Proof Comprehension Research Articles

Using Extant Proofs in the Classroom: A Comprehension Activity Structure

ABSTRACT Proof comprehension is an important skill for students to develop in their proof-based courses, yet students are rarely afforded opportunities to develop this skill. In this paper, we describe two implementations of an activity structure that was developed to give students the opportunity to engage with complex proofs and to develop their proof comprehension skills. We share one implementation in an introduction to proof course and one implementation in an abstract algebra course. In particular, we aim to elucidate the details of facilitating these days in class and offer suggestions on how other instructors can adapt this proof reading activity structure in their own classes.

The PRIUM qualitative framework for assessment of proof comprehension: a result of collaboration among mathematicians and mathematics educators

The PRIUM qualitative framework for assessment of proof comprehension: a result of collaboration among mathematicians and mathematics educators

Effects of Self-Explanation on the Expansion and Indulgent of Mathematics Teaching: Case of the Students of the University of Wachemo

The main goal of this study is to investigate the effects of self-explanation in improving proof comprehension in mathematics instruction: the case of Wachemo university undergraduate mathematics students of academic year 2020/2021. Moreover it is intended to find out interconnection between students achievement in mathematics and self-explanation in proof comprehension. To assess student’s self-explanation ability we applied self-explanation test and attitude test at the beginning and at the end to assess student’s character change to ward mathematics. In this study there are 48 students participated in both control and experimental groups. This study showed that the effectiveness of self-explanation in proof comprehension in mathematics courses. This also leading to recommend for curriculum developers to consider in self-explanation. One of the most strong findings of research is that conceptual indulgent is an important elements of expertise hence self-explanation is the tool to enhance students conceptual understanding, along with realistic knowledge and technical facility. The alliance of realistic knowledge, procedural expertise, and conceptual indulgent makes all three components operational in powerful ways. So that self-explanation has prominent effect in supporting students to achieve the intended objectives in proving theorems. Of these, two areas were measured quantitatively: student achievement in calculus and transformation geometry, and self-concept toward these courses. Those quantitative research findings are presented in chapter four with the details of the data collection procedures, the quantitative results, and an analysis of the outcomes.

“This cannot be”—refutation feedback and its potential affordances for proof comprehension

“This cannot be”—refutation feedback and its potential affordances for proof comprehension

Teaching and Learning Proofs of the Pythagorean Theorem during Distance Learning and in the Flipped Classroom

This paper presents the findings of two design research cycles on learning and teaching different proofs of the Pythagorean theorem in German secondary schools during distance learning in the first research cycle and a flipped classroom in the second research cycle. The study explores students’ difficulties when learning proofs in the different scenarios by analyzing students’ proof schemes, occurring error patterns, the students’ descriptions of their proof comprehension, and evaluations of the learning scenarios. The findings indicate that typical difficulties occur less in the flipped setting, especially for students with a low or average level of prior knowledge. Further analysis of the evaluation of the learning activities suggests that this can be attributed to the implemented adaptations of the learning scenario when changing it from a distance learning setting to a flipped learning setting, in particular, the opportunity to work collaboratively and direct support the teacher. These results have implications regarding the potential of the flipped classroom as a way for educators to integrate instructional approaches they developed during the COVID-19 pandemic into their in-person teaching in the future.

Assessing Proof Reading Comprehension Using Summaries

In this paper, we explore the role of mathematical proof summaries as a tool for capturing students’ reading comprehension of a given proof. We present an interview study based on mathematicians’ pairwise evaluations of student-produced summaries of a proof demonstrating the uncountability of the open unit interval. We present a thematic analysis, exploring features of mathematicians’ pairwise decision-making and their priorities in evaluating summaries. We argue that the students’ proof summaries shared several properties with traditional modes of proof-writing and were frequently evaluated against similar conventions. We consider the consequences for research and practice with proof comprehension and conclude that proof summaries have the potential to form the basis of a new approach to assessment in this area.

