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Fostering first-year pre-service teachers’ proof competencies
Abstract
In the study presented, as we report in this paper, we describe our theoretical and practical consideration to engage first-year pre-service teachers in proving activities in the context of a transition-to-proof course. We investigated how students argued to verify a claim of elementary number theory on entering university and compared the results to their performance in the final examination of the course. Subsequently, we elaborate on the following results: On entering university, students do not seem to be capable of using algebraic variables as a heuristic to engage in reasoning. However, after learning about different kinds of proofs and the symbolic language of mathematics, students give evidence of starting to value mathematical language and of enhancing their proof competencies.
... Fischbein 1982) as a "direct, self-evident" insight (Fischbein 1999, p. 29) is deeply tied to an operative proof, because it should immediately open an insight into the mathematical structure of the statement (Duncker 1935;Reusser 1984Reusser , 1993 and thus explain why the statement is correct. However, research also puts forward evidence that also operative proofs require specific prior knowledge about the underlying representations and are not always easily accessible for everyone (e.g., Kempen and Biehler 2019;Knuth 2002b;Leron and Zaslavsky 2009). ...
... Still, also operative proofs pose certain demands on students and it is unclear if students are sufficiently able to understand this type of proof. In particular, university students focusing on mathematics still have problems handling generic examples and operative proofs (e.g., Kempen and Biehler 2019). Thus, research needs to confirm that teachers' matching of class characteristics and selected proofs is suitable for actual classes. ...
... Moreover, it is questionable if and to what extent teachers are actually able to create high quality operative proofs themselves. As data from Kempen and Biehler (2019) highlights, creating generic examples and operative proofs is not easy. Teachers should thus either (i) receive according teaching resources with experimental, operative, and formal-deductive proofs (ideally with a constructive alignment that allows a purposeful combination of the proofs) or (ii) learn how to create according proofs themselves. ...
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.
... Because of its fundamental role in mathematics, proof has been a research focus in philosophy of mathematics and mathematics education for many decades, but the research area saw a particular rise of interest in the last 10-15 years (Sommerhoff & Brunner, 2021). Even though proof and argumentation are internationally seen as important learning goals in mathematics and are incorporated in many national curricula (e.g., Department of Basic Education, 2011; Kultusministerkonferenz, 2012;National Council of Teachers of Mathematics, 2000), students seemingly do not gain sufficient experience with proof during high school (e.g., Hemmi, 2008;Kempen & Biehler, 2019). Consequently, students of different school levels and forms seem to lack fundamental proof skills and understanding of proof (e.g., Dubinsky & Yiparaki, 2000;Harel & Sowder, 1998;Healy & Hoyles, 2000;Kempen, 2019;Recio & Godino, 2001;Weber, 2001). ...
... High drop out rates 2 1 Introduction in mathematics, in particular compared to other fields, seem to be one of the consequences (e.g., Dieter, 2012;Heublein et al., 2022). Research on university students' proof skills has therefore increased significantly over the past 30 years, especially at the transition (e.g., Alcock, Hodds, Roy, & Inglis, 2015;Gueudet, 2008;Kempen & Biehler, 2019;Moore, 1994;Rach & Ufer, 2020;Recio & Godino, 2001;A. Selden & Selden, 2003;Sommerhoff, 2017;Stylianides & Stylianides, 2009;Stylianou, Chae, & Blanton, 2006). ...
... 22), i.e., in other proofs. This is of direct importance for the teaching of proof at the transition from school to university, because students usually do not gain extensive experience with proof and proving during high school (e.g., Hemmi, 2008;Kempen & Biehler, 2019 The list above also emphasizes that the acceptance of an argument is indeed not a necessity for the acceptance of the truth of statement, but only one component. For instance, the fourth factor introduced by Hanna (1989), namely that of the authors reputation or authority, might also be of particular relevance regarding actual acceptance criteria of students. ...
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.
... The proof of arithmetical relationships is a mathematical content that causes difficulties especially for pre-service primary school teachers (Kempen & Biehler, 2019). They often encounter typical difficulties when carrying out proofs, formal transformation errors in terms as one example (Moore, 2016). ...
