Reasoning, Communication and Connections in Mathematics

The following sections are included:

• Contents

This introductory chapter provides an overview of the chapters in the book. The chapters are organised according to three broad themes that are central to reasoning, communication and connections. The themes are mathematical tasks, classroom discourse and connections within and beyond mathematics. It ends with some concluding thoughts that readers may want to be cognizant of while reading the book and also using it for reference and further work.

This chapter describes two aspects of knowledge work that are important in considering instructional tasks in classrooms: the epistemic focus of knowledge work and epistemic (knowledge) practices. We focus on four types of knowledge: epistemic, procedural, conceptual and metacognitive, and six types of knowledge practices: knowledge communication (use of syntax), knowledge communication (presentation), representation, generation, deliberation and validation. Using data from a survey of over 1000 Secondary 3 students in 30 schools drawn from a representative random stratified sample of secondary schools in Singapore, we report on: 1) the kind of mathematical knowledge(s) that teachers focus on in mathematics classrooms; 2) the kind of disciplinary knowledge practices teachers ask students to engage in; and 3) the relationships between different kinds of knowledge and practices. The students reported prevalence of metacognitive and procedural knowledge as well as knowledge communication (use of syntax) in the instructional tasks that teachers structured for them. However, they also indicated that teachers do not generally provide students with a rich and integrated array of opportunities to engage in knowledge practices at a level and in a sequence that emphasizes the disciplinary structure of mathematics work. We suggest that a clearer awareness of the epistemic nature of knowledge claims and instructional tasks that attends to the structure and logic of knowledge practices in mathematics could render instructional tasks more "authentic" — and doing mathematics more transparent, "visible" and accessible.

Reasoning and communication are critical processes in the mathematics curriculum. As students reason and communicate about mathematics, teachers gain insight into their students' thinking that can help guide instruction. This chapter illustrates a variety of specific and general strategies that can be used to modify typical textbook exercises to incorporate reasoning and communication and make student thinking visible. Sample items and modifications are presented.

Mathematical tasks with high cognitive demand often require students to make explicit their thinking. These tasks are necessary for the advancement of reasoning and communication in classrooms. This chapter focuses on specifically four "what" strategies that primary school teachers may use to develop such tasks to engage their students in reasoning and communication during mathematics lessons. The four "what" strategies explored in this chapter are: What number makes sense? What's wrong? What if? and What's the question if you know the answer? The chapter also features a primary one mathematics lesson during which the teacher used one of the strategies and engaged her students fruitfully in reasoning and communication.

Reasoning and justification are critical processes in the mathematics curriculum, yet students often struggle with these processes. This chapter uses a framework from textbook research to illustrate types of exercises that can be used to enhance the extent to which reasoning and justification are evident in the curriculum. Typical items from the algebra and statistics portion of the secondary curriculum are shared, with suggestions for modifications that highlight reasoning and justification.

This paper introduces the project-based mathematics learning through LOGO programming activities, which was designed for Korean 6^th grade (Year 6) promising students to improve their mathematical reasoning skills and to activate communication with peer students and teachers in various projects to connect mathematics and visual art. In this learning process, students were found to activate and promote reasoning strategies such as analogy, generalization, progressive and critical thinking and debugging based on visualization and empirical inference. Students could activate their communication with other students and their teacher in analyzing, debugging, comparing and contrasting their programming. This chapter shows that supporting with the peculiar characteristics of LOGO language, LOGO project-based learning is an effective environment to combine Piaget's epistemology on intelligence development, Polya's heuristics for improving mathematical problem solving abilities and Poincare's philosophy to emphasize mathematics education to nurture attitudes towards mathematics, intuition and esthetic sense.

After the curriculum review in 2006, the Singapore school mathematics framework expanded its scope to include reasoning, communication and connections as three main components of the Processes of problem solving. This new emphasis should pervade all levels of mathematics learning: primary, O-Levels and A-Levels. As A-Level students have learnt sufficient elementary mathematics at the O-Level and are preparing for tertiary education, this emphasis on reasoning, communication and connections is not less important than at the O-levels. In this chapter, sample activities of how reasoning, communication and connections can be infused into teaching various A-level mathematics topics are discussed. For mathematical reasoning at the A-levels, elementary mathematical proofs and derivation of results by first principles are good opportunities to transcend procedural emphasis of traditional teaching to higher level mathematical reasoning. Connections can be achieved by several means: facilitating students to connect across different mathematical ideas, across other disciplines and to daily life. Connections help students to better understand and appreciate various mathematical concepts and enable them to regulate their own thinking (meta-cognition). Teachers should provide opportunity for students to communicate their reasoning clearly, and formulate their argument during mathematics lessons.

Until recently, most mathematics tasks presented to primary-school students were exclusively word-based problems. Current practices, from both curriculum and assessment perspectives, have moved toward more visual and graphic forms of representation. This is unsurprising given the increased use of graphics in society and the ongoing challenge of representing burgeoning amounts of information in visual and graphic forms. From a young age students are exposed to visual forms of communication with more intensity and engagement, whether playing computer games or navigating web pages. This chapter highlights the important role visual and spatial reasoning plays in how mathematics is communicated. In particular, it considers the changing nature of mathematics representation in classroom practices, and an evolution in student engagement—where students are increasingly required to decode visual and spatial information. The chapter also considers the need for young students to employ encoding strategies which effectively encourage visual and spatial reasoning.

