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# Variation as a Pedagogical Tool in Mathematics PowerPoint PPT Presentation

Variation as a Pedagogical Tool in Mathematics. John Mason & Anne Watson Wits May 2009. Pedagogic Domains. Concepts Topics Arithmetic  Algebra Techniques (Exercises) Tasks. Topic: arithmetic  algebra. Expressing Generality for oneself

Variation as a Pedagogical Tool in Mathematics

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## Variation as a Pedagogical Toolin Mathematics

John Mason & Anne Watson

Wits

May 2009

### Pedagogic Domains

• Concepts

• Topics

• Arithmetic  Algebra

• Techniques (Exercises)

### Topic: arithmetic  algebra

• 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

=

What could be varied?

### What’s The Difference?

What then would be the difference?

What then would be the difference?

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

÷

=

What could be varied?

### What’s The Ratio?

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

### Counting & Actions

• 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?

### Construction before Resolution

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

• 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 ‘sharing’ actions

### Principle

• 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

+ 1

– 1

+ 2

+ 1

+ 3

+ 2

4

4

+ 6

+ 2

### Four Consecutives

• Write down four consecutive numbers and add them up

• and another

• and another

• Now be more extreme!

Alternative:

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

D of P V?R of P Ch?

### One More

• 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,

### Comparing

• 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 father’s years and divide the remainder by 8, the quotient is one third the son’s age; if you add two to the son’s age, multiply the whole by 3 and take 7 from the product, you will have the father’s age. How old are they? [Hill 1745 p368]

With the Grain

Across the Grain

### Tunja Sequences

-1 x -1 – 1 =

-2 x 0

0 x 0 – 1 =

-1 x 1

1 x 1 – 1 =

0 x 2

2 x 2 – 1 =

1 x 3

3 x 3 – 1 =

2 x 4

4 x 4 – 1 =

3 x 5

x2 + 5x + 6 = (x + 3)(x + 2)x2 + 5x – 6 = (x + 6)(x – 1)

x2 + 13x + 30 = (x + 10)(x + 3)x2 + 13x – 30 = (x + 15)(x – 2)

x2 + 25x + 84 = (x + 21)(x + 4)x2 + 25x – 84 = (x + 28)(x – 3)

x2 + 41x + 180 = (x + 36)(x + 5)x2 + 41x – 180 = (x + 45)(x – 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

### Up & Down Sums

1 + 3 + 5 + 3 + 1

=

22 + 32

=

3 x 4 + 1

See generalitythrough aparticular

Generalise!

1 + 3 + … + (2n–1) + … + 3 + 1

=

=

(n–1)2 + n2

n (2n–2) + 1

### Perforations

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?

### Differences

AnticipatingGeneralising

Rehearsing

Checking

Organising

### Tracking Arithmetic

• If you can check an answer, you can write down the constraints (express the structure) symbolically

• Check a conjectured answer BUT don’t ever actually do any arithmetic operations that involve that ‘answer’.

7

7 + 3

2x7 + 6

2x7 + 6 – 7

2x7 – 7

7

THOANs

Think of a number

Multiply by 2

Subtract 6

+ 3

2x + 6

2x + 6 –

2x –

Ped

Doms

### Concepts

• Name some concepts that students struggle with

• Eg perimeter & area;

• 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

### Comparisons

• 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 – (-232) or 437 + (-232)

• What variations can you produce?

• What conjectured generalisations are being challenged?

• What generalisations (properties) are being instantiated?

### Powers

• Specialising & Generalising

• Conjecturing & Convincing

• Imagining & Expressing

• Ordering & Classifying

• Distinguishing & Connecting

• Assenting & Asserting

### Teaching Trap

• Doing for the learners what they can already do for themselves

• Teacher Lust:

• desire that the learner learn

• allowing personal excitement to drive behaviour

### Mathematical Themes

• Doing & Undoing

• Invariance Amidst Change

• Freedom & Constraint

• Extending & Restricting Meaning

### Protases

Only awareness is educableOnly behaviour is trainableOnly emotion is harnessable

### Didactic Tension

The more clearly I indicate the behaviour sought from learners,

the less likely they are togenerate that behaviour for themselves

### Pedagogic Domains

• 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

• Varying DofPV; exposing RofPCh

### Variation

• Object(s) of Learning

• Key understandings; Awarenesses

• Intended; Perceived-afforded; Enacted

• Encountering structured variationVarying to enrich Example Spaces

• Actions performed

• Tasks  activity  experience

• Reconstruction & Reflection on Action (efficiency, effectiveness)

• Use of powers & Exposure to mathematical themes

• Affective: disposition

• Psyche

• awareness, emotion, behaviour

• DofPV & RofPCh