Eté 2016

Boero P., Fenaroli G., Guala E. **REASONING AND PROOF IN ELEMENTARY TEACHER EDUCATION: THE KEY ROLE OF THE CULTURAL ANALYSIS OF THE CONTENT**

Difficulties concerning the development of mathematical reasoning in elementary school in Italy, related to school culture and past teacher education, are described. A coordinate intervention in primary school teacher education at our university, aimed at getting prospective teachers free from those influences and able to perform autonomous professional choices, is outlined. In particular, in the case of mathematical argumentation and proof some results concerning the development, in one of our courses, of the competence of Cultural Analysis of the Content to be taught (CAC), assumed as a condition for professional autonomy, are briefly presented and discussed.

Hanna G. **REFLECTIONS ON PROOF AS EXPLANATION**

This paper explores the connection between two distinct ways of defining mathematical explanation and thus of identifying explanatory proofs. The first is the one discussed in the philosophy of mathematics, in which a proof is considered explanatory when it helps account for a mathematical fact, clarifying why it follows from others. It is concerned with intra-mathematical factors, not with pedagogical considerations. The second definition is the one current among mathematics educators, who consider a proof to be explanatory when it helps convey mathematical insights to an audience in a manner that is pedagogically appropriate. This latter view brings cognitive factors very much into play. The two views of explanation are quite different. The paper shows, however, citing examples, that insights from what are considered by philosophers of mathematics to be explanatory proofs can sometimes form a basis for explanatory proofs in the pedagogical sense and thus add value to the curriculum.

Harel G. **TYPES OF EPISTEMOLOGICAL JUSTIFICATIONS**

We distinguish among three types of epistemological justification: (1) Sentential epistemological justification (SEJ).This refers to a situation when one is aware of how a definition, axiom, or proposition was born out of a need to resolve a problematic situation. (2) Apodictic epistemological justification (AEJ).This pertains to the process of proving. It is when one views a particular logical implication, "a implies b", in causality, or explanatory, terms—how a causes b to happen. This can take place in two forms. One might observe a, asks what are its possible consequences, and finds out that b is a consequence of it. Or one might observe b, asks about its causes, and finds out that a is a cause of it. (3) Meta epistemological justification (MEJ).This refers to a situation when one not only possesses SEJ and AEJ, but also he or she is aware of how the sentence or the implication came into being. These three types will be illustrated with examples from the field of complex numbers.

Durand-Guerrier V. **WORKING ON PROOFS AS CONTRIBUTING TO CONCEPTUALIZATION – THE CASE OF IR COMPLETENESS PROLEGOMENA TO A DIDACTICAL STUDY**

In this communication, we propose a mathematical and epistemological study about two classical constructions of the real numbers system, and the associated proofs of its completeness. The general didactical issue pertains to the potential contribution of analyzing proofs as a means for deepening the understanding of the objects in play. This study falls within a larger project about the conceptualization of real numbers, taking into account the triad discreteness/density/continuity.

Stylianides A.J., Stylianides G.J. **CLASSROOM-BASED INTERVENTIONS IN THE AREA OF PROOF: ADDRESSING KEY AND PERSISTENT PROBLEMS OF STUDENTS' LEARNING**

While research has provided a strong empirical and theoretical basis about major difficulties students face with proof, it has paid less attention to the design of interventions to address these difficulties. We discuss the need for more research on classroom-based interventions in the area of proof, and we raise the question of what might be important characteristics of interventions that specifically aim to address key and persistent problems of students’ learning in this area. In particular, we draw on prior research to make a case for interventions with the following three characteristics: (1) they include an explanatory theoretical framework about how they “work” or “can work” in relation to their impact on students’ learning; (2) they have a narrow and well-defined scope, which makes it possible for them to have a relatively short duration; and (3) they include an appropriate mechanism to trigger and support conceptual change in students.

Mariotti M. A., Goiuzueta M. **CONSTRUCTING AND VALIDATING A MATHEMATICAL MODEL: THE TEACHER’S PROMPT**

Drawing on the hypothesis that an epistemology of school mathematics is interactively constituted in the classroom, we assume that different epistemological stances may lead students to get differently involved in the production and evaluation of arguments. In this paper we focus on how students exploit teacher’s interventions to produce arguments to validate their mathematical solving processes and different mathematical models. We show that, in spite of their potential to foster students’ reflection upon the adequacy of these models to the proposed empirical situation, teacher’s interventions do not have the intended effect. Instead, a particular interpretation of the situation emerges as a consequence of a model ultimately validated by the teacher; a phenomenon we call ex post facto modeling. We depart from this phenomenon to discuss some aspects of the mathematical culture of the classroom.

