Which might be the odd one out?

Teaching for Mastery What it isWhat it’s notWhat it looks like in the classroom

Robert WilneBPET Maths Advisor29 October 2018

All pupils must

The National Curriculum

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.

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.

REASON MATHEMATICALLY by following a line of enquiry,

conjecturing relationships and generalisations, and developing an argument,

justification or proof using mathematical language.

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 broadlythe 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 National Curriculum

What does “mastery” mean?

• I know how to do it.

• Doing it has become automatic: 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.

Everyday “mastery”

What does “mastery” mean?

• I know how to do it.

• Doing it has become automatic: 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.

Everyday “mastery”

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 WHEN-TO-APPLY (W-TO-A)

• recognise it in various guises and circumstances PF & W-TO-A

• see connections between it and other facts or ideas CU & PF & W-TO-A

Adapted from John Holt, How Children Fail

Maths lesson “mastery”

factual knowledge

procedural fluency

conceptual understanding

Confident, secure, flexible and connected

Teaching FOR mastery: ALL pupils develop …

I know that…

I knowwhy…

I knowhow…

I know when…

Counting FOR mastery

Counting for mastery

Counting FOR mastery• Count in directions / orders other than left to right (e.g. start in the

middle, or end on the pink one).

• Jumble, count, jumble: “How many are there”? Do the children say they need to re-count the objects?

• Count and then spread the objects further apart: “Are there more now?”

• Have 5 similar objects and 5 different ones: “Are there the same number?”

• Have 5 large objects and 5 small ones: “Are there the same number?”

• Counting the ‘clinks’ as marbles drop one at a time into an empty jar: “How many marbles are there in the jar?”

• Modelling ‘bad’ counting: “1, 2, 3, 5, 6, so there are 6 cars”; “5, 4, 3 so there are 3 buttons”, etc.: “Agree or Challenge?”

Counting FOR mastery• Using a tens frame: ”What’s the same, what’s different?”

Counting and reasoning• “If I know …, then I know …”

• 3 + 2 = ☐

• 5 = ☐ + 3

• 5 – 2 = ☐

• ☐ = 5 – 3

5

2 3

Concrete

Abstract

Pictorial

Counting and reasoning• “If I know …, then I know …”

• 30 + 20 = 50

• 20 + 30 = 50

• 50 – 20 = 30

• 50 – 30 = 20

50

20 30

Concrete

Abstract

Pictorial

ALL pupils should be confident to think and reason

• about the concrete

• with the pictorial

• in the abstract

and translate comfortably from one to another back and forth

Teaching FOR mastery: CPA

Different(iated) outcomes not tasks• Differentiation when teaching for mastery means that

teachers have DIFFERENT conversations and interactions across a group of pupils all of whom are thinking about the SAME mathematics and are progressing through the SAMEquestions and tasks.

• Different pupils need and get different support and challenge from their teachers, but everyone is literally on the same page.

0

Teachers for mastery

• expect

• enable

• support

their pupils to use correct mathematical vocabulary and terminology, and to give their answers in coherent full sentences. The teachers do so by

• modelling this themselves

• teaching vocabulary explicitly

• celebrating and reinforcing its correct use

• noticing and responding to its omission.

“I can do it but I can’t explain it”

“I’ve finished”

• “Now make up your own questions like the ones we’ve been doing”

• “Read each other’s reasoning and give feedback to improve it”.

• “I’m going to ask you to explain your answer to question 3 to the whole class. Write down what you’re going to say”.

• “Make up an example where this method we’ve been using won’t work well – and explain why not.”

Teaching FOR mastery: intelligent practice

Intelligent practice: “when designing exercises, the teacher is advised to avoid mechanical repetition and to create an

appropriate path for practising the thinking process with increasing creativity”

(Gu, 1991)

• 1 × 2 = ☐

• 2 × 2 = ☐

• 3 × 2 = ☐

• 4 × 2 = ☐

• 5 × 2 = ☐

• etc … zzzz ….

Times tables: ‘unintelligent’ practice

• 1 × 2 = ☐

• 2 × 2 = ☐

• 4 × 2 = ☐

• 8 × 2 = ☐

• 5 × 2 = ☐

• 10 × 2 = ☐

• 3 × 2 = ☐

• 6 × 2 = ☐

• 9 × 2 = ☐

Times tables: more intelligent practice

• 2 × 2 = ☐

• 5 × 2 = ☐

• 7 × 2 = ☐

• 10 × 2 = ☐

• 1 × 2 = ☐

• 9 × 2 = ☐

• 9 × 2 = ☐

• 3 × 2 = ☐

• 6 × 2 = ☐

Times tables: more intelligent practice

• 1 × 2 = ☐

• ☐ × 2 = 10

• 2 ×☐ = 16

• ☐ ×☐ = 12

• 6 ÷ 2 = ☐

• ☐ ÷ 2 = 7

• 8 ÷☐ = 2

• 8 ÷☐ = ☐

• ☐ ÷☐ = 10

Times tables: intelligent practice

Times tables: ‘unintelligent’ practice

× 1 2 3 4 5 6

1

2

3

4

5

6

Times tables: more intelligent practice

× 1 2 4 8 3 6

1

2

3

4

5

6

Times tables: more intelligent practice

× 1 2 4 8 3 6

1 1 2 4 8 3 6

2 2 4 8 16 6 12

3 3 6 12 24 9 18

4 4 8 16 32 12 24

5 5 10 20 40 15 30

6 6 12 24 48 18 36

Times tables: intelligent practice

× 5 2 6

10

5 50

20

11

90 45

100 400

Times tables: intelligent practice

× 5 10 2 4 6 3

2 10 20 4 8 12 6

5 25 50 10 20 30 15

10 50 100 20 40 60 30

11 55 110 22 44 66 33

15 75 150 30 60 90 45

100 500 1000 200 400 600 300

Times tables: intelligent practice

× 5 10 2 4 6 3

2 10 20 4 8 12 6

5 25 50 10 20 30 15

10 50 100 20 40 60 30

11 55 110 22 44 66 33

15 75 150 30 60 90 45

100 500 1000 200 400 600 300

Times tables: intelligent practice

× 5 10 2 4 6 3

2 10 20 4 8 12 6

5 25 50 10 20 30 15

10 50 100 20 40 60 30

11 55 110 22 44 66 33

15 75 150 30 60 90 45

100 500 1000 200 400 600 300

Times tables: intelligent practice

× 5 10 2 4 6 3

2 10 20 4 8 12 6

5 25 50 10 20 30 15

10 50 100 20 40 60 30

11 55 110 22 44 66 33

15 75 150 30 60 90 45

100 500 1000 200 400 600 300