1. Mastery in mathematics
Mentor and Link Tutor
Training
Tuesday 24th April 2018
Linda Mason
(Subject Lead for mathematics)

2. Start of our journey…
‘Teaching maths for mastery in ITE: Raise the water, raise the boats’ conference (8 December 2015)
UCET : Universities’ Council for the Education of Teachers
NCETM: National Centre Excellence in Teaching of Mathematics
NASBTT : The National Association of School-Based Teacher Trainers

3. NCETM & Maths Hubs Yorkshire Ridings Maths Hub
Helen Jones (NCETM mastery specialist)
Teacher Research Group (TRG) and Professional Development opportunities including lesson studies:
Shanghai Exchange
Year 2; Addition – bridging ten
Westgate Primary School
Year 1; Fractions – finding a half and a quarter of a shape
Year 5; Measures - perimeter
Robert Wilkinson Primary Academy
Year 4; Number - multiplication
Year 5; Geometry - symmetry

5. Mastery
The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. 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 with earlier material should consolidate their understanding, including through additional practice, before moving on.

6. Lesson Design A key teaching point of each lesson
Awareness of the tricky points

7. Representation & Structures
Remind our student teachers about the new Professional Development material being developed by the NCETM

8. What subtraction structures
can you see?

9. Representation and Structures
Example 1 use of two tens frames (twenty frame)
9 + 5 = ?
Is there another way to make a ten?

10. Video to explain this re: Fractions
CPA Approach
C – Concrete
P- Pictorial
A – Abstract
Video to explain this re: Fractions

11. CPA Approach
Evans (2017) continued: “…there doesn’t have to be a linear progression from concrete to pictorial to abstract. Instead, teachers should apply a cyclical approach. For example, even when a pupil has worked out the answer using an abstract method, it is worth asking them to use concrete manipulatives to convince others that they are correct.”

12. CPA approach supporting reasoning
Georgia says that: Do agree? Convince someone else…
8 + 5 is the same as

13. CPA approach supporting reasoning

15. Variation

16. Variation Variation offers a systematic way to look
at mathematical exercises in terms of what
is available for the learner to notice.
Marton, Runesson & Tsui (2003) cited by Morgan (2015)
By systematically keeping some things invariant while others are varied and then changing what is varied and what remains invariant, students are able to “see” (to directly perceive …) concept(s).
Lai (2003)

17. Conceptual Variation Examples and ‘non-examples’
…what is the same and what is different…?

18. Conceptual Variation – examples
Finding the area of:
Rectangle;
but what if…?;
what is the same and what is different…?
Right-angle triangle
Parallelogram (a non-rectangular example)
Scalene triangle
Trapezium
*SMART NOTEBOOK Link

19. Procedural Variation

26. Mathematical Thinking and Reasoning

27. KS2 Example

29. Fluency

31. Bridging/ Compensating
Adding 1
Adding 10
Bonds to 10
Bridging/ Compensating
Strategies
Adding 2
Adding 0
Doubles
Near Doubles

32. Principles of Mastery
If ALL pupils’ thinking and reasoning about the concrete is going to develop into thinking and reasoning with increasing abstraction, teachers need to consider:
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 lessons we design and prepare for the pupils to engage with.

33. Check List
A pupil really understands a mathematical concept, idea or technique if he/she can:
describe it in his/her own words;
represent it in a variety of ways (concrete, pictorial, abstract);
explain it to someone else;
make up his/her own examples (and non-examples of it);
see connections between it and other facts/ideas;
recognise it in a new situations and contexts;
make us of it in various ways, including new situations.

35. … today and beyond
Trainees have an equally strong understanding of how mathematical concepts develop. Training emphasises a ‘teaching for mastery’ approach to this aspect of the curriculum, placing greater importance on securing and applying pupils’ knowledge rather than moving them on to more difficult topics. Trainees and NQTs were observed putting this philosophy successfully into practice by using a range of practical resources and images to demonstrate more abstract concepts to pupils.
Ofsted (2017, p.9)

36. Mastery – Lesson Design
Including the use of textbooks

37. Warm-up and Review

38. Anchor Task
Exploring
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.

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

40. Anchor Task
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.

41. 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.

42. 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.

43. References
Lai, M. (2009) Teaching with Procedural Variation: A Chinese Way of Promoting Deep Understanding of Mathematics [Internet] (Accessed 6th Oct 2017) Morgan, D. (2015) Raise the water, raise the boats. Presented at the Teaching maths for mastery in ITE conference. London [8 December] Ofsted (2018)