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Toward comprehensive perspectives on the learning and teaching of proof
... Harel e Sowder (2007) consideram que as justificações dos alunos podem ser baseadas em convicção externa, evidência empírica ou argumentos dedutivos. As justificações por convicção externa dependem de uma autoridade como um professor ou um livro, da aparência do argumento, com foco na sua estrutura e não no seu conteúdo, ou de manipulações simbólicas, independentemente do significado dos símbolos (Harel;Sowder, 2007). As justificações empíricas são feitas a partir de evidência de casos particulares. ...
... Harel e Sowder (2007) consideram que as justificações dos alunos podem ser baseadas em convicção externa, evidência empírica ou argumentos dedutivos. As justificações por convicção externa dependem de uma autoridade como um professor ou um livro, da aparência do argumento, com foco na sua estrutura e não no seu conteúdo, ou de manipulações simbólicas, independentemente do significado dos símbolos (Harel;Sowder, 2007). As justificações empíricas são feitas a partir de evidência de casos particulares. ...
... Balacheff (1988) considera que estas justificações podem basear-se em exemplos aleatórios (empirismo ingénuo), exemplos cuidadosamente selecionados (exemplos cruciais) e exemplos que representam características presentes em uma classe de casos (exemplo genérico). As justificações dedutivas têm natureza analítica e podem ser por coerência lógica, baseada em princípios lógicos e resultados anteriores, e por prova transformacional, que se baseia em definições, teoremas, propriedades ou procedimentos (Harel;Sowder, 2007). No Quadro 2 sintetizamos estas classificações, com base nos autores referidos. ...
Resumo Este artigo tem como objetivo compreender como, no quadro de uma experiência de ensino que enfatiza a promoção do raciocínio matemático, se caracterizam as conjeturas, generalizações e justificações de alunos de 8.° ano, ao resolverem tarefas de cunho algébrico. Adotando uma abordagem qualitativa e interpretativa, utilizamos as resoluções escritas e orais de alunos de 8.° ano ao resolverem tarefas algébricas, no contexto de uma experiência de ensino baseada na promoção do raciocínio matemático. Como resultados, destacamos o surgimento de várias conjeturas e generalizações empíricas, com algumas generalizações teóricas/estruturais/simbólicas. Destacamos, também, o uso de justificações empíricas, por exemplos aleatórios, selecionados e genéricos, e poucos episódios de justificações por convicção externa, tanto por manipulações simbólicas como por aparência dos argumentos. Além disso, ressaltamos que a discussão em torno das generalizações empíricas favoreceu a mobilização de outras formas de justificação que não estavam baseadas apenas no uso de exemplos.
... Educators recognize it is crucial to expand the concept of proof to allow for a wider range of argumentation beyond the traditional two-column format (Knuth, 2002;Harel & Sowder, 1998). Harel and Sowder's (2007) transformational proof scheme permits less formal proof types while still addressing all three necessary criteria of a mathematically valid deductive proof: (1) It must be general, showing the argument must be true for all possible cases; ...
... Following Harel and Sowder's (2007) criterion for deductive proofs, participants' verbal responses were independently coded for three defining characteristics: (1) generality of the conjecture across the class of mathematical objects under consideration; (2) use of operational thinking, a systematic progression establishing a goal structure and anticipation of outcomes resulting from proposed transformations; and (3) exhibit a chain of logical inference with conclusions following from valid premises. Each verbalized response, including speech and gesture, was coded as 1 only if it met all three defining criteria, and zero (0) otherwise. ...
... To address this emergent question, we conducted a set of post hoc analyses. To narrow this investigation, post hoc analyses examined the relative contributions of each of the three essential criteria used for assessing mathematically valid proof production (Harel & Sowder, 2007): generalization, operational thinking, and logic inference. ...
Task‐relevant actions can facilitate mathematical thinking, even for complex topics, such as mathematical proof. We investigated whether such cognitive benefits also occur for action predictions. The action‐cognition transduction (ACT) model posits a reciprocal relationship between movements and reasoning. Movements—imagined as well as real ones operating on real or imaginary objects—activate feedforward mechanisms for the plausible predicted outcomes of motor system planning, along with feedback from the effect actions have on the world. Thus, ACT posits cognitive influences for making action predictions regardless of whether those actions are performed. Using a two‐by‐two factorial design, we investigated how generating task‐relevant action predictions or performing task‐relevant directed actions influenced undergraduates’ ( N = 127) geometry proof performance. As predicted, making action predictions significantly enhanced participants’ proof production. No evidence suggests that combining action predictions and directed actions provided additional benefits, supporting the claim that predicting and performing actions engage overlapping processes, as theorized by ACT. Gestural replays , reenactments of previously performed actions during explanations, were associated with significantly better insight and proof performance for both (actor‐generated) predicted actions and (investigator‐generated) directed actions. Prompting people to predict task‐relevant actions enhances mathematical cognition, possibly through simulated actions of transformations on imagined mathematical objects, as revealed by increased production of speech describing mathematical operations and increased production of gestural replays. We discuss the theoretical implications of these findings regarding the influences of embodied simulation of movements on cognition, and the educational implications of facilitating mathematical reasoning through interventions prompting students to perform and imagine performing task‐relevant body movements.
... There is a general agreement that the formal texts produced by mathematicians to communicate their results are proofs, but there is an open discussion about the texts produced by students which do not fit the requirement of mathematicians' proofs (Stylianides et al., 2017). We align with Balacheff (1988), Harel and Sowder (2007), Fiallo and Gutiérrez (2017), and other authors in considering as proofs any mathematical argumentation raised to justify the truth of a mathematical statement, not only the formal proofs made by mathematicians. ...
... In many studies, secondary school students were not successful completing deductive proofs, even with instruction (Clements & Battista, 1992). Even when students seem to understand the function of proofs and to recognize that they must be general, they prefer to rely on empirical methods (Hoyles & Küchemann, 2002) and on a few examples for proving a general claim (e.g., Balacheff, 1988, Harel & Sowder, 2007, Healy & Hoyles, 2000. ...
... Considering our specific context, we distinguish the following constructs: a conjecture is a mathematical statement the veracity of which is doubtful; an argument is a verbalization aimed to explain how a conjecture was identified, to convince that it is plausible, or to be part of a proof; a proof is a mathematical argumentation, not necessarily formal, produced to justify the truth or untruth of a conjecture (Balacheff, 1988;Harel & Sowder, 2007;and Fiallo & Gutiérrez, 2017); proof problems are problems asking to prove a conjecture which may be given in the statement or may have to be found by the solver as part of the solution (Polya, 1945). Marrades and Gutiérrez (2000) proposed a framework to classify students' proofs focusing on their production processes. ...
... In the accessible sources on proof and proving, there were studies conducted with students, teacher candidates or teachers. These studies were conducted to determine the views of students, teacher candidates and teachers about proof, their attitude, proving processes, learning difficulties related to proof, and proof schemes (Recio and Godino, 2001;Knuth, 2002;Almeida, 2003;Raman, 2003;Harel and Sowder, 2007;Keçeli Bozdağ, 2012;Güler, 2013;Karahan, 2013;Yılmaz, 2015;Çontay, 2017;Barak, 2018;Polat, 2018;Yıldız, 2019). In addition, there were limited number studies about proof with gifted students (Sriraman, 2005;Lee, 2005;Yim, Song and Kim, 2008;Lee, Park and Jung, 2009;Uğurel, Moralı, Karahan and Boz, 2016;Öztürk et al., 2017). ...
... So, the gifted students had deficiencies in pre-knowledge and mathematical knowledge. This situation is in line with result of Arslan (2007) (2016), Baker (1996), Boero (1999), Özer and Arıkan (2002), Harel and Sowder (2007), Güler et al. (2011) conducted on students with typical development and Sriraman (2005) conducted with gifted students. They also stated that the students did not prove when they used the trial method, but only confirmed the theorem. ...
... They also stated that the students did not prove when they used the trial method, but only confirmed the theorem. Although some study groups thought that they completed the proof by accepting the trial method (Almeida 1996;Boero, 1999;Harel and Sowder, 2007;Güler et al., 2011;Pekşen Sağır, 2013), gifted students realized that they did not. In addition, most of the time was spent in the stages of determining the appropriate strategies and performing the necessary actions. ...
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.
... This provides the required background to better understand and reflect the different views and usages of proof in recent literature, which I review in the following section. Furthermore, researchers have highlighted the potential relevance of historical developments of proof for the development of students' proof conceptions (e.g., Harel & Sowder, 2007). Lastly, main characteristics and acceptance criteria for mathematical proof are summarized and discussed. ...
... To assess students' evaluation of proof (and other proof-related activities), many researchers in mathematics education refer to mathematicians' conceptions of proof and their respective acceptance criteria as a benchmark (e.g., Dawkins & Weber, 2017;Harel & Sowder, 2007;Stylianides, 2007;Weber, 2013;Weber & Czocher, 2019). Thereby, proving practices in the mathematics classrooms are not expected 52 3 State of Research "to be exact replicas of professional mathematical communities" (Weber & Czocher, 2019, p. 253), but general standards for the acceptance and understanding of proof should be consistent with those of the mathematical community (Dawkins & Weber, 2017;Harel & Sowder, 2007) 7 . ...
