mastery approaches to maths ma2m+ · lesson phase outline anchor task. exploring: one problem or...

Mastery Approaches to Maths MA2M+

Monday 20th November 2017: Teaching for Mastery Maths: PrinciplesMonday 04th December 2017: Teaching for Mastery Maths: In your classroomMonday 05th February 2018: Teaching for Mastery Maths: Lessons from ShanghaiMonday 19th March 2018: Teaching for Mastery Maths: Greater depthMonday 23rd April 2018: Teaching for Mastery Maths: Next steps

Presenter

Presentation Notes

This is the session 2 twilight on mastery approaches to maths…

Because moving to mastery approaches:

• Requires whole school change through collective leadership…meanwhile need to support individual teachers

• Challenges embedded beliefs and strategies around intelligence, in-class grouping and differentiation by task

• Connects to social justice issues, basically by helping to stop labelling of children and raising expectations for all

• May be undertaken in a cautious pick ‘n’ mix approach by many schools

• Schools that commit to a mastery scheme still need to continue to question practice

You Cubed https://www.youcubed.org/

A tentative proposal…

NCETM ≈ principles and strategies of teaching for mastery

Singapore Maths ≈ contextualized problem-solving

Shanghai Maths ≈ small steps

Boaler (complex instruction) ≈ growth mindset classroom

A fact!?@#

All high scoring nations in maths tests use text book schemes

Presenter

Presentation Notes

Shanghai are currently looking at ways to improve pupil performance in problem-solving… So maybe Singapore Maths schools might need to consider how pupils are developing arithmetic fluency!

Lesson phase OutlineAn

chor

Tas

kExploring One problem or stimulus is presented to pupils (based on what is in the textbook) and they are

encouraged to explore it. The teacher uses this time to observe their responses and prompt further exploration with questioning to ensure that all pupils are challenged.

Structuring The teacher gathers together pupils’ ideas for solutions and the class discuss them as a whole group, often re-exploring new suggestions.

Journaling Pupils record what they have been doing in their maths journals – there is an emphasis on showing things in different ways and effective communication of thinking.

Reflect and refine The textbook is used and the teacher guides the class through the textbook solutions to the problem they have been discussing. There is a greater emphasis on teacher explanation during this phase.

Practice The teacher starts off by guiding the class through examples of similar problems to the one they have just done. Then, pupils work through more examples independently with the teacher supporting them if necessary. All questions are typified by their mathematical variation – they are designed to extend pupil’s thinking rather than just be lots of examples presented in the same kind of way.

Boyd, P. & Ash, A. (2018 in press) Teachers framing exploratory learning within a text-book based Singapore Maths mastery approach. Teacher Educator Advancement Network Journal.

Rosie’s Singapore Maths(Maths - No problem! )

lesson…

Presenter

Presentation Notes

Singapore Maths and the text book scheme Maths – No Problem! focus on contextualised problem solving…so this carefully selected and lesson study ‘tried and tested’ problem is the start of a year 4 lesson…

Rosie’s Singapore Maths(Maths - No problem! )

lesson…

Presenter

Presentation Notes

Carefully designed practice tasks include variation…

5 big ideas

Presenter

Presentation Notes

5 big ideas from Primary Mastery Specialist Programme Are these relevant to Secondary?

• A representation is used to pull out the concept being taught, exposing the underlying structure.

• In the end, pupils need to be able to do the maths without the representation.

• A stem sentence can be used to describe the representation and helps pupils move to working in the abstract.

Representation and Structure

Lily was asked to explain her answer. She said it was easy because she was taught in reception that the crocodile eats the biggest number.

Has this child got a deep understanding of the inequality symbols?

https://www.youtube.com/watch?v=W1eLt0Dz8Fk

Mathematics is an abstract subject, representations have the potential to provide access and develop understanding.

