The anxiety-performance link has two possible causal directions. They have been extended into the specific field of mathematics anxiety. The **first** direction is explained by the ** Deficit Theory**. Mathematics performance deficits, for example on mathematics tests, generate mild to extreme mathematics anxiety, which may lead to higher anxiety in similar situations. For example, students who have not mastered non-negotiable skills with efficient strategies at their grade level (number concept—Kindergarten, additive reasoning by the end of second-grade, multiplicative reasoning by the end of fourth-grade, etc.)[1] attempt fewer problems on tests, thereby lowering their score. For example, if a student does not have the mastery of (a) multiplication tables, (b) divisibility rules, (c) short-division, and (d) prime factorization before they do operations on fractions, they will have difficulty in mastering them; they will, therefore, be afraid of proportional reasoning (fractions, decimals, percents, ratio, proportion, scale factor, etc.) and then algebraic operations. To turn lower performance into high-level of math anxiety requires time. But, in the case of a vulnerable child (e.g., learning disability, lower cognitive and executive functions, etc.), anxiety may take less time to manifest and may escalate quickly. Thus, children with mathematical learning disabilities are often found to have disproportionately higher levels of mathematics anxiety than typically developing children, supporting the Deficit Theory.

The **second** causal direction is that anxiety, particularly math anxiety, reduces mathematics performance by affecting *any or all of these processes*:

- the
*pre-processing*(initiating or responding to mathematics tasks—attitudinally and cognitively, negative predisposition for mathematics in general and particular mathematics), *processing*(making sense of the problem—linguistically, conceptually, and/or procedurally, connecting multiple presentations of the problem—data into table, graph, or diagram, etc.),*retrieval of information*(relevant prior knowledge—formulae, definition, equations, concept, or skills),*comprehending*(understanding the problem, making connections between the incoming information and prior information and knowledge, translating the words and expressions into mathematical expressions and equations, etc.), and*perseverance*(engaging and staying with the problem and showing interest in the outcome of the problem),

thereby reducing the level of performance. This is referred to as the ** Debilitating Anxiety Model**.

**The Deficit Theory Model of Anxiety
**In at least some cases, having especially poor mathematics performance in early childhood could elicit mathematics anxiety. This poor performance could be the result of environmental factors such as poor math teaching, lack of resources and experiences.

Studies of developmental dyscalculia and mathematical learning disabilities indicate that specific cases of mathematics anxiety are related to poor performance, but that poor performance could be attributed to these deficits and then the resultant math anxiety. However, only 6-8% of the population suffers from developmental dyscalculia and such findings cannot be generalized to the typically developing child. It should also be noted that cognitive resources are not the only possible deficits that could cause poor mathematics performance and math anxiety. For example, self-regulation (one of the components of executive function) deficits have been associated both with mathematics anxiety and decreased mathematics performance. The condition of *acquired dyscalculia* (e.g., children without learning disabilities who show gaps in their mathematics learning will fall in this category) is a clear example of poor mathematics performance.

It has been found that significant correlations exist between a student’s mathematics performance, both at elementary and adolescent age, in one year and their mathematics anxiety in the following year. These correlations are stronger than those found between a student’s mathematics anxiety in one year and their academic performance in the following year, indicating that mathematics performance may cause mathematics anxiety, thus providing support for the Deficit Theory.

**The Debilitating Anxiety Model
**Mathematics anxiety can impact performance at the stages of pre-processing, processing and retrieval of mathematics knowledge. Recent research suggests that anticipation of mathematics tasks causes activation of the neural ‘pain network’ in high math anxiety individuals, which may help to explain why high math anxiety individuals are inclined to avoid mathematics. In young children, task-avoidant behaviors have been found to reduce mathematics performance. Similarly, many adolescents with mathematics anxiety avoid math-related situations, suggesting that mathematics anxiety influences performance by reducing learning opportunities.

Adults with high mathematics anxiety answer mathematics questions less accurately but more quickly than those with lower levels showing that mathematics anxiety is associated with decreased cognitive reflection during mathematics word problems. Because of poor numerical skills, adults do not have resources to check their answers for correctness. Such data suggest that adults with mathematics anxiety may avoid processing mathematical problems altogether, which could lead both to reduced mathematics learning and to lower mathematics performance due to rushing, lack of engagement, and lack of comprehension. Adults with mathematics anxiety are less likely to enroll in college or university courses involving mathematics.

The worry induced by mathematics anxiety impairs mathematics performance during mathematics processing by taxing processing resources and minimizing their impact. Worry reduces working memory’s processing and storage capacity, thus reducing performance. For instance, research shows a negative correlation between college students’ math anxiety levels and their working memory span. Further, there is an interaction between adults’ mathematics anxiety and their performance on high and low working-memory load mathematics problems, with high working-memory load questions being more affected by mathematics anxiety. Thus, mathematics anxiety appears to affect performance by compromising the working-memory functions of those with high math anxiety.

Mathematics anxiety also affects strategy selection, leading individuals to choose simpler and less effective problem-solving strategies and thus impairing their performance on questions with a high working-memory load. This is supported by evidence suggesting that those with high working-memory, who usually use working-memory intensive strategies, are more impaired under pressure than those who tend to use simpler strategies.

Mathematics anxiety may manifest as (a) lack of willingness to engage in the activity because of previous negative impressions of mathematics, (b) poor reception and information processing, therefore disposing individuals to avoid mathematics related situations, (c) poor comprehension of mathematics information in mathematics learning tasks, thereby abandoning the tasks prematurely and giving up too easily, and (d) later, at the stages of processing and recall, mathematics anxiety may influence performance by cognitive interference. Math anxiety, thus, may negatively tax executive function resources, such as working memory, which are vital for the processing and retrieval of mathematical facts and methods. All of these affected behaviors impact the *Standards of Mathematics Practice (SMP) [2]* identified and recommended by the framers of

On the other hand, positive emotions enhance learning by increasing the willingness to initiate tasks, develop persistence, use effective strategies and recruit cognitive resources. The idea that emotions have an effect on general achievement and particularly on math achievement is strongly supported by studies across all ages that manipulate anxiety to reveal either a decrement or improvement in performance. This effect of mathematics anxiety on performance is likely through executive function skills. This is particularly so in the case of working memory. The working memory functioning is impaired by the intrusive negative thoughts, negative talk, and poor self-esteem generated by math anxiety.

The mechanisms of influence of math anxiety, particularly cognitive interference, may be more immediate than from one academic year to the next. Since the effect of anxiety on recall would cause a fairly immediate performance decrement in those with high mathematics anxiety, this supports the debilitating anxiety model—the impact of math anxiety on performance.

To conclude, the evidence for the relationship between math anxiety and mathematics performance is mixed. Neither theory can fully explain the relationship observed between mathematics anxiety and mathematics performance. While some studies provide data, which fit the Deficit Theory, others provide more support for the Debilitating Anxiety Model. The mixture of evidence suggests a bidirectional relationship between mathematics anxiety and mathematics performance, in which poor performance can trigger mathematics anxiety in some individuals and mathematics anxiety can further reduce performance in a vicious cycle.

The belief about a causal relationship should prompt articulating educational policy, program planning in mathematics education, developing initial and then remedial mathematics instruction, assessment, particularly for those who suffer from math anxiety.

For example, if policy-makers share the belief that math anxiety is just another name for ‘bad at math,’ to reduce students’ math anxiety, effort and money will be targeted at courses to improve their mathematics performance. It will involve searching alternative teaching methods to mitigate this situation. In some cases, this may be (a) the development of computer-adaptive programs that may offer a way to ensure that students do not experience excessive failure in their math learning, (b) adjusting the difficulty level of mathematics tasks to an individual student’s ability, or (c) adapting remediation to student’s mathematics level and his/her mathematics learning personality[4].

If the relationship is in fact in the other direction, such efforts are likely to be ineffective and it would be better to focus on alleviating mathematics anxiety in order to improve mathematics performance. Then, it is important to understand the nature of classroom teaching that may produce math anxiety and focus on remediation of math anxiety. This will focus, particularly on methods, which may be undertaken in the mathematics classroom and during interventions. For example, writing about emotions prior to a math test has been seen to increase performance in those with high math anxiety. Because low mathematics self-concept is related to mathematics anxiety, when teaching, teachers should strengthen students’ academic self-concept, which has been identified as a factor related to academic performance.

The mechanisms proposed by the Deficit Theory are long-term, with the detrimental effect of poor performance on anxiety levels occurring over years. This may be why the Deficit Theory is often supported by

longitudinal studies. On the other hand, the Debilitating Anxiety Model, particularly cognitive interference, proposes some immediate mechanisms for anxiety’s interference with performance (e.g., taxing working memory resources). This could explain why the Debilitating Anxiety Model is best supported by experimental studies such as those investigating stereotype threats.

**The Reciprocal Theory
**The tension between the deficit and debilitating anxiety theories is indicative of the very nature of the mathematics anxiety-mathematics performance relationship. Whilst poor performance may trigger mathematics anxiety in certain individuals, mathematics anxiety lowers or further reduces the mathematics performance in others.

This relationship suggests a model in which mathematics anxiety can develop either from non-performance factors such as social, emotional, biological predisposition or from performance deficits. Mathematics anxiety may then cause further performance deficits, via avoidance and working-memory disruption, suggesting the bidirectional relationship of the ** Reciprocal Theory**. The question of whether the mathematics anxiety-mathematics performance relationship is in fact reciprocal is likely to be best answered by longitudinal studies across childhood and adolescence since only longitudinal data can determine whether mathematics anxiety or weak performance is first to develop.

