Spring Edition 2011

PERSPECTIVES on Language and Literacy

A Quarterly Publication of The International Dyslexia Association Volume 37, No. 2

Mathematical Difficulties in School Age Children

Perspectives comes with very generous permissions for sharing the knowledge. You should join IDA to get your own copy anyway :)

I have inserted my own annotations in green.  

by Daniel B. Berch

Some school-age children struggle with mathematics, routinely experiencing difficulty in learning or remembering basic arithmetic facts and carrying out even the seemingly most elementary numerical operations (Berch & Mazzocco, 2007). Such difficulties are compounded when students are expected to build upon these basic skills as they are introduced to increasingly abstract, mathematical content domains. Consider a letter published in the Washington Post written by a seventh-grade teacher not that long ago:

Many of the seventh graders I teach have a poor sense of numbers. They don’t understand that adding two numbers results in a larger number, that multiplication is repeated addition, that 5 × 6 is larger than 5 × 4 or that one quarter is smaller than one half. This lack of basic math facts detracts from their ability to focus on the more abstract operations required in math at a higher level” (Susan B. Sheridan, Washington Post, December 27, 2004).

Notice that she isn't a special ed teacher.  Many plain, ordinary students are in  7th grade with this level of confusion. No wonder they don't understand algebra!

What are the key factors contributing to this state of affairs? Is the problem due primarily to poor instruction, or is there something inherently difficult about learning even basic arithmetic because of the ways in which the developing child’s mind works? Have we been able to trace the origins of extremely low math performance that would warrant the diagnosis of a mathematical learning disability? And do effective remedial approaches exist for improving the mathematics achievement of such children?

As it turns out, definitive answers to these weighty questions still elude us. Nonetheless, progress is being made on a number of fronts, especially in the study of the fundamental cognitive processes that underlie mathematical thinking in general and those that are crucial for achieving proficiency in carrying out arithmetic calculations in particular. In this article, I will review what we have learned about the contributions of an especially important factor known as “working memory,” along with the difficulties that can arise for students who exhibit weaknesses if not outright deficits in the full complement of skills comprising this construct.

Introduction to the Concept of Working Memory

Precisely what do we mean when invoking the concept of working memory? As this cognitive construct actually encompasses several mental operations, definitions of working memory tend to vary considerably (Dowker, 2005; Shah & Miyake, 1999). Furthermore, although this concept seems comparatively straightforward at one level, it turns out to be much more complicated at another. Such a view is shared by many, including Susan Pickering, a leading researcher in this field who acknowledged that “The concept of working memory is both reassuringly simple and frustratingly complex” (2006, p. xvi).

As a consequence, it may prove instructive to present an example of how working memory can influence arithmetic problem solving before providing a definition. To begin with, consider the following quote taken from Lewis Carroll’s Through the Looking-Glass (1871) which Kaufman (2010) describes as “A working memory lapse in Wonderland”

(p. 153): “‘And you do addition?’ the White Queen asked. ‘What’s one and one and one and one and one and one and one and one and one and one?’ ‘I don’t know,’ said Alice, ‘I lost count.’”

Components of the working memory system. Reprinted from Working-Memory-and-Education – Introduction to Working Memory (WM), D. B. Berch, Retrieved November 17, 2010, from http://working-memory-and-education.wikispaces.com/ Introduction+to+Working+Memory+(WM). Copyright 2009 by Carren Tatton. Reprinted with permission.

Although it is doubtful that Alice’s failure to solve this problem is attributable to a mathematical learning disability, the example illustrates nicely some of the key components of working memory depicted in Figure 1. That is, in order not to lose count when attempting to solve such a problem, an individual would have to: a) focus attention on each new addend as it is presented, b) manipulate the information by mentally adding the “ones,” and at the same time, c) selectively maintain some of the information (in this case, the most recent prior sum)in temporary mental storage, and d) complete all of these tasks within the span of a few seconds. In other words, working memory is probably best defined as a limited capacity system responsible for temporarily storing, maintaining, and mentally manipulating information over brief time periods to serve other ongoing cognitive activities and operations. In essence, it constitutes the mind’s workspace.

