Patterns—a fundamental idea of mathematical thinking and learning

Taking advantage of patterns is typical of our everyday experience as well as our mathematical thinking and learning. For example a working day or a morning at school displays a certain structure, which can be described in terms of patterns. On the one hand regular structures give us the feeling of permanence and enable us to make predictions. On the other hand they also provide a chance to be creative and to vary common procedures. School students usually encounter patterns in math classes either as number patterns or geometric patterns. There are also patterns that teachers can find in analyzing the errors students make during their calculations (error patterns) as well as patterns that are inherent to mathematical problems. One could even go so far as to say that identifying and describing patterns is elementary for mathematics (cf. Devlin 2003). Practising good interacting with patterns supports not only the active learning of mathematics but also a deeper understanding of the world in general. Patterns can be explored, identified, extended, reproduced, compared, varied, represented, described and created. This paper provides some examples of pattern utilization and detailed analyses thereof. These ideas serve as “hooks” to encourage the good use of patterns to facilitate active learning processes in mathematics.     

Shifts of mathematical thinking in adolescence

This theoretical paper relates key features of the mathematics adolescents are expected to learn in school to other aspects of adolescent development. Difficulties in mathematical learning at that age include changes in perspective and in the actions that are mathematically productive. Commonly-recommended methods of trying to engage adolescents in mathematics do not necessarily enable students to shift to new perceptions and new ways of constructing mathematical understandings, yet the shifts students need to make are in accord with other aspects of adolescent development.     

Systems thinking and mathematical problem solving

Problem solving lies at the core of engineering and remains central in school mathematics. Word problems are a traditional instructional mechanism for learning how to apply mathematics to solving problems. Word problems are formulated so that a student can identify data relevant to the question asked and choose a set of mathematical operations that leads to the answer. However, the complexity and interconnectedness of contemporary problems demands that problem‐solving methods be shaped by systems thinking. This article presents results from three clinical interviews that aimed at understanding the effects that traditional word problems have on a student’s ability to use systems thinking. In particular, the interviews examined how children parse word problems and how they update their answers when contextual information is provided. Results show that traditional word problems create unintended dispositions that limit systems thinking.     

Qualities of examples in learning and teaching

In this paper, we theorise about the different kinds of relationship between examples and the classes of mathematical objects that they exemplify as they arise in mathematical activity and teaching. We ground our theorising in direct experience of creating a polynomial that fits certain constraints to develop our understanding of engagement with examples. We then relate insights about exemplification arising from this experience to a sequence of lessons. Through these cases, we indicate the variety of fluent uses of examples made by mathematicians and experienced teachers. Following Thompson’s concept of “didactic object” (Symbolizing, modeling, and tool use in mathematics education. Kluwer, Dordrecht, The Netherlands, pp 191–212, 2002), we talk about “didacticising” an example and observe that the nature of students’ engagement is important, as well as the teacher’s intentions and actions (Thompson avoids using a verb with the root “didact”. We use the verb “didacticise” but without implying any connection to particular theoretical approaches which use the same verb.). The qualities of examples depend as much on human agency, such as pedagogical intent or mathematical curiosity or what is noticed, as on their mathematical relation to generalities.     

Bridging second-grade children's thinking and mathematical recording

The focus of this article is how what students do and say to solve a problem may be recorded to mirror the student's actions or thoughts rather than portraying a common algorithm that is not connected to the student's thinking. It illustrates the multiple strategies used by a second-grade class that has solved a problem with three addends and how the teacher tries to faithfully map their thinking into the system of mathematical notation. Emphasis is placed on the step-by-step linking of action, thought, and symbol that must occur. The AFT Thinking Mathematics program that the teacher is using is briefly described as well as how the teacher developed number sense and a culture in which student thinking is respected. The article stresses that teachers need professional development experiences that help them understand how children best learn mathematics to be able to effectively address the needs of a class of heterogeneous learners and open doors to greater achievement.     

Accommodation in the formal world of mathematical thinking

ABSTRACT

In this study, we examined a mathematician and one of his students’ teaching journals and thought processes concurrently as the class was moving towards the proof of the Fundamental Theorem of Galois Theory. We employed Tall's framework of three worlds of mathematical thinking as well as Piaget's notion of accommodation to theoretically study the narratives. This paper reveals the pedagogical challenges of proving an elegant theory as the events unfolded. Although the mathematician was conscious of the students’ abilities as he carefully made the path accessible, the disparity between the mind of the mathematician and the student became apparent.     

Connecting multiplicative thinking and mathematical reasoning in the middle years

Mathematical reasoning and problem solving are recognised as essential 21st century skills. However, international assessments of mathematical literacy suggest these are areas of difficulty for many students. Evidenced-based learning trajectories that identify the key ideas and strategies needed to teach mathematics for understanding and support these important capacities over time are needed to support teachers and curriculum developers so that they do not have to rely solely on mathematics content knowledge. Given this goal and recent evidence to suggest a relationship between the development of multiplicative thinking and mathematical reasoning, this paper explores the processes involved in developing a single, integrated scale for multiplicative thinking and mathematical reasoning using data from a four-year design-based project to establish learning and assessment frameworks for algebraic, geometrical and statistical reasoning in the middle years of schooling.     

