Thesis (M.A.)--University of Wisconsin--Madison, 1971. / eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

This normative survey investigated the question, "What general heuristics are used by selected grade eleven academic algebra students in the solution of mathematical problems?" The investigator was interested in determining if students who had either A or B mathematics eleven grades used any heuristics. Forty-two students, who were enrolled in nine schools, were interviewed. Each student was given two mathematical problems to solve. These problems could be solved using two of nine general heuristics namely, cases, deduction, inverse deduction, invariation, analogy, symmetry, preservation of rules, variation, and extension. The researcher requested the students to think aloud. The student was encouraged to attempt the problems any way he chose. They were asked to be more concerned with revealing as much of their thought processes as possible, as with the accuracy of their solution. All the interviews were taped. The investigator found evidence that eight of the nine heuristics were used. The heuristics were cases, deduction, inverse deduction, invariation, analogy, preservation of rules, variation, and extension. Thirty-eight of the forty-two students interviewed showed evidence of using one or more of the heuristics. Eighteen of the students used cases, seven used deduction, three used invariation, two used inverse deduction, seven used analogy, two used preservation of rules, three used variation, and seven used extension. The investigator also found evidence that a heuristic which was not mentioned previously was used by eleven of the students. For the purpose of this investigation the heuristic was called "successive variation." When the heuristic of successive variation is used a possible resolution to the given problem is chosen at random. If the answer is not correct, the student determines what changes must be made. Then the possible solution is varied successively until the correct answer is found. The students’ command of the heuristics was not developed and therefore they could not use these techniques efficiently and effectively to solve the problems they were given. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

The structure of a classification appears to consist of two components: (a) relevant attributes and (b) the classification rule which combines the relevant attributes to describe the classification. An experiment was conducted to separate attribute identification (Al) from rule learning (RL) and compare these with the complete learning (CL) of a classification which requires learning both components. The comparison was conducted for two biconditional classification rules, a two attribute, simple biconditional rule (SB), and a three attribute, contingent biconditional rule (CB), and for two sets of solution strategy instructions, an intra-stimuli (RA) strategy involving classification according to the combination of relevant attributes on each card, and an inter-stimuli (ER) strategy involving classification according to the number of relevant attribute discrepancies between each stimulus card and an exemplar focus card. These three experimental factors were combined with two hypothesized control factors, sex and problem order, in a 3x2x2x2x2 factorial design. Each of 48 grade ten Ss (24 male and 24 female) completed both biconditional problems in one of two counterbalanced problem orders. Performance was recorded on six dependent variables: (1) Trials; (2) Errors; and (3) Seconds - all to a criterion of 27 consecutive correct responses; as well as the post-criterion variables, (4) Classifications, the number of correctly classified cards for a withheld subset of the stimulus population used for original learning; (5) Verbalization, a verbal response which describes a classification rule that separates the cards into two mutually exclusive and exhaustive categories; and (6) Strategy, the classification of the verbal response as implying the RA or ER strategy according to whether reference was made to relevant attribute combinations or relevant attribute discrepancies. In general, the results were as follows. First, the obtained order of paradigm difficulty for the Trials, Errors, Seconds, Classifications, and Verbalization variables was CL > Al > RL, This result was interpreted as support for the Al and RL component approach of Haygood and Bourne (1965), and as an extension of this approach to SB and CB rules. Second, the obtained order of rule difficulty for the Trials, Errors, and Seconds variables was CB > SB, This result was interpreted as support for the rule results of Shepard, Hovland, and Jenkins (1961), and as an extension of their results to include four dimension, bivariate stimuli. Third, the obtained order of difficulty for the strategy instructions was RA > ER, But, only for the Seconds variable was this result significant. Consideration of the results for the Strategy variable supported the conclusion that the instruction treatment was not sufficient to overcome the tendency of Ss to choose their own strategy. Consequently, several suggestions for a more effective instruction treatment were offered. Fourth, the obtained correlations between the Classifications and Verbalization variables were .89 and .79 for the SB and CB problems respectively. This result was interpreted as an indication that further investigation of the Classification variable as a method of determining concept attainment would be worthwhile. Finally, the educational emplications of this study were discussed. / Education, Faculty of / Educational and Counselling Psychology, and Special Education (ECPS), Department of / Graduate