... They often encounter typical difficulties when carrying out proofs, formal transformation errors in terms as one example (Moore, 2016). In addition, pre-service teachers have problems when changing representations and showing completeness or generality (Kempen & Biehler, 2019). ...
Digital support for proving arithmetic relationships - insights from the DigiMal.nrw project
... We believe generic examples accompanied by explicit clarification of their intended generality are appropriate forms of expression at the middle school level. The same proofs can also be represented using other notations, such as the 'geometric variables' used by Biehler and Kempen (2019;Kempen 2018) in their teaching of pre-service secondary school teachers. ...
... Similarly, they used ellipses to represent an arbitrary number of tiles. This is similar to the 'geometric variables' notation used by Biehler and Kempen (2019). ...
In this article we outline the role evidence and argument plays in the construction of a framing theory for Proof Based Teaching of basic operations on natural numbers and integers, which uses tiles to physically represent numbers. We adopt Mariotti’s characterization of a mathematical theorem as a triple of statement, proof and theory, and elaborate a theory in which the statement “The product of two negative integers is a positive integer” can be proved. This theory is described in terms of a ‘toolbox’ of accepted statements, and acceptable forms of argumentation and expression. We discuss what counts as mathematical evidence in this theory and how that evidence is used in mathematical arguments that support the theory.
... In the ZDM special issue Mathematical Evidence and Argument, published in 2019, a few papers deal with proof at the university level. Kempen and Biehler (2019) provide empirical evidence that pre-service teachers exposed to different types of proofs in multiple proof tasks and mathematical symbolic language in an innovative course entitled "Introduction to the Culture of Mathematics" begin to improve their proof competence. Aberdein (2019) "addresses from a philosophical perspective the relationships of evidence to proof, proof to derivation, argument to proof and argument to evidence, respectively," considering actual mathematical practices. ...
Recent research in university mathematics education has moved beyond the traditional focus on the transition from secondary to tertiary education and students' understanding of introductory courses such as pre-calculus and calculus. There is growing interest in the challenges students face as they move into more advanced mathematics courses that require a shift toward formal reasoning, proof, modeling, and problem-solving skills. This survey paper explores emerging trends and innovations in the field, focusing on three key areas: innovations in teaching and learning advanced mathematical topics, transitions between different levels and contexts of mathematics education, and the role of proof and proving in advanced university mathematics. The survey reflects the evolving landscape of mathematics education research and addresses the theoretical and practical challenges of teaching and learning advanced mathematics across various contexts.
... Noto et al. (2019) carried out a similar study with preservice teachers in Indonesia, although in the context of geometry, with similar findings. Moreover, there have been numerous attempts to improve student understanding of definitions in science (c.f., Zukswert et al., 2019) and mathematics (Zazkis and Leikin, 2008;Larsen, 2013) and to improve students' ability to use mathematical definitions in proof-writing (c.f., Jordan, 2019;Kempen and Biehler, 2019;Valenta and Enge, 2022). Yet, for all of the mathematics-educator-developed interventions to improve student understanding of definitions, we were unable to find any exploration of what mathematicians mean by student understanding of definitions and concepts. ...
Mathematics education research has long focused on students’ conceptual understanding, including highlighting conceptions viewed as problematic and looking for ways to develop more desirable conceptions. Nevertheless, limited research has examined how mathematicians characterize understanding of concepts and definitions or promote activities beneficial for students. Based on interviews with 13 mathematicians, we present thematic characterizations of what it means to understand a concept and definition, highlight activities mathematicians believe assist students’ learning, and examine their reasons for promoting these activities. Results include mathematically grounded descriptions of what it means to understand a concept but general descriptions of approaching and supporting learning. Implications include a need for attending to intended meanings for “understanding” in context and how this impacts appropriate activities for developing understanding, as well as a careful examination of the extant research literature’s claims about seemingly unified notions of conceptual understanding.
... Cho and Kwon (2017), for example, specify the ways that rigorously understanding the processes for making theorems and using and extending definitions is productive. This direction inspired research approaches which developed and examined course structures and activities which exposed students to different mathematical practices (e.g., Bauer & Kuennen, 2016;Christy & Sparks, 2015;Kempen & Biehler, 2019). ...