Some data suggest that previous achievement may affect a student’s mathematics levels of performance and that mathematics anxiety in turn affects future performance, and further proposes indirect feedback loops from performance to appraisals and emotions.

Mathematics anxiety in adults may result from a deficit in basic numerical processing (poor number concept, poor numbersense, and lack of mastery in numeracy skills), which would be more in line with the Deficit Theory. For instance, adults with high mathematics anxiety have numerical processing deficits compared to adults with low mathematics anxiety. Mathematics anxiety may result from a basic low-level deficit in numerical processing that compromises the development of higher-level mathematical skills. Highly mathematics anxious adults’ basic numerical abilities are impaired because they have avoided mathematical tasks throughout their education and in adulthood due to their high levels of mathematics anxiety, supporting the Debilitating Anxiety Model.** **

**Genetics, Environment and Mathematics Performance
**Genetic studies may help to elucidate whether mathematics performance deficits do in fact emerge first and cause math anxiety to develop. One such study suggests that some (9%) of the total variance in mathematics performance stems from genes related to general anxiety, and 12% from genes related to mathematics cognition. This may indicate that for some, mathematics anxiety is caused by a genetic predisposition to deficits in mathematics cognition. However, this does not preclude the possibility that the relationship between mathematics anxiety and performance is reciprocal.

Parental (and other authority figures in a child’s life) math anxiety could be transmitted to children; in other words, parents likely play an important role, either positive or negative. In that case, it is more of social transmission of attitudes towards mathematics rather than genetic.

Sometimes, some of the genetic factors are translated into or affected by stereotypical reactions. Stereotype threats also elevate anxiety levels, thereby affecting participation in and processing of math activities. Stereotype threat is the situation in which members of a group are, or feel themselves to be, at risk of confirming a negative stereotype about their group. Under stereotype threat, individuals are seen to perform more poorly in a task than they do when not under this threat. It is posited that this is due to anxiety elicited by the potential to confirm or disconfirm a negative stereotype about one’s group. This particularly applies to some minority and women’s groups.

The effect of increasing anxiety by stereotype threat can be seen in adults as well as in children. For example, research shows that 6–7 year-old girls showed a performance decrement on a mathematics task after they completed a task that elicited stereotype threat. Similarly, it has been observed that presenting women with a female role model who doubted her own mathematics ability reduced their performance in mathematics problems compared with a control group who were presented with a confident female role model.

Deficits in mathematics performance in women under mathematics stereotype threat appear because math anxiety coupled with the stereotype affect the working memory. This phenomenon supports the idea that mathematics anxiety taxes the working-memory resources and that reduces mathematics performance. The same phenomenon is active when mathematics anxiety affects mathematics performance as the compounding of stereotypes based on race, income level, and gender.

[1] See *Non-negotiable skills in mathematics learning* in previous posts of this blog.

**[2]*** Visit earlier posts on SMP on this blog. *

**[3]*** Visit earlier posts on CCSS-M.*

[4] See *The Math Notebook* on *Mathematics Learning Personality* by Sharma (1989).

]]>

**Math Anxiety of the Math-Type (or Specific Math Anxiety)**

Specific math anxiety is triggered by certain language, concepts, or procedures: for example, difficulty in memorizing multiplication tables when understanding is not there; long-division procedure; estimation when place value and facts have not been mastered; operations on fractions (why multiplications of two fractions may result in smaller numbers than the fractions being multiplied); understanding place value—particularly decimal places (where there is no one’s place); understanding and operations on negative numbers (how addition of two numbers is smaller than the numbers being added); algebraic symbols—the radical sign (one student declared how can a letter be a number, you cannot count with this); certain mathematical terms (how can a number have a value less than 4, e.g., p = 3.14159265358…, if it is going on for ever and it is not exact; *x* ≤ 4, how can any thing be equal to and smaller than something at the same time, etc.

These students are not able to come to terms with what their intuitive thinking tells them and what the new concept calls for. The conflict between their intuitive understanding of the mathematics ideas and the new mathematics concept creates a dilemma in their minds—a situation of cognitive dissonance. They may not have a strong conceptual framework and/or this particular concept to resolve the cognitive dissonance. The trigger for the resulting anxiety may be a symbol, a certain procedure, a concept, or a mathematical term. For some reason, that specific mathematics experience creates a mental block in the process of learning the new mathematics concept. Then, they doubt their competence in mathematics and, therefore, distrust mathematics. They find it difficult to go any further, give up or develop an antipathy towards the concept or procedure. Moreover, they declare incompetence in specific aspects of mathematics (self-diagnosis—I am terrible in fractions, equations, etc.). At this juncture of their math experience, fear of mathematics is the result and not the cause of their negative experiences with mathematics.

However, in some cases, since students remember the times they were successful and felt that they were good at mathematics; they do not fear **all** of mathematics. They have tried to understand that particular part of mathematics but now, as a result of unsuccessful and frustrating experiences, have developed anxiety about a specific aspect of mathematics. A particular concept becomes the locus of their math anxiety.

The reaction of persons with the specific math anxiety is also specific. When they seek help they have specific goals about mathematics and have specific need and their reactions about mathematics are also very specific. For example, they are apt to say:

“My teacher started doing geometry in class and I have always had difficulty in geometry. Can you help me go over this part of the course?”

“I have to take this exam. I always do poorly on exams, can you help me in passing this exam?”

“I used to be good in math up to sixth grade, but now with algebra I am lost.”

“I like geometry but I get lost in algebra, particularly the radical numbers and expressions.”

“I like arithmetic and algebra, but geometry is something else.”

“I had a really bad math teacher in eighth grade, it was all downhill after that.”

“I understand what you are saying, but I don’t see the meaning, I am sorry.”

“Calculus is so abstract. Can you show me this concretely?”

“Why can’t you explain the way my sixth grade teacher used to do?”

“If my sixth grade teacher had explained this material this way, I would have learned this material better.”

“I always got into arguments with the geometry teacher. I could not see the meaning of invisible points and lines.”

Key phrases by such students are: “Sorry!” and “I tried my best.” Their reactions are mild and of disappointment rather than of fear and inadequacy.

Many of these students are willing to try. They believe that if proper methods, materials, and examples are given, they can learn mathematics. These students complain about the teacher, the textbook, the class size, the composition of the class, anything outside of them. It could be anything related to their mathematics experience. As soon as that particular thing is changed, they feel they will be able to learn mathematics.

Whereas people with global math anxiety generally avoid taking mathematics courses, students with specific math anxiety will register for math courses, but if one of these conditions are not met, they may use that as an excuse for dropping the course. In that sense, they are easier to teach. They are looking for somebody to break the cycle of failure in that specific aspect of mathematics. They are eager to talk to math teachers willing to listen. They are not particularly afraid of math or math teachers, but they do not want to repeat the same experience of failure. They need help, and an effective math teacher can usually help them.

In the previous post we mentioned that social myths have created conditions for the prevalence of people with global math anxiety. It would seem that there are more people with global math anxiety, and that used to be the case only a few decades ago. Today however, specific math anxiety is much more prevalent than global math anxiety. There are several reasons for this phenomenon:

- A student may understand the concept on the surface level but may not truly understand the concept or procedure;
- A student may not practice the concept or procedure enough to the level of mastery so easily forgets the material. In the long run, the lack of mastery of nitty-gritty aspects of math is the source of the problem. Practicing problems of different types relating to the same concept helps students see the subtleties in the concept, and applications of the concept becomes easier. That builds stamina for mathematics learning.
- When important developmental concepts are not taught properly, students may not connect concepts properly, which means every new concept looks novel and unrelated, thereby creating mental blocks in the process of learning.
- When transitions of concepts are not handled properly, students may have difficulty learning concepts. For example, the transition from addition and subtraction (one dimensional—linear) to multiplication and division, is not just the extension of repeated addition/subtraction to skip counting for multiplication/division but is abstracting repeated addition to groups and developing it to a two-dimensional model of multiplication (as an array and area of a rectangle). It is a cognitive jump that requires effective and efficient concrete and pictorial models, language, and conceptual framework.
- Specific math anxiety can occur if mathematics is taught procedurally, without the proper base of language and conceptual development. Language serves as the container for concepts and concepts are the structure of a procedure. Without the integration of the three, students have to make extra effort to understand and master a concept. This takes a toll on their enthusiasm and motivation for mathematics learning.

These habits and inclinations do not help students learn mathematics easily and sufficiently well. They do poorly on examinations and tests and feel anxious about math because they lack the practice in integrating the language, concept and procedures. Timely help from a sympathetic mathematics teacher who uses efficient and effective methods of teaching that motivate these students to practice is key for improving math achievement and lowering math anxiety and thus breaking the cycle.

The ** first step**, in addressing specific mathematics anxiety is to identify the specific area of mathematics deficiency or where the students faced the first hurdle in mathematics.