This study developed a scale for use in assessing administrators' beliefs about problem formulation behaviour, examined selected aspects of its construct validity, and used the scale in an exploratory study to assess the problem formulation beliefs of educational administrators. Based on theoretical and empirical studies of problem formulation (Allal, 1973; Getzels and Csikszentmihalyi, 1976) and the theory of Cognitive Orientation (Kreitler and Kreitler, 1972; 1976) a conceptual framework was developed in which four kinds of beliefs could be held about each of four component behaviours of problem formulation. A set of statements which were consistent with this framework was developed. Screening and rating procedures yielded four equivalent sets of statements, one set for each belief domain. With the addition of questions about biodemographic characteristics these formed the instrument which was pilot tested and revised prior to being sent to 317 administrators in 12 Community Colleges and four Provincial Institutes in British Columbia. A 60% (189) return rate yielded the data for the study. Psychometric analyses indicated adequate internal reliabilities for the subtests. Hypotheses were tested by means of correlational analyses and showed that Normative, Goal and Self beliefs about problem formulation were moderately correlated with each other but not with General beliefs. Normative beliefs were positively and more highly correlated with Goal beliefs than with General or Self beliefs. A comparison of the responses of selected respondents (low scorers and high scorers) revealed that high scorers were more consistent than low scorers in the level and configuration of their responses. Training in problem solving was the only biodemographic characteristic found to distinguish significantly between low and high scorers. The results suggest some need for further examination of existing theory: the four belief domains may not be independent but organized in particular ways; computation of a summary "cognitive orientation" score is not well legitimized by the present data. Respondents' ability to recognize four component behaviours of problem formulation is confirmed by the study but their beliefs about the components are not equally consistent. The study concludes with speculations about the usefulness of the scale as a tool in administrative preparation. / Education, Faculty of / Educational Studies (EDST), Department of / Graduate

This study was undertaken to compare the effectiveness of two problem-solving strategies at the grade six and seven levels. Both strategies were designed to aid students in associating the verbal statement of a problem with its corresponding mathematical equation. One approach, the Translation Method, stressed literal, carefully structured translation of word problems, while the second, the Inductive Method, encouraged students to create their own problems, using mathematical equations given by the teacher. A control group practiced word problems without any instructional guidance. Forty-eight students from the sixth and seventh grades of a private elementary school in Vancouver, British Columbia were combined and assigned to the three treatment groups on the basis of their performance on a pretest in translation. For a period of four school days, all subjects used materials prepared by the investigator. Two criterion measures were used. Posttest One was composed of traditional word problems requiring only one mathematical operation for the correct solution. Posttest Two was constructed with novel or challenging word problems requiring more than one operation for the correct solution. Each test contained eight items and was designed for one forty-minute period. Scores of the tests were analyzed using multivariate analysis of variance for the two dependent measures. The three factors considered were Treatment, Sex, and Grade, and a simple main effects analysis was employed to examine male-female differences within each treatment level. Statistical comparisons among the three groups offered no evidence of superiority for one approach over another. In addition, no interaction was found between treatment and sex. Boys were found to be significantly superior to girls in performance on the posttests. Further analysis indicated that Posttest One scores for the Translation Group students differed significantly between boys and girls, with the girls' performance particularly weak for this measure. Subjective observation revealed differences in attitude. Students found the Translation Method burdensome. Students in the Inductive Group enjoyed that approach, and students in the Control Group seemed interested in the practice sequence of word problems. / Education, Faculty of / Graduate

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The thesis describes the theory and design of a general problem-solving system. The system uses a single general heuristic based on a formal definition of differences within the framework of means/ends analysis and employs tree search during problem solution. A comparison is made with two other systems using means/ends analysis. The conditions under which the system is capable of solving problems are investigated and the efficiency of the system is considered. The system has solved a variety of problems of varying complexity and the difference heuristic appears comparatively accurate for goal-directed search within certain limits.

No description available.

This study employed thinking aloud and semi-structured interviews to explore problem-solving representations, problem-solving processes, and problem-solving strategies of six grade seven students on word problems of linear equation in one variable. The instrument of the study was a researcher-designed test with literal, graphics and/or symbolic descriptions and was examined and revised by three senior secondary mathematics teachers. According to their mathematics scores of 3rd midterm exam last semester, students were divided into three achievement groups¡Ð¡Ðlow achievement group (the lowest 27%)¡Amiddle achievement group (46%) and high achievement group (the highest 27%). One subject was selected from each of middle and high achievement groups of three grade seven classes. Six subjects, in total, had taken thinking aloud training for three weeks, and then they took the paper and pencil test individually with a follow-up interview. All the processes of individual tests and interviews were video recorded. The videotapes were transcribed and provided the major evidence of the analyses of participants¡¦ performances of problem-solving processes, their problem-solving representations, and their problem-solving strategies. The results of problem-solving representation, problem-solving process, and problem-solving strategy were reported separately as follows: (1)Problem-solving representation. Participants applied literal, algebraic and numeral representations to solve one-variable leaner equation problems more often than used graphic one. (2)Problem-solving process. (a)When graphic representation was applied in this test, the time of problem solving could be shortened effectively. (b)The times that Participants repeat to read and analyze the topic increased relatively in the topics with more writing narration. (c)In more than one half of the fault problem-solving cases, the three stages of exploration, implementation, and planning were administered simultaneously. (d)The more verification was applied during participant¡¦s problem-solving process, his/her opportunity of success was higher. (e)Verification was often administered in problems with complex computations or questionable topics. (f)The relevance was higher between problem content and daily life, the opportunity of success was higher. (g)The time that the high achievement group used to solve problems was shorter than the middle achievement group used, and the opportunity of success was also higher than the middle achievement group. (3)Problem-solving strategy. (a)The problem-solving strategies applied by participants of high achievement group were more consistent, and the problem-solving strategies among participants of middle achievement group were more diverse. (b)The problem-solving strategies that participants often used to solve word problems of linear equations in one variable were translating the word problem into an equation, simplification of equation by collecting terms, using inverse operations, and properties of equality.