Getting back to the White Queen’s arithmetic problem, while adding single digits should be comparatively easy for most typically achieving seven-and-a-half-year-olds (Alice’s age), it is evident from this example that one can excessively tax working memory by requiring a learner to simultaneously attend, store, and mentally process a rather large amount of information (albeit elementary in some sense) within a relatively short period of time. As Susan Gathercole, another leading researcher in this field has pointed out, overloading this fragile mental workspace can lead to “complete and catastrophic loss of information from working memory” (Gathercole, 2008, p. 382).
Complete and catastrophic loss -- sadly, that's what happens all too often to math students.

. . . working memory is probably best defined as a limited capacity system responsible for temporarily storing, maintaining, and mentally manipulating information over brief time periods to serve other ongoing cognitive activities and operations.

Obviously, no teacher would deliberately choose to overload his or her students’ working memory capacity. Nevertheless, mathematical information can sometimes be presented in a manner (e.g., orally or in textbooks) that inadvertently strains the processing capacity of students. Practitioners can learn to readily avoid these situations if they are furnished with some basic information about the nature of working memory, its limitations, and the ways in which students can differ with respect to its constituent skills. Accordingly, the purpose of this article is to provide non-specialists with a succinct overview of the latest research on this topic, which I have organized in a way that I hope will shed light on some of the most important questions pertaining to the role of working memory in learning school mathematics, including: What are the ways in which working memory’s component skills can be measured? How do limitations in working memory contribute to the development of mathematical learning difficulties and disabilities? And finally, what kinds of instructional interventions or remedial approaches are available for mitigating the detrimental effects of working memory limitations on mathematics achievement?

So diplomatically stated!   I disagree about being able to "readily avoid" these situations given the current state of math instruction.  On the other hand, there are times when, really, the main issue is working memory.   I'm curious about the instructional aspect -- will motor memory stuff be included?

How Are Working Memory Skills Measured?

Children’s working memory skills are customarily assessed with a variety of what are referred to as “simple” and “complex” span tasks. Simple span tasks are used to measure the short-term storage capacity of two types of domain-specific representations: verbal and visuospatial. To appraise the former, a reading or listening span measure is usually employed that entails the recall of word or number sequences; when assessing the latter, either the recall of random block-tapping sequences or randomly filled cells in a visual matrix or grid is typically required.

In contrast, complex span tasks gauge domain-general, central attentional resources by imposing substantial demands both on mental storage and processing (Holmes, Gathercole, & Dunning, 2010). As I have described elsewhere (Berch, 2008), a particularly representative example of such a measure is the Backward Digit Span task in which a random string of number words is spoken by the examiner (e.g., saying “seven, two, five, eight . . .”), and the child must try to repeat the sequence in reverse order. Note that rather than simply having to recall the numbers in the same forward order (which is considered a measure of the short-term, verbal storage component per se), the backward span task requires that the child both store and maintain the forward order (i.e., verbal component) of the number words while simultaneously having to mentally manipulate this information to accurately recite the original sequence in the opposite order. It is this dynamic coordination and control of attention combined with the storing and manipulation of information in support of ongoing cognitive activities that I characterized earlier as being the sine qua non of working memory.

To carry out a comprehensive assessment of children’s working memory capacities, most researchers currently make use of one of two standardized batteries—the Working Memory Test Battery for Children (Pickering & Gathercole, 2001) or the Automated Working Memory Assessment (Alloway, 2007). As Holmes and her colleagues (2010) describe, each of these is comprised of several subtests, affording multiple assessments of different facets of working memory (e.g., central attentional resources as well as verbal and visuospatial short-term storage components). Additionally, these batteries permit the identification of children with poor working memory for their chronological age, based on existing norms.