Internationally, questions about the perceived utility of university mathematics for teaching school mathematics pose an ongoing challenge for secondary mathematics teacher education. This special issue is dedicated to exploring this topic and related issues in the preparation of secondary mathematics teachers—by which we mean teachers of students with ages, approximately, of 12–18 years. This article introduces this theme and provides a semi-systematic survey of recent related literature, which we use to elaborate and situate important theoretical distinctions around the problems, challenges, and solutions of university mathematics in relation to teacher education. As part of the special issue, we have gathered articles from different countries that elaborate theoretical and empirical approaches, which, collectively, describe different ways to strengthen university mathematics with respect to the aims of secondary teacher education. This survey paper serves to lay out the theoretical groundwork for the collection of articles in the issue.
... Performance in proof construction was rated by grading all 48 final student responses on a four-point scale. Leaning on existing category systems (Kempen & Biehler, 2019;Malone et al., 1980;Recio & Godino, 2001;Schoenfeld, 1982), a score of 4 described a complete and valid proof and a score of a 0 was given for proving attempts with no substantial progress (for the detailed category system, see Kirsten, 2021). If a response was graded with a score of 3 or 4, the corresponding proving process was classified as successful. ...
Proof construction and proof validation are two situations of high importance in mathematical teaching and research. While both situations have usually been studied separately, the current study focuses on possible intersections. Based on research on acceptance criteria and validation strategies, 11 undergraduates' proof construction processes are investigated in terms of the effective and non-effective validation activities that occur. By conducting a qualitative content analysis with subsequent type construction, we identified six different validation activities, namely reviewing, rating, correcting errors, reassuring, expressing doubts, and improving. Although some of these activities tend to be associated with successful or unsuccessful proving processes, their effectiveness depends primarily on their specific implementation. For example, reviewing is effective when accompanied by a knowledge-generating approach and based on structure-or meaning-oriented criteria to provide deeper understanding. Thus, the results suggest that difficulties in proof construction could be partly attributed to inadequate validation strategies or their poor implementation.
... In this study, earlier findings from TIMS were confirmed that German students show low performance in mathematical proof even in upper secondary school (Baumert et al., 1998). Kempen and Biehler (2019a) investigated first-year pre-service teachers' proof competencies when entering university. In this study, only 10% of the 71 students gave a coherent argument when verifying the claim that the sum of any two odd numbers is always even. ...
Actively engaging students in learning mathematics is crucial to student success and equitable teaching and learning. Yet, this practice requires instructors to shift teaching strategies, which is not easily accomplished, particularly by themselves. In this chapter, we report on a longitudinal study of mathematics departments in the process of shifting department norms and practices in support of active learning and inclusive teaching. The research-informed change efforts have drawn on theories of institutional change and Networked Improvement Communities (Bryk AS, Gomez L, Grunow A, LeMahieu P. Learning to improve: how America’s schools can get better at getting better. Harvard Education Publishing, 2015). Data include interviews with tertiary mathematics instructors, course coordinators, department chairs, deans, other campus administrators, and students, as well as document analyses. Analyses of the data from three institutions, through the lens of networked improvement communities, reveal that some of the most important drivers of change related to institutional change to enact active learning are shared tools and resources, professional development, policies and structures, and connections across a network to other mathematics departments engaged in similar efforts.KeywordsMathematics department changeDriver diagramsLeveraging institutional changeActive learning mathematicsEquitable tertiary mathematics outcomesNetworked improvement communities for institutional change
... In this study, earlier findings from TIMS were confirmed that German students show low performance in mathematical proof even in upper secondary school (Baumert et al., 1998). Kempen and Biehler (2019a) investigated first-year pre-service teachers' proof competencies when entering university. In this study, only 10% of the 71 students gave a coherent argument when verifying the claim that the sum of any two odd numbers is always even. ...
