1. Yorkshire Ridings Maths Hub Reasoning Work Group Strengthening Reasoning Date: 15/06/16 Venue: Harrogate Grammar School

2. Aims of session Recognise the importance of Reasoning in the new curriculum and understand its implications for teaching approaches and Schemes of Learning Know and be able to use some strategies for strengthening reasoning 2

3. Welcome Introduction to the project 3

4. I likeI don’t like Reasoning or not? What mathematical thinking was happening just then?

5. What is reasoning What do we understand by ‘reasoning’ Why is it important? 5

6. 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 have conceptual understanding and are able to recall and apply their knowledge rapidly and accurately to problems 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. New National Curriculum aims 6 Are there any aspects missing? Any that we didn’t just think about?

7. Understand situations and make connections Communicate Move to increasing formality 7 What if …

8. Assessment objectives from new GCSE (first teaching 2015) AO1 Use and apply standard techniques 40% Higher, 50% Foundation AO2 Reason, interpret and communicate mathematically 30% H, 25% F AO3 Problem solving 30% H, 25% F 8 Structure of the key stage 2 mathematics test The key stage 2 mathematics test materials comprise: Paper 1: arithmetic (40 marks) Paper 2: reasoning (35 marks) Paper 3: reasoning (35 marks)

9. Strategies to provoke reasoning I like – I don’t like Venn Diagram 9

10. Numbers with a factor of 3 Multiples of 9 Even numbers

11. Venn diagrams How might you use them with your students? How do they support the development of reasoning? What do you need to be wary of?

12. What is reasoning What do we understand by ‘reasoning’ Why is it important? What does it look like in the classroom? – Think of a (recent) lesson where you saw some high quality reasoning – What provoked it? – What qualities / behaviours were pupils showing? – What teacher behaviours are a pre-requisite for this? 12

13. "Put simply, the only point of asking questions is to raise issues about which the teacher needs information or about which the pupils need to think." [Wiliam and Hodgen, Working Inside the Black Box] Plenty more on building classroom dialogue in the teacher handbook

14. “students learn to think mathematically by being in the presence of a relative expert who makes their thinking processes explicit.” [VYGOTSKY, L., 1978, Mind in Society: The Development of the Higher Psychological Processes (London: Harvard University Press), paraphrased by John Mason] 14

15. What is reasoning What do we understand by ‘reasoning’ Why is it important? What does it look like in the classroom? How do we develop good reasoning? 15

16. nrich steps Step one: Describing: simply tells what they did. Step two: Explaining: offers some reasons for what they did. These may or may not be correct. The argument may yet not hang together coherently. This is the beginning of inductive reasoning. Step three: Convincing: confident that their chain of reasoning is right and may use words such as, ‘I reckon’ or ‘without doubt’. The underlying mathematical argument may or may not be accurate yet is likely to have more coherence and completeness than the explaining stage. This is called inductive reasoning. Step four: Justifying: a correct logical argument that has a complete chain of reasoning to it and uses words such as ‘because’, ‘therefore’, ‘and so’, ‘that leads to’... Step five: Proving: a watertight argument that is mathematically sound, often based on generalisations and underlying structure. This is also called deductive reasoning. 16

17. Strategies to build progress in reasoning Step one: Describing Step two: Explaining Step three: Convincing Step four: Justifying Step five: Proving 17 What can the teacher do to support progress through these stages? Where does questioning and classroom dialogue fit in?

18. Questions to prompt reasoning What’s happening here? Why do you think that? How do you know? Is that just a coincidence? How is that possible? What could you change? What effect would this have? Is there another way to explain this? Can you explain why that is right? Will that always happen? Why? Can we develop some that are more systematic?

19. Strategies to build progress in reasoning Step one: Describing Step two: Explaining Step three: Convincing Step four: Justifying Step five: Proving 19 Can we develop question stems to help progress through each of these stages?

20. Strategies to provoke reasoning 20 What additional insight does this activity bring?

21. Strategies to Provoke Reasoning I like – I don’t like Venn Diagrams Change one thing ….. Concept Cartoons 21

23. Strategies to provoke reasoning I like – I don’t like Venn diagrams Change one thing ….. Concept Cartoons Give me an example …. ….. of a rectangle with perimeter 40cm ….. and another ….. now give me an interesting one ….. and now one that you think nobody else will have thought of 23 What is the purpose of the last two questions?

24. Thank you m.stockton@easingwold.n-yorks.sch.uk jrt@harrogategrammar.co.uk