Asking “why?” without asking “why?” Reasoning: little and often Robert Wilne Director for Secondary(ish) NCETM 30 January 2016 Holmes or Watson? Which calculation is the odd one out? Explain your reasoning. 753 × 1.8 75.3 × 20 – 75.3 × 2 753 + 753 ÷ 5 × 4 750 × 1.8 + 30 × 0.18 (75.3 × 3) × 6 7.53 × 1800 Developing pupils’ reasoning is NOT … Why? Why? Why? Why? Why? Why? Why? Learning … through of through of through of through of through of through of through of … reasoning Odd one out 9 1 2 = + 20 4 5 13 1 2 = + 20 4 5 3 1 2 = + 9 4 5 3 1 2 = + 20 4 5 True or false? 5 ÷ 5 5 = 8 ÷ 8 8 5 × 2 5 = 8 ÷ 2 8 5 - 2 5 = 8 - 2 8 5 × 0 5 = 8 × 0 8 “Always, sometimes, never” a c a +c + = b +d b d a c a ×c × = b ×d b d a c a ÷c ÷ = b ÷d b d Mastery in mathematics (after Holt) I feel I have mastered something if and when I can • state it in my own words CONCEPTUAL UNDERSTANDING (CU) • give examples of it CU • foresee some of its consequences CU • state its opposite or converse CU • make use of it in various ways PROCEDURAL FLUENCY (PF) & KNOW-TO-APPLY (K-TO-A) • recognise it in various guises and circumstances PF & K-TO-A • see connections between it and other facts or ideas CU & PF & K-TO-A John Holt, How Children Fail “Mastery” … what? “Mastery teaching” “Mastery” “Mastery”does is not to what therefer hard-fought the TEACHER outcome for ALL knows and does the PUPILS “Mastery curriculum” “Teaching of mastery” Teaching FOR mastery: ALL pupils develop … Really? Low Yes! “Ability” ability as well? factual knowledge Confident, secure, flexible and connected procedural fluency conceptual understanding Rotten to the core or a one-off lapse? Fixed ability or attainment hitherto? I do believe that • some pupils grasp mathematical concepts more rapidly than their peers do; • some pupils get better marks in tests and exams than their peers do; • some pupils have the cognitive architecture that means they see patterns and structures more quickly, and remember them more readily, than others do. • But I don’t know reliably who will • and I can’t predict with certainty who will • and my pupils’ brain pathways will change over time, as they grow up and in response to their environment. I ONLY know a pupil’s attainment hitherto. I NEVER know a pupil’s ability or potential or “bright”-ness Conceptual understanding … 3, 5, 8: three numbers four sentences 3+5=8 5+3=8 8–5=3 8–3=5 … supports procedural fluency … • + 17 = 15 + 24 17 22 24 15 • 99 – = 90 – 59 90 9 9 59 ? … and leads pupils to generalisation • 90 – 59 ≡ 99 – 68 ≡ 102.4 – 71.4 3.4 9 3.4 9 90 59 ? • 10 – 4 ≡ 17 – 11 ≡ 6 – 0 • 10 – 4 ≡ 5 – -1 ≡ -3 – -9 Pupils can generalise subtraction into the negative numbers BEFORE being taught how to do so Generalisation … Pupils can generalise with confidence that • whenever B+A=C A+B=C it will always be true that C–A=B C–B=A … leads to abstraction Conceptual understanding … 3, 5, 15: three numbers four sentences 3 × 5 = 15 5 × 3 = 15 15 ÷ 3 = 5 15 ÷ 5 = 3 … and leads pupils to generalisation Pupils can generalise with confidence that • whenever A×B=C B×A=C it will always be true that C÷A=B C÷B=A A “good” representation … 6×⅓ 1 … support conceptual understanding 6 × ⅓ = 2, and what ELSE do I know? 1 1 A “good” representation … 4×⅗ 1 … support conceptual understanding 4 × ⅗ = 2⅖, and what ELSE do I know? 1 1 Pupils can consider non-integer rational divisors BEFORE being taught how to do so Conceptual understanding … ? Distance Speed Time ? Not long after … ? ? ? ... supports factual recall Snails and cheetahs Does +, –, × or ÷ capture the intuitive relationship of speed, distance & time? Distance Speed Time Ever after … “BODMAS”… • • • • 3238 + 5721 + 1762 – 5721 5a – 3b – 5a + 3b 731 ÷ 13 × 91 ÷ 7 8a ÷ 6 ÷ 4a × 9 ÷ 3 • (6a + 9b) ÷ 3 • (6a × 9b) ÷ 3 • (6a2 ÷ 9a) ÷ 3a Pupils read “BODMAS” left to right, and so make slow or no progress with these “BODMAS” is utterly silent here: it says NOTHING about distributivity “BODMAS” is kryptonite Because the mnemonic “BODMAS” • is unhelpfully (and inaccurately) dogmatic • is arbitrary (why not OBMDSA?) • often leads to cumbersome and inefficient calculating • masks the actual axiomatic structure