In higher education, the difficulties of implementing teaching sequences in which several academic and engineering disciplines, or even professional worlds, coexist have been widely documented. We hypothesize that these difficulties stem, especially, from a series of conditions and constraints that determine the knowledge life in these different universes. In this chapter, we propose using tools from the Anthropological Theory of Didactics (ATD) to analyze these epistemologies and illustrate their application with examples from land surveying, industrial, and computer science contexts.KeywordsAnthropological theory of the didacticEngineering educationInterdisciplinary mathematicsInter-institutional transpositionInstitutions’ epistemic activitiesIndustrial epistemology
... In this study, earlier findings from TIMS were confirmed that German students show low performance in mathematical proof even in upper secondary school (Baumert et al., 1998). Kempen and Biehler (2019a) investigated first-year pre-service teachers' proof competencies when entering university. In this study, only 10% of the 71 students gave a coherent argument when verifying the claim that the sum of any two odd numbers is always even. ...
Parmi les recherches en didactique des mathématiques consacrées à l’enseignement supérieur, ondistingue deux types d’initiatives : celles qui favorisent le développement des résultats de rechercheet celles qui promeuvent les réflexions sur l’enseignement des mathématiques. Dans les deux cas, desdidacticien.ne.s des mathématiques et des chercheur.e.s en mathématiques ont joué un rôle moteur,et parfois conjoint. En retraçant quelques éléments liés à l’histoire de ces initiatives, nous verronscomment la création de la revue EpiDEMES (Épijournal de Didactique et Epistémologie desMathématiques pour l’Enseignement Supérieur) se positionne dans la continuité des efforts entrepris.
... Kempen & Biehler (2014) show that a similar problem is difficult to solve for beginning preservice teachers. Kempen & Biehler (2019) reveal that advanced preservice teachers have extended skills in this field. ...
This paper presents the results of a qualitative study in which two newly developed pedagogical content knowledge (PCK) assessment tools in the fields of German and mathematics were used. In these instruments, preservice teachers (27 in German, 40 in mathematics) were presented with exemplary tasks for school students and with authentic student responses. Preservice teachers were asked to name the requirements of the tasks, to assess the quality of students’ answers and to formulate feedback to the students (i.e. preservice teachers dealt with authentic problems that are pivotal in the field of teaching). The data collected in the study were analysed from an interdisciplinary perspective. Findings showed that preservice teachers of both subjects followed comparable strategies for complexity reduction and encountered transfer problems. Interdisciplinary conclusions can be drawn for the design of learning environments in teacher education.
... Part of the students does not understand and master this largely implicit enculturation. In a joint project with Leander Kempen (Kempen & Biehler, 2019a, 2019b, we newly designed a first-semester course called "Introduction into the culture of mathematics". The accompanying research followed a design-based research paradigm, and we re-designed the course three times based on the previous research results. ...
The paper starts with discussing the transition problem by characterizing the type of mathematics that is characteristic of school calculus. General types of measures for supporting students in the transition phase will be briefly reviewed. An essential aspect of the transition problem is the new culture of mathematics that is underlying Analysis courses. Arguments for making this change in culture more explicit and some concrete suggestions will be provided. The paper discusses examples from two empirical studies to support this analysis. In the first study, some results of an Analysis 1 final examination course in the first semester are analyzed. One task of the examination is taken, where school-mathematical solution strategies conflict with university-based norms. A second example is taken from a design-based research study, where a workshop was designed for supporting the guided reinvention of the concept of convergence of a sequence before formally introducing it in the lecture. One lesson learned of this second study is to be much more explicit about which conditions mathematical definitions at university have to fulfil, which seldom is a topic of explicit instruction.
... Some examples of such activities can be found in modern textbooks. Kempen and Biehler (2019a) proposed some learning environment for first-year pre-service teachers to cope with different representational systems in the context of mathematical proof. ...
The use of figurate numbers (e.g., in the context of elementary number theory) can be considered a heuristic in the field of problem solving or proving. In this paper, we want to discuss this heuristic from the perspectives of the semiotic theory of Peirce (“diagrammatic reasoning” and “collateral knowledge”) and cognitive psychology (“schema theory” and “Gestalt psychology”). We will make use of several results taken from our research to illustrate first-year students’ problems when dealing with figurate numbers in the context of proving. The considerations taken from both theoretical perspectives will help to partly explain such phenomena. It will be shown that the use of figurate numbers must not be considered to be any kind of help for learners or some way of ‘easy’ mathematics. Working in this representational system has to be learned and practiced as another kind of knowledge is necessary for working with figurate numbers. The named findings also touch upon the concept of ‘proofs that explain.’ Finally, we will highlight some implications for teaching and point to a number of demands for future research.