Miyazaki M., Nagata J., Chino K., Fujita T., Ichikawa D., Shimizu S., Iwanaga Y. **DEVELOPING A CURRICULUM FOR EXPLORATIVE PROVING IN LOWER SECONDARY SCHOOL GEOMETRY**

Proving is explorative in nature. It means that proving involves producing statements, producing proofs, looking back (examining, improving and advancing) these productions, and their interactions among these aspects. We aim to echo the explorative nature of proving in curriculum development by mainly focusing on the planning aspects and constructing aspects in producing proofs. As the result we found two kinds of learning progressions as a framework, developed a curriculum of geometry for junior high school by corresponding the progressions with the units of “Course of Study” in Japan. We further refined the provisional curriculum by implementing lessons by expert teachers and reflecting on these lessons with them.

Komatsu K., Ishikawa T., Narazaki A. **PROOF VALIDATION AND MODIFICATION BY EXAMPLE GENERATION: A CLASSROOM-BASED INTERVENTION IN SECONDARY SCHOOL GEOMETRY**

This study addresses proof validation and modification in geometry wherein students find cases invalidating their constructed proofs and revise their proofs in response to this invalidation. We implemented a classroom-based intervention in grade 9, in which the students worked on assessment tasks designed to assess their performance in proof validation and modification at the end of the intervention. After reporting the intervention in brief, this study analyses the results of the tasks to examine the number of students who succeeded in proof validation and modification, the types of diagrams drawn by the students for invalidation of their proof, and their responses to this invalidation.

Karunakaran S. S. **ALLOWANCE BY EXPERTS FOR A BREAK IN “LINEARITY” OF DEDUCTIVE LOGIC IN THE PROCESS OF PROVING**

Mathematicians have long claimed that the proving process cannot be considered a “linear” process and that undergraduates may view the proving process to be necessarily “linear”. However, there is little empirical research that supports this familiar claim. Using grounded theory methods, “expert” provers of mathematics were examined in the process of proving novel mathematical statements. Expert provers of mathematics were willing to knowingly and temporarily interrupt the deductive logic of their proving process in order to make progress towards constructing an eventually complete deductive argument.

Pedemonte B. **HOW CAN A TEACHER SUPPORT STUDENTS IN CONSTRUCTING A PROOF?**

In this article we analyze the interaction between student and teacher when student is engaged in constructing a geometrical proof. This analysis shows that it is not easy for the teacher to modify student’s argumentation based on incorrect conceptions. Toulmin’s model, used to analyze student’s argumentation and teacher’s intervention, highlights that teacher’s intervention runs as an effective rebuttal in student’s argumentation only when it “acts” on the warrant and the backing of the student’s argumentation.

Kempen L. **HOW DO PRE-SERVICE TEACHERS RATE THE CONVICTION, VERIFICATION AND EXPLANATORY POWER OF DIFFERENT KINDS OF PROOFS**

In the opening session of a course for first-year pre-service teachers, the participants were asked to rate the conviction and explanatory power of four different kinds of proofs (a generic proof with numbers, a generic proof in the context of figurate numbers, a proof in the context of figurate numbers using “geometric variables”, and the so-called formal proof). The results may open the discussion about the concept of proofs that explain with regard to the explanatory power of the mathematical symbolic language.

Yan X., Hanna G., Mason J. **IDENTIFYING AND USING KEY IDEAS IN PROOFS**

The mathematics-education literature reveals an ongoing interest in fostering the students’ ability to construct and reconstruct proofs. One promising tool to this end is the notion of “key idea”. This study investigates (1) how students go about identifying key ideas in a proof and using them in reconstructing it, and (2) ways in which the notion of key idea is related to conceptual understanding and procedural fluency. The results show that though key ideas were highlighted and used by both students and instructor, many students were unable to capture them with sufficient precision.

Davis J. D. **IRISH TEACHERS' PERCEPTIONS OF REASONING-AND-PROVING AMIDST A NATIONAL EDUCATIONAL REFORM**

The syllabi driving the secondary mathematics education reform in Ireland expect students to engage in three components of reasoning-and-proving (RP) (Stylianides, 2008): pattern identification, conjecture formulation, and proof argument construction. This study examines the perceptions of these three RP processes by 22 Irish teachers with varying levels of teaching experience via semi-structured interviews. These teachers perceived pattern identification and conjecturing as disconnected from proof construction. Indeed, teachers struggled to define conjecturing and proof. There also appeared to be a bifurcation in students’ classroom experiences with RP processes. Teachers stated that students with perceived lower ability levels experiences with proof ended at pattern identification while higher-level students rarely engaged in pattern identification and focused on memorizing proofs due to the influence of high stakes assessments. The implications of these results are discussed.

Lekaus S., Askevold G. A. **MATHEMATICAL ARGUMENTATION IN PUPILS’ WRITTEN DIALOGUES**

In this article we present some results from a project about mathematical argumentation and proving in form of dialogues. Tasks are prepared in form of written dialogues between imaginary pupils discussing mathematical problems, and pupils are invited to write their own dialogues continuing the mathematical discussion. We analyse excerpts of dialogues about fractions written by pupils from two classrooms in Norway in grades 5 and 6. We show that many of the 5th grade pupils are strongly committed to visual representations of fractions in their argumentation, while the 6th graders used more rule-bound approaches.

