Variation as a Pedagogical Tool in Mathematics John Mason & Anne Watson Wits May 2009

Pedagogic Domains Concepts Topics Techniques (Exercises) Tasks Arithmetic  Algebra Techniques (Exercises) Tasks

Topic: arithmetic  algebra Expressing Generality for oneself Multiple Expressions for the same thing leads to algebraic manipulation Both of these arise from becoming aware of variation Specifically, of dimensions-of-possible-variation

What then would be the difference? What then would be the difference? What’s The Difference? – = 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 • 153 – 87 = ? Orally; visibly; •That is arithmetic; Now let’s do some mathematics! Two volunteers: think of a three digit number and a two digit number. • We’re going to subtract the smaller from the larger but before we do, lets add one to both. What’s the difference now?

÷ = What’s The Ratio? First, multiply each by 3 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 • 153 – 87 = ? Orally; visibly; •That is arithmetic; Now let’s do some mathematics! I am thinking of two numbers the first larger than the second • We’re going to divide the smaller int othe larger but before we do, lets multiply both by 3. What’s the rationow?

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. Notice that we don’t know how many things people have. We are working with actions not counts Use of mental imagery to imagine 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 I start with 12 and 8 12 8 12 8 11 9 13 7 10 10 14 4 15 5 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 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 …

Four Consecutives Write down four consecutive numbers and add them up – 1 + 1 + 2 4 Write down four consecutive numbers and add them up and another Now be more extreme! What is the same, and what is different about your answers? + 1 + 2 + 3 + 6 4 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] Stuck? See next slide!

With the Grain Across the Grain Tunja Sequences -1 x -1 – 1 = -2 x 0

Lee Minor’s Mutual Factors 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 44 45 46 47 48 49 50 49 42 21 22 23 24 25 26 25 41 20 7 8 9 10 27 9 40 19 6 1 2 11 28 1 39 18 5 4 3 12 29 4 38 17 16 15 14 13 30 16 37 36 35 34 33 32 31 36

36 37 38 39 40 41 42 43 44 35 14 15 16 17 18 19 20 45 64 34 13 2 3 4 21 46 33 12 11 10 1 5 22 47 32 31 30 9 8 7 6 23 48 81 29 28 27 26 25 24 49 50

Triangle Count Specialising; Generalising; Imagining; Expressing (to self, to others) When will crossings align?

See generality through a particular Up & Down Sums 1 + 3 + 5 + 3 + 1 = 22 + 32 = 3 x 4 + 1 See generality through a particular Generalise! Specialising in order to generalise Seeing things differently (Watch What You Do; Say What You See) Seeing this as 2^2 + 3^2 (inner and outer square on) leads to Draw the diagram for 1 + 3 + 5 + 7 + 5 + 3 + 1 and ‘see’ 3^2 + 4^2 = 5^2 1 + 3 + … + (2n–1) + … + 3 + 1 = (n–1)2 + n2 = n (2n–2) + 1

How many holes for a sheet of r rows and c columns Perforations If someone claimed there were 228 perforations in a sheet, how could you check? How many holes for a sheet of r rows and c columns of stamps?

Anticipating Generalising Differences Anticipating Generalising If n = pqr then 1/n = 1/pr(q-r) - 1/pq(q-r) so there are as many ways as n can be factored as pqr with q>r Rehearsing Checking Organising

Tracking Arithmetic THOANs Think of a number Add 3 Multiply by 2 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 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 – 2x – Ped Doms

Concepts 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

Comparisons Which is bigger? What variations can you produce? 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 educable Only behaviour is trainable Only emotion is harnessable

Didactic Tension The more clearly I indicate the behaviour sought from learners, the less likely they are to generate that behaviour for themselves

Pedagogic Domains Concepts Topics Techniques (Exercises) Tasks 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

Variation Object(s) of Learning Actions performed Key understandings; Awarenesses Intended; Perceived-afforded; Enacted Encountering structured variation Varying 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