of arithmetic • leads to thinking there are exceptions and “special cases” • doesn’t help pupils move from (concrete) arithmetic to (abstract) algebraic manipulation. “BODMAS” Pupils recall that × and ÷ are distributive over + and − because × and ÷ are over + and − Olympic Podium Learning maths through reasoning If pupils’ thinking and reasoning about the concrete is going to develop into thinking and reasoning with increasing abstraction, they need us their teachers to help make this happen. Crucial to this happening successfully are • the choice of the representation / model with which we introduce a (new) concept • the reasoning we cultivate and sharpen through the discussions we foster and steer • the misconceptions we predict and confront as part of the sequence of questions we plan and ask • the conceptual understanding we embed and deepen through the intelligent practice we design and prepare for the pupils to engage in and with. Convince yourself In each pair of fractions, what mathematical symbol could you insert between them? Explain your reasoning. 1 3 11 31 33 93 > > 9 3 33 11 1 5 2 7 < 2 5 Convince me In each pair of fractions, what mathematical symbol could you insert between them? Explain your reasoning. 1 3 > 1 5 • I have ⅓ of a bag of a sugar and ⅕ of a bag or flour. I notice that they weigh the THEsame. SAME.IsIsa afull fullbag bagofofflour flourheavier heavieroror lighter or the same weight as a full bag of sugar? “Which is a radius?” Ah-ha! The “top side” is not always the “top” side The choice of model matters a lot! If I SHARE 13 objects between 4 people, each gets 3 objects and there’s 1 object left over. 13 ÷ 4 = 3r1 If I GROUP 13 objects into groups of size 4, there are 3 full groups and one quarter-full group. 13 ÷ 4 = 3¼ Conceptual understanding … • 12 ÷ 3 = 4 … • … because four sticks of length 3 fit into a gap of length 12. • 12 ÷ 2.4 = 5, because … • 12 ÷ 8 = 1½ because … … supports procedural fluency … • 12 ÷ 3 = 4 (4 sticks of length 3 fill a gap of length 12) • … so 12 ÷ 1.5 = • 3÷⅔= … and supports generalisation • So 12 ÷ 0.3 = • and 120 ÷ 0.03 = • and 12 ÷ 0 = • and 24 ÷ 6 = • and 12x ÷ 3x = • and 12a ÷ 3/b = How would your pupils tackle this? • I share some sultanas between Alice and Bob in the ratio 3:5. Alice gets 28g fewer sultanas than Bob. How many grams of sultanas does Bob get? Alice Alice Alice Bob Bob Bob 28g Bob Bob 28g 14g Bob gets 70g Box / Bar representations 1. Sam has 9 fewer sweets than Sarah. They have 35 sweets altogether. How many sweets does Sam have? Sam 13 ? = 35 Sarah 13 ? 9 2. Sam and Tom share 45 marbles in the ratio 2:3. How many more marbles does Tom have than Sam? Sam 9? 9? = 45 Tom 9? 9? 9? Box / Bar going deeper In Year 4, 45% of the pupils are boys. There are 14 fewer boys than girls in Year 4. How many girls are there in Year 4? If you use different values other than 14, will the answers you get to the question “How many girls are there in Year 4?” always, sometimes or never make sense? Box / Bar challenge • On Saturday I opened a new packet of sultanas and gave Charlie 40% of them. On Sunday I gave him 25% of the remaining sultanas. Today I have given him the remainder of the sultanas, which weigh 27g. What was the weight of the packet when it was full? Full weight is 60g 12g 12g 12g 36g Sunday 9g 9g 9g Saturday Saturday “Good” models / representations … • can at first be explored “hands on” by ALL pupils irrespective of prior attainment • arise naturally in the given scenario, so that they are salient and hence “sticky” • can be implemented efficiently, and increase ALL pupils’ procedural fluency • expose, and focus ALL pupils’ attention on, the underlying mathematics • are extensible, flexible, adaptable and long-lived, from simple to more complex problems • encourage, enable and support ALL pupils’ thinking and reasoning about the concrete to develop into thinking and reasoning with increasing abstraction. But PF is not the same as CU • • • • I share some sultanas between Alice and Bob in the ratio 3:5. Alice gets 28g fewer sultanas than Bob. How many grams of sultanas does Bob get? I share some sultanas between Alice and