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We report findings from a longitudinal study of students' beliefs about empirical arguments and mathematical proof. We consider the influence of an 'Introduction to Proof (ITP)' course and the consequences of the observed changes in behaviour. Consistent with recent literature, our findings suggest that a majority of the thirty-eight undergraduate students in this study do not find empirical arguments convincing, even at the beginning of their ITP course. We use Sankey diagrams to show that, while many were unconvinced by these arguments at the start and end of the course, others began the course endorsing empirical arguments as similar to their own, shifting toward deductive-symbolic arguments by the end. Finally, we consider the value of Sankey diagrams for understanding changes in population behaviours, and the consequences of our work for future research on the role of empirical arguments in the classroom. [ABSTRACT FROM AUTHOR]

Copyright of International Journal of Educational Psychological Studies (EPS) is the property of Refaad for Studies, Research & Development and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This article focused on "Exploring students' misconceptions in learning mathematical proof techniques (MPT) at Debark University in Ethiopia." This research aimed to identify students' misconceptions in learning MPT and rank MPT based on the degree of students' misconceptions in learning MPT. Data were collected through assignments, follow-up, and structured interviews. Purposive sampling was used to select samples for follow-up interviews, whereas simple random sampling was used to select samples for structured interviews and assignments. The study collected data on the basis of mixed, case study, and pragmatic research approaches, designs, and paradigms from students. The results showed that the identified misconceptions of students in learning MPT were starting the proof with an inappropriate statement, using an ineffective MPT for their proof, providing incorrect symbolic representation, providing unacceptable reasons for each proof's step, reaching the conclusion without showing necessary steps clearly and neatly, showing non-sequential steps, incorrectly using technical aspects of mathematics, using the premise and conclusion parts interchangeably, incorrect perception regarding the technical concepts of proof, and misusing the pattern in the proof of a certain statement for the proof of another statement. [ABSTRACT FROM AUTHOR]

Research on the epistemic, logical, and discursive complexity of proof learning has generated a wealth of literature over the past two decades. The results contribute to a more precise understanding of the difficulties encountered by students and those encountered by teachers. They support the design of situations, especially validation situations in the sense of the Theory of Didactical Situations (TDS--Brousseau, 1998), in which proof is a problem-solving tool. However, there remains the difficulty of grasping proof as an object, recognizing its mathematical specificities and institutionalizing it as such. This is the topic of this text. This text complements those given at the Séminaire national de didactique des mathématiques in 2017 and at CORFEM in 2019. The common theme of these three presentations was the learning and the teaching of proof before the introduction of mathematical proof as the canonical form of proof in mathematics. After an introduction recalling the institutional and scientific context, the first part (sections 2 to 4) is devoted to a review of the state of research, taking into account the reports of outstanding work from different approaches. In the second part (section 5), proposals are made to provide a basis for future research. The conclusion considers the issues raised by the need for situation-specific engineering to encourage and support the genesis and recognition of proof norms in the mathematics classroom, prior to the explicit teaching of the introduction of mathematical proof. [ABSTRACT FROM AUTHOR]

This research aims to determine the influence of learning independence, gender, and mathematical beliefs on mathematical proof. This research uses a survey method with correlational analysis. The population in this study were students from one of the Ngawen 1 High Schools and sampling using the Purposive Sampling technique for this research was students in classes X.6, X.8, and XI.9. The instruments used are mathematics proof tests and non-test instruments in the form of independence learning questionnaires and mathematics beliefs questionnaires using the Linkert Scale. Data analysis was carried out using the SPSS 23.0 program, consisting of descriptive analysis, data analysis prerequisite tests, and hypothesis testing. This research concludes that together learning independence, gender, and mathematical beliefs have an effect on mathematical proofs, learning independence has a positive effect on mathematical proofs, gender has no effect on mathematical proofs, mathematical beliefs have no effect on mathematical proofs, students in class X.6, X.8 and XI.9 SMA N 1 Ngawen.

