The mathematics curriculum reform

1. The mathematics curriculum reform Dr Geoff Tennant g.d.tennant@reading.ac.uk

2. Approximate plan… Brief history of reforms in the UK; Comparisons with reform in Jamaica; Some reflections on the nature of reform – what are we trying to achieve? Some specific reforms in the UK – what do you think of these? • The three part lesson – the starter; • The three part lesson – the plenary: • Sharing lesson objectives; • Making use of assessment data; • ‘Rich tasks’; • Using learning theories (eg. VAK) to inform thinking. Closing thoughts.

3. Starting point: 1976a James Callaghan as Prime Minister in his Ruskin Speech (as reproduced in Ahier, Cosin and Hales 1996: 199): It is not enough to say that standards in {the general educational field} have or have not declined. With the increasing complexity of modern life, we cannot be satisfied with maintaining existing standards, let alone observe any decline. We must aim for something better. Ahier, J., Cosin, B. & Hales, M. (1996). Diversity and change: education, policy and selection. Routledge: London.

4. Starting point: 1976b James Callaghan as Prime Minister in his Ruskin Speech (as reproduced in Ahier, Cosin and Hales 1996: 202): Note: often quoted as calling for a National Curriculum and the starting point for what came next. It is not my intention to become enmeshed in such problems as whether there should be a basic curriculum with universal standards - although I am inclined to think there should be - nor about other issues on which there is a divided professional opinion such as the position and role of the Inspectorate, this being within the purview of the Secretary of State for Education. \ Ahier, J., Cosin, B. & Hales, M. (1996). Diversity and change: education, policy and selection. Routledge: London.

5. Note also White (1973) Argued for a compulsory curriculum according to the minimum as to what one would expect school leavers to be able to do, understand and know. Can be found in the UWI library! In any rational educational system… it is of paramount importance to determine [a] basic minimum. White, J. (1973). Towards a compulsory curriculum. London: Routledge and Kegan Paul. Page 1.

6. A flurry of documents on curriculum matters… From the Department for Education and Science and Her Majesty’s Inspectorate, eg. HMI (1977), DES (1980), DES (1982) DES (1980). A View of the Curriculum. London: HMSO. DES (1982). Mathematics Counts: report of the Committee of Inquiry into the teaching of mathematics in schools under the chairmanship of Dr W H Cockcroft. London: HMSO. HMI (1977). Curriculum 11-16. Working papers by HM Inspectorate: a contribution to the current debate. London: HMSO.

7. Introduction of National Curriculum from 1988 Defined what was to be taught in English and Welsh schools across all curriculum subjects except RE. 10 levels of difficult from level 1 (5 year olds beginning in school) to level 10 (highest attaining 16 year old). Attainment Targets (ATs) or topic headings reduced in mathematics from 14 (1988) to 5 (1991) to 4 (1995) Changes to framing (but not content in mathematics) in 2008. Kenneth Clarke as Education Secretary in 1991: "Questions about how to teach are not for government to determine“ (quoted in Watkins & Mortimore, 1999: 10) Watkins, C., & Mortimore, P. (1999). Pedagogy: what do we know? In P. Mortimore (Ed.), Understanding pedagogy and its impact on learning (pp. 1-19). London: Paul Chapman Publishing Ltd.

8. National Curriculum: framing as from 2008 In summary: Understand the ‘how’ and ‘what’ of mathematics; Engage with the ‘why’ of mathematics, including as a ‘service subject’ to other curriculum areas; Communicate in written and oral form; Understand the interconnections within, and power of, mathematics. “It is expected that if concepts rather than algorithms are properly taught, the need for re-teaching, remediation and revision will be seen to be much less than was previously the case.” Ministry of Education and Culture (1998). Curriculum guide grades 7-9 for career education, mathematics, language, arts, science and social studies. Kingston, Jamaica: MOEC. Page 86.

