Adventure and adolescence: learner-generated examples in

secondary mathematics Anne Watson

University of Oxford

Closer

• Find a number which is closer to 3/8 than it is to 3/16

• … and another• … and another

… and anotherTalk about

More ‘…and another’

• Make up a linear equation in x whose solution is x = 3

• … and another• … and another

Talk a bit about

• Example spaces

• Dimensions of variation

Even more ‘… and another’

• Make up a linear equation in x whose solution is 2.5

• … and another• … and another, but this one but be VERY

different from the previous one

Talk about

• Same

• Different

• Variation

summarise

• … and another

• Example spaces

• Dimensions of variation

• Ranges of change

Do you know this layout – important for the next bit

x + 3

x

- 2

Time to do this …. A bit• Use grid multiplication to find a pair of

numbers like a + √b which, when multiplied, have no irrational bits

c

√d

a √b

Talk about what happened when students did this..

• Age• Trying numbers• Systematic• Spotting fortuitous promising idea• Gossip method• Shifting to recognising structure as

important• 2 + root 2 mult. by 2 plus root 2

Talk about closed questions and open questions

: this is closed, but fairly powerful

• Find the equation of the straight line which goes through (1, 0) and (0, 1)

Relevance

• What is relevant for adolescents?– batting averages– journeys to school– divide n dollars between m people, etc.

• Modelling does not necessarily ‘lead’ to further knowledge of means, graphs, gradients, or ratio

Personal relevance

• pocket money? - they are interested in fairness, not ratio

• journeys to school? - their image is of the school gate, not time-distance relationships

What is relevant for adolescents?• identity• belonging• being heard• being in charge• being supported• feeling powerful• understanding the world• being able to argue in ways which make adults

listen • negotiating authority• sex

Adolescent learning - give examples (see screen)

• from ad hoc to abstract (power, understanding the world, being in charge)

• away from intuitive and everyday notions such as ‘multiplication makes things bigger’ or ‘the bigger the perimeter the bigger the area’ (lack of power, dependent, confusion, vulnerability)

• away from empirical approaches to mathematics (localised, generalised,not abstract)

Living in a complex world

• see abstract patterns and structures• verbal and kinaesthetic responses less

appropriate than considered, symbolic (in mathematics) responses

• learn to satisfy adults in new ways

Tasmanian essentials

• “identifying and clarifying issues, and gathering, organising, interpreting and transforming information …. the skills of inquiry can be used to clarify meaning, draw appropriate comparisons and make considered decisions.”

BUT, refer to previous Tasmanian slide

• mathematics is not an empirical subject at school level

• power is in abstraction, reasoning, and hypothesising about objects which only exist in the mathematical imagination

Problematic aspects which won’t get sorted with a purely experiential

approach, Vytgotsky says• probability• proportional relationships• non-linear sequences• symbolic representation• proving things• adding of fractions…..

My manifesto

• Mathematics, like some of the creative arts, can be an arena in which the adolescent mind can have some control, can validate its own thinking, and can appeal to a constructed, personal, authority.

Why powerful for adolesecnts?

• In mathematics there is always the possibility that learners can be absolutely sure they are right, and have grounds to argue with

• “People who have a sense of competence in their ability to think and learn … will be eager to pursue questions that really matter.”

Another one like this: equivalence

• Dots

• Give time to do this.

Exercises as objects – textbooks are full of these – how can they be

harnessed in this attempt to connect maths with adolescence

• do as many as you need to do to learn three new things• make up examples to show these three new things• at the end of this exercise you have to show the person

next to you, with an example, what you learnt• before you start, predict the hardest and easiest

questions and say why; when you finish, see if your prediction was correct

• make up harder ones and easier ones.• when you were doing question N, did you have to think

more about: method, negative signs, correct arithmetical facts, or what?

• can you make up examples which show that you understand the method without getting tied up with negative signs and arithmetic?

Tasmanian essentials: reflection• “Active reflection enables connections to

be made between different types of subject matter, and this enhances the likelihood of knowledge being transferable to new situations”

• Recognition of methods as structural, rather than operational, makes adaptable and transformable understanding more likely.

Mathematical methods

• “Learning is more effective, interesting and relevant when learners consciously choose and use particular methodologies, devise their own strategies to deal with challenges”

• Rules versus tools

Supermethods, e.g. enlargement

• Let them do if time …

Slide about choice

• … profusion of choice – fat, ill, confused, kids who give up easily – learn to make choices (not restrict to one thing)

• Supermethods … when to stick a 0 on the end? When do I need this posh method? Etc.

Adolescent sense of Adventure

• each starting out from the safe ground of their own knowledge-so-far and moving elsewhere within a mathematical community

Summary of task features in this presentation• … and another

• explore and extend example spaces• … and another (with constraints)• sameness/difference and variation• dimensions of variation/ ranges of change• old knowledge to make something new with a

particular property• exercise as object• find/construct • equivalent expressions• identify supermethods

Summary of adolescent concerns – how these task types relate to their

concerns• identity as active thinker• belonging to the class• being heard by the teacher• being in charge of own examples, own ideas, own

creations• being supported by inherent sense of mathematics• feeling powerful by being able to generate mathematics• understanding the world ??• being able to argue mathematically in ways which make

adults listen • negotiating the authority of the teacher through

mathematics• sex …??

Tasmanian Essentials

• Yes, great, human, understanding of learning – non-trivial how to apply this to mathematics so that students learn in ways which relate to the intellectual capabilities of adolescents – not just their short-term needs – hope I have done something towards that.

• Raising Achievement in Secondary Mathematics Watson (Open University Press)

• Mathematics as a Constructive Activity Watson & Mason (Erlbaum)