Psychological perspectives in assessing mathematics learning needs.

While the definition of learning disabilities has been the subject of controversy for decades, the current federal classification system identifies three specific areas of deficit: reading, written language, and mathematics and maintains the presumption that the disabilities are a result of a central nervous system dysfunction, in contrast to the expansive literature base in language arts, research on math disability is far less developed and continues to lack an empirically-based identification of core deficits. The purpose of this article is to review the current research base on math learning disabilities with the related literature in developmental, cognitive, social, and neuro- psychology in order to refine the reader's knowledge of relevant factors in mathematics learning and intervention planning for individual learners.

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The prevalence of formally classified learning disabilities (LD) has been increasing throughout the last two decades. As a result, there is a greater need for teachers to possess an extensive array of teaching techniques to address the needs of these students. Due to current emphasis on targeted interventions, it is important that educators have knowledge of the different learning disabilities and how they manifest in the children they teach. According to National Council of Teacher's of Mathematics "Students with special educational needs must have the opportunities and support they require to attain a substantial understanding of important mathematics." (NCTM, 2000 p.5).

Unfortunately, a consummate framework is dependent on the current state of research on math learning disabilities (MLD) which is far less developed than that of reading disabilities (Geary, 1993; Mazzocco & Myers, 2003). The need for more systematic research and unified theory surrounding MLD is highlighted by a major funding initiative through the National Institute of Health and the Education Department (Special Education Report, 2003) to propagate research in the areas of math cognition, learning, and MLD. A seminal component of the literature base, which remains lacking, is an empirically-based identification of core deficits, and thus, a consensus operational definition for math disability. For example, identification of phonological decoding deficits is evidenced across various subtypes of reading disabilities, persists over time, and serves as a basis for planned and efficient intervention. It may be the case that no core deficits exist for MLD, but rather, various subtypes coexist which lack a unifying core (Mazzocco & Myers, 2003).

The purpose of this article is to review the current research base in MLD with the related literature in developmental, cognitive, social, and neuro- psychology in order to refine the reader's knowledge of relevant factors in mathematics learning and intervention planning for individual learners.

Numerical Skills

Continued development of arithmetic skills, even at low levels of abstraction, is a complex process involving specialized arithmetic language, quantity, reasoning, and transcoding of words and symbols (Landerl, Bevan, & Butterworth, 2004). In studies predicting future academic achievement from skills measured at kindergarten screening, research has suggested that mathematics performance is predicted by a more complex set of skills than is reading performance (Augustyniak, Cook-Cottone, & Calabrese, 2004; Kurdek, & Sinclair, 2001). In addition to language processing skills which facilitate semantic understanding of quantitative concepts, even early levels of math curriculum require a multitude of cognitive activities, including counting knowledge, number production and comprehension, fact ability, procedural knowledge, and problem solving. Moreover, each of aforementioned cognitive demands commands a combination of inherent and experiential factors. For example, counting requires mastery of the principles of one-to-one correspondence, stable order, cardinality, and abstraction. Number production and comprehension requires the ability to transcode or translate numbers between verbal, visual and/or written form while comprehending the relative magnitude of place value (Geary, Hoard, & Hamson, 1999).

From a neuropsychological perspective, speedy and accurate retrieval of facts and procedural steps is dependent on a well organized neural network. The neural network is reinforced by repetitive experience (practice) and enriched by associative experience (generalizing concepts). From a practice/memory retrieval perspective, accuracy of the initial learning is paramount. It is helpful to think that neurons that wire together, fire together.