... To assess students' evaluation of proof (and other proof-related activities), many researchers in mathematics education refer to mathematicians' conceptions of proof and their respective acceptance criteria as a benchmark (e.g., Dawkins & Weber, 2017;Harel & Sowder, 2007;Stylianides, 2007;Weber, 2013;Weber & Czocher, 2019). Thereby, proving practices in the mathematics classrooms are not expected 52 3 State of Research "to be exact replicas of professional mathematical communities" (Weber & Czocher, 2019, p. 253), but general standards for the acceptance and understanding of proof should be consistent with those of the mathematical community (Dawkins & Weber, 2017;Harel & Sowder, 2007) 7 . Respective acceptance criteria have already been discussed in section 2.3.3. ...
In this open access book Milena Damrau investigates the understanding of generality of mathematical statements in first-year university students and its relation to other proof-related activities. Through an experimental study, she particularly analyses the effect of different types of arguments (empirical, generic, and ordinary proofs) and statements (familiar and unfamiliar, as well as true and false ones) on several proof-related activities. The results reveal students' struggles with the concept of generality, how their understanding of generality is related to proof reading and construction and how different types of arguments and statements impact students’ performance in other proof-related activities. The findings offer valuable insights for improving mathematics courses at the transition from school to university and highlight the need for more experimental studies in mathematics education.
... Students and preservice teachers tend to have different schemes of justification. We use the concept of personal proof scheme (hereafter, PS, Harel & Sowder, 2007), which consists of what constitutes ascertaining and persuading for a person about the validity of a mathematical statement. This is related to the knowledge of what constitutes a valid proof in mathematics. ...
... This is related to the knowledge of what constitutes a valid proof in mathematics. Harel and Sowder (2007) distinguish between three kind of proof schemes: external conviction PS (in which ascertaining and persuading come from reasons other than reasoning), empirical PS (which could be perceptual PS if are based in perception, or inductive PS in which conjectures are validated by the observation and check of one or more specific examples), and analytical PS (which are based on deductive reasoning). ...
... Research about proof and proving processes in prospective teachers found common difficulties like the persistent presence of inductive empirical PS (Harel & Sowder, 2007;Stylianides & Stylianides, 2009) and the knowledge about the role of examples and counterexamples in proving processes (Rodrigues et al., 2021;Stylianides et al., 2016). These difficulties could compromise the learning opportunities of reasoning and proving processes that EPT could design and implement. ...
This paper presents preliminary results of the implementation of a teaching experiment designed to promote reasoning and proving processes with elementary preservice teachers. The teaching experiment deals with the inscribed angle theorem. It has been designed following the principles of task design for conjecturing and proving of Lin et al. (2012), and it tries to generate the intellectual need of deductive reasonings. Using the Mathematics Teachers' Specialised Knowledge (MTSK) model as an analytical tool, we have analysed the productions of the participants to detect and characterise the mathematical knowledge deployed by them during the task. Then, we use these results to reflect about the potential and limitations of the teaching experiment. Here, we present the results of the analysis of a pair of preservice teachers, whose behaviour was more aligned with the objective of the teaching experiment.
... We also noticed that most of the reported instructional strategies are general teaching strategies instead of proof specific instructional strategies. Instructors' demonstration of the proof specific instructional strategies may help PSMTs learn the metacognitive strategies that could guide PSMTs' progress when they construct proofs (Harel & Sowder, 2007). Researchers (e.g, Aricha-Metzer & Zaslavsky, 2019; Leron & Zaslavsky, 2013) have documented the importance of exploring examples in proof constructions. ...
... However, only two instructors mentioned using examples to guide proof construction. Prior literature suggests engaging students in reasoning and proof as a process, rather than as a completed product (Harel & Sowder, 2007;NCTM, 2000). But purposefully allowing mistakes was reported by only three instructors. ...
This study examined opportunities provided for preservice secondary mathematics teachers (PSMTs) to learn reasoning and proof in algebra from the perspective of college instructors. We analyzed interview transcripts of 15 course instructors recruited from three teacher education programs in the United States. We examined the reported opportunities provided for PSMTs to engage in proving- related activities, including making conjectures, investigating conjectures, developing arguments, evaluating arguments, and disproving by using counterexamples. We also analyzed instructional strategies reported by the instructors. We found the inconsistency between instructors’ perceptions of the importance of reasoning and proof in algebra and instructor-reported opportunities to learn. Findings also indicated that developing arguments was reported the most frequently. In addition, instructors reported more pedagogy- focused general teaching strategies than proof-specific teaching strategies.
... Proof plays an important role in mathematics education at all levels, and this has been broadly acknowledged (e.g., Hanna & de Villiers, 2012;Harel & Sowder, 2007;Mariotti, 2006; National Council of Teachers of Mathematics [NCTM], 2000;Reid & Knipping, 2010;Stylianides & Stylianides, 2008Stylianides et al., 2023;. However, there are many difficulties and challenges students 1 face when engaged in proving-related activities. ...
... Some students might assume that examples can never prove or that a general (e.g., algebraic) proof is expected when proving universal statements (e.g., Barkai, et al., 2002;Dreyfus, 2000;Healy & Hoyles, 2000); some might assume that several confirming examples should be rejected as proof of an existential statement (e.g., Tabach, et al., 2010); others might assume that a general argument is needed even when disproving universal statements (e.g., Buchbinder & Zaslavsky, 2019); and some might make extreme assumptions such as there is nothing to be sure of in mathematics (e.g., Stylianides & Stylianides, 2009). These assumptions might be grounded in an external source of conviction (Harel & Sowder, 1998, 2007. They might have been established by an authority (e.g., the teacher, who might have taught students to reject examples as proof) or the appearance of the argument (e.g., algebraic arguments are usually regarded as proofs). ...
Examples play a variety of roles in proving and disproving. Buchbinder and Zaslavsky (2019) have produced an a priori mathematical framework for assessing students’ understanding of the role of examples when proving and disproving universal and existential statements. In this paper, I highlight three important aspects that suggest an extension of Buchbinder and Zaslavsky’s framework. These aspects arose a posteriori from research that had a focus on developing mathematically aligned assumptions related to proof and proving. To better approach students’ understanding of the role of examples in proving and disproving, these three aspects need to be explicitly included in the framework: (1) the size of the domain where universal and existential statements are defined, (2) the different forms of expression in which universal and existential statements can be presented, and (3) the quality of universal and existential statements. Illustrations of the development of in-service primary school teachers’ assumptions are given to show the emergence of these aspects and to support the extension of the existing framework.
... Justifications can be informal or formal (Harel & Sowder, 2007). Formal justifications are justifications that are typically referred to as mathematical proofs, and reflect the rigor and rules used by expert mathematicians when proving. ...
... Moreover, students were able to modify the reasoning when the shapes on the trains changed. Such reasoning is similar to what Harel and Sowder (2007) describe as anticipating transformations and using transformations in the justification process. ...
Standards for mathematics teaching require teachers to employ teaching practices that promote justification of mathematical ideas. This expected teaching practice is situated in substantial research on students’ and teachers’ difficulties with justifying mathematical ideas. This study shows different ways elementary school students in grades three through five may justify mathematical conjectures about pattern-finding activities. It also shows that even when teachers are capable of justifying particular tasks, enactment of such tasks in ways that encourage students to go beyond example-based justifications may be problematic. Video and audiotapes of class activities, students’ written work, and curriculum materials were sources of data.
... In conjunction with policy documents calling for an expanded role of reasoning-and-proving, international researchers have also identified reasoningand-proving as an area of focus (Hanna & de Villiers, 2012). Some researchers paid attention to teaching on reasoning-and-proving (Harel & Sowder, 2007;Krummheuer, 2007;Stylianides, 2016), and other researchers focused on textbook analysis (Bieda et al., 2013;Fujita & Jones, 2014;Stylianides, 2009). ...
... In China, in the mathematics subject, mathematics textbooks are teachers' primary references for their daily teaching activities, as well as the main resources from which students learn mathematics (Zhang & Qi, 2019). Previous studies have shown that many students in different countries face severe difficulties with reasoning and proving (Fujita & Jones, 2014;Harel & Sowder, 2007). Some international researchers thought that the main reason why elementary students did not perform well in reasoning-and-proving tasks was that they were not given enough opportunities to engage in reasoning-and-proving activities at the elementary level (Bieda et al., 2013;Stylianides, 2016). ...
Mathematics textbooks play an essential role in providing students with learning opportunities for reasoning and proving (RP). Researchers showed that tasks used by teachers were directly related to those provided by mathematics textbooks. Yet, little is known about the learning opportunities offered by elementary textbooks for students to learn RP in China. This study analyzed six volumes of New Century Mathematics (NCM) textbooks, focusing on RP’s opportunities in written tasks based on a revised analytical framework. The results showed that the textbooks we analyzed in this study provide students with more opportunities to engage in the RP of the algebraic domain, especially in narrative sets. In addition, in terms of sub-components and representation, the opportunities provided by textbooks were different in narrative sets and exercise sets.
... One of the methods to obtain answers to these problems is to examine the justifications students provide for their solutions to the problems. Mathematical justification is vital to determining and explaining the veracity of a mathematical assumption or claim (Balacheff, 1988;Harel & Sowder, 2007;Simon & Blume, 1996). Staples and Bartlo (2010) state that asking students for justifications encourages them to think more deeply about mathematical concepts, as they grapple with various ideas and look for mathematical connections to generate new insights. ...
... Açık uçlu sorular öğrencilerin yalnızca ne bildiklerinin değil, yaptıkları gerekçelendirmeler yolu ile eksiklerinin neler olduğu, sahip oldukları kavram yanılgıları ve yaşadıkları matematiksel zorlukların ortaya çıkarılmasına da olanak tanımaktadır. Matematiksel gerekçelendirme, matematiksel bir varsayımın veya iddianın doğruluğunu belirlemek ve açıklamak için önemli bir enstrümandır (Balacheff 1988;Harel & Sowder, 2007;Simon ve Blume, 1996). Staples ve Bartlo (2010) öğrencilerden gerekçelendirme talep edildiğinde, çeşitli fikirlerle boğuşarak yeni fikirler elde etmek için matematiksel bağlantılar araması gerektiğinden öğrencilerin matematiksel kavramlar üzerinde daha derin düşünmeye yönlendirildiğini ifade etmektedir. ...