This article focused on “Exploring students’ misconceptions in learning mathematical proof techniques (MPT) at Debark University in Ethiopia." This research aimed to identify students’ misconceptions in learning MPT and rank MPT based on the degree of students’ misconceptions in learning MPT. Data were collected through assignments, follow-up, and structured interviews. Purposive sampling was used to select samples for follow-up interviews, whereas simple random sampling was used to select samples for structured interviews and assignments. The study collected data on the basis of mixed, case study, and pragmatic research approaches, designs, and paradigms from students. The results showed that the identified misconceptions of students in learning MPT were starting the proof with an inappropriate statement, using an ineffective MPT for their proof, providing incorrect symbolic representation, providing unacceptable reasons for each proof’s step, reaching the conclusion without showing necessary steps clearly and neatly, showing non-sequential steps, incorrectly using technical aspects of mathematics, using the premise and conclusion parts interchangeably, incorrect perception regarding the technical concepts of proof, and misusing the pattern in the proof of a certain statement for the proof of another statement.

This research article is about "Introducing a Supportive Framework to Address Students' Misconceptions and Difficulties in Learning Mathematical proof techniques (MPT): A Case of Debark University". This study aims to develop, introduce, and implement a supportive framework to overcome students' misconceptions and difficulties in MPT. The framework, named IR2CP2CE, was developed, introduced, and implemented at Debark University in Ethiopia using various data-gathering instruments such as questionnaires, interviews, classroom observations, and document analysis from students and instructors. The study collected data over four months, including the implementation of a supportive framework using mixed, quasi-experimental, and pragmatism research approaches, designs, and paradigms respectively. The internal reliability of the data-gathering instruments was interpreted using Cronbach's coefficient, Spearman-Brown, Spearman correlations, Kuder-Richardson 20 and 21, and difficulty and discrimination indices. The results showed that the implementation of the supportive framework led to significant improvements in students' academic performance in MPT, regardless of factors such as gender, academic year category, and preliminary knowledge and proving skills. This study recommends additional imperatives for practice and future research. [ABSTRACT FROM AUTHOR]

Copyright of Cumhuriyet International Journal of Education is the property of Cumhuriyet University, Faculty of Education and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this study, we investigated undergraduate mathematics students' (N = 267) attitudes towards proving. The students were taking an introduction-to-proof type course that was situated at the beginning of the mathematics curriculum and lasted for one term. Four attitude variables were measured at the beginning and at the end of the course with a new self-report instrument: self-efficacy, anxiety, appreciation and motivation. The instrument was based on two existing instruments on mathematics attitudes and proof-related self-efficacy. We studied how these four attitude variables were related to the students' prior skills and their gender at the beginning of the course, how the attitude variables changed during the course, and how they affected the students' performance in the final course project. Our results indicate that students' prior performance is linked to their proof-related self-efficacy, anxiety and motivation at the beginning of the course. Female students exhibited lower efficacy and motivation levels than male students. During the focus course, students' self-efficacy increased and anxiety decreased. The gender gaps in self-efficacy and motivation persisted throughout the course. In addition, high motivation in the beginning of the course predicted good performance in the final project. Based on our results, we conclude that an introductory course on proving can enhance students' attitudes, and we suggest that these attitudes are taken into account in teaching as they can affect students' performance. Finally, we urge researchers and professionals to earnestly consider ways to mitigate gender differences in mathematics. [ABSTRACT FROM AUTHOR]

This research article is about "Introducing a Supportive Framework to Address Students' Misconceptions and Difficulties in Learning Mathematical proof techniques (MPT): A Case of Debark University”. This study aims to develop, introduce, and implement a supportive framework to overcome students’ misconceptions and difficulties in MPT. The framework, named IR2CP2CE, was developed, introduced, and implemented at Debark University in Ethiopia using various data-gathering instruments such as questionnaires, interviews, classroom observations, and document analysis from students and instructors. The study collected data over four months, including the implementation of a supportive framework using mixed, quasi-experimental, and pragmatism research approaches, designs, and paradigms respectively. The internal reliability of the data-gathering instruments was interpreted using Cronbach’s coefficient, Spearman-Brown, Spearman correlations, Kuder-Richardson 20 and 21, and difficulty and discrimination indices. The results showed that the implementation of the supportive framework led to significant improvements in students’ academic performance in MPT, regardless of factors such as gender, academic year category, and preliminary knowledge and proving skills. This study recommends additional imperatives for practice and future research.

