Teachers attending to students’ mathematical reasoning: lessons from an after-school research program

There is a documented need for more opportunities for teachers to learn about students’ mathematical reasoning. This article reports on the experiences of a group of elementary and middle school mathematics teachers who participated as interns in an after-school, classroom-based research project on the development of mathematical ideas involving middle-grade students from an urban, low-income, minority community in the United States. For 1 year, the teachers observed the students working on well-defined mathematical investigations that provided a context for the students’ formation of particular mathematical ideas and different forms of reasoning in several mathematical content strands. The article describes insights into students’ mathematical reasoning that the teachers were able to gain from their observations of the students’ mathematical activity. The purpose is to show that teachers’ observations of students’ mathematical activity in research sessions on students’ development of mathematical ideas can provide opportunities for teachers to learn about students’ mathematical reasoning.     

Classroom Interpreting Games with an Illustration

Classroom communication has been recognized as a process in which ideas become objects of reflection, discussion, and amendments affording the construction of private mathematical meanings that in the process become public and exposed to justification and validation. This paper describes an explanatory model named “interpreting games”, based on the semiotics of Charles Sanders Peirce, that accounts for the interdependence between thought and communication and the interpretation of signs in which teacher and students engage in mathematics classrooms. Interpreting games account both for the process of transformation (in the mind of the learner) of written marks into mathematical signs that stand for mathematical concepts and for the continuous and converging private construction of mathematical concepts. Teacher–student and student–student collaborative interactions establish a mathematical communication that shapes and is also shaped by the conceptual domains and the domains of intentions and interpretations of the participants. A teaching episode with third graders is analyzed as an example of a classroom interpreting game.     

Evaluating a web based intelligent tutoring system for mathematics at German lower secondary schools

The present study researches the implementation of a web based intelligent tutoring system for mathematics at lower secondary schools. In recent years, there is growing concern about the worrying situation at German lower secondary schools. Data from large scale educational assessments in the county of North Rhine-Westphalia (NRW) show that children at lower secondary schools have an embarrassing paucity of basic mathematical skills (Leutner et al., Lernstandserhebungen 9. Klasse 2004 in NRW: Erster Kurzbericht zur wissenschaftlichen Begleitung, 2004). In order to improve these basic mathematical skills in lower secondary school children, several schools implemented the web based intelligent tutoring system eFit. The aim of the present research was to investigate whether eFit constitutes an effective intervention of this target group. The results show that compared to a non-treatment control group, children in the eFit group significantly improved their arithmetic performance over a period of 9 months. As will be discussed, the findings have to be treated with cautions because eFit was specifically designed to alleviate mathematical difficulties and therefore “trained for the test” whereas traditional mathematics instruction followed the regular curriculum. The implications of this will be considered in the light of existing theory and research.     

Using culture as a resource in mathematics: the case of four Mexican–American prospective teachers in a bilingual after-school program

This paper explores Mexican–American prospective teachers’ use of culture—defined as social practices and shared experiences—as an instructional resource in mathematics. The setting is an after-school mathematics program for the children of Mexican heritage. Qualitative analysis of the prospective teachers’ and children’s interactions reveals that the nature of the mathematical activities affected how culture was used. When working on the “binder activities,” prospective teachers used culture only in non-mathematical contexts. When working on the “recipes project,” however, culture was used as a resource in mathematical contexts. Implications for the mathematics teacher preparation of Latinas/os are discussed.     

The Mathematical Concept of set and the 'Collection' Model

Analysing the various misconceptions held by students with regard to the mathematical set concept, the authors hypothesized that these misunderstandings may be explained by the initial ‘collection’ model. Even after learning the formal properties of a set in the mathematical sense, the students are still influenced in their reactions by the collection representation, which acts ‘from behind the scenes’ as a tacit model. If the mathematical concept is not continually reinforced through systematic use, it is the initial figural interpretation which will replace, as an effect of time, the formal one. The findings confirmed this hypothesis. This revised version was published online in July 2006 with corrections to the Cover Date.     

