published_or_final_version / abstract / toc / Psychology / Master / Master of Philosophy

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A Study of Problem-Solving Strategies and Errors in Inequalities for Junior High School Students The aim of this study is to investigate students in learning in inequalities with one unknown, as well as to collect corresponding strategies and errors in problem solving. The subjects of this study were nine-grade students from junior high school. Six classes were selected from three schools with total of 204 students. This investigator used a paper-and-pencil test in first round data collection. In the second round, some students were interviewed, to further understand students¡¦ way of thinking and reasons in errors produced in problem-solving procedures. Hopefully, results can be used as reference for junior high school math teacher to plan future teaching and to prepare teaching materials. The results of the study are three: students solved linear inequalities by using 12 different strategies; students¡¦ errors can be divided into 11 types; and, the reasons for errors are mainly understanding and transforming information from problems and the determination on solutions. The students also found it difficult to understand negative fractions and negative decimals relationships (no matter in word problems or in calculation problems). In this study, those who fail to solve problems involving inequalities with one unknown are those who cannot translate algebraic expressions or keywords. They produced errors 5 typical cases: determining objectives, integrating mathematics knowledge, using a problem solving method, calculating process, and, determining solution.

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This study reexamined ability differences on Luchin's (1942) water-jar test, a problem-solving task based on Gestalt theory. On this task a "set" or method of solution is assumed to be induced through the prior solving of a series of problems which require a common solution pattern. Researchers employing this task have generally discriminated between subjects who appear to have formed a "set" on the initial questions and subjects who do not appear to have formed the set. Subjects regarded as not forming the set have generally not been involved in all the analysis since the effect of the set on solution to subsequent questions cannot be analyzed if the set is not formed. / In a previous study using the water-jar task, some generalizations were made about the thinking processes of gifted and nongifted children without regard to whether or not they formed the set. The present study reexamined the relations among metacognitive knowledge, speed and giftedness taking into account set formation. (Abstract shortened by UMI.)

This thesis is about "authentic learning": learning from life-like contexts. The construct derived from the social situated approach (Lave & Wenger, 1991), has surprisingly no counterpart in cognitive psychology. The first objective of this thesis is to develop a cognitive formulation of authentic learning from classical cognitive works and recent neuroscience studies findings. The characteristically cognitive feature posited is "n-coding", the encoding of multimodal input (verbal, visual, kinesthetic, social...). To test quasi-experimentally the effectiveness of this cognitive definition, a review of the instructional literature identified Collaborative Group Problem Solving (Heller et al., 1992) as an appropriate candidate for authentic instruction in physics. / The study design was comprised of one control and three treatment conditions varying in degrees of n-coding: (high, medium and low) while controlling for each treatment group's "participatory framework". All students were assessed before and after instruction on the FCI (Hestenes et al., 1992). Confidence levels were measured with each FCI question resulting in four new measures (gain in mean: confidence, right answer confidence, wrong answer confidence and weighted FCI). Procedural problem solving skills were measured through final exam grades. / Two empirical questions are posed. First, does increasing n-coding enhance learning? Second, since cognitive n-coding is unaccountable from the social perspective, does the situated perspective "subsume" the cognitive (Greeno, 1998)? Here, a quasi-experiment was not only used to test interventions but paradigm effectiveness, a methodological first. / Results shows that high and medium n-coding groups were significantly more effective than the situated low n-coding group (p=0.003) showing the effectiveness of increasing n-coding and refuting the claim that social approaches must subsume cognitive ones. No significant difference was found between high and medium n-coding groups (p=0.74) whereas all treatment groups differed from the control (p=0.0497), replicating findings on the effectiveness of non-traditional instruction (Hake, 1998). / Competing cognitive and social perspectives (Schoenfeld, 1999) may be better replaced by cross-paradigm symbioses such as importing authentic learning from situated approach into cognition. A model for reflecting on cross-scale symbioses is developed through the presence of self-similar patterns across scales (from micro-cognitive to macro-social). The fractal is put forward as a metaphor for the field of education and may serve to unify paradigms and yield optimal pictures of learning.

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