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Learner Generated Examples in the Teaching of Mathematics

The Open University Maths Dept. University of Oxford Dept of Education. Learner Generated Examples in the Teaching of Mathematics. John Mason Grahamstown May 2009. Another & Another. Write down a pair of numbers whose difference is 2 and another pair and another pair.

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Learner Generated Examples in the Teaching of Mathematics

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  1. The Open University Maths Dept University of Oxford Dept of Education Learner Generated Examplesin theTeaching of Mathematics John Mason Grahamstown May 2009

  2. Another & Another • Write down a pair of numbers whose difference is 2 • and another pair • and another pair Write down a pairwhich obscure thedifference of 2 as much as possible What did you notice? What did you notice?

  3. Decimal Construction 1 • Write down a decimal number between 2 and 3 but which does not use the digit 5 and which does use the digit 7 and which is as close to 5/2 as possible

  4. more same less area No. of rectangles more more rectsless area more rectsmore area more rectssame area same rectsmore area same rectsless area same fewer rectsmore area fewer rectssame area fewer fewer rectsless area More Or Less Rectangles & Area Draw a rectilinear figure which requires at least 4 rectangles in any decomposition How many can have the same perimeter?

  5. more same less more same less More Or Less Percent & Value 50% of something is 20 Value % of 60% of 60 is 36 60% of 30 is 20 40% of 30 is 12 50% of 60 is 30 50% of 40 is 20 50% of 30 is 15 40% of 60 is 24 40% of 50 is 20 40% of 40 is 16

  6. Differences AnticipatingGeneralising Rehearsing Checking Organising

  7. Decimal Construction 2 • Write down a decimal number which has the property that every finite string of digits appears consecutively somewhere in the digits of your number • Write down a decimal number in which the string of digits for each whole number appears somewhere as a consecutive string in your number

  8. Fraction Construction • Write down a fraction which uses all of the digits from 0 to 9 and which lies between 3 and 4 and which is as close to 10/3 as possible

  9. Extremes • Write down a number which you think no-one else in the room is likely to write down • … which no-one is ever likely to have written down! • Write down a positive integer. The person writing down the smallest positive integer that no-one else writes down gets a prize!

  10. Interlude on Creativity • often identified with person, or product • often associated with novelty • these divert attention from the essence of creativity: • a flow of a particular kind of energy • Aha! Insight; construction; completion • Issue: how to encourage its appearance , and how to exploit the energy when it arises

  11. Quadrilateral Construction 1 • Draw a quadrilateral which has one pair of sides parallel, and one pair of sides equal, and one pair of angles equal How many different ones can you find?

  12. Quadrilateral Construction 2 • Draw a quadrilateral which has one pair of opposite sides equal, and one pair of opposite sides perpendicular, and a second pair of opposite sides perpendicular, and a second pair of sides equal

  13. Learner Choice • The more choices I make, the more likely I am to be engaged • Choices of: • special or particular cases, in order to comprehend • example (complexity, generality) • example meeting constraints • constraints to be met • distribution of activity • all contributing to • Sense of possible variation; generality; access to richer example spaces

  14. Lined Up • Write down the equations of two straight lines whose x-intercepts differ by 2 • and whose y-intercepts differ by 2 • and whose slopes differ by 2 • Now: find all such!

  15. Example Spaces • asking learners to construct objects • reveals something of their awareness of the scope of generality • promotes the extending and enriching of the examples available to them: their example spaces • The examples which come to mind and are available in a given situation form an example space (Watson & Mason 2002)

  16. Spinners • Colour the spinner so that the probability of getting a red is 1/4 and of a yellow is 3/8 • Colour the spinner so that a red is ¾ as likely as a yellow

  17. Seven Circles How many different size angles can you discern, using only the red points? How do you know you have them all? How many different quadrilaterals?

  18. Creativity as Energy Flow • moment of insight • requires preparation • entails perspiration and performance! • satisfaction of construction Feel creative when you go beyond habit/routine/expectation Energy flow enables you to take initiative, to respond freshly, to feel good

  19. Grid Squares • Draw a square with vertices on your grid; & A… • Draw one square inside another • Calculate the difference in their areas • Now multiply the sum of their edge lengths by the difference between their edge lengths

  20. Powers • Am I getting students to make significant mathematical choices for themselves? • Am I stimulating learners to use their own powers, or am I abusing their powers by trying to do things for them? • To imagine & to express • To specialise & to generalise • To conjecture & to convince • To stress & to ignore • To extend & to restrict

  21. More Resources • Questions & Prompts for Mathematical Thinking (ATM Derby: primary & secondary versions) • Thinkers (ATM Derby) • Mathematics as a Constructive Activity (Erlbaum) • Designing & Using Mathematical Tasks (Tarquin) • http: //mcs.open.ac.uk/jhm3 • j.h.mason @ open.ac.uk

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