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Proof schemes combined: mapping secondary students’ multi-faceted and evolving first encounters with mathematical proof
Abstract
In this paper, we propose an enriched and extended application of Harel and Sowder’s proof schemes taxonomy that can be used as a diagnostic tool for characterizing secondary students’ emergent learning of proof and proving. We illustrate this application in the analysis of data collected from 85 Year 9 (age 14–15) secondary students. We capture these students’ first encounters with proof and proving in an educational context (mixed ability, state schools in Greece) where mathematical proof is explicitly present in algebra and geometry lessons and where proving skills are typically expected, and rewarded, in key national examinations. We analyze student written responses to six questions, soon after the students had been introduced to proof and we identify evidence of six of the seven proof schemes proposed by Harel and Sowder as well as a further eight combinations of the six. We observed these combinations often within the response of the same student and to the same item. Here, we illustrate the eight combinations and we claim that a dynamic use of the proof schemes taxonomy that encompasses sole and combined proof schemes is a potent theoretical and pedagogical tool for mapping students’ multi-faceted and evolving competence in, and appreciation for, proof and proving.
... To this aim, we deploy a diagnostic tool, Proof-Schemes Combined (Kanellos et al., 2018), that we developed from Sowder's (1998, 2007) Proof-Schemes taxonomy. We developed Proof-Schemes Combined in the course of analysing data which was collected originally for Kanellos (2014). ...
... In what follows, first, we outline works on the learning and teaching of mathematical proof that have influenced our study and we present the Proof-Schemes (Harel & Sowder, 1998 and Proof-Schemes Combined (Kanellos et al., 2018) taxonomies that we deploy in our analysis. We then introduce the context, aims and methods of the study. ...
... The focus of this paper was motivated by observations in Kanellos (2014) and Kanellos et al. (2018) that, on many occasionseven when compliance with the conventions of mathematical writing was imperfect or when empirical, salient features (e.g. of geometrical shapes) proliferated in a piece of student writingthere were modicums of deductive reasoning, rudimentary but non-negligible attempts at generalisation and justification. Identifying and interpreting these modicums of deductive reasoning became the focus of the diagnostic work we report here. ...
... Learning (Q3) [20], [21],] [13], [22], [23]. 5 ...
... The Design Thinking methodology is used to improve the educational process of students, since this technique allows them to develop artistic development skills and content understanding [22], and express themselves in a more personal way [23]. ...
... Reference Questionnaires [20], [21], [13], [22] Analysis of data [20], [21], [13] Design Thinking [22], [23] D. Competence development Four tools were identified that are used in students for the development of competencies (TABLE V). An investigation affirms that the use of electronic devices contributes to student learning and allows for better results [23] [24] [25] [26], since it allows the use of tools such as evaluations and learning methodologies [24] [25]. ...
The quality of secondary education in Peru is one of the lowest in South America, as evidenced by the PISA 2018 evaluation. For this reason, we propose a model based on a B-Learning approach to monitor the competencies of the PISA test in Peru. The model is made up of 4 phases: (i) Selection of the methodology and technique, (ii) Design of the study material, (ii) Design of evaluations and (iv) Design of the web application. Three experiments were carried out to validate the proposal, where the ""usability"" was evaluated with a group of teachers and students, and with another group of students, the effect of the application on their ""performance"". The results showed that 73.3% of teachers and 80.7% of students found the application ""very good"". In addition, the results of the efficacy validation have shown that the application is effective in increasing the performance of students in the areas evaluated by at least 40%.
... Within these practices, the teacher was established as the "sole arbiter of correctness" (Harel & Rabin, 2010:156), which contributed to students' focus on the teacher's instruction of how to use the correct procedures, instead of making their own attempts to solve the problem and raise questions about their peers' solutions. However, several studies using the taxonomy of proof schemes as an analytical tool (e.g., Ellis, Lockwood, Dogan, Williams, & Knuth, 2013;Erickson & Lockwood, 2021;Housman & Porter, 2003;Kanellos, 2014;Kanellos, Nardi, & Biza, 2018;Lee, 2016;Liu & Manouchehri, 2013;Sears, 2019;Sen & Guler, 2015) have not taken classroom interaction into account. Instead, they have analyzed individual students' proving, based on interviews and/or written material. ...