The ** second step** is placement in an appropriate math class, instructional group, or matching with the right tutor with an individual educational plan. Then the teacher should develop a plan to attack first the student’s perceived and real incompetence/difficulty in mathematics. The perceived incompetence is often the result of negative experiences. Then the remedial help that they receive should begin with the focus on one’s deficient areas of mathematics and create success using

Vertical acceleration is taking a student from a lower level concept (where the student is functioning) to a grade level concept (where the student should be) by developing a vertical relationship (a direct path) from the lower concept to the higher concept. An example is when a student is having problems in fractions or solving algebraic equations because she does not have the mastery of multiplication tables. The teacher should focus on one multiplication table, say the table of 4, and, she should help the student to derive the entries on the multiplication table and learn the commutative, associative and distributive properties of multiplication using effective and generalizable model (area model) and efficient materials (e.g., Cuisenaire rods). Then she should help the student to practice the table of 4 using Multiplication Ladder[1] for 4 and then master the extended facts (×40, ×400, .4, etc.). Then the teacher should practice (a) multiplication of a multi-digit number by 4 (e.g., 12345×4, etc.) and division of a multi-digit by 4 (78695 ÷ 4, etc.), (b) form equivalent fractions and simplify them where the numerators and denominators are multiples of 4, and (c) solving one-step equations (e.g., *4x = 36; 40x = 4800*, etc.). When this skill/concept is mastered, it should be connected to the current mathematics. In the next session, the focus should be another table. Supplying students with multiplication tables and using calculators is not a solution.

When a student feels successful in one small area, then related metacognition helps manage learning and then math anxiety. Soon, it begins to disappear. When one provides successful experiences in mathematics at some level (even at a lower level than the chronologically expected mathematics complexity) to this type of student, he/she may lose the anxiety and feel better about mathematics and him/herself.

For this reason, I begin work with these students (say a ninth grader with gaps and anxiety) with simple algebraic concepts, integrating the corresponding arithmetic concepts or taking a simple arithmetic concept and relating it to algebraic concepts with the help of concrete materials and patterns. This process develops in students the feeling that they are capable of learning mathematics and begins to remove their fear. It is not uncommon to hear: “Is that all there is to algebra?” We then build on this newly acquired confidence by taking digressions to make-up for the arithmetic deficiencies by providing successful mathematics experiences using vertical acceleration techniques that result in further building of confidence and reduction in mathematics anxiety. Vertical acceleration is applicable in both global and specific math anxiety situations and in the case of all developmental mathematics concepts.

**Math Anxiety and Working Memory
**Working memory[2] is a kind of ‘mental scratchpad’ that allows us to ‘work’ with whatever information is temporarily flowing through our consciousness. It is of special importance when we have to do math problems where we have to juggle numbers, apply strategies, execute operations in computations, or conceptualizing mathematical ideas. For example, during computations (e.g., long-division, solving simultaneous linear equations, etc.), we have to keep some of the outcomes of these operations in our mind. These processes take place in different components of the working memory. Increased math anxiety with it demands on working memory reduces working memory’s functions that in turn affects performance. The cycle of poor performance and math anxiety ensues. However, the effect of math anxiety on working memory is limited to math intensive tasks. Thus, the role of working memory and its related component parts is a significant factor in accounting for the variance in math performance.

Just like general anxiety, math anxiety affects both aspects of working memory—visual and verbal, but there is no relationship between math anxiety and processing speed, memory span, or selective attention. However, in the case of mathematics, the effect on visual component of the working memory is more pronounced. Worries and self-talk associated with math anxiety disrupts and consumes a person’s working memory resources, which students could otherwise use for task execution.

Although there are similarities in the effect of general and math anxieties, math anxiety functions differently than general anxiety and other types of specific behavioral anxieties. Whereas general anxiety affects all aspects of human functions to differing degrees, there is no or only a limited relationship between math anxiety and performance on a non-math task.

There is an inverse relationship between math anxiety and performance on the math portion of working memory intensive math tasks. One reason for this is that math anxiety is directly related to the belief that mathematics seeks perfection (e.g., there is only one answer to a problem and there is one way of arriving at it) and there is a fear associated with the perceived negative evaluation when one gets a wrong answer.

It is true that people who are anxious in general often get test anxiety, but a lot of people who are not particularly anxious can still develop stress around tests in subjects like mathematics. What is actually going on when a student stresses out over a test? The moment an anxious student begins a test, the mind becomes flooded with concerns about the possibility of failure. Between the worry and the need to solve the problems on the test, a competition ensues for attention and working memory resources. That divided attention leads to a stalemate—called “choking.” The impact of this is the shutting down of the brain to that task.

This choking can be particularly visible in younger students. High school students may respond more like adults; they may find and use excuses for this shutting down—lack of preparation, poor teaching, irritability, lack of sleep, too early in the morning, too late in the day, etc. Young children just shut down—may start crying, won’t write much, withdraw from the activity, get angry, etc. They just get overwhelmed and don’t know how to deal with it.

Interestingly, due to anxiety, the fear response appears in both low- and high-performing students. However, the impact on students is different. It doesn’t matter how much the student actually knows, but rather how well he or she feels they have the resources to meet the demands of the test and how tightly the performance on the test is tied to the child’s sense of identity. Students who see themselves as “math people” but perform poorly on a math test actually repress their memories of the content of the class, similar to the “motivated forgetting” seen around traumatic events like death. The effort to block out a source of anxiety can actually make it harder to remember events and content around the event. So the student may feel, “I’m supposed to be a math person, but I’m really stressed out, so maybe I’m not as big a math person as I thought I was.” That stress becomes a major threat to the student. So, most surprisingly, math anxiety harms more the higher-achieving students who typically have the most working memory resources.

Changing a student’s mindset about the anxiety itself could boost test performance. For instance, students can be trained to reinterpret physical symptoms—a racing pulse or sweaty palms, say—as signs of excitement, not fear. Those students have better test performance and lower stress than students who interpret their symptoms as fear. Experiencing a sense of threat and a sense of challenge actually are not that different from each other. Ultimately, by changing one’s interpretation, one is not going from high anxiety to low anxiety but from high anxiety to optimal anxiety.

On mathematics tests and examinations, however, it is difficult to separate the effect of test taking anxiety from the mathematics anxiety; thus there is a compound impact. Specifically, for example, there is an effect of math anxiety on the SAT’s total score and individual SAT English, Math, and Science scores. In this case, the impact of test taking anxiety is a factor. A moderate amount of anxiety (irrespective of focus) has a positive impact on performance. For example, low math anxious individuals have higher SAT total and Math scores than both moderately and highly math anxious individuals. High math anxious individuals have low mathematics scores.

Although math anxiety begins to manifest more during the upper elementary school grades, studies show that younger children are beginning to demonstrate math anxiety. Some students report worry and fear about doing math as early as first grade. Research shows that some high-achieving students experience math anxiety at a very young age — a problem, if not treated, that can follow them throughout their lives, and they become underachieving gifted and talented students.

Studies have also found that among the highest-achieving students, about half have medium to high math anxiety. Still, math anxiety is more common among low-achieving students, but it does not impact their performance to the same levels, particularly on less demanding, simpler numeracy problems. Their performance is more affected by math anxiety on higher mathematics—multiplicative reasoning, proportional reasoning (fractions, decimals, percents, etc.), algebraic thinking (integers, algebra, etc.), and geometry.

A high degree of math anxiety undermines performance of otherwise successful students, placing them almost half a school year behind their less anxious peers, in terms of math achievement. High achieving students want to utilize efficient and multiple strategies that place higher demands on working memory and if these strategies are not properly taught, high achieving students begin to do poorly.

Less talented younger students with lower working memory are not impacted by math anxiety in the same way as it affects the students with high working memory. This is because less talented students develop (or taught, particularly, in remedial special education situations) simpler and inefficient ways of dealing with mathematics problems, such as counting on their fingers, on number line, or concrete materials. For example, they are taught that addition is counting up and subtraction is counting down, multiplication is skip-counting forward and division is skip-counting backward.

Counting is a less demanding mathematics activity on working memory when the counting objects are present—counting blocks, fingers, number-line, etc. However, when these students do not use these materials and want to do it without them in their head, then the same task is a heavily demanding working memory activity. However, counting mentally occupies the working memory completely and does not leave any space for higher order thinking or strategy learning. For example, to find the sum of 8 + 7 requires a student, whose only strategy, to hold two sets of numbers: 9, 10, 11, 12, 13, 14, and 15 and the matching numbers 1, 2, 3, 4, 5, 6, and 7. These 14 numbers fill the working memory space completely. Thus, these students have difficulty learning efficient strategies as they place more demands on working memory. In the absence of efficient strategies, they hardly achieve fluency without paying a heavy price on rote memorization.

Ironically, because these lower-performing students do not use working memory resources to solve math problems, their performance does not suffer when they are worried. However, their performance on demanding, complex and longer performance goes down as they demand the involvement of working memory and math anxiety undermines it. Because if these limitations, these students do not progress very high on the mathematics skill/concept continuum.

Academic abilities, size of working memory, and fear of mathematics interact with each other. Sometimes, due to mathematics anxiety even the higher cognitive ability and working memory are undermined. Such interaction affects the high achieving students more than low achieving students. Higher achieving students apt to apply higher order strategies in mathematics and these strategies demand more from working memory and math anxiety may undermine it.

Teachers who give choices in their classrooms lower the anxiety of students. Mathematics classrooms where students have the flexibility to choose some “must do” each day, as well as some “may dos” offer opportunities for them to succeed and make mistakes. Tasting success at the same time as learning to make mistakes is a sure way to improve learning skills. They should also have the opportunity to work with a group or alone. A more open-ended approach allows students to play to their strengths – choosing the problems that they are most comfortable with. This encourages them to stretch themselves a little, try out new things, and worry less.

[1] *Improving Fluency Using Multiplication Ladders (Sharma, 2008). *

[2] See previous posts on *Working Memory and Mathematics Learning*.

]]>

The myths that we inherit about a subject shape our thoughts and our journey in acquiring the knowledge and competence in it. Many times, individuals need courage to get out of those mythical ideas that we have formed to be truly open to learning.