Bob in the ratio 6:10. Alice gets 28g fewer sultanas than Bob. How many grams of sultanas does Bob get? I share some sultanas between Alice and Bob, so that Bob gets ⅝ of all the sultanas. Alice gets 28g fewer sultanas than Bob. How many grams of sultanas does Bob get? I share some sultanas between Alice and Bob, so that Alice gets 60% of what Bob gets. Alice gets 28g fewer sultanas than Bob. How many grams of sultanas does Bob get? Even good models have limitations • Drawing boxes / bars focuses pupils’ attention on the additive structure of the Alice and Bob problem: Alice has 2 boxes / bars fewer than Bob, Bob has 2 boxes / bars more. • The risk is that we don’t ensure that pupils also acquire deep conceptual understanding of the multiplicative structure of the problem: that A A A • Alice’s share is ⅗ of Bob’s share B B B B B • Bob’s share is ⅝ of the total • Alice’s share is Bob’s share reduced by 40% • the scale factor from Alice’s share to Bob’s is 1⅔. The CSMS project (Kath Hart et al.) Mr Short is as tall as 6 paperclips. He has a friend Mr Tall. When they measure their height with matchsticks, Mr Short’s height is 4 matchsticks and Mr Tall’s height is 6 matchsticks. What is Mr Tall’s height measured in paperclips? About ⅔ of pupils said “8”. They interpreted 4 to 6 as an additive increase “+ 2”, not a multiplicative increase “× 1.5” Aims and ethos of the UK curriculum All pupils must become FLUENT in the fundamentals of mathematics, through varied and SOLVE PROBLEMSincluding by applying frequent practice their mathematics to a variety of with increasingly REASON MATHEMATICALLY problems over time, so that routine andcomplex non-routine by following line of enquiry, pupils developaconceptual problems with increasing conjecturing understanding the ability to sophistication, including breakingand relationships and generalisations, and apply down problemsrecall into aand series of knowledge rapidly developing and accurately. simpler steps and persevering in an argument, justification or proof using seeking solutions. mathematical language. The vision of the new curriculum All pupils become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately; reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions. The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should language; always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent should consolidate their understanding, including through additional practice, before moving on. The vision of the new curriculum All pupils become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately; reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions. The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should language; always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent should consolidate their understanding, including through additional practice, before moving on. A Chinese takeaway You need to find out what the kites have each seen, and Teachingensure mathsthat successfully is like … they’ve all identified the key landmarks Reasoning deepens connections Reasoning deepens connections Learning amounts to being able to discern certain aspects of the phenomenon that one previously did not focus on or which one took for granted, and simultaneously bring them into ones focal awareness. (Lo, Chik & Pang, 2006) Comparison is considered the pre-condition to perceive the structures, dependencies, and relationships that may lead to mathematical abstraction. (Sun, 2011) A Chinese takeaway Homework With colleagues, choose a strand that runs across the curriculum, and discuss how to secure your pupils’ conceptual understanding and procedural fluency THROUGH reasoning about it. Topic Pupils Year N Year N+1 Year N+2 Year N+3 Year N+4 Content The model / representation, discussion, questioning and intelligent practice that will develop pupils’ … … factual knowledge … procedural fluency … conceptual understanding Agree or challenge? Bouquets to: Dan Abramson, Michael Davies, Tony Gardiner, Debbie Morgan, Matt Nixon, Maths Hub Leads, Team NCETM, et al. Brickbats to me! [email protected]

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