A coloured window on pre-service teachers’ conceptions of rational numbers

In undergraduate mathematics courses, pre-service elementary school teachers are often faced with the task of re-learning some of the concepts they themselves struggled with in their own schooling. This often involves different cognitive processes and psychological issues than initial learning: pre-service teachers have had many more opportunities to construct understandings and representations than initial learners, some of which may be more complex and engrained; pre-service teachers are likely to have created deeply-held–and often negative–beliefs and attitudes toward certain mathematical ideas and processes. In our recent research, we found that pre-service teachers who used a particular computer-based microworld, one emphasising visual representations of and experimental interactions with elementary number theory concepts, overcame many cognitive and psychological difficulties reported in the literature. In this study, we investigate the possibilities of using a similarly-designed microworld that involves a set of rational number concepts. We describe the affordances of this microworld, both in terms of pre-service teacher learning and research on pre-service teacher learning, namely, the helpful “window” it gave us on the mathematical meaning-making of pre-service teachers. We also show how their interactions with this microworld provided many with a new and aesthetically-rich set of visualisations and experiences.     

PANEL DISCUSSIONS

Representations of mathematical concepts play an important role in understanding: both in helping learners understand the to-be-learned material and in facilitating teachers’ understanding of pedagogical processes which, in turn, are involved in developing learners’ understanding. In this paper, we report on work with a cohort of pre-service primary teachers, with the aim of developing their understanding of mathematics, their confidence in their subject knowledge and their confidence in teaching mathematics. This was attempted through the introduction and use of a ‘representational approach’ to the teaching of the mathematical concepts required of teachers training to teach in primary schools in the UK. We present the results of attitude measures and a follow-up qualitative questionnaire in identifying whether and how the use of this representational approach supported pre-service teachers’ understanding and their confidence in teaching mathematics. The results suggest that the representational approach used had a positively significant impact on the attitudes towards studying and teaching mathematics.     

Understanding secondary mathematics: analysis of linguistic difficulties vis‐a‐vis errors

Misconceptions caused by misunderstanding mathematical language are of different types, e.g. interference of mathematical and non‐mathematical meanings, complexity or unfamiliarity of words, improper use of symbols, syntactical misunderstandings and redundancy or inadequacy of data. Fifteen mathematical words were presented to 84 students of 8th grade and the responses of students were analysed. Mathematical items of a Science Talent Search Test were also analysed from the answers of 100 candidates of 8th grade. The analysis casts light on the processes of the development of mathematical concepts which the students learn through the vicissitudes of the interaction of mathematical and non‐mathematical meanings of words which may be familiar or unfamiliar, relevant or irrelevant, and/or distinct or difficult, to the learners. Complications in understanding mathematical concepts for individual students are pointed out from these experiments. Acquaintance with such communication processes of learners can also help in detecting strategies of imparting mathematical instructions to the learners. Roles of the uses of common, mathematical, transformational, and story‐telling language have also been discussed. The paper concludes with some comments on the importance of guided discovery learning, error analysis by teachers, and the preparation of a register of mathematical language.     

Landscapes of Investigation

According to many observations, traditional mathematics education falls within the exercise paradigm. This paradigm is contrasted with landscapes of investigation serving as invitations for students to be involved in processes of exploration and explanation. The distinction between the exercise paradigm and landscapes of investigation is combined with a distinction between three different types of reference which might provide mathematical concepts and classroom activities with meaning: references to mathematics; references to a semi-reality, and references to a real-life situation. The six possible learning milieus are illustrated by examples. Moving away from the exercise paradigm and in the direction of landscapes of investigation may help to abandon the authorities of the traditional mathematics classroom and to make students the acting subjects in their learning processes. Moving away from reference to pure mathematics and in the direction of real life references may help to provide resources for reflection on mathematics and its applications. My hope is that finding a route among the different milieus of learning may provide new resources for making the students both acting and reflecting and in this way providing mathematics education with a critical dimension.     