In mathematics learning, as children we come to the subject matter without preconceived ideas, but very quickly we are shaped by the ideas and myths about mathematics learning our caregivers consciously and/or unconsciously share with us. Then we struggle between meaning making from our real experiences about learning math or forming ideas about it and reading these experiences in the shadow of imparted myths. Early on, many students, when they do not have positive experiences or do not have skills to make meaning from their own experiences, succumb to the prevailing myths about math learning.

In spite of many efforts by mathematics educators, psychologists and social reformers, the myths about mathematics learning and achievement persist. These myths color students’ mindsets about mathematics and its learning. Even administrators who are well-meaning and able but not well-versed in mathematics perpetuate these myths by emphasizing gimmicks and easy solutions to the problem of mathematics achievement. Moreover, even the experts in learning assumed that ability to learn (particularly mathematics) was a matter of intelligence and dedicated smarts and therefore did not study the issue. They assumed, it seems, that either people had the skill of learning or they did not. For them, intelligence –and thus the ability to gain mastery—was an immutable trait.

Yes, for learning mathematics, one needs some cognitive abilities, but one also needs to engage in the process of learning. The field of learning is rife with vague terms: studying, practice, know, mastery, etc. For example, does studying mean reading the mathematics textbook? Does it mean doing sample problems? Does it mean memorizing? Does practice mean repeating the same skill over and over again (like memorizing flash-cards, doing mad minutes)? Does practice require detailed feedback? What kind of feedback? Should practice be solving hard problems or easy problems? Should practice be intense or small chunks? There are so many imprecise terms, which feed into the myths a person selects. Effective teachers help students to achieve freedom from these myths.

**Myth 1
**

Learning math, like learning in general, takes knowledgeable teachers with high expectations, willing students, and, most importantly, a great deal of time and practice that result in success during each session. A growing number of studies shows that learning is a process, a method, a system of understanding. It is an activity that requires focus, planning, and reflection, and when people know how to learn, they acquire mastery in much more effective ways.

Learning math is much like learning a language—both need a great deal of exposure, “gestation” time, and with usage learners get better. Learning mathematics takes time, effective practice, and help in making connections. The symbols and notations make up the rules of grammar and the terminology is the vocabulary. Doing math homework is like practicing the conversation of math. Becoming fluent (and staying fluent) in math requires years of practice and continuous use. That is true about any field. To be good at anything we need to practice. Learning mathematics is a dedicated, engaged process; it is not a spectator sport. It is not about memorizing facts (*static data*); it is more about what we do with that data (*look for patterns*) and how we think better (*convert patterns into strategies*) by the help of that data (*learning*).

**Myth 2
**

Effective strategies and practice boost performance from baseball and tennis to balancing equations and proving theorems. On the other hand, when mathematics teaching is approached with an emphasis on procedures and memorization and when concepts and topics are taught in a fragmented manner, students see mathematics as a difficult subject.

Apart from effective teaching, the mathematics curriculum has to be well orchestrated at each grade level. To engage all students and take many more to higher levels, it is important that school systems emphasize (a) articulation and mastery of non-negotiable skills at each grade level, (b) common definition of knowing a concept or procedure by everyone concerned with mathematics education, (c) knowing the trajectory of developmental milestones in mathematics learning, and (d) the most effective and efficient pedagogy that respects the diverse needs of all children.

Many teachers feed into the myth about the difficulty of mathematics when they begin a topic with statements such as: “Fractions are difficult.” “Algebra is not for everyone.” “Irrational numbers are truly irrational, they generally do not make sense.” These kinds of statements make mathematics look difficult and then it becomes truly difficult.

Mathematics is the integration of language, concepts, and procedures. It is the study of patterns. As Godfrey Harold Hardy (mathematician) said: A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. Statements such as Hardy’s can serve as counter narratives to the myth of the difficulty of mathematics.

Another way to counter the myth is when teachers show students how it took many mathematicians a significant amount of time to develop the formulas and equations that they are attributed with creating and that they are studying now in middle and high schools. Teachers need to show that, as in any other field of endeavor, most mathematicians, even though they made mistakes, persisted in the process.

Students should be encouraged to learn that mistakes are part of learning and creating mathematics. Mistakes and efforts literally improve the cognitive abilities that are needed for any learning and, therefore, mathematics. Students struggle to find an immediate solution when solving problems and they give up so easily, but many mathematicians took many years to solve a single problem. This shows students that math is not about speed but rather devotion and perseverance. A teacher’s selection of a problem and the methods of attacking the problem should be well-thought out so that they begin to take interest.

**Myth 3
**

Can you be an artist, writer, or musician and be good at math too? Yes! Math is found throughout literature, art, music, film, philosophy, and it is essential to many “creative” fields. Although the structure of mathematics is created by man, every aspect of life can be modeled by mathematics principles. It is pervasive in nature, society, and all edifices. Mathematics is constantly making new tools that help all aspects of human endeavor. Its collection of tools shaped the imaginations of Leonardo DaVinci, Mozart, M.C. Escher, and Lewis Carroll. These are just a few of the artists who used math extensively in their works.

Geometry is the right foundation of all painting. – Albrecht Dürer (artist)

I am interested in mathematics only as a creative art. …Mathematics is the study of patterns. – GH Hardy (mathematician)

The mathematician’s best work is art, a high perfect art, as daring as the most secret dreams of imagination, clear and limpid. Mathematical genius and artistic genius touch one another. – G Mittag-Leffler (mathematician)

In addition to increasing student interest in learning math, a deeper understanding of how concepts were developed and which mathematicians were responsible for them adds content knowledge. Explaining and showing how the Babylonians and Greeks worked to get more precise values of pi only gives teachers additional credibility as masters of content.

Incorporating the history of math also allows for an interdisciplinary approach to teaching. As we move from a STEM philosophy to a STEAM philosophy (including the arts), the history of math shows us a relationship between music and mathematics, in particular with Pythagoras, as well as between art and mathematics, such as in geometry. There are many ways to incorporate the history of math into your classroom.

**Myth 4
**

The learning cycle in mathematics, as in other subjects, is predictable: it begins with information gathering, and we need some basic skills for this. Then, with discussion and exchange of ideas, we convert this information into knowledge. With application and usage in multiple settings (intra-mathematical, interdisciplinary, and extra-curricular applications) we convert our knowledge into insights. And over a period of time, with deeper engagement with the subject, we might even become wise (we may have *expert thinking skills* or *ability to solve unstructured problems*) in that area. For tough unstructured problems one begins to look for analogous thinking in other fields. Imagination and creativity become prominent as soon as we cross the threshold of information gathering to conversion into knowledge.** **

**Myth 5
**

**Myth 6
**

** **

]]>

I hated every minute of training, but I said, don’t quit. Suffer now and live the rest of your life as a champion. Muhammad Ali

Learning mathematics takes a special kind of courage and enthusiasm for the subject, and to be successful in it, one needs effective teachers. Effective teachers prevent failures while poor teaching may create them. Mathematics is a major hurdle for many elementary school students, particularly when teachers rely on drilling and memorization. Failures in early mathematics can develop into full-fledged anxiety and may result in avoidance of mathematics and more failure.

For a significant number people, young and adult, the failure to feel successful and consistent lowered achievement in mathematics turn into mathematics anxiety, ranging from mild to severe. Math anxiety is the strongest predictor of both applied and basic math performance. Math anxiety may continue at the work place and in higher education.

In addition to socio-cultural (e.g., educational) factors, differences in students’ mathematics achievement are also due to attitude, motivation, language and intellectual abilities (e.g., executive function skills). Learning difficulties, disorders, and mathematics disabilities, like dyscalculia, often arise as a result of issues with cognitive and executive functioning. However, some students experience failure in mathematics due to environmental/social factors only (e.g., inefficient teaching, lowered expectations).

A pertinent question related to mathematics anxiety is whether mathematics anxiety causes poor mathematics performance, or whether poor mathematics performance elicits mathematics anxiety.

**Executive Function
**Numerical skills are important for success with meaningful problem solving, higher order thinking and higher mathematics, but other cognitive factors and executive skills (domain-specific and domain-general) also play an important role. Some of these critical skills include: (a)

Executive function skills are critical at every stage of mathematics learning: from language processing for decision-making, monitoring, and controlling emotions, thought, and action—and then these abilities are required for planning and assessing one’s actions (*What mathematical operation to choose in this problem*? e.g., long division or prime factorization, etc.), attention (*What is the role of these symbols and words here? What is the relationship between them*?), and self-control and perseverance (*What do I know here? What else can I do here?*). These processes[1] allow us to respond flexibly to a mathematics problem or concept and to engage in deliberate, goal-directed, thought and action.

Executive functions also affect processing skills. The term *processing disorder* is an umbrella term covering a continuum from auditory processing to language processing—from acoustic/central auditory processing, to linguistic-phonemic, to decoding, to meaning within language.

An auditory processing disorder will impact language processing and therefore language related aspects of mathematics such as: conceptualization (e.g., 2 × 3 can be read as (i) 2 groups of 3, (ii) 2 repeatedly added 3 times, (iii) a 2 by 3 array, and (iv) area of a 2 by 3 rectangle; **⅔×¾ **is read as ⅔ of ¾, and the fraction, **¾** can be read as: (i) three-fourth/three-fourths, (ii) 3 out of 4 equal parts, (iii) 3 divided by 4, (iv) the ratio of 3 is to 4, (v) 3 groups of ¼, or (vi) ¼ repeated 4 times), of and problem solving. A child may have language processing difficulty in varying degrees of severity. Such a demand of language in mathematics creates difficulties, and with math anxiety the language processing is further compromised.