... However, neither Housman and Porter, Sears, nor Kanellos explicitly reported how proof schemes that co-occurred were related during the proving process, for instance with regard to their internal ordering, or if they built on one another. Kanellos et al. (2018) further analyzed grade-9 students' written solutions to mathematical problems where more than one proof scheme was identified. In these cases, the proof schemes were noted as "combined, " but the ordering of the demonstrations was not taken into account in the analysis. ...
... The taxonomy of proof schemes was a result of the analysis of students' proving when working individually, in small groups, and in whole class settings . However, previous studies that have used the taxonomy as a tool to analyze secondary-school students' proving (e.g., Kanellos, 2014;Kanellos et al., 2018;Lee, 2016;Liu & Manouchehri, 2013;Sen & Guler, 2015) have focused on written products in the form of answers to surveys and/or solutions to mathematical problems, which do not show the various steps taken during the proving process that resulted in the final product (Knipping, 2008). Our empirical material differs from the data of previous studies as we have analyzed video recordings of students' naturally occurring classroom interaction, which enabled the exploration of students' use of justifications during the interactional process of solving a mathematical problem. ...
In this study, we focused on grade-6 students’ justifications during mathematical problem solving in small group interaction. Video recordings from two classrooms were analyzed, using Harel and Sowder’s (2007) taxonomy of proof schemes as a tool to categorize students’ justifications, which showed that the frequency of various types of justifications corresponded to previous studies of adult students’ demonstrations of proof schemes. Results also showed that both Non-referential Symbolic and Transformational justifications contributed to students’ formulations of general arguments. In addition, agreements between proximately located justifications based on examples and calculations supported students’ improvements of solutions. However, in cases of disagreements, justifications based on calculations often had most influence on the proving process even when the calculation was incorrect. Our results imply that teachers should emphasize the importance of agreement between calculations and empirical examples, as well as the significance of counterexamples, as a support to students’ progression towards mathematical proving.
... Some researchers have used proof schemes to categorize students' proving activity. For example, Kanellos, Nardi, and Biza (2018) looked at the proof schemes employed by high-school students who were asked to provide proofs of statements the context of algebra and geometry. They found that students may use different proof schemes depending on the proposition they are given to prove, and they also identified eight combinations of proof schemes that emerged in their data. ...
... 277). Other researchers such as Kanellos, Nardi, and Biza (2018) have also used multiple proof schemes to categorize students' reasoning. After coding the data using Harel and Sowder's proof schemes, the first author checked the interview videos again to see if there were any other episodes that may warrant further analysis. ...
... While applying Harel and Sowder's (1998) proof schemes framework to students' proof production is common (e.g. Blanton & Stylianou, 2014;Healy & Hoyles, 2000;Kanellos et al., 2018;Stylianou, Chae, & Blanton, 2006), for our study it was not particularly insightful or interesting to categorize our participants' combinatorial proof production using proof schemes. This is simply because the students we interviewed were so successful producing the combinatorial proofs that the vast majority of their proof production work would have been categorized as using transformational analytical proof schemes. ...
Combinatorics is an area of mathematics with accessible, rich problems and applications in a variety of fields. Combinatorial proof is an important topic within combinatorics that has received relatively little attention within the mathematics education community, and there is much to investigate about how students reason about and engage with combinatorial proof. In this paper, we use Harel and Sowder’s (1998) proof schemes to investigate ways that students may characterize combinatorial proofs as different from other types of proof. We gave five upper-division mathematics students combinatorial-proof tasks and asked them to reflect on their activity and combinatorial proof more generally. We found that the students used several of Harel and Sowder’s proof schemes to characterize combinatorial proof, and we discuss whether and how other proof schemes may emerge for students engaging in combinatorial proof. We conclude by discussing implications and avenues for future research.
... Frequently, students use empiricalinductive arguments (Harel & Sowder, 1998;Healy & Hoyles, 1998;Klieme et al., 2003;Reiss, Hellmich, et al., 2002). Using the taxonomy of Harel and Sowder (1998) in a secondary context, Kanellos et al. (2018) identified empirical-inductive proof schemes mostly in responses to algebraic problems, whereas empirical-perceptual proof schemes were found mostly in responses to geometrical problems. ...
