1. Maths Hub Mastery
Jennifer Shearman, CCCU & Kent and Medway Maths Hub Lead Evaluator, Maths Hubs Secondary

2. LO: To be able to share a number by a given ratio
Example: Share 28 in the ratio 5:2
Add up the two ratios 5+2=7
Do Shared number/sum 28/7=4
Multiply each ratio by this number
5:2 (both x 4)
20:8
4) Now do another 50 examples.
Run through as a ‘boring’ lesson.
Then pose questions – fold paper so one person as twice as much as the other.
Fold paper so one person has three times as much as the other
If you had two pieces of paper, how would you fold so one person had twice as much as the other?
Use cubes to model 5:2 ratio. How many cubes do you need?

3. What is ‘Mastery’?
A mastery approach; a set of principles and beliefs.
A mastery curriculum.
Teaching for mastery: a set of pedagogic practices.
Achieving mastery of particular topics and areas of mathematics.
The essential idea behind mastery is that all children need a deep understanding of the mathematics they are learning.
A mastery approach; a set of principles and beliefs.
This includes a belief that all pupils are capable of understanding and doing mathematics, given sufficient time. Pupils are neither ‘born with the maths gene’ nor ‘just no good at maths.’ With good teaching, appropriate resources, effort and a ‘can do’ attitude all children can achieve in and enjoy mathematics.
2. A mastery curriculum
One set of mathematical concepts and big ideas for all. All pupils need access to these concepts and ideas and to the rich connections between them. There is no such thing as ‘special needs mathematics’ or ‘gifted and talented mathematics’. Mathematics is mathematics and the key ideas and building blocks are important for everyone.
3. Teaching for mastery: a set of pedagogic practices that keep the class working together on the same topic, whilst at the same time addressing the need for all pupils to master the curriculum. Challenge is provided through depth rather than acceleration into new content. More time is spent on teaching topics to allow for the development of depth. Carefully crafted lesson design provides a scaffolded, conceptual journey through the mathematics, engaging pupils in reasoning and the development of mathematical thinking.
4. Achieving mastery of particular topics and areas of mathematics.
Mastery is not just being able to memorise key facts and procedures and answer test questions accurately and quickly. It involves knowing ‘why’ as well as knowing ‘that’ and knowing ‘how.’ It means being able to use one’s knowledge appropriately, flexibly and creatively and to apply it in new and unfamiliar situations.

4. What is Mastery? Mastery means that learning is sufficiently: Embedded
Deep
Connected
Fluent
In order for it to be:
Sustained
Built upon
Connected to

5. Essence of Mastery All pupils can succeed.
Whole-class interactive teaching, where the focus is on all pupils working together on the same lesson content at the same time.
Pupils master concepts before moving to the next part of the curriculum sequence, allowing no pupil to be left behind.
Early intervention to ensure pupils keep up (not catch up).
Lesson design focuses on small steps through a carefully sequenced learning journey.
Typical lesson content includes questioning, short tasks, explanation, demonstration, and discussion.
NCETM document released in June 2016
Clear elements of TfM approach

6. Essence of Mastery
Procedural fluency and conceptual understanding are developed in tandem through intelligent practice.
Significant time is spent developing deep knowledge of the key ideas needed to underpin future learning.
Structures and connections are emphasised.
Key facts are learnt to avoid cognitive overload and to enable pupils to focus on new concepts.

7. What question is being asked?12 ? ? ? ? ?
How do we normally ask the question represented by the illustration?
Can you think of a worded question or an algebraic question that could result in this illustration?
The answer is really clear one blue box ie ? = 5
But if this was presented in algebra form 3x+12 =2x+17 how many of our students would stumble?
I suggest that the visual problem makes more sense intuitively than perhaps the way this type of problem often introduced 17

8. How many…?
Picking out a similar task (but different) to the one in Dave Hewitt’s article listed in the pre-reading.
Please consider and discuss between you the different ways children may arrive at 21 counters.
Find the sum of the yellows & reds separately.
5 x x 3 OR (5+5+5) + (3+3+3)
OR add the colours first then multiply (5 + 2) x 3 OR (5+2)= 7 so then = total
etc

9. What does the equals sign mean?
5 x x 3 = 21 (5 + 2) x 3 = 21 5 x x 3 = (5 + 2) x 3
There are a number of different aspects that can be derived from the previous slide, however one issue pertinent to our focus on algebra is that the equals sign can have more than one meanings – “makes” or the result on an operation AND “equivalent” two ways of saying the same thing? Or two ways of getting the same answer.
When do you think this should be pointed out to learners?

10. a b
What algebraic generalisation is being illustrated? 3a + 3b = 3(a+b)
How would you go further to represent n(a+b) or is this a step too far at this stage?
Is there a benefit in arriving at this through the arithmetical and visual, rather than presenting it as a “fait accomplis”?
Would there be any benefit in presenting the blocks in a different way or using concrete manipulatives?
Consider how this diagram how this diagram (and the previous one with counters) revealed the mathematical structure of key aspects of arithmetic.
Should we teach the rule and then test the students on it or should we ask how an arithmetic structure can be expressed, in words , numbers and eventually algebraic terms stressing this is an identity (or whatever equivalent words you choose to use) “another way of expressing something”

11. What do the letters mean?
For it to be more than just a nice explanation of this “rule” should we use things we’ve learned earlier (like variation & coherence)
Should we also ask them to illustrate expressions that don’t factorise or show partial factorisation?
What does the letters mean to a learner? – have we gone into this from an abstract point of view or a problem?
If we merely focus on the mechanics and not the meaning what do learners understand of the actual use of a letter?
3(a + b) = 3a + 3b

12. Primary maths
27 = 17 + ? a + b = c
The idea that an unknown number can be expressed as a letter, empty box, question mark is widely used in Primary schools at the moment.
Increasingly we may find use of letters as generalisations as they explore arithmetical structure in their developments towards teaching for mastery.
Do we want to make these distinctions and when?
If we do how find out what they know and have been introduced to so far either overtly or subliminally and whether they have a good understanding ?
Should we be making an effort to discuss this with our primary colleagues and feeder schools?

