{"title":"证明构建和过程验证——本科生构建数学证明的验证活动","authors":"Katharina Kirsten , Gilbert Greefrath","doi":"10.1016/j.jmathb.2023.101064","DOIUrl":null,"url":null,"abstract":"<div><p>Proof construction and proof validation are two situations of high importance in mathematical teaching and research. While both situations have usually been studied separately, the current study focuses on possible intersections. Based on research on acceptance criteria and validation strategies, 11 undergraduates’ proof construction processes are investigated in terms of the effective and non-effective validation activities that occur. By conducting a qualitative content analysis with subsequent type construction, we identified six different validation activities, namely <em>reviewing</em>, <em>rating</em>, <em>correcting errors</em>, <em>reassuring</em>, <em>expressing doubts</em>, and <em>improving</em>. Although some of these activities tend to be associated with successful or unsuccessful proving processes, their effectiveness depends primarily on their specific implementation. For example, reviewing is effective when accompanied by a knowledge-generating approach and based on structure- or meaning-oriented criteria to provide deeper understanding. Thus, the results suggest that difficulties in proof construction could be partly attributed to inadequate validation strategies or their poor implementation.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Proof construction and in-process validation – Validation activities of undergraduates in constructing mathematical proofs\",\"authors\":\"Katharina Kirsten , Gilbert Greefrath\",\"doi\":\"10.1016/j.jmathb.2023.101064\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Proof construction and proof validation are two situations of high importance in mathematical teaching and research. While both situations have usually been studied separately, the current study focuses on possible intersections. Based on research on acceptance criteria and validation strategies, 11 undergraduates’ proof construction processes are investigated in terms of the effective and non-effective validation activities that occur. By conducting a qualitative content analysis with subsequent type construction, we identified six different validation activities, namely <em>reviewing</em>, <em>rating</em>, <em>correcting errors</em>, <em>reassuring</em>, <em>expressing doubts</em>, and <em>improving</em>. Although some of these activities tend to be associated with successful or unsuccessful proving processes, their effectiveness depends primarily on their specific implementation. For example, reviewing is effective when accompanied by a knowledge-generating approach and based on structure- or meaning-oriented criteria to provide deeper understanding. Thus, the results suggest that difficulties in proof construction could be partly attributed to inadequate validation strategies or their poor implementation.</p></div>\",\"PeriodicalId\":47481,\"journal\":{\"name\":\"Journal of Mathematical Behavior\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematical Behavior\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0732312323000342\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"EDUCATION & EDUCATIONAL RESEARCH\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Behavior","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0732312323000342","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"EDUCATION & EDUCATIONAL RESEARCH","Score":null,"Total":0}
Proof construction and in-process validation – Validation activities of undergraduates in constructing mathematical proofs
Proof construction and proof validation are two situations of high importance in mathematical teaching and research. While both situations have usually been studied separately, the current study focuses on possible intersections. Based on research on acceptance criteria and validation strategies, 11 undergraduates’ proof construction processes are investigated in terms of the effective and non-effective validation activities that occur. By conducting a qualitative content analysis with subsequent type construction, we identified six different validation activities, namely reviewing, rating, correcting errors, reassuring, expressing doubts, and improving. Although some of these activities tend to be associated with successful or unsuccessful proving processes, their effectiveness depends primarily on their specific implementation. For example, reviewing is effective when accompanied by a knowledge-generating approach and based on structure- or meaning-oriented criteria to provide deeper understanding. Thus, the results suggest that difficulties in proof construction could be partly attributed to inadequate validation strategies or their poor implementation.
期刊介绍:
The Journal of Mathematical Behavior solicits original research on the learning and teaching of mathematics. We are interested especially in basic research, research that aims to clarify, in detail and depth, how mathematical ideas develop in learners. Over three decades, our experience confirms a founding premise of this journal: that mathematical thinking, hence mathematics learning as a social enterprise, is special. It is special because mathematics is special, both logically and psychologically. Logically, through the way that mathematical ideas and methods have been built, refined and organized for centuries across a range of cultures; and psychologically, through the variety of ways people today, in many walks of life, make sense of mathematics, develop it, make it their own.