## Thales' Theorem

Here is Thales' theorem. Every student in the UK must learn this theorem as part of the Maths GCSE.

You are explaining Thales' theorem, when one of the students in your class asks:

When will we ever need this in real life?

How might you respond? It of course depends on the context; here are some possible responses you might have tried:

- It's useful for solving geometrical problems, and solving problems is useful in itself. Solving problems teaches us how to think logically, and builds our resilience, perseverance, and so on.
- Knowledge is useful in itself, knowledge is power, and so on...
- It's part of maths, and maths is important. Maths describes the world around us, it's everywhere. Also, people who can do maths earn more money.
- We don't always learn things because we need them, sometimes we learn things because they are interesting or beautiful... Why do we learn art, or history, or anything?
- It's historically important (Thales was the first Greek mathematician), its part of Euclid's Elements which are an important part of human civilisation, and so on...
- It's useful for learning about proof (here are some proofs).
- You'll need it if you study maths at A-level.
- You won't need it in real life, but you'll need it to pass your GCSE

## My thoughts as a student

Do any of these responses answer the student's question? Are they even true? Would I, as student, be placated by these responses? My reactions to these responses might be something like:

It is hard to argue with [1], but why Thales' theorem? Couldn't one learn these skills in other more relevant or enjoyable settings? I don't subscribe to the acquisition of knowledge as an end in itself [2], if I am going to spend my time learning something I want it to have some meaning beyond the abstract. I don't believe that Maths is everywhere [3], but rather that maths *can be imposed* on many things due to its generality. I am suspicious of the idea that maths is more important than other studies; I personally don't believe it to be more important than (say) becoming politically aware. The economic argument is not of interest, this does not motivate me to learn. I agree we can learn things because they are interesting [4], or beautiful, but unfortunately I don't see the interest or beauty in this triangle inside a circle. Maths may be important in the development of human civilisation [5], but that doesn't mean I have to learn how to do it to be able to function in, or appreciate, the world. Arguments [6] and [7] are only relevant if I wish to continue to study maths, and finally, [8]... well, sure...

## What can you make sense of in this topic so far?

*for them*. What then?

## Identity-in-the-making

In her paper The Sociopolitical Turn in Mathematics Education, Rochelle Gutierrez describes the following view of mathematics education:

With this view of mathematics education as

*identity-in-the-making*, we could view the students question as a means of (publicly) positioning themselves (their identity)

*outside*of what is happening in the classroom. What exactly are they positioning themselves outside, and why? I think it is also important that they are choosing to do this publicly - who might be the 'we' in their question?

The reasons why a particular student is positioning themselves outside of what is happening will of course depend on the specific instance of the asking of the question. Perhaps our work as a teacher is then to ascertain why this student views themselves as outside what is happening, by talking to them.

Some thoughts I have had about general possible reasons for this positioning might be:

- A rejection of the authority, either of mathematics or the teacher (or other relational issues with the teacher or other students).
- An identification with what is not-mathematics - as viewed by that student. Perhaps they view themselves as a 'creative' person and view mathematics as not creative - a subscription to the artistic-scientific dichotomy that has been created by the segregation of the curriculum?
- Perhaps they view mathematics, or the teaching of mathematics, as cold, hard, unemotional, inhuman, where they respond more enthusiastically to (subjects) that are more strongly connected to the human or emotional?
- An inability to see how it will be useful in future employment, or future study. They might aspire to employment/study in a non-mathematical field, or perhaps can only see a future in low-paid work?
- A conditioning against mathematics and its relevance/usefulness, perhaps from parents, or friends, or...?
- A feeling of not being part of the mathematical community, perhaps due to gender, race, 'ability', and so on - a subscription to the narrative produced by history/society about who does maths, and who is successful at maths.

I am sure there are many other possible reasons for this question. There is only one way to find out what may be the case in each instance of the question - talk to the student who asked it.

Of course, the answer to the question is that it is very likely that they

*won't*need Thales' theorem in real life. But whilst it remains compulsory for all students in the UK to study it, all that is left is to work with the student who asked the question to try and find a way forwards...