Math anxiety compromises executive functions, contributing to difficulty in learning mathematics, thereby lowered achievement, and more mathematics anxiety. Math anxiety affects students’ intellectual factors such as learning styles, persistence, working memory, and organization.

Brain research shows that anxiety makes learning harder because it activates the amygdala and the limbic system and pulls processing away from the pre-frontal cortex. If a student who makes a mistake stays calm, his brain will stay in a mode to learn, whereas if a student freaks out with a stress response at each mistake or unfamiliar answer, his brain will be less able to process the incoming information or make connections between different pieces of information both new and old. This can create a spiral: *General anxiety —> making errors or lack of understanding —> math anxiety —> more mistakes or less understanding—> greater anxiety —> poor performance —> math anxiety.*

**Math Anxiety
**Mathematics anxiety is a state of discomfort and helplessness around the performance of mathematical tasks, and is generally measured using self-report trait anxiety questionnaires or can be observed by mathematics educators. On one hand, there is broad consensus that presence of mathematics anxiety is linked to poor math performance. On the other, poor mathematics performance is the result of math anxiety. Thus, debilitating emotional reaction to mathematics activity is either the cause of or an outcome of underachievement in mathematics.

Mathematics anxiety is a child’s emotional reaction to negative mathematics experiences, an intense feeling of helplessness about quantity. This is a global phenomenon. Even in countries where students on the average do better than most countries, there are students who are anxious about mathematics. Math anxiety is present in many students poorly performing in mathematics, but even many high performing students exhibit math anxiety. In cases of high-performing students, math anxiety makes a major impact on their performance. High achieving students are more fearful of poor performance, and this may translate into math anxiety. But, it is the feeling of helplessness about mathematics that grips a large percent of students from elementary school to adulthood. Math anxiety is a serious and pervasive problem, especially in the middle and high school and college setting.

In the earlier stage, math anxiety may indicate something is not right in a child’s mathematics learning. At this stage a child may not have internalized the mathematics anxiety. However, once math anxiety begins to be internalized, it undermines child’s self-esteem, and then it is a causative factor for poor mathematics achievement. Early math anxiety, if not treated, may lead to a cycle in which fear of math interferes with learning math that exerts an increasing cost to math achievement by changed attitude and motivation towards math, curtailing aspirations, increasing math avoidance, and ultimately reducing math competence. This may lead students to delay or stop taking math courses, limiting their educational opportunities.

Students experience math anxiety in many forms and degrees, from a feeling of mild tension and anxiety that interferes with the manipulation of numbers and solving mathematical problems in ordinary life and academic situations to freezing up during a math exam/test and avoiding anything having to do with quantity. Symptoms may be physical or psychological and may include:

- Physical: Nausea, shortness-of-breath, sweating, heart palpitations, increased blood pressure, fidgeting, lack of attention, avoiding direct eye contact, etc.
- Psychological: Short-term memory loss, feeling of panic, paralysis of thought, mental disorganization, loss of self-confidence, negative self-talk, helplessness, math task avoidance, isolation (thinking you are the only one who feels this way), etc.

In many cases, math anxiety is a unique kind of anxiety; for example, there is increased heart rate when people are tested on math but not on other academic areas. However, math anxiety is not restricted to tests or classroom settings, people develop severe avoidance of situations involving quantitative facts and/or reasoning and formal mathematics. They may not choose careers involving application of mathematics even if cognitively they are capable. There is overlap between math anxiety and other general types of anxiety, especially related to test taking.

Math anxiety has a variety of sources. Its development is tied to social factors such as a teachers’ and parents’ anxiety about their own math ability and cognitive skills or individual factors such as students’ own quantitative and spatial competencies. All those factors that have a negative influence on mathematics achievement are potential factors for the development of math anxiety.

Students can develop mathematics anxiety by the presence of teacher anxiety, societal, educational or environmental factors, innate characteristics of mathematics, failure and the influence of early-school experiences of mathematics.

**Types Of Math Anxiety
**Three types of math anxiety are identified by the factors that may cause them or by the nature of their manifestations. They are:

*Math type*(*Specific Math Anxiety*) is caused by mental blocks in the process of learning math and related to specific mathematics language, concept, or procedure. Generally, it relates to a difficulty or negative experience with one of the key developmental milestones—number concept, number relationships (arithmetic facts), place value, fractions, integers, algebraic or spatial thinking.*Socio-cultural type*(*Global Math Anxiety)*is the result of socio-cultural factors. It relates to socio-cultural conditions that may influence a learner in forming negative attitudes about and aversion towards mathematics once they experience difficulties in executing mathematics tasks.*Handicap type*is caused by some physical or mental handicaps.

These types are generalizations of cases. Sometimes one finds the presence of each, and other times the math anxiety may be because of the integration of more than one. To illustrate the three types of math anxieties, below one example of each reflects general principles of a type of math anxiety.

*Example of Math Type (Specific Math Anxiety)**.* Ms. Gamble had a master’s degree in humanities from a university and had an excellent academic record. After graduating she worked for an insurance company, taught at a school and finally decided to become a lawyer, so she wanted to take the law board examinations. She asked a friend, a professor of mathematics at the university[2] to help her with the math/quantitative reasoning portion of the law board exams. They had one or two sessions a week for tutorials. Most of these sessions were successful, but at times, she became irrational about math tasks. She would throw down her pencil and say things like, “You just don’t understand me. I can’t stand these things any more. It does not make any sense.” She would become visibly angry and upset. She would become quite anxious.

The professor started exploring the reasons behind her unpredictable behavior about mathematics. The professor ruled out any socio-cultural factor because she was very independent, assertive and knew what she was doing. She had not accepted female conditioning to the point of dependence, non-assertion and helplessness. She was intelligent, well educated and handled most of the math she was learning quite well. Finally, it became clear to him that anytime he tried to make her do problems involving the radical sign “√” and exponents, she had fits of math anxiety. Otherwise, she was quite comfortable with math and learned well. Somehow, she was uncomfortable with square roots and the radical sign. The radical sign and related concepts had become a mental block in her efforts to learn math. She had difficulty understanding how one could have an exponent less than a whole number, and she had difficulty conceptualizing the concept.

*Example of Socio-Cultural Type (Global Math Anxiety)**.* Ms. Cook needed to pass a required math course to graduate with a liberal arts degree. She put the math course off till her last semester at the university. She finally decided on taking an elementary course called *Topics in Mathematics*, designed especially for liberal arts majors. Soon she had problems in the course. She claimed: “Nothing made sense, I did not understand anything.” She dreaded coming to class and always felt inadequate—“stupid.” She did not take the first quiz, avoided the class and avoided the teacher. One day, the teacher saw her in the university cafeteria and asked her what was wrong, why did she not take the quiz. Rather hesitantly, she said, “Math makes me sick, I don’t need math, people are right math is not for girls. I cannot do math.”

Then, with tears in her eyes, she told the professor how her elementary school teacher used to tell her “You need to work hard. You should get help from your father every night to review the material and do the homework.” Anytime she had difficulty in arithmetic, the teacher would say: “Did you ask your dad?” She did not have a dad around. She lived with her mother—a single parent. When she asked her mother for help, her mother got angry with her and the teacher, and her mother told the teacher, “Why are you doing this to my daughter and me? I never learned this stuff in school. You know as well as I do that math is not for girls. She will be fine without it.” Since then Ms. Cook avoided math with the belief that math was not for girls.

** Example of Handicap Type. ** John, a bright young man, was crippled in a car accident. He registered in a linear algebra course at the university. John read the first test and thought, “This looks like a fair and easy test. I am going to do well.” But after about half an hour, his face became red, he began to perspire profusely as he tried one problem after another, and he looked sick. The professor asked if he was well and informed him that he could arrange a make-up test. John said he was all right and just made some mistakes somewhere and the answers he was getting were not checking out. The professor knew that even though John was confined to the wheelchair, he had enjoyed being in the class and worked hard.

When the professor corrected the test, he noticed that John had poor handwriting—a clear case of dysgraphia. He had difficulty rounding up letters and numbers like O, 0, 9, and 6. His writing showed that words, sentences, and expressions were a mixture of upper and lower case letters. Numbers were mixed and illegible; it was not possible to distinguish whether John was writing 6 or 0. Later in his office, the professor pointed this out to John and suggested that he should do his test on a computer. With this arrangement John did well in the course.

Handicaps may be of two types: *physical handicaps* and *cognitive handicaps. *Not every handicapping condition results in poor math performance or math anxiety. But, in some situations they may cause learning disabilities/difficulties in learning mathematics. Learning disabilities and some physical handicaps, generally, may affect some of the prerequisite skills necessary for mathematics learning. These prerequisite skills are: (a) the ability to follow sequential directions, (b) pattern analysis—pattern recognition, extension, creation and usage, (c) spatial orientation/space organization, (d) visualization, (e) estimation, (e) deductive reasoning, and (f) inductive reasoning. The absence of any or more of these skills affects mathematics achievement, and the lowered achievement may create math anxiety.

Today, the law requires that we do everything possible to facilitate the physically handicapped and learning disabled to provide access to a quality learning experience by providing appropriate accommodations and a least restrictive learning environment without compromising the standards of mastery and competence. To comply with the law, it is therefore important to recognize and address math anxiety.

