Working Memory: Role in Mathematics Learning (Part Two)

Markers of Working Memory Problems in Learning Mathematics
Because working memory is involved in so many activities, there are many different indications of working memory problems. Teachers, parents, and students should be aware of situations where working memory lapses may be evident. Here are some possible signs:

  • Trouble remembering the components of a multi-step task, like standard procedures (loss or delay in recall, failure to follow instructions, place-keeping errors, and frequent task abandonment)
  • Trouble remembering the sequence of the task elements
  • Trouble concentrating on a task or instruction
  • Easily distracted from a task
  • Forgets what s/he is doing in the middle of a task
  • Trouble staying on the same topic when talking
  • Forgets instructions easily
  • Trouble doing more than one thing at a time
  • Skipping or repeating steps in a task
  • Trouble with reading comprehension or understanding instructions
  • Difficulty prioritizing multiple demands posed by the task(s)

All or many of these signs may be implicated in a child having trouble with mental arithmetic and in mathematics problem solving.  It is important, however, to recognize that everyone experiences some of these difficulties some of the time; that is normal. But if these kinds of problems or incidents are frequent and/or severe, they may be an indicator of a working memory problem. Because it is difficult to determine a working memory problem informally, formal testing is required.

There are several reasons for a person’s working memory to be taxed: task complexity, lack of understanding of the task (due to poor reading skills, poor language of mathematics, lack of concepts and skills, etc.), poor teaching, and personal reasons (emotions, stress, and math anxiety). For example, when a person is stressed, the pressure of anxiety blocks the working memory; in such situations even the facts with which people are familiar cannot be recalled because the “mind has gone blank.” This is the impact of stress blocking the working memory. Even more importantly, math anxiety influences those with high rather than low amounts of working memory—precisely those students who have the greatest potential to take mathematics to high levels. When students who experience stress in timed conditions cannot access their working memory, they underachieve, which causes them to question their math ability and, in many cases, develop further stress and anxiety.  Several studies have demonstrated that the link between the working memory and mathematics difficulties is stronger in children than adults and therefore places greater limits on the computational performance of children than young adults.

Working Memory’s Influence on Arithmetic Problem Solving
Problem solving in any setting is a complex cognitive activity. However, it is more demanding and involved in mathematics settings as most of the activity is being accomplished at abstract and symbolic levels. A multitude of lower and higher order cognitive thinking skills (e.g., pattern analysis), language processing (decoding, comprehending, meaning making, etc.), visuo-spatial (e, g., space organization and orientation, visualization, etc.), and reasoning (deductive and inductive) skills are called upon. To be proficient in solving an arithmetic or mathematics problem, an individual would have to able to

(a) focus attention on each component of the information about and in the problem as it is presented (vocabulary, syntax, format, etc.) to understand it (as against reading in areas other than the sciences, where one can still have some understanding of the problem without close attention to the specific language, word for word, used in the problem);

(b) hold the relevant information in the working memory (to receive, comprehend the meaning and concept involved, and to translate from native language to math language—quantitative and spatial, etc.);

(c) scan the long-term memory to find related language, concepts, or procedures, formulas, definitions, and skills;

(d) mix, relate, and reformulate the new information with the relevant information from the long-term memory—reshaping existing conceptual schema by consolidating partial schemas, extending previous schemas, and even abandoning previous schemas in the light of new information and understanding;

(e) manipulate the problem information by mentally performing the required operations (attach the appropriate conceptual schema to the information);

(f) selectively maintain some of the information (most recent outcome of the operation) in a temporary mental storage (buffer between working and long-term memories); and

(g) complete all of these tasks within the span of a few seconds to minutes (each component of the operation is identified and acted upon).

All of this is performed multiple times during the problem solving process. Moreover, this process is iterative in every multi-step procedure/algorithm/ problem-solving situation in mathematics. For example, in executing the standard long division algorithm, the student needs to have a mastery (conceptual understanding, fluency, and applicability) of the related concepts and procedures (multiplication tables and procedure, place value, subtraction, and estimation), and the prerequisite skills (following directions, spatial orientation/space organization, visualization, following the pattern—estimate, multiply, subtract, bring-down, etc.).

In a world problem, understanding the language (vocabulary and syntax) for identifying the variables and the problem, seeing their relationships, forming expressions and equations (translating from native language to mathematics), then solving them (mastery of concepts and procedures) and finding the relation of the solution to the problem (translating from mathematics symbols to native language: does the answer make sense?) also involve an iterative process. The same process happens when we are finding the factors of a whole number or an algebraic expression (e.g., trinomial).