... Over the intervening years, research results may have reached mathematics classes, and therefore, the magnitude of this proof scheme may have scaled down. According to Kanellos et al. (2018), empirical arguments are still an issue and in relation to the empiricalperceptual proof scheme; most evidence of it was identified in the students' responses to geometrical problems, whereas empirical-inductive proof schemes were revealed in students' responses to algebraic problems. It seems that the dynamical element of the flag problems and the use of a DGE are decisive and redistribute the observed distribution. ...
This paper deals with a set of geometrical problems for mathematical problem-solving at different difficulty levels. All of these are presented as national flags and one has to investigate invariant area proportions when changing the locus of any corner of the flag. This dynamic element suggests the usage of dynamical geometry environments. Considering applications in mathematics classes, it allows training students' ability to find a proof above all in the phases of production of a conjecture and exploration of the conjecture. First, the paper introduces results of selected studies on reasoning and proof in mathematics classes. Then the formulation of the problems and their sample solutions are given. Finally, the paper reports on experiences teaching some of the posed problems.
... This framework has been used numerous times (e.g. Kanellos et al., 2018;Segal, 1999;Recio and Godino, 2001) to describe and categorise the types of arguments students make and the types of arguments they find convincing. In such research students often demonstrated an over-reliance on empirical evidence and appeals to authority. ...
Proof is a central concept in mathematics, pivotal both to the practice of mathematicians and to students’ education in the discipline. The research community, however, has failed to reach a consensus on how proof should be conceptualised. Moreover, we know little of what mathematicians and students think about proof, and are limited in the tools we use to assess students’ understanding. This thesis introduces comparative judgment to the proof literature via two tasks evaluated by judges performing a series of pairwise comparisons. The Conceptions Task asks for a written explanation of what mathematicians mean by proof. The Summary Task asks for a summary of a given proof, available to respondents as they complete the task. Having established robust evidence supporting the reliability and validity of both tasks, I then use these tasks to develop an understanding of the conceptions of proof held by mathematicians and students. I also generate insights for assessment, leading to an argument for the unidimensionality of proof comprehension in early undergraduate mathematics. In conducting this research I adopt a mixed methods approach based on the philosophy of pragmatism. By using a range of methodological approaches, from statistical modelling to thematic analysis of interviews with judges, I develop a multi-faceted understanding of both the validity of the tasks, and the behaviours and priorities of the participants involved. The Conceptions Task outcomes establish that mathematicians primarily think of proof in terms of argumentation, while students emphasise the arguably more philosophically naive notion of certainty. The Summary Task outcomes establish that references to the method of proof and key mathematical objects are most valued by mathematician judges. Further, from correlational analyses of various quantitative measures, I learn that the Summary Task scores are meaningfully reflective of local proof comprehension but are not related to more general measures of mathematical performance. Several open questions are identified. In particular, there is still much to learn about judges’ decision-making processes in comparative judgment settings, the dimensionality of proof comprehension, and the range of proofs for which the Summary Task is applicable. Future work on these questions is outlined in the final chapter, alongside the practical applications and theoretical implications of this work.
... Elaborar uma demonstração em matemática envolve processos mentais complexos com raciocínio dedutivo (Lannin, 2005), que pode ser precedido por raciocínio indutivo mediante a formulação e o teste de conjeturas, com exemplos e contraexemplos (Beites, 2015). apesar da sua importância, o raciocínio indutivo é negativamente apontado como emergente em esquemas de demonstração de álgebra, parecendo os tipos de tarefas facilitar o surgimento de certos tipos de esquemas de demonstração (Kanellos, nardi, & Biza, 2018). no presente trabalho encontramos raciocínio indutivo em esquemas de demonstração empírico indutivos e, ainda, em combinações destes esquemas com outros, mas tal não era expectável nem deveria suceder. ...
The purpose of the present work is to analyse proof schemes in students’ productions in a curricular unit of Linear Algebra in Higher Education. Concretely, we study the representations of three students, and their transformations, in tasks that contain propositions of Linear Algebra with logical value true, trying to access their mathematical reasoning. The conclusions of the study indicate that the level of performance in Mathematics of a student may not translate the level of performance of that student in the construction of proofs. They also suggest a possible connection between the contents, and the experience, associated to the first contact of the students with proof and the representations used by them in the proof schemes. Last but not least, the conclusions shed light on the meanings for the students of proof schemes which are not deductive. In a broad sense, the findings show the need for research-based interventions in the classroom to address the challenges of the teaching and the learning of proof in Higher Education.