9. Meanwhile introduction of National Strategies…. Introduced in primary schools in 1000 and secondary schools in 2001; Introduced against perception that “3Rs” were being neglected; Theoretically optional for schools, but in practice often understood to be compulsory ; Advocated structured lesson, often understood to be 3 part lessons; Brought detailed advice on how to go about teaching , eg….. DfEE (2001). The National Numeracy Strategy. Framework for teaching mathematics: years 7 to 9. London: HMSO.

10. Advice from the National Strategies • Some features of effective mathematics teaching • Lessons have clear objectives and are suitably paced • Teachers convey an interest in and enthusiasm for mathematics • A high proportion of lesson time is devoted to direct interactive teaching • Pupils are involved and their interest maintained • Regular oral and mental work • Whole-class discussion in which teachers question pupils effectively • Pupils are expected to use correct mathematical terms and notation • Written activities consolidate the teaching • Teachers make explicit for pupils the links between different topics in mathematics and between mathematics and other subjects • Manageable differentiation is based on work common to all pupils in a class

11. And also… Office of Standards in Education introduced in 1992, initially schools inspected every four years (previously very rarely), many changes to the model since introduced; City Technology Colleges introduced in 1988; Technology colleges (not necessarily in cities) 1993; Other subject specialist schools introduced in 1994, most secondary schools now specialist BUT additional funding now discontinued; More recently: teaching schools / training schools / high performing specialist schools / etc. etc. etc. Particularly under previous Government, huge concentration on hitting targets, eg:

12. Increase in examination grades…. Particularly under previous Government, huge concentration on hitting targets, for example, at the Piggott School, Wokingham, Berkshire, UK: GCSE Targets 2012 74.9 % of students achieving 5+ A*- C grades including both English and Maths 82.1% of students making expected progress in English 81.5% of students mMaking expected progress in Maths N.B. GCSE is the equivalent of CXC. But consider the following:

13. Is increase in GCSE (equivalent to CXC) a good thing? • “Currently the system gives us these stark outcomes: • Nearly half of all students “fail” GCSE Mathematics (ie. do not get grade C or above); • Only 15% of students take mathematics, in some form, beyond GCSE. • This systemic failure contrasts catastrophically with today’s economic reality. • (Vorderman et al. 2011: 3) Vorderman, C., Porkess, R., Budd, C., Dunne, R., Rahman-Hart, P., Colmez, C. & Lee, S. (2011). A world-class mathematics education for all our young people. London: The Conservative Party.

14. Is increase in GCSE (equivalent to CXC) a good thing? The analysis of 3000 secondary pupils’ performance in algebra, ratio and decimals tests conducted last year suggests that there has been little overall change in maths attainment since 1976. Exam pass rates, by contrast, have risen dramatically during that period. In the early 1980s, only 22 per cent of pupils obtained a GCE O level grade C or above in maths. Last year over 55 per cent gained a GCSE grade C or above in the subject. (Vorderman et al. 2011: 52) Vorderman, C., Porkess, R., Budd, C., Dunne, R., Rahman-Hart, P., Colmez, C. & Lee, S. (2011). A world-class mathematics education for all our young people. London: The Conservative Party.

15. Meanwhile in Jamaica ROSE stage 1: 1993-2000 ROSE stage 2: 2002-2009 Constant reference in Gleaner articles and elsewhere for the need to reform, eg. Knight, J. & Rapley, J. (2007). Educational reform in Jamaica: recommendations from Ireland, Finland and Singapore. Kingston, Jamaica: Caribbean Policy Research Institute.

16. But a wider search gives rise to….. Norris, E. (2012). Solving the maths problem: international perspectives on mathematics education. London: RSA. Comparison of England (negative) with Scotland and Hong Kong (positive). Mastrul, E. (2002). The mathematics education of students in Japan: a comparison with United States mathematics programs. San Jose, Costa Rica: MRCMM. Comparison of USA (negative) with Japan (positive). Noordin, A. (2009). Education in Singapore and Finland: a comparison part 5. Singapore: Singapore Educational Consultants Comparison of Sinagpore (negative) with Finland (positive). Jackson, B. (2009). Singapore and America: Worlds Apart? How The U.S. Can Raise the Bar in Math Instruction. Downloaded from http://www.thedailyriff.com/articles/producing-some-of-the-best-math-teachers-in-the-world-436.php 14th February 2012. Comparison of USA (negative) with Singapore (positive).

