During my yearly clinical course, where many of the participating children have learning problems in mathematics (for example, specific learning disabilities, dyscalculia, dyslexia, etc.), the question of the role of working memory and executive function in mathematics learning kept coming up from teachers and parents.
The terms working memory and executive function are seen as important components in human development and learning and are implicated in many learning problems. Working memory is at work not just in formal learning. Working memory is one of our most crucial cognitive capabilities, essential for countless daily tasks like following directions, making connections, and remembering information momentarily, complex reasoning, or staying focused on a project. In other words, it is the complex system of mental mechanisms responsible for the integration, manipulation, and temporary storage of information that is relevant and important to an individual’s focus of attention at the moment at the task at hand.
Understanding the importance of working memory can provide great hope to people who suffer from working memory deficits, including those with attention problems (ADD or ADHD), learning disabilities, or injury to the brain. Children with attention problems often have working memory deficits; however, in some cases, poor working memory may be the cause of certain second order attention problems. Deficits in working memory can affect an individual’s ability to focus attention, control impulses and solve problems. Someone with a working memory deficit or limitation can have difficulty attaining proficiency in mathematics, particularly problem solving as working memory load is a major factor in determining task difficulty. They may lose focus frequently when reading and solving a mathematics problem. While the connections between working memory deficits and mathematics performance seem clear, it is not certain whether these deficits cause attention deficits behaviors.
Executive functions and working memory differ between low achieving and typically achieving children not only in acquiring reading skills but also in mathematics achievement. The working memory and its functioning are heavily taxed in academic subjects such as reading and computational mathematics. Researchers have found strong involvement of working memory and executive function in mathematics learning and difficulties. For example, an increasing number of studies show executive functions as predictor of individual differences in mathematical abilities. Deficits in different components of executive function can be seen as precursors to math learning disabilities in children. Number concept, numbersense, and numeracy, implicated in dyscalculia and learning these concepts, are highly dependent on working memory and executive functioning. Similarly, math anxiety suppresses the working memory and executive functioning interfering normal calculations.
Numbersense is an intuitive understanding of numbers, their magnitude and inter-relationships. The cognitive mechanism that helps strengthen numbersense is working memory. Furthermore, when the predictive value of working memory ability is compared to preparatory mathematical abilities, there is a definite relationship between them. Performance on working memory tasks predicts math learning abilities and disabilities, even over and above the predictive value of preparatory mathematical abilities. Strong and efficient use of working memory has been linked to higher academic success, including mathematics.
There are many reasons children may fail to learn or experience difficulty in learning arithmetic—number concept, numbersense, and numeracy. And these arithmetic difficulties, in turn, contribute to difficulties in learning other mathematics concepts and procedures. Apart from environmental factors such as poor instruction (teaching ineffective and inefficient strategies), lack of skill experience and reinforcement/practice, poor expectations, other examples for difficulty in learning include anxiety about mathematics, lack of experience and poor motivation, reading difficulties, neuropsychological deficits and damage, and cognitive delay and deficits. Arithmetical learning difficulties can be associated with cognitive deficits.
The cognitive deficits have a long list; however, they are exemplified by poor memory—short-term, long-term and/or working memory, lack of flexibility of thought (centeredness), lower levels of abstract thinking, visual perceptual deficits, and inefficient language development.
One component of cognitive ability is the size and working of working memory. More working memory space and flexible and effective usage by individuals mean greater potential for academic success, including mathematics.
Working memory functioning improves throughout childhood, adolescence, and adulthood and can be strengthened through intensive practice and training. George Miller was the first psychologist to attempt to quantitatively measure the working memory’s capacity. Miller coined the term ‘magical seven’– the idea that working memory could hold seven plus or minus two items. These items can be digits, letters, words, phonemes, bits, or small groups.
We can fool this limitation by chunking the information. Chunking is grouping and organizing discrete pieces of information into smaller groups/clusters. Through chunking, we organize and collect information and relate ideas. By chunking related pieces of information we can fit more into our working memory. Our ability to chunk together different kinds of information allows us to carry out a complex practical task without being overwhelmed.
