A Review of Math Disability (MD) Literature
Submitted to Scott Berenson of the
California Community Colleges Chancellor‘s Office
Submitted by College in Focus
At present, postsecondary institutions face the pressing need to review and, in
many cases, reform their approach toward the teaching of mathematics. Mathematical
ability is an increasingly important determinant of success within the academic world and
beyond it; in a study on quantitative literacy, Rivera-Batiz (1992) asserted that after
reading ability and general intelligence, mathematical fluency is the single most
important factor for determining the level of success students achieve in school and their
income once they complete their education.
Despite its importance, college students—particularly at the community college
level—are often vastly ill prepared in terms of mathematical competency. A recent article
in the New York Times reported that ―the nation‘s 1200 community colleges are being
deluged with hundreds of thousands of students unprepared for college level
[mathematics] work‖ (Schemo, p. 1). Although the burden of this assessment falls on elementary and secondary educators, its repercussions for the community college system
are clear. Additionally, our ever increasing understanding of the cognitive processes
involved in mathematical thought and the many factors—psychological, cultural,
biological and neuroscientific—that may affect this understanding caution against a static view of mathematical instruction.
Because of the growing number of students with diagnosed learning disabilities, it
is essential that any educational reforms—if and when they are made—specifically
include appropriate provisions to address the needs of these students. Woodward &
Brown (2006) highlighted this need by stating that ―interventions need to be continually
refined to address the increasingly unique characteristics of students in special education‖
Since prevalence statistics depend on the very definition of math disability (MD), the reliability of these statistics is often brought into question. However, extensive and
frequent studies—among them those conducted by Badian (1983); Gross-Tsur, Manor, &
Shalev; Kosc (1974); and Lewis, Hitch, & Walker (1994)—estimate prevalence of MD at
between 4% and 7% in school-age individuals.
Regardless of the exact figure, many of the interventions intended to help students with MD would be to the benefit of all students, with or without MD. However, this does
not mean that changes in teaching technique or format intended solely to improve the
average student‘s understanding of mathematics will be sufficient for those students with
MD. If these students are to be expected to succeed in mathematics, additional attention
must be paid to them. Mazzocco & Myers (2005) summarized the relationship in this way:
―The RD [reading disability] field has demonstrated how modifications to the language
arts curriculum benefit readers of all levels. Thus, it is wholly conceivable that other
interventions will not only benefit many (but not all) children with MD, but also typically
achieving students‖ (p. 322).
This review of the pertinent literature concerning MD will cover a variety of issues, including definition and diagnosis, possible causes, intervention techniques, and
the efficacy of various tested programs.
Difficulties in effectively addressing the needs of MD students
Although the need for effective interventions to promote mathematical learning in
students with MD is clear, a number of factors make this goal particularly difficult to
achieve. First and foremost, it is nearly impossible to understand the workings of MD due
to a fundamental lack of understanding of the “normal” functioning of development and
learning in mathematics. Geary (2005) described the manner in which this lack of
understanding hinders the study of MD: “Unfortunately, in most mathematical domains,
such as geometry and algebra, not enough is known about the cognitive systems that
support the typical learning of the associated competencies to provide a systematic
framework for the study of MD‖ (Geary, ―Cognitive Theory,‖ p. 306). Mazzocco &
Myers (2003) pointed out that this deficiency in knowledge may stem from the
cumulative nature of mathematical development, especially in the elementary years.
In addition, the first step towards aiding students with MD is identifying them as
those needing special assistance. Since many students who may qualify as learning
disabled are often not identified until they reach college (or at all), ensuring that their
needs are met is a particular challenge (McGlaughlin, Knoop, & Holliday, 2005). This
problem is furthermore related to the utter lack of consistency in definitions of MD, a
topic that will be discussed in detail below.
These difficulties are compounded by the absence of appropriate research
conducted on MD, particularly when compared to the amount of studies dedicated to
understanding reading disability (e.g., Rasanen & Ahonen, 1995). Swanson & Jerman
(2006) described math disability research as in its ―early developmental stages‖
(Swanson, p. 270). Maccini & Gagnon (2006) noted that there existed no studies
addressing ―empirically validated and recommended instructional practices to assist
secondary students‖ with MD (Maccini, p. 219).