“Can any fraction be turned into a decimal?” A case study of a mathematical group discussion

This case study examines two days of teacher-led large group discussion in a fifth grade about a mathematical question intended to support student exploration of relationships among fraction and decimal representations and rational numbers. The purpose of the analysis is to illuminate the teacher’s work in supporting student thinking through the use of a mathematical question embedded in a position-driven discussion. The focus is an examination of the ways that the emergence of mathematical ideas is partially shaped by complex interactions among the mathematical contents of the question, the inherent properties of the discourse format and participant structure, and the available computational methods. The teacher’s work is conceptualized in terms of actions and practices that coordinate these diverse tools, in constant response to students’ concurrent use of them. This revised version was published online in July 2006 with corrections to the Cover Date.     

What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems

School students of all ages, including those who subsequently become teachers, have limited experience posing their own mathematical problems. Yet problem posing, both as an act of mathematical inquiry and of mathematics teaching, is part of the mathematics education reform vision that seeks to promote mathematics as an worthy intellectual activity. In this study, the authors explored the problem-posing behavior of elementary prospective teachers, which entailed analyzing the kinds of problems they posed as a result of two interventions. The interventions were designed to probe the effects of (a) exploration of a mathematical situation as a precursor to mathematical problem posing, and (b) development of aesthetic criteria to judge the mathematical quality of the problems posed. Results show that both interventions led to improved problem posing and mathematically richer understandings of what makes a problem ‘good.’     

Accounting for Students’ Schemes in the Development of a Graphical Process for Solving Polynomial Inequalities in Instrumented Activity

This paper provides an instrumental account of precalculus students’ graphical process for solving polynomial inequalities. It is carried out in terms of the students’ instrumental schemes as mediated by handheld graphing calculators and in cooperation with their classmates in a classroom setting. The ethnographic narrative relays an instrumental sociogenetic account of mathematical knowledge construction and foregrounds a progressive evolution of mathematical knowledge from the concrete to the abstract phase.     

Complementarity, sets and numbers

Niels Bohr's term‘complementarity' has been used by several authors to capture the essential aspects of the cognitive and epistemological development of scientific and mathematical concepts. In this paper we will conceive of complementarity in terms of the dual notions of extension and intension of mathematical terms. A complementarist approach is induced by the impossibility to define mathematical reality independently from cognitive activity itself. R. Thom, in his lecture to the Exeter International Congress on Mathematics Education in 1972,stated ‘‘the real problem which confronts mathematics teaching is not that of rigor,but the problem of the development of‘meaning’, of the ‘existence' of mathematical objects'. Student's insistence on absolute ‘meaning questions’, however,becomes highly counter-productive in some cases and leads to the drying up of all creativity. Mathematics is, first of all,an activity, which, since Cantor and Hilbert, has increasingly liberated itself from metaphysical and ontological agendas. Perhaps more than any other practice,mathematical practice requires acomplementarist approach, if its dynamics and meaning are to be properly understood. The paper has four parts. In the first two parts we present some illustrations of the cognitive implications of complementarity. In the third part, drawing on Boutroux' profound analysis, we try to provide an historical explanation of complementarity in mathematics. In the final part we show how this phenomenon interferes with the endeavor to explain the notion of number. This revised version was published online in July 2006 with corrections to the Cover Date.     

Promoting equity in mathematics teacher preparation: a framework for advancing teacher learning of children’s multiple mathematics knowledge bases

Research repeatedly documents that teachers are underprepared to teach mathematics effectively in diverse classrooms. A critical aspect of learning to be an effective mathematics teacher for diverse learners is developing knowledge, dispositions, and practices that support building on children’s mathematical thinking, as well as their cultural, linguistic, and community-based knowledge. This article presents a conjectured learning trajectory for prospective teachers’ (PSTs’) development related to integrating children’s multiple mathematical knowledge bases (i.e., the understandings and experiences that have the potential to shape and support children’s mathematics learning—including children’s mathematical thinking, and children’s cultural, home, and community-based knowledge), in mathematics instruction. Data were collected from 200 PSTs enrolled in mathematics methods courses at six United States universities. Data sources included beginning and end-of-semester surveys, interviews, and PSTs’ written work. Our conjectured learning trajectory can serve as a tool for mathematics teacher educators and researchers as they focus on PSTs’ development of equitable mathematics instruction.     

