Search

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]

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]

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]

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 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.

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.)

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.

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]

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.

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.

High-penetration wind power access to grid requires wind turbine generator (WTG) to provide frequency regulation service. Consequently, the frequency dynamics of wind power plants (WPPs) integrated system are changing; thus, it is necessary to investigate the dynamic frequency response of WPPs. In this paper, an analytical approach for an aggregated frequency response model for WPPs with primary frequency regulation service is presented and validated. First, different operation region of WTGs is fully taken into account, and a low-order wind power frequency response (WPFR) model with combined frequency control is deduced based on small signal analysis theory, which has been given in the form of symbolic transfer function. Afterwards, a system identification (SI) analytical method is proposed to aggregate a multi-machine WPFR model with heterogeneous parameters into a single equivalent model, which is called an aggregated WPFR (AWPFR) model, and this aggregation method is validated by the mathematical proof. Finally, the accuracy and effectiveness of the AWPFR model is verified through comparisons of simulation results obtained from the multi-machine WPFR model, detailed wind power plant (WPP) model and individual WPFR models, and the impact of the WTG parameters on the system frequency characteristics is analyzed and discussed. Such an aggregation model can provide a convenient way to describe the dynamic frequency response of WPPs by avoiding the need for modeling complex transient processes while maintaining a satisfactory level of accuracy.

Several studies related to mathematical proof have been done by many researchers on high-level materials, but not yet examined on the material of transformation geometry. The aim of this research is identification learning obstacles pre-service teachers on transformation geometry. This study is qualitative research; data were collected from interview sheets and test. There were four problems given to 9 pre-service mathematics teachers. The results of this research were as follows: learning obstacles related to the difficulty in applying the concept; related to visualize the geometry object; related to obstacles in determining principle; related to understanding the problem and related obstacles in mathematical proofs such as not understanding and unable to express a definition, not knowing to use the definition to construct the proof, not understanding the use of language and mathematical notation, not knowing how to start the proof.

An increasing number of learning goals refer to the acquisition of cognitive skills that can be described as ‘resource-based,’ as they require the availability, coordination, and integration of multiple underlying resources such as skills and knowledge facets. However, research on the support of cognitive skills rarely takes this resource-based nature explicitly into account. This is mirrored in prior research on mathematical argumentation and proof skills: Although repeatedly highlighted as resource-based, for example relying on mathematical topic knowledge, methodological knowledge, mathematical strategic knowledge, and problem-solving skills, little evidence exists on how to support mathematical argumentation and proof skills based on its resources. To address this gap, a quasi-experimental intervention study with undergraduate mathematics students examined the effectiveness of different approaches to support both mathematical argumentation and proof skills and four of its resources. Based on the part-/whole-task debate from instructional design, two approaches were implemented during students’ work on proof construction tasks: (i) a sequential approach focusing and supporting each resource of mathematical argumentation and proof skills sequentially after each other and (ii) a concurrent approach focusing and supporting multiple resources concurrently. Empirical analyses show pronounced effects of both approaches regarding the resources underlying mathematical argumentation and proof skills. However, the effects of both approaches are mostly comparable, and only mathematical strategic knowledge benefits significantly more from the concurrent approach. Regarding mathematical argumentation and proof skills, short-term effects of both approaches are at best mixed and show differing effects based on prior attainment, possibly indicating an expertise reversal effect of the relatively short intervention. Data suggests that students with low prior attainment benefited most from the intervention, specifically from the concurrent approach. A supplementary qualitative analysis showcases how supporting multiple resources concurrently alongside mathematical argumentation and proof skills can lead to a synergistic integration of these during proof construction and can be beneficial yet demanding for students. Although results require further empirical underpinning, both approaches appear promising to support the resources underlying mathematical argumentation and proof skills and likely also show positive long-term effects on mathematical argumentation and proof skills, especially for initially weaker students.

