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
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