Wong K. C., Sutherland R. **REASONING-AND-PROVING IN SCHOOL MATHEMATICS TEXTBOOKS: A CASE STUDY FROM HONG KONG**

To promote learning mathematics with understanding, mathematics educators in many countries recommend that proof (and proof-related reasoning) play a central role in school mathematics. In responding to this recommendation, this study examines the opportunities for students to learn reasoning-and-proving from a popular Hong Kong school mathematics textbook. The study adopts the methodology of Stylianides (2009). Results show that such opportunities are relatively limited. This suggests that proof plays a marginal role in school mathematics in Hong Kong.

Melhuish K. M., Thanheiser E. **TEACHER NOTICING OF JUSTIFYING IN THE ELEMENTARY CLASSROOM**

We present a data from a large-scale study evaluating the efficacy of a sustained professional development focused on increasing mathematically productive practices in elementary classrooms. We pair results from classroom observations of 56 teachers with their self-reports of student justification in their classroom. We found discrepancies between teachers’ self-evaluations and observed frequencies of student justifications in lessons. We explore this discrepancy by introducing a framework for teacher noticing of student justification. This framework was developed through careful study of teachers’ discussion of noticing justifying in a series of commonly observed lessons. Our analysis unveiled the complexity of noticing justification where a teacher must attend to both the underlying mathematical content context and the type of mathematical practice. We hypothesize that teachers may not have the underlying knowledge of justification to appropriately notice it within their classrooms.

Tsujiyama Y., Yui K. **USE OF EXAMPLES OF UNSUCCESSFUL ARGUMENTS TO FACILITATE STUDENTS’ REFLECTION ON THEIR PROVING PROCESSES**

Proving is an essence of mathematical activities but difficult topic for many students. One reason for this might be that unsuccessful arguments during proving processes do not appear in a completed proof, and students cannot see how these arguments influenced the proof. If students can reflect on these arguments, they would be able to learn more about what proving is. In the previous studies, worked examples that show successful processes of deriving a proof are intended to deepen students’ understanding of proving. On the other hand, such examples do not include unsuccessful arguments. This study examines how examples of unsuccessful arguments can facilitate students’ reflection on their proving processes by designing, implementing, and analysing an eighth-grade geometry lesson. This study found that an example of unsuccessful arguments enabled the students to clarify the reason why the unsuccessful arguments failed, and why successful arguments worked.

Rapke T. K., Allan A. **WHAT MAKES A GOOD PROOF? STUDENTS EVALUATING AND PROVIDING FEEDBACK ON STUDENT-GENERATED “PROOFS”**

In this paper, we report findings of a research project that involved students evaluating and offering feedback on mathematical arguments claimed by the author, usually a fellow student, to constitute a proof of a given statement. The focus of this study is on thenorms understood and enactedby first-year post secondary mathematics students around whatmakes a “good” student-written “proof”.We conclude that our research participants could identify qualities of a good proof and enact their perceived norms through revisions. Furthermore, students should not be expected to merely identify simple errors or shortcomings of “proofs”, as they should be able to offer more complex analysis, suggestions, and revisions.

Buchbinder O. **SYSTEMATIC EXPLORATION OF EXAMPLES AS PROOF: ANALYIS FROM FOUR THEORETICAL PERSPECTIVES**

This paper offers a multi-layered analysis of one specific category of students’ example-based reasoning, which received little attention in research literature so far: systematic exploration of examples. It involves dividing a conjecture’s domain into disjoint sub-domains and testing a single example in each sub-domain. I apply four theoretical perspectives to analyze student data as a way to deepen and broaden insights gained from the analysis of this phenomenon. Implications for teaching and learning of proof in school mathematics are discussed.

Knuth E., Ellis A., Zaslavsky O. **THE ROLE OF EXAMPLES IN PROVING RELATED ACTIVITIES**

While example-based reasoning has typically been viewed as a stumbling block to learning to prove, we view example-based reasoning as an important object of study and posit that examples play both a foundational and essential role in the development, exploration, and understanding of conjectures, as well as in subsequent attempts to develop proofs of those conjectures. In this paper, we describe our project whose goals are to (a) investigate the nature of middle school and high school students’, undergraduate students’, and mathematicians’ thinking about the examples they use when developing, exploring, and proving conjectures; and (b) investigate ways in which thinking about and analyzing examples may facilitate the development of students’ learning to prove.

Reid D., Vallejo Vargas E. A. **WHEN IS A GENERIC ARGUMENT A PROOF?**

We consider 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.

