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Mathematics and Dyslexia

Perspectives, Fall 1998 International Dyslexia Association
reprinted with permission

Not all individuals with dyslexia have problems with mathematics, but many do. There are those who have a good memory for sequences and can execute procedures in a "recipe" style, i.e., step-by-step. They are able to remember formulas, but may not understand why the formula makes sense. They prefer to do paper and pencil tasks and are attentive to the details, but do not see the big picture. Then, there are those who see the big picture and have insight into the patterns of mathematics, but are poor at computation and have problems with remembering step-by-step procedures. They also understand mathematical concepts and like to solve problems mentally and quickly, yet their answers may be inaccurate. These individuals may have difficulty in verbalizing and explaining their answers.

Too frequently and too readily, individuals with dyslexia who have difficulty with mathematics are misdiagnosed as having dyscalculia - literally trouble with calculating, a neurologically based disability. True dyscalculia is rare (Steeves, 1983).1 We know that for individuals with dyslexia, learning mathematical concepts and vocabulary and the ability to use mathematical symbols can be impeded by problems similar to those that interfered with their acquisition of the written language (Ansara, 1973).2 Additionally, we know that the learning of mathematical concepts, more than any other content area, is tied closely to the teacher's or academic therapist's knowledge of mathematics and to the manner in which these concepts are taught (Lyon, 1996).3 Therefore, there are individuals with dyslexia who will exhibit problems in mathematics, not because of their dyslexia or dyscalculia, but because their instructors are inadequately prepared in mathematical principles and/or in how to teach them.

In addition, we know that individuals with dyslexia may have problems with the language of mathematics and the concepts associated with it. These include spatial and quantitative references such as before, after, between, one more than, and one less than. Mathematical terms such as numerator and denominator, prime numbers and prime factors, and carrying and borrowing may also be problematic. These individuals may be confused by implicit, multiple meanings of words, e.g., two as the name of a unit in a series and also as the name of a set of two objects. Difficulties may also occur around the concept of place value and the function of zero. Solving word problems may be especially challenging because of difficulties with decoding, comprehension, sequencing, and understanding mathematical concepts. In understanding the complex nature of dyslexia, Ansara (1973)4 made three general assumptions about learning, in particular, for individuals with dyslexia. These assumptions affect the way one needs to provide instruction. They are:

• learning involves the recognition of patterns which become bits of knowledge that are then organized into larger and more meaningful units;

• learning for some children is more difficult than for others because of...deficits that interfere with the ready recognition of patterns; (and)

• some children have difficulty with the organization of parts into wholes, due to ... a disability in the handling of spatial and temporal relationships or to unique problems with integration , sequencing or memory.

Therefore, teachers and academic therapists who provide remedial instruction in mathematics to these individuals must have an understanding of the nature of dyslexia and how it affects learning, not only in written language, but also in mathematics. Additionally, the instructor needs to have an understanding of the mathematics curriculum; the ability to use a variety of instructional techniques that are simultaneously multisensory and which provide for explicit instruction that is systematic, cumulative, diagnostic, and both synthetic and analytical- as well as a knowledge of current research in mathematical instruction.

Simply just being good at mathematics is not enough. The teacher and academic therapist need to understand that mathematics is problem-solving which involves reasoning and the ability to read, write, discuss and convey ideas using mathematical signs, symbols and terms. This requires an understanding of mathematical knowledge, both conceptual (relationships constructed internally and connected to already existing ideas) and procedural (knowledge of symbols used to represent mathematics, and the rules and procedures that are used to carry out mathematical tasks). Both are important and need to be understood. For procedural knowledge, the most important connection is to the conceptual knowledge that supports it; otherwise, procedural knowledge will be learned rigidly and used narrowly. Usually, when there is a connection to a conceptual basis, the procedure is not only understood, but the learner will have access to other ideas associated with the concept (Van de Walle, 1994).5 For individuals with dyslexia, this linkage is critical and language plays an important role.