There has been significant growth in the literature base exploring questions of teaching in undergraduate mathematics. In this paper we synthesize the literature on the teaching of proof-based undergraduate mathematics, drawing on 104 published reports from a range of countries and research traditions. We primarily differentiate the papers into those which explore lecture-based pedagogy and student-centered pedagogy. For each type of instruction, we focus on three categories of findings from the literature: description of instruction, instructor beliefs and rationales, and the relationship between instruction and students (cognitive, participatory, affective, and equity oriented). Much is known about the enactment of lecture-based teaching, including instructors’ cognitive and affective goals. The student-centered literature focuses on tensions and challenges implementing curricula with a greater focus on participatory goals. Overall, there are few studies that attempt to link instructors’ classroom activity and students’ learning. Similarly, attention to equity is relatively lacking in the extant research.
Wie lassen sich Fachdidaktik und Sprachbildung im Lehramtsstudium so verknüpfen, dass zukünftige Mathematiklehrkräfte optimal auf sprachbewussten Fachunterricht vorbereitet werden? Dieser Frage geht dieses Open Access Buch aus interdisziplinärer Perspektive nach – fachlich fundiert, forschungsorientiert und anschaulich:
• Die vorgestellten Leitideen und Design-Prinzipien für eine sprachbewusste Hochschullehre bilden den konzeptionellen Rahmen, der auch für die Weiterentwicklung eigener Lehrveranstaltungen genutzt werden kann.
• An zahlreichen empirischen Beispielen aus Schule und Hochschule werden Synergien zwischen Mathematikdidaktik und Sprachbildung aufgezeigt und Vorschläge für die konkrete Ausgestaltung der Hochschullehre gemacht.
• Einen besonderen Schwerpunkt bildet das Forschende Lernen in Praxisphasen an der Schnittstelle von Fach und Sprache.
• Impulse aus anderen Fachdidaktiken (Informatik, Chemie) eröffnen Transfermöglichkeiten.
Mit seinen vielfältigen Einblicken in das hochaktuelle Thema Sprachbildung im Fachunterricht als Gegenstand einer sprachbewussten Hochschullehre ist das Buch für Hochschuldozierende und Studierende in der Lehramtsausbildung Mathematik gleichermaßen von Interesse.
Das Buch
• bietet Konzepte für die Verknüpfung von Mathematikdidaktik und Sprachbildung in der Lehrkräfteausbildung Mathematik,
• ist interdisziplinär und forschungsorientiert,
• ist geeignet für Hochschuldozierende und Studierende.
Dieses Buch ist unter den Bedingungen der Creative Commons Attribution 4.0 International License über SpringerLink (link.springer.com) frei zugänglich.
Die Autor*innen
Prof. Dr. Florian Schacht studierte Mathematik und Musik an der TU Dortmund. Er ist Professor für Didaktik der Mathematik an der Universität Duisburg-Essen. Bei der Erforschung von Sprachbildung adressiert er einerseits den Mathematikunterricht, andererseits die Frage, welche Konsequenzen sich für die Hochschullehre im Rahmen der Lehramtsausbildung sowie für die Fort- und Weiterbildung im Fach Mathematik ergeben. Dr. Susanne Guckelsberger studierte an der LMU München Deutsch als Fremdsprache, Anglistik und Linguistik. Sie ist wissenschaftliche Mitarbeiterin am Institut für Deutsch als Zweit-/Fremdsprache an der Universität Duisburg-Essen. Ihre Schwerpunkte liegen u.a. in der Entwicklung von Konzepten für die interdisziplinäre Lehramtsausbildung sowie für die Qualifikation von DaF-Lehrkräften an Hochschulen international.
This paper gives an account of the conceptual and practical approach to “operative proofs” that has been developed in the Mathe 2000 project. By means of some typical learning environments, this notion and its theoretical background are explained.
We discuss whether a generic argument can be considered a proof. Two positions on this question have recently been published which focus on the fussiness of an argument as a deciding criterion. We take a third view that takes into account psychological and social factors. Psychologically, for a generic argument to be a proof it must result in a convincing deductive reasoning process occurring in the mind of the reader. Socially, for a generic argument to be a proof it must conform to the social conventions of the context. For classroom settings, we suggest two kinds of evidence that should be reflected in written work in order for a generic argument to be accepted as a proof. These kinds of evidence reveal the linkage between the psychological and social factors.
