsidiary *slave systems* for the short-term storage of verbal and visuo-spatial information (the phonological loop and visuo-spatial sketchpad, respectively).

Working memory accounts for unique variance in written and verbal calculation, as well as mathematical word problems, across different age groups. Importantly, it is the ability to manipulate and update, rather than simply maintain, information in working memory that seems to be critical for mathematics proficiency (e.g., a partial product is added to the previous information in a multi-digit multiplication problem; keeping track of different elements and sequence of arguments in the development of a geometrical and algebraical proof). The role of working memory is so important that the variance in the rate of learning and difference in achievement in fact mastery and procedure proficiency cannot all be explained by other factors such as age, IQ, mathematics ability, processing speed, reading and language skills.

The role of executive functions is related to different domains of mathematics skills and age. For example, at the start of school, inhibition and working memory contribute to performance in tests of both mathematics and reading. For example, in 5-year-olds, EF skills explain more variance in mathematics than in reading. In later years, working memory and inhibition skills predict performance on school exams in English, mathematics and science at both 11 and 14 years of age. EF skills predict both mathematics and reading scores across development. However, the role of working memory is reduced with age because students begin to rely on written forms of mathematics, rote procedures, and aids to calculations (e.g., multiplication tables, graphic organizers, number line, concrete models, calculators, etc.).

Just like children’s reliance on working memory changes over a greater developmental range, executive function also changes with age. For example, when 10–12-year-olds solve arithmetical problems while performing an active concurrent task designed to load the central executive, their performance is impaired by the demands of the dual task for all strategies that children use. This effect is greater for a decomposition strategy than for retrieval or counting. The amount of impairment decreases with age for retrieval and counting but not for decomposition as the decomposition strategies are consistent in their demands. When 9–11-, 12–14-year-olds and adults solve addition problems by counting, decomposition and retrieval strategies while performing either a concurrent working memory or a control task, it was found that the load on working memory slowed 9–11-year-olds′s performance on the addition problems for all three strategies, 12–14-year-olds for the two procedural strategies but adults only for counting. This suggests that children do rely on working memory to a greater extent than adults when solving arithmetic problems, most likely due to the fact that all arithmetic strategies are less automatic and efficient in children and therefore rely more on general processing resources.

The role of EF becomes even more evident as arithmetical processing involves multiple tasks in the same problem. As a result, a student may face difficulty with tasks that require the manipulation of information within the central executive component of working memory. There may be manifestation of impairment. However, the central executive component is not impaired when the tasks require only storage of verbal information. In most assessments and problem solving, the arithmetic calculations involve three tasks: *arithmetic verification* (“Is it a multiplication or a division problem?” “Is it a linear or quadratic equation?) and *constructing* (e.g., setting or recalling appropriate form of the operation, equation, algorithm, formula— “How do I write the equation?” “Is it an application of Pythagoras theorem?” “Should I solve this system by method of elimination or substitution?”), and *generating an answer* (actually performing that operation— “How do I convert this improper fraction into a mixed fraction?”).

Arithmetic poses extra complexity: people use different strategies to solve even the simplest of problems (8 + 6 = ?), such as *rote* *retrieval* (respond: 14), sequential *counting* (respond: 9, 10, 11, 12, 13, 14. It is 14) or *decomposition strategies* (respond: 8 + 2 + 4 = 14, 7 + 1 + 6 = 7 + 7 = 14, 8 + 8 – 2 = 16 – 2 = 14, 2 + 6 + 6 = 2 + 12 = 14, etc.). Each one of these strategies place different demands on the working memory. For example, *retrieval from memory* (automatized facts—learning by flash cards, mad minutes, Apps, etc.) and *generating facts by counting* (whether counting both addends or counting up from bigger addend or from smaller addend) are not affected by EF and working memory deficits.