Cultivating Computational Thinking Practices and Mathematical Habits of Mind in Lattice Land

There is a great deal of overlap between the set of practices collected under the term “computational thinking” and the mathematical habits of mind that are the focus of much mathematics instruction. Despite this overlap, the links between these two desirable educational outcomes are rarely made explicit, either in classrooms or in the literature. This paper presents Lattice Land, a computational learning environment and accompanying curriculum designed to support the development of mathematical habits of mind and promote computational thinking practices in high-school mathematics classrooms. Lattice Land is a mathematical microworld where learners explore geometrical concepts by manipulating polygons drawn with discrete points on a plane. Using data from an implementation in a low-income, urban public high school, we show how the design of Lattice Land provides an opportunity for learners to use computational thinking practices and develop mathematical habits of mind, including tinkering, experimentation, pattern recognition, and formalizing hypothesis in conventional mathematical notation. We present Lattice Land as a restructuration of geometry, showing how this new and novel representational approach facilitates learners in developing computational thinking and mathematical habits of mind. The paper concludes with a discussion of the interplay between computational thinking and mathematical habits of mind, and how the thoughtful design of computational learning environments can support meaningful learning at the intersection of these disciplines.     

Paradox,programming, and learning probability: A case study in a connected mathematics framework

Formal methods abound in the teaching of probability and statistics. In the Connected Probability project, we explore ways for learners to develop their intuitive conceptions of core probabilistic concepts. This article presents a case study of a learner engaged with a probability paradox. Through engaging with this paradoxical problem, she develops stronger intuitions about notions of randomness and distribution and the connections between them. The case illustrates a Connected Mathematics approach: that primary obstacles to learning probability are conceptual and epistemological; that engagement with paradox can be a powerful means of motivating learners to overcome these obstacles; that overcoming these obstacles involves learners making mathematics—not learning a “received” mathematics and that, through programming computational models, learners can more powerfully express and refine their mathematical understandings.     

What is the opposite of cat? A gentle introduction to group theory

This paper has originated from our interest in approaching mathematical concepts starting from people's common-sense intuitions and building up from there. This goal is challenging both in designing the didactical transposition and sequencing of the mathematical subject matter, and in adopting the open and interactive teaching approach needed to achieve students' active participation and reflection. To demonstrate these challenges, and our experience of trying to cope with them, we have chosen the concept of ‘inverses’ as used in group theory, and its common-sense precursor ‘opposites’. We present our approach via a series of workshop iterations, which summarizes our experience in the many actual workshops we ran in Israel and in Denmark.     

Teachers creating mathematical models to fairly distribute school funding

Our study aims to investigate what teachers do as they draw on their mathematical understanding and personal experiences to engage in social justice-oriented mathematical modeling. We analyze what ideas were expressed by teachers regarding their mathematical identities while they explore, wrestle with, and reconcile the underlying societal values that support mathematical models. We invited groups of teachers to make mathematical models for distributing school funding given real data from diverse, anonymized schools. Our results show that teachers created and refined diverse mathematical models to connect the mathematical world and societal space and these models reflected different societal values. Drawing on their own experiences, teachers expressed a sense of agency and critical consciousness while making decisions about school funding. This study delineates mathematical contents and processes necessary for advancing a societal goal of fairly distributing funds and we explore how teachers connect to this context as learners and members of society.     

The role of examples in the learning of mathematics and in everyday thought processes

The purpose of this paper is to present a view of three central conceptual activities in the learning of mathematics: concept formation, conjecture formation and conjecture verification. These activities also take place in everyday thinking, in which the role of examples is crucial. Contrary to mathematics, in everyday thinking examples are, very often, the only tool by which we can form concepts and conjectures, and verify them. Thus, relying on examples in these activities in everyday thought processes becomes immediate and natural. In mathematics, however, we form concepts by means of definitions and verify conjectures by mathematical proofs. Thus, mathematics imposes on students certain ways of thinking, which are counterintuitive and not spontaneous. In other words, mathematical thinking requires a kind of inhibition from the learners. The question is to what extent this goal can be achieved. It is quite clear that some people can achieve it. It is also quite clear that many people cannot achieve it. The crucial question is what percentage of the population is interested in achieving it or, moreover, what percentage of the population really cares about it.     