Current research on the neural network of the brain suggests that each learning experience creates a specific pattern of neuronal circuit firing. The more the pattern has been stimulated, or fired in the past, the higher the probability of future activation (Siegel, 1999). Thus, a child who experiences high practice rates and high success rates (taught correctly, errors are immediately identified and corrected) with manipulation of numbers is mathematically advantaged over a child who, with equal neuronal integrity (i.e. equivalent general intelligence, no specific learning disability), practices math less often and with more undetected errors. Efficient number skills can be fostered by a student's ability to apply and generalize various cognitive strategies or problem solving principles, which deepen associations between numerical concepts and, in turn, shorten processing time. Children experiment with and adopt many strategies in math problem solving even prior to formal schooling when their skills grow increasingly symbolic and abstract (Klein & Bisanz, 2000; Rasmussen, Ho, & Bisanz, 2003). For example, principles of number quantification, inversion, and estimation emerge from a combination of inherent and experiential factors. Among young children, more successful counters are more likely to employ the counting-on strategy (e.g. when given 3 + 7, will start with highest number (7), then count up 3) rather than the counting-all method (Simon & Hanrahan, 2004). With cognitive maturity, estimation skills increase in complexity from eyeballing and guessing to more complex benchmark and decomposition-recomposition strategies (Montague & van Garderen, 2003). Research suggests (Rasmussen, Ho, & Bisanz, 2003) that, absent prior instruction, many preschool and primary grade children and the majority of older children use these cognitive strategies to short-cut problem solving. Use of cognitive strategies is thought to utilize (and therefore reinforce) visual-spatial working memory, presumably leading to more efficient problem solving (e.g. 7 + 9 - 8 = 7+1). However, young children and students with learning disabilities are more likely to overextend cognitive strategies, use them inconsistently, or struggle with maintaining number sets. Thus many children require direct instruction and monitoring to benefit fully from such cognitive exercises.

When not accompanied by additional processing problems (e.g. those highlighted in the following sections), procedural ability and arithmetic fact ability are likely to improve with high quality, individualized teaching and supervised practice (Geary, 1993; Levine, 2002). If the aforementioned conditions are met and specific disability in basic numerical processing continues, educational practitioners may consider a classification of specific learning disability or diagnosis of developmental dyscalculia (Landerl, Bevan, & Butterworth, 2004). To allow for monitoring of individual progress and progress relative to peers, curriculum-based measures of response time and response consistency are good measures of the quality of a child's response to these interventions. If curriculum-based norms are not available, individual baselines alone can still be useful.

If numerical skill deficits are hypothesized to be the primary deficit, the following strategies may prove helpful.

* Scaffold computational and conceptual skills by increased exposure to basal math curriculums involving frequent teacher questioning and student response.

* Use daily timed tests to review and monitor progress. Tests are 2-3 minutes long and give the teacher immediate feedback to adjust instruction for the lesson of the day.

* Use small group instruction during cooperative learning time to give individual instruction to certain students. During this time the teacher can simplify language and instructions for students with those needs. While in a small group, similar students can make their own math dictionaries. They can add new math vocabulary words to their math dictionaries as additional reinforcement. These math dictionaries can contain the terms reviewed prior to new lessons for further practice.

* Students with MLD require over-learning to retain the skills required in math. Games at home and school reinforce basic math skills in a pleasant manner. Board games, Math Jeopardy, puzzles, dot-to-dots, color by numbers where numbers are obtained by computing math problems all provide drill without the traditional worksheet approach.

Visual Spatial Deficits

When assessing a student for MLD, current practice often assesses visual perceptual skills. A perceptual or visual--spatial impairment is often evidenced as problems in discriminating between similar letters, copying shapes and figures, using computerized answer sheets, making sense of graphs and charts, and lining up numbers in math problems. Therefore, a common characteristic of a child referred for evaluation of MLD is that s/he has trouble imitating and copying important information, or is prone to inverting, reversing or misaligning important information or has trouble with spatial aspects of maintaining place value.

Example: 971-91

In other words, if brain injury was involved in the mathematics problems denoted above, the student would likely exhibit evident impairment in other areas of cognitive functioning. For example, alexia and agraphia (difficulty reading and writing numbers) is almost always associated with lesions to the left hemisphere which also frequently involve neurological systems associated with aphasia (word retrieval difficulties) and reading disorders. Rarely are mathematical processes solitarily involved (Geary, 1993). Spatial acalculia is a distinct disorder characterized by difficulties in the spatial representation of numbers which manifest as number rotation, number omission, misreading of arithmetical signs, difficulty with column alignment and placing of decimals, while leaving number reading and number formation skills intact (Geary, 1993).

