Teaching Secondary School Mathematics

Research and practice for the 21st century

Merrilyn Goos, Gloria Stillman, Sandra Herbert, Vince Geiger

• Publisher: Routledge
• Year: 2020
• ISBN:
  9781000247770
• Edition: 2
• Topic: Education
• Subtopic: Teaching Mathematics

Since its first publication, Teaching Secondary School Mathematics has established itself as one of the most respected and popular texts for both pre-service and in-service teachers. This new edition has been fully revised and updated to reflect the major changes brought about by the introduction of the Australian Curriculum: Mathematics, as well as discussing significant research findings, the evolution of digital teaching and learning technologies, and the implications of changes in education policies and practices.The mathematical proficiencies that now underpin the Australian curriculum -- understanding, fluency, problem solving and reasoning -- are covered in depth in Part 1, and a new section is devoted to the concept of numeracy. The chapter on digital tools and resources has been significantly expanded to reflect the growing use of these technologies in the classroom, while the importance of assessment is recognised with new material on assessment for learning and as learning, along with a consideration of policy development in this area. Important research findings on common student misconceptions and new and effective approaches for teaching key mathematical skills are covered in detail.As per the first edition readers will find a practical guide to pedagogical approaches and the planning and enactment of lessons together with enhanced chapters on teaching effectively for diversity, managing issues of inequality and developing effective relationships with parents and the community.This book is the essential pedagogical tool for every emerging teacher of secondary school mathematics.'The text offers an excellent resource for all of those involved in the preparation of secondary mathematics teachers, with links to research literature, exemplars of classroom practices, and instructional activities that encourage readers to actively examine and critique practices within their own educational settings.' Professor Glenda Anthony, Institute of Education, Massey University 'A rich and engaging textbook that covers all of the important aspects of learning to become an effective secondary mathematics teacher. The second edition of this text... is further enhanced with updated references to the Australian Curriculum, NAPLAN, STEM, current Indigenous, social justice and gender inequity issues, and the place of Australian mathematics curricula on the world stage.' Dr Christine Ormond, Senior Lecturer, Edith Cowan University

PART 1
Introduction

CHAPTER 1
Doing, teaching and learning mathematics

Excellent teachers of mathematics are purposeful in making a positive difference to the learning outcomes, both cognitive and affective, of the students they teach. They are sensitive and responsive to all aspects of the context in which they teach. This is reflected in the learning environments they establish, the lessons they plan, their uses of technologies and other resources, their teaching practices, and the ways in which they assess and report on student learning. (Australian Association of Mathematics Teachers (AAMT), 2006)

This statement appears in a professional standards framework that describes the unique knowledge and skills needed to teach mathematics well. It reflects findings from a multitude of research studies that show how students’ mathematics learning and their dispositions towards mathematics are influenced—for better or for worse—by the teaching that they experience at school (see Mewborn, 2003). While it is sometimes difficult for researchers to untangle the complex relationships that exist between teaching practices, teacher characteristics and student achievement, it is clear that teachers do make a difference to student learning.

This chapter discusses what it means to be a teacher of secondary school mathematics and the requirements and challenges such a career choice entails. We first consider the mathematical beliefs of teachers and students, as well as students’ perceptions of mathematics teachers, reflecting on how teachers communicate powerful messages about the nature of mathematics and mathematics learning to the students they teach. Next, we turn our attention to the secondary school mathematics classroom by examining recent research on mathematics teaching practices, and identify the types of knowledge needed for effective teaching of mathematics. Finally, we review some of the challenges for mathematics curriculum and teaching arising from this and other research.

Mathematical beliefs

Whether we are aware of it or not, all of us have our own beliefs about what mathematics is and why it is important. In summing up findings from research in this area, Barkatsas and Malone (2005) conclude that ‘mathematics teachers’ beliefs have an impact on their classroom practice, on the ways they perceive teaching, learning, and assessment, and on the ways they perceive students’ potential, abilities, dispositions, and capabilities’ (p. 71). However, the relationship between beliefs and practices is not quite so straightforward as this, and many researchers would agree that rather than being related in a linearly causal way in either direction, beliefs and practice influence one another and develop together (Beswick, 2007). Context also contributes to the relationship between beliefs and practice. Raymond’s (1997) model of the relationships between beliefs and practices and the factors influencing them is informative in this regard (see Figure 1.1). The model suggests some of the complexity involved in understanding how beliefs shape, and are shaped by, teaching practices, and why inconsistencies sometimes exist between the beliefs that teachers might espouse and those they enact through their practice. Beswick (2012) also identified inconsistencies between mathematics teachers’ beliefs of mathematics as a discipline and as a school subject, to further confirm the complexity of the relationship between beliefs and practice.

Beliefs about the nature of mathematics

Given that teachers communicate their beliefs about mathematics through their classroom practices, it is important to be aware of one’s beliefs and how they are formed.

REVIEW AND REFLECT: In your own words, write down what you think mathematics is and why it is important for students to learn mathematics at school. Compare your thoughts with a fellow student and try to explain why you think this way.

Look up the definition or description of mathematics provided in the Australian Curriculum: Mathematics (Australian Curriculum, Assessment and Reporting Authority (ACARA), 2015) and by notable mathematicians or mathematics educators (Gullberg, 1997; Hogan, 2002; Kline, 1979). Compare these with your own ideas.

Discuss with a partner some of the possible influences on the formation of your beliefs, using Raymond’s (1997) model as a guide (see Figure 1.1).

Figure 1.1Raymond’s model of the relationships between teachers’ mathematical beliefs and their teaching practice

Source: Raymond (1997).