... What is relevant to our purposes in this paper is that a rather large body of research has investigated students' justification schemes and has tended to use one or both of two kinds of tasks to elicit and document students' justification schemes (see, e.g. Housman & Porter, 2003;Kanellos, Nardi, & Biza, 2018;Lee, 2016): (1) proof construction tasks, i.e. tasks that ask students to formulate a proof for the truth or falsity of a mathematical claim (usually a mathematical generalization); and (2) proof evaluation tasks, i.e. tasks that ask students to indicate whether they think given arguments for a mathematical claim (usually researcher-generated arguments of various mathematical qualities) meet the standard of proof. The operational hypothesis has been that a student's purported proof (in the first kind of task) or a student's evaluation of a given argument as meeting the standard of proof (in the second kind of task) both reflect, or indicate, the student's justification scheme. ...
In this paper, we argue that posing new researchable questions in educational research is a dynamic process that reflects the field’s growing understanding of the web of potentially influential factors surrounding the examination of a particular phenomenon of interest. We illustrate this thesis by drawing on a strand of mathematics education research related to students’ justification schemes that has evolved rapidly during the past few decades. Also, we reflect on the possible boundaries of the domain of application of the thesis, and we hypothesize that it would apply equally to other strands of educational research. To support this hypothesis, we briefly consider how the thesis would be applicable in two additional research strands. We conclude by elaborating on three important implications of the thesis: (1) as new potentially influential factors about the phenomenon of interest are identified, findings from past research that had not accounted for those factors might prove to be insufficient or be put into question; (2) there are increased methodological challenges for researchers as they seek to design new studies that pay due regard to research advances about all relevant and potentially influential factors surrounding the phenomenon of interest; and (3) as a wider range of potentially influential factors get discovered and considered about a particular phenomenon, research knowledge becomes not only more refined, and presumably more accurate, but possibly more fragmented too.
... This framework has been used numerous times (e.g. Kanellos et al., 2018;Segal, 1999;Recio and Godino, 2001) to describe and categorise the types of arguments students make and the types of arguments they find convincing. In such research students often demonstrated an over-reliance on empirical evidence and appeals to authority. ...
Proof is a central concept in mathematics, pivotal both to the practice of mathematicians and to students' education in the discipline. The research community, however, has failed to reach a consensus on how proof should be conceptualised. Moreover, we know little of what mathematicians and students think about proof, and are limited in the tools we use to assess students' understanding.
This thesis introduces comparative judgment to the proof literature via two tasks evaluated by judges performing a series of pairwise comparisons. The Conceptions Task asks for a written explanation of what mathematicians mean by proof. The Summary Task asks for a summary of a given proof, available to respondents as they complete the task.
Having established robust evidence supporting the reliability and validity of both tasks, I then use these tasks to develop an understanding of the conceptions of proof held by mathematicians and students. I also generate insights for assessment, leading to an argument for the unidimensionality of proof comprehension in early undergraduate mathematics.
In conducting this research I adopt a mixed methods approach based on the philosophy of pragmatism. By using a range of methodological approaches, from statistical modelling to thematic analysis of interviews with judges, I develop a multi-faceted understanding of both the validity of the tasks, and the behaviours and priorities of the participants involved.
The Conceptions Task outcomes establish that mathematicians primarily think of proof in terms of argumentation, while students emphasise the arguably more philosophically naive notion of certainty.
The Summary Task outcomes establish that references to the method of proof and key mathematical objects are most valued by mathematician judges. Further, from correlational analyses of various quantitative measures, I learn that the Summary Task scores are meaningfully reflective of local proof comprehension but are not related to more general measures of mathematical performance.
Several open questions are identified. In particular, there is still much to learn about judges' decision-making processes in comparative judgment settings, the dimensionality of proof comprehension, and the range of proofs for which the Summary Task is applicable. Future work on these questions is outlined in the final chapter, alongside the practical applications and theoretical implications of this work.