Regardless of the origin of the performance deficit, in order to avoid perpetuating failure and to encourage self-monitoring behavior, the teacher and the student need to check problems copied by the student for accuracy prior to problem solving. Utilizing graph paper may provide the necessary visual organization for the student having trouble with alignment. Another option is to provide the problems on worksheets to eliminate the problems associated with copying and the resultant fatigue often experienced by students with visual motor and visual spatial impairments. One would expect that attention-based problems would require time-limited remediation, whereas true neuropsychologic impairment would require longer-course compensation.

If visual spatial deficits are hypothesized to be primary, the following strategies may prove helpful:

* Provide copies of problems to be computed rather than having the student copy problems from the chalkboard. The student with MLD will save valuable thinking time that would otherwise be invested in copying the problems, perhaps inaccurately.

* When testing or completing word problems, provide grid paper, or instruct student to lay paper lengthwise to "supply" columns for correct place value. Once the student learns place value remove the grid paper to return the student to regular paper as soon as possible.

* Use the overhead projector or other visual aides to ensure that students receive both auditory and visual reinforcement of concepts.

* Use manipulatives wherever possible. Again, visual representations help the learner see the problem as s/he attempts to analyze and make sense of the material. Blocks, counting sticks, rulers, calculators, etc. Assist the student to spend more time on strategy than basic computation skills.

Cognitive Skill Development: Memory, Retrieval, Working Memory, Speed of Processing, Attention Regulation, Planning and Problem Solving

Neuropsychological theories are based on the assumption that efficient new learning and retrieval of previously learned information is dependent on cognitive functions that are well organized and become more elaborate with increasing experience. This conceptual relationship between achievement and the inherent cognitive demands of curriculum assumes reciprocity between the learner (i.e. adequate general intelligence, intact neurological system) and the environment (e.g. rich experience, good instruction, timely feedback). Cognitive education emphasizes the abilities of the learner to perceive, sustain attention, organize & remember (i.e. encode, store, retrieve), and monitor information (e.g. distinguish between essential v. nonessential detail) (Jepson & VonThaden, 2002).

Except in rare cases of anarithmetria (difficulty retrieving previously mastered basic arithmetic facts), which is associated with damage to the posterior region of the left hemisphere usually acquired via trauma/ illness (Geary, 1993), one would expect that students whose MLDs are associated primarily with attention, organization, or recall would verify those difficulties not only on standardized assessments (e.g. WRAML: Wide Range Assessment of Memory and Learning, (Adams & Sheslow, 1990)) but in other academic areas as well. In fact, it is estimated that the comorbidity of reading and math disabilities between 40-50% (Geary 1993; Lewis, Hitch, & Walker, 1994).

Because the assessments of specific neuropsychological abilities (e.g. working memory) is less reliable than for general intelligence, one must not assume that identified deficits in attention, encoding, or memory retrieval are reflective of a specific learning disability. Indeed, there is a great deal of variability in normative development of these skills. However, identification of both the student's cognitive and academic skills profile at the time of intervention planning can be extremely valuable. For example, if a student performs within the average range on general measures of memory but has difficulty recalling new arithmetic learning, one must consider the synergistic relationship of new learning and memory structures. Memory representations arise from and are reinforced by previously executed problems. With each execution, the probability of direct retrieval increases. Therefore, guided practice (continued accurate execution of a computational or procedural strategy) in and of itself will improve long-term memory representation and retrieval. However, it has been asserted when students are also encouraged to actively identify familiar aspects, or themes of math problems, memory and retrieval skills are further reinforced (Levine, 2002). Short-term memory difficulties may include problems remembering numbers, symbols, or words in correct order (Nolting, 1991) and erring when "carrying" numbers. Students exhibiting short-term memory problems will have difficulty in math due to the sequential nature of the subject. Intervention for short-term memory deficits should involve rehearsing step-wise procedure in increasing longer intervals.

