Mastery and Theory of Mathematics

1. Mastery and Theory of Mathematics PGCE Primary 2019-20 Mark Harris

2. To develop a greater understanding of how to develop a ‘Mastery of Maths’. • To consider strategies to promote children’s depth of knowledge in mathematics. • To explore theory that underpins teaching mathematics. • Session Overview

3. TS 3 Demonstrate Good Subject and Curriculum Knowledge • Have a secure knowledge of the relevant subject/s and curriculum areas, foster and maintain pupils’ interest in the subject and address misunderstandings • Demonstrate a critical understanding of developments in the subject and curriculum areas and promote the value of scholarship • If teaching early mathematics, demonstrate a clear understanding of appropriate teaching strategies • Standards Addressed

4. Boaler, J (2016) Mathematical Mindets • NCETM (2014) Mastery Approaches to Mathematics and the New National Curriculum • Skemp, R (1978) Relational understanding and instrumentalunderstanding. Arithmetic Teacher, 26 (1978), pp. 9–15. • Research

5. Maths Curriculum – Historical Context • 1982 - Cockcroft Report: Identified principles for teaching maths • 1989 – National Curriculum Introduced • 1999 – National Numeracy Strategy introduced • 2003 – Primary National Strategy • 2014 – Current National Curriculum Introduced

6. The National Curriculum – An international Perspective Many countries are paying attention to the quality of their mathematics education A common theme is Depth and Mastery • Singapore • Shanghai China • Japan

7. The National Curriculum – An international Perspective Singapore: To develop deep understanding of mathematical concepts, to make sense of various mathematical ideas as well as their applications, students should be exposed to a variety of learning experiences including hands on activities and use of technological aids to help them relate abstract mathematical concepts with concrete experiences. Japan: Japanese teachers do not move on to a new topic until everyone has mastered the material.

8. The National Curriculum 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. (NC 2014 p3).

9. The aims of the new curriculum The national curriculum for mathematics aims to ensure that 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 are able to recall and apply their 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 • can 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.

10. The principles underlying the new curriculum • Close the gap andraise attainment • Providing access to mathematical conceptsforallchildren • Userepresentationsto support learning (concrete and pictorial) • Pupils should make connectionsin mathematics • Deeprather than superficial learning • Calculatingwith confidence • Longer time on key topics

11. What is Mastery of Maths? What is your understanding of the term ‘Mastery of Maths’ What have been your experiences in school of ‘teaching for mastery’? What does it mean to ‘master’ something?

12. What does the research say? Mastery is based on the idea of children not moving on until they are secure in their understanding of a particular concept. The whole class is taught the same thing, at the same time, with children learning at an appropriate level through support and enrichment. Challenge is offered through higher order questioning and activities that develop deeper understanding, problem solving and reasoning skills. STEM (2018)

13. What does the research say? Inspired by teaching approaches developed in Singapore and Shanghai, mastery is an inclusive way of teaching that is grounded in the belief that all pupils can achieve in maths. A concept is deemed mastered when learners can represent it in multiple ways, can communicate solutions using mathematical language and can independently apply the concept to new problems. TES

14. TIMSS Data (2015)

15. How is the term ‘mastery’ used in Mathematics? NCETM (2015)

16. A Mastery Approach A belief that all pupils are capable of understanding and doing mathematics, given sufficient time. Pupils are neither ‘born with the mathsgene’ 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. How can you ‘use’ mistakes in the classroom?

17. A Mastery Curriculum Key building blocks and mathematical ideas are fundamental for all children. One set of mathematical concepts and big ideas for all. Small carefully sequenced steps for children to master before moving on (NAMA, 2015) States that mastery involves a curriculum: • that is flexible • employs problem solving as a integral part • aims for fluencywith understanding • supports the development of mathematical reasoning

18. A Mastery Curriculum – Steps of Progression Example from Year 3 – Subtraction 58 - 4 658 – 4 658 – 40 658 – 500 975 – 723 831 – 26 608 – 135 520 – 269 300 - 125 “If practice is just repeating the same procedure with different numbers, chosen randomly, then it has no purpose. Some appear to think that such practice is like training a muscle, where repeated exercise builds up some kind of inner mental strength and speed. In fact it usually results in boredom. Variation theory tells us that by systematically changing significant aspects of a task, keeping the rest fixed, we can focus the students’ attention on those aspects and conceptual change can result. But the emphasis in making such variations is not to develop speed but to develop an awareness of pattern, leading to conjecture, generalisation, explanation and deeper understanding.” Professor Malcolm Swan. (NAMA, 2015)

19. Teaching for Mastery Enhancement not acceleration Teaching is focused, rigorous and thorough – CPA and modelling Long term gaps in learning are prevented through speedy teacher intervention. More time is spent on teaching topics to allow for the development of depth and sufficient practice to embed learning. Carefully crafted lesson design provides a scaffolded, conceptual journey through the mathematics, engaging pupils in reasoning and the development of mathematical thinking.