Resumo As produções escritas de estudantes podem ser objeto de análise com vista à investigação e ao ensino. Neste trabalho, um dos poucos sobre erros e números complexos, analisam-se as produções, elaboradas por estudantes do ensino superior, para uma tarefa de demonstração de um caso particular da identidade do paralelogramo. Num trabalho que pode dar pistas para a prática letiva de professores do ensino secundário e do ensino superior, procura-se classificar os esquemas de demonstração nessas produções, identificar os erros neles cometidos e relacionar estas duas vertentes de análise. A natureza da investigação é qualitativa, recorrendo à análise de conteúdo empregando, por um lado, critérios prévios de categorizações de esquemas de demonstração e, por outro, critérios novos de categorizações de erros. Constatou-se que os estudantes utilizaram, principalmente, esquemas de demonstração de convicção externa, incluídos numa subcategoria nova designada por não válidos no universo, e esquemas de demonstração dedutivos, incluídos num nível novo que se caracteriza por discernimento incipiente da compreensão da tarefa, do contexto matemático, de hipóteses e tese, e dos conhecimentos prévios a mobilizar. Os erros dominantes nas produções estão relacionados à compreensão dos conceitos de linearidade de uma função e de módulo de um número complexo, apresentando-se possíveis explicações para os mesmos. Apesar de não se identificar uma relação propriamente significativa entre os esquemas de demonstração produzidos pelos estudantes e os erros cometidos, obteve-se material para construção e discussão de questões conceituais no âmbito do tópico números complexos.
This study was conducted to examine the effectiveness of proof tasks in the transition from conjecture to proof in Euclidean geometry on freshmen’s proof schemes. In line with this aim, the proof schemes of the freshmen who performed conjecture-proof and theorem-proof tasks were compared. The freshmen were composed of 109 pre-service middle school mathematics teachers who are enrolled in their first year of undergraduate education. The study was modeled as a multiple-case study. Fifty-three freshmen performed conjecture-proof tasks in Case-1, and fifty-six freshmen performed theorem-proof tasks in Case-2. The video recordings, including the written proof reports, reflection papers, and proof explanations of the freshmen, were used as data collection tools in the tasks. The proof schemes were used as construct maps to evaluate the proofs of freshmen and analyzed using Winsteps Rasch software. The proof schemes in freshmen’s proof tasks were evaluated by the Wright Map and supported with direct quotations from the proofs. In the study, it was observed that the proof schemes of freshmen who made proofs based on their own conjectures were mostly empirical and analytical proof schemes, while the proof schemes of freshmen who made proof of the presented theorem were generally external and empirical proof schemes.
Nous étudions les relations entre preuves et contradictions dans la résolution d'un problème de mathématiques. Cette étude montre la nécessité d'une approche à la fois situationnelle et cognitive, notamment en référence au fonctionnement des connaissances dans l'apprentissage des mathématiques. Ceci nous conduit à distinguer différents stades dans l'évolution des preuves pragmatiques aux preuves intellectuelles. Enfin nous montrons que le dépassement d'une contradiction ne constitue pas nécessairement un progrès cognitif, en particulier nous examinons le traitement d'un contre-exemple par des élèves de quatrième.
This paper is a commentary on the classroom interventions on the teaching and learning of proof reported in the seven empirical papers in this special issue. The seven papers show potential to enhance student learning in an area of mathematics that is not only notoriously difficult for students to learn and for teachers to teach, but also critically important to knowing and doing mathematics. Although the seven papers, and the intervention studies they report, vary in many ways—student population, content domain, goals and duration of the intervention, and theoretical perspectives, to name a few—they all provide valuable insight into ways in which classroom experiences might be designed to positively influence students’ learning to prove. In our commentary, we highlight the contributions and promise of the interventions in terms of whether and how they present capacity to change the classroom culture, the curriculum, or instruction. In doing so, we distinguish between works that aim to enhance students’ preparedness for, and competence in, proof and proving and works that explicitly foster appreciation for the need and importance of proof and proving. Finally, we also discuss briefly the interventions along three dimensions: how amenable to scaling up, how practicable for curricular integration, and how capable of producing long-lasting effects these interventions are.