If the student demonstrates a weakness in attention span or working memory resources there are implications for how much information can be rehearsed (Geary, 1993). Students with working memory deficits may need arithmetic problems restructured as to not overtax their ability to simultaneously process the essential elements. Working memory can be thought of as activating the appropriate codes to process new information so it can be stored efficiently. For example, when hearing a list of words such as: ball, apple, truck, pencil, cheese, doll, skate, bat, one might employ a strategy of associating the majority of words with toy and food groups. When attempting to recall a strand of digits: 2,4,7,9,5,3,1; one might group the digits into segments similar in manner to telephone numbers: 247,9531. Children who score higher on measures of working memory perform better on both processing speed and accuracy of both reading comprehension and arithmetic tasks, presumably because efficient encoding releases more cognitive resources for other activities (Ransdell & Hecht, 2003). In neuropsychological theory, working memory resources are related to how much information can be rehearsed in a 2-to- 3 second span and is often assessed on standardized assessments via digit or word spans, picture memory, or counting speed tasks. Research suggests that students with MLD show, on average, a significantly slower counting speed than non-disabled children (Geary, Brown, & Samaranayake, 1991). Therefore, counting speed may be a valuable skill to address and monitor during intervention.

Attention Deficit Hyperactivity Disorder is a diagnostic term given to clinically referred children with a specific spectrum of behavioral manifestations of inattention and/or hyperactivity-impulsivity that cause impairment across multiple settings (e.g. home, school, work) (American Psychiatric Association, 1994). Neuroimaging (CT scan, MRI) and neuropsychological studies have implicated the structural and metabolic abnormalities in the prefrontal and frontal regions (presumed responsible for executive functions such as planning, self-regulation, metacognition, etc.) of the brain with symptoms typifying ADHD (for review see Barkley, 1998). However, as with many other mental disorders, difficulties with regulating attention occur on a continuum across the population. Many students experience attention problems at levels below the defined thresholds of clinical diagnosis and, though the symptoms may be circumscribed, they are troublesome at times. For example, non-clinical problems with attention manifested in academic tasks may present as (a) a preference for the most salient and/or novel features of stimuli, (b) greater difficulty focusing on relevant stimuli if it is not made obvious (e.g. embedded in nonessential detail), (c) shorter perceptual cycles (Kercood, Zentall & Lee, 2004). Though students with these processing difficulties may not meet the diagnosis for ADHD, they are susceptible to poor achievement.

Poor skill in allocating attention must be distinguished from computational deficits and may become apparent when a student (a) has intermittent difficulty or more frequent difficult with carrying or borrowing than with solitary calculation, (b) struggles with word problems containing unessential details or distracters, (c) is able to self-correct computational errors on cue. Because consistent (preferably accurate) computation increases memory retrieval, students with attention problems alone may reach optimal benefit from rehearsal when given guided practice with a reduced number of problems and immediate corrective feedback.

From a cognitive-behavioral perspective, becoming a self-regulated learner requires mindful engagement in a recursive cycle of analyzing tasks, defining presenting problems, planning and implementing strategies, monitoring outcomes, and making necessary/logical adjustments (Butler. 2003). The self-regulated process is contingent upon the interplay of lower-order cognitive and academic skills (previously discussed) and higher order concepts (operational and organizational strategies) and is facilitated by social cognition (motivational beliefs, self-concept, social modeling, etc.--discussed in following section). Good problem-solvers tend to focus on deeper structures of problems (e.g. notices the problem requires regrouping or describes a scenario associated with Pythagorean Theorem, Newton's Law, etc.). Good problem-solvers and those with adequate mental age (i.e. between 9 and 11 years) can benefit from being allowed to generate their own strategies, presumably because it reinforces both their conceptual and procedural knowledge (Butler, 2003). However, some direct instruction of active categorization strategies can be useful as it has been demonstrated to improve problem solving ability in even average learners (Kercood, Zentall, & Lee, 2004).

If it is believed that underdevelopment of higher order cognitive skills are impeding mathematics performance, the following strategies may prove helpful:

* Review terms prior to new instruction or testing. This provides review and memory reinforcement for the whole class as well as the student with MLD.

* Write or illustrate critical information and directions to locus attention on key concepts.

