Teaching Of Proof Research Articles

Investigation of Prospective Mathematics Teachers’ Proof Completion Processes Supported By Key Ideas

The aim of this research was to examine prospective mathematics teachers’ completion processes of mathematical proofs supported by key ideas in the field of abstract algebra. Prospective teachers were asked to complete the parts of a proof left incomplete using those key ideas. It was decided together with three academics who are experts in the field of algebra which theorems to use and which proof sections to leave missing in the proof completion form. The proof completion form consisted of 5 proofs and was applied to 5 participants. The participants' processes of completing the proofs were evaluated in semi-structured interviews supported by the think-aloud method. Interviews were recorded by video and transcribed, and descriptive analysis was performed. The findings obtained were analyzed with think-aloud protocols. According to the results, key ideas can be said to play an active role in the proof process and the teaching of proof. In addition, developing activities consisting of proofs supported by key ideas can contribute to prospective teachers’ abilities to prove without memorization and internalize proof processes.

Human-Machine Collaboration in the Teaching of Proof

This paper argues that interactive theorem provers (ITPs) could play an important role in fostering students’ appreciation and understanding of proof and of mathematics in general. It shows that the ITP Lean has three features that mitigate existing difficulties in teaching and learning mathematical proof. One is that it requires students to identify a proof strategy at the start. The second is that it gives students instant feedback while allowing them to explore with maximum autonomy. The third is that elementary formal logic finds a natural place in the activity of creating proofs. The challenge in using Lean is that students have to learn its command language, in addition to mathematics course content and elementary logic.

Visualization of Algebraic Identities in R × R × R = R3 (in Space) at Secondary Education

Algebraic identities, which are considered as one of the important subjects of mathematics and mathematics education, have important areas of use such as physics, chemistry, many branches of engineering, as well as calculating the distances among the earth, sun and stars and planning space travels. Algebraic identities play an important role in the mathematics curriculum and in mathematics in general. However, it is seen as one of the subjects that are difficult to learn by students in formal education. In this context, this study aims to convey the visual proofs of the identities to the researchers in an explanatory way by using the visualization approach in order to make it easier for students to understand some of the important identities in R3. In the teaching of proofs of algebraic identities in secondary education, teachers only give formulas or make algebraic proofs in lessons rather than showing visual proofs and relating them to daily life. This situation poses a problem for students with visual learning tendencies. For this reason, it is thought that showing the proofs of identities by making visual models or visualizations in lessons can be a solution to this problem. In this study, document analysis technique was used within the scope of descriptive analysis and it was designed as a compilation study. Visual proofs of some identities were made by using squares and rectangles from geometric shapes. In this context, in the light of the study, some suggestions were made to guide interested researchers in this regard.

Investigating Students’ Proof Reasoning: Analyzing Students’ Oral Proof Explanations and their Written Proofs in High School Geometry

Constructing formal geometry proofs in is an important topic in the mathematics curriculum. But students’ difficulties with proof are well documented. This article focuses on proofs that use triangle congruence postulates in US high school geometry. Examining students’ proof reasoning in one-on-one task-based interviews, we analyzed students’ oral planning/explanations and two-column written proofs. We found that much of the reasoning students conveyed in their oral planning/explanations was not incorporated into their written proofs, resulting in formalization and fatal logical reasoning gaps. We found that the majority of the students utilized sound reasoning in their oral explanations, but struggled to capture that reasoning in their written two-column proofs. We link our findings to learning progression research, further elaborating the van Hiele levels. Our research contributes to understanding the learning and teaching of proof and to the ongoing controversy over the use of two-column proofs in the high school geometry curriculum.

Teaching mathematical proof at secondary school: an exploration of pre-service teachers’ situative beliefs

Despite the recognized importance of mathematical proof in secondary education, there is a limited but growing body of literature indicating how preservice secondary mathematics teachers (PSMTs) view proof and the teaching of proof. The purpose of this survey research was to explore PSMTs’ knowledge and beliefs about proof and proof teaching in the context of secondary mathematics teaching and learning. Six cases (PSMTs’ survey responses) were purposively selected from an entire data set comprising participants from the United States and Australia. Using a situative learning perspective as the theoretical framework, the authors developed a sensible belief system [Leatham, K. R. (2006). Viewing mathematics teachers’ beliefs as sensible systems. Journal of Mathematics Teacher Education, 9(1), 91–102. https://doi.org/10.1007/s10857-006-9006-8] for sampled PSMTs and inductively conceptualized how their knowledge of and orientations toward proof acted as a function of their teaching and learning contexts. Furthermore, an analysis of data revealed that PSMTs’ knowledge and beliefs about proof are multifaceted, dynamic and evolving as they engage with proof in their various contexts.