We can easily overfill our working memory. When we do, we induce cognitive overload. Students who struggle with chunking new information become overloaded and cannot fit more information in their working memory, not without discarding something else. Cognitive overload can create misconceptions and muddy previously clear concepts.
Definition of Working Memory
Working memory (WM) refers to the capacity to store information for short periods of time when engaging in cognitively demanding activities. Whereas the short-term memory is like a relay station—the information is constantly coming and going, WM plays a more influential role in learning and academic performance, including mathematics. This is because mathematics tasks involve multiple steps with intermediate solutions, and children need to remember those intermediate solutions as they proceed through the tasks.
One of the areas affected by poor working memory is attention. But what we rate as inattention has nothing to do with actual attention. Actual attention is hard to assess (you have to control for motivation, competence, reward, relevance, etc.). We generally settle for sustained engagement as an alternative and call it attention. The prevalent thinking is that if you didn’t keep going, you probably lost attention, which is usually untrue. Most kids stop trying because the task is difficult (they do not understand it and do not have the skills for it), it has no intrinsic or extrinsic value (it is boring or unrewarding for them), or they do not see the purpose or connection of it with anything they know (the teaching was not engaging). It usually has nothing to do with attention even though many parents and teachers consider boredom and switching from one task to another as a sign of inattention. The question is: Did they just drift away or were they looking for something more fun and interesting? In other words, is their attention poor or is their tolerance for boredom and frustration poor?
Most current research explores the dependence of mental calculation on working memory and how the limited-capacity system of working memory affects keeping track of temporary information during ongoing processing of mental calculations. Empirical studies tend to support the view that it is the limited capacity of the working memory that is responsible for inattention.
Not all models and pedagogy in use when planning activities in schools pay enough attention to the role of working memory in learning arithmetic. In this post, we explore answers to questions such as:
- How is working memory related to learning arithmetic, therefore, mathematics?
- How does working memory support numeracy, particularly, calculations?
- What can we do to help children with poor working memory?
- What teaching/intervention strategies and models can enhance and support working memory for all children?
Components of Working Memory
In the late 19th century, the American psychologist William James first proposed the distinction between a “primary” memory with a limited capacity and a long-term memory. British psychologists Baddeley and Hitch added a third element—working memory, to the learning cycle consisting of short-term and long-term memories. They postulated working memory as a temporary storage of information between the short- and long-term memories.
This model has three components: a central executive component, a phonological loop, and a visuo-spatial sketchpad. The phonological and visual components are referred to as ‘slave’ systems given that they hold specialized information for short periods of time. Working memory, thus, is a multifaceted function that captures visual, spatial, kinesthetic, and auditory information, directs attention to it, and coordinates processes to deal with its components, nature, and functioning. Much of the research in the cognitive psychology of working memory has been influenced by this multi-component model of working memory. It is time to bring it to the classroom and tutorials.
The central executive component has five capacities: to (1) coordinate and monitor input from the two slave systems, (2) shift attention, (3) focus on one stimulus, inhibit and/or enhance it, (4) hold and manipulate information from short- and long-term memories, and (5) update information.
The phonological loop is further divided into two sub-processes: a phonological input store and an articulatory rehearsal process. The articulatory rehearsal process refreshes verbal input. It focuses on the auditory and linguistic input.
The visuo-spatial sketchpad is devoted to visuo-spatial input. It pays attention to color, shape, texture, size, patterns, etc. The phonological loop is devoted to processing verbal speech input and is part of the rehearsal process for visual input as well.
Working memory is the ability to maintain and manipulate information temporarily. Despite its limited capacity, with effective materials and efficient strategies, an individual is able to perform complex cognitive tasks. It is the core of high-level cognitive activities and an essential component in the processes of learning, comprehending, reasoning, problem solving and intelligent functioning.
Whereas the short-term memory is a unitary storage and a passive place, the working memory is a multi-modal, multi-component, and multi-function place where temporary storage takes place before the information is intentionally transferred to long-term memory – if it is not transferred, it escapes. It is an active system that provides the basis for complex cognitive abilities. In working memory, we consciously process selective information; therefore, working memory is linked to attention control.
