Higher order reasoning produced in proof construction: how well do secondary school students explain and write mathematical proofs?

The culture of mathematical explanations and writings based on conceptual understanding in proof construction is on the focus of the paper. We explore students’ attempts to explain construction of mathematical proofs after reading them and write mathematical proofs after working out their own constructions. Two examples of proofs, by induction and by contradiction, are discussed in detail to highlight students’ difficulties in proving and possible ways for their resolving

Computer theorem proving in math

We give an overview of issues surrounding computer-verified theorem proving in the standard pure-mathematical context. This is based on my talk at the PQR conference (Brussels, June 2003)

Mathematical Understanding and Proving Abilities: Experiment with Undergraduate Student by Using Modified Moore Learning Approach

This paper reports findings of a post test experimental control group design conducted to investigate the role of modified Moore learning approach on improving students' mathematical understanding and proving abilities. Subject of study were 56 undergradute students of one state university in Bandung, who took advanced abstract algebra course. Instrument of study were a set test of mathematical understanding ability, a set test of mathematical proving ability, and a set of students' opinion scale on modified Moore learning approach. Data were analyzed by using two path ANOVA. The study found that proof construction process was more difficult than mathematical understanding task for all students, and students still posed some difficulties on constructing mathematical proof task. The study also found there were not differences between students' abilities on mathematical understanding and on proving abilities of the both classes, and both abilities were classified as mediocre. However, in modified Moore learning approach class there were more students who got above average grades on mathematical understanding than those of conventional class. Moreover, students performed positive opinion toward modified Moore learning approach. They were active in questioning and solving problems, and in explaining their works in front of class as well, while students of conventional teaching prefered to listen to lecturer's explanation. The study also found that there was no interaction between learning approach and students' prior mathematics ability on mathematical understanding and proving abilities, but there were quite strong association between students' mathematical understanding and proving abilities

Students' mathematical argumentation ability when proving mathematical statements based on self-efficacy

Argumentation as an aspect of problem-solving has been studied in mathematics education. However, mathematical proof still needs to be resolved further. This study investigates students' mathematical argumentation skills when proving mathematical statements based on their self-efficacy. The research subjects were 43 mathematics education students at a university in Aceh Province who had taken a number theory course. The study used a qualitative approach with a case study design: students’ mathematical proving self-efficacy. Data was obtained using self-efficacy questionnaires and mathematical proof test instruments that experts have validated, while the data triangulation used was an in-depth interview. The results of this study reveal that students' mathematical argumentation skills in proving mathematical statements have differences based on their self-efficacy. The mathematical argumentation ability of students with high self-efficacy involves all aspects of argumentation well so that the construction of the proof is scientifically correct. Meanwhile, the argumentation ability of students with moderate or low self-efficacy still does not involve essential aspects of argumentation. So, the proof results are not scientifically correct because they have not arrived at the proper conclusion

Mathematics and language

This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes that we view mathematics as a system of conventions and norms that is designed to help us make sense of the world and reason efficiently. Like any designed system, it can perform well or poorly, and the philosophy of mathematics has a role to play in helping us understand the general principles by which it serves its purposes well

Performance Assessment to Measure Student’s Mathematical Proving Ability Based on the Abductive-Deductive Approach

This study provides an overview of performance assessment instruments to measure students’ mathematical proving abilities based on the abductive-deductive approach. This is descriptive qualitative research of performance assessment instruments and their rubrics in measuring students’ mathematical proving abilities. The research method was literature study. The performance assessment instrument consists of essays designed to identify the mathematical proving abilities of students in mathematics courses. In this article, the examples are given for the Real Analysis class. The items of performance assessment were arranged referring to the abductive-deductive reasoning approach which has a pattern containing three main questions: 1) “What conditions can be obtained from the conclusion?” which was answered with abductive reasoning, 2) “What are the consequences that can be obtained from known facts?” which can be answered with deductive reasoning, and 3) “What conditions connect the conditions of conclusions and the implications of premise?” which can be answered with a key process of the mathematical statement proving process. Keywords: performance assessment, mathematical proving ability, abductivedeductive approac

Mathematical Understanding and Proving Abilities: Experiment With Undergraduate Student By Using Modified Moore Learning Approach

This paper reports findings of  a  post test experimental control group design conducted to investigate the role of modified Moore learning approach  on improving students’ mathematical understanding and proving abilities. Subject of study were 56 undergradute students of one state university in Bandung, who took advanced abstract algebra course. Instrument of study were a set test of mathematical understanding ability, a set test of mathematical proving ability, and a set of students’ opinion scale on modified Moore learning approach. Data were analyzed by using two path ANOVA. The study found that proof construction process was more difficult than mathematical understanding  task  for all students, and students still posed some difficulties on constructing mathematical proof task.  The study also found there were not differences  between students’  abilities on mathematical understanding and on proving abilities of  the both classes, and both abilities were classified as mediocre. However, in modified Moore learning approach class there were more students who got above average grades on mathematical understanding than those of conventional class. Moreover, students performed positive  opinion toward  modified Moore learning approach. They  were  active in questioning and solving problems, and in explaining their works in front of class as well, while students of conventional teaching prefered to listen to lecturer’s explanation. The study also found that there was no interaction between learning approach and students’ prior mathematics ability on mathematical understanding and proving abilities,  but  there were  quite strong  association between students’ mathematical understanding and proving abilities.Keywords:  modified Moore learning approach, mathematical understanding ability, mathematical proving ability. DOI: http://dx.doi.org/10.22342/jme.2.2.751.231-25