Teaching for mastery Robert Wilne Director for Secondary NCETM 8 December 2015

robert.wilne@ncetm.org.uk @NCETMsecondary

Everyday “mastery” What does “mastery” mean? I know how to do it. It becomes automatic and I don’t need to think about it. I do it confidently. I do it well. (Does “well” mean “quickly”? Sometimes?) I can do it in a new way, or in a new situation. I can now do it better than I used to. I can show someone else how to do it. I can explain to someone else how to do it.

“Mastery” … what? “Mastery teaching” “Mastery” does not refer to what the TEACHER knows and does “Mastery” is the hard-fought outcome for ALL the PUPILS “Teaching of mastery” “Mastery curriculum”

Mastery of 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

Evidence of mastery?

Insufficient mastery? 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”

Teaching FOR mastery: ALL pupils develop … Really? Low ability as well? Yes! “Ability” factual knowledge Confident, secure, flexible and connected conceptual understanding procedural fluency

I ONLY know a pupil’s attainment hitherto. 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

Rotten to the core or a one-off lapse?

Teaching FOR mastery If ALL 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.

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 language; 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 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 language; 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 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.

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.

A good model … 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 …

… leads to 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 =

“BODMAS”… waste of oxygen! 3238 + 5721 + 1762 – 5721 5a – 3b – 5a + 3b 731 ÷ 13 × 91 ÷ 7 8a ÷ 6 ÷ 4a × 9 ÷ 3 44 × 175 37 × 17 + 17 × 63 (6a + 9b) ÷ 3 (6a × 9b) ÷ 3 Pupils read “BODMAS” left to right, and so make slow or no progress with these “BODMAS” is utterly silent here: it says NOTHING about associativity or distributivity

A Shanghai lesson When you design a lesson, what do you think about? What is the sequence of your thinking? Where do you get ideas from? Where do you get help from? How do you decide what’s the correct design?

A Shanghai lesson

A lesson from Shanghai Often, teachers start by planning what their Shanghai teachers always started with different groups will DO in the lesson what ALL pupils must THINK in the lesson

… confident, secure, flexible and connected conceptual understanding Teaching FOR mastery: N (all!) QTs need … factual knowledge … confident, secure, flexible and connected conceptual understanding procedural fluency

And their self- commitment to do so through … Time to think Time to be observed Time to prepare And their self- commitment to do so Time to discuss Time to develop Time to observe Time to reflect

Raise the water, raise the boats … and what’s in the tap? What’s the boat? What’s the water?