A concept map for teaching-learning logic and methods of proof: Enhancing students' abilities in constructing mathematical proofs (original) (raw)
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Concept Maps to Assess Student Teachers’ Understanding of Mathematical Proof
This study reports the analysis of concept maps dealing with -Mathematical Proof,‖ as generated by a group of student teachers. The researcher examined the type of concept map generated, the number of key terms utilized in the construction of the map, and the multiplicity of relationships indicated among those key terms/concepts. The conceptual understanding represented within the concept map was then mapped onto Balacheff's (1988) taxonomy of proofs. The lack of sophistication in the concept maps produced may point towards limitations in student teachers' understanding of mathematical proof. Since teacher's conceptions of proof inevitably influences both the role and nature of the instruction of mathematical proof within a mathematics classroom, limited knowledge in this core area of mathematics may typically prompts feelings of uncertainty and a lack of confidence when it comes to teaching this concept.
Uniciencia, 2022
The objective of this study is to characterize the knowledge of mathematics teachers in initial training (MTITs) at the Universidad Nacional (Costa Rica) on the logic-syntactic and mathematical aspects involved in proving, when evaluating mathematical arguments. The research is positioned in the interpretive paradigm and has a qualitative approach. It consists of two empirical phases: in the first, a questionnaire regarding logic-syntactic aspects was applied to 25 subjects, during the months of September and October 2018 and; in the second phase, a second questionnaire covering mathematical aspects was applied to 19 subjects, during the months of May and June 2019. For the analysis of the information, knowledge indicators were proposed. Knowledge indicators are understood as phrases to determine evidence of knowledge in the responses of the subjects. It was appreciated that the vast majority of future mathematics teachers show knowledge to discriminate when a mathematical argument corresponds or not to a proof by virtue of the logic and syntactic aspects, and of mathematical elements associated with propositions with the structure of universal implication. In general, subjects displayed greater evidence of knowledge on the logic-syntactic aspects than on the mathematical aspects. Specifically, they evidenced that consideration of a particular case or the proof of the reciprocal proposition does not prove the result; likewise, subjects evidenced knowledge about the direct and indirect proof of the universal implication. In the case of the mathematical aspects considered as hypotheses, axioms, definitions and theorems, it was appreciated that subjects could have different levels of difficulties to understand a proof.
Exploration of Student Thinking Process in Proving Mathematical Statements
2020
A mathematical statement is not a theorem until it has been carefully derived from previously proven axioms, definitions and theorems. The proof of a theorem is a logical argument that is given deductively and is often interpreted as a justification for statements as well as a fundamental part of the mathematical thinking process. Studying the proof can help decide if and why our answers are logical, develop the habit of arguing, and make investigating an integral part of any problem solving. However, not a few students have difficulty learning it. So it is necessary to explore the student's thought process in proving a statement through questions, answer sheets, and interviews. The ability to prove is explored through 4 (four) proof schemes, namely Scheme of Complete Proof, Scheme of Incomplete Proof, Scheme of unrelated proof, and Scheme of Proof is immature. The results obtained indicate that the ability to prove is influenced by understanding and the ability to see that new ...
Uniciencia
The objective of this study is to characterize the knowledge of mathematics teachers in initial training (MTITs) at the Universidad Nacional (Costa Rica) on the logic-syntactic and mathematical aspects involved in proving, when evaluating mathematical arguments. The research is positioned in the interpretive paradigm and has a qualitative approach. It consists of two empirical phases: in the first, a questionnaire regarding logic-syntactic aspects was applied to 25 subjects, during the months of September and October 2018 and; in the second phase, a second questionnaire covering mathematical aspects was applied to 19 subjects, during the months of May and June 2019. For the analysis of the information, knowledge indicators were proposed. Knowledge indicators are understood as phrases to determine evidence of knowledge in the responses of the subjects. It was appreciated that the vast majority of future mathematics teachers show knowledge to discriminate when a mathematical argument...
Rigorous Mathematical Thinking: Conceptual Knowledge and Reasoning in the Case of Mathematical Proof
Kreano, Jurnal Matematika Kreatif-Inovatif
This study aims to analyze in-depth students' conceptual knowledge and reasoning when solving problems using mathematical proof as a rigorous mathematical thinking paradigm. The research uses a qualitative method with a case study approach that analyzes the mathematical proof ability of nine students who represent different cognitive functions from each level of rigorous mathematical thinking. The results showed that each level of rigorous mathematical thinking meant other indicators according to their ability to master conceptual knowledge and implement mathematical ideas through reasoning. This research has an impact on the treatment that the teacher must give in determining the learning model and evaluation instrument that can raise students' conceptual knowledge and reasoning.Penelitian ini bertujuan untuk menganalisis secara mendalam pengetahuan konseptual dan penalaran siswa ketika memecahkan masalah menggunakan pembuktian matematis sebagai paradigma berpikir matematis...