Another technique for identifying children with poor working memory is derived from ratings provided by a child’s teacher, a prominent example being the Working Memory Rating Scale (Alloway, Gathercole, & Kirkwood, 2008). This measure consists of approximately 20 statements of problem behaviors such as “She lost her place in a task with multiple steps” and “The child raised his hand but when called upon, he had forgotten his response.” Teachers rate how typical each of these behaviors is of a given child using a four-point scale. Although this technique affords a fast and efficient method for initial identification of working memory problems in a school setting (Holmes et al., 2010), it is probably best used as one component of a comprehensive evaluation by the school psychologist. Furthermore, if need be, teachers can choose to make supplementary, informal observations for guiding adjustments to their instructional approaches with particular children.

How Do Working Memory Limitations Contribute to Mathematical Learning Difficulties?

As noted earlier, measures of working memory are usually designed to assess one or more of three presumed subsystems comprising what is known as a multicomponent model: a domain-general, limited capacity central executive that governs the storage and temporary maintenance of information in two domain-specific representational subsystems—the phonological loop and visuospatial sketchpad—by means of attentional control (Baddeley, 1990, 1996; Baddeley & Hitch, 1974). To date, the vast majority of investigations aimed at determining particular relationships between various working memory skills and mathematics learning or performance have been based on this model.

Such relationships have been studied in children ranging from preschool age to adolescence, and for math skills extending from the very basic (e.g., numerical transcoding—writing an Arabic numeral in response to hearing a number word, counting, numerical magnitude comparison, and single-digit addition and subtraction) to more complex mathematical operations and content domains, such as multidigit arithmetic, rational numbers, and algebraic word problem solving. Furthermore, according to Raghubar, Barnes, and Hecht (2010), numerous other factors may influence and therefore complicate the interpretation of findings pertaining to the relations between working memory and math performance, including but not limited to skill level, language of instruction, how math problems are presented, the type of math skill at issue, whether that skill is just being acquired or has already been mastered, the type of working memory task administered, and the kinds of strategies that different-aged children operating at diverse skill levels may employ for a given task.
Comment from me:  Emotional state isn't mentioned -- but anxiety wreaks havoc on working memory!

Consistent with this perspective, Geary and his colleagues (Meyer, Salimpoor, Wu, Geary, & Menon 2010) highlighted the importance of their findings that the contributions of particular components of working memory to individual differences in mathematics achievement can vary with grade level or the type of math content being assessed. More specifically, these researchers showed that the central executive and phonological loop play a more important role in facilitating mathematics performance for second graders, while the visuospatial sketchpad does so for third graders. Furthermore, they provide a compelling argument that this grade-level difference is attributable to instruction and practice rather than a developmental change in working memory capacity.

Fascinating! What happens later?   Is this what I see when I watch students imitate what a problem looks like instead of discerning what it means?

All this being said, earlier reviews of research on this topic (DeStefano & LeFevre, 2004; Swanson & Jerman, 2006) along with more recent ones (Geary, 2010; Raghubar et al., 2010) have yielded reasonably clear evidence of a generally strong association between working memory capacity and mathematics performance. *Indeed, even the leading proponent of the view that the development of mathematical learning disabilities is attributable to a deficit in a domain-specific, inherited system for coding the number of objects in a set has recently acknowledged that the domain-general, central executive functions of working memory are at the very least associated (i.e., correlated) with arithmetic learning and performance (Butterworth, 2010). What is the nature of this relationship? As Geary (2010) concludes after reviewing the findings, the greater the capacity of the central executive, the better the performance both on cognitive mathematics tasks and math achievement tests (Bull, Espy, & Wiebe, 2008; Mazzocco & Kover, 2007; Passolunghi, Vercelloni, & Schadee, 2007). Furthermore, Geary notes that the phonological loop seems to be important for verbalizing numbers, as in counting (Krajewski & Schneider, 2009) and in solving math word problems (Swanson & Sachse-Lee, 2001).

*Is this because "performance" is measured in how well students perform symbolic procedures? As in, Math is abotu memorizing symbol manipulation, so the kiddo who actually is mathematically gifted is going to perform poorly because that's not how we teach it?

. . . factors (that) may influence . . . the relations between working memory and math performance (include) skill level, language of instruction, how math problems are presented, the type of math skill at issue, whether that skill is just being acquired or has already been mastered, the type of working memory task administered, and the kinds of strategies that different-aged children operating at diverse skill levels may employ for a given task.

