Developing a reasoning classroom

1. Developing a reasoning classroom Written by Tim Handley (Mathematics Specialist Teacher, NCETM Professional Development Lead and author of Problem Solving and Reasoning)

2. Aims • Understand what reasoning ‘is’ and why it is important. • Explore some reasoning activities. • Explore ways in which reasoning can become part of day-to-day teaching.

3. Running order • Introduction: What do our children already do? • Taking a slice: a non-numerical activity • What is reasoning and why is it important? • Reasoning in the past week • Graffiti Maths with Chicken Nuggets: a numerical activity Break • Children’s mathematical superpowers • Key strategies to make reasoning part of day-to-day life in our classrooms • Planning a reasoning activity • Next steps

4. What do children in our school already do? • Have a look at the statements on the progression cards from http://nrich.maths.org/6105. • Highlight them green if children in your year group do these regularly and independently. • Highlight them orange if children in your year group do these regularly but require support. • Leave them blank if children in your year group struggle to do these, or only do them infrequently. • Then identify the 4 statementswhich you consider to be age appropriate, but which children in your year group struggle with the most and mark these with a star.

5. What do they find difficult? What can children do independently?

6. Take a slice: a non-numerical activity Here is one way of splitting a 4 x 4 board into 2 identical pieces: How many other ways can you find?

7. Take a slice: a non-numerical activity • So, what were you doing? • What key phrases were you using? • What were you thinking? Things to think about We were all reasoning!

8. So, what is reasoning? Some things reasoning can be seen as: • Thinking about mathematics • Making connections • Application of facts and knowledge – either explicitly or implicitly • Justifying • Convincing yourself and others

9. Why is reasoning important …? What do you think?

10. What OTHERS say ...

11. Academics “In our understanding, we have two components – the range of mental representations and the reasoning linking them together. Therefore we must also develop the reasoning that we carry out between representations.” Barnaby et al, Primary Mathematics, Teaching for understanding. And lots, lots more…

12. Examiners Just some of the questions that require children to reason from the 2009 KS1 + 2012 KS2 SATs tests.

13. Ofsted The reports Mathematics: made to measure (Ofsted, 2012) and Good practice in primary mathematics: evidence from 20 successful schools (Ofsted, 2011) make clear that: • Real mathematical thinking lies at the heart of learning mathematics. It involves making sense of mathematics through engagement, reasoning and making connections. • Sometimes this may involve significant challenge in terms of the requirement to think. But it is only through their own thinking that children will develop real depth of learning.

14. Ofsted Maths subject-specific criteria Outstanding • Pupils understand core concepts and make connections. • They think for themselves. • They show exceptional independence when solving problems in a range of contexts. • They think for themselves and persevere with challenges, knowing they will succeed. • They embrace the value of learning from mistakes and false starts. • They reason, generalise and make sense of solutions. Requires improvement • Children use techniques correctly. • They have an insecure understanding of concepts. • They solve routine problems in known contexts. • They follow the teacher’s example. • They avoid making mistakes.

15. The DfE: 2014 National Curriculum Aims It is expected that these aims are applied throughout the programmes of study.

16. How have our children been reasoning over the past week? Share with your table!

17. Graffiti maths • Graffiti maths is an approach to problem solving and reasoning tasks which encourages children to think and work ‘big’. • See page 12 of the Introduction in the Problem Solving and Reasoning books for an explanation of how it works. • See the Key Strategies Maths Stories section on the DVD for footage of Graffiti maths in action. Now use the graffiti maths approach to try out the Chicken nuggets problem from Problem Solving and Reasoning Year 5.

18. BREAK!

19. All children have them! Mathematical Superpowers

20. Mathematical superpowers Conjecturing and Convincing • John Mason identified a set of 8 ‘Mathematical Powers’ that all children posses. They come in pairs: • See page 8 of the Introduction in the Problem Solving and Reasoning books for more details. Specialising and Generalising Imagining and Expressing Organising and Classifying

21. Watch the Maths Stories clip on the DVD. The Story of a Number An easy way to get children reasoning, pattern spotting and generalising.