El idrissi A., Rouan O. **AREA AS A TOOL IN MATHEMATICAL PROOFS SOME HISTORICAL CASES**

In this paper, we are interested in the concept of area and mathematical proof. We will present and analyze some historical situations and problems where the area is used not as a subject but as a tool to prove and justify results a priori not related to the area. Area as a school content, has been a subject of many researches where authors was interested in the difficulties of the counting or measuring areas, relations between areas and perimeters, area units, etc. Some important scientific results are built upon the area, for example, Kepler’s Laws for the planets motion.

Solar H. C., Deulofeu J. **CONTINGENCY IN MATHEMATICS LESSONS THROUGH ARGUMENTATIVE ORCHESTRATION**

We consider that argumentative orchestration fosters the emergence of contingent moments in lessons. In order to provide evidence for this view, we describe one Mathematic lesson designed to include argumentative orchestration and which involve contingent situations triggered by students' mistakes. Two High Leverage Practices (HLP) were used as a strategy for analyzing how contingency is addressed: “Eliciting and interpreting each student's thinking” and “Identifying specific common patterns in students' thinking”. It was observed that these HLPs are an effective tool for addressing contingent moments

Rahimi Z., Talaee E, Reihani E., Fardanesh H. **DESIGNING AN INSTRUCTIONAL MODEL FOR REALIZATION OF MATHEMATICAL THINKING IN SECONDARY SCHOOL STUDENTS**

Our concern is changing the deviated path of teaching mathematics from merely teaching the procedures, methods and techniques to the main road of improving the students` thinking cap and reasoning in order for the students to get familiar with mathematical thinking. In this study, the researcher tries to design a model which leads to creating and improving mathematical thinking in secondary school students. She claims that the teachers` utilization of multiple solutions in teaching will help students to increase their mathematical thinking and their inclination and interest toward mathematics. The broad area of concepts in the nature of mathematical thinking made the researcher to choose among numerous mental process of mathematical thinking. She selected four elements as the key elements of mathematical thinking, which the specialists of this field of study believe them to be more comprehensive and applicable, namely: specializing, generalizing, conjecturing and convincing.

Brunner E., Pauli C. **DO TEACHERS TAKE FULL ADVANTAGE OF THE POTENTIAL PROVIDED BY DIFFERENT TYPES OF MATHEMATICAL PROOF?**

Drawing on data from a video study conducted in 32 classes of the 8th and 9th grade, the research to be presented aimed to investigate what types of mathematical proof teachers opt for when working on a specific proving problem in class. As the individual procedures make different cognitive demands on the students, the analyses also pursued the question as to what extent the use of a certain type of proof depends on the average achievement level of the participating classes. Moreover, the study was interested in whether the teachers take full advantage of the opportunities for active student involvement, because the different types of proof vary considerably in their potential for participation. The results point to a purposeful use in terms of an adaptation to the mean mathematical knowledge of the classes. What is still underused, by contrast, is the potential with respect to student participation that is peculiar to different types of proof.

Zhang G. **FROM ORIGINAL INDUCTION TO NUMERICAL REASONING**

In this paper, we discuss the issue that numerical reasoning in more essential and valuable for the development of the logical inference for students. Many teaching cases are analyzed to explain the teaching processes.

Jazby D. **TEACHER DISCURSIVE PRACTICES WHICH SUPPORT PRIMARY STUDENTS’ DEVELOPMENT OF DEDUCTIVE REASONING**

Six classes of grade 5 and 6 students were asked to prove a statement regarding the interior angles of triangles. Results showed that the number of students who could provide plausible deductive proofs for this task varied greatly between the six classes. This study seeks to try to identify what causes this variation in rates of deductive reasoning between classes. Variance in teacher discursive practices provides an economical explanation for this variance. Teacher discursive practices which encouraged students to publicly make generalizations and evaluate each other’s conjectures were present in classes in which students displayed higher rates of deductive responses to proof tasks.

Brown S. **TO BE OR NOT TO BE: STUDENTS' REASONING ABOUT THE CONSTRUCTIVE DILEMMA**

This study examines: (1) students’ preferences during comparative selection tasks with a constructive proof and a non-constructive proof, involving the law of the excluded middle form of a constructive dilemma; and (2) students’ metatheoretical difficulties with non-constructive proofs. Survey data did not confirm a preference for the constructive proof, with students’ selections indicating difficulties interpreting the logical structure of dilemma arguments. Analyses of students’ rationales provide further evidence that metatheoretical difficulties are prevalent and that constructive preferences may not be as common as anticipated by prior research on indirect proofs.

Zhou C. **A SURVEY OF 94 ELEMENTARY MATHEMATICS TEACHERS ABOUT MAY MATHEMATICAL REASONING BE TAUGHT AT ELEMENTARY SCHOOL PERIOD**

In the recent compulsory education mathematics curriculum standard put forward specific requirements of reasoning and proof for all grades. In the questionnaire and partial interview, 94 Chinese elementary school mathematics teachers have their understanding of mathematical reasoning and proving, a large proportion of them endorse pupils can reasoning and proving, They mentioned the main learning difficulties of elementary school students in mathematical reasoning and proving more than 9 types, and except several(7of 94) elementary school mathematics teachers, they do not feel they have the mathematical background to teach proving ,other informants said some teaching strategies in teaching of mathematics reasoning and proving in the elementary school with their experience.