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.

Proof facilitates conceptual and meaningful learning in mathematics education rather than rote memorization. In this study, incorrect theorems and proofs are used to assess secondary school pre-service mathematics teachers’ proof assessing skills. Using the case study method, the study is conducted on pre-service mathematics teachers studying at the Department of Mathematics Education. There were eight pre-service mathematics teachers selected from each grade, resulting in 32 participants in total. A semi-structured proof form containing 13 questions was used to collect data, which was analyzed using content analysis. As the analysis reveals, pre-service mathematics teachers are highly likely to make incorrect decisions regarding theorems and proofs, and the margin of error is unaffected by grade level. Moreover, pre-service mathematics teachers tend to use proving terms incorrectly and, at times, are unable to differentiate between terms that are commonly used in proving. The pre-service mathematics teachers are believed to have learned proofs by rote rather than understanding how proofs work. With the help of interviews and tests created for different proof methods, it has been suggested that pre-service mathematics teachers should be tested on their proof evaluation skills in more detail.

Mathematical culture is an essential part of general culture, and mathematical proof is the essence of mathematical culture, which plays an essential role in education for sustainable development. The presented research focuses on the following issue: Can the use of visual tools contribute to a better understanding of mathematical proofs presented within the framework of courses in graph theory? The research focuses on Czech university students of computer science. The presented results were achieved in a pedagogical experiment based on a pre-test, treatment and post-test design carried out during three academic years before the COVID pandemic period. In the pre-test and post-test phases, students were tested to determine their level of mathematical logic knowledge. Visual applications were used in teaching in the treatment phase. The research results clearly showed that the use of visual tools supporting formal explanation of mathematical proofs results in better understanding of the abstraction of the presented process and thus contributes significantly to sustainability in mathematical education. Thanks to the use of the mentioned visual tools during the COVID pandemic period, when schools had to switch to on-line education, the students' results were comparable to those from face-to-face classes. [ABSTRACT FROM AUTHOR]

The "proving process" was considered as the stages that should be exist in a proof and in this study, it was aimed to investigate the mathematical proving processes and opinions about proof of gifted students. Case study, one of the qualitative research methods, was used in the study. The research was carried out with the students determined by criterion sampling method. The attendance of students to Science and Art Education Centre, the data obtained from the Proof Interview Form and the opinions of mathematics teachers constituted the criteria of the criterion sampling method. As a data collection tool, the Proof Interview Form and the Proof Clinical Interview Form were used. "Proof Clinical Interview Form" prepared by the researcher was applied to examine the students' proving processes. The data obtained from the clinical interviews were analysed with the descriptive analysis. Students were generally able to examine the problem situation and formulate the conjecture, but they did not determine the appropriate strategies, perform the necessary actions and summarize clearly while proving. [ABSTRACT FROM AUTHOR]

This paper puts forward a new account of rigorous mathematical proof and its epistemology. One novel feature is a focus on how the skill of reading and writing valid proofs is learnt, as a way of understanding what validity itself amounts to. The account is used to address two current questions in the literature: that of how mathematicians are so good at resolving disputes about validity, and that of whether rigorous proofs are necessarily formalizable. [ABSTRACT FROM AUTHOR]

Proof facilitates conceptual and meaningful learning in mathematics education rather than rote memorization. In this study, incorrect theorems and proofs are used to assess secondary school pre-service mathematics teachers' proof assessing skills. Using the case study method, the study is conducted on preservice mathematics teachers studying at the Department of Mathematics Education. There were eight preservice mathematics teachers selected from each grade, resulting in 32 participants in total. A semistructured proof form containing 13 questions was used to collect data, which was analyzed using content analysis. As the analysis reveals, pre-service mathematics teachers are highly likely to make incorrect decisions regarding theorems and proofs, and the margin of error is unaffected by grade level. Moreover, pre-service mathematics teachers tend to use proving terms incorrectly and, at times, are unable to differentiate between terms that are commonly used in proving. The pre-service mathematics teachers are believed to have learned proofs by rote rather than understanding how proofs work. With the help of interviews and tests created for different proof methods, it has been suggested that pre-service mathematics teachers should be tested on their proof evaluation skills in more detail. [ABSTRACT FROM AUTHOR]

