Blogging from The Tenth Southern Hemisphere Conference on the Teaching and Learning of Undergraduate Mathematics and Statistics

Prof David Wagner from The University of New Brunswick.

Mathematics educators: including all people who teach mathematics at any level.

What is mathematics?

Thought experiment: What is football? Translate these ideas into the question regarding mathematics:

- Football is defined by the rules? But surely it’s much more than the rules.
- You don’t know football unless you’ve been in a crowd of 10,000 with vuvuzelas!
- You don’t know football unless you play football – rules, and the culture of fans is different from the practice of it. Many rules which are not in the rule book – informal rules.
- There’s also a whole culture of marketing.
- Organisations…and so much more.

You learn things when you watch the game about how to play it.

How does this relate to mathematics? Mathematics is much more than the usual definitions, has so many facets… What we do as teachers do affects how others see mathematics.

**Discourses**

- Mathematical exploration: The research that mathematicians do.
- Scholarly publications – independent of the above but connected to it.
- Scholarly communities
- Universities and their politics
- Undergraduate mathematics taching
- Mathematics teacher groups
- Mathematics teaching in schools
- Textbooks and other media
- Mathematics curriculum bodies
- Mathematics in public policy
- Everyday mathematics
- Mathematics for business interests
- Mathematics in the media

In each of these there are individual intentions/goals.

**These things may be related to:**

- Gender
- Social/political issues
- Family
- Religion
- Romance

Nature is somehow behind all of this.

Tools for thinking about what we do when we’re teaching mathematics.

**Positioning theory: **

- Every communication act sets up a positioning between people – creates a relationship. Every relationship affects what we would say.
- Radical focus on the imminent (and a rejection of the transcendent). Only la parole is present. La Langue is a myth.
- Positioning theorists would say that there’s no such thing as mathematics!
- Positioning of a point A in the xy-plane. We position us in relation to mathematics.
- You can’t change the discourse that’s outside of you.
- However, even if mathematics is a myth – it’s a powerful myth.

When instructing a mathematics class, you are a representative of mathematics – it only exists in the communication between the teacher and those learning.

Any interaction that we look at is affected by the position that we see it from.

How does mathematics develop its power/authority. Mathematics is not taking power – people give it power. Are we giving power up, as humans?

**How and why do we give mathematics power**

- Abstraction
- Depersonalisation
- Detemporalisation
- Decontextualisation
- Abstraction is a human action
- Danger in obscuring agency
- Justice value of trumping status

There’s value in having a system that’s fair and objective so that people who are beneath authority can argue against those above them in status.

Theodore Porter: Objectivity lends authority to officials who have very little of their own.

**Lexical bundles exercise:**

*Stance Expressions: Stance bundles express attitudes or assessments that provide a frame for the interpretation of the following proposition, such as I don’t know if and it is necessary to. They convey two major kinds of meaning: epistemic and attitude/modality. Epistemic stance bundles comment on the knowledge status of the information in the following proposition: certain, uncertain, or probable/possible (e.g.I don’t know what, I don’t think so, the fact that the). Attitudinal/modality stance bundles express speaker attitudes towards the actions or events described in the following proposition (e.g. I don’t want to, I’m not going to). We found four types of attitudinal/modality bundles—focused on desire (e.g. I don’t want to), obligation/directive (e.g. you don’t have to, it is necessary to), intention/prediction (e.g. I was going to, it’s going to be), and ability (it is possible to). Stance bundles are also classified by whether they convey the stance in a personal or impersonal way. Personal stance bundles overtly attribute the stance to the speaker/writer or addressee (you or I). Impersonal stance bundles express similar meanings without being attributed directly to an individual (e.g. it is possible to).*

**Personal authority**

- I want you to
- Want you to do
- What I want you
- You to do is
- I would like you
- You want me to

**Discourse as authority**

- Do you have to
- Do we need to
- We have to do

**Personal latitude:**

- You want to do
- What do we do
- Are we going to

**Discursive inevitability:**

- You are going to
- We are going to
- So I’m going to

Personal authority is much more prevalent than in all other settings. Interesting as personal authority should not, naively have a place in mathematics.

How do we construct mathematics in our classroom?

How can I, as a mathematics instructor or teacher educator, acknowledge and deal with the range of authorities in tension with the kinds of truth claims that ware supposedly inherent to the discipline of mathematics?

Pay attention to mathematisation and ethnomathematics

**Mathematisation:**

Noticing the roots and fruits of mathematical thinking:

**Fruits:**How do our mathematical obsessions shape the things we construct in the world?**Roots:**How do natural phenomena inspire our mathematical obsessions.

Things that we build: Reflections of our mathematical obsessions.

**Ethnomathematics**

Look for mathematics in a cultural context. Recognizes that all mathematics is situated in particular cultural contexts.

“Show me your math” – children look for the mathematics in their communities.

Numbers are verbs in Mi’kmaq culture. Research paper on verbification: The ‘verbification’ of mathematics: Using the grammatical structures of Mi’kmaq to support student learning.

There will be tensions: It’s very difficult to decide what is mathematics when you’re doing ethnomathematics.

**NB The danger of ghettoisation – “this is not mathematics, this is my culture!”**

**Also the danger of tokenism/stereotyping.**

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