17. So…. Beware ‘policy tourism’ (Professor Dylan Wiliam as quoted in SFS Group 2009). BUT: The message from right across the world seems to be: everybody is doing better than us, we are doing very badly and something URGENTLY must be done NOW about it!!!!!! SFS Group (2009). Policy tourism should be avoided in education. Downloaded from http://www.sfs-group.co.uk/policy-tourism-should-be-avoided-in-education/ on 14th February 2012.

18. Chance for me to have some water…. ….while you discuss with the person next to you: What are the desired end results of educational reform? What is it ultimately we are trying to achieve? Why?

19. Some answers I prepared earlier…. Highly skilled, highly paid workforce Highly respected teaching force Autonomous, questioning, problem solving individuals Compliant work force Able to access world of work now Able to access world of work for next 50 years Stretch all children as far as they will go Look to shore up minimal standards for all Look to share a love of learning and cultural heritage (eg. music and art) Ensure youngsters entering the workforce have the knowledge and skills to do so.

20. Some reforms in the UK Three part lesson, ie. Starter Main part Plenary As teacher trainer, consider this to be a good starting point for planning lessons.

21. The starter activity: characteristics • is lively and challenging, engaging all pupils for the whole of the time; • is interactive, drawing responses from pupils with a skilful mix of open and closed questions, and uses these responses well to assess pupils’ learning; • is well planned, making effective use of resources to illustrate and model mathematical operations and structures ; • consolidates pupils’ knowledge, skills and understanding and then moves them on, using what they know to derive new facts or introducing new ideas that build upon what is already secure ; • expects pupils to explain how they arrive at their solution, and to learn and use mathematical vocabulary correctly when doing so. DfEE (2001). The National Strategy. Framework for teaching mathematics: years 7, 8 and 9. London: HMSO.

22. The starter activity: purpose • rehearse previously taught skills in a variety of lively ways • focus on skills needed in the main part of the lesson • ensure the lesson gets off to a clear purposeful start and sets a brisk pace • enable informal assessment of pupils’ progress to direct the next part of the lesson DfEE (2001). The National Strategy. Framework for teaching mathematics: years 7, 8 and 9. London: HMSO.

23. The starter activity: the uses • develop and explain mental calculation strategies, including figuring out new facts from known facts and explaining the strategies used • apply number facts to real life situations • develop estimation skills • develop links between the laws of number and those of algebra • develop mental imagery of shapes, movements and constructions • develop inference skills from data in a variety of forms • develop the use of correct mathematical vocabulary • develop the ability to generalise, reason and prove DfEE (2001). The National Strategy. Framework for teaching mathematics: years 7, 8 and 9. London: HMSO.

24. The starter activity: some examples • Tarsia activity; • Follow on cards; • Spider diagram; • Countdown; • 4 4s; • Multitudinous other activities. DfEE (2001). The National Strategy. Framework for teaching mathematics: years 7, 8 and 9. London: HMSO.

25. Plenary: from the original KS3SM document (1) The plenary is an opportunity to round off and summarise the lesson, so that pupils focus on what was important, what they have learned and the progress they have made. It is a time when you can relate mathematics to their work in other subjects: for example, how their work on calculation will be used in science, or how their measuring skills will be practised in physical education or design and technology. DFEE (2001). The National Strategy. Framework for teaching mathematics: years 7, 8 and 9. London: HMSO. (part 2 page 30)

26. Plenary: from the original KS3SM document (2) • You can use this part of the lesson to: • draw together what has been learned, summarise key facts, ideas and vocabulary, and stress what needs to be remembered; • generalise some mathematics from examples generated earlier in the lesson; • go through a written exercise pupils did individually during the lesson, so that you can question them about it, assess it informally and rectify any remaining misconceptions or errors; • make links to other work and what the class will go on to do next; • highlight the progress made and remind pupils about their personal targets; • set homework to extend or consolidate class work and prepare for future lessons. DFEE (2001). The National Strategy. Framework for teaching mathematics: years 7, 8 and 9. London: HMSO. (part 2 page 30)