This phenomenology research aims to examine prospective elementary mathematics teachers' proving and proof evaluation and their thoughts on convincing according to proof type and argument type. The participants were eight prospective teachers. The data collection tools were semi-structured group interviews, interviews video recordings and the participants' written proof documents. The participants were expected to prove different mathematical statements presented to them with different proof types, to convince each other, and to identify the convincing arguments in the interviews. The results revealed that prospective mathematics teachers had absolute conviction about empirical arguments, while their level of convincing about deductive arguments increased as a result of discussions on convincing regardless of the proof type. In addition, the unconvincing for induction and visual proof types' arguments have emerged and this category has changed to convincing over time. Accordingly, suggestions about increasing the convincing of deductive and visual arguments have been presented. [ABSTRACT FROM AUTHOR]

Experiences at the educational personnel education institution, Department of Mathematics Education Institute of Teacher Training and Education Pontianak (IKIP) shows that students often had difficulty in proving propositions (theorems in mathe-matics). The alternative offered through this study was to develop students' abilities in proof through mathematical communication. The study method used in this study was qualitive research, case study. The subjects used in this study consisted of two un-dergraduate students who had been taught real analysis introductions and. The data collection tool used was tests and interview. Data analysis techniques used was de-scriptions: data reduction, data display and conclusion. Based on the results of re-search, study of theory and discussion, it could be concluded in this study that the student obstacles in answering mathematical proof questions are the students have difficulty in writing mathematical symbolic and inability in mathematical proof. But after being given didactic anticipation by mathematical communication, the ability of students to answer mathematical proof questions has increased. Thus, mathematical communication can be used as an alternative to overcome obstacles or difficulties for students in solving mathematical proof problems.

The work done so far on the understanding of mathematical (proof) texts focuses mostly on logical and heuristical aspects; a proof text is considered to be understood when the reader is able to justify inferential steps occurring in it, to defend it against objections, to give an account of the “main ideas”, to transfer the proof idea to other contexts etc. (see, e.g., Avigad in The philosophy of mathematical practice, Oxford University Press, Oxford, 2008). In contrast, there is a rich philosophical tradition dealing with the concept of understanding and interpreting texts, namely philosophical hermeneutics, represented, e.g., by Schleiermacher, Dilthey, Heidegger or Gadamer. In this tradition, “understanding” generally refers to the integration in a comprehensive (historical, existential, life-worldly,...) context. In this article, we take some first steps towards exploring the question how the ideas from philosophical hermeneutics presented in Gadamer’s “Truth and Method” apply to mathematical texts and what (if anything) can be learned from these for the didactics and presentation of mathematics. [ABSTRACT FROM AUTHOR]

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.

The article analyzes the role of the concept of understanding in mathematical proof. Understanding seems to be a natural and necessary characteristic of proof, interpreted as an argument in favor of the established result. It is shown that in general two traditions in the treatment of mathematical proofs can be distinguished, going back to Descartes and Leibniz. It arguments for conceptual treatment of category of understanding which is not connected with individual mental acts are resulted. The prospect of achieving conceptual understanding in the computational interpretation of mathematical proof is problematized.

Mathematical proof is the primary form of justification for mathematical knowledge, but in order to count as a proper justification for a piece of mathematical knowledge, a mathematical proof must be rigorous. What does it mean then for a mathematical proof to be rigorous? According to what I shall call the standard view, a mathematical proof is rigorous if and only if it can be routinely translated into a formal proof. The standard view is almost an orthodoxy among contemporary mathematicians, and is endorsed by many logicians and philosophers, but it has also been heavily criticized in the philosophy of mathematics literature. Progress on the debate between the proponents and opponents of the standard view is, however, currently blocked by a major obstacle, namely, the absence of a precise formulation of it. To remedy this deficiency, I undertake in this paper to provide a precise formulation and a thorough evaluation of the standard view of mathematical rigor. The upshot of this study is that the standard view is more robust to criticisms than it transpires from the various arguments advanced against it, but that it also requires a certain conception of how mathematical proofs are judged to be rigorous in mathematical practice, a conception that can be challenged on empirical grounds by exhibiting rigor judgments of mathematical proofs in mathematical practice conflicting with it. [ABSTRACT FROM AUTHOR]