In the working memory we store small amounts of information in order to use that information to complete a task at hand—e.g., create a new conceptual schema and learn or form a new idea. In the working memory we bring information from the long-term memory and mix it with the incoming information from the short-term memory to comprehend, to learn, to solve problems, complete tasks, manipulate and see relationships, and make connections.
Short-term memory is a relay station—information enters involuntarily and leaves. Either involuntarily or voluntarily information is transferred to either long-term memory or working memory. It goes to long-term memory when (a) it connects to some related information that overlaps with it, (b) we connect it with what we already have in store, (c) it has novelty, or, (d) it is emotionally charged.
The information from the short-term memory goes to the working memory when we consciously begin to work on it—(a) mix it with information from long-tem memory, (b) reorganize or represent it to construct new information, and (c) rehearse it.
Construction of Concepts
The comprehension of incoming information takes place in the working memory. We classify it, represent it, organize/reorganize it; we transform it into a word(s), a graphic, a conceptual schema, a strategy, or a procedure and then by understanding it and rehearsing it send it to the long-term memory. The process of construction, in the working memory, can be self-initiated by reflection either on a recent event or in the past. It may also be instigated by concrete or visual models, words, diagrams, metaphors, similes, analogies, or some information that can be accessed from the long-term memory. For example, a child is asked:
What is 6 + 8?
Situation 1: Answer: 14. The problem 8 + 6 (presented orally or symbolically) invokes the response instantly, if the child has mastered it before (it already resides in the long-term memory). If the answer is affirmed, the memory traces: 8 + 6 = 14 is strengthened.
Situation 2: 6 + 8? Child manipulates the information: I take 4 from 8 and give it to 6 and then I have 10 and 4. That is equal to 14. This is happening in the working memory. He has made connections and the working memory and long-term memory both are strengthened.
Situation 3: Oh 6 + 8 = 8 + 6. I take 2 from 6 and give it to 8 and then I have 10 and 4. That is equal to 14. This is again happening in the working memory. He has made connections and the working memory and long-term memory both are strengthened.
Situation 4: 6 + 8. I think of 6 as 8 and I know 8 and 8 is 16 then I take 2 away from 16 and I have 16 – 2 = 14. This is happening in the working memory. He has made connections and the working memory and long-term memory both are strengthened.
Situation 5: 6 + 8. I take 1 from 8 and I have 7 and 7. That is 14. So 6 + 8 = 14. This is happening in the working memory. He has made connections and the working memory and long-term memory both are strengthened.
Situation 6: 6 + 8. I can find the answer in several ways. I have found it before and now I know it right away. This is where we want to ultimately arrive in learning: understanding, fluency, and applicability.
Teaching students effective and efficient strategies using effective instructional models makes lower demands on their working memory; it also facilitates recalling information from the long-term memory, holding the information, and manipulating it in the working memory. In the process they improve their working memory.
Notice all the components of the working memory system are being called upon and strengthened.
Situation 7: When the child does not know the fact, the responses are: (a) repeats the question, once or several times. Let me see: 6 + 8? A child with a poor strategy and/or poor teaching (adding is counting up) counts 7, 8, 9, 10, 11, 12, 13, and 14 either in his head or on his fingers. Let me see I have counted 8 or not? And he verifies by counting. Notice, he needs to maintain 16 numbers (1-7, 2-8, 3-9, 4-10, 5-11, 6-12, 7-13, 8-14) simultaneously in his head (working memory) and that is difficult. In this process none of the components of the working memory system are used and strengthened. The activity will not leave the trace of the final result in the memory, and the same process will be repeated next time the same problem is presented.
Ineffective strategies place a higher demand on the working-memory and create frustration that further diminishes the functioning of the working memory. When facts are not mastered using effective strategies, situations 1 to 6 do not happen.
When effective strategies are learned, one can extend them to develop and strengthen mental math—an activity that is dependent on a strong working memory. For example, when one tries to find the sum: 58 + 17, the child may think of the following operations in his mind. 58 + 2 is 60 by taking 2 from 17 so 2 less than 17 is 15, so I add 15 to 60 and I get 75, so 58 + 17 is 75. All of this is taking place in the working memory. However, to come to this level of mental calculation, efficient concrete materials and pictorial representations help. Here the child had experience in using Cuisenaire rods (58 = 5 orange rods and the brown rod—the 8-rod and 17 = 1 orange rod and the black rod—7-rod.). Similarly, using the Empty Number Line (ENL), one can create images in the working memory and, therefore, develop mental math—holding numerical information in the working memory and manipulate it.