Using Math Journals to Enhance Second Graders’ Communication of Mathematical Thinking

As an action research project, using mixed methodology, this study investigated how the use of math journals affected second grade students’ communication of mathematical thinking. For this study, math journal instruction was provided. The data gathering included pre- and post- math assessment, students’ math journals, interviews with the students, and teacher’s reflective journal. Findings of the study indicated that the use of math journals positively influenced the students’ communication of mathematical thinking and the use of math vocabulary. Additionally, math journals served as a communication tool between the students and teacher and an assessment tool for the teacher. The implications of this study regarding students’ writing ability and time constraints issues were also discussed.     

A mathematical experience involving defining processes: in-action definitions and zero-definitions

In this paper, a focus is made on defining processes at stake in an unfamiliar situation coming from discrete mathematics which brings surprising mathematical results. The epistemological framework of Lakatos is questioned and used for the design and the analysis of the situation. The cognitive background of Vergnaud’s approach enriches the study of freshmen’s processes at university. The mathematical analysis and the results specifically underscore the in-action definitions and the zero-definitions and highlight the need of similar mathematical experiences in education, particularly focused on defining processes and the exploration of a research problem.     

Using the onto-semiotic approach to identify and analyze mathematical meaning when transiting between different coordinate systems in a multivariate context

The main objective of this paper is to apply the onto-semiotic approach to analyze the mathematical concept of different coordinate systems, as well as some situations and university students’ actions related to these coordinate systems. The identification of objects that emerge from the mathematical activity and a first intent to describe an epistemic network that relates to this activity were carried out. Multivariate calculus students’ responses to questions involving single and multivariate functions in polar, cylindrical, and spherical coordinates were used to classify semiotic functions that relate the different mathematical objects.     

Conceptualizing sound as a form of incarnate mathematical consciousness

Much of the evidence provided in support of the argument that mathematical knowing is embodied/enacted is based on the analysis of gestures and bodily configurations, and, to a lesser extent, on certain vocal features (e.g., prosody). However, there are dimensions involved in the emergence of mathematical knowing and the production of mathematical communication that have not yet been investigated. The purpose of this article is to theorize one of these dimensions, which we call incarnate sonorous consciousness. Drawing on microanalyses of two exemplary episodes in which a group of third graders are sorting geometric solids, we show how sound has the potential to mark mathematical similarities and distinctions. These “audible” similarities and distinctions, which may be produced by incarnate dimensions such as beat gestures and prosody, allow children to objectify certain geometrical properties of the objects with which they transact. Moreover, the analysis shows that sonorous production is intertwined with other dimensions of students’ bodily activity. These findings are interpreted according to the “theory of mathematics in the flesh,” an alternative to current embodiment/enactivist theories in mathematics education.     

Mathematical practices in a technological workplace: the role of tools

This paper investigates the role of tools in the formation of mathematical practices and the construction of mathematical meanings in the setting of a telecommunication organization through the actions undertaken by a group of technicians in their working activity. The theoretical and analytical framework is guided by the first-generation activity theory model and Leont’ev’s work on the three-tiered explanation of activity. Having conducted a 1-year ethnographic research study, we identified, classified, and correlated the tools that mediated the technicians’ activity, and we studied the mathematical meanings that emerged. A systemic network was generated, presenting the categories of tools such as mathematical (communicative, processes, and concepts) and non-mathematical (physical and written texts). This classification was grounded on data from three central actions of the technicians’ activity, while the constant interrelation and association of these tools during the working process addressed the mathematical practices and supported the construction of mathematical meanings that this group developed from the researchers’ perspective. Technicians’ emerging mathematical meanings referred to place value, spatial, and algebraic relations and were expressed through personal algorithms and metaphorical and metonymic reasoning. Finally, the educational implications of the findings are discussed.     

