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Using Theoretical Constructs to Inform Teaching

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Using Theoretical Constructs to Inform Teaching

John Mason

IMEC9

Sept 2007

- Teaching Mathematics
- Tasks, activities, experience, reflection

- Teaching People To Teach Mathematics
- Consistency
- Awareness of the role of
- Tasks, activities, experience, reflection

The canal may not itself drink, but it performs the function of conveying water to the thirsty

- Experiential
What you get from this session iswhat you noticehappens inside you you, and how you relate that to your own situation

- Reflection
- Linking to theories
- Preparing to notice more carefully in future
- Brief-but-vivid accounts

- The square of the larger added to the smaller?

- The square of the smaller added to the larger?

Don’t calculate!!!

Conjecture!

Only then Check!

- I have two numbers which sum to 1
- Which will be larger:

a

a

1

(1-a)2

1

1-a

a2

Anticipating,not waiting

2.499…97

2.497

2.479…9

2.479

- Write down a decimal number between 2 and 3
- and which does NOT use the digit 5
- which DOES use the digit 7
- and which is as close to 5/2 as possible

2.47

2.49…979…

- Write down two numbers which differ by 2
- And another pair
- And another pair
- And another pair which obscure the fact that the difference is 2

Fractions?

Decimals?

9999 & 10001

Negatives?

… ?

- What numbers can be two more than the sum of four consecutive whole numbers?

What numbers can be one more than

the product of four consecutive numbers?

What did you do first?

Do you encourage your learners to do this?

How often do you set tasks for themwhere they need to do this?

Sketch the graphs of a pair of straight lines whose y-intercepts differ by 2

Sketch the graphs of a pair of straight lineswhose x-intercepts differ by 2

Sketch the graphs of a pair of straight lineswhose slopes differ by 2

Sketch the graphs of a pair of straight linesmeeting all three conditions

area

more

same

less

altitude

more

more altless area

more altmore area

more altsame area

Same altmore area

same altless area

same

less

less altmore area

less altsame area

less altless area

Draw a scalene triangle

area

more

same

less

perimeter

more

more perimless area

more perimmore area

more perimsame area

Same perimmore area

same perimless area

same

less

less perimmore area

less perimsame area

less perimless area

Draw a rectangle

When can it be done? When can it not be done?

In childhood we strove to go to school, Our turn to teach, joyous as a rule The end of the story is sad and cruel From dust we came, and gone with winds cool.

Pursuing knowledge in childhood we rise Until we become masterful and wise But if we look through the disguise We see the ties of worldly lies

Myself when young did eagerly frequent Doctor and Saint, and heard great ArgumentAbout it and about: but evermore Came out by the same Door as in I went

Doing Talking Recording

- Specialising & Generalising
- Conjecturing & Convincing
- Imagining & Expressing
- Ordering & Classifying
- Distinguishing & Connecting
- Assenting & Asserting

- Doing & Undoing
- Invariance Amidst Change
- Freedom & Constraint
- Extending & Restricting Meaning

Habit forming can be habit forming

One thing we do not often learn from experience, is that we do not often learn from experience alone

Absence of evidenceis NOTevidence of absence

A sequence of experiences does not add up to an experience of that sequence

- If learners ‘do’ the tasks they are set, then they will ‘learn’ what is required
- Contrat didactique

- The more clearly and specifically the teacher specifies the behaviour sought, the easier it is for learners to display that behaviour without encountering mathematics, without thinking mathematically
- Didactic tension

- A task is what an author publishes, what a teacher intends, what learners undertake to attempt.
- These are often very different

- What happens is activity
- Teaching happens in the interaction made possible by activity: performing familiar actions in new ways to make new actions
- Learning happens through reflection and integrating new actions into functioning

Teaching takes place in timeLearning takes place over time

Inner World of imagery

Worldof Symbols

Material World

enactive

iconic

symbolic

- Enactive-Iconic-Symbolic
- Three modes; three worlds

- Manipulating–Getting-a-sense-of–Articulating
- Doing–Talking–Recording

- Mathempedia (http://www.ncetm.org.uk)
- Fundamental Constructs in Mathematics Education, RoutledgeFalmer, London (2004).
- Designing and Using Mathematical Tasks. St. Albans: Tarquin.

http://mcs.open.ac.uk/jhm3