**Global Math Anxiety Caused by Socio-Cultural Factors
**Math anxiety of the type experienced by Ms. Cook is caused by socio-cultural factors. This type of math anxiety is not about any specific or particular concept or procedure in mathematics. There is a global negative emotional reaction to any thing mathematical. Such a student may have general anxiety to start with—may have an overall anxious personality, worrying about things in general. But, the anxiety is exacerbated and manifested more profoundly in mathematics classrooms if the teaching is not appropriate and other sociological factors come into play.

Some of these individuals may experience failure in mathematics and also in other academic areas, but they admit to having anxiety only in mathematics because in mathematics students feel they are required (student perception) to give exact answers. This perceptual demand brings out the underlying anxiety to the forefront and converts into math anxiety. The anxiety is triggered whenever a student is asked to respond to specific information—facts, formulas. The subject could be grammar, a foreign language, etc. But, it happens more often in mathematics classes, particularly because mathematics teaching is driven by executing procedures and based on memorization. It happens when the teaching is compartmentalized and no connections are made.

Some students have the belief that mathematics problems always have exact answers; therefore, they may feel especially pressured (e.g., many students get upset when they are solving an equation and get a fraction answer). To these students mathematics is “solving.” The only instruction they know is “solve.”

When he/she makes mistakes, a person with general anxiety is much more susceptible to socio-cultural factors related to mathematics learning and performance. They search for a rationale for their negative reactions and socio-cultural factors come into play. Common causes of this type of anxiety are:

- distrust of one’s intuition, especially as applied to math
- illusion that math practices are an exact science
- common myths about mathematical ability
- the myth that boys naturally do better in math than girls
- the female fear of competing on man’s turf
- negative experiences of people one trusts and respects, in school, at home or society in general

When exposed to any one or more of these negative experiences consistently, individuals may develop a negative attitude towards mathematics and perform poorly, reinforcing the anxiety.

People with this type of anxiety have difficulty in most aspects of mathematics, so their response to mathematics is more global in nature and more intense. Their self-diagnosis is: “I am just very bad in mathematics. Mathematics is beyond me.” Therefore, we call this kind of anxiety global mathematics anxiety.

When referred to my Center for evaluation or remediation, even before we begin the evaluation process, persons with global math anxiety volunteer comments about their incompetence in mathematics and their feelings about it. They are apt to say:

- “I’m not smart enough to learn mathematics.”
- “I can’t do math.”
- “I just don’t understand math.”
- “Math makes me sick.”
- “Complete blackout when I see a math problem.”
- “So and so is super bright, he is really good in math.”
- “You would not believe it, but math makes me throw up.”
- “I will always stay away from math. I hate math.”

Key phrases used by these people are “I Hate Math”; “Math Makes Me Sick.” Such a person is likely to show irrational behavior towards math or mathematicians.

Much of the anxious, blocking, fear-stricken behavior, and helplessness that many students experience and exhibit in mathematics often is not primary to mathematics as a subject but is caused by other factors. People experiencing this type of anxiety will usually talk about their moment of disinterest in math, about friends and relatives who are good in math and about tricks they played on math teachers whom they hated. There is usually a moment of “sudden death syndrome” when they felt that as far as math was concerned, they were through. These math anxious people stay away from mathematicians and anybody who is good at math. For his reason, it is almost impossible to get them to talk to a math teacher or to take them near a math building.

Persons with global math anxiety are less willing to deal with their math deficiencies because they are not aware of what they are. They spend a great deal of time and energy in making decisions about how to avoid math.

Global math anxiety is on a decline due to the efforts of organizations involved in improving math education for all and making special efforts to recruit women and minorities into mathematics courses and programs. But, due to the new efforts of education leaders to improve skill levels by making the math curriculum more rigorous and because many teachers are not yet prepared to teach effectively, we are seeing more students experiencing specific math anxiety.

[1] See previous three posts on *Executive Function* and previous three posts on *Working Memory. *

[2] Most mathematics teachers in high schools and colleges report cases of this type; however, these particular cases were reported by Dr. Dilip Datta, Professor of Mathematics, University of Rhode Island.

]]>

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

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

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

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

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

Organization skills deal with:

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

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

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

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

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

**How Do Executive Functions Work?**

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

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

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

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

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

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

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

**Concepts and Understanding
**

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

**Computational Procedures**

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

**Fluency**

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

**Flexibility in Thought and Action**

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

]]>

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

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.

** **

]]>

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[1]. 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—

The most prominent EF skills called upon in different components of mathematics learning are: ** inhibition**,

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

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[1]—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 4^{2}, 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.

* *

]]>

- 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.

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:

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.

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., *x ^{2} +6x +5)*, the use of algebra tiles or Base Ten blocks helps students to see

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—

(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),

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

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:

** **

**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).

]]>

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 19

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
**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 a^{3}×a^{4}). 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.

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)

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)**

]]>

In the previous standard (*Look for and use the structure of mathematics*), the framers of CCSS-M and SMP placed an emphasis on students learning to observe and appreciate the structure of mathematics—paying attention to its unique properties and systems, big ideas, and organizing principles, appropriate to their age and grade level. The goal of this standard is to help students to see the nature and appreciate the power of mathematics.

The eighth standard: *Look for and express regularity in repeated reasoning* promotes the ability to recognize quantitative (numeric) or spatial (geometric) patterns, make connections, and use those patterns to arrive at an understanding of mathematics structures—the general methods, the reasons behind the efficiency and elegance of algorithms/procedures, work out effective strategies, properties of numbers and operations at their grade levels.

In the previous standard, students looked for and used the structures in mathematics to apply them to learn more mathematics and solve problems. In this standard, the aim is to develop those structures from concepts and even construct new ones—new to students or even new to mathematics at their grade level. This process provides skills for students to own, retain and communicate, efficiently, accurately, and with rigor, the understanding and mastery of mathematics content. It also means developing the awareness of their own learning and the nature of mathematics conceptualization—the patterns. But most importantly, the objective is to help students to create interest in mathematics and to continue it with accuracy, confidence, and enthusiasm beyond school.

Mastery of this standard allows students to complete the cycle of learning mathematics: try problems, recognize a pattern in the solution process, discern a pattern(s), see the mathematics structure in the pattern, generalize the terms of that structure, generate a conjecture, test the conjecture, and then arrive at a key concept, procedure, strategy, or relationship between different elements of mathematics.

**What is a pattern?
**A pattern is a relationship between many of pieces of data, that is on going, something that is repeating—a data or a relationship between data, and can be predicted either explicitly or implicitly. Can one have a pattern when there are one, two, or three pieces of data? One piece of data is an incident, two are a coincident, with three there is a possible beginning of a pattern. If there is definite relationship between these pieces of data, then one needs a fourth one to verify that pattern.

Observing patterns and regularity in processes is the key to learning and using the prerequisite skills for mathematics learning. Students experience and develop affinity for mathematics when they also develop the prerequisite skills for mathematics learning. These prerequisite skills include: *ability to follow sequential directions, classification, organization, quantitative* *and* *spatial reasoning*, *visualization, deductive* and *inductive reasoning*. In designing lessons, therefore, teachers must also include the development of these prerequisite skills in addition to introducing interesting and relevant content. However, they should be integrated rather than isolated exercises.

A central idea here is that mathematics is drawing general results (or at least good conjectures for the moment) by trying examples and looking for regularity in the results generated by these patterns. In this standard, we want students to experiment with examples and arrive at patterns and then generalize those patterns to arrive at workable rules and procedures.

When students have explored several problems related to a concept, a few inevitably say, “I see a pattern here, so do I have to do all this work?” This is where the teacher moves the learning into generalizing mode—using the pattern to arrive at a conjecture, testing it and then arrive at the procedure or rule. She can’t stop at the pattern level. Students need to test their pattern as well as to recognize it in diverse situations and construct counter examples or non-examples. For example, instead of teaching just the rules for combining integers, students should look at patterns of combining numbers to generalize and come up with the rule. In the number -3, the digit 3 is the numeral and -3 is the number. Thus, a number is a directed numeral.

Mathematically proficient students develop two opposing but intertwined abilities: persevering in solving problems and looking for generalizable, efficient methods (i.e., short-cuts). These students notice patterns in calculations, use those patterns to create shortcuts, and learn multiple algorithms for the same idea and choose the appropriate one for the context. This helps them to do more interesting mathematics with less effort, thus generating an inclination for mathematics.

On the other hand, low achieving students lack both these qualities. As a matter of fact, low achieving students work harder in mathematics classes as they persevere using inefficient and long-drawn out methods, for example, they will use counting for learning addition and subtraction facts rather than efficient strategies based on decomposition/recomposition. Partly, it is not their fault; they have been instructed to do so. Because of these laborious methods, they give up easily. For them mathematics becomes hard, uninteresting, and a collection of unrelated procedures because they are not given help to develop efficient strategies and practice in seeing patterns and mastering strategies.

Solving many problems develops students’ inductive reasoning, and then justifying and communicating their thinking develops deductive reasoning. Inductive and deductive thinking are essential components of the mathematical way of thinking. The SMP classrooms are expected, through these practices, to develop habits of higher order thinking skills. Synthesizing concrete models and examples into symbolic and abstract ideas, at the elementary school, are good examples of applications of inductive reasoning. In contrast, most middle school and high school concepts and procedures are based on deductive reasoning or analysis.

Habits of mathematical thinking (observing patterns, looking for structure and regularity, repeated reasoning, making conjectures, providing examples, and counter examples, etc.) enrich learners’ problem solving experiences. By looking for patterns, developing conjectures, making discoveries and inferences, generalizing results, monitoring the solving process, checking the reasonableness of answers, and devising new avenues to explore to make learning mathematics engaging, they develop the habit of persevering (increasing stamina) for problem solving and communicating what they know.