First-year students are supposed to be able to handle the deductive, axiomatic system of mathematics, to learn the formal symbolic language and to master different methods of proving. In this paper, we report on our findings from a redesigned bridging course lecture for preservice teachers, in which the students were asked to construct generic proofs to complete their transition to formal proof, using their mathematical knowledge from school. The students first assignment was collected and their handling and use of examples, generic proofs, formal proofs and variables were analysed.
This technical report presents an overview of students' difficulties with learning to understand and construct proofs. There are sections on: The Curriculum and Students’ and Teachers’ Conceptions of Proof; Understanding and Using Definitions and Theorems; Understanding the Structure of a Proof and the Order in which it Might be Written; Knowing How to Read and Check Proofs; Knowing and Using Relevant Concepts; Bringing Appropriate Knowledge to Mind; Knowing What's Important and Useful; and Teaching Proof and Proving;
This article discusses forms of proof and proving in the learning and teaching of mathematics, including different representations used in proof production, different ways of arguing mathematically, different degrees of rigour in proving, and multiple proofs of the same statement. First, we focus on external forms of proof. We report research on students’ and teachers’ beliefs about visual aspects of proving and discuss the importance of visibility and transparency in mathematical arguments, particularly those using visualisation. We highlight the pedagogical potential of proving activities involving visualisation and reflect on its limitations. Next, we discuss the importance of various mathematical, pedagogical, and cognitive aspects of different forms of proof in multiple-proof tasks. We then examine which forms of proof might support students’ transition from empirical arguments to general proofs, using examples from the history of mathematics and discussing the roles of operative and generic proofs. We conclude by indicating potential future research agendas.
When a topologist colleague was asked to teach remedial geometry, he used Schaum's Outline of Geometry and also wrote proofs on the blackboard. One day a student, who was familiar with two-column proofs having statements such as ΔABD ≅ ΔBCD and reasons such as SAS, blurted out in utter surprise, “You mean proofs can have words!” This geometry student's previous experience had led him to an unfortunate view of proof. Other students experience epiphanies about themselves and proof. Asked what she (personally) got out of a transition-to-proof course, one of our students answered, “I learned that I could wake up at 3 A.M. thinking about a math problem.” What do responses like this tell us? Almost all undergraduate mathematics courses are about the concepts and theorems of mathematics — when a matrix has an inverse, how to find it, and when to use it; when a series converges; the distinction between continuous and uniformly continuous; the meaning of compact. However, students in courses like abstract algebra, real analysis, and topology normally demonstrate their competence by solving problems and proving theorems. And, if students go beyond a few lower-division courses such as calculus or first differential equations, this usually involves constructing original proofs or proof fragments. But, often not much time can be devoted to helping students learn how to construct proofs. This might not lead to difficulties, if only students came to university understanding something about the nature of proof and already had some experience constructing simple proofs. © 2008 by The Mathematical Association of America (Incorporated).
For a variety of reasons, students often find the transition to tertiary level mathematics study difficult. These reasons can include an expectation of greater student autonomy, a change in the nature of the objects studied, and the seeming unapproachability of professors. However, one of the major reasons for students’ difficulties seems to be the requirement that they understand and construct proofs. A variety of difficult aspects confront tertiary students struggling with proof and proving: the proper use of logic; the necessity to employ formal definitions; the need for a repertoire of examples, counterexample, and nonexamples; the requirement for a deep understanding of concepts and theorems; the need for strategic knowledge of important theorems, and the important ability to validate one’s own and others’ proofs.
In addition, there are ways to teach proof other than by presenting proofs in a finished, linear top-down fashion. One can use generic proofs or employ structured proofs. There are a variety of courses and strategies that may help: transition-to-proof courses, communities of practice, the Moore Method, the co-construction of proofs, and the method of scientific debate. There are also resources in the form of videos, DVDs, and books to help university teachers.