This study aims to explore the strategies that secondary school students employ and the difficulties they encounter when solving area measurement problems. The participants consist of 75 seventh and eighth-grade students from southeast Turkey. Data were obtained through a form comprising six open-ended problems, designed to uncover the “nature of justifications”. Analysis of the students' responses revealed 11 distinct strategies and 11 difficulties. The most frequently employed strategies for solving area problems were reasoning through drawing shapes and applying the area formula (axb). Students struggled the most with distinguishing changes in the area from changes in the perimeter. It was observed that the root of the difficulties experienced by the students was challenges in measuring length. Notably, when presented with contextual problems, students focused on the context and justified their solutions based on cultural factors. As such, it is recommended that the process should be designed while considering cultural factors (both facilitators and inhibitors) in teaching subjects such as area measurement, which are closely related to real life.
... This description is very different from the way mathematics is traditionally taught as a collection of terms, rules and formulae that need to be memorized (Thompson, 1985), and that exists in a finalized form. Knowing is closely related to deductive proofs (Harel & Sowder, 2007;Weber, 2004), which play the role of technical tools for verifying the correctness of statements proposed by the teacher (instead of proofs that explain), which are already known to be correct. ...
... The core ideas and principles that serve as a basis for warranting or connecting claims in mathematics are tree-like structure rooted in axioms and basic (undefined) concepts (Harel & Sowder, 2007). Progressive mathematics classrooms value vertical mathematization, by which students undertake a reorganization within the mathematical system resulting in insights based on connections between concepts and strategies. ...
... Pedersen et al. (2021) suggest that, to address such issues, task designers could require students to predict changes in these representations when using digital technology. Moreover, students' justifications tend to rely on empirical knowledge (Harel & Sowder, 2007) or phenomenological evidence (Baccaglini-Frank, 2019). This tendency is enhanced by the dynamic properties of environments, which allow students to interact and observe representations that appear as real virtual objects that can be experienced phenomenologically (Baccaglini-Frank, 2019; Leung & Chan, 2006). ...
... In all mathematical reasoning, arguments are put forward to change the epistemic value (the degree of certainty) of a claim (Duval, 2007). The epistemic value can be considered from the perspective of the reasoner or the general mathematics community (Duval, 2007;Harel & Sowder, 2007;Jeannotte & Kieran, 2017;Knuth et al., 2019). ...
The study advances the instrumental approach to mathematics education (Drijvers et al., 2013; Trouche, 2003), aiming to elucidate the interplay between students’ reasoning competency, conceptual knowledge and tool utilisation in dynamic digital geometry and algebra environments. The dynamic properties of these environments pose a nuanced predicament, as the outsourcing of translation between visual and algebraic representations raises concerns regarding students’ conceptual development and reasoning competency. To mitigate this issue, a prediction task is proposed, focusing on the dynamic behaviour of variable points in GeoGebra. I introduce a comprehensive framework adapting Toulmin’s argumentation model into the instrumental approach, emphasising processes of justification. This is complemented by the application of components of Vergnaud’s (1998) scheme concerning generative and epistemic ways to approach how students’ conceptual knowledge has played a part in these processes. Through a case study of a student pair solving a prediction task, I explore the links between instrumented justification, students’ mathematical reasoning competency and conceptual understanding, and how students’ use of GeoGebra tools is intertwined with their justification processes. The analysis reveals the intricate interplay between data production and interpretation, and it is grounded in inference drawn regarding students’ implied theorems about concepts, dynamic behaviour and progression in terms of techniques. The results indicate that the progression of technique is driven by the experience of the inefficiency of techniques and artefacts related to the goal of justification. Essentially, the framework links students’ reasoning competency to their use of tools and conceptual knowledge, as well as demonstrates that predicting dynamic behaviour can enhance knowledge-based justification.
... Harel and Sowder [7] offer a thorough analysis of the proof. The development of pupils' capacity to "reason deductively" is an aim shared by mathematics curricula around the globe. ...
... From the reasoning perspective of the introduction of the Pythagorean theorem in each version, the PEP involves "reasoning that relies on the area of a special case" and "based on general model area inference". They fall under the categories of "empirical reasoning" and "deductive reasoning," respectively, according to Harel and Sowder [7]. The PEP of the Pythagorean Theorem is introduced in a way that moves from the specific to the general, first examining the relationship between the three sides of isosceles right triangles before moving on to right triangles in general. ...
Education has always been a hot topic of discussion. From the results of junior high school mathematics assessments in recent years, there are differences in mathematical abilities between Eastern Europe and Europe and America. Since reasoning is crucial to understanding mathematics, we compare the "Pythagorean Theorem" in textbooks from China, the United States, and Singapore, examining the theorem's different types of reasoning and how it relates to previously acquired knowledge. The reasoning categories proposed by Shi Ningzhong, Harel, and Sowder serve as the fundamental basis for categorizing the varieties of reasoning in this research. The introduction of proofs for the hook and Pythagorean theorem is investigated for each edition of the textbook, and concerning the deductive reasoning ability, the People's Education Press edition (PEP) and Zhejiang Education edition (ZJE) editions are in the first tier, followed by Singaporean textbooks and American textbooks. In terms of the correlation between the Pythagorean theorem and what has been learned, the ZJE involves the most points of correlation. Finally, we discuss the implications of these results for teachers' explanations of the Pythagorean theorem: teachers can briefly introduce the geometric deductive proof process of the Pythagorean theorem to stimulate students' interest in learning mathematics.
... En este último subdominio, KPM, es donde se sitúa el conocimiento de la práctica de demostrar, que incluye las formas de generar y validar conocimiento en matemáticas, como el establecimiento de patrones, la construcción de conjeturas, el rol de los ejemplos y contraejemplos, los métodos y tipos de demostraciones o las funciones de la demostración (Delgado-Rebolledo et al., 2022). En general, el aprendizaje de estos procesos es complejo, para alumnos y, también, para profesores en formación (Harel y Sowder, 2007;Stylianides y Stylianides, 2009). ...
... Un prerrequisito para ello es que el docente de matemáticas posea un conocimiento sólido sobre los mismos lo que, como hemos comentado, no es un hecho sencillo puesto que tanto la experiencia docente como la investigación atestiguan la complejidad que, en muchos casos, tienen estos procesos. Una de las dificultades más habituales es la presencia de un esquema de prueba inductivo, es decir, la consideración (errónea) del razonamiento inductivo (la comprobación en ejemplos concretos) como generador de pruebas válidas en matemáticas y como proceso que genera convencimiento y persuasión a una persona sobre la veracidad y validez de un enunciado (Harel y Sowder, 2007). Esta dificultad también aparece como muy arraigada entre los futuros profesores de matemáticas (Arce y Conejo, 2019; Stylianides y Stylianides, 2009), siendo necesario trabajar en la formación inicial para intentar su superación. ...
Una de las competencias específicas en matemáticas que establece el nuevo decreto LOMLOE es la de razonamiento y prueba. Diseñar y gestionar situaciones de aprendizaje donde los procesos de razonamiento y prueba tengan una presencia relevante precisa de que los docentes tengan un conocimiento sólido de los mismos. Estos procesos, que suelen resultar complejos, involucran dos grandes fases, conjeturar y demostrar. Se presentan aquí una serie de tareas diseñadas e implementadas con estudiantes para maestro de Educación Primaria, pero que son también aplicables en aulas de últimos cursos de Primaria y de Secundaria, para fortalecer y reconstruir su conocimiento sobre estos procesos. Estas tareas están centradas en contenidos geométricos, como las relaciones angulares o las propiedades de los polígonos, y hacen uso de las potencialidades de GeoGebra tanto para conjeturar como para transitar hacia la elaboración de demostraciones.
... Bununla birlikte Bartlo (2013), yaptığı çalışma sonucunda De Villiers'ın ispat işlevlerinden beşini Tablo 1'de gösterildiği gibi alt işlevleri ile tanımlayarak detaylandırmıştır: Tablo 1. İspat İşlevleri ve Alt İşlevleri (Bartlo, 2013) Öğrencilerin ispatın işlevlerini deneyimlemeleri, bir yandan ispatı anlamalarını ve ispat uygulamalarına katılımlarını desteklemekte (Hemmi, 2010) öte yandan ispatın matematik biliminde oynadığı rolün matematik sınıflarına yansımasını sağlayarak bu sınıflarda ispatın daha anlamlı bir etkinlik olmasının önünü açmaktadır (Hanna, 1995;Knuth, 2002). İspatın bir topluluğu matematiksel bir ifadenin doğruluğuna ikna etmede bir araç olarak kullanıldığı ve ikna edici olabilmesi için o topluluğunun normlarıyla uyumlu olması gerektiği belirtilmektedir (Harel ve Sowder 2007;Zaslavsky, Nickerson, Stylianides, Kidron ve Winicki-Landman, 2012). Benzer şekilde sınıf topluluğunun bir iddianın doğruluğunu onaylayabilmesi için yapılan ispatın sınıfın sosyal ve sosyo-matematiksel normlarına uygun olması gerekmekte ve zaman içinde ispat, bu normları değiştirmekte ve şekillendirmektedir. ...
