Learning exceptionalities in mathematics

Summary: There are a variety of ways in which students can perform especially well or poorly in particular areas of mathematics and ways for schools to address their needs.

Learning exceptionalities in mathematics include both difficulties in learning mathematical concepts and mathematical giftedness. In both cases, exceptionalities take many forms and may manifest themselves at nearly any stage of life. In some cases, students may even excel in one area while displaying a deficit in another. The neurobiology of learning mathematics is not yet fully understood. Research in these areas is ongoing, often using sophisticated medical imaging to identify and map mathematical associations and processes, such as calculations, visualization of polyhedrons, proving theorems, or pondering number theory problems. There are also some difficulties in devising tests to reliably identify specific types of exceptionalities, and many people, especially those with difficulties, may not be diagnosed until very late in their academic careers.

Educational institutions often struggle with appropriate ways to serve students with exceptionalities so that all students may reach their maximum potential. These range from specific classroom instruction techniques all the way up through broader policies or legislation that addresses the needs of these subgroups of students. There are currently many formal systems in place by which students are assessed and accommodation plans are developed, most of which require periodic reassessment and revision. Plans for students with disabilities typically fall under Section 504 of the Americans with Disabilities Act and are commonly referred to as “504 Plans.” In recent years, the term “dyscalculia” has emerged as a broad term to encompass the set of mathematics learning disabilities.

Mathematical Disabilities

Sadly, unlike reading disabilities, many mathematical disabilities go undiagnosed, primarily because of social acceptance of the idea that certain people either have or have not mathematical abilities. For many students, the perception that they are “not good at math” provides a reason for them not to strive for success in the mathematics classroom. Researchers and practitioners alike indicate that if teachers truly want all students to succeed, a new perspective must be adopted: “all students can do math, should do math, and will do math.”

With this new perspective in mind, the focus shifts to identifying mathematical disabilities. According to David Geary, researchers in the field of mathematical disabilities have attempted to identify disabilities by studying normal mathematical development theories and using those theories to study children who demonstrate difficulties in mathematics despite having average or better IQs. Since most of the research has focused on students in the elementary grades, it becomes even more important for teachers and parents to be alert for mathematical difficulties early in school.

Currently, children with mathematical disabilities are defined as children with at least average IQ scores who also score at or below the 10th percentile on mathematics achievement exams. Research indicates that 6% to 7% of elementary school children demonstrate persistent mathematical difficulties in the area of number and arithmetic. It is important to note that current research studies indicate these difficulties persist regardless of IQ, motivation, and other factors that influence learning. What makes this area of research perplexing is that these children may have very specific deficits that make only certain aspects of mathematics difficult. For example, a child may have difficulty with counting but show a strong ability in geometry. Because standardized tests, which are frequently used for making decisions about whether a child should be recommended for special services, assess a wide variety of mathematical skills, a child’s particular mathematical disability may not be immediately identified. Adding to the difficulty is that children who score at approximately the same level on standardized tests may have vastly different mathematical deficits. Unfortunately, current methods of assessing mathematics knowledge are not sufficient for identifying mathematical disabilities, as assessments that focus on specific number and arithmetic skills are needed.

Learning Basic Numbers

Several mathematical disabilities have been identified. First, children may have a disability in learning basic number skills. Geary states that the “learning of basic number skills is much more complicated than many adults would assume.” In order to learn basic number skills, children must learn the English number words (known as “word tags”) and the Arabic numbers in the correct sequence, and learn to translate between the two. Children must then learn the quantities associated with the number words and number symbols, as well as develop an understanding that numbers can be decomposed into smaller numbers or combined into larger numbers.

The learning of place value in the base-10 system is a key component of developing number sense, and children with this particular type of mathematical disability may not be able to comprehend that 12 is actually 10 + 2, leading to later difficulty with basic arithmetic skills.

Counting Skills

A second mathematical disability is in the area of counting. While children do not typically have difficulty learning the basic counting sequence, they may have difficulty learning the basic concepts that enable them to count objects effectively. Geary identifies these concepts as the following:

• One–one correspondence: When counting, one does not count and tag the same item twice
• Stable order: The order of the word tags remains constant across counted sets
• Cardinality: The value of the final word tag represents the quantity of the items in the counted set
• Abstraction: The concept that objects of any kind can be collected together and counted
• Order-irrelevance: Items within a set can be counted in any sequence

Geary notes that having children count does not provide an indication of a child’s understanding of the counting rules, as children may learn the sequence of counting without developing the understanding of applying the word tags to objects. An additional complexity to this mathematical disability is that children may have difficulty remembering information during the act of counting; therefore, they may understand the counting rules but may forget numerical information during the counting process.

Arithmetic Skills

A third area of mathematical disability is that of arithmetic skills. Children with arithmetic disabilities typically have difficulty remembering as many basic arithmetic facts as other children and may not recall basic facts as quickly. This memory difficulty may be the result of children having trouble storing basic facts in long-term memory, or it may be the result of other arithmetic facts inhibiting the child’s ability to recall. For example, a child may see a problem like 4 + 5, and the child may correctly remember 9, but also may remember 20 (or 4 × 5), causing the child to take longer to recall the correct fact. Children with arithmetic difficulties also may not use highly developed problem-solving procedures to solve arithmetic problems but may rely on procedures typically used by younger children.

