Jo Randhawa Carolyn Hume Developing departmental practice and culture to improve teachers’ confidence when addressing the reasoning component of the new secondary maths curriculum Jo Randhawa Carolyn Hume

Research Question 3 “How does comparing and discussing others’ proofs to a problem develop a student’s own ability to reason?” RW

Programme 9.00- 9.30 Arrival and coffee 9.30-10.00 Feedback on gap task 1 10.00 – 10.45 Always a multiple? 10.45 – 11.15 Multiplicative reasoning 11.15 – 11.30 Coffee 11.30– 12.00 Working in your department 12.00 – 1.00 Time to plan and preparation for evaluation (lunch available) RW

Session 1: Reflecting on gap task 2… “Does regular use of a chosen structured activity impact on students’ articulation of their mathematical reasoning?”

National Curriculum Aims The national curriculum for mathematics aims to ensure that all pupils: become fluent …… reason mathematically ……. can solve problems ……. jd Aims 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 the ability to recall and apply 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. The aims of the National Curriculum are to develop fluency and the ability to reason mathematically and solve problems. Reasoning is not only important in its own right but impacts on the other two aims. Reasoning about what is already known in order to work out what is unknown will improve fluency; for example if I know what 12 × 12 is, I can apply reasoning to work out 12 × 13. The ability to reason also supports the application of mathematics and an ability to solve problems set in unfamiliar contexts.

Session 2: Always a multiple? Comparing different proofs Videos - always a multiple

Further problems…. Choose 1 or 2 of these to explore Take any two-digit number. Reverse the digits, and subtract your answer from your original number. What do you notice?   2)Take any two-digit number. Add its digits, and subtract your answer from your original number. What do you notice?   3) Take any five-digit number. Reverse the digits, and subtract your answer from your original number. What do you notice?

http://www.flashlightcreative.net/swf/mindreader/

Discussion Do you do this sort of task already? What skills might this problem develop in pupils? In what way does this task differ to reasoning activities in previous workshops? How might your department use the NRICH resources to support this type of learning? What support would you need? Do we need videos?

Session 3: Multiplicative Reasoning Comparing different strategies The following slides show some examples from the NCETM Multiplicative Reasoning project. Unit 1 : ‘Fair Shares’

This cheddar cheese weighs 900 grams. Show where to cut off 600 grams Stage 1: Catering Ruby works in the kitchen of a café. She has responsibility for ordering food and portioning the food out for the various recipes once it arrives. Ruby finds it is quicker to portion food out by eye rather than measuring accurately. This cheddar cheese weighs 900 grams. Show where to cut off 600 grams Use worksheet 1 to show where to cut each item

Compare your strategies for cutting up the food: This block of butter weighs 250 grams. Show where to cut off 50 grams. The smoked cheese weighs 450 grams. Show where to cut off 200 grams

The hot water urn Sometimes Ruby’s café provides outside catering for business lunches. They supply hot water urns, so that people can have access to hot drinks. The hot water urns are similar to the one shown in the picture. Each urn has a vertical gauge that shows how much water is in the urn. When it is full the urn can supply 120 cups. For each picture, estimate what fraction of the urn is full. For each picture, say how many cups of water there are left in the urn.

Sharing Food The café where Ruby works specialise in making large rectangular pizzas. These are popular with lunch time diners who often order one of the large pizzas to share. Masood and Tim are students. They meet for a pizza in the café once a week. On one occasion they order the rectangular cheese and tomato pizza shown below costing £10.35. Tim is not as hungry as Masood, so they cut the pizza into 9 slices, Tim eats 4 slices and Masood eats 5 slices Draw a picture to show how they might share out the pizza Share out the cost of the pizza between Masood and Tim in the ratio 5 : 4

Summary Questions A water tank can hold up to 60 litres of water. The gauge on the front of the tank shows that it is 5/8ths full. How much water is currently in the tank? A straight jug can hold up to 300 ml of liquid. Jo wants to make a fruit crush to fill the jug. The recipe says mix 2 parts of orange juice to 3 parts of pineapple juice. How many millilitres of each juice will Jo need? One of the raffle prizes at a school fair is a gigantic bar of chocolate measuring 120cm in length. The prize is won by two friends, Eliza and Shola. They wonder how to share out the prize. Shola thinks it is not right to split it in half because there are only three people in her family whereas there are five people in Eliza’s family. Suggest a fair way that they could share out the prize.

Example question: KS2 Reasoning

Session 4: Working in your department Lesson Study Spend 5 mins discussing how you could collect evidence for the research question. “How does comparing and discussing others’ proofs to a problem develop a student’s own ability to reason?”

Session 4: Working in your department Introducing the tasks more widely You will need to think about how you introduce the tasks you have chosen to your department. Remember your schools have signed up to this project so can you get time to plan for these activities?

Session 4: Working in your department Time to plan We now have time for you to plan your lessons lesson study department meeting

Extract from NRICH article by Malcolm Swan: -  Improving Reasoning: Analysing Different Approaches …how often do we let pupils see us reasoning? One final challenge then! Choose a problem from the NRICH website that you have never seen before. Try to tackle it with a class, thinking aloud. Try different approaches. Ask the class to help with ideas when you get stuck. Go down blind alleys. If you get nowhere, then park the problem overnight and return to it later. If students are to learn what thinking like a mathematician is like, let them hear your thinking!

Session 5: Evaluating this project The final session is on: Wednesday 29th June 9.30am-3.00pm. You will be asked to complete the NCP4 KS3 Reasoning Workgroup Schools Report You don’t need to complete this now but do look at the form and begin to think about your responses.

Useful links Always a multiple (and further problems) http://nrich.maths.org/7208 Mind reader http://www.flashlightcreative.net/swf/mindreader/ Multiplicative reasoning https://www.ncetm.org.uk/resources/48755 Educreations (quick way to make videos on ipad) https://www.educreations.com/ Mr Barton’s Rich tasks – low barrier, high ceiling’ http://www.mrbartonmaths.com/richtasks.htm#tasks

Useful links – suggested at meeting http://www.resourceaholic.com/ http://www.cimt.plymouth.ac.uk/projects/mepres/allgcse/unitpbk.pdf https://colleenyoung.wordpress.com/