Advances in Cognition, Education, and Deafness

Analysis

Analysis Robert Lee Williams In their paper for this volume, Hillegeist and Epstein state that "apparent cognitive-processing problems may be linguistic-conceptual differences rather than an inherent inability to think abstractly." It is a measure of how far we have come since the first Symposium on Cognition, Education and Deafness (1984) that such caveats are now scarcely necessary. According to Tsui, Rodda, and Grove in their description of the view of Soviet psychologists, "Rather than from a deficiency perspective, the research perspective is [now] more akin to that of cross-cultural variation." The papers in this chapter are both descriptive and prescriptive. Explanations for the disparity in performance between deaf and hearing students are explored, and possible remedies are offered. The topics in this chapter flow easily from mathematics to cognition to reading and to the complex interplay among them. Those who are unfamiliar with deaf persons and deafness often assume that deaf persons will excel in the area of mathematics in the same way that foreign students often do in American universities. The common misconception is that mathematics is "language-free" or at least "language-fair." Unfortunately, this notion is not the case. The papers in this chapter by Kemp and by Hillegeist and Epstein address the difficult topic of mathematical reasoning in deaf students. Both point out the difficulty that deaf students have with geometry. Kemp focuses her attentions specifically on the development of geometric thought in deaf undergraduate students . She indicates that van Hiele describes five levels through which one must progress towards full understanding of geometry: visualization, analysis, informal deduction, formal deduction, and rigor. It would appear that van Hiele's system 320 Analysis 321 follows the necessary protocol to be called a true stage theory. Stage theories must show an invariant sequence that is more than just an innate unfolding, an underlying structure that accounts for a variety of behaviors at a specific stage (no matter how varied they may be), and a clearly presented description of how each stage prepares the way for succeeding stages. As Kemp says, "the structure of a certain level is constructed from and incorporates the prior levels...." Clearly, van Hide's stages go hand-in-hand with Piaget's stages, and the role of formal operations is critical. However, neither van Hide nor Piaget clearly explained the role of formal instruction. Piaget believed that children move through the stages in a process of active discovery, but at the same time formal operations seem to be more common in students who have had formal training in, for example, the sciences. Even with formal training it appears that students who are at the first or second van Hiele level are doomed to fail when they take geometry. Is this result due to the courses being taught at the third or fourth level and the gap being too great for the students to close? If these reasons apply, then one would expect a tremendous difference in an Individualized Education Plan (IEP) in which a teacher could locate a student's level of reasoning and then adapt the course, as opposed to a lecture format in which the teacher presents the information at a fixed and predetermined level. Hillegeist and Epstein point out that three languages are involved in learning mathematics, "the language of mathematics, the language of instruction, and the language of learning ... success in learning mathematics depends on how successful each student is in integrating [these languages]." A number of research possibilities are raised in this paper. Clearly, an interesting idea would be to compare classes in which both teacher and student are communicating in American Sign Language with those classes in which both are using spoken English as well as with those classes in which there is a mismatch. This process would allow us to ferret out the validity of the later statement: "Relatively good language skills appear to be necessary to successfully learn mathematics ..." as opposed to the problems of "limited life experiences upon which to draw to derive meaning from new concepts or to fully understand example problems." Additionally, one would wonder what proportion of the problems that deaf university students encounter with mathematics (see article by Kemp) could be attributed to a longterm problem with mathematics at the arithmetic level (a difficulty that will be difficult to solve at the university level). Hillegeist and Epstein further suggest that "... errors ... appear to indicate that the student focused on the surface structure of a mathematical statement rather than...