Adolescence and secondary mathematics: possible shifts of perspective Anne Watson Nottingham, November 2007

Stanislav Stech, PME Prague Although formalized learning of decontextualized ‘scientific’ knowledge makes use of spontaneous learning (and indeed is based on it), the important thing is that it transforms the substance of the knowledge thus acquired (ibid.p.22)

What does this mean in secondary mathematics? What is the intellectual activity involved in doing secondary school mathematics? Does the notion of ‘transforming the substance of knowledge’ make sense? Are classical sites of difficulty in school mathematics related to transformation of intellectual activity? Is adolescence the right time to be making these changes?

Classic difficulties in secondary mathematics probability proportion & ratio non-linearity symbolic representation proving things trigonometry graphical representation equality/equivalence … and more…..

e.g. Trigonometry measure; angle similarity ratio; proportionality equivalent relationships use of letters: labels, unknowns, givens function; inverse

Characteristics of successful ‘understanding’ in the secondary mathematics curriculum familiar - unfamiliar tangible, observable - imagined, abstract (quasi) intuitive – reasoned (quasi) spontaneous – scientific immediate – mediated objects – elements procedure – application sense-making – procedures

Key ‘learnable-teachable’ shifts in secondary mathematics Discrete – continuous Additive – multiplicative - exponential Rules – tools Procedure – encapsulated meaning Example – generalisation Perception – conception Operations – inverses Binary operations – distributivity and order Syntactic reading – semantic reading Pattern – relationship – properties Assumptions of linearity- expectations of relationship Conjecture – proof Results – reflection on method and results Inductive – deductive Other ….

So, yes, classic difficulties of secondary mathematics are associated with shifts in the nature of knowledge which can be described generically (cf. Tall, Dreyfus, Dubinsky, van Hiele, Mason, Biggs and Collis) Problems there seem to be rather a lot of nameable shifts! is adolescence a good time?

The project of adolescence identity belonging being heard being in charge being supported feeling powerful understanding the world negotiating authority arguing in ways which make adults listen

Adolescent learning from ad hoc to abstract and predictive attunement from imagined fantasy to imagined actuality with constraints and consequences from intuitive notions to ‘scientific’ notions from empirical approaches to reasoned approaches from doing to controlling

Mathematics from ad hoc to abstract and predictive attunement from imagined fantasy to imagined actuality with constraints and consequences from intuitive notions to ‘scientific’ notions from empirical approaches to reasoned approaches from doing to controlling

A task-type and teaching situation in which such shifts can be made by all students Year 9, all levels of prior attainment; groups working on A3 paper; calculators available; used to exploratory tasks

Enquiring about surds/irrationals Use grid multiplication to find a pair of numbers like a + √b which, when multiplied, have no irrational bits c √d a √b

The work Most started using small positive integers and the calculator Some explored by generating systematically varying examples (helpful because it led to fortuitously informative examples) Some ‘made it harder’ by using larger numbers (not a helpful move!) A significant number in all groups realised that this was a structural problem rather than ‘find the number’ Two main findings it is easier to have square numbers inside the root sign! It is something to do with having the same, or mutliplicatively-related, numbers inside the root sign

Affordances of the task Enquiry Choice; action (agency) Conjectures; perspectives (identity) Ownership Discussion (collaboration) Reflection Changes in the nature of mathematical activity

Changes in mathematical activity afforded by this task: Discrete – continuous Additive – multiplicative Rules – tools Procedure – encapsulated meaning Example – generalisation Syntactic reading – semantic reading Relationship – properties Conjecture – proof Results – reflection on results Result – reflection on procedure/method Inductive – deductive Other ….

Adolescent self-actualisation in and through mathematics identity as active thinker belonging to the class being heard by the teacher understanding the world negotiating the authority of the teacher through mathematics being able to argue mathematically in ways which make adults listen controlling personal example space being supported by inherent structures in mathematics feeling powerful by being able to generate mathematics thinking in new ways …

Alternative … Limitations due to inappropriate approaches and images Dependency on mnemonics, cues, tricks, routine questions Limited understanding of the shifts which adolescents have to make to learn mathematics Pathologise the student instead of analysing the intellectual nature of mathematics

What do I know about making intellectual ‘shifts’? Observation of mathematical behaviour in successful and unsuccessful students in secondary classrooms Personal experience of doing mathematics Teachers’ vivid metaphors, e.g. ‘crashing through…’ Finding more: concurrent eye-tracking and brain imaging; expert/novice differences; effects of slight differences in task demands

Watson (2006) Raising Achievement in Secondary Mathematics (Open University Press) Watson & Mason (2006) Mathematics as a Constructive Activity (Erlbaum) Stech (2007) School Mathematics as a Developmental Activity, in Winbourne and Watson (eds.) New Directions for Situated Cognition in Mathematics Education (Springer)