The criteria by which a proof is judged in mathematics seem, on the face of it to be quite different when considered by sophisticated mathe-maticians as compared with learners. The sophisticated mathematician may concern himself with logical structure, mathematical style, the degree of generality, the aesthetic quality of the proof, and so on. The learner may lack the sophistication to appreciate these criteria fully and may concern himself more with the manner in which the proof explains the result and demonstrates why it must be true, based on his current state of development. A basis for the cognitive development of proof is already available in the psychology of learning in terms of the meaningful learning of Ausubel (1978) or the relational understanding of Skemp (1976). At any stage in development the ideas presented need to be potentially meaningful (in Ausubel's terminology), which may mean, in the short term, presenting proofs in a radically different form from that ultimately desired. In this paper we will see an example of this phenomenon. The initial proof may be more cumbersome, less aesthetically pleasing, yet prove more meaningful to the learner at the particular stage under consideration. Even so, the long term desire for full sophistication must be kept in mind, yielding two complementary but, at times, conflicting, principles:
This paper explores some of the ambiguities inherent in the notions of generality and genericity, drawing parallels between natural language and mathematics, and thereby obliquely attacking the entrenched view that mathematics is unambiguous. Alternative ways of construing 2N, for example, suggest approaches to some of the difficulties which students find with an algebraic representation of generality. Examples are given to show that confusion of levels is widespread throughout mathematics, but that the very confusion is a source of richness of meaning.
Although studies on students’ difficulties in producing mathematical proofs have been carried out in different countries,
few research workers have focussed their attention on the identification of mathematical proof schemes in university students.
This information is potentially useful for secondary school teachers and university lecturers. In this article, we study mathematical
proof schemes of students starting their studies at the University of Córdoba (Spain) and we relate these schemes to the meanings
of mathematical proof in different institutional contexts: daily life, experimental sciences, professional mathematics, logic
and foundations of mathematics. The main conclusion of our research is the difficulty of the deductive mathematical proof
for these students. Moreover, we suggest that the different institutional meanings of proof might help to explain this difficulty.
Preservice mathematics teachers are entrusted with developing their future students' interest in and ability to do mathematics effectively. Various policy documents place importance on being able to reason about and prove mathematical claims. However, it is not enough for these preservice teachers, and their future students, to have a narrow focus on only one type of proof (demonstration proof), as opposed to other forms of proof, such as generic example proofs, pictorial proofs, and so on. This article examines the effectiveness of a course on reasoning-and-proving on preservice teachers' awareness of and abilities to recognize and construct generic example proofs. The findings support assertions that such a course can and does change preservice teachers' capability with generic example proofs.
In dem Beitrag werden verschiedene didaktische Ansätze zum Beweisen (intuitive Beweise, inhaltlich-anschauliche Beweise, operative Beweise, präformale Beweise, generische Beweise etc.) aufgearbeitet und ihre speziellen Charakteristika und Zielsetzungen herausgestellt. Weiter wird hinterfragt, inwieweit die verschiedenen Konzepte als „intellektuell ehrlich“ (Kirsch) vereinfachte Beweise gelten können. In diesem Kontext wird weiter erörtert, welche Rolle der Darstellung der Argumente und der Sprache zukommt.
While development of a teacher’s expertise includes continuous incorporation of innovations throughout his/her career, teachers
are often reluctant to adopt and implement new practices when challenged by innovative teaching approaches. This paper presents
an analysis of the development of teachers’ expertise in relation to the implementation of novel (for them) instructional
material. The study examines the ways in which teachers implement multiple-solution tasks (MSTs) (as an example of instructional
tools new to the teacher) in their classes, following a professional development course in which they participated. The analysis
focuses on the nature of MSTs implemented by the teachers and of the subsequent class discussion. The nature of MSTs is analyzed
focusing on the goals with which MSTs were implemented, mathematical connections embedded in the MSTs, scaffolding provided
to the learners and the learning settings. This analysis has led to the identification of four main implementation styles:
straightforward, simple, adaptive and inventive. Concluding discussions are examined with respect to elevating and framing
elements. Two lessons by mathematics teachers are described in the paper to explain how lessons were analyzed, and to exemplify
adaptive and inventive implementation styles.