27. Plenary: from the original KS3SM document (3) The plenary part of the lesson will be more effective if you have a clear idea of its purpose and what you want to achieve in it. DFEE (2001). The National Strategy. Framework for teaching mathematics: years 7, 8 and 9. London: HMSO. (part 2 page 30)

28. Possible plenary activities include… teacher exposition; Getting a child up to explain something they’ve done in the lesson; Game or quiz; Question and answer, possibly with view to informing next lesson; Go through a question which in general caused problems; Other formats? Overlap with activities suitable for starter.

29. Interesting to note…. An early OFSTED review from the National Numeracy Strategy indicated that in: 6/10 lessons the oral and mental work was good, with 5/10 lesson main teaching activity and 4/10 plenary Discussion point: why might the plenary be a more difficult part of the lesson to ‘get right’ than others? OFSTED (2000). The National Numeracy Strategy: an interim evaluation by HMI. London: HMSO.

30. Possible difficulties with the plenary • Depending on purpose (eg. picking up on misconceptions through the lesson), not possible to plan rigorously in advance; • May be difficult to get attention to front again; • Other activities may have taken longer than you anticipated; • Danger of ‘going through the motions’ rather than fully thinking through what the purpose of the plenary is; • Any others? OFSTED (2000). The National Numeracy Strategy: an interim evaluation by HMI. London: HMSO.

31. Consider the lesson finishing with this worksheet…. What might the plenary be?

32. Possible plenaries set up homework (but what homework?); do further question with a view to dealing with misconceptions which arose (what misconceptions? How do we deal with them?); ask a question from with a ‘real life’ context – painting a triangular wall????? play a game – matching pairs? Dominoes? Get pupils thinking about area of parallelograms; Have students come up and talk through one of the questions; Summarise key ideas (which are what? Exactly what would you say under this heading?) Get pupils to write down their own explanations in words as to how to find the area of a right angle triangle and why; Many others!

33. Sharing lesson objectives Many schools in UK insist that lesson objectives be shared; Within spirit of National Strategies. So lesson will start, “In this lesson we will…” May insist that pupils copy these ideas down. Example: “today we’re going to learn how to find the area of right angled triangles” Discuss with the person next to you: • Is this a good idea? Why (not)? • If we are to share lesson objectives, when might we do this? And how? • What might the pitfalls be? How might we overcome these pitfalls?

34. Sharing aims: when? • Beginning • Middle • End • Other times as well?

35. Sharing aims: how? • By writing them on the whiteboard; • By asking children to copy them down (combined with heading or separately?); • By asking children to put the aims into their own words (as homework?) • Discussing them at the beginning, eg. by giving an example of a question they will be able to do at the end of the lesson which they can’t do at the beginning; • Discussing them at the end, to establish whether the objectives have been met; • Anything else?

36. Sharing aims: why? • To bring clarity to our work in the classroom; • To provide motivation / interest; • To round off lessons with a sense that something has been achieved; • Anything else?

37. Sharing aims: why not? • Occasionally this spoils the point of the lesson (eg. investigation leading to Pythagoras); • For children to copy aims down can be time consuming for them; • Finding ‘child friendly’ language to express aims can be time consuming for us; • Can easily become ritualistic; • Anything else?

38. Using assessment data (1) Sources of data available to UK teachers include: • Test data from primary schools, including compulsory examinations at the end of year 6 (equivalent to grade 4); • (in later years) test data from earlier years; • Test data from screening tests undertaken when pupils enter secondary schools, similar to old style IQ tests; • Home data such as parental occupation, number of brothers / sisters / number of rooms per person in the house, etc.; • Also informal data through questioning etc. (see assessment for learning). This data is used to form complex models which gives target grades for pupils in their GCSE examinations. There follow some scenarios whereby you have gained information from assessment evidence. The question then arises: what do you do with the information you’ve gained?