Cramer J. C. **ANALYZING OBSTACLES FOR MATHEMATICAL ARGUMENTATION**

The difficult processes of teaching and learning argumentation have been examined from several different perspectives in the past decades. This paper presents an integrated theoretical approach based on Habermas’ works on language and communicative action. It takes into account obstacles for argumentation from the perspectives of discourse ethics, rationality and language. A paraphrased example analysis is provided to underpin the benefits of this integrated view.

Müller-Hill E. **ASPECTS OF OPERATIONAL MATHEMATICAL EXPLANATION**

This paper outlines a didactically motivated and philosophically informed theoretical approach towards an operational concept of mathematical explanation. In particular, according to this conception mathematical explanation in classroom does not start with and is not restricted to mathematical proof, but can be regarded as identifying a particular type of reasons for the occurrence of mathematical phenomena. Nevertheless, mathematical proofs are still within the scope of the conception, as explanations for sufficiently skilled epistemic subjects. As a by-product of the underlying operational perspective, the conception brings mathematical explanation, explanation in the sciences and life-world explanations closer together

Mata-Pereira J., Ponte J.P. **ENHANCING STUDENTS' MATHEMATICAL REASONING IN WHOLE CLASS DISCUSSIONS**

The aim of this paper is to understand how teacher’s actions may enhance students’ mathematical reasoning. We take mathematical reasoning as making justified inferences, and assume generalizing and justifying as central reasoning processes. The intervention, based on a design research, is carried out on lessons about equations in a grade 7 class. Data is analyzed considering reasoning processes and a model about teacher’s actions. The results show that a central challenging action followed by guiding and other challenging actions may be a path to promote students’ mathematical reasoning

Iwata K, Miyazaki M, Makin T, Fujita T. **LEARNING OF APPLICATION OF FUNCTIONS THROUGH CONSTRUCTIING PROOFS**

In Japanese lower secondary schools, constructing a proof is explicitly learned in geometry and algebra, but, in the learning of functions proving is not explicitly taught, even though students are often given opportunities to justify their thoughts or answers. We consider that this is due to the fact that we do not have a consensus view as to what constructing a proof in the teaching and learning of functions would mean. In this paper we provide suggestions to help conceptualize proofs in the context of the teaching and learning of functions. Our approach is to use mathematical modeling process, i.e. reasoning around formulating, employing, and interpreting, and we will give consideration as to why such an approach would be useful for conceptualizing proofs in functions.

Ferreira F. A., Santos C. A. B. **MATHEMATICAL PROOFS: INTERPRETATIVE ANALYSIS OF RESEARCHES PRESENTED AT ICME BETWEEN 2003 AND 2013**

The aim of this paper is to present a State of the Art of papers regarding the teaching and learning of mathematical proof and proving. To do so, we have mapped out papers presented in Mathematical Education events from 2003 to 2013. In this article, we elucidate data from three editions of the International Congress on Mathematical Education (ICME), i.e. ICME 10, ICME 11 and ICME 12. The results about the mapped out papers that exposed reflections about in-class proof and proving and the role of the teacher in this environment will be presented, from a hermeneutic approach, taking teachers’ knowledge and didactic-pedagogic choices into account. Furthermore, the results presented may contribute to the teaching activity regarding proof and proving, even though there is little research about this theme focusing on teaching practice.

Dogan M. F. **NATURE OF TEACHERS’ ENGAGMENTS IN PROVING ACTIVITES**

Although reasoning and proof in learning and teaching mathematics is crucial, both students and their teachers face great difficulties when engaging in proving activities. One potential cause for such difficulties might be due to teachers’ conception of proof. This study examines secondary school in-services teachers’ engagement in proving activities by providing observational data from a master’s level professional development course that focuses on teaching reasoning and proof. The results show that teachers were very successful at engaging in exploration of the proving tasks, but they fail to produce complete-deductive arguments and evaluate arguments.

An T. **PRESERVICE SECONDARY MATHEMATICS TEACHERS’ CONCEPTION OF APPLICATION OF THEOREMS IN GEOMETRY**

It is widely accepted by the field of mathematics education that proof and reasoning should be integrated into students’ mathematical experiences across all grades and across a breadth of content areas. However, studies have shown that both secondary students and teachers have encountered difficulties with learning and teaching geometry proofs. By adopting a collective case study design, this study looks in-depth at preservice secondary mathematics teachers’ (PSMTs) conception of application of geometry theorems in order to help them develop their proof writing ability more effectively. Three task-based interviews are designed to assess each of the six PSMTs’ participatory and anticipatory stages of conception of three aspects of application of theorems respectively. Results of the study will provide suggestions for the design of undergraduate level geometry courses and other proof training programs for in-service teachers.

