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Variation as a Pedagogical Tool in Mathematics

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Variation as a Pedagogical Toolin Mathematics

John Mason & Anne Watson

Wits

May 2009

- Concepts
- Topics
- Arithmetic ïƒ Algebra

- Techniques (Exercises)
- Tasks

- Expressing Generality for oneself
- Multiple Expressions for the same thingleads to algebraic manipulation
- Both of these arise from becoming aware of variation
- Specifically, of dimensions-of-possible-variation

â€“

=

What could be varied?

What then would be the difference?

What then would be the difference?

First, add one to each

First, add one to the larger and subtract one from the smaller

Ã·

=

What could be varied?

First, multiply each by 3

What is the ratio?

What is the ratio?

First, multiply the larger by 2 and divide the smaller by 3

- If I have 3 more things than you do, and you have 5 more things than she has, how many more things do I have than she has?
- Variations?

- If Anne gives me one of her marbles, she will then have twice as many as I then have, but if I give her one of mine, she will then be 1 short of three times as many as I then have.

Do your expressions express what you mean them to express?

Working down and up, keeping sum invariant, looking for a multiplicative relationship

- I start with 12 and 8
- 128128
- 119137
- 1010144
- 155

- So if Anne gives John 2, they will then have the same number; if John gives Anne 3, she will then have 3 times as many as John then has
- Construct one of your own
- And another
- And another

Translate into â€˜sharingâ€™ actions

- Before showing learners how to answer a typical problem or question, get them to make up questions like it so they can see how such questions arise.
- Equations in one variable
- Equations in two variables
- Word problems of a given type
- â€¦

+ 1

â€“ 1

+ 2

+ 1

+ 3

+ 2

4

4

+ 6

+ 2

- Write down four consecutive numbers and add them up
- and another
- and another
- Now be more extreme!
- What is the same, and what is different about your answers?

Alternative:

I have 4 consecutive numbers in mind.They add up to 42. What are they?

D of P V?R of P Ch?

- What numbers are one more than the product of four consecutive integers?
- Let a and b be any two numbers, one of them even. Then ab/2 more than the product of any number, a more than it, b more than it and a+b more than it, is a perfect square, of the number squared plus a+b times the number plus ab/2 squared,

- If you gave me 5 of your things then I would have three times as a many as you then had, whereas if I gave you 3 of mine then you would have 1 more than 2 times as many as I then had. How many do we each have?

- If B gives A $15, A will have 5 times as much as B has left. If A gives B $5, B will have the same as A. [Bridges 1826 p82]

- If you take 5 from the fatherâ€™s years and divide the remainder by 8, the quotient is one third the sonâ€™s age; if you add two to the sonâ€™s age, multiply the whole by 3 and take 7 from the product, you will have the fatherâ€™s age. How old are they? [Hill 1745 p368]

With the Grain

Across the Grain

-1 x -1 â€“ 1 =

-2 x 0

0 x 0 â€“ 1 =

-1 x 1

1 x 1 â€“ 1 =

0 x 2

2 x 2 â€“ 1 =

1 x 3

3 x 3 â€“ 1 =

2 x 4

4 x 4 â€“ 1 =

3 x 5

x2 + 5x + 6 = (x + 3)(x + 2)x2 + 5x â€“ 6 = (x + 6)(x â€“ 1)

x2 + 13x + 30 = (x + 10)(x + 3)x2 + 13x â€“ 30 = (x + 15)(x â€“ 2)

x2 + 25x + 84 = (x + 21)(x + 4)x2 + 25x â€“ 84 = (x + 28)(x â€“ 3)

x2 + 41x + 180 = (x + 36)(x + 5)x2 + 41x â€“ 180 = (x + 45)(x â€“ 4)

43

45

46

47

48

49

50

44

49

42

21

22

23

24

25

26

25

41

20

10

27

9

40

19

1

1

8

7

6

5

4

3

2

9

11

28

39

18

12

29

4

38

17

16

15

14

13

30

16

35

34

33

36

37

36

32

31

43

45

46

47

48

49

50

44

42

21

22

23

24

25

26

41

20

10

27

40

19

1

8

9

7

6

5

4

3

2

11

28

39

18

12

29

38

17

16

15

14

13

30

35

34

33

64

81

37

36

32

31

1 + 3 + 5 + 3 + 1

=

22 + 32

=

3 x 4 + 1

See generalitythrough aparticular

Generalise!

1 + 3 + â€¦ + (2nâ€“1) + â€¦ + 3 + 1

=

=

(nâ€“1)2 + n2

n (2nâ€“2) + 1

If someone claimedthere were 228 perforationsin a sheet, how could you check?

How many holes for a sheet ofr rows and c columns

of stamps?

AnticipatingGeneralising

Rehearsing

Checking

Organising

- If you can check an answer, you can write down the constraints (express the structure) symbolically
- Check a conjectured answer BUT donâ€™t ever actually do any arithmetic operations that involve that â€˜answerâ€™.

7

7 + 3

2x7 + 6

2x7 + 6 â€“ 7

2x7 â€“ 7

7

THOANs

Think of a number

Add 3

Multiply by 2

Subtract your first number

Subtract 6

You have your starting number

+ 3

2x + 6

2x + 6 â€“

2x â€“

Ped

Doms

- Name some concepts that students struggle with
- Eg perimeter & area;
- slope-gradient;
- annuity (?)
- Multiplicative reasoning
- Algebraic reasoning

- Construct an example
- Now what can vary and still that remains an example?Dimensions-of-possible-variation; Range-of-permissible-change

- Which is bigger?
- 83 x 27 or 84 x 26
- 8/0.4 or 8 x 0.4
- 867/.736 or 867 x .736
- 3/4 of 2/3 of something, or 2/3 of 3/4 of something
- 5/3 of something or the thing itself?
- 437 â€“ (-232) or 437 + (-232)

- What variations can you produce?
- What conjectured generalisations are being challenged?
- What generalisations (properties) are being instantiated?

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

- Doing for the learners what they can already do for themselves
- Teacher Lust:
- desire that the learner learn
- allowing personal excitement to drive behaviour

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

Only awareness is educableOnly behaviour is trainableOnly emotion is harnessable

The more clearly I indicate the behaviour sought from learners,

the less likely they are togenerate that behaviour for themselves

- Concepts
- What do examples look like?What in an example can be varied? (DofPV; RofPCh)

- Topics
Learners constructing examples (Solving as Undoing of building)

Learners experiencing variation (DofPV, RofPCh)

Learners constructing variations (Doing & Undoing)

- Techniques (Exercises)
- See above!
- Structured exercises exposing DofPV & RofPCh

- Tasks
- Varying DofPV; exposing RofPCh

- Object(s) of Learning
- Key understandings; Awarenesses
- Intended; Perceived-afforded; Enacted
- Encountering structured variationVarying to enrich Example Spaces

- Actions performed
- Tasks ïƒ activity ïƒ experience

- Reconstruction & Reflection on Action (efficiency, effectiveness)
- Use of powers & Exposure to mathematical themes
- Affective: disposition

- Psyche
- awareness, emotion, behaviour

- DofPV & RofPCh