Assessment of Students’ Mathematical Proof Comprehension: Gender and Year Level Background

The ability to understanding proof is essential before students can write their proof. A literature study shows a few interests in investigating the students’ knowledge from reading proof. This study aims to assess students’ comprehension when they were reading proof using a developed instrument based on the previous study on the same field. Three groups of students from a different level of years from a preservice teacher training program in an Indonesian university are taken as the participants. This paper reports the preliminary finding of students’ comprehension score comparison by gender and year level. Further analysis also involved the correlation of students’ basic knowledge to their comprehension scores.

Comparative judgement, proof summaries and proof comprehension

Proof is central to mathematics and has drawn substantial attention from the mathematics education community. Yet, valid and reliable measures of proof comprehension remain rare. In this article, we present a study investigating proof comprehension via students’ summaries of a given proof. These summaries were evaluated by expert judges making pairwise comparisons, which were used to generate a score for each summary. This approach, known as comparative judgement, has been demonstrated to generate reliable and valid scores when assessing other mathematical constructs. Our findings suggest that comparative judgement can produce valid and reliable assessments of the quality of student-produced proof summaries. We also explored which features of students’ proof summaries were most valued by the expert judges, and found that high-scoring summaries referenced a proof’s logical structure and the mechanism by which it reached a contradiction.

Do Generic Proofs Improve Proof Comprehension?

Students often have difficulty understanding the proofs that they read. Some mathematics educators have suggested using generic proofs to improve students’ proof comprehension. However, few empirical studies about students’ perceptions of generic proofs or whether students understand generic proofs better than traditional proofs have been conducted. In this paper, we describe a qualitative interview study and a quantitative experimental study exploring these issues. Our first finding is that generic proofs are popular with students, who believe they have the potential to improve proof comprehension. Our second finding is that in a randomized controlled experiment, students did not learn more from reading a generic proof than a traditional proof. The significance of these findings is discussed at the end of the paper.

The effect of quantitative reasoning on prospective mathematics teachers’ proof comprehension: The case of real numbers

We report a mixed-methods research study investigating the effect of quantitative reasoning on prospective mathematics teachers’ comprehension of a proof on real numbers. Nineteen prospective mathematics teachers engaged in quantitative reasoning while developing real numbers as rational number sequences in a series of instructional activities. All participants completed a proof comprehension assessment prior to and upon completion of the instruction. Six of the prospective mathematics teachers also participated in semi-structured interviews after the post-test. Results showed a significant difference in proof comprehension performance between the pre- and post-tests. Moreover, results from the interviews showed that prospective teachers reasoned quantitatively on the proof comprehension dimensions. Results suggest that engaging in quantitative reasoning during instruction may help to develop proof comprehension, particularly in situations involving the analysis of proofs entailing properties of the real number system. We recommend embedding quantitative reasoning in teacher education and professional development programs to facilitate mathematics teachers’ proof comprehension and proving activities.

Exploring students’ proof comprehension of the Cauchy Generalized Mean Value Theorem

AbstractUndergraduate students majoring in mathematics often face difficulties in comprehending mathematical proofs. Inspired by a number of studies related to students’ proof comprehension, and Mejia-Ramos et al.’s study in particular, a test was designed in relation to the proof comprehension of the Cauchy Generalized Mean Value Theorem (CGMVT). The test mainly focused on (a.) investigating students’ understanding of relations between the statements within the CGMVT proof and (b.) the relations between the CGMVT and other theorems. Thirty-five first-year university students voluntarily participated in this study. In addition, 10 of these students were subsequently interviewed to seek their opinion about the test. Test results indicated that most of the students lacked an understanding of the relations between the mathematical statements within the CGMVT proof, and the relations between the CGMVT and other theorems. The results of interviews showed that this type of assessment was new to students and helped them to improve their insights into mathematical proofs. The findings suggested such a test design could be used more frequently in assessments to aid instructors’ understanding of students’ proof comprehension and to teach students how mathematical proofs should be learned.

