STRATEGIES FOR GENERATING MATHEMATICAL TALK IN A KS2 CLASSROOM

1. Lyn Wickham STRATEGIES FOR GENERATING MATHEMATICAL TALK IN A KS2 CLASSROOM

2. Bowland Maths • Estimating and interpreting: There are about 60 million people in the UK. • About how many school teachers are there? • About how many dentists are there? Estimate some other facts and check them out! How many toilet rolls would it take the Andrex puppy to wrap the world around the equator?

3. What type of talk did you use? Dialogic (helpful) talk is: • Collective: teachers and children address learning tasks together, as a group or as a class, rather than in isolation • Reciprocal: teachers and children listen to each other, share ideas and consider alternative viewpoints • Cumulative: teachers and children build on their own and each others' ideas and chain them into coherent lines of thinking and enquiry • Supportive: children articulate their ideas freely, without fear of embarrassment over 'wrong' answers and they help each other to reach common understandings • Purposeful: teachers plan and facilitate dialogic teaching with particular educational goals in view (Alexander, 2006) Cumulative talk Speakers build positively, but uncritically on what each other has said. This is typically characterised by repetitions, confirmations and elaborations. Disputational talk This consists of disagreement and individualised decision making. It is characterised by short exchanges consisting of assertions and counter-assertions. Exploratory talk Speakers work on and elaborate each other’s reasoning in a collaborative, rather than competitive atmosphere. Exploratory talk enables reasoning to become audible and knowledge becomes publicly accountable. It is characterised by critical and constructive exchanges. Challenges are justified and alternative ideas are offered. (Mercer, 1995, 2000)

4. Bowland Maths • www.bowlandmaths.org.uk • Reasoning (substantial problems) • Connections (maths and real life) • Professional development support – videos, case studies, advice • Resources – Problems for discussion, Recognising helpful and unhelpful talk Ten ground rules for pupil-pupil discussion Planning for pupil-pupil discussions What is the teacher's role during discussion?

5. Context • Two form entry Junior School • Catchment area • Behaviour • SATs results • Teaching styles

6. Hiebert et al, 1997, p12

7. My Beliefs • Mathematics will be used with a purpose • Tasks will leave behind something of value • Mathematical talk will be present in most lessons • Children will use equipment and resources as they choose to support their learning • Mathematical tasks are accessible to all children

8. My Previous Research • Flanders Interaction Analysis Categories (FIAC) • Questioning • Raiker • Mathematical vocabulary

9. Early Task • p and q each stand for whole numbers • p + q = 1000 • p is 150 greater than q • Calculate the numbers p and q

10. Purposeful Talk? • “600 and 400” • “No, 600 and 50” • “That’s what I said” • “No, 675 and…” • “No not 600” • “No that’s right they both add to 1000” • “No its 150 greater” • “It IS 150 greater look” • “900 take away… could be…”

11. My Research Focus • Precise mathematical vocabulary? • Opportunities to talk • Choice of activity (Swan, 2005,p3)

12. Better? • “750 and 250” • “Yep, make 1000” • “Yeah, but if you put another 100 over here” • “Make 850 and make that 150” • “That means that 150 greater, no its not, it has to be 150 greater” • “So?” • “So they need to be more like the middle of 1000, near 500 each, so 500 and 500 make 1000” • “So this one has to be bigger than that one” • “Yeah so we take 150 off that one and put it over there” • “So that would now be 650 and make that?” • “350” • “350... so that plus that is 1000, and p is ...” • “Not 150 more than q…”

13. Why Talk in Class? • “Children solve practical tasks with the help of their speech as well as their eyes and hands,” • “Sometimes speech becomes of such vital importance that, if not permitted to use it, young children cannot accomplish the given task” • (Vygotsky, 1978, p26)

14. Why Talk in Maths? (Swan, 2005, p31) (Hiebert et al, 1997, p6) • You need discussion in mathematics in order to learn: • What words and symbols mean; • How ideas link across topics; • Why particular methods work; • Why something is wrong; • How you can solve problems more effectively” • “Students who reflect on what they do and communicate with others about it are in the best position to build useful connections in mathematics.”