Cognitive Demands and Second-Language Learners: A Framework for Analyzing Mathematics Instructional Contexts

The issues involved in teaching English language learners mathematics while they are learning English pose many challenges for mathematics teachers and highlight the need to focus on language-processing issues related to teaching mathematical content. Two realistic-type problems from high-stakes tests are used to illustrate the complex interactions between culture, language, and mathematical learning. The analyses focus on aspects of the problems that potentially increase cognitive demands for second-language learners. An analytical framework is presented that is designed to enable mathematics teachers to identify critical elements in problems and the learning environment that contribute to increased cognitive demands for students of English as a second language. The framework is proposed as a cycle of teacher reflection that would extend a constructivist model of teaching to include broader linguistic, cultural, and cognitive processing issues of mathematics teaching, as well as enable teachers to develop more accurate mental models of student learning.     

Cognitive Demands and Second-Language Learners: A Framework for Analyzing Mathematics Instructional Contexts

The issues involved in teaching English language learners mathematics while they are learning English pose many challenges for mathematics teachers and highlight the need to focus on language-processing issues related to teaching mathematical content. Two realistic-type problems from high-stakes tests are used to illustrate the complex interactions between culture, language, and mathematical learning. The analyses focus on aspects of the problems that potentially increase cognitive demands for second-language learners. An analytical framework is presented that is designed to enable mathematics teachers to identify critical elements in problems and the learning environment that contribute to increased cognitive demands for students of English as a second language. The framework is proposed as a cycle of teacher reflection that would extend a constructivist model of teaching to include broader linguistic, cultural, and cognitive processing issues of mathematics teaching, as well as enable teachers to develop more accurate mental models of student learning.     

Utilization of mathematical programming for public systems: An application to effective formation of integrated regional information networks

This paper concerns a methodological reflection on the multiobjective approach to public systems which involve group decision processes. Particular attention is given to an integrated program of regional systems which include value trade-offs between multiple objectives. Our intention is to combine the judgmental processes with the optimization processes in the soft public systems. A two-layer approach is applied. At the first layer, each regional program is formulated in mathematical programming based on a utility assessment with different regional characteristics. Each subsystem independently reflects its particular concern as a single agent. The dual optimal solutions obtained for each subsystem are treated as an index, or the theoretical prices, representing the value trade-offs among the multiple objectives. At the second layer, an effective formation of interregional cooperation for compromising the conflicting regional interests is examined. Ann-person cooperative game in the characteristic function form is used to evaluate the effectiveness of the cooperation. The characteristic function for the game is derived on the incremental value of the regional benefit after the formation of a cooperation. The nucleolus and the augmented nucleolus as the solution concepts of the cooperative game are used for indicating the effectiveness of the cooperation. Finally using alternative criteria, the results in assessing the best decisions are examined comparatively.     

Construction of a geometrical concept within a dialectical learning environment

The goals of the study were to design and investigate a teaching-learning environment that encourage freedom and autonomy of pre-service teachers in constructing their own new geometrical concept, and to analyze the dialectic process of the concept construction in the designed environment. A dialectic process of the participants’ defining activity emerged from the necessity to resolve the tensions between hypothesizing the concept’s examples and the appropriate critical attributes. Such a process created a vivid learning trajectory, in which learners examined logically the ways in which the examples, the critical attributes and the definition match. In this way, the three elements of the mathematical concept cannot play a passive role, or be neglected, and it appeared that prototypical examples were not created, so that no example is more dominant than others. The groups constructed different concepts, but with full harmony between its definition, example space, and concept-critical attributes.     

MATHEMATICS LEARNING TRAJECTORIES: CLASS,CAPITAL AND CONFLICT

This is the story of Marie and Edward as they approach the time of transfer from the primary to the secondary school. They both consider themselves to be successful mathematicians and have shared common classroom experiences throughout their time in the primary school. However, as they approach this critical relocation point it becomes clear that their future mathematical careers are set on two distinct trajectories. This paper explores the impact of the class-formed family habitus on them as learners of mathematics, and describes how family transferred cultural capital propels one of them whilst notions of conflict threaten to restrict the progress of the other.     