In this paper proving ability - according to the Lakatosian sense of proof - refers to the ability to make something evident. A test of proving ability was administered to 2572 students in Hungary. A dichotornous evaluation system allows for both hierarchical and non-hierarchical evaluation of proving ability. An hierarchical evaluation can be based on Harel and Sowder's proof-categorization. The results show that there is a positive correlation between proof types of dffirent content domains (using the above-mentioned hierarchical model), suggesting that Harel and Sowder' taxonomy can be a powerful tool for measuring proving ability. AIMS Proofs are used in mathematics, in philosophy, in jurisprudence, and in everyday life. This study addresses the evaluation of proving ability. In this paper the term 'proof will be used in the Lakatosian sense of 'making something evident' (Lakatos, 1976), and proving ability will refer to the ability to construct proofs.
This paper explores secondary teachers' views on the role of visualisation in the justification of a claim in the mathematics classroom and how these views could influence instruction. We engaged 91 teachers with tasks that invited them to: reflect on/solve a mathematical problem; examine flawed (fictional) student solutions; and, describe, in writing, feedback to students. Eleven teachers were also interviewed. Here we draw on the interviews and the responses to one Task in order to explore the influence on the teachers' feedback to students of: persistent images of the tangent line; beliefs about the sufficiency of a visual argument; and, beliefs about the role of visual arguments in student learning. We focus particularly on the influence on the didactical contract regarding mathematical reasoning that teachers with a variation of beliefs about the role of visualisation are likely to offer their students.
This paper explores the role of proof in mathematics education and providesjustification for its importance in the curriculum. It also discusses threeapplications of dynamic geometry software – heuristics, exploration andvisualization – as valuable tools in the teaching of proof and as potentialchallenges to the importance of proof. Finally, it introduces the four papers in this issue that present empirical research on the use of dynamicgeometry software.
We report some findings of the Longitudinal Proof Project, which investigated patterns in high-attaining students' mathematical reasoning in algebra and in geometry and development in their reasoning, by analyses of students' responses to three annual proof tests. The paper focuses on students' responses to one non-standard geometry item. It reports how the distribution of responses to this item changed over time with some moderate progress that suggests a cognitive shift from perceptual to geometrical reasoning. However, we also note that many students made little or no progress and some regressed. Extracts from student interviews indicate that the source of this variation from the overall trend stems from the shift over the three years of the study to a more formal approach in the school geometry curriculum for high-attaining students, and the effects of this shift on what students interpreted as the didactical demands of the item.
Yehuda Rav’s inspiring paper “Why do we prove theorems?” published in Philosophia Mathematica (1999, 7, pp. 5–41) has interesting
implications for mathematics education. We examine Rav’s central ideas on proof—that proofs convey important elements of mathematics
such as strategies and methods, that it is “proofs rather than theorems that are the bearers of mathematical knowledge”and thus that proofs should be the primary focus of mathematical interest—and then discuss their significance for mathematics education in general and for the teaching of proof in particular.
Literature suggests that the type of context wherein a task is placed relates to students' performance and solution strategies. In the particular domain of logical thinking, there is the belief that students have less difficulty reasoning in verbal than in logically equivalent symbolic tasks. Thus far, this belief has remained relatively unexplored in the domain of teaching and learning of mathematics, and has not been examined with respect to students' major field of study. In this study, we examined the performance of 95 senior undergraduate mathematics and education majors in symbolic and verbal tasks about the contraposition equivalence rule . The selection of two different groups of participants allowed for the examination of the hypothesis that students' major may influence the relation between their performance in tasks about contraposition and the context (symbolic/verbal) wherein this is placed. The selection of contraposition equivalence rule also addressed a gap in the body of research on undergraduate students' understanding of proof by contraposition . The analysis was based on written responses of all participants to specially developed tasks and on semi-structured interviews with 11 subjects. The findings showed different variations in the performance of each of the two groups in the two contexts. while education majors performed significantly better in the verbal than in the symbolic tasks, mathematics majors' performance showed only modest variations. The results call for both major- and context- specific considerations of students' understanding of logical principles, and reveal the complexity of the system of factors that influence students' logical thinking. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/42655/1/10649_2004_Article_5150044.pdf
Classification of students' proof schemes offers insights into students' mathematical proving. However, previous studies revealed little about finer progressions in students' proving and barely examined their proof schemes involving counterexamples. This study examined students' proof schemes through their considerations of examples and counterexamples and reasoning in Proof Constructions. From 480 proofs constructed by 60 Singapore secondary three (14-15 years old) students, classifications of schemes for Deductive-proof Construction and Proof-by-counterexample Construction were established as a progression through four phases. The former consisted of seven levels, spanning from irrelevant inferences to deductive proofs using formal representations. The latter consisted of six levels, spanning from irrelevant inferences to construction of a general set of counterexamples. The refined classifications revealed students' progression through their sophisticated use of examples, counterexamples, and deductive inferences and that Proof Construction was knowledge-driven. Comparisons to previous classifications and the role of counterexamples in Proof Construction were discussed.