Douek N. **PROMOTING EXPLORATION IN THE PERSPECTIVE OF TEACHING AND LEARNING PROVING PRACTICES IN MATHEMATICS**

This paper concerns students' approach to conjecturing and proving in the perspective of a working hypothesis about filling the gap, evidenced in the literature, between exploration in the conjecturing phase, and access to a proof of the conjectured statement. Some evidence is provided about the potential richness of students' exploration. We also deal with how the teacher may help students to develop their exploration skills, at the same time building links between exploration and proof.

Isler I. **WHAT ARE ELEMENTARY TEACHERS’ EXPECTATIONS REGARDING REASONING AND PROOF IN SCHOOL MATHEMATICS?**

Reasoning and proof play critical roles in the discipline of mathematics, and recent reform initiatives in the United States have elevated the status of reasoning and proof in school mathematics, advocating that it be a continuous part of K-12 education (Common Core State Standards Initiative [CCSSI], 2010; National Council of Teachers of Mathematics [NCTM], 2000). This study investigated elementary teachers’ expectations regarding reasoning and proof in school mathematics through an online survey and follow-up interviews. The results indicated that elementary teachers were favorable towards visual generic example arguments but saw deductive arguments as too complex and inappropriate for their students.

Griffiths B. J. **A COMPARISON OF SYLLOGISTIC REASONING SKILLS AMONG AMERICAN UNDERGRADUATES**

It has long been argued that the study of mathematics promotes logical reasoning. As a result, the foundations of formal logic are incorporated into most undergraduate programs in the United States, usually as part of general education mathematics courses for non-science majors. This study of 203 undergraduates compares the effect that these courses have to see if the inclusion of formal logic in the curriculum has the desired effect of increasing the reasoning skills of liberal arts majors, or whether those majoring in engineering and mathematics – even those who have not been trained in formal logic – still maintain an advantage.

Magiera M. T., Zambak V. S. **ANALYSIS OF ARGUMENTS FORMULATED BY GRADES 1-8 PROSPECTIVE TEACHERS IN “CONSTRUCTING” AND “CRITIQUING” PROBLEM SITUATION**

This study examined data from a mathematics course for grades 1-8 prospective (PST) teachers. In this course the instructor routinely required students to justify and analyze each other’s solution methods with a goal of supporing their own understanding of mathematical argumentation. Over the course of the semester, we examined changes in the overal quality of PSTs’ arguments and the processes by which their individual and collective notions of mathematical argumentation and argument strength and coherence developed. In this paper, we compare PSTs’ arguments in two types of problem situations: problems that asked them to construct mathematical arguments, and problems that asked them to analyze and critique mathematical arguments of elementary and middle school students. We examine PSTs’ responses to both types of tasks with a focus on four indicators that collectively contribute to argument’s strength and coherence.

ZhiLing W. **CASES STUDY ON EIGHT GRADE STUDENTS’ PSYCHOLOGICAL MODEL OF GEOMETRIC REASONING AND PROOF - IN CASE OF CONGRUENT TRIANGLES**

Mathematical reasoning and proof is an important ability, it is the element of the mathematics core literacy. However, it is difficult for high school students to learn. This is one of factors that the geometric reasoning and proof is weakened in mathematical curriculum. In this paper, we attempt to investigate and analyze eight grade students’ psychological model of geometric reasoning and proof from the mathematical learning psychological perspective, to figure out major mental activities of reasoning and proof. We believe that this paper could provide useful references for the studies about students' cognitive impairment of geometric reasoning and proof.

Hohenwarter M., Kovács Z., Recio T. **DECIDING GEOMETRIC PROPERTIES SYMBOLICALLY IN GEOGEBRA**

It is well known that Dynamic Geometry (DGS) software systems can be useful tools in the teaching/learning of reasoning and proof. GeoGebra 5.0 was recently extended by an Automated Theorem Prover (ATP) subsystem that is able to compute proofs of Euclidean geometry statements. Free availability and portability of GeoGebra has made it possible to harness these novel techniques on tablets, smartphones and computers. Then, we think it is urgently necessary to address the new challenges posed by the availability of geometric ATP’s to millions of students worldwide.

Park H. **ONE COLLEGE STUDENT’S PERCEPTIONS OF PROOF METHODS AND CHARACTERISTICS OF CHOOSING PROOF METHODS IN CONSTRUCTING PROOFS**

This case study examines a college student’s perceptions of proof methods with respect to direct proof, proof by contrapositive, proof by cases, proof by contradiction, and proof by induction and her rationales for choosing certain methods of proving. The results from this research show Camilla’s proof method preferences are largely aligned with the order in which she tried to use the proof methods with some exceptions. Also, the characteristics of proof methods she had constructed affected her choice of proving method.