Representation and structure•“Mathematical tools should be seen as supports for learning. But using tools as supports does not happen automatically. Students must construct meaning for them. This requires more than watching demonstrations; it requires working with tools over extended periods of time, trying them out, and watching what happens. Meaning does not reside in tools; it is constructed by students as they use tools” (Hiebert 1997 p 10) Cited in Russell (May, 2000). Developing Computational Fluency with Whole Numbers in the Elementary Grades

http://investigations.terc.edu/library/bookpapers/comp_fluency.cfm

http://investigations.terc.edu/library/bookpapers/comp_fluency.cfm

Concrete Pictorial (iconic) Abstract

The C-P-A approach isn’t about getting the answer quickly and isn’t just for the less able children, its about giving children the tools to understand the problem in front of them. Even when a child has answered the question in the abstract method, it is worth getting them to use concrete manipulatives to convince others that they are correct.

C-P-A is for everyone, mastery teaching encourages the use of concrete manipulatives in any lesson and suggests there is value in the children having a variety of equipment to aid their thinking and deepen their understanding by explaining in different ways.

Finally, CPA is a way to deepen and clarify mathematical thinking. Students are given the opportunity to discover new ideas and spot the patterns, which

will help them reach the answer. From the start of KS1, it is a good idea to introduce CPA as three interchangeable approaches, with pictorial acting as

the bridge between concrete and abstract.When teaching for mastery, the CPA approach helps learners to be more secure in their understanding, as they have to prove that they have fully grasped an idea. Ultimately, it gives pupils a firm foundation for future

learning.

Resources and Representations of Mathematics• Resources to help build concepts

Used during the learning of new concepts or when building further onto learnt conceptsIt allows teachers to gain a greater understanding of where misconceptions lie and the depth a child exhibits. allows children to develop their ability to communicate mathematically and to reason.It gives children a deep understanding of mathconcrete resources give pupils time to investigate a concept first - and then make connections when formal methods are introduced

Presenter

Presentation Notes

Explain that the concrete-pictorial-abstract approach is based on Jerome Bruner’s important psychological research. Bruner suggested that pupils need to take three vital steps in order to develop their understanding of a concept. This systematic approach is designed to give pupils the time to make connections, notice mathematical patterns and really understand what’s going on.

Using the structure of thetens frame

• Using your tens frames illustrate this• Calculation:

There are 7 daffodils and 5 rosesHow many flowers are there altogether?

Bridging 10

How can we use 10 to solve the addition problem?

Presenter

Presentation Notes

Is there more than one way to solve this? Option 1: some pupils might count all. Option 2: others might move three yellow counters to make 10. Option 3: a group of pupils might move one red counter down to make 5 because they know that 5 + 5 = 10. It’s important to allow your pupils to explore different ways of getting the answer, but a teacher will always have a preferred method they want the pupils to use. How can this resource be used in other year groups?�In KS2 this resource could be adapted eg the ten frame is now worth 1, in which case each concrete item is worth 0.1 (similarly, it might also be worth 100, 2, 20 etc). How can this resource be represented pictorially?

Spine Materials – NCETM

Mastery Professional Development

www.ncetm.org.uk/masterypd© Crown Copyright 2017

Autumn 2017 pilot

1.2 Introducing 'whole' and 'parts’:part–part–whole

•Representations | Year 1

• Number, Addition and Subtraction• https://www.ncetm.org.uk/resources/50719

http://www.ncetm.org.uk/masterypd

https://www.ncetm.org.uk/resources/50719

© Crown Copyright 2017www.ncetm.org.uk/masterypd

The following slides contain the representations described in the teacher guide, and are intended to accompany the teacher guide. They do not represent complete lessons and should not be used as such.

However, you may wish to use the slides in conjunction with the teacher guide to support the planning of lessons, in combination with other resources such as high-quality textbooks that follow a teaching-for-mastery approach.

Autumn 2017 pilot

• How to use this presentation

You can find the teacher guide 1.2 Introducing 'whole' and 'parts': part–part–whole by following the link below.

https://www.ncetm.org.uk/masterypd

© Crown Copyright 2017www.ncetm.org.uk/masterypd Autumn 2017 pilot

• 1.2 Part–part–whole – step 1:1




5




3




2

1

3




5

3 2




5

2 3



1.2 Part–part–whole – Steps 4:4• 1.2 Part–part–whole – step 4:4

5

1 4




5

4 1


Multiple representations

“If we do not use concrete manipulations, then we can not understand mathematics. If we only use concrete manipulations, then we are not doing mathematics.”