In general, children with mathematical disabilities use less mature strategies in their approach to mathematics, resulting in more errors and delayed acquisition of advanced mathematical thinking. Finally, children may verbally show an excellent grasp of mathematical concepts but have difficulty translating that understanding into paper and pencil assessments. These children struggle with paying attention to operations and sequencing steps in complex operations. Interestingly, many students who show difficulty with arithmetic skills in the elementary grades become “good math students” in the higher grades where conceptual understanding is emphasized more heavily.

The Language of Mathematics

Fourth, some children may have difficulty with the language of mathematics. These children easily confuse mathematics terminology and struggle with verbally communicating their mathematical thinking. This deficit can inhibit students from making progress in advanced mathematics, as they may not have the verbal skills necessary to track the steps needed for complex calculations.

Visual-Spatial Skills

Finally, children may be disabled in their visual-spatial skills. These students frequently have difficulty with complex problems, as they may not be able to maintain a logical, coherent sequence of steps on a piece of paper. Additionally, these students have difficulty with pictorial representations, making mathematical topics such as graphing and trigonometry especially challenging.

The National Council of Supervisors of Mathematics (NCSM) offers several recommendations for teachers of students with mathematical disabilities in a 2008 position paper. First and foremost, teachers must reconsider their perceptions of what students with mathematical disabilities can and cannot do and maintain high expectations for all students. Teachers need to be better educated about mathematical disabilities, particularly about the diagnostic language that is used to describe the needs of the mathematically disabled student. If teachers develop a conceptual framework for what students with mathematical disabilities need, they can incorporate effective interventions and accommodations in the classroom. NCSM also suggests that mathematics teachers should establish collaborative relationships with special education teachers. Mathematics teachers should focus on using teaching strategies that enable students to move from the concrete to the abstract and that allow students to demonstrate understanding through a variety of methods. Mathematics education activities should be meaningful and connected to a number of mathematical topics, thereby enabling struggling students to make connections between mathematical concepts.

Mathematical Giftedness

When learning exceptionalities are mentioned, most people automatically think of learning disabilities. However, there is another group of students that has exceptional needs: gifted mathematics students. These students are typically described as having “natural mathematics ability” and frequently are left to their own devices as teachers spend the majority of their time and attention on struggling students. While the reality of the classroom is that teachers focus more on students with difficulties, the needs of the gifted students are just as important.

M. Katherine Gavin points out that three main issues exist regarding gifted mathematics students. First, just as with students with mathematical disabilities, gifted mathematics students demonstrate a wide variety of aptitude, and abilities. Some students learn concepts quickly, which makes mathematics easier to learn and apply. Other students show great persistence in problem-solving, while still others demonstrate an ability to apply mathematical concepts in new ways.

Second, elementary teachers typically do not have specialized training in mathematics and may not know how to address the gifted student’s needs in the elementary grades. The response of many elementary teachers is to keep gifted mathematics students occupied with puzzles or advanced curricular materials, which typically do not advance the gifted student’s mathematical ability.

Third, current grade-level curricula are lacking in materials that are challenging and substantial enough for the gifted mathematics student. Therefore, gifted mathematics students may be given materials that do not allow for the development of critical thinking skills and the conceptual understanding of complex mathematics concepts.

For classroom teachers, it can be difficult to meet the needs of gifted mathematics students. Dana Johnson offers the following suggestions:

• Pre-assess students to determine which students already have mastered the material. For students who demonstrate mastery, provide instructional materials with advanced content and a problem-solving focus.
• Utilize a variety of assessment techniques, providing students with opportunities to show differences in understanding, creativity, and accomplishment.
• Choose textbooks with a variety of enriched opportunities. Use multiple resources to meet the needs of gifted mathematics students.
• Be flexible in expectations about pacing. A student may be gifted in one area of mathematics but struggle in another.
• Use hands-on, discovery-based teaching strategies as well as higher level questions.
• Provide opportunities for students to participate in mathematics contests, such as Mathematical Olympiads and Math Counts.

According to Gavin, the implementation of such strategies in the classroom will allow gifted mathematics students to develop their cognitive skills while maintaining the joy of doing mathematics.

Bibliography

Emerson, Jane, Brian Butterworth, and Patricia Babtie. Dyscalculia Assessment. New York: Continuum, 2010.

Garnett, K. “Math Learning Disabilities.” LD Online. http://www.ldonline.org/article/5896?theme=print.

Gavin, M. K. “Meeting the Needs of Talented Elementary Math Students.” Project M3. http://www.gifted.uconn.edu/projectm3/meeting%20the%20needs.html.

Geary, D. C. “Mathematical Disabilities: What We Know and Don’t Know.” LD Online. http://www.ldonline.org/article/5881?theme=print.

Hannell, Glynis. Dyscalculia: Action Plans for Successful Learning in Mathematics. London: David Fulton Publishers, 2005.

National Council of Supervisors of Mathematics. “Improving Student Achievement in Mathematics for Students with Special Needs.” The National Council of Supervisors of Mathematics Improving Student Achievement Series 4 (Winter 2008).

Sousa, David. How the Brain Learns Mathematics. Thousand Oaks, CA: Corwin Press, 2007.