Mathematical thinking

is of course

very special

Why should we care about whether it’s

special?

Because we’re asking society to fund us to teach it.

Because we want to be able to recognise mathematical thinking when we see it.

Because somebody might ask us – at a party or in a classroom.

Because a teacher’s political position +

her general educational philosophy +

her views about nature of mathematics and numeracy

= (sort of)her approaches to teaching, and to curriculum and accreditation issues

Based on Ernest, P., 1991, The Philosophy of Mathematics Education, Basingstoke, Flamer Press

What do you really hope or believe about the “specialness” of maths?

Hopes and beliefs exercise

What makes maths special?

•Content?•Style of thinking?•Style and standards of proof?

Maths is ABOUT something

It’s about numbers orshapes orsymbols ormental objects or........

Bain, I. 1986. Celtic Knotwork. London: Constable

Bain, I. 1986. Celtic Knotwork. London: Constable

Zaslavsky, C. 1973 Africa Counts. Westport. Lawrence Hill & Co.

Zaslavsky, C. 1973 Africa Counts. Westport. Lawrence Hill & Co.

Zaslavsky, C. 1973 Africa Counts. Westport. Lawrence Hill & Co.

Zaslavsky, C. 1973 Africa Counts. Westport. Lawrence Hill & Co.

Deal or No Deal.

Any mathematical thinking going on there?

OK...... it’s not about things......

it’s about FACTS about the things.

Maths is really a set of facts about the world...

like 1 + 1 = 2

Or.......

“for every line, L, and point, P, which is not on that line, there exists a unique line, M, through P that is parallel to L.”

Is that a fact? A mathematical fact?

Ok, forget content, forget facts.

Maths isn’t a noun, it’s a verb.

It’s about a style of thinking.

style of thinking.....

logical objective challenging integrated stuck but happy knitting ideas together deductive consistent compartmentalised

creative questioningstep-by-step disciplined rule-generating speculating generalising enquiring practical abstract well-organised

rule-following proof refutation algorithmic

structured by leaps and bounds intuitive

How about proof?

If you prove something, you’ve been doing mathematical thinking........?

And if you haven’t proved something, you haven’t .......?

When is a proof really a proof?

Formal ? Algebraic? Computer-generated? Visual? Intuition? Consensus?

Proof-building by “incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations”Lakatos, I. (1976). Proofs and Refutations. Cambridge: Cambridge University Press.

XXXX

XXXX

XXXX

XXXX

Mathematicians as enquirers

• Visual - thinking in pictures, often dynamic

• Analytic - thinking symbolically, often formalistically

• Conceptual - thinking in ideas, classifying

Burton, L. (2004). Mathematicians as Enquirers - Learning about Learning Mathematics. Dordecht: Kluwer Academic Publishers.

And finally.......

1

2

3

4

0

56

7

8

9