20. Teaching for Mastery – Structure of a Lesson

21. Lets have a go …

22. Dienes – Exploration before Structure Manipulative materials support children’s learning of mathematics Learning should be playful and informal at the initial stage

23. Bruner’s Theory People learn in three basic stages: • Enactive Representation (action-based) • Iconic Representation (image-based) • Symbolic representation (language-based) Bruner (1966) Concrete Pictorial Abstract

24. Implications for teaching mathematics Some of the implications of Bruner's theory for the teaching of mathematics are: • children's 'readiness' to learn is not linked to age (unlike Piaget's theory); • development of language is important to concept formation; • adults are important in structuring and supporting children's developing ideas (compare this with Piaget's theory); • new concepts (regardless of the age of the learner) should be taught enactively, then iconicallyand, finally, symbolicallyas ways of capturing experiences in the memory; • it is important to include practical activities and discussion as an integral part of mathematics.; • the use of pictorial recording and the classroom environment are important.

25. Discuss … What concrete and pictorial resources have you observed being used and why? How often are they used? Which pupils use them? Who decided whether a child used a particular resource or not?

26. Differentiation in Maths In a typical class of Year 6 children there is likely to be a seven year gap between achievers in Maths. How have the activities been differentiated in any maths lessons you’ve observed? How have the children been organised?

27. Differentiation in Maths Skilful questioning within lesson to promote conceptual understanding (Drury, 2014, Jones, 2014, Guskey, 2009) Identifying and rapidly acting on misconceptions which arise through same day intervention (Stripp, 2014, MathsHubs, 2015a) Challenging, through rich and sophisticated problems, those pupils who grasp concepts rapidly (NCETM, 2014) Use of concrete, pictorial and abstract representations according to levels of conceptual development (Jones, 2014, Drury, 2014)

28. How to Challenge Advanced Learners – Achieving ‘Greater Depth’ How would you describe an advanced mathematician? How could you challenge the advancedlearners for these questions? 6 + 7 + 4 = 9 + 0 + 4 = 8 + 5 + 9 = 7 + 9 + 6 =

29. Discuss … There are 23 children in a class. How many children are there in 4 classes 20 3 20 3

30. I know what to do • Skemp – Relational Understanding • Instrumental Understanding • I know why I’m doing what I do • Relational Understanding Calculate the answer to: ½ x ¾ Particularly relevant to learning written methods.

31. Times Tables - Investigation What do you notice? Does this happen always, sometimes, never? How can you prove it?

32. Achieving Mastery Mastery is not just being able to memorise key facts and procedures and answer test questions accurately and quickly.

33. Vygotsky – Social Constructivism Experience information at a social level and internal level Need for collaboration and interaction for learning

34. Experience creates a schema (mental map) • New information is added to this map • Piaget – Assimilation and Accommodation • Assimilation • Schema is challenged • Adjustment of schema to encompass contradictory information • Accommodation Implications for maths teaching: students need processing time to accommodate new ideas.

35. Underpinning Theories in Mathematics

36. Plenary How will you adapt your maths teaching to develop Mastery? What one thing would you like to develop in your maths teaching?

37. Further Reading STEM: https://www.stem.org.uk/news-and-views/opinions/mastery-primary-mathematics NCETM (2014) Developing Mastery in Mathematics https://www.ncetm.org.uk/public/files/19990433/Developing_mastery_in_mathematics_october_2014.pdf NCETM (2015) Teaching for Mastery: Questions, Tasks and Activities to Support Assessment Drury, H (2014) Mastering Mathematics, Oxford University Press Boaler, J (2016) Mathematical Mindsets, San Francisco: Jossey-Bass Mccrea, P (2018) Memorable Teaching NAMA (2015) Five Myths of Mastery in Mathematics http://www.nama.org.uk/Downloads/Five%20Myths%20about%20Mathematics%20Mastery.pdf https://www.tes.com/teaching-resources/teaching-for-mastery-in-primary-maths