Section II: Opening spaces of learning with mathematics classroom research on language

7

The role of mathematical vocabulary in moving from the particular to the general with visual representations

Nick Andrews, Lucy Dasgupta and jenni Ingram

Introduction

For many teachers, the accurate and precise use of mathematical vocabulary is a key aim of mathematics lessons. Mathematical problems that afford student explanation can make a positive contribution to developing a classroom climate in which this appropriate use of mathematical vocabulary is encouraged. The actions of the teacher, through what they say and from the images they offer, can also contribute to such a climate. Throughout this chapter we will be offering a categorisation of teacher utterances that may arise in such a context.

In this chapter, we will examine two episodes from mathematics lessons where the teachers are asking questions to encourage and support their students to use vocabulary, yet in very different ways. In one extract the pursuit of the precise use of vocabulary is intricately related to the mathematical ideas being discussed, whilst in the other it is specific to the prototypical example being discussed. We propose that supporting students to use vocabulary is more than a goal in itself, it can also be purposefully part of the development of the concepts under consideration.

We offer the following brief lesson episode as a useful starting point because it succinctly captures several of the aspects on which we will focus. Tamsin, an experienced mathematics teacher, had been working with a class of 11- and 12-year-olds on a task that involved generating examples of shapes with area 24 cm2. Examples of rectangles and parallelograms had already been considered with the whole class when a small group of students had claimed that a triangle of area 24 cm2 could be formed by constructing a triangle of base 8 cm and height 6 cm within a rectangle of base 8 cm and height 6 cm. A student from the small group had drawn Figure 7.1 on the board so that it could be shared publicly.

Extract 1 is the public talk that followed on from this. Tamsin posed a question to the rest of the class and selected Simon to respond.

Triangle inscribed inside a rectangle with area 24 cm

FIGURE 7.1 Triangle inscribed inside a rectangle with area 24 cm2

Extract 1. Whole-class discussion of Figure 7.1 in Tamsin’s lesson

1 TAMSIN: Put your hand up if you understand the idea of triangles being half of the

area of the rectangle. So just explain it in your own words please, Simon.

2 SIMON: The area of the triangle. Um, so like you draw a square, in this case six

by eight.

  • 3 TAMSIN: Is that a square?
  • 4 SIMON: Sorry, rectangle.
  • 5 TAMSIN: Okay, why isn’t it a square?
  • 6 SIMON: ’Cause not all the sides are equal.
  • 7 TAMSIN: That’s it, yes, okay.
  • 8 SIMON: Okay. And you just draw a diagonal of the rectangle to get the triangle.

There are several aspects to this extract that are worthy of note at this stage and will be expanded on in the chapter. Firstly, the student uses mathematical vocabulary in their turns, although not necessarily accurately. Secondly, the teacher inviting an explanation of a mathematical phenomenon initiates the extract and a sequence of student then teacher turns follows the initiation turn. The teacher turns include a question about the student’s use of vocabulary (turn 3), a question about their understanding (turn 5), and an acceptance of what the student has said (turn 7).Thirdly, the mathematical phenomenon has already been represented visually through a publicly shared figure. We consider first how each of these aspects are considered in the research literature before explaining how they will shape the analysis of the two episodes from mathematics lessons.

Learning the vocabulary of mathematics

There is a lot more to learning mathematics than just knowing the specialised vocabulary (Morgan, 2005), but this specialised vocabulary carries meaning and through its use students can develop their knowledge and understanding of mathematical concepts, structures and relationships.Yet the knowledge, understanding and use of this specialised vocabulary is also something that students often find difficult and can also be a barrier to learning mathematics (Pimm, 1987).We are interested in the interrelationships between the ways in which mathematical vocabulary is introduced and used, and the nature of the examples used, focused on visual representations. Particular words can be associated with the generalised class of mathematics objects that they name, or they can refer to a specific mathematical object being used in an example. When we use these specialised words this distinction between the general and the particular is not always explicit in teaching, which may have consequences on the meaning students associate with these words as they learn to recognise and use them. The use of specialised vocabulary is consequently a form of both meaning making and meaning representation (Leung, 2005, p. 128).

Leung argues that there are three related processes involved in the learning of mathematical vocabulary: it involves both formal and semantic features of words in a variety' of contexts; it involves thinking with and through the concepts associated with the word; and it is an incremental activity' where the meanings associated with a word can develop and expand as part of meaning making (2005, pp. 133-134). In classroom practice, Ingram and Andrews (2018) showed that the teaching of specialised vocabulary can also be incremental, often focused on using words to label or name something before shifting to more meaningful uses such as using them to communicate mathematical ideas or relationships when the students demonstrated that they had some appropriate meanings and other ways of expressing the objects to which the words were referring.

The particular and the general

Examples of concepts used in mathematics classrooms are specialisations, but a distinction can be made between generic examples, which are “a specialization which nonetheless speaks the generality” (Mason & Pimm, 1984, p. 277), and particular examples. Examples are also always examples of something, and it is this something that we name, but it may not always be clear to students (and potentially teachers) what this something is even when the name is used. For instance, many geometrical images, such as the one in Tamsin’s lesson below, could be examples of many properties or relationships, not all of which are mathematically valid. Particular examples can offer students the opportunity to apprehend or conjecture an emergent structure (Venkat, Askew, Watson, & Mason, 2019), but it is not until students are thinking about general relationships within a particular instance of a relationship that they begin to apprehend mathematical structure. In the case of vocabulary, in mathematics the names we give mathematical objects or relationships refer to general classes of objects or relationships, yet in the classroom they are often introduced through particular (often prototypical) examples.

Structures of interaction

One of the most prevalent structures of interaction within the mathematics classroom is that of the IRE sequence, with a teacher initiating the exchange followed by a student responding to this initiation and then the teacher evaluating or giving feedback to this response (Mehan, 1979a). The ways in which this structure is described and used by researchers vary considerably, from a view of it as a sequence of turns that is tightly controlled by the teacher, to a view where the focus is on the content or nature of the initiation such as being a procedure-bound pattern (Franke, Kazemi, & Battey, 2007), to a view where IRE refers solely to the turn-taking structure of the interactions. There can also be considerable variation within this structure in terms of the types of initiation made, the nature and content of student responses and the wide variety of ways in which teachers handle this response in their next turn (Wells, 1993). It is this latter view that we use in this chapter, focusing in particular on the response and evaluation turns. From a conversation analytic perspective, these turns are reflexive. The evaluation or third turn responds to the student’s turn and can include an evaluation of this response, but also offers recognition that the response is acceptable in terms of being a response to the teacher’s initiation, and can also demonstrate that the teacher has comprehended the students’ response (Schegloff, 2007). Most research in mathematics education has focused on the evaluative or assessment role of this sequence, where a teacher asks a question that they already know the answer to and therefore the student’s response is a demonstration that they also know the answer, followed by a teacher evaluation or assessment as to whether this response is ‘correct’ (Mehan, 1979b).Yet this structure also occurs in interactions where teachers are asking questions to which they do not know the answer, or where it would be difficult or inappropriate to evaluate the ‘correctness’ of the student’s response. In these situations, the teacher’s evaluation or feedback deals with the appropriateness of the response within the interactional context, does it allow the interaction to continue unproblematically, does it contribute to the topic being discussed and explored. In the analysis below the appropriateness of the turn is considered by the teachers in terms of both the language used by the students and also the meaning of the students’ turn as understood by the teacher. Specifically, we focus on occasions where the teacher ‘problematises’ the student’s response either in terms of the vocabulary used by the student, their understanding of the concept, or both.

 
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