An increasing number of learning goals refer to the acquisition of cognitive skills that can be described as 'resource-based,' as they require the availability, coordination, and integration of multiple underlying resources such as skills and knowledge facets. However, research on the support of cognitive skills rarely takes this resource-based nature explicitly into account. This is mirrored in prior research on mathematical argumentation and proof skills: Although repeatedly highlighted as resource-based, for example relying on mathematical topic knowledge, methodological knowledge, mathematical strategic knowledge, and problem-solving skills, little evidence exists on how to support mathematical argumentation and proof skills based on its resources. To address this gap, a quasi-experimental intervention study with undergraduate mathematics students examined the effectiveness of different approaches to support both mathematical argumentation and proof skills and four of its resources. Based on the part-/whole-task debate from instructional design, two approaches were implemented during students' work on proof construction tasks: (i) a sequential approach focusing and supporting each resource of mathematical argumentation and proof skills sequentially after each other and (ii) a concurrent approach focusing and supporting multiple resources concurrently. Empirical analyses show pronounced effects of both approaches regarding the resources underlying mathematical argumentation and proof skills. However, the effects of both approaches are mostly comparable, and only mathematical strategic knowledge benefits significantly more from the concurrent approach. Regarding mathematical argumentation and proof skills, short-term effects of both approaches are at best mixed and show differing effects based on prior attainment, possibly indicating an expertise reversal effect of the relatively short intervention. Data suggests that students with low prior attainment benefited most from the intervention, specifically from the concurrent approach. A supplementary qualitative analysis showcases how supporting multiple resources concurrently alongside mathematical argumentation and proof skills can lead to a synergistic integration of these during proof construction and can be beneficial yet demanding for students. Although results require further empirical underpinning, both approaches appear promising to support the resources underlying mathematical argumentation and proof skills and likely also show positive long-term effects on mathematical argumentation and proof skills, especially for initially weaker students. [ABSTRACT FROM AUTHOR]

The use of figurate numbers (e.g., in the context of elementary number theory) can be considered a heuristic in the field of problem solving or proving. In this paper, we want to discuss this heuristic from the perspectives of the semiotic theory of Peirce (“diagrammatic reasoning” and “collateral knowledge”) and cognitive psychology (“schema theory” and “Gestalt psychology”). We will make use of several results taken from our research to illustrate first-year students’ problems when dealing with figurate numbers in the context of proving. The considerations taken from both theoretical perspectives will help to partly explain such phenomena. It will be shown that the use of figurate numbers must not be considered to be any kind of help for learners or some way of ‘easy’ mathematics. Working in this representational system has to be learned and practiced as another kind of knowledge is necessary for working with figurate numbers. The named findings also touch upon the concept of ‘proofs that explain.’ Finally, we will highlight some implications for teaching and point to a number of demands for future research.

The use of figurate numbers (e.g., in the context of elementary number theory) can be considered a heuristic in the field of problem solving or proving. In this paper, we want to discuss this heuristic from the perspectives of the semiotic theory of Peirce ("diagrammatic reasoning" and "collateral knowledge") and cognitive psychology ("schema theory" and "Gestalt psychology"). We will make use of several results taken from our research to illustrate first-year students' problems when dealing with figurate numbers in the context of proving. The considerations taken from both theoretical perspectives will help to partly explain such phenomena. It will be shown that the use of figurate numbers must not be considered to be any kind of help for learners or some way of 'easy' mathematics. Working in this representational system has to be learned and practiced as another kind of knowledge is necessary for working with figurate numbers. The named findings also touch upon the concept of 'proofs that explain.' Finally, we will highlight some implications for teaching and point to a number of demands for future research. [ABSTRACT FROM AUTHOR]

The article presents the results of a qualitative study that aims to characterize the knowledge of future mathematics professors at the National University of Costa Rica (UNA) on the logical and syntactic aspects of the mathematical proof. Knowledge indicators are proposed to characterize that related to the mathematical proof practice, focusing on the logical and syntactic aspects. It was appreciated that most of the subjects showed knowledge of the logical aspects of the proof and that the proposed analysis indicators allowed to classify most of their responses. [ABSTRACT FROM AUTHOR]