22. Money boxes • If we know that half of the coins are £1, what else do we know? • Which of these clues are hard/easy to solve? Why? • Which of the clues is the odd one out why? • Try the Money boxes problem from Problem Solving and Reasoning Year 3. • Can you work out how much money could be in each money box? Add in questions to draw out the reasoning…

23. Key strategies Watch the clips on the DVD. To help include reasoning as part of our day-to-day teaching. The Problem Solving and Reasoning books include 14 key strategies.

24. Always, sometimes, never • Always, sometimes, never can also be used as part of shared learning. • Can you come up with some ‘always, sometimes, never’ statements for your year group? • Look at the statements from the Number Knowledge problem Resource Sheet 13.1 from Problem Solving and Reasoning Year 6. • Discuss with your partner. Are they always, sometimes or never true? • Make sure you give examples.

25. Other examples for always, sometimes, never… • The picture on a pictogram represents one piece of data. • If the answer is in cm2 you are measuring area. • The diagonals of a quadrilateral bisect each other. • Two negatives make a positive. • All multiples of 3 have a digital root of 3, 6 or 9.

26. Another, another, another… • An even number… • Number with the product of 60 • Factors of 36 • A triangle • A prime number Draw out the links and generalisations between the sets generated.

27. Convince me ... • All angles in a triangle always add up to 180 degrees. • That the next whole number after 119 will be 120. • That coordinates are useful for identifying points. • An odd number plus an odd number always equals an even number. • That the area of a rectangle divided by the length of one side gives the length of the other side.

28. Hard and easy • A calculation whose answer is 7 • A three digit subtraction • A word problem • A region made up of rectangles whose area and perimeter are to be found • A calculation to be carried out on a calculator

29. If this is the answer, what’s the question? • 24 • Impossible • No • Square numbers • 15.3

30. Odd one out (and why?) • 3, 6, 9 … • 3 x 10 = 30, 31 x 10 = 310, 423 x 10 = 4230, 0.3 x 10 = 3, 1111 x 10 = 11 110 • 1/4 , 8/36, 6/32, 10/40 • 11, 101, 1001, 101

31. POGs: Peculiar, Obvious, General • A quadrilateral • A multiple of 2 • A factor of 36 • A fraction equivalent to ¾ • A number • A set of 3 numbers with a product of 10

32. What do you notice? • 2, 4, 6, 8, 10, 12, 14, 16 • 2, 3, 5, 7, 11,13 (Can you give me more in this set?) • 1/2 , 2/4 , 3/6 , 4/8 • 4, 9, 16, 25, 36, …

33. What else do we know? (and why?) • 5 x 8 = 40 • 3 + 4 =7 • 1 cm = 10 mm, 2 cm = 20 mm • All even numbers end in an even digit: give me an even number over 1000. • 50 is 5 lots of 10, 40 is 4 lots of 10, … • 1/2 = 2/4 = 4/8 = 3/6

34. What’s the same? What’s different? • One and 1 • 9 and 18 • Multiples of 3 and Multiples of 6 • 1/2, 0.5, 50%, 4:8, 3/6, 1:2 • Square numbers and numbers that are not square

35. What’s the same? What’s different ? Choose any two of the three. In what way are they the same as each other and different from the third?

36. Zooming in Write down a: • Multiple of 4... • ... that is greater than 30... • ... that is also square... • ... where both digits are even. Draw a: • Quadrilateral... • ... that has at least one line of symmetry... • ... and only one pair of parallel sides... • ... and at least one acute angle.

37. Is it a good explanation of…? • Is “having three sides” a good explanation of a triangle? • Is “adding a zero” a good explanation of multiplying by 10?” • Is “a shape with 4 sides” a good explanation of a square? • Is “moving the digits 3 places to the left” a good explanation of multiplying 1.3 by 1000?

38. Planning a reasoning activity • In your year group, plan a reasoning activity that you could use next week. • This could be a key question, it could use one of the strategies we’ve just looked at, or be a full lesson activity.

39. Next steps • Encourage and support our children to reason! • Over the next 2 weeks, ensure there are at least 6 opportunities for your children to reason. • Try also to include reasoning and some of the key strategies in other subjects.

40. Resources to help • 14 key strategies with example questions for each year • 18 activities and investigations per year level