Lampen E. **PROBLEMATISING THE CIRCLE: MATHEMATICS EDUCATION STUDENTS’ CONSTRUCTION REASONING**

Logical reasoning based on the circle as a construction tool rests on information about distance relationships between points and line segments. The question why a construction works is fundamentally answered by referring to such relationships constructed with a compass. Leaving off explanations of the logic behind constructions by using the circle as a construction template, presents spatio-graphic reasoning rather than theoretical reasoning. Our ongoing analysis of mathematics education students’ construction-based reasoning suggests that the circle is understood at a purely visual level, evident from their written definitions of a circle and their explanations of its use in construction. This is problematic for the development of logical reasoning in Euclidean Geometry. We describe the prevalent construction-reasoning of a group of 50 primary school mathematics education students when they solve a construction problem and the implications for logical reasoning.

Mobarakeh F. A., Fadaee M. R. **THE STATUS OF REASONING AND PROOF IN IRANAIN SEVENTH-GRADE MATHEMATICS TEXTBOOK**

The aim of this study is to investigate the status of reasoning and proof in Iranian seventh-grade mathematics textbook. It is used a content analysis methodology. We classified the units of analysis according to the mode of reasoning used. It was indicated a significant class of propositions that needs a mode of reasoning that there is no in the Stacy and Vincent's category (2009).

Stenger C. L., Jerkins J. A., Jenkins J. T., Stovall J. E. **USING COMPUTER PROGRAMMING TO TEACH GENERALIZATION**

As colleagues in a Mathematics/Computer Science department, we found that many of our undergraduates were not able to participate successfully in the full range of STEM course offerings. In response to this need, we developed a strategy for explicit instruction in mathematical generalization. Our instructional design is grounded in a theory of mathematical learning that uses computer programming to induce students to build the mental frameworks needed for understanding a math concept. The design includes writing mini programs to explore a mathematical concept, finding general expressions in the code, making conjectures about the relationships among general expressions, and writing logical arguments for the conjectures. We share results from a longitudinal study of 106 middle and high school math teachers attending professional development workshops employing this teaching method over a period of 3 years.

Fahse C. **DIFFERENT TYPES OF ARGUMENTATION IN A QUASI-LONGITUDINAL STUDY IN A SECONDARY SCHOOL**

In this study we explore students’ ways of argumentation concerning division by zero. The answers of 365 students of four different grades in a German secondary school were analyzed on the basis of a filled in questionnaire asking the students to explain their results of 7:0. Applying qualitative content analysis (Mayring, 2000) we were able to distinguish three different types of argumentation. The relative frequencies of these different types vary with the progressing age of the students.

Prusak N., Swidan O., Schwarz B. **FROM PEER ARGUMENTATION TO DEDUCTIVE REASONING AND PROOFS**

This study aims at examining how a meticulous design of a series of consecutive learning tasks in which a synchronous collaborative DGE tool is at disposal, may lead Grade 9 students to productive peer argumentation. By productivity, we mean here both a shift from reasoning based on visualization to reasoning moved by logical necessity and proving, as well as the emergence of geometrical principles. We report on a case study in which a dyad solved a geometrical task on the properties of diagonals and on the hierarchical classifications of quadrilaterals. We show that the characteristics of the environment we used – the Virtual Math Teams (VMT)) which includes a Geogebra applet shared by all participants and offers students the opportunity to work together collaboratively are crucial: VMT encourages generating assumptions and students' co-construction of meaning, and the chat rooms support collaboration.

Garcez Palha S. A., Spandaw J. **HOW COLLABORATIVE REASONING CONTRIBUTES TO STUDENT’S UNDERSTANDING OF INTEGRALS?**

Small group work can elicit students to think mathematically by providing a natural context for students to communicate about their ideas, to listen to each other’s contributions and to adapt, modify or improve their conceptions. In this paper we discuss the results of a qualitative study on students’ reasoning in small groups. The context of the study was an 8-week teaching experiment on the learning of integrals. By a case study, we investigate how the students’ discussion, while solving reasoning tasks in a small group, contributes to their understanding of the notion of ‘integral’.

Dhlamini Z. B., Chuene K. **MATHEMATICAL REASONING STRATEGIES THAT ARE CHALLENGING FOR LEARNERS IN THE ANA IN SOUTH AFRICA**

The purpose of this study was to elucidate mathematical reasoning that was tested in South Africa’s grade 9 Annual National Assessments (ANA) as well as those exhibited by learners’ in their response to the tests. The study used the first three consecutive ANA test questions n=182, 2012 with 61 questions, 2013 with 59 questions, 2014 with 62 questions and n=1000 learners responses to the 2014 paper. Findings on the analysis of the question paper revealed that on the four themes of mathematical reasoning, inductive, deductive, analogical, and conjecturing very little was tested by the ANA tests. On the limited reasoning tested by the tests, learners performed extremely poor on these questions and in most instances they did not even respond to such questions.