z Anahtar Kelimeler İspat ile matematiksel önermelerin doğruluğunun gösterilmesi işlevinin yanı sıra açıklama, keşif, sistematikleştirme ve iletişim gibi işlevleri de ortaya çıkar. İspat öğretiminin sınıflarda anlamlı ve derin bir matematik etkinliğine dönüşmesi için tüm işlevlerinin işe koşulması önemlidir. Matematiksel ispat ve muhakeme sosyal bir süreç olduğundan özellikle ortaokul düzeyindeki öğrencilere ispatın önemi ve gerekliliği gösterilirken öğretim birtakım normlardan etkilenir. Dolayısıyla öğretmenlerin hem ispat işlevlerini hem de sosyal ve sosyo-matematiksel normları dikkate alması öğrencilerin muhakeme süreçlerini destekler. Bu doğrultuda bu araştırmanın amacı; öğrencilerin etkileşim kurarak ispat problemlerini çözebilmelerine imkân tanıyan bir öğrenme ortamında ispat işlevlerini, sosyal ve sosyo-matematiksel normları ve ispat işlevleri ile normlar arasındaki ilişkiyi ortaya çıkarmaktır. Çalışmanın katılımcıları İç Anadolu Bölgesi'nde yer alan bir devlet ortaokulunda öğrenim gören 7. sınıf öğrencilerinden oluşmaktadır. İspat işlevlerinin ortaya çıkarılması ve normların oluşturulması gerçek bir öğrenme ortamında uzun süreli etkileşimi gerektirdiği için bu çalışmada öğretim deneyi yöntemi benimsenmiş ve 12 haftalık öğretim sürecinden elde edilen video ve ses kayıtları ile çalışma kâğıtları analiz edilerek bulgular sunulmuştur. Araştırmanın sonuçları sınıf topluluğunun tartışma, problem çözme, gerekçelendirme ve iş birliği konularında ortak normlar geliştirdiğini, bu sosyal ve sosyo-matematiksel normların öğrencilerin tartışmalara katılımını yönlendirdiğini ve katkılarının kalitesini artırdığını göstermiştir. Ayrıca araştırmada normlar ile ispat işlevlerinin diyaloglarda iç içe geçtiği, hem farklı temalardaki normların hem de ispat işlevlerinin birbirleri ile ilişkili oldukları görülmüştür. Araştırmada tartışma ve iş birliği temasındaki normların özellikle ispatın iletişim işlevini desteklediği, gerekçelendirme ve problem çözme temasındaki normların ise İspat işlevleri Matematiksel ispat Matematiksel muhakeme Sınıf mikrokültürü Sosyal normlar Sosyo-matematiksel normlar
... Incorporating AI-powered tools into informatics education can also help address some of the challenges facing informatics educators in keeping the curriculum up-to-date, as discussed in previous literature [28,29]. For instance, it can help bridge the gap between abstract informatics concepts and their application to real-world problems, which is a key area of concern in informatics education [30]. Overall, this study highlights the potential of artificial intelligence in enhancing informatics learning and logical thinking skills of future informatics teachers. ...
Despite the increasing integration of artificial intelligence (AI) in education, its potential to enhance cognitive thinking skills—particularly logical thinking skills—among future informatics teachers remains underexplored. Addressing this gap, the present study examines the effectiveness of AI-based instruction in developing logical thinking and problem-solving abilities. Conducted within the broader context of educational modernization in Kazakhstan, the study investigates how AI tools influence students' intuitive understanding of abstract informatics concepts. A randomized controlled trial was conducted with 48 future informatics teachers from a university in Kazakhstan, divided into experimental and control groups. The experimental group received training using AI tools, while the control group followed traditional informatics instruction. The results indicate that the experimental group outperformed the control group in problem-solving tasks and exhibited significantly greater logical thinking skills. These findings highlight the potential of AI-based instruction in improving cognitive competencies essential for future educators in informatics. The study emphasizes the importance of integrating artificial intelligence in informatics education to promote logical thinking skills in future educators. It advocates for targeted support and training initiatives that will enable informatics teachers to use artificial intelligence tools effectively. This study contributes to the ongoing discourse on improving the quality of education in Kazakhstan and similar contexts through innovative pedagogical approaches. Finally, the study highlights the need for further research to investigate the long-term impact of artificial intelligence on teachers' pedagogical practices and student learning outcomes.
... Strategic competence refers to the ability to formulate and solve problems while adaptive reasoning involves intuitive and inductive thinking, causal explanations and justifications by deductive approach. The work done on these strands show their relevance in frameworks that seek to understand students' ideas about evidence [9] as well as research that examines practices that sustain reasoning from justification [24]. Learning in higher education develops mathematical thinking skills to make students capable of presenting their ideas and arguments effectively. ...
This study explores the impact of digital textbooks, specifically Kognity, on enhancing Mathematical Thinking Skills among Grade 12 students in Shanghai, China. Conducted within International Baccalaureate Diploma Programme (IBDP) Mathematics: Analysis and Approaches – Higher Level class, the research involved 44 students who participated in a Calculus course. A quasi-experimental design was employed, with pretest and posttest assessments comparing the performance of a control group receiving traditional lecture-based instruction and a treatment group utilizing Kognity. The pretest results in the control group showed that 47.62% achieved a grade of 5, while the posttest revealed 47.62% earned a grade of 6. In contrast, the treatment group exhibited remarkable improvement: pretest results showed 30.43% attaining a grade of 6 or higher, compared to 100% in the posttest. Mean pretest scores and standard deviations for the control group were 40.79 and 14.90, respectively, increasing to 60.95 (SD = 11.98) in the posttest. Meanwhile, the treatment group’s mean pretest score of 49.57 (SD = 16.55) improved significantly to 83.77 (SD = 13.72) post-intervention. Statistical analysis revealed highly significant differences, with a calculated t-value of –12.12 (p < 0.001) within the treatment group and –22.82 (p < 0.001) between the groups. These results suggest that integrating Kognity not only enhances students’ problem-solving and abstract reasoning skills but also significantly outperforms traditional methods. However, limitations such as non-randomized sampling, contextual variables, and the absence of long-term retention data are acknowledged. Recommendations include further investigation using randomized trials and longitudinal studies to better understand the broader implications of digital learning tools on mathematics education.
... This seemed to increase Ella's motivation for engaging in mathematical reasoning. Looking at the literature about reasoning in mathematics education, it has mainly focused on characterising the different types of argumentation and reasoning (Brousseau & Gibel, 2005;Harel & Sowder, 1998, 2007 and less so about what makes students engage in reasoning processes and how to make the transition from making an argumentation based on rationales and intuition to more deductive reasoning (EMS, 2011). We suggest, however, that Ella's inclination towards more deductive reasoning was spurred by a cognitive conflict and that deeper and more articulated conflicting understandings were the larger the internal motivation for addressing this conflict with mathematical reasoning. ...
Students’ independent mathematical inquiry is often endorsed as a valuable teaching method. In this article, we scrutinise in what ways these independent situations entail the students’ development of mathematical reasoning. We study the cognitive conflict in one fifth-grade class participating in an inquiry-based intervention study. The findings indicate that cognitive conflicts can support the students’ reasoning processes and that the environment has an important role in retaining the conflicting positioning by making the cognitive conflicts available for discussion and scrutiny. The students’ processes of resolving cognitive conflicts are stretched over time and involve different routes and exploring approaches and understandings.
... The finger is often pointed to the high school curriculum not providing sufficient preparation for students in this area. Studies indicate that students in high school have great difficulty with proof; see for example, Harel and Sowder (2007); Stylianou et al. (2009) for reviews. Further, high school teachers find teaching proof challenging (Knuth, 2002;Wu, 1996). ...
... Some very important issues are consensually clarified. One of these issues concerns the concept of proof in mathematics research and education which has stimulated a wide range of perspectives (e.g., Harel and Sowder 2007;Knipping et al. 2019;Mariotti et al. 2018;A. J. Stylianides and Harel 2018;) that cannot be displayed in this paper. ...
Selecting a proof for teaching is a frequent task for teachers. However, it is so far unclear, which factors are considered by teachers when selecting proofs. Is the selection based on task and proof characteristics such as the didactical type of proof? Or based on class characteristics such as students’ algebraic skills? Or do teachers’ characteristics such as their proof skills govern their decision? Or is the selection too non-generic for these characteristics to show a meaningful impact?
To address these questions, a quasi-experimental study with N = 183 pre-service teachers was conducted to evaluate the influence of each of these factors on their selection of proofs for teaching. Results highlight several significant effects of the abovementioned characteristics and underline that—even at the pre-service level—the selection of proofs is more nuanced than often assumed in prior research and that teachers deliberately and adaptively select proofs for their teaching based on these factors.
... The results highlight the importance of establishing socio-mathematical norms that support argumentation, such as making it normative that students present warrants and agreeing that empirical arguments are not acceptable proof. In this study, justification was expected, and students started to move away from empirical to more deductive arguments, an important goal of instruction (e.g., Harel & Sowder, 2007;Inglis et al., 2007). Teachers may need to promote more empirical reasoning in classrooms. ...
Studies show that supporting argumentation in mathematics classrooms is challenging for teachers. This study reports on the experience of seven elementary and middle-school mathematics teachers in an afterschool classroom-based program where they had the opportunity to facilitate research sessions on students’ development of mathematical ideas and different types of reasoning and justification. The study examines the teachers’ support of argumentation as students work collaboratively on challenging mathematical tasks. The results describe pedagogical strategies teachers can use to support students’ development of valid mathematical arguments and types of mathematical knowledge for teaching (MKT) that influence teachers’ supporting efforts.
