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#### PhD Series – Speaking Mathematically

Speaking Mathematically: Communication in Mathematics Classrooms – Chapter 4 (from this book) (1987), by David Pimm

This is a blog post I did as part of my PhD studies that I am sharing here on Educating Now, it is more academic than my typical blog posts – Nikki

Chapter Summary:

In this chapter Pimm explores different linguistic aspects of mathematics; mathematics register, words and expressions that may cause confusion but mostly he focuses on metaphors including extra-mathematical metaphors and structural metaphors. Pimm builds on Halliday’s explanation of mathematical register by summarizing that “registers have to do with the social usage of particular words and expressions, ways of talking but also ways of meaning” (p. 108). He discusses what mathematical register can include as well as provides examples of the different ways mathematical registers are created, including borrowing from everyday English (for example: face, degree, relation, power, etc.). he goes on to share examples of where ambiguity can create confusion and then further explores this idea by examining the effects of borrowing terms but using them in grammatically different ways. In this section he uses the example of diagonal and how it can be interpreted as a line that is not horizontal nor vertical versus the accepted definition of a diagonal of a polygon. Pimm further explores register confusion by more closely examining how and when misunderstandings arise in children and provides more examples that might be beneficial for teachers to know. Pimm argues that metaphor is “central to the development of the mathematics register” (p. 109) and that understanding of the processes will benefit math education.

Below are the mathematical definitions of diagonal:

When asking students how many diagonals they wrote this because they interpreted diagonal to mean how many sides are slanted (neither horizontal nor vertical):

This is an important reminder that we need to ensure students are understanding our intended meanings when we talk about math (visuals can help with this!).

My reflections:
1) Metaphors in mathematical language – I feel embarrassed to admit that I hadn’t ‘crossed over’ my own understanding of metaphor from literature to math. I also realized that I don’t do a good job of articulating the fact that I’m using metaphors. The example that resonated with me was in thinking about an equation as a balance. I use this imagery, especially that of a teeter totter, often but have never gone further to accentuate that this is a metaphor and that an equation isn’t literally a balance. Pimm suggests that we must be explicit in explaining that a metaphor is being used as a conceptual bridge. I appreciate that he suggests having students create their own metaphors for mathematical ideas as this would make it easier for students to see the metaphor as a tool. I also was thinking about English language learners and how our use of metaphors, similar to using more informal language, is culture-based and therefore not necessarily transferable to all/most students. Pimm does state that using our own (teacher’s own) metaphor may not be understood by students but it can still benefit students by modelling the process of ‘image-making’ in mathematics.

2) I made many connections with this article and started creating a list in the margins of words and phrases that are confusing for students (improper fraction = there’s something wrong with it, even numbers = whole numbers) as well as the metaphors I use (many had gone unexamined until reading this article). I feel that the use of metaphors is certainly powerful but am now encouraged to go the extra step in explaining that it is in fact a metaphor.