FLAWS IN PROOF CONSTRUCTIONS OF POSTGRADUATE MATHEMATICS EDUCATION STUDENT TEACHERS

Intending to improve the teaching and learning of the notion of mathematical proof this study seeks to uncover the kinds of flaws in postgraduate mathematics education student teachers. Twenty-three student teachers responded to a proof task involving the concepts of transposition and multiplication of matrices. Analytic induction strategy that drew ideas from the literature on evaluating students’ proof understanding and Yang and Lin’s model of proof comprehension applied to informants’ written responses to detect the kinds of flaws in postgraduates’ proof attempts. The study revealed that the use of empirical verifications was dominant and in situations. Whereby participants attempted to argue using arbitrary mathematical objects, the cases considered did not represent the most general case. Flawed conceptualizations uncovered by this study can contribute to efforts directed towards fostering strong subject content command among school mathematics teachers.

The ability of senior high school students in comprehending mathematical proofs

Mathematical proving ability is an ability which has function to improve mathematical high order thinking and reasoning skill. This ability consists of two sub-abilities, which are mathematical proof comprehension and mathematical proof construction. Both of the sub-abilities are related each other. To construct mathematical proofs, students need to have good mathematical proof comprehension. Due to the importance of mathematical proof comprehension, a qualitative descriptive study was conducted to describe senior high school students’ ability in mathematical proof comprehension. A test was given to 10 students of 11th grade in one of senior high schools at Bandung, West Java, Indonesia. The students’ works were analyzed for each proof comprehension indicator. Overall, the research found four results, the students: (i) were good in giving a reason for each step of proofs; (ii) got difficulties in validating a step of proof; (iii) were good in applying the steps of a proof to prove similar statement; and (iv) failed in determining and proving a notion based on pattern from the given statements. This study concludes that the senior high school students’ ability in comprehending mathematical proofs has not been satisfied. In several parts, it needs to be improved during mathematics learning process.

The impact of peer solution quality on peer-feedback provision on geometry proofs: Evidence from eye-movement analysis

Providing feedback on peer solutions to geometry proofs can support preservice mathematics teachers' assessment skills of such complex tasks. However, the quality of peer solutions may influence cognitive processing during peer-feedback provision, learning from providing peer-feedback, and peer-feedback content. To investigate this effect, we recorded the eye-movements of fifty-three preservice mathematics teachers while providing feedback on a near-correct or an erroneous peer solution to a geometry proof, and we measured their proof comprehension and peer-feedback content. Results show that the absence of errors earlier in the peer solution facilitated reliance on a figure-based approach, whereas encountering errors earlier in the peer solution was associated with more focus on the text of the proof. Students who provided peer-feedback on the near-correct peer solution had better comprehension of the proof, and they provided more accurate peer-feedback. Errors in peer solutions thus appear to hinder positive peer-feedback outcomes.

High School Students’ Performances on Proof Comprehension Tests

This study is a part of a large scale project in which an action research design is used to teach proof to 11th grade students. This part of the project aims to identify students’ comprehension level through five proof comprehension tests developed by the researchers based on the National Geometry Curriculum. Data were analyzed by considering the framework of Yang and Lin’s (2008) multilevel model. Results showed none of the students were successful at the most sophisticated level of the proof comprehension tests which requires conducting a proof in various ways or proving different theorems by using the same proof methods. Moreover, the highest proof comprehension was obtained from the level containing knowledge about definition, properties, and meanings of symbols. Achievement and comprehension decreased for components of a proof needing higher level mathematical skills. Based on the study’s results, suggestions about teaching proof are provided.