* Use the computer for drill and practice and as reinforcement for appropriate behavior. Students with disabilities are deficient in basic computational skills. Rather than spending instructional time needed for higher order thinking skills on basic drill and practice, let the computer be the opportunity for students to practice on their own. Teachers need to monitor the drill and practice to check for progress and on-task behavior. Teachers need to preview software to determine if the program is challenging enough and provides the drill desired.

* Teach students to underline cue words. The fast recognition of important words will acquaint students with terminology critical to the solution of the problem.

* For students who have problems remembering to look for the operation of the problem, highlight each sign in a different color (e.g. addition in yellow; subtraction in blue). Many students look at the first operation and use the same throughout the paper regardless of change of operation (e.g. addition throughout despite having both addition and subtraction problems present).

* As previously indicated, students with MLD frequently require additional time to process information and respond. Allow additional time on tests and in-class assignments. Remember that it can be a painful experience when a student is called upon by a teacher and loses opportunity to respond successfully simply because they cannot answer quickly enough. Encourage active participation and increase academic serf-esteem by showing understanding and making allotments for students needing extra time. One successful technique is to create a "private signal" between student and teacher. For example, the teacher promises not to cold-call the student to answer if the student agrees to make a good faith attempt to volunteer at least once per week/day (intervals can be increased with student confidence). The student is instructed to raise her/his hand with a closed fist to signal the teacher that s/he is willing to attempt a problem but needs some additional processing before being called upon. The teacher will bide time until the student opens her/his palm fully indicating readiness (Lavoie, 1990).

Social Cognition

From the psychological perspective of cognitive behaviorism, both emotions and behaviors (e.g. poor problem solving strategies, low frustration tolerance, lack of motivation to persevere, etc) are expressions of beliefs which often develop in a biased or dysfunctional manner and over time become automatic core beliefs (Beck, 1995; Ellis & Dryden, 1997). Vygotskian theory informs us that students' beliefs (and knowledge) are inextricably determined by the sociocultural environment (i.e. mathematics classroom) with language and social modeling being the vehicles of shared knowledge. For example, while the pedagogical mainstays of direct instruction, modeling, and guided practice are often adequate to build procedural proficiency and increase probability of fact retrieval, they have generally been found to be insufficient to promote transfer of independent strategy performance to new/novel tasks among a significant percentage of students. In fact, many theorists and researchers assert that traditional model s of teaching mathematics have produced an inhibitory effect on mathematics learning by promoting sell-limiting belief systems. Schoenfeld (1988) and DeCorte, Op't Eynde, and Verschaffel (2000, 2002) provide heuristic categorizations of mathematical beliefs. Based on empirical research, they also identify common mathematically related, socially shared belief systems and their respective influence. The following have implications for development or inhibition of self-regulated mathematics learning:

Beliefs about mathematics education

* "Mathematics learning is memorization."

* "There is only one correct way to solve any mathematics problem."

* "All math problems can be solved in a few minutes."

Possible corollaries: Students engage only superficially and/or for a limited period of time. They exhibit limited persistence and tolerance for frustration.

Beliefs about the self in relation to mathematics

* Goal beliefs: "It is personally satisfying for me to master the concepts in this math course." v. "I just have to learn enough, long enough to pass the exam."

* Task Value beliefs: "This course material is important and useful to me." v. "The material in this course has nothing to do with the real world."

* Locus-of-Control beliefs: "If I study appropriately, then I will be able to learn this material well." v. "It doesn't matter how hard I study, I will never learn the hardest material in this course." or "It doesn't matter how hard I study, a lot of how you score on the exam comes down to chance."

Possible corollaries (of negative beliefs): Students fail to recognize their achievements, thus do not actively build upon previous successes such as volitional behaviors (e.g. studying, problem solving) or previously mastered concepts.

Beliefs about the social context of mathematics learning and problem solving

* "The teacher will point out all the important aspects of what I am supposed to learn."

* "One succeeds in math by performing tasks exactly the way the teacher wants."

* "Math knowledge is passed down from above, only geniuses are capable of discovering, creating or really understanding mathematics."

Possible corollaries: Students believe learning is incidental to getting the work done. They lack the expectation that they can make sense of it themselves, so they do not intrinsically engage in discovery, invention, or problem-solving.