Copyright of Necatibey Faculty of Education Electronic Journal of Science & Mathematics Education is the property of Balikesir University, Necatibey Faculty of Education and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of the College Of Basic Education is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Although traditionally neglected, mathematical diagrams have recently begun to attract attention from philosophers of mathematics. By now, the literature includes several case studies investigating the role of diagrams both in discovery and justification. Certain preliminary questions have, however, been mostly bypassed. What are diagrams exactly? Are there different types of diagrams? In the scholarly literature, the term “mathematical diagram” is used in diverse ways. I propose a working definition that carves out the phenomena that are of most importance for a taxonomy of diagrams in the context of a practice-based philosophy of mathematics, privileging examples from contemporary mathematics. In doing so, I move away from vague, ordinary notions. I define mathematical diagrams as forming notational systems and as being geometric/topological representations or two-dimensional representations (or both). I also examine the relationship between mathematical diagrams and spatiotemporal intuition. By proposing an explication of diagrams, I explain (away) certain controversies in the existing literature. Moreover, I shed light on why mathematical diagrams are so effective in certain instances, and, at other times, dangerously misleading. [ABSTRACT FROM AUTHOR]

This study aims to comprehensively analyze the improvement of students ' mathematical development ability through the concept of three zone learning, with the type of research is quantitative and quasi experiment design. Examples of research as many as two classes amounted to 82 pre service teacher. Data analysis by calculating gain normalization. Results: based on the Post Hoc test, the level of the category that has a positive score of 0.012 with the upper category and the middle category means that the average score of students from the upper category is better than the middle category, then also for the level that has a positive score of 0.028 with the upper category and the lower category which means that the average score of students from the upper category is better than the lower category. As for the positive value of 0.016 with the middle category and the lower category means the average of the middle category is better. From the results showed that for levels with upper, middle and lower categories, the effect is greater in improving students.

The Euclidean ideal of mathematics as well as all the foundational schools in the philosophy of mathematics have been contested by the new approach, called the “maverick” trend in the philosophy of mathematics. Several points made by its main representatives are mentioned – from the revisability of actual proofs to the stress on real mathematical practice as opposed to its idealized reconstruction. Main features of real proofs are then mentioned; for example, whether they are convincing, understandable, and/or explanatory. Therefore, the new approach questions Hilbert’s Thesis, according to which a correct mathematical proof is in principle reducible to a formal proof, based on explicit axioms and logic.

In practice, mathematical proofs are most often the result of careful planning by the agents who produced them. As a consequence, each mathematical proof inherits a plan in virtue of the way it is produced, a plan which underlies its "architecture" or "unity." This paper provides an account of plans and planning in the context of mathematical proofs. The approach adopted here consists in looking for these notions not in mathematical proofs themselves, but in the agents who produced them. The starting point is to recognize that to each mathematical proof corresponds a proof activity which consists of a sequence of deductive inferences—i.e., a sequence of epistemic actions—and that any written mathematical proof is only a report of its corresponding proof activity. The main idea to be developed is that the plan of a mathematical proof is to be conceived and analyzed as the plan of the agent(s) who carried out the corresponding proof activity. The core of the paper is thus devoted to the development of an account of plans and planning in the context of proof activities. The account is based on the theory of planning agency developed by Michael Bratman in the philosophy of action. It is fleshed out by providing an analysis of the notions of intention—the elementary components of plans—and practical reasoning—the process by which plans are constructed—in the context of proof activities. These two notions are then used to offer a precise characterization of the desired notion of plan for proof activities. A fruitful connection can then be established between the resulting framework and the recent theme of modularity in mathematics introduced by Jeremy Avigad. This connection is exploited to yield the concept of modular presentations of mathematical proofs which has direct implications for how to write and present mathematical proofs so as to deliver various epistemic benefits. The account is finally compared to the technique of proof planning developed by Alan Bundy and colleagues in the field of automated theorem proving. The paper concludes with some remarks on how the framework can be used to provide an analysis of understanding and explanation in the context of mathematical proofs. [ABSTRACT FROM AUTHOR]