The Demands of Procedures on Working Memory
Iterative processes heavily tax the working memory. However, concrete and pictorial models used in arriving at these iterative processes help minimize the impact on the working memory. For example, the use of area model in arriving at multiplication and division procedures whether involving whole numbers, fractions, decimals, or binomial expressions helps see the sequence of steps and connections between concepts. In the case of finding the factors of a number the proper use of empty number line helps.

Similarly, to find the factors of 72, if we represent the factors on the number line by their location, it is easier to understand the interrelationship between the different factors and can extend to finding the greatest common factor (GCF) or least common multiple (LCM). For example, the factors 1 and 72 are located on the number line. Then we ask what the next number that divides 72 is. Using the divisibility rules, we find 2 divides 72 as 72 is even. And factors 2 and 36 are identified (by performing the short division). Now, 2 is placed next to 1, and 36 is placed in the middle, in its right location, on the number line. The empty space (between 36 and 72) indicates that there are no factors between 36 and 72. The divisibility rule is used again and we find 3 and 24 are the new factors.  They are located in their appropriate places.  By the same argument we arrive at the conclusion that there are no factors between 24 and 36, between 18 and 24, and 9 and 18.

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To find the greatest common factor (GCF) of two numbers (say, 48 and 72), it becomes visually clear to a student as the common factors will occupy the common locations, thus, the greatest of the common factors of 48 and 72 can be found easily, as:

capture2

On the other hand, to find the least common multiple (LCM) of the two numbers we write the multiples of the two numbers, identify the common multiples, and then identify the least of them.  This means, we need to employ two processes (one for GCF and the other for LCM with finding the multiples of the numbers may involve several calculations); these procedures are cumbersome and not generalizable to algebraic expressions. This puts more demands on the working memory.

However, the successive prime division (SPD) method connects GCF and LCM better and makes fewer demands on the working memory. For example, the first prime factor is found by asking the question: “What is the first prime number that divides 72 and 48?” By the help of the divisibility test one knows the first prime number is 2 and the corresponding factors (36 and 24) by the help of short-division are found. The same process of successive prime division is repeated to find all the common factors.

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The first column gives us the common factors of 72 and 48, therefore, the greatest common factor (GCF) of 72 and 48 = 2×2×2×3 = 24.Similarly, applying the definition of the least common multiple of 72 and 48 (LCM), we find that it should have factors 2, 2, 2, 3, 3, 2 and removing any one of them will not satisfy the property of multiples of 72 and 48. Thus, the LCM of 72 and 48 = 2×2×2×3×2×3 = 144.

The visual representation of the successive prime division method visually reduces the demands on the working memory and GCF and LCM are found together. This process reduces the work involved in adding and subtracting fractions more effectively using the successive prime division to find the least common denominator (LCD).

Similarly, in the case of finding the factors of a trinomial (e.g., x2 +6x +5), the use of algebra tiles or Base Ten blocks helps students to see (x + 3) and (x +2) as the factors, represented by the sides of the rectangle and its area as their product—the trinomial, x2 +6x +5.

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The model facilitates the visualization of the multiplication of binomials, and even the division of the trinomial by a binomial becomes clear—the vertical side as the divisor, the area as the dividend and the horizontal side as the quotient. This visual modeling helps students to hold the visual representation in the working memory’s sketchpad. For this reason, the procedures we use and how we derive them play an important role in the task-load on the working memory.

Engagement and Its Role on Working Memory
Engaging the learner, particularly those who experience difficulty in learning mathematics because of ADD or ADHD and working memory issues, in a mathematics learning or problem-solving task is the most important goal of a teacher, tutor, or interventionist. To effectively and meaningfully engage students in the learning tasks requires proficiency in:

(a) knowing the trajectory of the development of the mathematics concept or procedure at hand, that means knowing: (i) what secondary and primary concepts and prerequisite skills are involved, (ii) how much and the level of language required, (iii) how to establish the sequence of the tasks, and, (iv) to determine the cognitive complexity of each task involved, etc.,

(b) understanding the learner characteristics—cognitive preparation, mastery of language, reading and comprehension levels, basic skills level, presence or absence of conceptual schemas, nature of mathematics learning problem/disability, limitations imposed by the nature of the learning disability (e.g., the level and condition of the learning disability places limitations on the prerequisite skills—following directions, pattern recognition, spatial orientation/space organization, visualization, estimation, deductive, or inductive reasoning); and

(c) acquiring competence in effective, efficient, and elegant (generalizable to abstract concepts and procedures) pedagogical approaches—models, questioning, instructional materials, sequencing of tasks, tool building, appropriate and timely reinforcement.