After surveying high-attaining 14- and 15-year-old students about proof in algebra, we found that students simultaneously held 2 different conceptions of proof: those about arguments they consid- ered would receive the best mark and those about arguments they would adopt for themselves. In the former category, algebraic arguments were popular. In the latter, students preferred arguments that they could evaluate and that they found convincing and explanatory, preferences that excluded algebra. Empirical argument predominated in students' own proof constructions, although most students were aware of its limitations. The most successful students presented proofs in everyday language, not using algebra. Students' responses were influenced mainly by their mathematical competence but also by curricular factors, their views of proof, and their genders.
There are currently increased efforts to make proof central to school mathematics throughout the grades. Yet, realizing this goal is challenging because it requires that students master several abilities. In this article we focus on one such ability, namely, the ability for deductive reasoning, and we review psychological research to enhance what is currently known in mathematics education research about this ability in the context of proof and to identify important directions for future research. We first offer a conceptualization of proof, which we use to delineate our focus on deductive reasoning. We then review psychological research on the development of students' ability for deductive reasoning to see what can be said about the ages at which students become able to engage in certain forms of deductive reasoning. Finally, we review two psychological theories of deductive reasoning to offer insights into cognitively guided ways to enhance students' ability for deductive reasoning in the context of proof.
The process of observing and analyzing studentsÕ behaviors is interesting and complex but also unstable. It is unstable because it involves countless variables, many of which are uncontrollable. Despite this, what we learn from this process is useful, even essential, in designing and implementing mathe-matics curricula for both students and teachers. This presentation is about studentsÕ behaviors in relation to justi®cation and proof. Some of these be-haviors are assumed to be due to faulty instruction in school; others seem to be unavoidable, in the sense that they are of human developmental nature. An-alyzed from a historical perspective of mathematical development, these stu-dentsÕ understandings of proof can be classi®ed into three categories: · Category 1: In this category, studentsÕ understandings of proof (viewed in re-lation to those of their instructors) seem to parallel the Greek conception of mathematics (viewed in relation to that of modern days). · Category 2: In this category, studentsÕ understandings of proof are reminis-cent of the 16±17th century conception of mathematics. · Category 3: In this category, studentsÕ understandings of proof seem, to a large extent, to be a result of faulty instruction in the elementary and second-ary schools.
In this paper, we distinguish between two ways that an individual can construct a formal proof. We define a syntactic proof production to occur when the prover draws inferences by manipulating symbolic formulae in a logically permissible way. We define a semantic proof production to occur when the prover uses instantiations of mathematical concepts to guide the formal inferences that he or she draws. We present two independent exploratory case studies from group theory and real analysis that illustrate both types of proofs. We conclude by discussing what types of concept understanding are required for each type of proof production and by illustrating the weaknesses of syntactic proof productions.
What patterns can be observed among the mathematical arguments above-average students find convincing and the strategies these
students use to learn new mathematical concepts? To investigate this question, we gave task-based interviews to eleven female
students who had performed well in their college-level mathematics courses, but who differed in the number of proof-oriented
courses each had taken. One interview was designed to elicit expressions of what students find convincing. These expressions
were categorized according to the proof schemes defined by Harel and Sowder (1998). A second interview was designed to elicit
expressions of what strategies students use to learn a mathematical concept from its definition, and these expressions were
classified according to the learning strategies described by Dahlberg and Housman (1997). A qualitative analysis of the data
uncovered the existence of a variety of phenomena, including the following: All of the students successfully generated examples
when asked to do so, but they differed in whether they generated examples without prompting and whether they successfully
generated examples when it was necessary to disprove conjectures. All but one of the students exhibited two or more proof
schemes, with one student exhibiting four different proof schemes. The students who were most convinced by external factors
were unsuccessful in generating examples, using examples, and reformulating concepts. The only student who found an examples-based
argument convincing generated examples far more than the other students. The students who wrote and were convinced by deductive
arguments were successful in reformulating concepts and using examples, and they were the same set of students who did not
generate examples spontaneously but did successfully generate examples when asked to do so or when it was necessary to disprove
a conjecture.
