Mathematical thinking in adolescence: possible shifts of

perspective

Anne WatsonUniversity of Oxford

Nottingham, November 2007

Mathematical thinking

• Thinking about mathematics• The thinking that is required in order to

understand ‘hard’ concepts• The thinking that is required to work

mathematically• The thinking that ‘real’ mathematicians do

Mathematical thinking in adolescence

• The thinking that is required in order to understand the essential conceptual shifts in secondary school mathematics

• The thinking that is required to adapt and apply mathematical knowledge at school level

Shifts to empowerment in mathematics

• Discrete – continuous

• Additive - multiplicative

• Rules – tools• Linear – non-linear• Procedure –

meaning• Example – generality• Percept – concept• Operations –

inverses

• Pattern – relationship• Relationship –

properties• Conjecture – proof• Result – objectify

result• Result –objectivify

procedure/method• Intuitive – deductive• Inductive – deductive

Who were they?

• Year 9 class, above average prior attainment, mixed comprehensive

• Summer term after SATs

Task

• To find pairs of numbers of the form a + √b which, when multiplied together, give integer answers

What they knew

• ‘grid’ multiplication for numbers and algebra

• squares and square roots in simple cases, and use of √

Grid multiplication

X z +3

2z

-1

2z2 6z

-z -3

What did they do?Reach for the calculator!

(7+ √19) (√17 + 3)(7 + √18) (√18 + 3)(7 + √18) (√17 + 3)(7 + √17) (√17 + 3)

(4 + √4) (5 + √5)(√8 + √8) (√8 + √8)

(12 + √69) (8 + √12)(10 + √6) (10 + √6)

(2 + √3) (√2 + √3)(2 + √3) (3 + √2) (2 + √2) (3 + √3)

(a + √2) (b + √8)

(2 + √2)(2 + √2)

Other classes• Year 9 average and below average prior

attainment

• Average were better at using negative signs, so several ‘found’ answers

• Below average ‘found’ that square numbers were more useful than ‘unsquare’ numbers

Adolescence• identity• belonging• being heard• being in charge• being supported

• feeling powerful• understanding the

world• negotiating authority• arguing in ways which

make adults listen

• sex

Adolescence• identity• belonging• being heard• being in charge• being supported• feeling powerful• understanding the world • negotiating authority• arguing in ways which make

adults listen

» My examples:» shared with group» choice of recording

method» generate their own

characteristics» friends; calculator» calculator; my examples» can check answers; don’t

need teacher» can justify answers

Further features• The grid as domain, support, authority• Grid has syntactic and semantic function

– Tells you what to do symbolically– Also has mathematical meaning as physical model of

distributivity in 2 dimensions• Shift from empirical view of examples to

structural view happened, for some, without teacher intervention

• Grid provides scaffold for example generation AND window on examples generated

Shifts to empowerment in mathematics

• Discrete – continuous√

• Additive - multiplicative√

• Rules – tools√• Linear – non-linear• Procedure –

meaning√• Example –

generality√• Percept – concept√• Operations –

inverses √

• Pattern – relationship√

• Relationship – properties√

• Conjecture – proof• Result – objectify

result√• Result –objectivify

procedure/method√• Intuitive – deductive√• Inductive –

deductive√

Mathematical thinking• There is a need to become more articulate about

specific kinds of shifts in thinking which are required to learn secondary mathematics

• There is a need to identify methods-in-classrooms which seem to ensure these shifts are made by a large majority of students

• There is a need to understand such methods to identify common characteristics

Future plans

• Continue fine-grained classroom work• Continue fine-grained analysis of

mathematical activity• Connecting very fine-grained differences

with brain-and-eye function to understand more about expert/novice response to task layout and sequencing

• anne.watson@education.ox.ac.uk