Working memory functions require the learner, in mathematics problem solving and conceptualizing mathematics ideas, to simultaneously attend, store, and mentally process a rather large amount of information within a relatively short period of time. These functions demand a higher level of active involvement on the part of the child. This is particularly difficult if a child has organic/neurological reasons for his/her ADD or ADHD. It requires the teacher or interventionist to be extra vigilant, creative and aware of the child’s strengths and weaknesses.

Information Load on Working Memory
Information load on working memory in the problem solving process is a major factor in determining task complexity and difficulty and completion of the task by the child. The task difficulty arises, for example:

(a) in elementary school, in the use of effective strategies for learning facts (additive and multiplicative—sight facts, decomposition/ recomposition, development and execution of sequence of steps in procedures and mastering them)[1],

(b) in upper elementary and middle school, in discerning and using patterns in arriving at conceptual schemas and standard procedures (multi-digit multiplication and division) and in pre-algebra (fractions, integers, concept of equations, etc.)

(c) during high school, in deductive and inductive reasoning in understanding concepts and procedures in algebra, and in deriving definitions, relationships between different geometrical entities, development and execution of proofs—use of spatial, deductive, and syllogistic reasoning, in geometry.

In all of these complex concepts and procedures, the working memory is highly involved, and this involvement is essential for understanding and applying concepts and ideas.

Whenever a child faces a new concept, particularly a secondary concept (a concept involving several primary concepts, multi-directions, etc.), s/he faces an overload on the working memory. The more a child is free from constructing basic arithmetic facts when needed, the more the child is able to devote the limited working memory resources on learning the new language, concept, or procedure and their relationships. Automatizing basic arithmetic facts, therefore, is important for two reasons: (i) the student is able to discern patterns in number relationships and therefore make more connections and (ii) the working memory is freed from constructing these facts every time there is need for them.

On the other hand, using ineffective strategies such as “counting up or down” for arithmetical operations (counting up addition, counting down for subtraction, skip counting forward for multiplication, skip counting backward for division) takes up all the available space in the working memory.

Similarly, giving children the multiplication tables or addition charts does not reduce the load on the working memory. The students will not comprehend the problem without understanding the language and conceptual schemas related to the problem and the isolated facts retrieved from these sources or generated on calculators do not provide the opportunity of making connections. These approaches, therefore, are not “real” answers to the problem. However, when the students have acquired the language, concepts, and strong numbersense, then giving them calculators or computer for executing procedures or problem solving is good use of assistive technology.

How to manage the workload on working memory is the joint work of the teacher and the learner. Better understanding of task analysis (by the teacher and then by students) and the development of metacognition in students are essential. Effective task analysis is dependent on understanding the concept. The teacher should share how to do the task analysis (e.g. “notice to multiply 124×8,one should know place value—expanded form, multiplication table of 8, distributive property of multiplication over addition, and then adding) and then during the “tool building time” of the lesson, she should be developing these tools and their integration. Each lesson should have three components: tool building (25% of the lesson time), main concept/ procedure (50% of the lesson time), and reinforcement—guided, differentiated and supervised practice to master the concept/procedure in order to convert it into skill-set.

The teacher should anticipate the level of information load on children’s working memory before assigning a task. When a teacher has a better understanding of the task analysis of concepts, procedures, and problem solving process and the student has a higher level of metacognition, the student is better able to manage information. Therefore, helping students to become aware of their own understanding of their learning, strategies, and awareness of the task analysis prepares them for better engagement in the tasks and enables them to manage the task-load on the learning system, including the short-term, working and long-term memories.

As children develop, they use their working memory in different ways and for different purposes. It is critical for learning the alphabet and number concept in the first few years (Pre-K, K, and 1), for reading comprehension and mental arithmetic in elementary school, and for completing homework independently, solving multi-step problems, and completing projects in upper elementary and middle school. In high school, working memory is essential for writing essays and reports, proving theorems, solving problems. Working memory performance is crucial on tests such as the SAT and ACT: one has to keep all four multiple-choice options in mind and decide which is best! In college, working memory helps students maintain their focus during long lectures, complete papers, lab reports, and study for exams.