Grade 11 Students’ Proof Construction Ability in Relation to Classroom Resources

Despite proof being fundamental to the mathematics discipline and its role as a means to convey mathematical content, little is known about the effect of resources on influencing students’ proof construction ability. The purpose of this study was to compare two didactic environments, one regarded as resourced (favored) and the other under-resourced (disadvantaged), in relation to the construction of a mathematical proof. Motivated by the discrepancies in the literature on the influence of school resources on students’ performance and the unfortunately prevalent view that the sole function of proof in mathematics is verification by using confirmatory cases, this study sought to examine the differences (if any) between resourced and under-resourced classrooms in relation to students’ proof construction ability. To this end, data were drawn from a proof-related task performed by 78 Grade 11 students in the Ethekwini Metropolitan area, South Africa. A modified version of the Proof Construction Assessment tool showed that students in resourced schools significantly performed better than those in under-resourced schools in relation to proof construction. In addition, there was an observable and noticeable effect of this. Specifically, at an alpha = .05, the t-test for independent means revealed a significant difference between the two groups, t(76) = 2.749, p < .01, d = .624 SD. The practical significance of the results emphasizes the importance of taking into account the role of resources when investigating the learning and teaching of proofs. Further, preliminary results also suggested that most students struggled to even begin to prove the proposition. Recommendations and implications for the students’ careers and future research are raised and discussed.

PEMBUKTIAN DALAM SENGKETA TATA USAHA NEGARA

The purpose of this study is to find out and analyze the evidentiary process at the trial in a State Administration dispute involving evidence, the burden of proof and evaluation of the evidence carried out in the State Administrative Court. This type of research is normative legal research. Proof is a procedure for establishing evidence of a fact in a State Administration case to be used as a basis for consideration in passing a decision. In general explanation of Law Number. 5 of 1986 stated that the teaching of proof used in the State Administrative Court is the teaching of free proof, in the State Administrative Court, the judge plays a more active role in the trial process to obtain material truth. An administrative suit does not in essence delay the implementation of the disputed State Administration Decree. The judge has the authority to provide an assessment of the results of evidence in examining, deciding, and resolving State Administration Disputes based on Article 107 of Law Number 5 of 1986.

Key ideas in a proof: The case of the irrationality of √2

The notion of key idea is a crucial component of mathematical explanation and a promising tool in the teaching of proof. This study investigates how undergraduate mathematics students interpret the notion of key idea in proof and how they identify key ideas in a particular proof of the irrationality of √2. Drawn from a questionnaire and students’ work samples, the findings show that the students varied in their understanding of what “key idea” might mean. These variations stemmed in part from the different interpretations the students gave to the notion of key idea and in part from why a key idea is judged to be one. To help students succeed in proof construction and reconstruction, the findings suggest that mathematics instructors need to actively intervene to draw students’ attention to features of proofs that are candidates for key ideas.

Perceptions on proof and the teaching of proof: a comparison across preservice secondary teachers in Australia, USA and Korea

Despite the recognised importance of mathematical proof in secondary education, there is a limited but growing body of literature indicating how preservice secondary mathematics teachers (PSMTs) view proof and the teaching of proof. The purpose of this survey research was to investigate how PSMTs in Australia, the USA and Korea perceive of proof in the context of secondary mathematics teaching and learning. PSMTs were able to outline various mathematical and pedagogical aspects of proof, including purposes, characteristics, reasons for teaching and imposed constraints. In addition, PSMTs attended to differing, though overlapping, features of proof when asked to determine the extent to which proposed arguments constituted proofs or to decide which arguments they might present to students.

Conjecturing, generalizing and justifying: Building theory around teacher knowledge of proving

The purpose of this study was to detail teachers' proving activity and contribute to a framework of Mathematical Knowledge for Teaching Proof (MKT for Proof). While working to justify claims about sums of consecutive numbers, teachers searched for key ideas and productively used examples to make, test and refine conjectures. Analysis of teachers' mathematical activity revealed knowledge of the proving process that would be useful for and useable in the teaching of proof. This includes knowledge of the interconnections among empirical exploration, conjecturing, generalizing, and justifying as well as an understanding of the characteristics of examples and conjectures that could support the proving process. The central premise of this paper is that delineating aspects of teacher knowledge is a first step to supporting teachers' efforts to engage all students in fundamental mathematical practices of conjecturing, generalizing and justifying.