One of the central aims of the philosophical analysis of mathematical explanation is to determine how one can distinguish explanatory proofs from non-explanatory proofs. In this paper, I take a closer look at the current status of the debate, and what the challenges for the philosophical analysis of explanatory proofs are. In order to provide an answer to these challenges, I suggest we start from analysing the concept understanding. More precisely, I will defend four claims: (1) understanding is a condition for explanation, (2) unificatory understanding is a type of explanatory understanding, (3) unificatory understanding is valuable in mathematics, and (4) mathematical proofs can contribute to unificatory understanding. As a result, in a context where the epistemic aim is to unify mathematical results, I argue it is fruitful to make a distinction between proofs based on their explanatory value. [ABSTRACT FROM AUTHOR]

This article documents differences between novice and experienced undergraduate students' processes of reading mathematical proofs as revealed by moment-bymoment, think-aloud protocols. We found three key reading behaviors that describe how novices' reading differed from that of their experienced peers: alternative task models, accrual of premises, and warranting. Alternative task models refer to the types of goals that students set up for their reading of the text, which may differ from identifying and justifying inferences. Accrual of premises refers to the way novice readers did not distinguish propositions in the theorem statement as assumptions or conclusions and thus did not use them differently for interpreting the proof. Finally, we observed variation in the type and quality of warrants, which we categorized as illustrate with examples, construct a miniproof, or state the warrant in general form. [ABSTRACT FROM AUTHOR]

This essay discusses the instructional value of mathematical proofs using different interpretations of the analysis cum synthesis method in Apollonius' Conics as a case study. My argument is informed by Descartes' complaint about ancient geometers and William Thurston's discussion on how mathematical understanding is communicated. Three historical frameworks of the analysis/synthesis distinction are used to understand the instructive function of the analysis cum synthesis method: the directional interpretation, the structuralist interpretation, and the phenomenological interpretation. I apply these interpretations to the analysis cum synthesis method in order reveal how the same underlying mathematical activity occurs at different levels of scale: at the level of an individual proof, at the level of a collection of proofs, and at the level of a single line within a proof. On the basis of this investigation, I argue that the instructive value of mathematical proof lies in engendering in the reader the same mathematical activity experienced by the author themselves. [ABSTRACT FROM AUTHOR]

The emerging of formal mathematical proof is an essential component in advanced undergraduate mathematics courses. Several colleges have transformed mathematics courses by facilitating undergraduate students to understand formal mathematical language and axiomatic structure. Nevertheless, college students face difficulties when they transition to proof construction in mathematics courses. Therefore, this descriptive-explorative study explores prospective teachers' mathematical proof in the second semester of their studies. There were 240 pre-service mathematics teachers at a state university in Surabaya, Indonesia, determined using the conventional method. Their responses were analyzed using a combination of Miyazaki and Moore methods. This method classified reasoning types (i.e., deductive and inductive) and types of difficulties experienced during the proving. The results conveyed that 62.5% of prospective teachers tended to prefer deductive reasoning, while the rest used inductive reasoning. Only 15.83% of the responses were identified as correct answers, while the other answers included errors on a proof construction. Another result portrayed that most prospective teachers (27.5%) experienced difficulties in using definitions for constructing proofs. This study suggested that the analytical framework of the Miyazaki-Moore method can be employed as a tool to help teachers identify students' proof reasoning types and difficulties in constructing the mathematical proof.