As students reach higher grades, the complexity of mathematics content and the related higher order reasoning skills needed place extra demands on the learning systems. Mathematics arguments in proving theorems in geometry, multi-step problem solving, and multi-step equations invite reasoning about relationships that involve two or more concepts and ideas, for example, “all rectangles are closed figures” and “all squares are rectangles.” The student needs to hold this information to derive an inference from these two statements.  If this information is provided only orally, it is difficult to hold that much information in the working memory and also to focus on each statement simultaneously. It taxes the working memory and overwhelms the thinking process for those who have limited working memory space. In such a situation, however, if the same information is visually presented (e.g., Venn diagrams, graphic organizers, flow-charts, etc.), then the two premises are constantly available for examination and therefore the load is reduced. Overloading this fragile mental workspace can lead to significant loss of information from the working memory and a feeling of being lost and bewildered.

Teachers and textbooks sometimes inadvertently present information in a manner that may unduly strain the processing capacity of a student’s working memory. To help children learn arithmetic and mathematics, teachers and interventionists need to understand:

  • the role of working memory in learning and teaching of arithmetic and mathematics,
  • how the working memory components can be formally and informally assessed,
  • how limitations in the working memory contribute to the development of mathematics difficulties and disabilities,
  • what kinds of instructional interventions or remedial approaches are available for mitigating the detrimental effects of the working memory limitations in mathematics achievement, and,
  • what kinds of interventions can improve the working memory space and its components.

Memory Functioning and Learning
Research points towards the feasibility of working memory strategy training in enhancing numbersense. In fact, all three types of memories can be improved with the use of efficient models, effective strategies, proper practice, and an enabling sequence of questions that help a child to see patterns and relationship. A strong positive relationship between the working memory and mental arithmetic competence is evident.

Of course, memory (short-, working, and long-term, explicit and implicit, episodic and procedural, etc.) plays important roles in any learning. Poor long-term memory has impact on learning. If a child is a fluent reader and has good comprehension, s/he is capable of mastering the basic arithmetic facts (addition and subtraction—sums up to 20 and decomposition/recomposition, multiplication and division—tables up to 10 and distributive property of multiplication over addition). However, the child’s ability is dependent on whether effective strategies have been used in teaching them. Poor strategies (addition is counting up and subtraction is counting down) do not leave residue from a learning experience. Most of the time, therefore, it is not poor long- term memory, but poor strategies of learning a concept or skill and lack of proper and efficient practice that are responsible for children not remembering their facts.

There are ways that long-term memory can be improved. The two most effective learning techniques are distributed practice and practice testing. Distributed practice is spreading out practice sessions over time. So instead of spending a long time on the same subject/topic/same procedure, one studies for small segments of time each day. For example, rather than reviewing the previous grade’s material for the first two months of the new grade, the teacher devotes a few minutes each day as a tool building session in each lesson. Or, rather than devoting a great deal of time on mastering facts, the student masters a strategy and applies it right away. For example, learn the table of 8 and then simplify fractions that involve only the multiples of 8, divide a multi-digit number by 8, or multiply a six-digit number by 8.

Practice testing is trying to answer a question without looking at the solution. That means doing a new math problem to learn a math concept or answering a fact related question to remember a fact rather than just repeatedly reading that material or repeating that fact. Arriving at facts by counting again and again does not help children to automatize them. Using properties of operations and definitions to deriving the facts: For example, using distributive property and decomposition/recomposition to derive facts.

Distributed practice is effective because of the spacing effect. Spreading exposure to multiple sessions, separated in time, will have a better long-term impact on memory.  In order to turn an experience into long-term memories, the short-term exposures to information need to be consolidated. Spending more time with the material, but not allowing space between for consolidating the information may mean some of the extra exposure time is wasted. Each time we instigate the old information as we start using the information, our brain needs to activate the context of memories it is a part of. Each time we activate this context, we strengthen our ability to do it in the future. Studying in one batch only needs to load the context once, so it doesn’t strengthen as much as having to recall from scratch multiple times.

Distributed practice does not mean cutting practice sessions into tiny slivers to maximize the spacing effect, which has the unrelated downside of making focus very difficult to accomplish.

A better way to implement distributed practice is to review older units, chapters, tests, facts, or definitions on a regular basis. The teacher should not make learning the current lesson as the only goal for the children. By devoting a little extra time to review, the accumulated reviews will make far more impact than a cram session. For example, each homework assignment should include problems from previous topics on a regular basis.