... Dies könnte aber daran liegen, dass viele Studierende keinen Beweis benötigen, um von der Allgemeingültigkeit einer Aussage überzeugt zu sein. Meist reicht bereits die Aussage einer Autoritätsperson, wie sie es aus der Schule kennen, oder die Aussage anhand von einigen Beispielen zu überprüfen (Brunner, 2014;Harel & Sowder, 2007). ...
Vorgegebene Beweise bieten eine wichtige Lerngelegenheit im Mathematikstudium, die aber vermutlich nur wenige Studierenden ausreichend nutzen. Daher wurden Studierende gefragt, inwiefern sie vorgegebene Beweise als nützlich empfinden, und mithilfe einer qualitativen Inhaltsanalyse ermittelt, wie die Studierenden die Nützlichkeit begründen. Eine Mehrheit der Studierenden empfindet vorgegebene Beweise als nicht nützlich oder beschreibt eine extrinsische Motivation für die Auseinandersetzung mit diesen. Diese Ergebnisse sowie die teilweise sehr oberflächlichen Begründungen zeigen die Schwierigkeit von Studierenden auf, Nützlichkeit von Beweisen und generell Hochschulmathematik eigenständig erkennen zu können.
... This leitmotiv is reflected in some curricula (e.g., NCTM, 2000 for the US; Department for Education, 2013 for the UK; Leikin & Livne, 2015 for Israel), which cannot be taken for granted considering the marginalization of the activity in the curricula of other countries (e.g., Hanna, 2000;Knox & Kontorovich, 2022). The other leitmotiv pertains to how challenging proof is for newcomers (e.g., Harel & Sowder, 2007). This finding comes from many studies, including those involving high achievers in school and future mathematics majors (e.g., Kontorovich & Greenwood, 2023;Stylianides & Stylianides, 2022;Weber, 2010). ...
Coming from the commognitive standpoint, we consider proof-based mathematics as a distinct discourse, the transition to which requires special rules for endorsement and rejection of mathematical statements. In this study, we investigate newcomers’ learning of these rules when being taught them explicitly. Our data come from academically motivated high-school students who took a special course in undergraduate mathematics. The course teacher dedicated three academic hours to introducing and explaining selected rules of proof to support students’ shift to the new discourse. The homework assignment consisted of typical proof-requiring problems and a scriptwriting task, asking students to compose a dialogue between fictional characters about a proof-related mistake of their choice. We analyzed the differences and similarities between the rules discussed in the classroom and those that students addressed and implemented in their proofs. The analysis showed that while students’ solutions to proof-requiring problems required rule implementation, fictitious dialogues opened the space for rule formulation and substantiation. In many cases, the students discussed the rules presented in the classroom, extending, elaborating, and specifying the teacher’s formulations. Furthermore, while the students’ proofs were mainly consistent with the teacher’s expectations, some of their rule formulations were more radical and overgeneralized than expected. These findings suggest that newcomers’ communication about the rules of proof may lag behind their capability to implement those rules to prove mathematical statements.
... As a topic of advanced mathematics, proof is steeped in abstractions and formalizations intended to make universal claims about shape and space that go well beyond any one person's range of personal experiences. Furthermore, developing proof performance does not rest on learning of wellrehearsed procedures or algorithms; rather, it requires the development of one's mathematical intuition along with logical, generalizable, and goal-directed, operational thought (Harel & Sowder, 2005). ...
... We excluded one case of non-compliance, where the participant was prompted to use gestures but did not, leaving the total observations 159. As outcome measures of geometric reasoning, one of the authors transcribed and coded for intuition (The immediate assessment of the conjecture's truth value is correct), insight (The explanation is only composed of the key mathematical ideas that are relevant to prove the conjecture correctly) and proof (The explanation is general, logical, and operational (Harel & Sowder, 2007)) as binary variables. The three codes captured different levels of rigor in proof. ...
... Considering the roles of proof in mathematics, Hanna (2000) asserted that mathematicians prefer proofs that explain to proofs that "just prove", and that the main role of proof in mathematical classrooms is explanation, which is strongly related to the notion of conviction. Convincing and removing doubt in the truthfulness of a statement have been considered by many as important, even primary goals of proving (Brown, 2018;Harel & Sowder, 2007). Still others emphasize that achieving psychological conviction is not a necessary requirement of proof; there are convincing arguments that are not proofs, and valid deductive proofs that bestow little conviction. ...
... En varias de estas investigaciones se entiende a la argumentación como el discurso o medios retóricos utilizados por un individuo o un grupo para eliminar sus dudas o convencer a otros acerca de la verdad o falsedad de una afirmación. En este sentido, la argumentación se sitúa en una actividad matemática que puede incluir la exploración de ejemplos, la generación o el refinamiento de conjeturas y la producción de argumentos para estas conjeturas, que no necesariamente califiquen como pruebas, pero que apoyen su desarrollo (Harel y Sowder, 2007;Stylianides et al., 2016;Zaslavsky, 2014). ...
(Objetivo)
Este trabajo tiene como objetivo indagar los significados sobre conjetura en estudiantes de la escuela secundaria, poniendo de manifiesto algunas disparidades entre estos significados personales y el que adquiere en la institución matemática.
(Metodología)
La metodología utilizada es de tipo cualitativo. Se efectúa un estudio de casos, a fin de profundizar los significados personales de conjetura en estudiantes del ciclo orientado de escuelas secundarias de la ciudad de Río Cuarto, Argentina (15-16 años). Para ello, se analizan los sistemas de prácticas que ponen a funcionar un grupo de estudiantes ante tres problemas que involucran la elaboración, contrastación, reformulación o validación de conjeturas, incluyendo entrevistas realizadas a ellos mismos. Estos análisis se llevan a cabo utilizando herramientas del enfoque ontosemiótico, con el cual se determinan configuraciones de objetos primarios y procesos cognitivos en estas prácticas personales.
(Resultados)
Se logran identificar conflictos semióticos en dichas prácticas, particularmente, disparidades entre significados personales de conjetura y el significado pretendido desde la institución matemática, hecho que pone en evidencia, a través de indicadores empíricos, el valor epistémico otorgado por los estudiantes a las proposiciones emergentes de sus prácticas argumentativas. Esto obstaculiza la posibilidad de dudar sobre el alcance general de sus afirmaciones y la necesidad de plantear otro tipo de argumentación, tal como se pretende desde la institución matemática.
(Conclusiones)
Esta investigación revela la importancia de desvelar la complejidad ontosemiótica de las prácticas personales relacionadas con la formulación y validación de conjeturas y de poner al descubierto los conflictos semióticos cognitivos vinculados a dichos significados personales.
... Proving (can also be referred to as justifying or convincing) is defined as finding and expressing why something is true (Öztürk, 2013). It is the mental activity that individuals engage in to eliminate doubts about the accuracy of a claim (Harel & Sowder, 2007). ...
Mathematical thinking is a higher-order thinking style specific to mathematics that allows the solving of problems. In this context, it is inevitable to consider mathematical thinking in determining giftedness specific to mathematics. How superior mathematical thinking should be measured and the potential of different environments to elicit this mathematical thinking are a matter of debate. In this study, it was investigated how mathematical thinking in gifted students differed between using a paper and a pencil and using dynamic geometry software. Three gifted students’ solutions for given tasks in the paper-and-pencil and GeoGebra environments were compared within the scope of sub-dimensions (specializing, generalizing, conjecturing, and proving) of mathematical thinking. As a result of this study, the work undertaken by the students in the specializing step were seen to be similar in both the P&P and GeoGebra environments. On the other hand, it can be said that GeoGebra had the potential to reveal high-level work at the generalizing step. Different environments seemed to be important in revealing the ability to make assumptions. And it was seen that higher-order thinking skills for proof can be revealed with GeoGebra.
Sekizinci sınıf öğrencilerinin kanıt şemalarının matematiksel bilginin türü açısından incelendiği bu araştırmanın modeli durum çalışması olarak belirlenmiştir. Çalışma grubunu, 2020-2021 yılında Afyonkarahisar merkez ilçesinde bulunan iki okulda öğrenim gören 36 sekizinci sınıf öğrencisi arasından ölçüt örneklemesi yöntemiyle seçilen 6 öğrenci oluşturmaktadır. Elde edilen verilerin analizinde, betimsel yöntem kullanılmıştır. KİBS' ten edilen verilere göre; kavramsal bilgiye sahip öğrencilerin denklem çözme performanslarının daha yüksek olduğu ve öğrencilerin çoğunun kavramsal bilgiye sahip olamadıkları belirlenmiştir. Yarı Yapılandırılmış Mülakatlar sonucunda öğrencilerin denklemler çözümlerini kanıtlamakta yetersiz kaldıkları ve gerekçe olarak otoriter kanıt şemalarını kullandıkları görülmüştür. Sonuç olarak öğrencilerin denklemlerde eşitlik bilgisine kavramsal olarak sahip olsalar da bu bilgilerini ispat yaparken kullanamadıkları belirlenmiştir. Öğrencilerin kavramsal bilgilerini geliştirmek ve matematiksel olarak geçerli kanıt yapabilmelerini sağlamak için problem çözümünde buldukları sonuçların ve yaptıkları işlemlerin nedenleri üzerine tartışmaları sağlanabilir.