Azrou N. **PROOF TEXT WRITING AT THE UNDERGRADUATE LEVEL: NEW FINDINGS FROM STUDENTS' INTERVIEWS**

In this paper we analyse third year university students' writing of a proof text as the final step of the proving process. In particular, we focus on students' difficulties to write their proof text when answering open questions for which the proof process has to be built up, as opposite to traditional tasks of calculation or direct use of a mathematical result. We tried to identify the reasons behind writing an unclear, messy draft instead of a clear readable proof text. This paper concerns some results emerging from the comparison between interpretative hypotheses derived from the analysis of students’ proof texts, and their answers during subsequent interviews.

Johansson H. **REAL-LIFE CONTEXT AND MATHEMATICAL REASONING – INFLUENCES ON STUDENTS’ SUCCESS ON MATHEMATICS TASKS**

Within the field of mathematics education there seem to exist common assumptions concerning the value of introducing real-life contexts in mathematics tasks. In this study, the influence of real-life context on upper secondary students’ success in test tasks is explored. The study also departs from earlier studies concerning mathematical reasoning requirements when students solve national mathematics tests. Data consist of tasks from six Swedish national tests, as well as students’ results (number of students 829 ≤ n ≤ 3481). Each task is categorised as having a real-life context or not, and if creative mathematical reasoning is required or not. Both descriptive statistics and significance testing have been used for the analyses. The results indicate that real-life context had a positive effect on students’ success if the task required creative mathematical reasoning and that this effect was higher for students with lower grades.

Vanegas Diaz J. A. **RECONSTRUCTION OF AN ABDUCTIVE STRUCTURE: THE CASE OF EQUAL AREAS IN GEOMETRY**

In this investigation we highlight the role of collective argumentation and the arguments’ socialization phase like key scenarios of social interaction for understanding structural aspects of a proof. In particular, we exemplify an abductive structure. A group of undergraduate students were asked to prove that a line through the center of a square is divided into two regions of equal area. Students used arguments that included at least one assumption to validate their conjecture. This suggest that they recruited an abductive hypothesis to construct their claims. We discuss possible relationship between abductive structures and proof in mathematics.

Kondratieva M. F. **WHAT CAN BE LEARNER BY TEACHERS THROUGH THE PROCESS OF COLLECTIVE PRODUCTION OF MULTIPLE PROOFS?**

This paper describes a teacher education intervention based on participants’ engagement in production of multiple proofs in a mixed group setting. The data extracted from self-reports and group communications reveal many instances indicating improvements in teachers’ understanding of the proving and explanation processes through participation in these activities.

Fernández-León A., Toscano R., Gavilán-Izquierdo J. M.
**A MODEL TO CHARACTERIZE THE ACTIVITIES OF PROVING AND CONJECTURING OF PROFESIONAL MATHEMATICIANS**

Higgins A. L., Karunakaran S. S. **AN INQUIRY-BASED APPROACH TO TEACHING AN INTRODUCTION TO PROOF COURSE**

Ozgur Z. **AN INVESTIGATION OF PROOF CONCEPTIONS IN A HIGH SCHOOL MATHEMATICS CLASSROOM**

Dongwi B. L., Schafer M.
**EXAMINING MATHEMATICAL REASONING THROUGH ENACTED VISUALIZATION WHEN SOLVING WORD PROBLEMS**

Conner K., Cirillo M., Otten S. **LAUNCHING PROOF: A MULTI-LEVEL ANALYSIS OF SEVEN TEXTBOOKS**

Ripoll C. C. **MATHEMATICAL REASONING AND PROOF IN SCHOOL**

Gholamazad S. **PROOF AS A LITERATE MATHEMATICAL DISCOURSE**

Tebaartz P. C. **PROVING IN MATHEMATICAL OLYMPIADS – A TASK ANALYSIS**

Krieger M., Paravicini W., Panse A. **SELF-EXPLANATION TRAINING FOR ENHANCING PROOF COMPREHENSION AT UNIVERSITY – AN EMPIRICAL ANALYSIS**

Çontay E. G., Paksu A. D., Kazak S. **THE PROOF SCHEMES OF PROSPECTIVE ELEMENTARY MATHEMATICS TEACHERS**

Hofmann A., Ali S., Gustavsen T. S. **UNDERSTANDING AND DEVELOPING PRACTICES OF REASONING IN MATHEMATICS AMONG PRE-SERVICE AND IN-SERVICE MATHEMATICS TEACHERS**

Krumsdorf J. **VISUAL REASONING**

Editors-in-chief –
Bettina Pedemonte,
Maria-Alessandra Mariotti

Associate Editors –
Orly Buchbinder,
Kirsti Hemmi,
Mara Martinez

Redactor –
Bettina Pedemonte

Scientific Board –
Nicolas Balacheff,
Paolo Boero,
Daniel Chazan,
Raymond Duval,
Gila Hanna,
Guershon Harel,
Patricio Herbst,
Celia Hoyles,
Erica Melis,
Michael Otte,
Philippe Richard,
Yasuhiro Sekiguchi,
Michael de Villiers,
Virginia Warfield