Many teachers and students do not see the difference between recall and recognition. For example, when we see some material again and want to answer questions related to it, we are testing only recognition.  But, recognition is usually easier than recall. The ability to recognize information can give a false sense of confidence about the subject. One may feel that one “knows” but cannot recall it when one needs it on a test or in a real situation, leading to frustration and failure. Recall is an indication of mastery, whereas recognition shows only superficial mastery. If one knows something, one should be able to recognize that it is the right answer and should be able to recall the answer if someone asks the question. Properly designed multiple-choice questions can test both recognition and recall.

Memory research gives evidence that recognition and recall may involve different cognitive and psychological processes. And even if the two processes do share a common mechanism, it is not the case that being able to recognize a piece of knowledge is equivalent to or results in being able to recall it. Being able to remember something has two parts: first you need to have the knowledge represented in your brain. This representation, initially calls for visualizing that information and then constructing a conceptual schema for that information. Understanding the language associated with the concept and then constructing the schema (a working memory activity) helps to send it to the long-term memory. But then, crucially, you also need to be able to find it at the right moment and in acceptable time.

Practice testing gets around this since it forces the student not just to store information but also to develop strategies to search for it at the correct time. Practice and reinforcement using effective, efficient strategies help place the information in multiple places. Recall is facilitated when there are multiple contexts to search in the long-term memory.

The best way to apply practice testing is to avoid its opposite: passive learning. Students should not re-read again and again the notes unless they are searching for the answer to a particular question. Instead, they should cover up the notes and see if they can describe the concept or repeat the definition or reproduce the procedure without looking at the solution. This also means that after the students have understood the concept or procedure, they should have access to large sets of practice questions on that topic, and they should be encouraged and helped to practice them. Nothing will serve the students better than doing large numbers of practice questions as the foundation to learning a subject.

Let us consider an example: learning the long-division algorithm. At first, the student might not get the procedure correct all the time. There are several concepts, skills, and steps and a lot to think about, so she may forget to apply a step as the memory for it as a whole may not be that strong yet. To master this procedure (estimation, multiplication, and subtraction), the teacher should write 10 to 20 problems and make the first step to estimate the answer in all of these problems and check it by actually multiplying the estimated quotient with the divisor, and then the next task for all of these problems, and so on. If we sampled the student performance on a test with different component tasks/questions, we could measure how much better she gets at doing the test over time. The score will steadily get better as she gets more practice.

Eventually, however, the student will reach a plateau. After this point, she mostly stops getting better at the test. This could be because she is scoring 100% every time as the familiarity of the tests sets in. Now, we could make the test difficult enough so that it is impossible not to make a few silly mistakes, leaving her with 95% or something similar. The question is: Should she keep on practicing beyond this point? Is there any benefit in continuing to practice? The answer, surprisingly, is yes.

Continuing to practice after this point has a different effect. This effect, called overlearning, doesn’t affect her test scores (that’s already at a maximum). Instead, it helps with the longevity of the memory.

In the beginning, practice improves performance. Later, when performance is maximized, it continues to improve longevity of the memory, and in the process fluency is also achieved. Overlearning something, therefore, is the strategy for remembering it permanently.

Is Overlearning Worth It?
The answer is, no doubt, yes. It is essential for automatization, fluency, and freeing the working memory to acquire higher order thinking. In every facet of our lives we do it, so how do we achieve it?

Strategy #1: Learning +1
The first approach comes from studies where the effects of learning a skill are assessed at different intervals. For example, in a study in which people were given an algebra test after a class, and then tested again decades later, to see how much they had remembered.

Interestingly, the people who did best on the first test didn’t have more durable memories than those who did poorly. Of course, if you remembered more for the first test, you’d remember more for the second. But the *rate* of forgetting in both cases was the same. But one group of students did not see their memories decay: students who went on to study calculus. In the process of learning calculus they used algebraic skills routinely and therefore reinforced the algebraic skills. Learning a subject above one’s current level forces one to overlearn the basics of the previous subject. If one wants to make memories last longer, one should apply the skills to problem situations and to more advanced topics, and in the process one learns the previous skills and retains them longer and stronger.