Dynamic hyperbolic geometry: building intuition and understanding mediated by a Euclidean model

This paper explores a deep transformation in mathematical epistemology and its consequences for teaching and learning. With the advent of non-Euclidean geometries, direct, iconic correspondences between physical space and the deductive structures of mathematical inquiry were broken. For non-Euclidean ideas even to become thinkable the mathematical community needed to accumulate over twenty centuries of reflection and effort: a precious instance of distributed intelligence at the cultural level. In geometry education after this crisis, relations between intuitions and geometrical reasoning must be established philosophically, rather than taken for granted. One approach seeks intuitive supports only for Euclidean explorations, viewing non-Euclidean inquiry as fundamentally non-intuitive in nature. We argue for moving beyond such an impoverished approach, using dynamic geometry environments to develop new intuitions even in the extremely challenging setting of hyperbolic geometry. Our efforts reverse the typical direction, using formal structures as a source for a new family of intuitions that emerge from exploring a digital model of hyperbolic geometry. This digital model is elaborated within a Euclidean dynamic geometry environment, enabling a conceptual dance that re-configures Euclidean knowledge as a support for building intuitions in hyperbolic space—intuitions based not directly on physical experience but on analogies extending Euclidean concepts.     

The stability of learners’ choices for real-life situations to be used in mathematics

One of the efforts to improve and enhance the performance and achievement in mathematics of learners is the incorporation of life-related contexts in mathematics teaching and assessments. These contexts are normally, with good reasons, decided upon by curriculum makers, textbook authors, teachers and constructors of examinations and tests. However, little or no consideration is given to whether students prefer and find these real-life situations interesting. There is also a dearth of studies dealing explicitly with the real-life situations learners prefer to deal with in mathematics. This issue was investigated and data on students’ choices for contextual issues to be used in mathematics were collected at two time periods. The results indicate that learners’ preferences for contextual situations to be used in mathematics remained fairly stable. It is concluded that real-life issues that learners highly prefer are not normally included in the school mathematics curriculum and that there is a need for a multidisciplinary approach to develop mathematical activities which take into account the expressed preferences of learners.     

THE NATURE OF PARTICIPATION AFFORDED BY TASKS,QUESTIONS AND PROMPTS IN MATHEMATICS CLASSROOMS

This paper reports on the development of an analytical instrument which identifies mathematical affordances in the public tasks, questions and prompts of mathematics classrooms. The aim is to become more articulate about mathematical activity. I have explored the use of several frameworks which identify learning outcomes, structures of knowledge, mental actions, teaching actions and intentions and found that none of them give me access to the detail of what makes one mathematics lesson different from another for learners. From the experience of using these I devised a new analytical tool which unfolds patterns of participation afforded in mathematics lessons. This tool has been tested on several videos of lessons, and has been used by pre-service teaching students to analyse their own lessons.     

Mathematisches Denken in der Linearen Algebra

How can first years students learn to think and act mathematically by learning Linear Algebra? We want to present an approach that considers reflection of mathematical acting and its connections to general thinking to be an important part of learning. By understanding mathematics as a specific conventionalization of general thinking, patterns of general thinking can become the starting point for learning mathematics. This points out the specific contribution that mathematics can give to describe reality. By example of Linear Algebra, we discuss the common ground and differences between thinking in mathematics and in non-mathematical subjects. Based on this discussion, we analyse why and how these reflections can be objects of learning.     

A model for developing students’ example space: the key role of the teacher

Our research addresses the role of examples to foster the students’ development of the mathematical concepts, and of their mathematical ways of thinking. We consider the notion of example space introduced by Watson and Mason (Mathematics as a constructive activity: learners generating examples, 2005), particularly when it is not formed by a simple juxtaposition of examples, rather it is endowed by a certain structure. Such a structure is provided by the semiotic actions and by the theoretic and logical dimensions of the mathematical activities. However, the formation of structured example spaces is far from being an automatic process. In this paper, we focus on the genesis of examples and on the role of the teacher in helping the students to structure their examples spaces through the so-called cognitive apprenticeship method. We point out that the genesis of examples is often accomplished within a complex cyclic dynamics, the “cycle of examples production and modification”. We illustrate it by means of two emblematic episodes from a classroom discussion. We show that the teacher’s intervention can be crucial in helping the students to modify a wrong example, to generate the right one for the task and to start the long-term process of building up the structure of their own space of examples.