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Variation as a Pedagogical Tool in Mathematics John Mason & Anne Watson Wits May 2009 1 Pedagogic Domains Concepts Topics – Arithmetic Algebra Techniques Tasks 2 (Exercises) 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 3 What’s The Difference? – = First, add one to each First, add one to the larger and subtract one from the smaller 4 What then would be the difference? What could be varied? What’s The Ratio? ÷ = First, multiply each by 3 First, multiply the larger by 2 and divide the smaller by 3 5 What is the ratio? What could be varied? 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? 6 Construction before Resolution Working down start with 12 and 8 and up, – 12 8 12 8 keeping sum invariant, – 11 9 13 7 looking for a – 10 10 14 4 multiplicative relationship – 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 Translate into I – And another – And another 7 ‘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. – – – – 8 Equations in one variable Equations in two variables Word problems of a given type … Four Consecutives 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? –1 Write Alternative: I have 4 consecutive numbers in mind. They add up to 42. What are they? 9 +1 4 +2 +1 +3 +2 +6 4 D of P V? R of P Ch? +2 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, 10 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] 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] If 11 Tunja Sequences 12 -1 x -1 – 1 = -2 x 0 0x0–1= -1 x 1 1x1–1= 0x2 2x2–1= 1x3 3x3–1= 2x4 4x4–1= 3x5 With the Grain Across the Grain Lee Minor’s Mutual Factors x2 + 5x + 6 = (x + 3)(x + 2) 2 x = (x + 6)(x – + x2 + + 5x 13x–+630 = (x + 10)(x 1) 3) 2 x x2 + + 13x 25x – + 30 84 = = (x (x + + 15)(x 21)(x – + 4) 2) x2 + 25x – 84 = (x + 28)(x – 3) x2 + 41x + 180 = (x + 36)(x + 5) x2 + 41x – 180 = (x + 45)(x – 4) 13 14 43 44 45 46 47 48 49 42 21 22 23 24 25 26 41 20 7 8 99 10 27 40 19 6 1 2 11 28 39 18 5 4 3 12 29 38 17 16 15 14 13 30 37 36 35 34 33 32 31 50 64 36 37 38 39 40 41 42 43 44 35 14 15 16 17 18 19 20 45 34 13 2 3 4 21 46 33 12 11 10 1 5 22 47 32 31 30 9 8 7 6 23 48 29 28 27 26 25 24 49 50 15 81 Triangle Count 16 Up & Down Sums 1+3+5+3+ 1 22 + 3 2 = = 3x4+1 See generality through a particular Generalise! 1 + 3 + … + (2n–1) + … + 3 + 1 = 17 (n–1)2 + n2 = n (2n–2) + 1 Perforations How many holes for a sheet of r rows and c columns of stamps? 18 If someone claimed there were 228 perforations in a sheet, how could you check? Differences 19 1 1 1 1 1 1 7 6 42 2 1 2 1 11 1 1 1 1 1 11 3 2 6 8 7 56 6 24 4 8 Anticipating 1 1 1 1 1 Generalising 4 3 12 2 4 Rehearsing 1 1 1 5 4 20 Checking 1 1 1 1 1 1 1 1 1 Organising 6 5 30 2 3 3 6 4 12 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’. THOANs Think of a number Add 3 Multiply by 2 Subtract your first number Subtract 6 You have your starting 20 number 7 7+3 2x7 + 6 2x7 + 6 – 7 2x7 – 7 7 +3 2x 2x 2x +6 +6– – 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-ofpermissible-change 21 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? 22 Powers Specialising & Generalising Conjecturing Imagining Ordering & Convincing & Expressing & Classifying Distinguishing Assenting 23 & Connecting & 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 24 Mathematical Themes Doing & Undoing Invariance Freedom & Constraint Extending 25 Amidst Change & Restricting Meaning Protases Only awareness is educable Only behaviour is trainable Only emotion is harnessable 26 Didactic Tension The more clearly I indicate the behaviour sought from learners, the less likely they are to generate that behaviour for themselves 27 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 Tasks – Varying DofPV; exposing RofPCh 28 Variation Object(s) of Learning – 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 29 DofPV & RofPCh