Strategy #2: Immersive Overlearning
Immersion method of learning languages makes a great use of overlearning. It can only partly be used in learning mathematics. Complete immersion is possible if the students’ learning is project-based learning where they see mathematics and all other skills for extended periods, integrate mathematics ideas with language, representation, discerning and forming relationships as mathematics expressions and equations, and then solve them. True and complete immersion in mathematics is practiced only by professional students of mathematics (mathematicians), physicists, engineers, technologists and mathematics teachers. However, the new crop of STEM courses is a good attempt to have the kind of immersion that helps students to engage in learning.

Strategy #3: Practice Makes Perfect
Practicing a test repeatedly is not a good idea, but practicing a different test on the topic is good even after scoring 100%, particularly those concepts and skills that you want to make permanent. For example, mastery of arithmetic facts should be made permanent as they are used in all aspects of life. The key is that one needs to be selective—overlearning every possible fact will limit time to learning new things. However, if a certain set of knowledge is essential for students to have at their fingertips, we should make sure that they overlearn it. We should practice it until they get it perfect and then practice it some more.

Working Memory Can be Strengthened
Though working memory has been studied for decades, it has only recently been proven to be a plastic function of the brain, able to be strengthened through rigorous training and effective and efficient teaching strategies. The brain is capable of enormous change through experience. This capacity of the brain is called neuroplasticity. With appropriate activities and exercises, we can improve our working memory to be better equipped to meet challenges. Like a muscle, it can be improved through certain types of exercises.

Researchers have used neuroimaging techniques to explore the neural basis of working memory plasticity. Their results show that, through working memory training, the activity of the brain areas related with working memory can be enhanced. For example, although the working memory capacity of seven-year-olds is smaller than that of older children and adults, their attentional processes are just as efficient—so long as their smaller working memory capacity is not exceeded by overloading it with extraneous and irrelevant information. When their working memory is overloaded, attentional efficiency declines, suggesting that intervention aimed at enhancing working memory will in turn improve attentional efficiency. In general, children’s attention to relevant information can be improved by (a) minimizing irrelevant objects or information cluttering working memory, (b) training them in efficient learning strategies, and (c) using effective and efficient learning models and materials.

Studies show a positive relationship between brain plasticity and learning. For example, just like the reading brain is different than the non-reading brain, the experience of learning math facts actually changes the memory patterns and neural connections and in turn aids in learning more facts. These connections become more stable with skill development. So, learning addition and multiplication tables and having them in rote memory (of course, after understanding them) helps develop the capacity to learn not just the content but the ability to learn more.

Nevertheless, the memorization of facts should be practiced first with the simple facts children already know (e.g., table of 1, 10, 5, 2, 4, 9 must be mastered before others and the 45 sight facts of addition, commutative property, making ten, and teens numbers should be learned before any other addition or subtraction facts)[2], then one should show them how to construct new ones using decomposition/recomposition for addition and multiplication facts. Similarly, in preparation for mastering tables, we should quiz children their multiplication tables in different order, for example, ask: 8×1, 8×10, 8×5, 8×2, 8×9, and then 8×6, etc. If the child does not respond within a few seconds, help her to break it into two multiplication facts: 8×6 = 8×5 + 8×1, then help the child to combine them. Then practice them so they really remember and do not have to think it through. Similarly, to find 8 + 7, we should ask what number will make 8 as 10. The answer: 2 (application of the strategy of making 10 or sight facts of 10). Then, what is left in 7 after 2 is used up to make 8 as 10. Answer: 5 (Sight facts of 7). Then what is 10 + 5. The answer: 15 (making teen’s numbers).  So what is 8 + 7? The answer: 15. So, the arithmetic facts should be automatized using efficient strategies. For some children, initially, the construction of facts in their minds may still be slow, but it will be faster than if it is not automatized or they derive them by counting.

With effective strategies (e.g., for addition—decomposition/recomposition, mastering making ten, teens’ numbers), rehearsals, and usage, facts become automatic.

The feeling of automaticity is a result of brain circuitry that’s been strengthened through repetition.  When we have automatized basic facts, the brain doesn’t have to work as hard on simple math. It has more working memory free to process the teacher’s new lesson on more complex math, and more patterns can be seen and more connections are made. The novice and the person who has not mastered facts use, for example, the memory system differently. For instance, most adults don’t use their memory-crunching hippocampus in the same way as novices. Retrieving six plus four equals 10 from long-term storage for them has become almost automatic, instead of a great effort.

Learning, for example the 45 sight facts of addition, calls for the integration of information and the function of the four components of the working memory—executive function (allocator of resources like attention and focus), phonological memory (holding linguistic information and its rehearsal), the visual/spatial sketchpad and memory (holding visual spatial information), and the episodic buffer. It is important, therefore, to use materials and strategies that help the child to enhance the working memory and learn, in this case, the sight facts.