Development Process of a Praxeology for Supporting the Teaching of Proofs in a CAS Environment Based on Teachers’ Experience in a Professional Development Course

This paper presents the development process of a praxeology (theory-of-practice) for supporting the teaching of proofs in a CAS environment. The characteristics of the praxeology were elaborated within the frame of a professional development course for teaching analytic geometry with CAS. The theoretical framework draws on Chevallard’s anthropological approach to the didactics of mathematics and Duval’s analysis of transformations within and between registers of semiotic representations. The teachers (n = 43) were asked (a) to draw conjectures regarding unfamiliar behavior of tangents to hyperbola, before and after exploration using given slider bars; and (b) to prove their conjectures after being exposed to the algebraic expressions underlying the slider bars. The teachers were also asked twice, before and after (b), to rate the need to ask students for an algebraic proof in similar tasks. Three types of proofs are presented in an increasing order of the level of mathematical maturity exhibited in each proof. Based on results coming from the empirical study, we propose a praxeology for preparing teachers to teach proofs consisting of Task design, Techniques, and Didactical discourse.

KNOWLEDGE USE IN THE CONSTRUCTION OF GEOMETRY PROOF BY SRI LANKAN STUDENTS

Within the domain of geometry, proof and proof development continues to be a problematic area for students. Battista (2007) suggested that the investigation of knowledge components that students bring to understanding and constructing geometry proofs could provide important insights into the above issue. This issue also features prominently in the deliberations of the 2009 International Commission on Mathematics Instruction Study on the learning and teaching of proofs in mathematics, in general, and geometry, in particular. In the study reported here, we consider knowledge use by a cohort of 166 Sri Lankan students during the construction of geometry proofs. Three knowledge components were hypothesised to influence the students’ attempts at proof development: geometry content knowledge, general problem-solving skills and geometry reasoning skills. Regression analyses supported our conjecture that all 3 knowledge components played important functions in developing proofs. We suggest that whilst students have to acquire a robust body of geometric content knowledge, the activation and the utilisation of this knowledge during the construction of proof need to be guided by general problem-solving and reasoning skills.

Beliefs and beyond: hows and whys in the teaching of proof

In this paper, we report on a study aimed at describing the way secondary school teachers treat proof and at understanding which factors may influence such a treatment. This study is part of a wider project on proof carried out for many years. In our theoretical framework, we combine references from research on proof with those from research on teachers in relation to their beliefs. The study was carried out through interviews with secondary school teachers aimed at learning how they describe their work with proof in the classroom, and to elicit beliefs and other factors that shape this work. Through the interviews we were able to detect reasons behind teachers’ choices in planning their work in the classroom. In the present paper, we concentrate on four cases that, among other factors, offer elements suitable to unravel the problem of inconsistencies using the construct of leading beliefs, i.e., beliefs (whose nature may vary from teacher to teacher) that seem to drive the way each teacher treats proof.

The teaching of proof at the lower secondary level—a video study

Teaching mathematical proof is one of the most challenging topics for teachers. Several empirical studies revealed repeatedly different kinds of students’ problems in this area. The results give support that students’ abilities in proving are significantly influenced by their specific mathematics classrooms. In this paper we will present a method for evaluating proof instruction and some results of a video study that describe proving processes in mathematics classrooms at the lower secondary level from a mathematical perspective.

PROOF, EXPLANATION AND EXPLORATION: AN OVERVIEW

This paper explores the role of proof in mathematics education and provides justification for its importance in the curriculum. It also discusses three applications of dynamic geometry software - heuristics, exploration and visualization - as valuable tools in the teaching of proof and as potential challenges to the importance of proof. Finally, it introduces the four papers in this issue that present empirical research on the use of dynamic geometry software.

A Study on Teaching of Proof by Constructivistic Approach (5) : Through a teaching experiment

A Study on Teaching of Proof by Constructivistic Approach (5) : Through a teaching experiment

Some pedagogical aspects of proof

1. Formal proof: proof as a theoretical concept in formal logic (or metalogic), which may be thought of as the ideal which actual mathematical practice only approximates. 2. Acceptable proof: proof as a normative concept that defines what is acceptable to qualified mathematicians. 3. The teaching of proof: proof as an activity in mathematics education which serves to elucidate ideas worth conveying to the student.