To assist individuals with dyslexia in making this linkage, it is essential that teachers and academic therapists provide instruction that allows the learner to work through the following cognitive developmental stages when teaching mathematical concepts at all grade levels: concrete, pictorial, symbolic, and abstract. Individuals with dyslexia will learn best when provided with concrete manipulatives with which they can work or experiment. These help build memory as well as allowing for revisualization when memory fails. The next stage, pictorial, is one which may be brief, but is essential for beginning the transition away from the concrete. This is where individuals recognize or draw pictures to represent concrete materials without the materials themselves. Symbols, i.e., numerals, plus signs, etc., are introduced when individuals understand the basic concept, thereby making the connection to procedural knowledge. Finally, the abstract stage is where individuals are able to think about concepts and solve problems without the presence of manipulatives, pictures, and symbols. (Steeves & Tomey, 1998a).6

According to Steeves and Tomey (1998a),7 it is important that the four developmental stages are linked through language for these individuals. There are three kinds of language which allow one to fully integrate mathematical learning. First, is the individual's own language. No matter how imperfect this language is, it is important that the individual discusses, questions, and states what she/he has learned. Second, is the language of the instructor, or standard English, which clarifies the learner's own language, and links to the third language, the language of mathematics. The language of mathematics is not just the vocabulary but the use of sign, symbols, and terms to express mathematical ideas, such as 2 + 4=6. Also, language allows the instructor to determine if the learner understands the concept and is not just following steps demonstrated by the instructor to complete a process, even at the concrete stage.

For these reasons, teachers and academic therapists who, in mathematics, work with individuals with dyslexia, must be well-trained in multisensory structured techniques both in language and mathematics instruction and remediation. They must not only demonstrate competencies in knowledge and skills in teaching language to these individuals, but also demonstrate the following competencies in mathematics (Steeves and Tomey, 1998b)8:

1. Understanding of the mathematics and the use of appropriate methodology, technology, and manipulatives within the following content:

• Number systems, their structure, basic operations and properties;

• Elementary number theory, ratio, proportion and percent;

• Algebra;

• Measurement systems - U.S. and metric;

• Geometry: geometric figures, their properties and relationships;

• Probability;

• Discrete mathematics: symbolic logic, sets, permutations and combinations; and

• Computer science: terminology, simple programming, and software applications;

2. Understanding of the sequential nature of mathematics, and the mathematical structures inherent in the content strands;

3. Understanding of the connections among mathematical concepts and procedures and their practical applications;

4. Understanding of and the ability to use the four processes - becoming mathematical problem- solvers, reasoning mathematically, communicating mathematically, and making mathematical connections at different levels of complexity;

5. Understanding the role of technology, and the ability to use graphing utilities and computers to teach mathematics;

6. Understanding of and ability to select, adapt, evaluate, and use instructional materials and resources, including technology;

7. Understanding of and the ability to use strategies for managing, assessing, and monitoring student learning, including diagnosing student errors; and

8. Understanding of and the ability to use strategies to teach mathematics to diverse learners.

The editors thank Harley A. Tomey, III (VA) and Joyce Steeves, Ed.D. (MD) for their suggestions for and review of this article, and especially Mr Tomey for his help in its preparation.

References and Endnotes

1Steeves, K.J. (1983). Memory as a factor in the computational efficiency of dyslexic children with high abstract reasoning ability. Annals of Dyslexia, 33,141-152. Baltimore: International Dyslexia Association.

2, 4 Ansara A. (1973). The language therapist as a basic mathematics tutor for adolescents. Bulletin of the Orton Society, 23, 119-138.

3 Lyon, G.R. (1996). State of Research. In Cramer, S. & Ellis, W. (Eds.), Learning disabilities: Lifelong issues (pp. 3-61). Baltimore: Brooks Publishing.

5 Van de Walle, J. A. (1994). Elementary school mathematics: Teachi ng developmentally (2nd ed.). White Plains, NY. Longman.

6,7 Steeves, K. J., & Tomey, H.A. (1998a). Mathematics and dyslexia: The individual who learns differently may still be successful in math. Manuscript in preparation.

8 Steeves, K. J., & Tomey, H.A. (1998b). Personal written communications to the editors.