Argumentation is fundamental and essential in mathematics education. It promotes deep mathematical understanding and helps students connect abstract ideas logically. However, teachers and students in the traditional teacher-centered classroom face difficulties teaching and learning math argumentation. In this article, a cooperative conversation-based tutoring system to foster argumentative skills in learning Pythagorean Theorem was designed. The study analyzed results from an experiment involving 118 middle school students in Taiwan who engaged in formulating, validating, generalizing, and justifying learned mathematics concepts with the help of virtual tutors and peers. The control group included 82 middle schoolers who received the same learning content in the teacher-centered classroom setting. Results confirmed that this agent system promoted significantly better learning outcomes, and the learners’ argumentative experience was also enhanced. Moreover, the unique interactions that took place between each student and the agent system resulted in numerous opportunities for learning mathematical argumentation through an adaptive learning mode.
Mathematical induction is a powerful method of proof, taught in most undergraduate programs involving mathematics and in secondary schools in some countries. It is also commonly known to be complex and difficult to comprehend. During the last five decades, mathematics education research has produced numerous studies on the learning and teaching of mathematical induction. In this systematic literature review, we consider important features of this body of research. First, we investigate how the nature of mathematical induction is presented in the research literature. Then, we synthesize the findings of the research. In particular, we pay attention to the approaches and findings of teaching interventions and establish the current state of the art.
The significance of mathematical reasoning skills is often highlighted in national and international curricula. In recent years, the process aspect of mathematical reasoning has been examined through comparison, generalization, and justification. Emphasizing these process abilities is crucial for creating learning settings that develop mathematical thinking and enhance teacher's understanding. This study assessed middle school students' comparation, generalization, and justification within reasoning activities. The participants were 27 sixth-grade students engaged in a mathematical reasoning workshop. The research data were gathered via a reasoning activity including three open-ended sub-problems addressed by the students. The data were analyzed using content analysis. The results showed that middle school students were capable of comparison, although they had difficulties in generalization and justification. Upon comprehensive evaluation, it was concluded that the number of students who completed these three steps cohesively was considerably low.
Bu çalışmanın amacı ilköğretim matematik öğretmeni adaylarının cebir ve geometri alanında yaptıkları ispatların özelliklerini ortaya çıkarmaktır. Bu bağlamda öğretmen adaylarının cebir ve geometri alanında yaptıkları ispatların; ispat yapılarına ve şemalarına odaklanılmıştır. Araştırmanın çalışma grubunu Türkiye’deki bir devlet üniversitesinin ilköğretim matematik öğretmenliği bölümü dördüncü sınıfında öğrenim gören 29 öğretmen adayı oluşturmaktadır. Araştırmanın verileri Cebir-Geometri İspat Formu aracılığı ile toplanmıştır. Formda öğretmen adaylarının cebir ve geometri alanından ispat yapmaları gereken iki açık uçlu soru yer almıştır. Bu ispatların çözümlenmesinde betimsel analiz kullanılmıştır. İspat yapılarının analiz sonuçları ilköğretim matematik öğretmen adaylarının hem cebir hem de geometri alanındaki ispatlarda çoğunlukla tümevarımsal ve yapısal-sezgisel yapıda ispat üretebildiklerini, her iki alanda da çok sınırlı sayıda tümdengelimsel yapıda geçerli ispat üretebildiklerini ortaya koymuştur. İspat şemalarının analizi öğretmen adaylarının cebir alanındaki ispatlarda çoğunlukla tümevarımsal ve referanssız-sembolik ispat şemalarını kullandıklarını, geometri alanındaki ispatlarda ise çoğunlukla tümevarımsal ve algısal ispat şemalarını kullandıklarını ortaya çıkarmıştır. Bu sonuçlar öğretmen adaylarının hem cebir hem de geometri alanında ispat yapmada başarısız olduklarını gözler önüne sermiştir. Cebir ve geometri alanlarındaki ispatlar arasındaki yapısal bütünlük durumları incelendiğinde bu iki alandaki ispatlar arasında yapısal sürekliliğin olmadığı, çoğunlukla yapısal mesafenin olduğu bunu da spantone sürekliliklerin takip ettiği belirlenmiştir.
Like in proving, teachers have an important role in critical thinking because critical thinking is not an innately given skill, rather, it is acquired later in life. However, at this point, educators have a lot of responsibilities because if the individual can transfer critical thinking to his life at school age, he can live his life more meaningfully, be inquisitive and find solutions to problems after school. For this reason, the opinions of teachers and hence prospective teachers concerning both critical thinking and doing proof are important. In this study, survey and correlational research methods were used together since it was intended to explore the views of pre-service elementary mathematics teachers regarding critical thinking and doing mathematic proof and to understand whether these two correlate. The study was carried out on 158 pre-service elementary mathematics teachers enrolled in a state university in the Eastern Black Sea Region of Turkey. According to the results, the pre-service teachers in this study have a low level of critical thinking disposition. service teachers depending on the grade level at which they were studying. When the participating pre-service teachers’ opinions on mathematical proof were examined, it was noticed that mental process was the highest while self-confidence was the lowest
In this paper, we present a qualitative study on what values students perceive in their abstract algebra course. We interviewed six undergraduates early in their abstract algebra course and then again after their course was completed about what motivated them to learn abstract algebra and what value they saw in the subject. The key finding from the analysis was that participants found intrinsic value (i.e., their enjoyment of the subject) to be essential to learning abstract algebra. While participants desired utility value in the form of mathematical applications, they ultimately did not find this necessary to learn abstract algebra. Finally, some participants had different motivations for learning abstract algebra than for learning other branches of advanced mathematics, such as real analysis, suggesting that motivation research in mathematics education should not treat mathematics as a unitary construct. We offer analysis about how the nature of advanced theoretical proof-oriented mathematics may have contributed to these findings.
Fredriksdotter, H. 2024. Young students' mathematical argumentation in social interaction. Video-based observations of student-student interaction during everyday work in the mathematics classroom. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Educational Sciences 34. 110 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-513-2099-1. Previous research indicates that students benefit from engaging in mathematical problem-solving activities together with peers. The aim of this thesis was to increase the knowledge of how social interaction can contribute to shaping young students' mathematical argumentation. The analysis was based on a dialogical perspective on communication. In particular, an ethnomethodological approach was applied to the analysis of students' social interaction while engaging in discussions about solutions to mathematical tasks. Students' contributions to interaction were analysed using Conversation Analysis and multimodal analysis. In addition, the contents of students' explanations, justifications and generalisations were analysed according to procedures of qualitative content analysis. The empirical material consisted of video recordings of naturally occurring interaction during mathematics lessons in two grade-6 classrooms (i.e., among students who are 11-12 years old). Findings were presented in four studies. Study I indicated that the mathematical argumentation among students working in the same classroom can orient towards very different social and sociomathematical norms. Study II focused on students' use of different types of justifications, showing that their general arguments consistently built on (and agreed with) results of preceding examinations of particular examples. In Study III, students' strategies of handling differing proposals were analysed, which showed that students often solicited explanations of peers' proposals by commenting on or asking questions about them without explicitly criticising them. Moreover, when students conceded to someone else's proposal and rejected their own, concessions and rejections were marked by affect-laden and/or embodied acts, indicating an urgency to display a change of state. In addition, marking their concessions may be part of students' ways of displaying independent epistemic access to the mathematical task as well as to the differing proposal. Focusing on students' methods of co-constructing general arguments, Study IV confirmed the importance of having access to and building on others' arguments. In addition, Study IV showed how the use of linguistic resources can indicate that students have identified regularities and/or transferred known mathematical facts into a new context. The detailed analysis of students' argumentation while engaging in mathematical problem solving with peers emphasised the reflexive relation between "social" and "mathematical" aspects of interaction in the mathematics classroom. The analysis also exemplified how young students' use of justifications can be a first stage in developing an understanding of formal mathematical proof.
Penelitian ini bertujuan untuk mengembangkan buku ajar struktur aljabar berbasis pendekatan deduktif bagi mahasiswa program studi Tadris Matematika di UIN Syahada Padangsidimpuan. Pendekatan deduktif dimulai dari prinsip-prinsip umum yang kemudian diterapkan pada situasi khusus, memungkinkan mahasiswa untuk memahami hubungan logis antar konsep matematika secara mendalam dan sistematis. Buku ajar yang dikembangkan diharapkan dapat meningkatkan pemahaman dan keterampilan berpikir kritis mahasiswa, memfasilitasi keterlibatan aktif dalam pembelajaran, serta mempersiapkan mereka menghadapi tantangan matematika yang kompleks. Metode penelitian ini mencakup tahap-tahap pengembangan buku ajar, pemilihan konten relevan, desain pembelajaran berbasis pendekatan deduktif, serta evaluasi efektivitas melalui tes pengetahuan, observasi kelas, dan wawancara. Hasil penelitian menunjukkan peningkatan signifikan dalam pemahaman dan kemampuan mahasiswa setelah menggunakan buku ajar ini. Pendekatan deduktif terbukti efektif dalam membantu mahasiswa mengaitkan konsep abstrak dengan aplikasi praktis, meningkatkan kualitas pembelajaran, dan memenuhi kebutuhan beragam mahasiswa.
This research aims to develop a textbook on algebraic structures based on a deductive approach for students in the Mathematics Education program at UIN Syahada Padangsidimpuan. The deductive approach starts from general principles which are then applied to specific situations, enabling students to understand the logical relationships between mathematical concepts in depth and systematically. The developed textbook is expected to enhance students' understanding and critical thinking skills, facilitate active engagement in learning, and prepare them to tackle complex mathematical challenges. The research methodology includes stages of textbook development, selection of relevant content, design of deductive approach-based learning, and evaluation of effectiveness through knowledge tests, classroom observations, and interviews. The research findings demonstrate a significant improvement in students' understanding and abilities after using this textbook. The deductive approach proves effective in helping students connect abstract concepts with practical applications, enhancing learning quality, and meeting diverse student needs.
