Students' Ideas of Covariation in Realistic Problem Situations in an Introductory Algebra Program in Taiwan
Chih-Cheng Hung
educch@ccunix.ccu.edu.tw
Center for Teacher Education
Chung-Cheng University
Ming-Hsiung, Chia-Yi 62117 Taiwan
Abstract
The purpose of this study was to document how students reasoned about fu nctions when they use their informal knowledge that were related to ideas of co variation (Confrey, 1994) or of joint-change. The general premise was that, prio r to receiving formal instruction in functions, students have covariation-relate d informal intuitions, including, rate-of-change (Confrey, 1994), dependency ide as ( Romberg et al., 1993), joint-changes (Abels, et al., 1994), one-dimensiona l variation ( Hung ,1995). To be more specific, this study aims to (1) investiga te junior-high school students' use of covariation-related ideas in their daily- life situations; (2) to elaborate and to transform those covariation-related i deas in terms of mathematical structures and perspectives. Two theories adopted include social constructivism and realistic mathematics; Han Freudenthal origin ally proposed the latter, and the author revised by adding the facet of the app lication of algorithm to the concrete problem situation in this study. Several research methods will be adopted in this study, including survey, clin ic-interviews, teaching experiments(Cobb, 1983) , and action research. Twenty-fi ve volunteer 7th-graders in Taiwan participated in an innovative field-test curr iculum on functions for twenty sessions, four sessions a week during July-August 1996. Students were assigned to different groups during problem solving and w ere encouraged to communicate and defend their ideas in a learning community. A researcher/instructor with two researchers assistant worked with these students .
CONCLUSION
1. Students most common strategies for interpolating and extrapolating alge braic problems were partially rooted in the ideas of one-dimensional variation a nd those of rate-of-change, which could be classified into 3 and 5 major categor ies, respectively. Furthermore, several levels of these categories could be impl ied. 2. As a source of premature understanding of covariation, ideas of one-dime nsional variation seemed to resist changing toward a more advanced level of idea s associated with covariation. 3. Students exhibited an increased reliance on ideas of rate-of-change, particu larly adopting quantitative as well as global perspectives, even though they te nd to shift among/between their intuitions 4. The concrete problem situations tended to support students when they applied their invented algorithm to solving different concrete problems.
EDUCATIONAL IMPORTANCE
Although this study's conclusions was limited by the small sample and non-randomized ability range, the findings of this study will be especially important as information may help educators understand how to teach students in a way that (1) build a framework to use students' prior knowledge about function a starting points; and (2) makes algebra reverent and understandable by focusing on material that could related to students' dependency notions in the introductory level; (3) rethink the needs to design algebra curriculum that build upon students limited ideas by focusing on one-dimensional-variation to reasoning about function and that allow students to make a transition from those informal ideas about functions to more formal ideas.
REFERENCE
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