Conceptual coherence

1. Conceptual coherence • In mathematics, new ideas, skills and concepts build on earlier ones. • If you want build higher, you need strong foundations. • Every stage of learning has key conceptual pre-cursors which need to be understood deeply in order to progress successfully. • When something has been deeply understood and mastered, it can and should be used in the next steps of learning.

2. What is ‘Mastery’? • A mastery approach; a set of principles and beliefs. • A mastery curriculum. • Teaching for mastery: a set of pedagogic practices. • Achieving mastery of particular topics and areas of mathematics.

3. What image is helpful? Procedural Fluency Conceptual Understanding INTEGRATION 3

4. Conceptual vs procedural knowledge • Mathematical knowledge, in its fullest sense includes significant, fundamental relationships between conceptual and procedural knowledge. Students are not fully competent in mathematics if either kind of knowledge is deficient or if they both have been acquired but remain separate entities. • When concepts and procedures are not connected, students may have a good intuitive feel for mathematics but not solve the problems, or they may generate answers but not understand what they are doing. Heibert and LeFevre - Conceptual and procedural knowledge in mathematics: an introductory analysis (Chap 1 of ‘Conceptual and procedural knowledge: the case of mathematics’ (1986).

5. conceptual understanding - comprehension of mathematical concepts, operations, and relations • procedural fluency - skill in carrying out procedures flexibly, accurately, efficiently, and appropriately • strategic competence - ability to formulate, represent, and solve mathematical problems • adaptive reasoning - capacity for logical thought, reflection, explanation, and justification • productive disposition - habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

6. 5 big ideas

7. Coherence • Messages • Small steps are easier to take. • Focussing on one key point each lesson allows for deep and sustainable learning. • Certain images, techniques and concepts are important pre-cursors to later ideas. Getting the sequencing of these right is an important skill in planning and teaching for mastery. • When something has been deeply understood and mastered, it can and should be used in the next steps of learning.

8. Representation & Structure • Messages • The representation needs to pull out the concept being taught, and in particular the key difficulty point. It exposes the structure. • In the end, the children need to be able to do the maths without the representation • A stem sentence describes the representation and helps the children move to working in the abstract (“ten tenths is equivalent to one whole”) and could be seen as a representation in itself • There will be some key representations which the children will meet time and again • Pattern and structure are related but different: Children may have seen a pattern without understanding the structure which causes that pattern

9. Variation • Messages • The central idea of teaching with variation is to highlight the essential features of a concept or idea through varying the non-essential features. • When giving examples of a mathematical concept, it is useful to add variation to emphasise: • What it is (as varied as possible); • What it is not. • When constructing a set of activities / questions it is important to consider what connects the examples; what mathematical structures are being highlighted? • Variation is not the same as variety – careful attention needs to be paid to what aspects are being varied (and what is not being varied) and for what purpose.

10. What is the value of the ? 3/5 of 30 = ? 3/4 of 30 = ? 3/? of 30 = 9 3/5 of 60 = ? 3/10 of 30 = ? 3/? of 30 = 1.5 Find 3/5 of 30 Find 4/7 of 28 Find 6/11 of 99 Find 12/15 of 60 Find 7/25 of 300 Find 23/40 of 1000

11. Mathematical Thinking • Messages • Mathematical thinking is central to deep and sustainable learning of mathematics. • Taught ideas that are understood deeply are not just ‘received’ passively but worked on by the learner. They need to be thought about, reasoned with and discussed. • Mathematical thinking involves: • looking for pattern in order to discern structure; • looking for relationships and connecting ideas; • reasoning logically, explaining, conjecturing and proving.

12. Fluency • Messages • Fluency demands more of learners than memorisation of a single procedure or collection of facts. It encompasses a mixture of efficiency, accuracy and flexibility. • Quick and efficient recall of facts and procedures is important in order for learners’ to keep track of sub problems, think strategically and solve problems. • Fluency also demands the flexibility to move between different contexts and representations of mathematics, to recognise relationships and make connections and to make appropriate choices from a whole toolkit of methods, strategies and approaches.