There are a lot of arguments for the inclusion of applications and modelling in mathemat- ics teaching: so-called pragmatic, formal, cultural, and psychological arguments. Among the psychological arguments, the most often mentioned is the role of applications for mo- tivating and introducing new topics and for practising and consolidating them. What is rarely mentioned is another psychological aspect: applications provide contexts for what I call reality-related proofs. This is the topic of my paper. It has three aims: 1) to explain the concept of reality-related proof by means of four examples, 2) to elaborate the role of Grundvorstellungen in these proofs, 3) to show why all this can be important for mathematics teaching. I concentrate deliberately on theoretical considerations and do not refer to empirical as- pects. 1. An introductory example Example 1: Let us presuppose a pupil knows the definition of
Viewed internationally, the proof aspect of mathematics is probably the one which shows the widest variation in approaches. The present French syllabus adopts an axiomatic treatment of geometry from the third secondary school year (age 14), Papy's Mathdmatique Moderne is axiomatic from the age of 12, early American developments based primary school number work on the laws of algebra. In England, proofs of geometrical theorems have been steadily disappearing from O-level syllabuses for thirty years, and 'it continues to be the policy of the SMP to argue the likelihood of a general result from particular cases'. (Preface to Book 5). Underlying this divergence in practice lies the tension between the awareness that deduction is essential to mathematics, and the fact that generally only the ablest school pupils have achieved understanding of it. The purpose of the work described in this paper is to analyse pupils' attempts to construct proofs and explanations in simple mathematical situations, to observe in what ways they differ from the mature mathematician's use of proof, and thus to derive guidance about how best to foster pupils' development in this area. In a previous paper (Bell, 1976), I have shown that pupils' attempts at making and establishing generalisations, and at supporting these with reasons, can be interpreted in terms of a number of identifiable stages of attainment which are loosely related to age. Two of these stages were fairly well-defined - Stage 1
The starting point of our reflections is a classroom situation in grade 12 in which it was to be proved intuitively that non-trivial solutions of the differential equation f' = f have no zeros. We give a working definition of the concept of preformal proving, as well as three examples of preformal proofs. Then we furnish several such proofs of the aforesaid fact, and we analyse these proofs in detail. Finally, we draw some conclusions for mathematics in school and in teacher training.
Argumentation and mathematical proof: A complex, productive, unavoidable relationship in mathematics and mathematics education
- P Boero
Boero, P. (1999). Argumentation and mathematical proof: A complex,
productive, unavoidable relationship in mathematics and mathematics education. International Newsletter on the Teaching and
Learning of Mathematical Proof, July/August.
Forms of proof and proving in the classroom
- T Dreyfus
- E Nardi
- R Leikin
The quality of argumentations of first-year pre-service teachers
- L Kempen
- R Biehler
Kempen, L., & Biehler, R. (2014). The quality of argumentations of
first-year pre-service teachers. In S. Oesterle, P. Liljedahl, C.
Nicol, & D. Allen (Eds.), Proceedings of the 38th Conference
of the International Group for the Psychology of Mathematics
Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education (Vol. 3, pp.
425-432). Vancouver, Canada: PME.
Innermathematisches Beweisen und Argumentieren in der Sekundarstufe 1. Mögliche Erklärungen für systematische Bearbeitungsunterschiede und leistungsförderliche Aspekte
- E Brunner
Brunner, E. (2013). Innermathematisches Beweisen und Argumentieren
in der Sekundarstufe 1. Mögliche Erklärungen für systematische
Bearbeitungsunterschiede und leistungsförderliche Aspekte. Münster: Waxmann.
Begründen und Beweisen im Übergang von der Schule zur Hochschule
- L Kempen
Kempen, L. (2019). Begründen und Beweisen im Übergang von der
Schule zur Hochschule. Wiesbaden: Springer Spektrum.
Qualitative Inhaltsanalyse
- U Kuckartz
Kuckartz, U. (2012). Qualitative Inhaltsanalyse. Methoden, Praxis,
Computerunterstützung. Weinheim/Basel: Beltz Juventa.
Developing thinking in Algebra
- J Mason
- A Graham
- S Johnston-Wilder
Mason, J., Graham, A., & Johnston-Wilder, S. (2005). Developing
thinking in Algebra. London: The Open University.