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This is a report on a naturalistic study of the role mathematical paradoxes can play in the preservice education of high school mathematics teachers. The study examined the potential of paradoxes as a vehicle for: (a) sharpening student-teachers' mathematical concepts; (b) raising their pedagogical awareness of the constructive role of fallacious reasoning in the development of mathematical knowledge. Course material development and data collection procedures are described. Results obtained in parts of the study through written responses and class-videotapes are analyzed and discussed. The findings indicate that the model of dealing with paradoxes as applied in this study has relevance to such aspects of mathematics education as cognitive conflicts, motivation, misconceptions and constructive learning.This is a second report on this study. The first report (Movshovitz-Hadar, 1988) was a brief one, and focused on the problem, the procedures and findings in a general way. The present report is focused in details on one task. The authors intend to present an across tasks analysis in a third report to be presented at AERA 90. 相似文献
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Nitsa Hadar 《Educational Studies in Mathematics》1977,8(4):413-438
This study stemmed from a desire to redress the distorted view of mathematics in the elementary curriculum, created by the current imbalanced emphasis on computational rules and some applications, but very little logical analysis. The study is intended to show that fifth-grade students can significantly improve their use of logical analysis through a suitable instructional unit taught under ordinary classroom conditions.Concrete teaching materials were developed, through several trials and revisions, to familiarize students with the distinction between the valid inference patterns — Modus Ponendo Ponens and Modus Tollendo Tollens (AA, DC), and the fallacious ones — Affirming the Consequent and Denying the Antecedent (AC, DA). No formal rules were taught.The experimental unit was implemented four to five times a week for 23–25 sessions, by 4 fifth-grade teachers in their ordinary classes. The teachers participated in a twelvehour pretraining workshop.A pretest/posttest treatment/no-treatment design was applied to assess resulting improvement in students' conditional reasoning ability. The sample consisted of 210 fifth graders in a suburban area, 104 in 4 experimental classes and 106 in 4 control classes.A written group test was developed, through trials and revisions. Test items are formulated with a reasonable hypothetical content. Each item includes two premises: the first a conditional sentence, and the second either its antecedent, its consequent, or the negation of one of these, thus determining the logical form: AA, DC, AC, or DA. The question following the premises is stated positively. AA and DC are answered correctly by yes or no: AC and DA by not enough clues (NEC).The test contains 32 randomly-ordered three-choice items, eight in each logical form (two of the eight in each of the four possible modes in which negation may or may not occur in the antecedent or consequent). No sentential connective other than negation and conditional appears in the premises. Test/retest reliability was 0.79.Experimental and control group pretest performance levels did not differ (=0.05). More than 78% of the answers on AA and DC, and fewer than 33.1% on AC and DA, were correct. Overall pretest mean scores were 54.3% and 53.8% for the experimental and control groups respectively.There was a significant difference (=0.01) between the experimental and control groups' posttest overall performance—74.7% and 55.4%, respectively.There was no significant change in the control group's pretest and posttest performance levels on any logical form, or for the experimental group's on AA and DC. However, on AC and DA the two groups' gain scores were found significantly different.Negation mode, unlike logical form, was not found to be independently influential in analyzing test scores, but interacted with logical form.There was a pretest/posttest increase of 3.5 in experimental group frequency (percentaged) of incorrect NEC answers (AA and DC). As NEC appeared infrequently on the pretest, this increase was interpreted as learning that NEC is an acceptable answer. Separating out this effect from the percentaged frequency of correct NEC answers (AC and DA) left a pretest/posttest average increase of 37.8. This increase was attributed to learning when NEC is correct.Teachers were excited at the beginning, frustrated in the middle, and felt competent and involved in the project at the end. They felt the teaching should be less condensed. The majority of the students reacted positively to most parts of the experimental unit. However, some thought the unit as a whole was too repetitive and boring.No correlation was found between learning logic through the experimental unit and standard school achievement as measured by the Stanford Achievement Test (SAT). High, average, and low SAT achievers of the experimental group did not differ significantly in their pretest and posttest gain scores.Results of the study call for further investigation of the value and usefulness of teaching various parts of logic as an ordinary part of the elementary mathematics curriculum.This study was supported by NSF grants Nos. GW 7659 and PES 74-018450 and by an AAUW International Fellowship. It was carried out at the University of California, Berkeley, in 1974/75 under supervision of Professor Leon Henkin. 相似文献
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