In order to develop student independence, emerging models for strategies training have become increasingly constructivist. The goal would be to teach children to think math, not just do math. Students must be given a math problem to solve and by working in small groups or individually using manipulatives, students are encouraged to come up with what they feel is the correct answer. Through public sharing, students share the answers and thought process for confirmation and comparison. It is important to keep in mind that while there is only one answer in math there are many ways to get there.

A constructivist approach to math provides students with opportunities to think through a math problem and learn how to represent their thinking in mathematical symbolism and ultimately be instructed on the standard mathematical algorithm. One of the important things they learn through this process is to learn the language of mathematics and the need to have the math language to express the mathematical concepts. Some practical interventions include:

* Make math relevant to the students' lives. Though word problems and story-based narratives can enliven math thinking, some students with MLD often experience difficulty solving word problems and find abstract language confounding. Use familiar names and places when creating written word problems to provide familiarity to the student attempting to interpret the text of the problem. The contributions and limitations of current pedagogical practices include: Problems that require analysis of the unknown

--Problems that provide too much, too little, or incorrect data

--Problems that can be solved in more than one way

--Multi-step problems

--Problems with more than one correct answer

--Problems that require an extended effort

* A good source to use to make math fun and show the students that math is involved with real-life experiences is Math for Every Kid by Janice VanCleave (1991). The book is filled with exciting ideas and projects adding personal relevance to four main topics areas; basic fractions, averages, multiples, and measurements. Combining math with everyday life should increase on-task behavior in math while decreasing undesirable behaviors.

* Allow for cooperative learning so that students can learn from each other and encourage students to take on different roles within the cooperative groups. Allow flexibility in problem solving and maintain frequent contact with the group to ensure that misinformation is not being disseminated.

* Have students create their own problems. Problems can be devised individually or in small groups. This helps the student identify with information and allows her/him to take ownership of the problem. This does not ensure successful solution of the problem, since solving word problems is found to be one of the most difficult areas in mathematics for students.

Conclusion

Though this past decade has evidenced many advances in our understanding of the diagnostic features of the major LD types, our current knowledge base related to understanding causal factors, our ability to identify specific cognitive/information processing deficits, and development of successful targeted interventions remains largely relegated to the area of reading disabilities. Much less is known about the developmental and corrective trajectories associated with learning disabilities in mathematics (Lyon & Cutting, 1998). Not only is there a wide range of variation in the characteristics of students who struggle in the area of mathematics, there is also a wide range of expression in their performance deficits. To date, there has been insufficient research to link specialized instruction modalities with specifically defined impairments in mathematics learning. Nevertheless, time devoted to discerning variations in performance deficits and learning impairments displayed by an individual student can be invaluable in informing effective individualized interventions. A teacher is limited only by her or his own perceived limitations in finding reasonable, creative approaches to instruction and ability to gather valid and applicable information about students' needs.

Collaboration with one's peers and designated intervention teams, which may include special education teachers, school psychologists, and others with specialized knowledge of child development, can be a vital means to sharing ideas and forming solutions to instructional dilemmas but is dependent on a shared knowledge base. Therefore, it is essential for educators and other specialists to share a common conceptual framework.

Due to the range of variation in performance deficits, one would be remiss to construe mathematics learning disabilities as a unidimensional impairment. Instead, it is more helpful to maintain the perspective that MLD is a manifestation of more generalized functional problems. Therefore, students with MLD have a heightened need for a flexible educational environment, consistent application of appropriate pedagogical principles, and the attention of experienced, sensitive teachers (Lyon & Cutting 1998). Consistent with best practice treatments for other developmental issues, early interventions that are closely monitored for efficacy and adjusted accordingly are associated with better outcomes.

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Kristine Augustyniak, Ph.D., Associate Professor of School Psychology, Department of Education; Jacqueline Murphy, Ph.D, Associate Professor of Education; and Donna Kester Phillips, Ph.D., Assistant Professor of Education, Department of Education, Niagara University, Niagara University, NY 14109.

Correspondence concerning this article should be addressed to Dr. Kristine Augustyniak at [e-mail].