The relative silence of the scientific community is especially noticeable in regard
to the lack of research conducted specifically on MD in postsecondary students. In their
review of twenty-three studies, Bryant & Dix (1999) only found two that focused on
algebra. In point of fact, the majority of studies are concentrated on arithmetic
computation (Mastropieri, Scruggs, & Shiah, 1991). According to McGlaughlin, Knoop,
& Holliday (2005), ―empirical evidence for effective interventions for postsecondary
students is, at best, scarce‖ (McGlaughlin, p. 225).
As alluded to above, an agreed-upon definition of MD has proven to be elusive
(Mazzocco & Myers, 2003; Swanson & Jerman, 2006). Mazzocco (2005) indicated that
not only are the definitions and measurements of MD inconsistent, but the terminology is
likewise erratic (i.e., mathematics difficulties vs. mathematics disabilities). This variation
in definitions necessarily complicates the design and implementation of effective
Another element that presents a challenge is the complex and varied nature of
mathematical understanding, and therefore of MD. Batchelor, Gray & Dean (1990) have shown that arithmetic performance is affected by a variety of verbal and nonverbal
neuropsychological factors. It is fair to assume then, as is discussed below, that students
with MD can have deficits in any one of these specific areas or in a potentially endless
combination of them. Furthermore, mathematical performance has been linked to
affective factors, such as math anxiety (Ashcraft & Ridley, 2005), which indicates a need
for a holistic approach when helping students with MD. The infinite possible
interactions between these areas reveal the need for an approach to intervention that is
tailor-made for each student.
Although these factors represent an impressive obstacle in effectively addressing MD, the necessity to do so is clear. Fortunately, the research, though it may be limited,
does present some important conclusions.
―In the field of MD, work toward establishing a consensus definition is in its early stages‖ (Mazzocco & Myers, p. 219). Indeed, many of the relevant studies point to a need
for consistent definitions, criteria and testing (e.g., Fuchs, et al., 2005; Geary, 2005). On
the one hand, there is little debate about the most basic standards, particularly the
requirement that the student in question be of at least average intelligence. Of course, the
question then becomes how one defines and quantifies average intelligence. This is
beyond the scope of this analysis, but should be noted as an example of how difficult
standardization of MD diagnosis can be. It is clear, however, that students must at least
be diagnosed as having a learning disability according to whatever established federal,
state, and institutional guidelines apply (e.g., Title V regulations).
Various researchers have identified general features of MD and what specific manner of problems tend to be the most difficult for MD students. Nolting (2000)
asserted that most students with LD will show average abilities in arithmetic, but will be
unable to learn algebra without assistance. He believes that students therefore often learn
to compensate for their disability, often going undiagnosed until reaching college, where
self-taught strategies cease to work. According to Nolting‘s research, trouble with multi-
step problems and browning errors are the two best methods for predicting MD. Dehaene
& Cohen (1991, 1995, 1997) researched the neuroscientific base of math disability and believe that inaccurate retrieval of arithmetic facts is related to abnormal functioning of the left basal ganglia, the thalamus, and the left parieto-occipito-temporal areas.
Sometimes, math disability is defined as a single entity with various deficits. In
these cases, it is most commonly referred to as dyscalculia or developmental dyscalculia. Kosc (1974) defined it in broad terms, stating that dyscalculia has its basis in a genetic or congenital disorder that affects the specific parts of the brain involved in the development of normal mathematical ability. Recent research (Oliver, et al., 2004) illustrated the genetic basis for mathematic ability. Kosc also divided dyscalculia into two categories: practognostic dyscalculia and ideognostical dyscalculia. According to his differentiation, practognostic dyscalculia involves difficulty with the manipulation of objects, either real or mental, during mathematical work. People with ideognostical dyscalculia, on the other hand, have trouble connecting mathematical concepts to allow for calculation. It should be noted that dyscalculia is often confused with acalculia; the difference between the two being that acalculia is the direct result of brain damage.
Mahesh Sharma (1986) also chose to divide dyscalculia into two groups:
quantitative and qualitative. In this model, quantitative dyscalculia involves difficulty with sequential directions and spatial organization, whereas qualitative dyscalculia is characterized by problems with inductive and deductive reasoning, estimation, pattern recognition and/or visualization.