Mathematical practices such as constructing reasoning and critiquing the reasoning of others are examples of deductive reasoning; however, this particular standard is best developed when students have extensive experiences in inductive reasoning. Observation of regularity, patterns, structure, and symmetry develops inductive reasoning. Further, practice of formal mathematics logic and reasoning such as supporting arguments and formal proofs develops rigor and a deeper understanding of mathematics. Deeper understanding in mathematics is an integration of deductive and inductive reasoning: analysis and synthesis in the process of solving problems.

Being a student of mathematics means balancing both qualities. Sometimes one needs to persevere and to stick with a set of strategies that will definitely work, given time. Putting in the effort, even though it is hard, is necessary. But, the effort must be put in developing and using efficient methods and strategies. Getting stuck doing the same inefficient methods is not what being a mathematician is about. Mathematicians explore new approaches and look for efficient and elegant solutions. They want methods to be powerful and generalizable. They effectively use what they know and have done to become efficient, more powerful problem solvers. They look for and represent regularity in repeated reasoning in order to develop efficient shortcuts or notations. This standard urges us, as teachers, to prepare and remind students that mathematicians move from painstaking but reliable problem-solving methods to efficient procedures via looking for regularity in their repeated reasoning.

**Look for and express regularity in repeated reasoning.
**Regularity and repeated reasoning abound in mathematics. Many facts, diverse terms, concepts and processes reoccur in the development of mathematical ideas. In this standard, the key words are: “look for” and “express” regularity, which means the realization on the part of students that learning mathematics is

For example, the idea of decomposition and recomposition of numbers recurs whenever we add natural numbers (58 + 64 = 50 + 8 + 60 + 4 = 50 + 60 + 8 + 6 = 50 + 60 + 8 + 2 + 4 = 50 + 60 + 10 + 2 = 50 + 50 + 10 + 10 + 2 = 100 + 20 + 2 = 122, 58 + 2 + 62 = 60 + 62 = 60 + 60 + 2 = 122 or 58 + 64 = 50 + 60 + 8 + 4 = 110 + 12 = 122) to rational numbers. Consider the example,

*A recipe calls for 1***⅞ ***cups of milk. Larissa has 3***⅛ ***cups of milk. How much milk will be left for the cake recipe, she plans to make?*

Using the decomposition/recomposition, one observes:

*1***⅞ +⅛ =2 **and then** 2 + 1****⅛ ***= 3***⅛.
**Therefore, to go from

Both of these examples can be translated into a powerful method called *Empty Number Line (ENL).* Several other methods have emerged in the same way (e.g., the *Bar Model*, used in Singapore and other Asian countries).

The area of a rectangle is used in arriving multiplication procedures from two-digit by two-digit numbers to multiplication of complex numbers. This recurrence is the basis of making connections between concepts and procedures.

For example, consider the instructional sequence, in a problem in which students study the concept of rate and ratio in a mixture of juice (NCTM Practice Illustration, 2013).

For every 5 cups of grape juice, mix 2 cups peach juice. Students are asked to develop a general mathematical relationship between the amounts of juices (say *x *and *y*, representing the two juices) needed to create a particular amount of mixture with the same taste.

To graph this data, we begin with (0,0) and one finds that for each 1 unit one moves to the right, one moves up **⅖** of a unit. Thus, when one goes 2 units to the right, one goes up 2·**⅖** units. Similarly, when one goes 3 units to the right, one goes up 3·**⅖** units and when one goes 4 units to the right, one goes up 4·**⅖** units. Observing this regularity, one can generalize this to:

* When one goes x units to the right, one goes up x·**⅖**or **⅖**(**x)** units.
*Therefore, starting from (0, 0), to get to a point

While students may immediately notice some regularity, it is the process of expressing—from noticing a pattern in the observations about the data in the table—the pattern into statements like “*if we increase the grape juice by 1 cup, we must increase the peach juice by **⅖** of a cup to have the same”* in language and ultimately to translating this statement into writing the equation y = **⅖****(***x) *which constitutes the bulk of the mathematical work of the task. The formulation of CCSSM, at appropriate places, provides language to discuss this kind of expressive mathematical work—a result from the regularity of the data.

Mathematically proficient students notice if calculations are repeated. They look for general methods (by observing specific examples, they extend to generalized situations) and for shortcuts (from general cases, they arrive at efficient applications to specific cases). The observation of regularity and repeated reasoning is an important aspect of both deductive and inductive reasoning.

In the early grades, students notice repetitive actions in counting and computation, etc. For example, all addition and subtraction facts can be derived by using the strategies that employ the property of making ten and decomposition and recomposition of numbers (9 + 7 = 9 + 1 + 6 = 10 + 6 = 17, 9 + 7 = 6 + 3 + 7 = 6 + 10 = 16). Similarly, when children have multiple opportunities to add and subtract―ten and multiples of ten, they notice the pattern and gain a better understanding of place value. And then decomposition/recomposition is extended to multiplication in the form of distributive property of multiplication over addition or subtraction.

For example, an addition fact-practice exercise might result in seeing patterns and generalization: ask children to choose three consecutive whole numbers (e.g., 5, 6, and 7) and compare the double of the middle number to the sum of the two outer numbers. In this example, the two sums are 12 and 12. After they do this for several such triplets of numbers, they are likely to conjecture a pattern that allows them to state that when the two numbers are two apart, their sum is double of the middle number, 6 + 8 = 7 + 7, 19 + 21 = 20 + 20, etc. In later grades, they will realize that the average (mean) of three consecutive numbers is the middle number and later they will generalize the result to the triplet: (N -1), N, and (N+1) with the equation: (N-1) + (N+1) = N – 1 + 1 + N = N + N = 2N.

After several examples with addition, some students may ask what will happen when we ask the same question about multiplication about this triplet. Teacher asks children to compare the middle number times itself to the product of the two outer numbers in the triplet of the consecutive whole numbers. The two products are 36 and 35. After they do this for several triplets of whole numbers, they are likely to conjecture a pattern that allows them to multiply 9 × 11, 29 × 31, 99 × 101, etc. mentally because they expect it to be one less than 10 × 10, 30 × 30, 100 × 100 which they can do in their heads. Seeing this regularity is typically easy for fourth and fifth graders, but expressing it clearly and formally is harder. They do not have the language for communicating.

In later grades, students are able to understand the idea of *naming* the numbers. A simple non-algebraic “naming” scheme can be used to describe the pattern: the numbers are named “middle/inner” and “outer.” In this form, the triplet can be seen as “middle minus one”, middle, “middle + 1” or (middle – 1), middle, (middle + 1). Then they can attempt to state the result as: (middle – 1) × (middle + 1) = (middle)^{2} – 1. The step from this statement to standard algebra is just a matter of adopting algebraic conventions: naming numbers with a single letter *m* instead of a whole word like “middle,” and omitting the × sign: m^{2} – 1 = (m + 1)(m – 1). It is revisited in algebra as we teach this pattern in factoring. It is generalized into: *x ^{2} – 1 = (x + 1)(x – 1)*. Let’s take the first problem: if

Upper elementary students, for example, might notice that when dividing 25 by 3, after the initial step, they are repeating the same calculations and conclude they have a repeating decimal. In order to arrive at these general observations, the teacher needs to give examples of this kind of division to show that the repetition of digits is dependent on the number used for the divisor. For example, when the divisor’s prime factorization involves only the prime numbers *2* and *5*, the resulting decimal will be terminating as *10* and *10 ^{n}*, for every positive integer

On the other hand, when prime factorization of the divisor involves some other prime numbers or does not have *2* and/or *5 *as prime factors only, the division results in a non-terminating, repeating decimal. The repetition will occur before the number of digits is less than the divisor. For example, in the case of ⅔, the repetition will take place with less than three digits as the divisor is 3. In the case of these fractions the repetition will take place before in number of digits is 7 or less than 7 and 11 or less than 11 digits, respectively.

Middle school students might abstract the equation (*y* – 2)/(*x* – 1) = 3 by paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3. Let (*x, y*) be any point on the line passing through (1, 2) with slope 3, then applying the definition of slope formula using two points *(x, y)* and *(1, 2)*, one has the equation:

This converts into the form*: y *−* 2 = 3(x − 1), *or by simplifying the equation, we get the equation in the *y = 3x −1 *in the slope-intercept form. Therefore, in general, the equation of the line passing through the point *(x _{1}, y_{1})* and slope

Earlier we observed that *(x – 1) (x + 1) = x ^{2} – 1*. Similarly, noticing the regularity in the way terms cancel when expanding (

However, if we continued the repetition and the pattern leads to the general formula for the sum of a geometric series:

Let *S = 1 + x + x ^{2 }+ x^{3 }+… + x^{n-1} + x^{n}.*

Regularity also means looking for similarity and symmetry. The development of divisibility rules is a good example of regularity, symmetry, and repeated reasoning. For example, when we consider the divisibility test for 2, we look for the one’s place being even as the value of digits in other places (10’s, 100’s, etc.) are already divisible of 2 and will not leave any remainder when divided by 2 as its value is a multiple of *10 ^{n}* for every natural number

Similarly, when we consider the divisibility test for 4, we look for the number formed by the ten’s and one’s place digits being divisible by 4 as all the other digits in the number will not yield any remainder as their value is a multiple of *10 ^{n}*, for every natural