Although studies have also shown that children with either math learning difficulties or disabilities exhibit deficits in all three working memory subsystems, Geary (2010) concludes that impairment in their central executive appears to be particularly troublesome (Bull, Johnston, & Roy, 1999; Swanson, 1993). However, Geary also observes that the interpretation of these findings is complicated by the fact that at least three purported subcomponents of the central executive (i.e., inhibition, updating, and attention shifting) have been found to influence math learning in different ways (Bull & Scerif, 2001; Murphy, Mazzocco, Hanich, & Early, 2007; Passolunghi, Cornoldi, & De Liberto, 1999; Passolunghi & Siegel, 2004).

In summing up what researchers have learned about associations between working memory and math learning disabilities, Geary (2010) affirms that: “At this point, we can conclude that children with MLD have pervasive deficits across all of the working memory systems that have been assessed, but our understanding of the relations between specific components of working memory and specific mathematical cognition deficits is in its infancy” (p. 62).

What Kinds of Interventions or Remedial Approaches Exist for Improving Working Memory?

In a review of techniques used to date for mitigating the difficulties encountered by children who have poor working memory, Holmes and her colleagues (2010) grouped these under three main approaches: 1) a classroom-based intervention that consists of encouraging teachers to adapt their instructional approaches in ways that minimize working memory loads; 2) training designed to teach children to make use of


Working Memory and Mathematics Learning continued from page 23

TABLE 1. Principles of the Classroom-Based Working Memory Approach
Principles Further Information
Recognize working memory failures Warning signs include recall, failure to follow instructions, place-keeping errors, and task abandonment
Monitor the child Look out for warning signs, and ask the child
Evaluate working memory loads Heavy loads caused by lengthy sequences, unfamiliar and meaningless content, and demanding mental processing activities
Reduce working memory loads Reduce the amount of material to be remembered, increase the meaningfulness and familiarity of the material, simplify mental processing, and restructure complex tasks
Repeat important information Repetition can be supplied by teachers or fellow pupils nominated as memory guides
Encourage use of memory aids These include wall charts and posters, useful spellings, personalized dictionaries, cubes, counters, abaci, Unifix blocks, number lines, multiplication grids, calculators, memory cards, audio recorders, and computer software
Develop the child’s own strategies These include asking for help, rehearsal, note-taking, use of long-term memory, and place-keeping and organizational strategies

Note. Adapted from “Working memory in the classroom,” by S. E. Gathercole, 2008, The Psychologist, 21, 382–385. Copyright 2008 by The British Psychological Society. Adapted with permission.

memory strategies for improving the efficiency of working memory; and 3) training aimed directly at improving working memory through the use of an adaptive computerized program that involves repeated practice on working memory tasks.
Again, assuming it's a neurological deficit would be, I believe, a mistake because if you understand what's going on, you don't *need* as much working memory.

The classroom-based intervention is founded on a set of seven principles that originated from both classroom practice and cognitive theory (Gathercole, 2008) and are summarized in Table 1. Recently, a research team carried out an evaluation over a one-year period of two forms of this intervention aimed at primary school children with poor working memory (Elliott, Gathercole, Alloway, Holmes, & Kirkwood, 2010). Although there was no evidence that the intervention programs directly improved either working memory or academic performance, the extent to which teachers implemented these seven principles was predictive of their students’ mathematical (and literacy) skills. Furthermore, teachers were reportedly very pleased about the ways in which the intervention had improved their own understanding and practice (which exemplifies the kind of mathematics knowledge enhancement that Dr. Murphy and her colleagues (this issue) promote for all teachers). Additional studies exploring how best to implement this kind of intervention are clearly warranted if we are to determine the optimal ways for practitioners to enhance children’s mathematics achievement through the strengthening of working memory skills.

TEachers had their own understanding improved... there's rather an important key.