Drawing space: mathematicians’ kinetic conceptions of eigenvectors

This paper explores how mathematicians build meaning through communicative activity involving talk, gesture and diagram. In the course of describing mathematical concepts, mathematicians use these semiotic resources in ways that blur the distinction between the mathematical and physical world. We shall argue that mathematical meaning of eigenvectors depends strongly on both time and motion—hence, on physical interpretations of mathematical abstractions—which are dimensions of thinking that are typically deliberately absent from formal, written definition of the concept. We shall also show how gesture and talk contribute differently and uniquely to mathematical conceptualisation and further elaborate the claim that diagrams provide an essential mediating role between the two.     

A Cross-Analysis of the Mathematics Teacher’s Activity. An Example in a French 10th-Grade Class

The purpose of this paper is to contribute to the debate about how to tackle the issue of ‘the teacher in the teaching/learning process’, and to propose a methodology for analysing the teacher’s activity in the classroom, based on concepts used in the fields of the didactics of mathematics as well as in cognitive ergonomics. This methodology studies the mathematical activity the teacher organises for students during classroom sessions and the way he manages¹ the relationship between students and mathematical tasks in two approaches: a didactical one [Robert, A., Recherches en Didactique des Mathématiques 21(1/2), 2001, 7–56] and a psychological one [Rogalski, J., Recherches en Didactique des Mathématiques 23(3), 2003, 343–388]. Articulating the two perspectives permits a twofold analysis of the classroom session dynamics: the “cognitive route” students are engaged in—through teacher’s decisions—and the mediation of the teacher for controlling students’ involvement in the process of acquiring the mathematical concepts being taught. The authors present an example of this cross-analysis of mathematics teachers’ activity, based on the observation of a lesson composed of exercises given to 10th grade students in a French ‘ordinary’ classroom. Each author made an analysis from her viewpoint, the results are confronted and two types of inferences are made: one on potential students’ learning and another on the freedom of action the teacher may have to modify his activity. The paper also places this study in the context of previous contributions made by others in the same field.     

Forms of mathematical interaction in different social settings: examples from students’, teachers’ and teacher–students’ communication about mathematics

The study presented in this article investigates forms of mathematical interaction in different social settings. One major interest is to better understand mathematics teachers’ joint professional discourse while observing and analysing young students mathematical interaction followed by teacher’s intervention. The teachers’ joint professional discourse is about a combined learning and talking between two students before an intervention by their teacher (setting 1) and then it is about the students learning together with the teacher during their mathematical work (setting 2). The joint professional teachers’ discourse constitutes setting 3. This combination of social settings 1 and 2 is taken as an opportunity for mathematics teachers’ professionalisation process when interpreting the students’ mathematical interactions in a more and more professional and sensible way. The epistemological analysis of mathematical sign-systems in communication and interaction in these three settings gives evidence of different types of mathematical talk, which are explained depending on the according social setting. Whereas the interaction between students or between teachers is affected by phases of a process-oriented and investigated talk, the interaction between students and teachers is mainly closed and structured by the ideas of the teacher and by the expectations of the students.

Mathematical thinking of kindergarten boys and girls: similar achievement, different contributing processes

The objective of this study was to examine gender differences in the relations between verbal, spatial, mathematics, and teacher–child mathematics interaction variables. Kindergarten children (N = 80) were videotaped playing games that require mathematical reasoning in the presence of their teachers. The children’s mathematics, spatial, and verbal skills and the teachers’ mathematical communication were assessed. No gender differences were found between the mathematical achievements of the boys and girls, or between their verbal and spatial skills. However, mathematics performance was related to boys’ spatial reasoning and to girls’ verbal skills, suggesting that they use different processes for solving mathematical problems. Furthermore, the boys’ levels of spatial and verbal skills were not found to be related, whereas they were significantly related for girls. The mathematical communication level provided in teacher–child interactions was found to be related to girls’ but not to boys’ mathematics performance, suggesting that boys may need other forms of mathematics communication and teaching.