Tujuan penelitian ini adalah untuk mendiskripsikan penerapan strategi REACT (Relating, Experiencing, Applying, Cooperating, Transferring) untuk mengembangkan kemampuan mahasiswa dalam mengkonstruksi bukti. Jenis penelitian yang digunakan adalah deskriptif dengan pendekatan kualitatif. Subjek penelitian adalah mahasiswa program studi pendidikan matematika yang sedang menempuh mata kuliah Analisis Real. Hasil peneltian adalah strategi yang diberikan berupa memberi kesempatan kepada mahasiswa untuk menghubungkan teorema satu dengan teorema lain (Relating), menuliskan kembali bukti sesuai dengan bahasa sendiri dan penggunaan bukti non formal sebagai bantuan (Experiencing), memanipulasi bentuk pernyataan berdasarkan prinsip logika dasar (Applying), saling memeriksa kebenaran proses pembuktian (Cooperating), dan membuktikan pernyataan yang dikembangkan (Transferring). REACT berdampak pada berkembangnya kemampuan mahasiswa dalam mengonstruksi bukti yaitu mahasiswa mampu menggunakan teroema sebelumnya untuk digunakan dalam proses pembuktian, melakukan manipulasi bentuk logika matematika sebagai strategi awal memulai proses pembuktian, menggunakan bantuan bukti non formal untuk menentukan langkah pembuktian selanjutnya, memeriksa kevalidan proses pembuktian dan mampu melakukan pembuktian kontradiksi. The concern of this study is to describe the application of REACT (Relating, Experiencing, Applying, Cooperating, Transferring) strategies to develop students' abilities in constructing evidence. This type of research used is descriptive with a qualitative approach. The research subjects were students of mathematics education study programs who were taking Real Analysis courses. The result of the research is a strategy given in the form of allowing students to connect one theorem with another (Relating), rewriting evidence according to their language and using non-formal evidence as an aid (Experiencing), manipulating statement forms based on basic logic principles (Applying), checking the correctness of the verification process (cooperating), and proving the statements developed (Transferring). REACT has an impact on the development of students' abilities in constructing evidence, namely students can use the previous theorem to be used in the proving process, manipulate mathematical logic forms as an initial strategy to start the proof process, use non-formal evidence to determine the next step of evidence, check the validity of the proof process and be able to do a proof of contradiction.

Abstract: Various mathematical abilities require a high level of struggle to be achieved, one of which is the ability to solve mathematical proof problems. Several factors are also associated with this ability, including adversity quotient and resilience. This research aimed to look at the effect of adversity quotient and resilience on the mathematical proof problem-solving ability. This research used a quantitative approach with a correlational method. The sample in this study were 31 students randomly selected from all students of the Mathematics Education Study Program at Universitas Sulawesi Barat who programmed Abstract Algebra course in the academic year of 2019/2020, amounting to 71 students. The analysis technique used is multiple linear regressions. The results of hypothesis testing indicate that adversity quotient and resilience affect the mathematical proof problem-solving ability positively both individually and simultaneously. Abstrak: Berbagai kemampuan matematis memerlukan daya juang tinggi untuk berhasil dicapai, salah satunya kemampuan pemecahan masalah pembuktian matematis. Beberapa faktor pun dikaitkan dengan kemampuan tersebut, antara lain adversity quotient dan resiliensi. Penelitian ini bertujuan untuk melihat pengaruh adversity quotient dan resiliensi terhadap kemampuan pemecahan masalah pembuktian matematis. Penelitian ini menggunakan pendekatan kuantitatif dengan metode korelasional. Sampel pada penelitian ini sebanyak 31 mahasiswa yang dipilih secara acak dari seluruh mahasiswa Program Studi Pendidikan Matematika Universitas Sulawesi Barat yang memprogramkan mata kuliah Struktur Aljabar tahun akademik 2019/2020 yang berjumlah 71 mahasiswa. Teknik analisis yang digunakan adalah regresi linear berganda. Hasil pengujian hipotesis menunjukkan bahwa adversity quotient dan resiliensi berpengaruh positif terhadap kemampuan pemecahan masalah pembuktian matematis baik secara sendiri-sendiri maupun secara simultan.

Group Isomorphism is one of the sub topic in the Algebra Structure. This sub topic required prerequisite material about the bijective function. If this prerequisite material is not mastered, it will be difficult to study Isomorphism Group material. There are still many students who found difficulties to solve probative questions about Group Isomorphism. The major problem are they were forgetting and not understanding the prerequisite material. Therefore, it should be a further research on the ability of students to prove the questions of proof and errors in preparing evidence about Group Isomorphism.