Mathematics and numeracy

In recent years, the idea of numeracy has gained prominence in discussions about the essential knowledge and competencies to be developed by school students for participation in contemporary society. Indeed, it is included in the Australian Curriculum: Mathematics (ACARA, 2014) as a general capability, meaning that it applies to and can be developed across the curriculum, and is not confined to mathematics or the sole responsibility of mathematics teachers. Thus, it is important for mathematics teachers to have a clear understanding of the nature of numeracy and its relationship with mathematics (Queensland Board of Teacher Registration, 2005). The term ‘numeracy’ is common in Australia, New Zealand, Canada and the United Kingdom (UK), but in other parts of the world, the terms ‘quantitative literacy’, ‘mathematical literacy’ or ‘statistical literacy’ are used. These different names convey different meanings that may not be interpreted in the same way by all people. For example, some definitions of quantitative literacy focus on the ability to use quantitative tools for everyday practical purposes, while mathematical literacy is understood more broadly as the capacity to engage with mathematics in order to act in the world as an informed and critical citizen (Organisation for Economic Cooperation and Development [OECD], 2000). Steen (2001) offers the following distinction between mathematics and numeracy:

Mathematics climbs the ladder of abstraction to see, from sufficient height, common patterns in seemingly different things. Abstraction is what gives mathematics its power; it is what enables methods derived in one context to be applied in others. But abstraction is not the focus of numeracy. Instead, numeracy clings to specifics, marshalling all relevant aspects of setting and context to reach conclusions … Numeracy is driven by issues that are important to people in their lives and work, not by future needs of the few who may make professional use of mathematics or statistics. (pp. 17– 18)

These definitions suggest that numeracy is broader than, and different from, the way that mathematics traditionally has been viewed by schools and society. These meanings and approaches to teaching numeracy are discussed in Chapter 5.

REVIEW AND REFLECT: Consider your beliefs about numeracy and revisit your beliefs about the nature of mathematics and compare these with the distinction between mathematics and numeracy proposed by Steen (2001).

How is numeracy described in the Australian Curriculum? To what extent does this description incorporate ideas about mathematics being used for practical purposes, in real- world contexts and for developing critical citizenship?

Beliefs about mathematics teaching and learning

Just as important as mathematics teachers’ beliefs about the nature of mathematics are their beliefs about mathematics teaching and learning. Beswick (2005, 2007, 2012) shows the connections between these types of beliefs by drawing on categories developed by Ernest (1989) and van Zoest et al. (1994), as shown in Table 1.1.

REVIEW AND REFLECT: Researchers usually obtain information about teachers’ mathematical beliefs via questionnaires (e.g., Barkatsas & Malone, 2005; Beswick, 2005; Frid, 2000a; Perry et al., 1999). Obtain a copy of one of these beliefs questionnaires and record your responses. Discuss your answers with a partner in the light of the classifications in Table 1.1.

Table 1.1 Relationship between beliefs about mathematics, teaching and learning

Beliefs about the nature of mathematics (Ernest, 1989)	Beliefs about mathematics teaching (Van Zoest et al., 1994)	Beliefs about mathematics learning (Ernest, 1989)
Instrumentalist: mathematics as a tool kit of facts, rules, skills	Content-focused with an emphasis on performance	Skill mastery, passive reception of knowledge
Platonist: mathematics as a static body of absolute and certain knowledge comprising abstract entities	Content-focused with an emphasis on understanding	Active construction of understanding
Problem solving: mathematics as a dynamic and expanding field of human creation	Learner-focused	Autonomous exploration of own interests

Student beliefs about mathematics

Thus far, we have given our attention to teachers’ mathematical beliefs, but what do students believe about the nature of mathematics? A subtler way to investigate this than to ask a direct question involves using metaphors for mathematics, such as:

If mathematics were a food, what kind of food would it be? If mathematics were a colour, what colour would it be? If mathematics were music, what kind of music would it be? (See Frid, 2001; Ocean & Miller-Re illy, 1997 for more ways of using metaphors for mathematics.)

Pre-service teachers who tried this activity with their junior secondary students during a practice teaching session were surprised, and somewhat disturbed, by the results. If mathematics were a food, most students agreed that it would be a green vegetable such as broccoli, brussels sprouts or zucchini. According to them, these vegetables taste terrible but we have to eat them because they are good for us, thus implying that mathematics is a necessary but unpleasant part of their school diet. Others who were more favourably disposed towards mathematics compared it with bread (a staple food), fruit salad (because it contains a variety of ingredients) or lasagne (different layers are revealed as you eat it). These responses perhaps suggest that students had varying perceptions of mathematical knowledge as either necessary, diverse or sequenced in layers of complexity. Students thought that if mathematics were a colour it would be either black (depressing, evil), red (the colour of anger and pain) or brown (boring). The few who admitted to liking mathematics often said it would be blue because this colour is associated with intelligence or feelings of calm and peacefulness. There was more variety in metaphors for mathematics as music. Many students said that mathematics was like classical music because they found it difficult to understand; some likened it to heavy metal music because ‘it hurts your brain’; while one responded that it was like the theme from the movie Jaws— because ‘it creeps up on you’. Writing in her practice teaching journal, one pre- service teacher lamented, ‘There was not one person in the class who admitted to liking maths and compared it with McDonald’s or Guy Sebastian!’

REVIEW AND REFLECT: Try the mathematical metaphors activity with some school-aged children and some adults (if possible, with mathematics teachers, non-mathematics teachers and non- teachers). Analyse the results and compare them with those of a partner.

Investigating students’ views about mathematics and comparing these with teachers’ beliefs might lead us ...