Decomposition demands more from the working memory as it involves strategies and holding keeping track of intermediate steps. Therefore, many teachers take the easier route of teaching arithmetic facts by counting and memorization or giving children multiplication tables, facts charts, and calculators. However, counting (addition: counting up, subtraction: counting down, and multiplication and division: skip counting forward and backward, respectively, on a number line) do not help students for mastering arithmetic facts easily. These are not generalizable strategies and they neither develop mathematical way of thinking or strengthen EF skills. And, when facts are not mastered effectively (with understanding, fluency, and applicability), students find operations on fractions, decimals, algebra and higher mathematics difficult.

Studies found that the effects of working memory load are greater when participants use counting and less for retrieval. But, the effect is the greatest in the case of strategies that rely on decomposition/recomposition, which are the most efficient strategies for mastering arithmetic facts—addition and subtraction and then extended to multiplication over addition or subtraction for learning multiplication tables (e.g., 8×7 = 8(5 + 2) = 8×5 + 8×2 = 40 + 16 = 56). Rather than abandoning this fundamental strategy because it taxes the working memory, we should use efficient and effective instruction models to teach decomposition/recomposition. Decomposition/recomposition strategies at different grade levels can be learned efficiently with instructional materials such as: *Visual Cluster cards*, *Cuisenaire rods*, *fraction strips*, *algebra tiles*, and *Invicta balance*.

A second* executive function* is *switching retrieval strategies *(see Executive Function Part I where I discuss the first *executive function of cognitive inhibition*). This is clearly necessary for problems such as multi-digit multiplication or long division algorithm, which typically involves place value, multiplying, regrouping, adding and subtracting. Switching from one sub-task to another is essential for carrying/regrouping operations in all algorithms in arithmetic and mathematics. For example, the process of long division (estimate, multiply, subtract, bring down) or solving simultaneous linear equations (scanning the different methods available to solve the system, selecting the most efficient method, arithmetic operations involved, algebraic manipulations, attending to several variables, keeping the process alive in the brain) are difficult for many students as the number of subtasks is so large and involves frequent task and concept switching.

In addition to the *executive functions *of* cognitive inhibition *and

*a third*

*switching retrieval strategies,**executive function*is identifying, activating and bringing the relevant information from long-term memory to the working memory and then manipulating the incoming information in from the short-term memory. Executing relevant strategies and procedures in a situation such as using the equivalence of two relationships to find a new fact: what is 5 + 7? (e.g., thinking of decomposition/recomposition in adding: 5 + 7 = 5 + 5 + 2 = 10 + 2 = 12), extending one’s knowledge: what is

*3 × 400*? (the student retrieves

*3*×

*4 = 12*from long-term memory, and then attempts

*3 × 40 =120,*receives a feedback

*—it is right*, and then extends it

*to 3 × 400 = 1200,*etc.), or in order to simplify a calculation or thinking of factors of

*x*and recognizing this as a difference of squares (

^{2}– 16*16 = 4*and

^{2}*a*)), therefore, the factors of

^{2 }– b^{2 }= (a – b)(a – b*x*:

^{2}– 16*(x – 4)(x + 4)*.

Both central executive measures of working memory, as well as composite EF measures, predict improvements in mathematical competency – and they can be improved. The important finding from research is that progress in mathematics is related to improvement in executive working memory and vice-versa.

Effective teaching focuses both on the development of mathematics content and strengthening EF skills. Executive function is strengthened when there

(a) is information in the long-term memory—vocabulary, conceptual schemas, efficient strategies and procedures,

(b) is immediate feedback to students’ attempts in applying a strategy or solving a problem,

(c) are “good” “scaffolded” questions from the interventionist, and

(d) is enough supervised practice to automatize skills.

Nearly all the components of executive function are involved in arithmetical calculations and in creating conceptual schemas, each playing a somewhat different role. Working memory, as a whole, is the cognitive function responsible for keeping information online (the screen and the sketch-pad of the mind), manipulating it, and using it in our thinking; it is truly responsible for thinking. It is where we delegate the things we encounter to the parts of our brain that can take immediate action. In this way, working memory is necessary for staying focused on a task, blocking out distractions, and keeping our thinking updated and aware about what’s going on around us. Working memory is intrinsically related to executive function. No matter how smart or talented a child, he or she will not do well without the development of key capacities of working memory and executive function.

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