The color and size of the Cuisenaire rods engage the slave systems—articulatory loop (as we read the rods as numbers), articulatory rehearsal (the presence of the rods keeps the information alive in the memory), and visuo-spatial memory (the size and color of the rods). Even children with poor working or short-memories are able to achieve more. In the process they improve their working memory and create and leave a residue of the experience in the long-term memory. When concrete models are supported by efficient and elegant representations, the images are further strengthened. For example, one can go easily from Cuisenaire rods to Empty Number line.
Cuisenaire rods, Visual Cluster cards, and Empty Number Line help children to acquire mental math competence.
Recognition and Comprehension
Many studies have examined the relationships between working memory and word recognition. The same system is involved in the recognition of visual clusters and large numbers (place value)—essential elements for the development of numeracy. However, a problem like the one mentioned above also involves comprehending and understanding that system and manipulating the numbers using strategies. Good readers allocate more working memory resources to text comprehension than to word recognition when compared to poor readers. Good readers produce more integrative inferences than poor readers, who are constrained by their working memory processing capacities when building mental models of texts. Similarly, low achieving students, because of poor strategies (addition is counting up and subtraction is counting down) and ineffective instructional models (counting objects, number line, and lack of patterns and color) face heavier loads on their working memories. They allocate more working memory space for deriving arithmetic facts by counting, and little space in working memory is left for seeing patterns, relations, and making connections. As a result, they miss developing understanding, fluency, and mental math and have difficulty in applying their knowledge and skills to problem solving.
Working memory, effective strategies, efficient models, and residues in the long-term memory interact with and influence each other throughout the learning process.
We use the working memory as the sketchpad and working place for thinking—as the brain’s conductor. It allows us to hold onto information, for a short time, and then to work with and manipulate that information. So for example, when we speak or do mental calculations, the working memory brings the words, arithmetic facts, concepts, and relevant procedure that we know together and connects them into a coherent sentence or outcome—a calculated answer.
The phonological loop is specialized for the storage and rehearsal of speech-based verbal information (notice the language used in the six situations of 6 + 8 above) whereas the sketch-pad is specialized for holding visual and spatial material (the equation formed by the Cuisenaire rods, graphic organizers, Empty Number lines, Invicta Balance, Algebra Tiles, etc.). They constantly interact with each other.
Most of the time, the working memory is the connecting link between the short- and long-term memories and plays a crucial role in sending information to the long-term memory. What goes in the long-term memory is dependent on what is being worked on in the working memory. One can greatly enhance the capacity, the nature, and functioning of the long-term memory by improving the working memory system. For example, when we want to calculate the product 222×3, we bring the relevant information from the long-term memory, such as 222 = 200 + 20 + 2 (understanding of place value), 2×3 = 6, 20×3=60, 200×3=600 (the facts and the distributive property of multiplication over addition) to the working memory and then mixing this information, we get 232×3 = (200 + 30 + 2)×3 = 200×3 + 20×3 + 2×3 = 600 + 60 + 6 = 666.
Different types of information are brought from the long-term memory to the working memory—from specific to general (recognizing 5 as a prime number) and from general to specific (which rule of exponents to apply in evaluating a3×a4). Children compute with mathematics facts—such as those required in timed tests—by recalling them from the long-term to working memory and using them in computations/procedures in paper-pencil situations or mental math.
Formation of Memories
In living and learning, we rely on two types of long-term memories: explicit and implicit memories. Explicit memory is the “conscious” memory for specific facts and events, as opposed to “subconscious” implicit memory. Remembering a fact (8 + 6 = 14) or concept one relies on explicit memory (multiplication means repeated addition, groups of, an array, or the area of a specific rectangle). Remembering how to add (8 + 6 = 8 + 2 + 4 = 10 + 4 = 14), in general, relies on implicit memory. Similarly, remembering the concept of multiplication as repeated addition relies on implicit memory whereas the ability to remember that 7 × 8 is 56 or am × an = am+n relies on explicit memory.
