Mathematical Literacy

Developing Identities of Inclusion

Yvette Solomon

• Publisher: Routledge
• Year: 2008
• eBook ISBN: 9781135620493
• Topic: Education
• Subtopic: Education General

Why do so many learners, even those who are successful, feel that they are outsiders in the world of mathematics? Taking the central importance of language in the development of mathematical understanding as its starting point, Mathematical Literacy explores students' experiences of doing mathematics from primary school to university - what they think mathematics is, how it is presented to them, and what they feel about it. Building on a range of theory which focuses on community, knowledge, and identity, the author examines two particular issues: the relationship between language, learning, and mathematical knowledge, and the relationship between identity, equity, and processes of exclusion/inclusion.

In this comprehensive and accessible book, the author extends our understanding of the process of gaining mathematical fluency, and provides tools for an exploration of mathematics learning across different groups in different social contexts. Mathematical Literacy's analysis of how learners develop particular relationships with the subject, and what we might do to promote equity through the development of positive relationships, is of interest across all sectors of education—to researchers, teacher educators, and university educators.

Part I Inclusion and Exclusion

DOI: 10.4324/9780203889275-1

1 Formulating the Problem

Identifying Mathematical Literacy

DOI: 10.4324/9780203889275-2

Attempts to understand success—and more often failure—in mathematics education have drawn on a number of different approaches in the fields of psychology, sociology and linguistics. The range is vast: it includes cognitive modeling of arithmetic algorithms; cognitive constructivist theories of concept development; social-psychological analyses of motivation patterns; psychoanalytic explanations of emotional responses to mathematics such as fear and anxiety; feminist and discourse-based analyses of exclusion and inclusion; and social constructivist approaches which focus on cultural mediation and apprenticeship. Here, I take the position that mathematics is a social practice, a standpoint that I developed in an earlier book (Solomon, 1989). My argument then was that Piagetian cognitive constructivist conceptions of mathematics and learning could not explain how mathematical ideas could develop in an individual learner alone, and that mathematical knowledge had to be seen as intrinsically social in order to make sense of the learning we do. My aim now is to explore this latter issue further by looking more closely at the ways in which learners develop identities of participation or marginalization within formal learning contexts, and how they consequently gain—or fail to gain—access to the central meaning-making practices of mathematics—that is, how they become mathematically literate. In this chapter, I set the scene for this exploration by reviewing theory and research which describe mathematics and mathematics learning as inherently social and cultural activities, and introducing the initial theoretical framework which underpins my analysis of what it means to become mathematically literate.

Mathematics as a Social and Cultural Activity

Theory and practice which recognizes mathematics as a social and cultural activity draws on three major strands of research involving the role of language. The first of these focuses on the language of mathematics itself, ranging from observations about its basic vocabulary to the ways in which it makes meaning in its grammatical forms. Specific issues are uncovered by this research, such as the difficulties for learners presented by the complex relationship between everyday and mathematical language and the role of metaphor in mathematical representations and modes of argument. The second strand is closely connected to the first: it concerns the apprehension of discipline-based ground rules as carried within the written and spoken mathematics discourse, and the extent to which learners are able to recognize and control the discourse of mathematics with respect to its relationship with everyday practices, for example. It also concerns the role of discourses about mathematics: these discourses are key to beliefs that individuals hold about mathematics and how it is learned and by whom. The third strand draws on Bernstein’s identification of the sociological context of classroom cultures, and on the neo-Vygotskian tradition in terms of a concern with the role of language as a tool in mathematics learning. This research points to issues in classroom cultures and the ways in which interactions between teachers and pupils shape access to mathematical understandings. Clearly, these three strands are intertwined, and I move between them in this book.

I will begin this section by recounting an episode from my observations in a British state primary school classroom of 10-year-olds—they are in Year 5, their penultimate year in this school. It illustrates many aspects of the social and cultural nature of mathematics learning and I will use it to identify a number of the issues that I address in this book, and to outline its theoretical framework.