Drawing upon the imperative to foster argumentation within the mathematics classroom, this paper integrates two key areas of mathematics education research – argumentation and teachers’ noticing. It proposes a model of mathematics teachers’ noticing of argumentation that was designed and piloted in professional development courses for teachers in Israel dedicated to argumentation in mathematics teaching. The paper discusses the theoretical and empirical construction of the model and presents empirical evidence regarding the model’s
potential to foster and analyze teachers’ noticing of argumentation. The paper concludes with a discussion of the model’s contributions and limitations.
Bu araştırmada matematik öğretmen adaylarının ortaklaşa argümantasyon sürecinde oluşturdukları argümanlarının ve ispat şemalarının incelenmesi ve söz konusu argümanların ispat şemaları dikkate alınarak yorumlanması amaçlanmaktadır. Tekli durum çalışması olarak desenlenen bu araştırmanın katılımcılarını dört matematik öğretmenliği lisans öğrencisi oluşturmaktadır. Bu araştırmanın veri toplama araçlarını ortaklaşa argümantasyon sürecinin video kayıtları, araştırmacının gözlem notları ve öğrencilerin çözüm kağıtları oluşturmaktadır. Bu araştırmada veri analizi iki aşamada gerçekleştirilmiştir. İlk olarak video kayıtlar transkript edilerek Toulmin argümantasyon şemaları oluşturulmuş daha sonra öğrencilerin kullandıkları ispat şemaları belirlenmiştir. Araştırmanın bulguları öğrencilerin argümantasyon sürecine aktif olarak katıldıklarını ve bu süreçte dört farklı alt argüman ortaya çıktığını göstermektedir. Söz konusu alt argümanlarda dışsal ve deneysel ispat şemalarının kullanılmış ve bu alt argümanların tamamı öğrenciler tarafından çürütülmüştür. Araştırmanın verileri öğrencilerin fikir birliği sağladıkları ana iddiada ağırlıklı olarak analitik ispat şemalarının kullanıldığını göstermektedir. Argümantasyon sürecinde veri, iddia, gerekçe, çürütücü ve destekleyici bileşenleri ortaya çıkmıştır. Özellikle öğrencilerin birbirlerinin açıklamalarına karşılık çürütücüler sunmaları ve bunlardan bazılarının diğer öğrenciler tarafından çürütülmesi dikkat çekici bir bulgu olmuştur. Ayrıca argümantayson sürecinde niteleyici bileşeninin kullanılmadığı belirlenmiştir.
When justifying a mathematical statement, textbooks often interweave several instances of justification, creating, what we term, paths of justification. This study aims to characterize the paths of justification of mathematical statements that are offered in eight 7th grade Israeli mathematics textbooks. The study attended to context and compared paths of justification in different mathematical domains (algebra and geometry) and in textbooks intended for students with different achievement levels. Analysis focused on the number of instances of justification in a path, the types of justification of those instances, and their order. Comparing paths of justification between algebra and geometry, and between textbooks intended for students with different levels of achievements, revealed several similarities and differences. The findings portray a complex picture of textbook justifications, underlining the need to be sensitive to context when studying textbook justifications. The study results are discussed in light of relevant literature, highlighting potential factors involved in the way a textbook path of justification of a specific mathematical statement is structured.
Activities related to reading and understanding mathematical proofs are notoriously challenging for college students. On top of applying advanced logico-mathematical knowledge, these activities require the reader to process and operate on a substantial amount of information. Failure to successfully navigate the incoming stream of information may result in cognitive overload and impede one's ability to make progress on the task. This case study investigates how an undergraduate student, David, used his hands to overcome the cognitive load he experienced when reading a proof of the Two-Color Theorem. The findings suggest that in the absence of other modes of offloading (e.g., using pen and paper, figurative materials, or technology), hand gesturing may serve as a powerful and convenient offloading mechanism.
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Rethinking Proof harnesses the power of Dynamic Geometry to engage students in proving conjectures and thinking about proof in different ways. Rethinking Proof offers other motivations for proof besides verification, including explanation, discovery, challenge, and systemization. In the activities, students work with SketchpadTM sketches to make conjectures. Carefully constructed questions guide students to develop explanations of why their conjectures are true. Comes with extensive teacher's notes, and includes a download link containing pre-made Sketchpad sketches. A link to download Sketchpad for free is also provided.
Teaching mathematical proof is a goal in most mathematical curricula. This target is sometimes explicitly stated, that is “mathematical proof” can appear as a chapter in the textbook, or it is sometimes expressed as a general aim of the teaching. In this latter case, the aim is the training in the construction and the formulation of deductive reasoning, “mathematical proof” not being named as such. Actually, that is the situation at the present time in France: the last French programmes1 (1985) for the eighth grade state that students should be trained progressively to construct deductive reasoning, and this training must be intensive in the ninth grade. The situation is the same in other countries, like in Québec:
“To develop pupils’ logical thinking by bringing them to structure their reasoning [...] It is therefore advisable to demand that the pupils’ problem-solving is based on logical and structured reasoning.”
(Objectif du programme2 du secondaire V, Québec, 1984)
This longitudinal case study presents a sequence of episodes that document the mathematical thinking of one child, Stephanie, over a 5-year period. Her development of the idea of mathematical justification spans grades 1-5. Stephanie worked on several combinatorics tasks in small-group, whole-class, and individual interview settings. The documented events indicate Stephanie's progress in classifying, organizing, and reorganizing data. The study provides some significant insight into the process by which Stephanie learned to make proofs, within a setting that encouraged the development of her ideas.
This paper sets forth a way of interpreting mathematics classrooms that aims to account for how students develop mathematical beliefs and values and, consequently, how they become intellectually autonomous in mathematics. To do so, we advance the notion of sociomathematical norms, that is, normative aspects of mathematical discussions that are specific to students' mathematical activity. The explication of sociomathematical norms extends our previous work on general classroom social norms that sustain inquiry-based discussion and argumentation. Episodes from a second-grade classroom where mathematics instruction generally followed an inquiry tradition are used to clarify the processes by which sociomathematical norms are interactively constituted and to illustrate how these norms regulate mathematical argumentation and influence learning opportunities for both the students and the teacher. In doing so, we both clarify how students develop a mathematical disposition and account for students' development of increasing intellectual autonomy in mathematics. In the process, the teacher's role as a representative of the mathematical community is elaborated.
Ten second-grade classes participated in a year-long project in which instruction was generally compatible with a socioconstructivist theory of knowledge and recent recommendations of the National Council of Teachers of Mathematics. At the end of the school year, the 10 project classes were compared with 8 nonproject classes on a standardized achievement test and on instruments designed to assess students' computational proficiency and conceptual development in arithmetic, their personal goals in mathematics, and their beliefs about reasons for success in mathematics. The levels of computational performance were comparable, but there were qualitative differences in arithmetical algorithms used by students in the two groups. Project students had higher levels of conceptual understanding in mathematics; held stronger beliefs about the importance of understanding and collaborating; and attributed less importance to conforming to the solution methods of others, competitiveness, and task-extrinsic reasons for success. Responses to a questionnaire on pedagogical beliefs indicated that the project teachers' beliefs were more compatible with a socioconstructivist perspective than were those of their nonproject colleagues.
We consider the research question: Can lower-division undergraduates tell if the (purported) proof of a theorem actually proves it -- as opposed to some other theorem?
This article reports on an exploratory study of the way that eight mathematics and secondary education mathematics majors read and reflected on four student-generated arguments purported to be proofs of a single theorem. The results suggest that such undergraduates tend to focus on surface features of arguments and that their ability to determine whether arguments are proofs is very limited-perhaps more so than either they or their instructors recognize. The article begins by discussing arguments (purported proofs) regarded as texts and validations of those arguments, that is, reflections of individuals checking whether such arguments really are proofs of theorems. It relates the mathematics research community's views of proofs and their validations to ideas from reading comprehension and literary theory. Then, a detailed analysis of the four student-generated arguments is given and the eight students' validations of them are analyzed.
School geometry is the study of those spatial objects, relationships, and transformations that have been formalized (or mathematized) and the axiomatic mathematical systems that have been constructed to represent them. Spatial reasoning, on the other hand, consists of the set of cognitive processes by which mental representations for spatial objects, relationships, and transformations are constructed and manipulated. Clearly, geometry and spatial reasoning are strongly interrelated, and most mathematics educators seem to include spatial reasoning as part of the geometry curriculum. Usiskin (Z. Usiskin, 1987), for instance, has described four dimensions of geometry: (a) visualization, drawing, and construction of figures; (b) study of the spatial aspects of the physical world; (c) use as a vehicle for representing nonvisual mathematical concepts and relationships; and (d) representation as a formal mathematical system. The first three of these dimensions require the use of spatial reasoning.
This article reports on an exploratory study of the way that eight mathematics and secondary education mathematics majors read and reflected on four student-gener-ated arguments purported to be proofs of a single theorem. The results suggest that such undergraduates tend to focus on surface features of arguments and that their ability to determine whether arguments are proofs is very limited—perhaps more so than either they or their instructors recognize. The article begins by discussing argu-ments (purported proofs) regarded as texts and validations of those arguments, that is, reflections of individuals checking whether such arguments really are proofs of theorems. It relates the mathematics research community's views of proofs and their validations to ideas from reading comprehension and literary theory. Then, a detailed analysis of the four student-generated arguments is given and the eight students' vali-dations of them are analyzed. Proofs, that is, arguments 1 that prove theorems, have been viewed from a wide variety of perspectives. Previous studies have considered the structure of proofs (Leron, 1983), their explanatory power (Hanna, 1989), errors and misconceptions in college students' proofs (Selden & Selden, 1987), the logic involved (Dubinsky & Yiparaki, 2000; Selden & Selden, 1995), university students' "proof schemes" (Harel & Sowder, 1998), and generic proofs (Rowland, 2002). In addition, one might describe the genre of proofs or discuss their rhetoric. Considerably more could also be done in examining the process of proof construction.