University Undergraduate Science Students’ Validation and Comprehension of Written Proof in the Context of Infinite Series

The reported study explored undergraduate science students’ validation and comprehension of written proofs, reasons given either to accept or reject mathematical procedures employed in the proofs, and the difficulties students encountered in reading the proofs. The proofs were constructed using both the Comparison and the Integral tests in the context of infinite series. One-hundred and thirty-one second year science students majoring in mathematics were the subjects of the study. Data were collected through a written assignment and interviews. An inductive approach was adopted in developing categories of proof validation by students. The categories were then compared with Weber’s classification on proof validation. Proof comprehension was analysed using the model by Meija-Ramos and colleagues differentiating between understanding proof locally and holistically. The findings of the study show that students validated the proofs according to what in their view constituted proof or what they viewed as the most important characteristic(s) of proof from both their lecture notes and instruction. The validation of the mathematical procedures was mainly based on whether or not the Comparison Test and the Integral Test theorems were considered to be applied appropriately. The difficulties students encountered with proof were attributed to the way the proofs were presented. The majority of students required that each step of the proofs be explained or justified. Implications for teaching are presented.

Developing and validating proof comprehension tests in undergraduate mathematics

ABSTRACTIn this article, we describe and illustrate the process by which we developed and validated short, multiple-choice, reliable tests to assess undergraduate students’ comprehension of three mathematical proofs. We discuss the purpose for each stage and how it benefited the design of our instruments. We also suggest ways in which this process could be employed by other researchers to develop and validate their own reliable proof comprehension tests.

Multimedia resources designed to support learning from written proofs: an eye-movement study

This paper presents two studies of an intervention designed to help undergraduates comprehend mathematical proofs. The intervention used multimedia resources that presented proofs with audio commentary and visual animations designed to focus attention on logical relationships. In study 1, students studied an e-Proof or a standard written proof and their comprehension was assessed in both immediate and delayed tests; the groups performed similarly at immediate test, but the e-Proof group exhibited poorer retention. Study 2 accounted for this unexpected result by using eye-movement analyses to demonstrate that participants who studied an e-Proof exhibited less processing effort when not listening to the audio commentary. We suggest that the extra support offered by e-Proofs disrupts the processes by which students organise information, and thus restricts the extent to which their new understanding is integrated with existing knowledge. We discuss the implications of these results for evaluating teaching innovations and for supporting proof comprehension.

Students’ Difficulties with Definitions in the Context of Proofs in Elementary Set Theory

In this paper we explore first-year students’ difficulties with the use and interpretation of definitions of mathematical objects as they attempt proof construction exercises in the area of elementary set theory. The participants are students at a historically disadvantaged university in South Africa. In this study the activities and utterances of 10 students who took part in consultative group sessions were observed and analysed. Consultative sessions were organised so as to encourage and develop students’ active participation while engaging in the task of proof construction. The framework that was used to analyse students’ proof comprehension and construction actions and contributions, particularly their interpretation and use of definitions, is described in the paper. The findings of the study resonate closely with those of researchers in the developed world. Students’ difficulties with definitions of mathematical objects include their misinterpretation of definitions of objects such as the union of sets and the Cartesian product and their association of mathematical objects with a word or symbol contained in their definitions.

Investigating and Improving Undergraduate Proof Comprehension

Undergraduate mathematics students see a lot of written proofs. But how much do they learn from them? Perhaps not as much as we would like – every professor knows that students struggle to make sense of the proofs presented in lectures and textbooks. Of course, written proofs are only one resource for learning; students also attend lectures and work, independently or with support, on problems. But, because mathematics majors are expected to learn much of their mathematics by studying proofs, it is important that we understand how to support them in reading and understanding mathematical arguments. This observation was the starting point for the research reported in this article. Our work uses psychological research methods to generate and analyse empirical evidence on mathematical thinking, in this case via experimental studies of teaching interventions and quantitative analyses of eye-­‐movement data. What follows is a chronological account of three stages in our attempts to better understand students’ mathematical reading processes and to support students in learning to read effectively.