The concrete models and particular learning activities are the means for students to create conceptual schemas, associated visual representations, verbal discussions (development of language containers, and words and instructions for language rehearsals), and cues (mnemonic devices). These processes help us remember the information later. They are responsible for the formation of explicit memories. However, activities such as projects, explorations, experimentation, and problem solving situations are helpful in forming implicit memories. Proficiency in mathematics calls for both. Implicit memories build understanding and comprehension and context for a particular concept and explicit memories help in developing fluency, skills, and procedural competence. The integration of the two types of memories helps us in the applications and problem solving processes.
Components and Related Tasks
Different components of the working memory have specialized roles in learning arithmetic. For example, the phonological loop appears to be involved in arriving at facts by using a variety of strategies: (a) counting: when a child tries to find 8 + 6 by counting up and says: 9, 10, 11, 12, 13, and 14 and at the same time keeps track of the addend 6 as 1, 2, 3, 4, 5, and 6; (b) decomposition/recomposition: (i) when the child says: 8 + 2 is 10 and then 4 more is 14; (ii) when he says: 6 + 4 is 10 and 4 more is 14; (iii) using properties of mathematical entities in holding information involved in complex calculations: to find the answer for 23 × 7 mentally, the child first thinks of 23 as 20 + 3, calculates 20 × 7 and thinks 2 × 7 = 14 so 20 × 7 = 140 and 3 ×7 = 21, and 140 + 21 = 161, so 23 × 7 = 161 (also known as distributive property of multiplication over addition); and verbal rehearsal of the problem to keep it current: to find 23 × 7, the child keeps repeating the problem or the components of the problem. The actions in (b) rely heavily on visual/spatial working memory.
Children with poor arithmetic have normal phonological working memory but have impaired spatial working memory and some aspects of executive processing. Compared to ability-matched controls, they are impaired only on one task designed to assess executive processes for holding and manipulating information in the long-term memory. These deficits in executive and spatial aspects of working memory seem to be important factors in poor arithmetical attainment.
The visuo-spatial sketch pad appears to be involved in operations involving multi-digit problems where visual and spatial knowledge of column positioning is required, relationships between positions of digits in multi-digit numbers, and location and position of objects in visual clusters. For example, mentally locating numbers on the Empty Number Line as in when we find the difference 91 – 59, one thinks of 59 on the number line and then takes a jump of 1 to reach 60 and then a jump of 31 to reach 91, arriving at the answer of 91 – 59 = 32, and spatial representations of individual numbers. All of these actions take place in the working memory’s visual-spatial sketch-pad.
The role of the central executive is noted in many situations in learning and mastering arithmetic language, concepts, operations, and procedures. The central executive processor is responsible for identifying, initiating and directing processing, symbol and word recognition, comprehension and understanding, and retrieval of relevant information from the long-term memory. For example, all the decisions in estimating and computing the answer for 23 ´ 7 mentally are executive functions of the central processor and are being processed in the working memory. To estimate, the student first thinks of 23 as about 20 and 7 as either 5 or 10. And converts the problem mentally as 20 ×10. Then he thinks 20 × 1 = 20, so 20 × 10 = 200, or he thinks of 20 × 7. This is possible, of course, if he knows the concept: What happens when you multiply a number by 10, place value, and the table of 2. Of course, there are other routes. However, a child who thinks of 23 is made up of 2 and 3 will never be able to estimate the answer. This is an interplay between executive function in the working memory and the information being brought from the long-term memory. To compute 23 × 7, the child first thinks 23 as 20 + 3, therefore, 23 ×7 is thought as (20 + 3) × 7, and then to calculate 20 ×7 he thinks 2 ×7 = 14 so 20 × 7 = 140 and 3 × 7 = 21, and 140 + 21 = 161, so 23 × 7 = 161. All of these decisions involve the central executive. When a student has mastered the operation of multi-digit multiplication with understanding, he can apply the procedure mentally. That will again take place in working memory with help from long-term memory.
Appropriate and precise language, effective concrete and pictorial models, and efficient strategies are important not only for learning quality content but also for improving student learning capacity, including the working memory.
(Part Two: How to Enhance Working Memory for Mathematics Learning)