15. Group Work • How group work enable talk • How to organise and manage groups • Choice of group • Advantages and disadvantages of group work • The role of the teacher

16. Types of talk • Transmissional • Formal/Presentational • Informal • Disputational • Cumulative • Exploratory

17. Exploratory Talk • “in which partners engage critically but constructively with each other’s ideas. Statements and suggestions are offered for joint consideration. These may be challenged and counter challenged, but challenges are justified and alternative hypothesis are offered. Compared with the other two types, in exploratory talk knowledge is made more publicly accountable and reasoning is more visible in the talk. Progress then emerges from the eventual joint agreement reached” (Mercer, 1995, p104).

18. Rich Mathematical Task • “It must be accessible to everyone at the start. • It needs to allow further challenges and be extendible. • It should invite children to make decisions. • It should involve children in speculating, hypothesising making and testing, proving or explaining, reflecting, interpreting. • It should not restrict pupils from searching in other directions. • It should promote discussion and communication. • It should encourage originality/invention. • It should encourage ‘what if?’ and ‘what if not’ questions. • It should have an element of surprise. • It should be enjoyable.” (Ahmed, 1987, p20)

19. Progression • “The effectiveness of their talk, including adaptation to purpose, context and audience; • Contributions that show positive and flexible work in groups; • Clarity in communicating, including the use of reason, clear sequences of ideas... (DfES, 2003, p30). • Growing confidence, independence, involvement. • More experience or knowledge about language. • Greater ability to reflect on own talk and interaction with others… • Greater readiness to explore new ways using talk” (Baddeley, 1991b, p21). • Mathematical Understanding • Type of talk

20. Methodology • Casual comparative • Survey research • Action research • Observation • Formal/Structured • Informal/Unstructured • Field notes • Video recordings • Ethics

21. PHSE Lessons Do discussions need rules? If so, what should they be? (Fisher, 1995, p50) What good can come out of discussing things with others? (Fisher, 1995, p50) • When you need quiet, too many rules means it is not a discussion. Sometimes you need rules, e.g. not talking when others are talking. • Understanding, getting to know people, make new friends, solve issues, learn new things and develop a liking for each other, learn things you didn’t know before.

22. Does talking with others help you think and learn? Sometimes? Always? Never? Why? (Fisher, 1995, p45) • “It helps if the teacher talks to you, a smart person can help you in mathematics. If you share ideas it can help you to make decisions. You can learn different things from each other. But sometimes it makes it confusing. You can have things explained to you. If you do get it, it may help to share opinions (you may have made a silly mistake and someone else might pick it up)”

23. What do you want to do in mathematics lessons? (Do you want to talk? Work in silence etc?) • Work in groups, as when you work in silence the teacher can’t see everyone. The smarter children can help you if you get it wrong. You can ask other people and share ideas, then work silently. Work in silence so that you can concentrate. Others can explain things you are not sure about. People remember things better when it’s talked about. Helping people (not just giving the answers).

24. Diamond 9

25. Talk Rules 1 • Everyone should have a chance to talk – “So that no one is left out.” • Cooperate and work together – “It is about being a team!” • Respect each other’s opinions – “It’s important not to say ‘I don’t like that’ but to say ‘its ok but and so on,’” • Talk one at a time – “We should take turns because if we all talked at the same time we wouldn’t be able to understand what everyone was talking about.”

26. Talk Rules 2 • After discussion the group should agree on an idea • Give reasons to explain our ideas • Everyone’s ideas should be listened to – “Because it is only fair,” • We also thought it was important to: look at the person who is talking “so its not unfair on the other people, because you could ignore them and they could get very frustrated and there idea could be the winning one or something,” and to use a calm voice “so we don’t end up arguing.”