2014
The topic of the thesis is proof. At Year 9 Greek students encounter proof for the first time in Algebra and Geometry. Thus the principal research question of the thesis is: How do students’ perceive proof when they first encounter it? The analysis tool in order to obtain an image of students’ perception of proof, the Harel and Sowder’s taxonomy, is itself a research question in what concerns its applicability under Greek conditions. Its applicability, of which there is strong evidence, provides the space to shape an image of students’ proof fluency, proof appreciation, proof readiness etc. In order to collect data with regard to answering the research questions in collaboration principally with the class teacher I constructed the two tests on proof that are presented in this thesis. The first test was administered to the students of Year 9 at the beginning of the school year 2010-2011 before the teaching of proof. The second was administered after the teaching of proof of the same ...
The project summary for this grant proposal was as follows: This empirical research proposal is in the STEM learning strand. It will produce transformational, fundamental knowledge about: How can advanced university students learn, and be taught, to construct mathematical proofs autonomously? The emphasis will be on what students are capable of, in contrast to the literature’s focus on their difficulties. The proposal is also partly in the cognitive underpinnings strand because it will develop a new theoretical view of how the mind controls its own deductive reasoning. The above question (and more tractable subquestions) will be answered through a naturalistic study (Lincoln & Guba, 1985) using a generative/integrative analysis (Clement, 2000) of data arising from an unusual experimental “proofs” course. This course is for beginning graduate and some advanced undergraduate mathematics students who need help with proving. Its sole purpose is to provide that help, and classes consist very largely of students presenting their proofs and receiving critiques and advice. The course is consistent with a constructivist point of view, in that the teachers attempt to help students reflect on, and learn from, their own proof writing experiences. It is also somewhat Vygotskian (1978) in that the teachers represent to the students how the mathematics community writes proofs. That is, they are instruments in the cultural mediation of community norms and practices. Data will be collected mainly in four ways: (1) videoing and analyzing all class meetings; (2) videoing extensive one-on-one tutoring of students constructing proofs; (3) task-based interviews of former students; and (4) for the first time, using tablet computers to record and observe students independently constructing proofs. The theoretical framework used in a data analysis of a naturalistic study emerges from the data itself. From pilot project data a framework has already started emerging including: (1) kinds of proofs – needed to keep track of student progress; and (2) aspects of a proof – mathematicians fail to prove a theorem because of its “problem-oriented aspect”, but students (mainly) fail to prove a theorem because they cannot write its “formal-rhetorical aspect” (which is teachable). The pilot project data have also yielded a view of much of the proof construction process as a sequence of mental (or physical) actions in response to situations that arise in the partially completed proof construction. Similar situations tend to yield similar actions that can become permanently linked with those situations. The smallest of such linked, automated situation-action pairs are regarded as persistent mental structures, called behavioral schemas. An example of a behavioral schema, that students often take a long time to develop, links theorems of the form “For all numbers x, P(x)” [the situation] with writing into the proof “Let x be a number,” meaning x is fixed but arbitrary [the action]. Such schemas are stored in procedural memory and can be activated without requiring conscious recollection, thus reducing the demands on working memory. The idea of behavioral schemas in this project is not derived from other schema theories in psychology. However, it is reminiscent of social psychologists’ view of automaticity in everyday life, except that the linking of situations to actions is reified into mental objects. Intellectual merit: (1) The idea of a course and a teaching experiment devoted to teaching proving at this level is novel, and the project will produce much detailed information on what students can do, rather than focusing mainly on their difficulties. (2) One of the proposed data collection methods is new and will provide insights into students’ actions and thinking when alone. (3) The emerging framework will allow one to see that an aspect of a proof that blocks a mathematician’s progress is different in nature from the aspect that blocks most students. Thus teaching students to mimic mathematicians may not help. (4) The emerging view of behavioral schemas should prove very helpful in teaching, and elucidate how procedural knowledge can guide the use of conceptual knowledge. Such schemas may also illustrate the utility of converting System 2 cognition to System 1 cognition (Stanovich & West, 2000). Broader impact: (1)This project will provide information for designing courses for teaching students to prove theorems on a variety of topics and for students at a variety of levels – beginning graduate, mid-level undergraduate, and even high school. (2) It is common knowledge among teachers of beginning graduate students that significantly many need help constructing proofs, and some, who do not improve, drop out of mathematics. The experimental course in this project could be conveniently adapted to other universities to alleviate this situation. (3) When students first learn to construct proofs, they often find the experience empowering, and raise their career expectations and their interest in mathematics. This can be particularly helpful for minority populations whose expectations are not high.
Students' Semantic-Proof Production in Proving a Mathematical Proposition
Journal of Education and Learning (EduLearn), 2018
Proving a proposition is emphasized in undergraduate mathematics learning. There are three strategies in proving or proof-production, i.e.: proceduralproof, syntactic-proof, and semantic-proof production. Students" difficulties in proving can occur in constructing a proof. In this article, we focused on students" thinking when proving using semantic-proof production. This research is qualitative research that conducted on students majored in mathematics education in public university in Banten province, Indonesia. Data was obtained through asking students to solve proving-task using thinkaloud and then following by interview based task. Results show that characterization of students" thinking using semantic-proof production can be classified into three categories, i.e.: (1) false-semantic, (2) proof-semantic for clarification of proposition, (3) proof-semantic for remembering concept. Both category (1) and (2) occurred before students proven formally in Representation System Proof (RSP). Nevertheless, category (3) occurred when students have proven the task in RSP then step out from RSP while proving. Based on the results, some suitable learning activities should be designed to support the construction of these mental categories.