Various elements of college mathematical curriculum prove to be difficult for
students with dyscalculia, including timed tests (Landerl, Bevan, & Butterworth, 2004) and higher-level reasoning (Sullivan, 2005). Nolting (2000) also suggested that some
students‘ math disability may only be related to a particular mathematical domain, specifically referring to dysalgebria and dysgeometria.
It is clear that even when mathematical disability is defined as a single entity, a
variety of subgroups become necessary to further qualify the problems that an individual student has. For this reason, the majority of researchers choose to define MD as a composite disability and highlight various cognitive areas where problems have been identified in students with mathematics difficulty or disability. These areas differ depending on the model, but they often overlap or are interconnected. Judging by the complex nature of mathematical thought, it is a distinct possibility that all of these elements are involved.
In their recent study on college students with MD, McGlaughlin, Knoop, &
Holliday (2005) concluded that the most significant contributors to these students‘ difficulties were working memory, mathematical fluency, reading comprehension, and visual/spatial/nonverbal weaknesses. Other researchers specifically mention some of these elements and others are related or affected by areas included in different models.
One of the most commonly referenced models is that of Geary (1993), which
divides MD into three categories: semantic retrieval, executive-procedural and visuospatial MD. In this model, semantic memory MD is characterized by difficulty in retrieving mathematical facts and variation in response time. People with procedural MD have more conceptual difficulty, involving problems with strategizing, execution, and the acquisition of mathematical concepts. Finally, visuospatial MD, the least understood of the three (Mazzocco & Myers, 2003), is identified by sign confusion (x vs. +), improper
numerical alignment, or any other difficulty visually processing numerical information
incorrectly and then presenting in similarly.
Cirino, Morris & Morris (2007) recently tested the efficacy of using Geary‘s three
categories to predict MD among 337 college students. They found that ―overall predictive
power for [calculation and mathematical reasoning] was high,‖ but also alluded to the possibility that other cognitive factors may be involved (Cirino, p. 103). They concluded
that these factors had the same ability to predict MD in college students as they did in
children, which is significant, considering that the majority of studies on MD focus on
younger children and not adults.
Nolting (2000), in his work with college students with MD, proposed another model that is supported by other research (McGrew & Hessler, 1995; Batchelor, Gray, &
Dean, 1990; McGrew, 1994; Meyers, 1987; etc.). In this model, students‘ specific
cognitive deficits, in combination with their content problem (e.g., algebra), are used to
further categorize their specific difficulties. For example, ―a student may have
dyscalculia with a visual processing speed deficit‖ (Nolting, p. 27). In his model, the
cognitive clusters tested on The Woodcock Johnson Psycho-Educational Battery –
Revised: Test of Cognitive Ability (WJTCA-R) (Woodcock & Johnson, 1989) were used
to differentiate specific areas where a student with MD may have problems. These
clusters are: processing speed, auditory processing, visual processing, short-term memory,
long-term retrieval (working memory), comprehension-knowledge (long-term memory),
and fluid reasoning. Fluid reasoning was found to have the highest correlation to
mathematical performance, but it remains unknown how students can improve in this
area (Hessler, 1993; McGrew, 1995). This points again to the need for further research in the understanding of MD and successful interventions.
One area that has received particular attention in the literature is memory,
specifically short-term memory. Nolting (2000) divided memory into eight stages (sensory input, sensory register, short-term memory, working memory, long-term memory, abstract reasoning and memory output) and asserts that MD can affect any one of them. In their comparison study between children with and without MD, Swanson & Jerman (2006) observed significant differences in performance between the two groups when testing verbal working memory, visual-spatial working memory, and long-term memory. They found that verbal working memory was the main cognitive factor differentiating MD students and their non-MD peers. In their study on the arithmetic word problem solving of children, Passolunghi, et al. (1999) found that MD children had deficits in working memory, but not short-term memory. Geary & Hoard (2002) also underlined the importance of working memory, proposing that it may be the central deficit of MD.
Nolting (2000) stated that working memory is especially important in math since
mathematical problem solving often involves multiple steps and keeping track of different information throughout the full process. Also, if students have difficulty memorizing arithmetic facts (such as the multiplication table), they will have to dedicate more working memory to computation, leaving less available for more complex problem solving. This is particularly evident in higher-level math, but can be seen even in children with MD learning to count (Carr & Hettinger, 2002).