This paper presents preliminary results from the first year of a multi-year research project i exploring the development of middle school students' competencies in justifying and proving and the conditions and pedagogy necessary to promote that development. In particular, in this paper we present and discuss results from a written assessment completed by approximately 350 6 th through 8 th grade students. The assessment focused on students' abilities to both evaluate and generate mathematical arguments, their understanding of various aspects of deduction,
Our epistemology of proof—in short, the acquaintance we have with truth and validity—first shapes our research framework, even before the choice of a problématique (the choice of the relevant questions and research problems), and the choice of a theoretical framework and its related methodology. I do not see that this issue has been addressed, in my best knowledge, although to be aware of it and to explore it systematically may be what conditions both the quality of what we produce and the possibility to exchange results. My claim is that our epistemology of proof is the first deadlock to figure out and to cope with when entering our research field. One may easily agree that this is especially crucial for young researchers, who generally enter the field with a naïve or intuitive problématique. My claim is that this is a deadlock for the whole field. Unless we have clarified precisely what this deadlock is like and how it limits our capacity to share research outcomes, it will be hardly possible to make significant progress in the field.
This talk will be organised around snapshots at some piece of research I consider as being representative of the main currents in our field. To sketch in the way I will do it, colleagues research is a bit risky, since there is much more in these works than what I kept for the
purpose of my talk. Fortunately, several of the authors I quote participate in this conference, so they will be able to react and correct me if necessary. The presentation of the snapshots will be organised so as to insist on the contrasts between the different approaches. In any case, this talk intends to open a collective reflection on our research, it will end on a call for opening a workshop on the impact of researchers epistemology on their own work. That is, a call for a
Taipei manifesto for educational research on mathematical proof.
Our overall intent is to clarify relations between the psychological constructivist, sociocultural, and emergent perspectives. We provide a grounding for the comparisons in the first part of the article by outlining an interpretive framework that we developed in the course of a classroom-based research project. At this level of classroom processes, the framework involves an emergent approach in which psychological constructivist analyses of individual activity are coordinated with interactionist analyses of classroom interactions and discourse. In the second part of the article, we describe an elaboration of the framework that locates classroom processes in school and societal contexts. The perspective taken at this level is broadly sociocultural and focuses on the influence of indlividuals' participation in culturally organized practices. In the third part of the article, we use the discussion of the framework as a backdrop against which to compare and contrast the three theoretical perspectives. We discuss how the emergent approach augments the psychological constructivist perspective by making it possible to locate analyses of individual students' constructive activities in social context. In addition, we consider the purposes for which the emergent and sociocultural perspectives might be particularly appropriate and observe that they together offer characterizations of individual students' activities, the classroom community, and broader communities of practice.
Simple elements of mathematical proof were analyzed by considering the making of a proof as a complex problem-solving task. Quantitative measures of performance on certain problem-solving tasks were obtained in order to determine the relation between problem-solving ability and chronological age (6-18). A consideration of all variables suggested that, even though problem-solving ability does increase with age, certain aspects of problem solving that approximate mathematical proof can be dealt with successfully in the upper elementary grades. An analysis of problem-solving strategies is recommended as a further step in the investigation of problem solving. IQ, reading ability, and mathematical aptitude are suggested as possible covariates or blocking variables in future research.
This 3-year longitudinal study investigated the development of 82 children's understanding of multidigit number concepts and operations in Grades 1-3. Students were individually interviewed 5 times on a variety of tasks involving base-ten number concepts and addition and subtraction problems. The study provides an existence proof that children can invent strategies for adding and subtracting and illustrates both what that invention affords and the role that different concepts may play in that invention. About 90% of the students used invented strategies. Students who used invented strategies before they learned standard algorithms demonstrated better knowledge of base-ten number concepts and were more successful in extending their knowledge to new situations than were students who initially learned standard algorithms.
To test the feasibility of the Cambridge Conference on School Mathematics recommendation that proof be taught in the elementary school, a unit on mathematical proof was developed using an iterative procedure. The basic objective of the unit was to teach the students to prove and understand a sequence of 6 theorems in number theory. After 2 formative pilot studies were conducted, a Nonequivalent Control Group Design was used to test the revised unit with 10 capable sixth-grade students. The students displayed an understanding of the theorems and were able to learn the proofs.
Recent reform efforts call on secondary school mathematics teachers to provide all students with rich opportunities and experiences with proof throughout the secondary school mathematics curriculum-opportunities and experiences that reflect the nature and role of proof in the discipline of mathematics. Teachers' success in responding to this call, however, depends largely on their own conceptions of proof. This study examined 16 in-service secondary school mathematics teachers' conceptions of proof. Data were gathered from a series of interviews and teachers' written responses to researcher-designed tasks focusing on proof. The results of this study suggest that teachers recognize the variety of roles that proof plays in mathematics; noticeably absent, however, was a view of proof as a tool for learning mathematics. The results also suggest that many of the teachers hold limited views of the nature of proof in mathematics and demonstrated inadequate understandings of what constitutes proof.
The research literature dealing with critical thinking vis-à-vis formal logic implies that (a) an individual's reasoning ability seems to improve with age and (b) reasoning from negatively worded hypotheses is more difficult than reasoning from positively worded hypotheses. The purpose of this study was to investigate these two implications. A test consisting of 30 items in sentential logic was constructed and given to elementary school children and prospective elementary school teachers. The findings indicate that only on certain forms of questions is maturation a factor in logical reasoning ability. Indeed, with certain types of items in sentential logic, maturation--and, inseparably, education--seems to be a hindrance to logical reasoning.
The purpose of this study was to investigate the relationship between the frequency of seventh-grade mathematics teachers' use of the language of conditional logic and their students' growth in conditional reasoning ability. Children of twenty classrooms were tested in September and again at the end of January using the Cornell Conditional Reasoning Test-Form X. Frequency of teacher use of conditional logic language was determined by analyzing the transcriptions of five 42-minute lessons for each teacher. Adjusted posttest performance on the conditional reasoning test was correlated significantly with these frequencies exhibited by the teachers.
Contemporary philosophy of mathematics offers us an embarrassment of riches. Among the major areas of work one could list developments of the classical foundational programs, analytic approaches to epistemology and ontology of mathematics, and developments at the intersection of history and philosophy of mathematics. But anyone familiar with contemporary philosophy of mathematics will be aware of the need for new approaches that pay closer attention to mathematical practice. This book is the first attempt to give a coherent and unified presentation of this new wave of work in philosophy of mathematics. The new approach is innovative at least in two ways. First, it holds that there are important novel characteristics of contemporary mathematics that are just as worthy of philosophical attention as the distinction between constructive and non-constructive mathematics at the time of the foundational debates. Secondly, it holds that many topics which escape purely formal logical treatment - such as visualization, explanation, and understanding - can nonetheless be subjected to philosophical analysis. The Philosophy of Mathematical Practice comprises an introduction by the editor and eight chapters written by some of the leading scholars in the field. Each chapter consists of short introduction to the general topic of the chapter followed by a longer research article in the area. The eight topics selected represent a broad spectrum of contemporary philosophical reflection on different aspects of mathematical practice: diagrammatic reasoning and representation systems; visualization; mathematical explanation; purity of methods; mathematical concepts; the philosophical relevance of category theory; philosophical aspects of computer science in mathematics; the philosophical impact of recent developments in mathematical physics.
This research reviewed literatures on proof in mathematics education. Several views of proof can be classified (and identified) such as psychological approach (Platonism, empiricism), structural approach (logicism, formalism, intuitionism) and social approach (ontology, axiomatic systems). All these views of proof are valuable in mathematics education society. The concept of proof can be found in the form of analytic knowledge not of constructive knowledge. Human beings developed their knowledge in the sequence of constructive knowledge to analytic knowledge. Therefore, in mathematics education, the curriculum of mathematics should involve the process of cognitive knowledge development.
In Deduction, Johnson-Laird and Byrne have made an extremely important contribution to the study of cognition by providing a unified theory of all types of deduction, including modal inferences. Their theory is that, contrary to the traditional view, deduction is not rule-based but model-based reasoning, and the central cognitive process involves a model-based search for counterexamples to conclusions. Although the author believes that Johnson-Laird and Byrne are right about the significance of mental models when it comes to certain types of deduction, their quest for a single cognitive process underlying all types of deductive inference has forced them to adopt the logician's notion of a model rather than a more psychological one. As a result, they are open to criticism. In this article, the author shows how mental models actually work in the case of deduction. However, the mental models discussed are a far cry from the mental models found in Deduction.
Editors' preface Acknowledgments Author's introduction 1. A problem and a conjecture 2. A proof 3. Criticism of the proof by counterexamples which are local but not global 4. Criticism of the conjecture by global counterexamples 5. Criticism of the proof-analysis by counterexamples which are global but not local: the problem of rigour 6. Return to criticism of the proof by counterexamples which are local but not global: the problem of content 7. The problem of content revisited 8. Concept-formation 9. How criticism may turn mathematical truth into logical truth Appendices Bibliography Index of names Index of subjects.