With respect to the strategy training approach, the kinds of memory strategies children have been taught to use include repetitively rehearsing information, generating sentences from words or making up stories based on them, or creating visual images of the information (Holmes et al., 2010). Strategy training incorporating all of these techniques was recently administered to children ranging in age from five to eight years old in two sessions per week over a six-to-eight-week period using a computerized adventure game (St. Clair-Thompson, Stevens, Hunt, & Bolder, 2010). Although training significantly enhanced both verbal short-term memory and working memory, there were no gains in visuospatial short-term memory. More relevant to the focus of this article, performance on a mental arithmetic task improved significantly. Furthermore, all of these gains were evidenced by children with poor working memory as well as those with average working memory. Nevertheless, no significant changes emerged on standardized assessments of arithmetic or other mathematical domains either immediately following training or five months afterward.

Yo. this is probably more imporant than we think.  *Everybody* improved, not just the people with poor working memory ... and nobody actually did better on math tests. Hmmm....

. . . the extent to which teachers implemented these seven principles (of working memory intervention) was predictive of their students’ mathematical (and literacy) skills.

Finally, according to Holmes and her colleagues (2010), the most impressive gains in working memory obtained thus far have resulted from a direct training program developed originally for use with children with attention deficit hyperactivity disorder (ADHD; Klingberg et al., 2005; Klingberg, Forssberg, & Westerberg, 2002). Children undergoing this intensive training regimen participate in a variety of computerized tasks designed to repeatedly tax their working memory capacity (i.e., requiring simultaneous storage and manipulation of information) to the greatest extent possible without exceeding a level they can still manage effectively. This is achieved by matching the difficulty of each successive task to a child’s current memory span on a trial-by-trial basis. Holmes, Gathercole, and Dunning (2009) administered this so-called adaptive training program to 9- and 10-year-olds with poor working memory skills in 20 training sessions, each 35 minutes long, over a period of five to seven weeks. Not only did the children exhibit sizeable improvements in verbal and visuospatial working memory, but six months later these gains had still not declined. And even though no gains were found on a standardized mathematics reasoning test immediately after training, a small but significant improvement emerged on the six-month follow-up assessment.

Fascinating -- for the computerized training, having the good working memory had 'em doing better six months later.  I bet this one would have worked for the normal kiddos, too, perhaps???  

In sum, although these three types of interventions have been shown to improve working memory skills, evidence of their impact on academic performance in general and on mathematics abilities in particular is as yet rather limited (Holmes et al., 2010). However, it is our hope that continued study of ways to enhance such outcomes will yield stronger proof regarding whether such training can transfer to students’ mathematics performance.

One final investigation is worth describing here, primarily because even though it was a cognitive laboratory study, it has important implications for improving classroom instruction. Briefly, this investigation revealed that 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 (Cowan, Morey, AuBuchon, Zwilling, & Gilchrist (2010). However, when their working memory is overloaded, attentional efficiency declines, suggesting that interventions aimed at enhancing working memory will in turn improve attentional efficiency. As these researchers put it, “In general, children’s attention to relevant information can be improved by minimizing irrelevant objects or information cluttering working memory” (p. 131).


Taken together, the research reviewed in this article shows that we are making significant progress toward achieving a more complete understanding of the nature of working memory, its typical course of development, and the best methods for assessing its various features. We have also made important advances in discerning how working memory limitations and impairments can hinder the attainment of proficiency in mathematics, and we have just begun to explore the most promising strategies that can be implemented to enhance the working memory skills most relevant for improving students’ mathematical learning and performance. Finally, I hope that the information provided here will be of some use to those of you who teach in identifying working memory limitations in your students, modifying the instructional environment to minimize extraneous or distracting information that might interfere with efficient selective attention, and designing strategies for enhancing your students’ working memory skills.


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Daniel B. Berch, Ph.D., is a Professor of Educational Psychology and Applied Developmental Science at the University of Virginia’s Curry School of Education. He has authored assorted articles and book chapters on children’s numerical cognition and mathematical learning disabilities, and is senior editor of the book (co-edited by Dr. Michèle Mazzocco), Why Is Math So Hard for Some Children? The Nature and Origins of Mathematical Learning Difficulties and Disabilities. Dr. Berch served on the National Mathematics Advisory Panel commissioned by President George W. Bush and is a member of the National Center for Learning Disabilities Professional Advisory Board.