The class is arranged, as is typical in British primary school classrooms, in tables shared by children with similar attainment levels, and they are just about to embark on their daily mathematics lesson. The teacher, who I will call Mr. Wilson (all the names of people and places in this book are pseudonyms), introduces the topic. He tells me that he has picked a set of word problems (and for later lessons, a sequence problem and geometric problems) as a way of fostering a “feel for mathematics.” He also has a more specific agenda, which is to encourage the children to gain marks for method in their mathematics Standard Assessment Tests (SATs). Although they still have a year to go before they take their Key Stage 2 statutory SATs, they will take an internal SATs test in the current year, and one of the props for this lesson is a Level 6 “extension paper”; this is a paper given at the time of this observation to higher attainers identified by the school, taken in addition to the standard paper, which only assesses Levels 3 to 5. To the class he says:

We are using words in arithmetic and recognizing different processes: note the difference between a question which asks you to ... cut up 39 cm of ribbon into 6 equal pieces, share 39 marbles between 6 people, or work out how many egg boxes you need for 39 eggs if each box holds 6 eggs. We use the same arithmetic process but there are three different answers.

He rehearses with the pupils the different answers to each question type and how these are justified. Then he enters into a question and answer routine with the whole class:

Mr. Wilson: Multiply six and another number to get fifty-four.

Pupil: Nine.

Mr. Wilson: Why nine? What was the process?

Pupil: Divide.

Mr. Wilson: Yes, divide. Look at the Level 6 SATs paper marks boxes. (He holds up a Level 6 SATs paper.) It says “show your working, you may get a mark.” SHOW YOUR WORKING!

Having introduced the topic in this way, with considerable emphasis on “showing working” in order to gain test marks, Mr. Wilson sets the children to work on some word problems. He presents the class with a typed sheet of some 20 problems, and they get on with these alone or in pairs. An early question on the sheet asks: If a family of 4, eating three meals a day, eats 12 meals a day between them, how many meals will the family eat in a leap year? A leap year has 366 days. Julie, who is one of the lower attainers in the class, turns to me for help.

Julie: The question is wrong—a leap year has 360 days.... It’s twelve divided by four.

YS: How do you know?

Julie: Because the last two [questions] were “divides.”

YS: Well, it doesn’t have to be divides does it? Read the question again carefully, what does it ask you to work out?

Julie: (reads out question, slowly) How many! It says “how many,” that means ... (looks up at posters on wall featuring the symbols for division, addition, multiplication and subtraction and their associated words—“share,” “difference,” “sum,” etc) ... Look that means “lots of,” “many” means “lots” so it’s a “times”.. .

YS: So what are you going to multiply to answer the problem?

Julie: Twelve times four.

YS: How do you know that?

Julie: Well, the middle number is the one you multiply.

Paul, on the other hand, speeds through the problems, getting to the following question very quickly: Pencils come in packets of 6. There are 36 children in Lisa’s class. If the teacher buys 6 packets, will she have enough for one pencil for each of them? I ask him some questions about his answer.

Paul: (writes “36 × 6 = 216” in his exercise book.)

YS: Can you tell me how you decided it was 36 times 6?

Paul: It just is.

YS: Well, have another look at the question and see if you can talk it through ... (he changes his written answer to 36/6)

YS: Think about what Mr. Wilson said, show your working, can you do that? What’s the first thing you have to work out?

Paul: (no response)

YS: Well, how many pencils does the teacher need?

Paul: Six times six. She needs thirty-six.

YS: Yes, so what do you have to do to show that she has enough pencils, to answer the question?

Paul: (no response—looks puzzled)

YS: You could write down, couldn’t you, that she has 36 pencils and so she has enough for the 36 children in the class, to show your working?

Paul: (changes written answer to 6 × 6 = 36)

Julie and Paul provide contrasting and familiar pictures of mathematics learners. Julie presents the well-known phenomenon of the learner who cannot identify irrelevant information and who struggles with the interpretation of mathematical symbols and sentences; she quickly asks for help. Paul, on the other hand, epitomizes the student who wants to move quickly through the problem sheet and get the answers right; he is not very interested in talking about what he is doing. Clearly, Julie is not as competent in answering the questions as Paul, but, as Sfard (2006, p. 156) notes, “knowing what children usually do not do is not enough to account for what they actually do.” Rather than seeing Julie’s approach to mathematics in terms of absences in her acquisition of knowledge, a more “participationist” view (see Sfard, 2006, p. 153) seeks to understand how she engages in mathematics as a social practice in terms of the various cultural resources and discursive positionings that she brings to the situation. Sfard articulates this strong participationist view thus:

According to this vision, learning to speak, to solve mathematical problem[s] or to cook means a gradual transition from being able to take a part in collective implementations of a given type of task to becoming capable of implementing such tasks in their entirety and on one’s own accord. Eventually, a person can perform on her own and in her unique way entire sequences of steps which, so far, she would only execute with others. (Sfard, 2006, p. 157)

Identifying with mathematics in this view is a question of movement from collective action towards a position of individual action, but still with regard to its discourse rules—what Sfard calls its “endorsed narratives” (p. 163)—since participation in mathematical communication is central to its practice. On this reading, then, we may see Julie and Paul not as disconnected individuals, one “good at mathematics,” one not, but as actors within the same community of practice, taking up differing positions within it.

The next step, then, is to explore the nature of that practice in terms of the enactment of mathematics within the classroom community that Julie and Paul find themselves in. Returning to my observations in the classroom itself, it is perhaps surprising that, given Mr. Wilson’s emphasis on the upcoming tests, the instruction to “show your working” did not change the children’s problem-solving behavior. However, if, as Sfard (p. 162) suggests,“Learning mathematics may now be defined as individualizing mathematical discourse, that is, as the process of becoming able to have mathematical communication not only with others, but also with oneself,” we may see Paul’s “it just is” as part of the same phenomenon as Julie’s search for clues: in their different ways, they struggle to take control of the discourse. We can also see other discourses at play in this classroom: part of their resistance to the idea of breaking down the problems either verbally or in writing was that the children expected to either “see” the answer or not; consequently some of them simply marked out certain questions as “too hard for them.” Paul’s emphasis on speed, and his impatience with the pencils problem, appears to be part of this equation of being able to do mathematics with finding answers quickly; he is one of “the bright boys” identified by Mr. Wilson when he talks to me after the lesson, in danger of disaffection if they are allowed to get bored—his solution is to start teaching mathematics in “ability groups.”

The institutional context and discursive positionings of the mathematics classroom come to the fore here, bridging the issues of identification and practice. My major concern in what follows will be to explore the ways in which individuals come to develop identities of participation and non-participation, and how they draw on cultural models and resources to develop their own particular narratives of doing mathematics. In this sense I am concerned to understand the interplay of identity and agency in the “figured world” of the classroom (Holland, Lachiotte Jr., Skinner, & Cain, 1998). So, for example, Mr. Wilson’s reference to the “bright boys” invokes familiar discourses of gender and ability which may operate as powerful constraints on what positions are available to learners, as Gee (2001) suggests. The possibility of gendered and classed “designated identities” as mathematics learners (Sfard & Prusak, 2005) and children’s reaction to “hard” problems are similarly aspects of their self-positioning within the available discourses, which include powerful discourses about mathematics. There is also an emotional aspect to these positionings of self: identifying as “good at mathematics” involves particular investments which take place within the interpersonal context of learning, and the role of teacher–pupil relationships is a crucial aspect of this socio-emotional background. In terms of pedagogic practice itself, it is possible to see how Mr. Wilson’s own perception of the nature of mathematics influences his teaching; despite his advocacy of teaching a “feel for mathematics,” he casts the children’s difficulties in terms of a lack of learned facts, to be addressed by “going back to basics.” Against the explicit context of the audit culture and its impact on teaching and assessment (see for example Morgan, Tsatsaroni, & Lerman, 2002), his desire to pass on his own liking for mathematics, and his identity as a good mathematics teacher are compromised.

My aim in this book is to show how these multiple issues contribute to the development of learners’ relationships with mathematics and their access to it as a social practice. I begin in this chapter with an examination of the discursive nature of mathematics and the links between identity and practice.

The Language of Mathematics: Making Meaning Through Discourse

The language of mathematics is complex and dense, and its relationship to everyday language is far from straightforward. When embedded w...

Table of contents

• Cover Page
• Half-Title Page
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Acknowledgements
• Part I: Inclusion and Exclusion
• Part II: Developing Inclusion
• Appendix A: The Education System in England and Wales
• Appendix B: The Students, the Data and the Analysis
• Notes
• References
• Author Index
• Subject Index