39. Using assessment data (2) Case study 1(a) You pick up a new year 9 class (13-14 year olds) in a school with a very rigid scheme of work. You are supposed to be teaching them straight line geometry work. You ask questions about basic substitution of algebra in the starter, to find that the class en masse denies all knowledge of ever having done this before. What do you do?

40. Using assessment data (3) Case study 1(b) You pick up a newly setted year 9 class in a school with a very rigid scheme of work. You are supposed to be teaching them straight line geometry work. You ask questions about basic substitution of algebra in the starter, to find that those from one previous teacher can do the questions, those from another can’t. What do you do?

41. Using assessment data (4) Case study 2(a) National (ie. UK) data indicates that children from working class backgrounds attain at a lower level than middle class children. You are at a school with a roughly equal number of each. How might: • i) you as an individual teacher; • ii) the mathematics department; • iii) the whole school staff; use this information to inform your teaching?

42. Using assessment data (5) Case study 2(b) National data indicates that girls do better at GCSE (equivalent to CXC) overall than boys. You are at a mixed school. How might: • i) you as an individual teacher; • ii) the mathematics department; • iii) the whole school staff; use this information to inform your teaching?

43. Using assessment data (6) Case study 2(c) National data indicates that some ethnic minority groups in the UK hold their own in tests when entering school at age 5, with boys particularly doing progressively less well as they move up towards GCSE. You are at a school with about 30% of the children of minority ethnic origin. How might: • i) you as an individual teacher; • ii) the mathematics department; • iii) the whole school staff; use this information to inform your teaching?

44. Using assessment data: some reflections • The nature of statistical data is that it tells us about large populations, can’t then conclude that small groups or individuals will share that characteristic (see ‘ecology fallacy’); • Good overall maxim: ‘understand the differences, treat them the same’; • Good monitoring systems can lead to positive interventions, eg. “Last year you were doing well in maths and now you’re falling behind. Is there anything we can do to help?”

45. ‘Rich tasks’ Part of the 2008 reframing of the National Strategy approach, schemes of work should embed rich tasks. A rich task: • must be accessible to everyone at the start • needs to allow further challenges and be extendible • should invite children to make decisions • should involve children in speculating, hypothesis making and testing, proving or explaining, reflecting, interpreting • should not restrict children from searching in other directions • should promote discussion and communication • should encourage originality/invention • should encourage “what if ?” and “what if not?” questions • should have an element of surprise • should be enjoyable.

46. Rich task: example (1) Choose any four consecutive numbers and place them in a row with a bit of a space between them, like this: 4 5 6 7When you've chosen your consecutive numbers, stick with those same ones for quite a while, exploring ideas before you change them in any way. Now place  and  signs in between them, something like this :4 + 5 - 6 + 74 - 5 + 6 + 7and so on until you have found all the possibilities. You should include one using all ‘+’s and one that includes all ’-'s.Now work out the answers to all your calculations (e.g. 4 - 5 + 6 + 7 = 12 and so on). Are you sure you've got them all?

47. Rich task: example (2) Now work out the answers to all your calculations (e.g. 4 - 5 + 6 + 7 = 12 and so on). Are you sure you've got them all?If so, try other sets of four consecutive numbers and look carefully at the sets of answers that you get each time. It is probably a good idea to write down what you notice. This can lead you to test some ideas out by starting with new sets of consecutive numbers and seeing if the same things happen in the same way.You might now be doing some predictions that you can test out... Good place to start for further ideas is http://nrich.maths.org

48. Learning theories: opening exercise Think of as many ways as possible of completing the sentence: Children learn in the mathematics classroom when…. …they are not too hungry ….the teacher tells them something

49. Learning theories Opening position: All teaching draws upon a model or models of learning, whether explicit or implicit.

50. Learning theories • Brief look at: • Vygotsky’s zone of proximal development (ZPD); • Piaget’s stages of development; • Gardner’s theory of multiple intelligences; • Lave’s situated cognition; • Skemp’s relational rather than instrumental understanding; • Von Glaserfeld’s constructivism; • Bruner’s spiral of learning.