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

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

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

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.

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

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

What subtraction structures can you see?

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

Video to explain this re: Fractions CPA Approach C – Concrete P- Pictorial A – Abstract Video to explain this re: Fractions https://mathsnoproblem.com/en/the-maths/teaching-methods/concrete-pictorial-abstract/ https://www.tes.com/teaching-resources/blog/examining-cpa-approach-primary-maths

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.” https://www.tes.com/teaching-resources/blog/examining-cpa-approach-primary-maths

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

CPA approach supporting reasoning

Variation

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)

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

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

Procedural Variation

Mathematical Thinking and Reasoning

KS2 Example

Fluency

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

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.

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.

… 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)

Mastery – Lesson Design Including the use of textbooks

Warm-up and Review https://vimeo.com/177991762

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.

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.

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.

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.

References Lai, M. (2009) Teaching with Procedural Variation: A Chinese Way of Promoting Deep Understanding of Mathematics [Internet] https://scholar.google.co.uk/scholar?hl=en&q=Lai+-+Teaching+with+Procedural+Variation:+A+Chinese+Way+of+Promoting+Deep&btnG=&as_sdt=1,5&as_sdtp (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) https://reports.ofsted.gov.uk/inspection-reports/find-inspection-report/provider/ELS/70118