Visual Cluster Cards (VCC) and Cuisenaire Rods (CR), because of their color, shape, size and patterns, force the executive function to attend to and direct it to the task. Further, they engage the visual-spatial sketchpad. The language used by the teacher is supported by the ability of the VCC and CR to attract the attention of the phonological loop. Thus, the child can hold that information in the working memory, manipulate it, and send it to the long-term memory.

The teacher’s questions, commentary and associated language with VCC and CR activities, student decision making, the patterns of the VCC, lengths, and the colors of the CRs enhance the phonological loop, visual/spatial sketchpad, and executive functions. As a result, these materials not only help children to learn the sight facts but also integrate them with existing concepts and schemas. Thus, these materials strengthen the different components of the working memory. The auditory information gains obligatory access to the phonological store: we do not have to do anything to create a phonological record. However, nameable visual inputs such as pictures, written letters or written words, must first be ‘recoded’ into a phonological form in order to gain access. Concepts presented through these materials provide access (as the color, length, and pattern invoke words) to the phonological store and, therefore, the learning is optimized.

Since VCC and CR have quantitative information, it is easy to create a phonological and visuo-spatial record as efficient and elegant concrete models invoke language naturally. The mix of visual-spatial information from VCC and CR and vocabulary forms number relationships and creates a higher possibility of a stay in the episodic buffer before searching for the related facts and concepts, making the number relationships and connections in the long-term memory.

Improving Working Memory
It is possible to significantly improve working memory through training, practice, and the effective use of efficient models and materials.

Using brain research, gaming experts have created computerized programs to improve working memory through exercises and training. Some examples are computerized working memory-training programs such as: Cogmed Working Memory Training and Lumosity. These programs include a series of engaging working memory games and challenges. Like most popular videogames, these games get progressively harder as players’ skills and capacity improve. The programs claim that working memory improves and that the improvements last long after the training ends. Their research seems to show that the programs are effective for working memory deficits that accompany ADHD, stroke, and aging. They advise that an adult oversee the training process when children are doing the training – sitting with the child, encouraging him or her when the tasks get difficult, witnessing and appreciating his or her efforts, hard work, and successes.

In general, certain principles about this research on learning are useful in day-to-day interactions in the classroom and at home:

Learning means new connections in the brain
The brain works like an electrical circuit. Just as an electrical current travels through a circuit, signals are transmitted from one group of neurons to another. Every time one learns something new connections form between neurons in the brain. The stronger the electrical signal, the stronger the connection between the neurons. The stronger the connections, the greater an individual’s ability to form and retrieve facts from memory easily and fluently.

More connections mean more effective learning
Just as the strength of neural connections plays a role in learning, so does the quantity and quality of these connections. Learning is not just acquiring knowledge or facts; it is linking them and freely connecting old and new knowledge. An isolated fact can be tough to remember or recollect, unless it is overlearned, connected, or accompanied with a strategy.

One can make learning a fact easier by relating it to other networks of information in one’s brain. This is the idea of constructivism – that we are able to place Lego bricks of knowledge into our long-term memory, use them, and build on them when and where we need to. The more ways we construct (this happens in the working memory), the more places we place (this happens in the long-term memory), the more connections we make (more residue in the long-term memory), the more is the flexibility of thought, and the easier it is to retrieve the information.

For example, when we derive a fact (e.g., 8 + 6) in multiple ways (e.g., (a) 8 + 6 = 8 + 2 + 4; (b) 4 + 4 + 6; (c) 2 + 6 + 6; (d) 8 + 8 – 2; (e) 7 + 1 + 6; and (f) 3 + 5 + 5 + 1) and with multiple strategies such as: making ten [(a), (b), and (f)]; doubles [(c), (d), (e), and (f)] not only do we make many more neural connections and reinforce previous facts, but we also place the facts in different files, by making more connections, in the long-term memory. The quantity, quality, and strength of these connections are the memory traces. Memories can be procedural, emotional, semantic, and some other kinds. Thus, to summarize:

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What Can Teachers Do to Enhance the Working Memory?
The teacher must monitor the child’s performance and be on a lookout to recognize working memory related failures and lapses. She should ask parents, the child’s other teachers, and the child for examples of such working memory lapses. For example: “she lost her place in a task with multiple steps,” “raised his hand to answer a question, but when called upon he had forgotten his response,” “raises his hand, but then responds by saying never mind,” etc.