Gu (2015)

Presenter

Presentation Notes

Concrete manipulatives are essential learning tools, but if pupils use only concrete resources, then they will become mechanics of maths, rather than mathematicians. With this in mind, the final quote powerfully reinforces and summarises to your staff the importance of switching through the three different modes to develop and deepen understanding.

Representation and Structure

The Bar Model

• a way of revealing the mathematical structure within a problem

• enables children to gain insight and clarity as to how to solve it

• supports the transformation of real life problems into a mathematical form

• bridges the gap between concrete mathematical experiences and abstract representations

• It can be used to represent problems involving

o the four operationso ratio and proportiono unknowns in a problem

The Bar Model

Part – Whole: Addition and Subtraction

The Bar Model

whole

part

5

23

5 = 3 + 2 5 = 2 + 3

3 = 5 - 2 2 = 5 - 3

part

Part – Whole: Addition and Subtraction

The Bar Model

whole

partpart

a

cb

a = b + c a = c + b

b = a - c c = a - b

Comparison: Addition and Subtraction

The Bar Model

part

part

whole 5

2

3

Part – Whole: Multiplication and Division

The Bar Model

whole

part

24

6

24 = 6 x 4 6 = 424

4 = 62424 = 4 x 6

Comparison: Multiplication and Division

The Bar Model

whole

part

24

6

95 – 79 = 16

16

17 + 23 = 40

75 - 40 = 35355

= 735

21 x 3 = 63842

= 42 422

= 21

42 4242

21 21

21

42 x 3 = 126126 + 63 = 169

£4.80 £4.80

£4.80 x 2 = £9.60

£9.60 £9.603

= £3.20£3.20 £3.20 £3.20

£3.20 x 7 = £22.40£3.20 £3.20 £3.20 £3.20

22 22 9

n n 922 + 22 + 9 = 53

q q 4 100 – 4 = 96

10096

962

= 4848 48

q = 48

sheep

goats

28

242 + 28

242 + 28 = 270270

5= 54

54 x 3 = 162

John KotterKurt Lewin

Andy Hargreaves & Michael Fullan

Hargreaves, A. & Fullan, M. (2012) Professional Capital: Transforming Teaching in Every school. New York: Teachers’ College Press.

Presenter

Presentation Notes

In the last session we noted that moving to mastery approaches is a curriculum development project that requires collective leadership – teachers alone working bottom up will not succeed and a head teacher top down is also likely to fail…so some useful models are available to inform change leadership…

The characteristics of effective professional learning for teachers:

• A collective sustained focus on learning• Increasing ownership, collaboration and trust• Classroom inquiry and experimentation• Focus on pedagogy within curriculum subjects• Focus on formative assessment and impact• Critical engagement with external knowledge

Developing a community of practice - shared language, purpose, ways of working and of learning…

Wenger (1998)

Professional inquiry (ten steps) Lesson study

Developing Great Teaching (2015) http://tdtrust.org/about/dgt/

• Dialogue• Modelling• Inquiry

Presenter

Presentation Notes

And the TDT review of effective CPD for teachers is a useful foundation for leading professional development… if it can be combined with the principles and strategies suggested by the change leadership models… professional inquiry, perhaps in the form of lesson study could be useful…and if designed and facilitated effectively could match the TDT characteristics of effective CPD…

http://tdtrust.org/about/dgt/

Mastery Approaches to Maths MA2M+

Monday 20th November 2017: Teaching for Mastery Maths: PrinciplesMonday 04th December 2017: Teaching for Mastery Maths: In your classroomMonday 05th February 2018: Teaching for Mastery Maths: Lessons from ShanghaiMonday 19th March 2018: Teaching for Mastery Maths: Greater depthMonday 23rd April 2018: Teaching for Mastery Maths: Next steps

Presenter

Presentation Notes

This is the session 2 twilight on mastery approaches to maths…

mastery approaches to maths ma2m+ · lesson phase outline anchor task. exploring: one problem or...

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