Reflected acting in mathematical learning processes

Acting and thinking are strongly interconnected activities. This paper proposes an approach to mathematical concepts from the angle of hands-on acting. In the process of learning, special emphasis is put on the reflection of the own actions, enabling learners to act consciously. An illustration is presented in the area number representation and extensions of number fields. Using didactical materials, processes of mathematical acting are stimulated and reflected. Mathematical concepts are jointly developed with the learners, trying to address shortcomings from own experiences. This is accompanied by reflection processes that make conscious to learners the rationale of mathematical approaches and the creation of mathematical concepts. Teaching mathematics following this approach does intent to contribute to the development of decision-making and responsibility capabilities of learners.     

Supporting mathematical literacy: examples from a cross-curricular project

Mathematical literacy implies the capacity to apply mathematical knowledge to various and context-related problems in a functional, flexible and practical way. Improving mathematical literacy requires a learning environment that stimulates students cognitively as well as allowing them to collect practical experiences through connections with the real world. In order to achieve this, students should be confronted with many different facets of reality. They should be given the opportunity to participate in carrying out experiments, to be exposed to verbal argumentative discussions and to be involved in model-building activities. This leads to the idea of integrating science into maths education. Two sequences of lessons were developed and tried out at the University of Education Schwäbisch Gmünd integrating scientific topics and methods into maths lessons at German secondary schools. The results show that the scientific activities and their connection with reality led to well-based discussions. The connection between the phenomenon and the model remained remarkably close during the entire series of lessons. At present the sequences of lessons are integrated in the European ScienceMath project, a joint project between universities and schools in Denmark, Finland, Slovenia and Germany (see www.sciencemath.ph-gmuend.de).     

Versatile thinking and the learning of statistical concepts

Statistics was for a long time a domain where calculation dominated to the detriment of statistical thinking. In recent years, the latter concept has come much more to the fore, and is now being both researched and promoted in school and tertiary courses. In this study, we consider the application of the concept of flexible or versatile thinking to statistical inference, as a key attribute of statistical thinking. Whilst this versatility comprises process/object, visuo/analytic and representational versatility, we concentrate here on the last aspect, which includes the ability to work within a representation system (or semiotic register) and to transform seamlessly between the systems for given concepts, as well as to engage in procedural and conceptual interactions with specific representations. To exemplify the theoretical ideas, we consider two examples based on the concepts of relative comparison and sampling variability as cases where representational versatility may be crucial to understanding. We outline the qualitative thinking involved in representations of relative density and sample and population distributions, including mathematical models and their precursor, diagrammatic forms.     

Cross-cultural issues in linguistic, visual-quantitative, and written-numeric supports for mathematical thinking

An in-depth analysis of the major early numerical aspects (single-digit and multidigit addition and subtraction) in a representative Chinese textbook series and a US textbook series (Math Expressions) with major East Asian components illustrated how linguistic issues create different teaching and learning tasks for the same mathematical topic and how additional meaning-making supports may be needed in the US. Analyses of multidigit methods in several East Asian textbooks revealed a wide range of written-numeric support of the steps in these operations. Coherence and learning paths in both programs were identified. A framework that identifies elements of a coherent learning path of meaning-making supports is proposed to facilitate future cross-cultural analyses.     

Assessing and understanding U.S. and Chinese students' mathematical thinking

If the main goal of educational research and refinement of instructional program is to improve students' learning, it is necessary to assess students' emerging understandings and to see how they arise. The purpose of this paper is to address issues related to assessments of students' mathematical thinking in cross-national studies and then to discuss the lessons we may learn from these studies to assess and improve students' learning. In particular, the issues related to assessing U.S. and Chinese students' mathematical thinking were discussed. Then, this paper discussed the findings from two studies examining the impact of early algebra learning and teachers' beliefs on U.S. and Chinese students' mathematical thinking. Lastly, the issues related to interpreting and understanding the differences between U.S. and Chinese students' thinking were discussed.     

On mathematical games

In this paper we survey most of the games that, throughout history, can be classified as mathematical games. This paper is based on a talk given at the conference ‘Numeracy: historical, philosophical, and educational perspectives’ at St Anne's College, Oxford, December 2009.     

On uniqueness of Riemann's examples

We prove that a properly embedded minimal annulus with one flat end, bounded in a slab by lines or circles, is a part of a Riemann's example.

     

Features of mathematical activities in interdisciplinary, project-based learning

Project-based learning in mathematics education leads to mathematical activities that are uncommon in regular lessons at school. Among these activities, the following are identified and examined more closely:

⊙ the elaboration and formulation of relevant mathematical problems, including necessary definitions.

⊙ the search for the mathematically feasible, and

⊙ the recognition of opportunities to apply mathematical methods.

Also, implications for the design of project-based learning environments are developed.