13. a b 3a + 3b 3(a + b) Making the connection or
So to emphasise this point – there is power in putting these two things together
Diagram & abstract , is important as we would expect Dienes blocks & calculations to be alongside each other in early KS2
Small things matter - does it makes more sense to write the equivalence on one line?
What do you make of using the word “or” instead of “=“ ?

14. Explain why …
5 + 2 = 4 + 3
Another examples to consider, which reveals a concept that is addressed at quite a young age in Shanghai
How would children explain this feature of equivalence?

15. a c b (a + c) + (b) = (a) + (c + b) Generalising
The use of letters as a generalisation is being introduced at a much early age
What do we think our students & teachers can cope with?
Does this suggest we should be talking with our Primary colleagues to find out what they are doing prior to KS3.
Schools adopting a TfM style may be using algebra much earlier than the Nat Curriculum suggests and for good reason

16. Multiplication
Before making the jump to area (and later volume) learners need to be fully fluent with the concept of multiplication
This diagram (and the alternative –see animation) only touches the surface of what is a major topic to conceptually and fluently be understood, but once this is mastered…

17. What do you see?
…we can start to ask much more interesting and challenging questions for a KS3/4 learner.
Reflect on what they “see” and how they might use such a diagram?
How would they assess the success of using such a visualization with learners?
Even though we may have come across this kind of interesting diagram before we have a lot to learn and explore about the impact of such use of diagrams and the best way of employing this with learners in the classroom.

18. Solutions
John is 8 years older than Sally. Together their ages total 30 years. How would we expect learners to use bars to express an equation rather than an equivalence?
Up until now we have focused on equivalence of expressions but what about solving equations.
Ask the delegates to jot down some ideas of how they might illustrate this equation.
You could use the equal bars or in this case Yeap Ban Har would suggest the difference model

19. Difference model
John 30
Sally
Go through with the logical sequence 8

20. 8 30 x + (x + 8) = 30 2x + 8 = 30 Sally = x 2x = 22 John = x + 8
How powerful are the bars alongside the abstract solution in linking the visual representation and the mechanism of the solution?
Personal experience has indicated a striking impact on some learners I work with – is this a consistent feature?
Is this an example of Bruner’s CPA being illustrated beyond Primary?
2x = 22
John
= x + 8
x = 11

21. More examples
I think of a number multiply it by 3 and then add 7. I find that when I add 6 first and then multiply by 2 I get the same result. Three consecutive odd numbers total 33
Discuss how learners might tackle these by modelling them as bars first.

22. I think of a number multiply it by 3 and then add 7
I think of a number multiply it by 3 and then add 7. I find that when I add 6 first and then multiply by 2 I get the same result. 7 ?
Do we need worked solutions? 12

23. Three consecutive odd numbers total 33? 2 33 2 2

24. Illustrating a relationship
a + b = 10
We’ve looked at developing an understanding of equivalence from very simple to more complex and moved on to how solving worded problems can be modelled firstly in bars and then linked to an algebraic solution but what about a relationship between two variables?
How might this relationship be illustrated in a bar?

25. Illustrating a relationship
a + b = 10
Can this lead to a better understanding of a graphical representation of this relationship?
What do the axes represent?
Does illustrating a relationship through a variety of means lead to deeper understanding MULTIPLE REPRESENTATION
Bars limit themselves to positives (unless you delve into hills and holes) but the graph takes it further

26. Structural Arithmetic
= = ? How would you continue?
From our previous work on variation take 2 mins to discuss where you could go with this.
Tony Gardiner’s article (in the reading list) touches upon aspects that can be drawn from this

27. Starting with a model 97
100
Are teachers (never mind students) fully comfortable with the fact that if they add the same value to the minuend and the subtrahend you get the same difference?
Playing with these ideas can reveal methods to create easier calculations for example – 2.37 could become 9.99 – 2.36 and subsequently no need for decomposition/partitioning 48 51

28. 9999 + 999 + 99 + 9 + 5 = 0.62 x 37.5 + 3.75 x 3.8 = Variation theory
( ) + ( ) + ( ) + ( ) + ( ) =
Seen in a Y5 lesson in Birmingham

29. 0.62 x x 3.8 =
( ) + ( ) + ( ) + ( ) + ( ) = 5 =

30. Variation Theory vs Variety
Compare the two sets of calculations
What’s the same, what’s different?
Consider how variation can both narrow and broaden the focus
Taken from Mike Askew, Transforming Primary Mathematics, Chapter 6

31. Variation leads to
Intelligent Practice
Shanghai Practice Book

32. Intelligent Practice Noticing things that stay the same, things that change, providing the opportunities to reason make connections

33. When teaching…. Look out for the 5 big ideas: Coherence
Representation & Structure
Variation
Fluency
Mathematical Thinking
Lesson 1 in comparing fractions – only considering those with same denominator or same numerator
Restrict learning to this key point
Reflect on lesson – how does it differ to what would normally be taught?

34. What next?
Thinking Blocks App NCETM website – Teaching for Mastery (Kent and Medway Hub Mastery site) Join a Maths Hub workgroup (Primary and Secondary Mastery groups)

35. …. And finally
And not a hungry crocodile in sight!