Despite increased appreciation of the role of proof in students’ mathematical experiences across all grades, little research has focused on the issue of understanding and characterizing the notion of proof at the elementary
school level. This paper takes a step toward addressing this limitation, by examining the characteristics of four major features
of any given argument – foundation, formulation, representation, and social dimension – so that the argument could count as
proof at the elementary school level. My examination is situated in an episode from a third-grade class, which presents a
student’s argument that could potentially count as proof. In order to examine the extent to which this argument could count
as proof (given its four major elements), I develop and use a theoretical framework that is comprised of two principles for
conceptualizing the notion of proof in school mathematics: (1) The intellectual-honesty principle, which states that the notion of proof in school mathematics should be conceptualized so that it is, at once, honest to mathematics
as a discipline and honoring of students as mathematical learners; and (2) The continuum principle, which states that there should be continuity in how the notion of proof is conceptualized in different grade levels so that
students’ experiences with proof in school have coherence. The two principles offer the basis for certain judgments about
whether the particular argument in the episode could count as proof. Also, they support more broadly ideas for a possible
conceptualization of the notion of proof in the elementary grades.
In diesem Beitrag wird über eine Interviewstudie mit zehn Schülerinnen und Schülern der Jahrgangsstufe 8 berichtet, die als
qualitative Ergänzung zu einer quantitativen empirischen Untersuchung mit 659 Probanden durchgeführt wurde. Die Probanden,
die in der 7. und 8. Klasse an schriftlichen Tests teilgenommen hatten, wruden beim Lösen geometrischer Beweisaufgaben videografiert
und anschließend befragt. Es zeigt sich, dass Schülerschwierigkeiten bei diesen Aufgaben im Wesentlichen auf das Faktenwissen,
das Methodenwissen zum mathematischen Beweisen und die Entwicklung und das Verfolgen einer Beweisstrategie zurückgeführt werden
können. Während schwächere Schüler in allen drei Bereichen Defizite aufweisen, liegen die Schwierigkeiten der stärkeren Probanden
vor allem in der Entwicklung einer Beweisstrategie.
In this article we report on an interview study involving ten grade 8 students. These interviews served as a qualitative supplement
for a large-scale quantitative study on proof and argumentation (N=659). During videotaped interviews the students were asked
to solve geometrical proof problems. The results indicate that students’ difficulties with proof and logical argumentation
may be explained by insufficient knowledge of facts, deficits in their methodological knoledge about mathematical proofs,
and a lack of knowledge with respect to developing and implementing a proof strategy. Low-achieving students show difficulties
with respect to all these three aspects, whereas high-achieving students’ difficulties are mainly based on deficits of developing
an adequate and correct proof strategy.
ZDM-ClassificationE53-C33
Secondary students’ proof schemes during the first encounters with formal mathematical reasoning: Appreciation, fluency and readiness (Unpublished doctoral thesis)
- I Kanellos
Kanellos, I. (2014). Secondary students' proof schemes during the first encounters with formal mathematical reasoning: Appreciation, fluency and readiness (Unpublished doctoral thesis). University of East Anglia, UK. Retrieved
from https://ueaeprints.uea.ac.uk/49759/1/2014KanellosIEdD.pdf
The interplay between fluency and appreciation in secondary students’ first encounter with proof
- I Kanellos
- E Nardi
- I Biza
Kanellos, I., Nardi, E., & Biza, I. (2013). The interplay between fluency and appreciation in secondary students' first
encounter with proof. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th conference of the
international group for the psychology of mathematics education (Vol. 5, pp. 84). Kiel, Germany: PME.
Theorems in school: From history, epistemology, and cognition to classroom practice
- G Hanna
Hanna, G. (2007). The ongoing value of proof. In P. Boero (Ed.), Theorems in school: From history, epistemology, and
cognition to classroom practice (pp. 3-16). Rotterdam, The Netherlands: Sense Publishers.
Theorems in school: From history, epistemology, and cognition to classroom practice
- G Harel