Developing the ability to observe, discern, expand, create, and apply patterns is an anti-dote to working memory failures and the means to enhance the working memory. For example, asking children to solve addition and subtraction problems (any level from whole numbers to algebraic expressions) using visual-spatial (continuous) materials that emphasize patterns, color, size, shape, etc. (Visual Cluster Cards, Cuisenaire rods, Base-Ten blocks, TenFrames, Fraction strips, Algebra Tiles, Invicta tiles, etc.) rather than just sequential discrete (discontinuous) materials (e.g., fingers, counting blocks, number line, hash marks, etc.). Compared to discrete discontinuous material the visual-spatial (continuous) materials because of their pattern forming ability linger in the working memory longer and their representations easy to recall from the long-term memory.  Discrete discontinuous models should be used only initially to introduce an idea, but should move to more efficient continuous models.

In planning their lessons, teachers should aim at reducing unnecessary working memory loads and include activities that enhance the working memory. Working memory load should be varied according to the task either verbally or visually (e.g., long definitions; premises in syllogistic reasoning tasks are presented visually so that the information is continuously available for inspection). Arithmetic procedures should be arrived at by using efficient models and materials (e.g., multiplication and division of numbers, from whole numbers to algebraic expressions should be arrived by using Cuisenaire rods, fraction strips and BaseTen blocks and employing the area model) and appropriate language and questioning. Heavy loads are caused by lengthy sequences of information, unfamiliar and meaningless content, and demanding mental processing activities. The teacher should evaluate working memory loads in tasks before assigning them to children.

The teacher should repeat important information to help children process and should ask diagnostic questions to ascertain whether the information has been received correctly. The teacher or students nominated as memory guides can supply the repetition. Load on the working memory can also be minimized if important information can be identified with the student who makes/arrives at the conjectures. Conjectures named after the students are remembered better as they serve as “pegs” for memory. Similarly, the teacher should name each strategy derived and used (e.g., making ten, divisibility rules, successive prime division method, vertex form of the equation, etc.) and should remind them when they forget the name (e.g., missing double strategy, etc.).  When they use a strategy, students should use the name of the strategy, why they chose to use it, and describe it (e.g., in combining integers, if the signs are same, then keep the sign and add the numerals and if the signs are different, then keep the sign of the larger numeral and subtract the smaller numeral from the larger numeral).

Teachers should encourage use of memory aids only after understanding of the concept or procedure so that memory aids do not tax the memory. These include wall charts and posters, word wall, display of important definitions and formulas, use of graphic organizers, frequent use of visualization exercises, etc. Complex concepts place heavier loads, but use of efficient materials mitigates that load. For example, use of Visual Cluster cards and Cuisenaire rods is more efficient in developing decomposition/recomposition, properties of numbers, and properties of operations—commutative, associative, distributive properties; properties of equality and inequalities. Efficient use of multi-sensory materials helps minimize the working memory loads.

The load on working memory and retrieval from long-term memory are reduced when children develop their own effective and efficient strategies. These include asking for help, frequent rehearsals, productive self-testing, and effective note-taking—designated math notebook (graph papers—where left pages is for note-taking and the right side page for student work, the right page is divided vertically into two parts—2/3 vs. 1/3, left part for problems and right part for calculations and subsidiary work), efficient organizational strategies, graphic organizers, mnemonic devices, etc. Visual and auditory information is presented for a subject-determined time and material.

Emotion and Working Memory
Emotional control, or the ability to regulate one’s emotional responses and staying focused on a goal, helps determine success. Research shows that emotional control and working memory rely upon some of the same brain areas, including the fronto-parietal area and the amygdala. Since these two functions share brain pathways, strengthening one could strengthen the other. At the same time, certain parental and teacher behaviors—meaningful praise, affection, sensitivity to the child’s needs, and encouragement, along with intellectual stimulation, support for autonomy, help in organization, task analysis, and well-structured and consistent rules—can help children develop well functioning working memory and robust executive function skills.

[1] See an earlier blog on Sight Facts and Sight words.

[2] Sequence of strategies for teaching addition and multiplication facts in How to Teach Arithmetic facts Effectively and Easily (Sharma, 2008). List of strategies is available free from the Center (www.mathematicsforall.org).

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Working Memory: Role in Mathematics Learning (Part Two)