Now, we can develop a test for divisibility by 8 by following this pattern, e.g., 111112 is divisible by 2 (look at one’s place); 111132 is divisible by 4 (look at 32—the number formed by the ten’s and one’s place); and 111128 is divisible by 8 (look at 128—the number formed by hundred’s, ten’s, and one’s place). Thus, for the divisibility by 8, we look at the number formed by the 100’s, 10’s, and 1’s digits as all other digits in the number will have values as multiples of *10 ^{n}* for every natural number

For elementary and middle school children above results are achieved by observing the nature of many numbers and actually performing the division or the teacher playing a game of seeing the pattern in the divisibility and then arriving at the tests.* *

Many middle school and almost all high school students should be given experiences in deriving these results. In general, the number N = a_{n} a_{n-1} a_{n-2 …….} a_{0 }(In the expanded form it is N =10^{n} a_{n} +10^{n-1} a_{n-1 }+ ……+ 10^{2} a_{2} +10^{1} a_{1 }+ 10^{0 }a_{0} ) is divisible by 2, if a_{0 }is an even digit and a_{n,} a_{n-1, } a_{n-2, ….., }a_{1 }can be any digit. N is divisible by 4 if the number the number a_{1}a_{0 }(which in the expanded form is 10^{1} a_{1 }+ 10^{0} a_{0}) is divisible by 4. Similarly, N is divisible by 8 if the number a_{2} a_{1} a_{0} (which in the expanded form is 10^{2} a_{2} +10^{1} a_{1 }+ 10^{0} a_{0) }is divisible by 8 as 10^{n} a_{n}, 10^{n-1} a_{n-1, }10^{n-2} a_{n-2, }…10^{3} a_{3 }are all divisible by 2, 4, 8 for any n ⫺ 3.

The divisibility by 5 and 10 is quite clear: the one’s place should have 0 or 5 in the case of divisibility by 5 and 0 in the case of divisibility by 10 as all other places are multiples of 10^{n}, where n ⫺ 1.

Let us consider the divisibility by 3. We can see a pattern in the numbers that are divisible by 3. For example, in the numbers 33; 24; 234; 111; 232323 the sum of the digits in these numbers is a multiple of 3 and we observe that the numbers are divisible by 3. Once again, the teacher gives the examples of numbers that are divisible and not divisible by 3. And create two columns. One for the numbers divisible by 3 and the other with those not divisible by 3. By observing these numbers and share their observations. The teacher provides examples, counter examples, and non-examples to test their conjectures. Students arrive at a pattern. Thus, by observation they arrive at a conjecture (a conjecture is an observed pattern) for the divisibility of 3: *A number is divisible by 3 when the sum of its digits is a multiple of 3*.

In a general case, we can use mathematical reasoning to prove this statement. The proof for high school students is a very good example of understanding place value, laws of exponents, and the concept of division. Let us consider a six digit number: N = *abcdef *where *a, b, c, d, e, f *are digits in the number and can assume values between 0 and 9, except a ≠ 0, otherwise it will not be a six digit number (we could easily extend it to a more general number and any number of digits, N = a_{n} a_{n-1} a_{n-2 …….} a_{0 }=10^{n} a_{n} +10^{n-1} a_{n-1 }+ ……+ 10^{2} a_{2} +10^{1} a_{1 }+ 10^{0 }a_{0, }where the coefficients are digits ).

N = *abcdef* = *a×10 ^{5} + b×10^{4} + c×10^{3} + d×10^{2} + e×10^{1} + f×1*=

The number 6 is a product of *2* and *3*. The test for *2* and *3* combined works for *6*. Thus, *a number is divisible by 6 if it is divisible by 2 and 3*. This test works because the numbers *2* and *3* have an interesting relationship. *6 = 2×3* and *2* and *3* are *relatively prime* (two numbers are called relatively prime when their greatest common factor is *1*).

This suggests that this divisibility test can be generalized to any whole number. For example, a number is divisible by *24*, if it is divisible by *8* and *3*; since, *24 = 8×3* and *8* and *3* are relatively prime number. For example, *144* is divisible by *24* because *144* is also divisible by *8* and *3*. However, a number may not be divisible by *24* if the number is divisible by *12* and *2* as *12* and *2* are not relatively prime numbers (*2* is their greatest common factor). For example, *36* is not divisible by *24* although *36* is divisible by*12* and *2* and *24 = 12×2*. The condition of two factors being relatively prime is important. Thus, in general, *a number n is divisible by another number c, if n is divisible by a and b, where c = a×b, and a and b are relatively prime. *Thus, for most purposes—learning operations on fractions, simplifying radical expressions, etc., the divisibility test for up to ten are adequate. However, there is not a “good” test for the divisibility by 7. There is a procedural test for the divisibility by 7 developed in the 7^{th} century.

A great deal of research shows that one can predict which students will do well in higher mathematics are those who are well-versed and fluent in long division and operations on fractions. This is particularly so in a course on algebra. And the four major prerequisite skills for understanding and mastering the concept of fractions and operations on them are:

- Multiplication tables
- Divisibility rules
- Short-division, and
- Prime factorization.

These are the tools of success in mastering fractions and then algebraic operations. These skills are best derived by patterns and then practiced using these patterns.

To make the observation of regularity, pattern, repeated reasoning, and structure possible, both teachers and students need to pose a series of questions such as:

- Is there a process—operation, property, condition that I am repeating?
- Can I use this repeated idea to create a pattern and then to develop a conjecture?
- Can I make a generalization of the conjecture?
- Can I express this abstractly based on this repeated reasoning?
- Does this abstraction result into a procedure, method, or strategy?
- Are there exceptions to this generalization/abstraction?
- Can I make this method, procedure more efficient and elegant (this is like editing a piece of writing)

Teachers can ask questions such as:

- You claim that you observed a pattern, how do you know it is a pattern?
- Can you describe your pattern, conjecture, or method?
- Explain why your method makes sense?
- Can you describe your method in formal mathematics terms and reasoning?
- How would you explain why it works?
- Have you checked whether the answer seem reasonable?
- What have you learned about ….?
- How would this work with other numbers or situations? Does it work all the time? How do you know?
- What do you notice when …?

Mathematics is about constructing and solving problems, and interesting problems must be the focus of students’ mathematical experiences; they should be the content of daily mathematics lessons. Problems should be accessible to children, yet they should be modestly challenging and at times creatively frustrating. These kinds of problems should be the focus when students and their teachers are engaged in sharing their thinking process. These thinking processes— having ideas, not having ideas, seeing relationships, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other’s work are the back-bone of the standards of mathematics instruction.

Standards of mathematics practice—techniques and methods, when routinely practiced, will implement the content advocated in the CCSSM. Arising naturally out of practice, content will not be in isolated from; it will be an organic outgrowth of the problem background. When a concept and procedure are learned, they should be practiced till they convert into skills and tools for doing meaningful mathematics. Doing mathematics is different than knowing mathematics. However, practice is a good reinforcement tool if students have learned the content first.

The Standards of Mathematics Practices (SMP) aim to develop mathematics content, the mathematical way of thinking, and mastery of the content with deeper understanding. As students work on solving problems, they are guided to observe regularity and symmetry, discover relationships, and make conjectures. They are asked to evaluate the reasonableness of their intermediate results and arrive at mathematical concepts. They articulate these results and then prove or disprove their results using formal reasoning and their prior knowledge of related concepts. The success of this evaluation process is dependent on several skills that can be developed with the following activities:

- Encourage students to share multiple methods for solving the same problem and then request that students use someone else’s method to solve a similar problem.
- After sharing and naming multiple methods for a problem, tweak the parameters and conditions of the problem and ask students “Which method would you use to solve the problem if it were like this?”
- Playfully and explicitly add an element of time to students’ problem solving, once they have had experience developing methods for a particularly problem structure. For example, if students have painstakingly drawn pictures and calculated the sum of the interior angles of rectangles, rhombuses, pentagons, and hexagons, and shared their different methods for doing so, ask them to think about ways they could get more efficient. Then ask them to make a bid for how many polygons they could find interior angle sums for in 5 minutes. Pass out the requested number of puzzles to each team, and start the clock. While you’ll have to pay attention to group dynamics and support students to be good team members even under pressure, you’ll also hear good ideas about mathematical shortcuts that didn’t come up when students didn’t have any reason to need more efficient methods.

Another purpose of this standard is to make sense of the formulas we use by finding a pattern or relationship in numbers generated during an exploration of the topic. It is to understand that there are purposeful connections between procedures and concepts or between the manipulation of tools and a procedure. For example, students can explore the area of a triangle by building rectangles, drawing the diagonal, and realizing that the area of a triangle is half of the area of a rectangle formed with the same base and same height—thus the formula is developed and then relate to the parallelograms of the same base and height.

It is important to design lessons where patterns are revealed and lessons that encourage students to make generalizations. Some of these tasks involve students working with prior knowledge in a non-routine way.

The learning that takes place through these lessons is incredible. Using the best-designed lessons is not the only necessary component in developing this practice in students, so be sure to facilitate conversations that include comments and questions such as:

- Notice repeated calculations and reasoning
- What generalizations can you make?
- Look for general methods and shortcuts
- What mathematical consistencies do you notice?
- How would you prove that …?
- What would happen if …?
- What predictions or generalizations can this pattern support?
- How would this strategy work with another number? Does it work all the time? How do you know?
- Maintaining over sight of process while looking at details evaluate reasonableness of results
- Finding new structures/methods, generalizing
- Can you find a shortcut to solve the problem? How would your shortcut make the problem easier?

]]>