Spring Edition 2011
A Quarterly Publication of The International Dyslexia Association Volume 37, No. 2
Mathematical Difficulties in
by Daniel B. Berch
ome 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).
Conclusions
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.