13. ComparingLessons • Look out for the 5 big ideas: • Coherence • Representation & Structure • Variation • Fluency • Mathematical Thinking

14. Which diagram shows of the whole is shaded? 1 5 5 1 1 6 6 6 6 6 1 0

15. What fraction is shaded? （ ） （ ） 2 7 7 2 Compare and fill in the blank with “<,>,=” 10 10 10 10 <

16. Compare these fractions. Think of the reason and fill in the blank. > < 3 b 4 6 a 2 c 7 5 7 5 c > （a>b>0 c≠0 ）

17. Comparing fractions: When the denominatorsare the same,we can compare the numerators. When the numerator is bigger, the fraction is bigger. b c b c a a a a > （when b > c and a≠0, b≠0, c≠0）

18. What fraction of each shape is shaded? What do you notice?

19. Compare these fractions using < > or = > > 1 2 1 2 2 2 1 1 3 3 3 3 5 5 5 5 2 lots of > 2 lots of >

20. Compare these fractions: > Reasoning： 3 lots of> 3 lots of 1 3 3 3 3 1 0 1 10 10 10 8 8 8 0 1

21. Compare these fractions: < Reasoning： 5 lots of< 5 lots of 5 5 1 1 5 5 12 9 9 12 9 12 0 1 0 1

22. The comparison of fraction: When the numeratorsare the same,we can compare the denominators. When the denominator is bigger, the fraction is smaller. a a a a c c b b < （when b > c and a≠0, b≠0, c≠0）

23. Compare these fractions using inequalities, from biggest to smallest 11 11 11 8 8 8 8 8 8 8 > 18 18 15 15 15 15 18 15 15 15 > > >

24. Compare the following groups of fractions, and write “>,<,=” > > > > 3 23 23 1515 8 6 3 9 8 2 2 3 2 7 3 14 28 7 16 16 28 6 47 16 9 3 5 7 8 14 = > > >

25. Fill in the blanks with fractions: ▲▲▲ △△△ △△△ ▲▲▲ △△△ ★★★☆☆ ★★★☆☆ 6 6 > （ ） （ ） 15 10

26. Compare these fractions using inequalities: <

27. Mathematical modelling

28. Modelling & models • Provide a visual representation of mathematics to help learners “see” the underlying structure of a concept or problem. • Provide a frame of reference through which learners can articulate their reasoning and explanations.

29. Algebra & Arithmetic • To fully understand & achieve procedural fluency, learners would benefit from developing a “sense of the structure” behind the way mathematics works. • Representing that structure in ways other than numerically or a generalised format can support that (Hewitt 2010)

30. Column addition using manipulatives 24 + 17 41 1

31. Column subtraction using manipulatives 32 - 14 1 2

32. If two-thirds of a number is 90. What’s the number?

33. If two-thirds of a number is 90. What’s the number?

34. What’s three-quarters of 60?

35. What’s three-quarters of 60?

36. Bar Models also … • allow learners to “translate” worded problems into a mathematical diagram and so more easily decide on appropriate strategies to take.

37. I’m thinking of two whole numbers. The larger is three times the smaller number.

38. If the smaller number is 12, what is the sum of the two numbers? If the larger number is 12, what is the sum of the two numbers? If the difference between the two numbers is 12, what is the larger number? If the sum of the two numbers is 12, what is the smaller number?

39. If two numbers that total 30 are in the ratio 2:3, what’s the difference between these two numbers?

40. If two numbers that total 30 are in the ratio 2:3, what’s the difference between these two numbers?

41. Five cans of drink cost £1.20What do three cans cost?

42. Five cans of drink cost £1.20What do three cans cost?

43. Combining “lots of” What is the cost of two pens & three pencils?

44. Left over “change” What is the change from £10 If you buy two pens & three pencils?

45. Chocolate bars are on an offer buy two get a third bar at half price. If seven bars cost £4.20 , what would four bars cost?

46. Chocolate bars are on an offer buy two get a third bar at half price. If seven bars cost £4.20 , what would four bars cost?

47. If it took 5 hours to drive 300 km. How far could you drive in 7 hr at the same speed?

48. If it took 5 hours to drive 300 km. How far could you drive in 7 hr at the same speed?

49. SpeedWho is going faster? Car travels 200 km in 4hrs Motorbike travels 40km in 30mins

50. Best value for money? 300 ml £1.50 500 ml £2.40