27. We talked about what we thought a good talker and a good listener looked like: • Someone that understands • Someone who concentrates • Someone who works hard • Someone who listens to other peoples ideas and waits for their turn to talk • Someone that thinks what they say first and then talk about it • Someone that is clear and bold • Someone that listens but also joins in • Someone who understands you • Someone that listens to your problems and helps you sort them out • Someone that listens, thinks and then says the answer

28. What is a good mathematical explanation? • Give reasons • Give examples • Be logical and be systematic • Clarify meaning

29. Class Display

30. Discuss the Activities • What type of talk do you think the activities would allow/encourage? • Look at the transcripts – what types of talk were there?

31. Evaluating Their Own Talk • There was lots of noise and the group were not listening to ideas. • When a child said, “I know what to do” they didn’t explain. • They kept saying they wanted to use calculators instead of working things out. • They were being silly around the camera. • Only one child tried to take control and they were not working together.

32. Evaluating Their Own Talk 2 • The group in the video understand why it’s the answer. • They were discussing not arguing. • If they went wrong they said why it went wrong and then helped. • They explained. • They gave reasons and talked about it. • They read the question out loud. • They corrected each other. • They kept looking back at the question to see what it was asking and if it made sense.

33. Learning From Their Experiences • When asked why the later example was better the class replied that the group were: • Talking about the work. • Giving reasons why it’s the answer, not saying “it’s that one”. • Working well as a group. • Using better mathematical vocabulary. • The group asked questions “is it..?” and “why is it…?” • They checked it over 3 or 4 times, then “let’s go on to the next one”.

34. NCETM Blog

35. Good To Be Involved • If we make the rules we keep them instead of being told them • Learnt better by being involved? • Involved – if you just tell us we are not really going to learn a lot, we need to get involved to learn from our mistakes • Includes everyone; makes it more exciting than when you just tell us to do it. • We might have different ideas, you might have lots of ideas and forget some of them but we might have better ideas than you have and if we get involved you get our ideas and find out what we think. • You listened to what we said and put it up on a poster • Not exactly fun but you can understand each other and you can tell each other in a child form.

36. Identifying features • What features do you think need to be present for a task to encourage purposeful mathematical discussion?

37. Criteria to use to choose a task that can be used to generate purposeful mathematical talk • The activity needs to: • Be accessible and intriguing • Be enjoyable and contain an element of surprise • Require the children to make decisions • Be challenging • Allows the use of mathematical vocabulary • Encourage collaboration • Give opportunities to generalise • Give opportunities to make connections • Encourage the children to ask questions • Encourage the children to give reasons

38. Use the Criteria to Select an Activity Now complete the activity: What type of talk are you using? Are any of the criteria: Essential? Unnecessary? What would you change about the task or the criteria? • The activity needs to: • Be accessible and intriguing • Be enjoyable and contain an element of surprise • Require the children to make decisions • Be challenging • Allows the use of mathematical vocabulary • Encourage collaboration • Give opportunities to generalise • Give opportunities to make connections • Encourage the children to ask questions • Encourage the children to give reasons

39. N9 Lesson (Swan, 2005)

40. Transcript for N9 Lesson • If you added that would it go to 0? • No that’s not right, it said... • -5 + -6 • Sometimes, because if you add the same number that would take it to 0 • No, that would be –11, • -11 • Yes it would, because the two minuses, I think it’s always true • Because you are plussing minuses together, it’s the same as adding plusses together... • Explain more? • Well like 5 and 6 would be 11, -5 and –6 would be –11. Add the minuses together… • It’s pretty much exactly the same as that... (Draws a number line) –9..-10..-11 add minus 6 it goes… • That’s always true!

41. Limitations / Problems • Video equipment / user error • Permission • Restrictions of curriculum due to SATs • Beliefs of other parties

42. Further Research • Developing the PHSE lessons • Mathematical vocabulary • Mind maps • Include drama, and jig sawing