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Article Excerpt
Abstract. This article reports on action research that took place in one section of a college general education mathematics course in which all three students who were enrolled had diagnosed learning disabilities related to mathematics. The project emerged in response to a question about performance in a mathematics course in which making sense of mathematics would be a primary focus, explaining one's work would be expected, and discourse among members would be a routine occurrence. Implications for teaching similar courses to students who have a mathematics-related learning disability are discussed.
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Literature related to postsecondary students who have mathematics-related learning disabilities (LD) is scarce. As a result, there are few content-related teaching suggestions to guide student-centered college mathematics faculty who have students with diagnosed LD in their classes. Some faculty rely on campus learning centers to assist these students, others help the students themselves. Generally, college-level students with LD do not have access to the degree of support that existed for them at lower grade levels. Some students with LD become discouraged when they cannot keep up in their mathematics class, and withdraw. Others persevere; they expend great amounts of effort and time and take advantage of college tutoring services and faculty office hours, yet, fail the course. Some institutions have course waiver or substitution policies or offer special sections of their required mathematics courses, but many do not.
This article reports on an action research project situated in a section of a general education mathematics course that enrolled three students with diagnosed LD related to mathematics. All students had a history of multiple attempts to satisfy the college's mathematics requirements and, with the exception of mathematics and science, all had performed at or above average levels in their courses.
The author sought suggestions from the literature for teaching course topics to the enrolled students. While unsuccessful in locating teaching suggestions, the author noted that many LD specialists do not favor current reform efforts in mathematics (Jones & Wilson, 1997; Maccini & Ruhl, 2000; Miller & Mercer, 1997), preferring the more traditional "present, practice, and test" approach. Mathematics reform efforts, based on the premise that students must make sense of mathematics, have been central to the author's professional practice. She considered traditional forms of instruction guided by behaviorist psychology, the norm in previous studies, as necessary but not sufficient for mathematics instruction.
The question naturally arose: How would college students who have mathematics-related LD perform in a course where making sense of mathematics is a critical component, where explaining one's work is expected, and where discourse among members is common practice? This question provided the impetus for the project. In teaching the course, the author intended to utilize reform methodology, including use of manipulatives, journal writing, and multiple forms of assessment. In order to contribute to the literature on teaching college-level mathematics to students who have mathematics-related LD, she planned to document her course modifications and chronicle students' efforts in making sense of the mathematics in an environment based on constructivist principles.
This article reports the results of the project, which utilized qualitative methodology. After reviewing the literature related to characteristics of students who have mathematics-related LD and teaching strategies suggested therein, the institution and its mathematics course requirement are described. In the methodology section, the author relates the impact of prior research on planning the course, completed prior to knowledge of student profiles. Next, she describes the students, based on data they provided during the first class, and presents one unit of the course, the mathematics of finance, in some detail. She describes modifications of original plans and includes examples of student work as evidence of instruction broadly based on constructivist principles. In the final section, the author discusses problems that might have been averted with additional information, as well as implications for practice.
Mathematics-Related Learning Disabilities and Teaching Suggestions
Some college students who possess average to above-average intelligence but are less successful in particular academic areas are described as having LD (Miles & Forcht, 1995). The characteristics of LD related to mathematics are diverse and can be connected to issues related to language, information processing and cognition (Daley, 1994; Strawser & Miller, 2001).
Vocabulary and reading issues impact mathematics performance. For example, words in English whose meaning in mathematical contexts differs can cause confusion. In algebra, the terms "reduce" or "cancel" are used when the goal is to simplify expressions, but the value does not change. In statistics, the term "mean" differs from either common context in English. Small words ignored by some students while reading can drastically alter meanings. Interpreting "x is less than y" as "x less than y" and using "y - x" instead of "x < y" will likely result in an incorrect solution that may be unrelated to understanding of mathematical concepts.
Students who have information-processing difficulties (Miller & Mercer, 1997) may not understand what the professor is saying or may not be able to listen and take notes at the same time. Others may copy notes from the overhead or blackboard incorrectly or they may leave out numbers when copying answers from calculators to paper. For example, they might interpret the number 98 as 89, or 86 as 68 in processing, even if the number is written correctly on the paper. Students who have motor difficulty may have poor or slow handwriting. They often have "holes" in their notes, resulting in gaps that interfere with content understanding. Further, attention deficits affect processing of mathematics problems that require multistep solutions: students lose the problem focus partway through a solution. Memory issues appear in students who do well on daily tasks but fail exams. Others can memorize and retrieve information on demand, but may not be able to connect mathematics concepts or know where to begin or end a task.
A specific LD subtype that primarily affects mathematics, dyscalculia or nonverbal learning disability (Strawser & Miller, 2001), is not language based and can be traced to the right hemisphere of the brain. Characteristics include selective impairment in mathematics, visual-spatial disturbances, and difficulties with social perception and development of social skills (Fleischner & Manheimer, 1997). Generalizations and abstract rules that characterize secondary and postsecondary mathematics courses are difficult for students with this diagnosis. While they can memorize definitions and state them when asked on tests, they are usually unsuccessful when asked to explain their understanding of the concepts. Similarly, they can perform a calculation on a test when it is similar to others completed in class and on assignments, but are unable to verbalize their reasoning.
Recent reviews of studies involving teaching mathematics to students with LD reveal that the amount of research in this content area has increased (Miller, Butler, & Lee, 1998), but is still underrepresented (Bryant & Dix, 1999). Both reviews build on the work of Mastropieri, Scruggs, and Shiah (1991), whose review of 30 studies found that mathematics interventions primarily addressed arithmetic computation. In the 23 studies that Bryant and Dix cited in their review spanning 1988-1997, only 2 were at the algebra level. Miller et al. cited all but three of the Bryant and Dix studies, adding 32 more; of these, only 4 were at the high school level. The three studies reviewed by Hughes and Smith (1990) that described mathematics beyond test score results provided descriptive, not empirical, research results. No studies were found that discussed mathematics content at the college level.
In light of the paucity of research on teaching college-level mathematics content to students with LD, faculty must consider research results from studies conducted at lower grade levels. Whether strategies found to be effective at lower levels transfer to older learners remains to be empirically validated. However, effective strategies cited in the above reviews go further than "present, practice, and test." They often appear in the repertoires of effective college mathematics instructors, particularly those who teach general education level courses:
1. Make the mathematics content relevant and authentic (Witzel, Smith, & Brownell, 2001).
2. Employ a concrete-to-abstract sequence (Fleischner & Manheimer, 1997; Maccini & Ruhl, 2000; Miles & Forcht, 1995; Witzel et al., 2001) that starts with a demonstration or activities using manipulatives, moves to a representational phase with specific examples and diagrams, and ends with an abstract generalization, rule, or proven theorem.
3. Provide opportunities for guided practice in solving problems prior to independent practice (Witzel et al., 2001), perhaps...
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