This chapter discusses the interactive opportunities for learning that teachers provide to their students through one key element of classroom talk — teacher questions. We study three broad categories of teacher questions: performative, procedural and conceptual. We also subdivide conceptual questions into clarifying, connecting, explanatory and epistemic questions. Using data from a survey of over 1,000 Secondary 3 students in 30 schools drawn from a representative random stratified sample of secondary schools in Singapore, we report on: 1) the kind of teacher questions that teachers focus on in mathematics classrooms; and 2) the relationships between different kinds of teacher questions. The relative prevalence of performative questions arising from the highly prevalent and mundane triadic talk structure: teacher initiate (I), student respond (R) and teacher evaluate (E) [IRE]suggests that Singapore mathematics classrooms provide limited opportunity for students to engage in rich classroom conversations. However, deeper structural analyses of teacher questions indicate that not all performative questions lead to a cul de sac, and that a substantial proportion of them lead on to procedural and explanatory talk. We think this helps explain the success of Singaporean students in international assessments despite relatively low levels of classroom dialogue. It also suggests that we need to be extremely cautious in judging "closed" teacher questions and performative talk, including IRE, as a hum-drum ritualized exchange that has little or no capacity to prompt students to work on understanding.

Analogical reasoning is recognized as an efficient means of problem solving. However, few research studies have examined how to integrate analogical reasoning into mathematics lessons and study the relationship between use of analogical reasoning and discourse in mathematics classroom. This study presents an analysis of discourse produced during a mathematics lesson that involved analogical reasoning. Observation analysis of the class and students' final constructs supported the notion that analogical reasoning facilitates rich discourse, interactions between the class teacher and the students themselves.

Although mathematics is essentially deductive, students are exposed to various other types of reasoning when studying mathematics in school. They do not always understand the subtle differences in the types of reasoning that they come across and whether a particular statement based on a type of reasoning is valid or not. This invariably leads to reasoning errors. In this chapter, we explore some of the errors made by students in writing proofs by mathematical induction from a sample of 69 Junior College (Year 12) students. The data collected included students' responses to a test and post-test interview with selected students. The common errors were then further analysed in the light of the literature to understand the mathematical reasoning behind these errors. Semi-structured interviews were conducted with 11 selected students based on the solutions they had provided to either, confirm, refine or reject the initial understandings.

The pressure to complete syllabus within a limited time in the work schedule of a teacher can sometimes lead one to view mathematics "content" as comprising points from a check list to tick against. This can in turn lead to teaching as "covering" isolated pieces of skills practice. Against this portrait of instructional practice is one where mathematics is presented as skills and concepts that are closely connected to each other. In this conception of classroom practice, students are encouraged to see the mathematics that they are learning as situated and connected to a wider context of mathematical knowledge. In this chapter, we discuss ways in which this alternative portrait of mathematics instruction can be enacted, drawing upon examples that include findings from a project that places a high value on presenting mathematics as being connected.

This chapter discusses the nature of numeracy, which is concerned with the use of mathematics in context, reflecting some recent Australian work concerned with numeracy across the curriculum, and some Australian initiatives to highlight mathematics in wider contexts than the mathematics classroom. Australian national curriculum developments, in which numeracy is identified as a General Capability to be addressed in all school curricula, not only in mathematics curricula, are described. In the primary years, situations that demand some level of mathematical thinking arise in all aspects of the curriculum, sometimes unexpectedly, and thus provide opportunities for numeracy to be developed providing we are alert to the possibilities, while in the secondary years, mathematics is important for learning in other subjects, although this is not always well recognized. Students need to develop a capacity to use mathematics in their everyday lives, their school lives and eventually in the world of work. The chapter explores examples of numeracy demands and opportunities across the curriculum, and considers the roles of teachers to develop the necessary mathematical, contextual and strategic expertise to address these.

Curriculum documents often refer to the use of mathematics in the everyday world, clearly of importance to them as students in classes other than mathematics and also eventually as adults. Mathematics teachers often find that resources for teaching and learning, such as standard textbooks, do not connect well to the everyday worlds of the students in their classes. There are many opportunities in the classroom to make connections between school mathematics and the everyday world, however, and this chapter illustrates some of these for the particular context of health. Mathematics is connected with official health syllabus expectations, although this is not evident at first glance, and it is also needed for students to adopt healthy lifestyles in their everyday world. A progression of suitable activities is illustrated in this chapter, highlighting some of the ways in which mathematics is needed to understand key aspects of health. These activities include the mathematics associated with describing diets, measuring what is eaten, understanding public health advice about balanced diets, as well as understanding and using nutritional information on packaged foods. For some activities of these kinds, sound use of technology allows mathematics to be connected to health issues more efficiently. Of necessity, teachers will need to adapt these ideas to the mathematics and cultural backgrounds of their students, as well as recognising that connections of mathematics to the everyday world require specific attention to both mathematics and to the everyday world of their students, of which health connections represent only a single example.

In this chapter some interesting mathematics content from undergraduate mathematics courses that shows the connection and relevance of mathematics to daily life is presented. The philosophy behind these mathematics courses is also discussed. Hopefully, teachers will be inspired to develop exciting material for their mathematics classroom teaching.

The following sections are included:

• Contributing Authors