Supporting professional development in algebraic and fractional thinking

1. Supporting professional development in algebraic and fractional thinking Workshop presented at National Numeracy Facilitators Conference February 2007 Teresa Maguire and Alex Neill and Jonathan Fisher

2. P.D. on the ARB website Research Concept maps Assessment strategies … and the Teacher information pages

3. Research Research The ARCT (Assessment Research for Classroom Teachers) project promotes classroom-based research as an integral part of developing ARB resources. ARB resources are based on our own, national, and international research. We present research information in formats relevant to a variety of audiences.  We provide evidence-based information to support teachers' assessment practices. Research by curriculum bank

4. Journalling in Mathematics* What is journalling? The benefits of journalling Teaching tips for journalling Writing prompts for journalling References What is Journalling? Journalling involves students writing about their learning in mathematics. What they write …

5. Research Self-regulated learning The focus of the ARCT project's research for 2004 was self-regulated learning. This was used to inform the development of new resources for the Assessment Resource Banks. Support material for teachers is published as it becomes available. Research conducted into self regulated learning in 2004 follows: Reflecting on reflective journalling (maths) Self-regulated learning in the mathematics class**

6. Researchers: Charles Darr and Jonathan Fisher** Context: proportional reasoning (Year 7) Using thinking models to represent proportional relationships Thinking models help students to form a representation of a problem situation. They can involve concrete objects, or be more abstract. Thinking models used included double number lines, geometrical shapes, cuisenaire rods, and decimal pipes. The double number line One of the most successful models was the double number line. The double number line allows the elements in a proportional relationship to be modelled on a two-sided scale. To read more, see this PDF document.

7. Assessment strategies(under ARBs & Assessment) Assessment strategies

8. Mathematical Classroom Discourse*** When to use Discourse can be used ……… The theory While classroom discussions are nothing new, the theory behind classroom discourse stems from constructivist … How the strategy works Well-designed distractors provide alternatives that identify particular misconceptions. Providing …… What to do In order for discussion to take place, classroom (sociomathematical) norms need to be ….. Examples of ARB resources NM1199, NM1201, and NM1221 ask students to identify which cartoon characters are estimating and … Selected references Burns, M. (2005). Looking at How Students Reason. Educational Leadership, 63 (3), pp. 26-31.

9. Teacher information pages  To do • Task administration • Answers • Calibration easy (60-79.9%) • Diagnostic information (common wrong answers and misconceptions) • Diagnostic and formative information • Next steps • Links to other resources/information and to concept maps

10. Concept maps This is a series of framework statements that have been developed in significant areas of mathematics. They:  • provide information about the key mathematical ideas involved • link to relevant ARB resources, and  • suggest some ideas on the teaching and assessing of that area of mathematics. Computational estimation**** Fractional thinking

11. Concept Map - example

12. Algebra is… Love is…. 2 + 2 = 3 + 1. “The language of arithmetic is focussed on answers. The language of algebra is focussed on relationships.” MacGregor, M & Stacey, K. (1999) “A flying start to algebra. Teaching Children Mathematics, 6/2, 78-86. Retrieved 17 May 2005 from http://staff.edfac.unimelf.edu.au/~Kayecs/publications/1999/MacGregorStacey-AFlying.pdf

13. Equality problem 7 + = 10 + 2

14. Equality =

15. Solve this equation 2x – 6 = x + 4

16. Additive Identity Who am I? What am I? What happens when I’m added to a number?

17. Conjectures about zero When you add zero with another number it doesn’t change the number you started with. a + 0 = a When you take away zero from a number it doesn’t change the number you started with. a – 0 = a If you take away the same number from the one you started with you get zero. a – a = 0

18. Wagons and hand-holding 7 + 28 + 13 + 12 = 7 + 13 + 28 + 12 = (7 + 13) + (28 + 12) = 20 + 40 = 60

19. Commutativity and Associativity (a + b) + c = a + (b + c) a + b = b + a

20. Student’s rules It doesn’t matter if the numbers are swapped around on each side of the number sentence. If the numbers are the same, the number sentence will still balance If you add up numbers in a different order you still get the same answer. When you add three numbers it doesn’t matter whether you start by adding the first pair of numbers or the last pair of numbers.1 1 Carpenter, Franke & Levi Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School.

21. Relational thinking 3 + 16 = + 18 36 + 78 = + 74 57 + = 63 + 51 For each equation, work out what number goes in the box to make the number sentence true. Be prepared to explain how you worked out the answer.

22. Introduce pronumerals For each equation, find the number that replaces the letter. 68 + b = 57 + 69 82 – 47 = g – 46 a + 38 = 36 + 59 87 + 45 = 86 + 46 + t 234 + 578 = 232 + 576 + e 92 – 57 = 94 – 56 + h

23. Repeated pronumerals Find the Variables Find the number that each variable is replacing in the following number sentences. z + z + z + z = 20 11 = t + t b = 8 – b j + j + 3 = 7 d + d – 5 = 13 2 × d – 5 = 13 (5 × b) – 6 = 14

24. Fractional Thinking is … … Not just about Pizza

25. Fractional Thinking Concept Map

26. Partitioning: examples Share this shape evenly amongst 4 people

27. Partitioning: examples Share this shape evenly amongst 3 people

28. Partitioning: examples Share this shape evenly amongst 3 people

29. Partitioning: examples Share this shape evenly amongst 7 people

30. Partitioning: examples Example: Share 2 cakes equally amongst 3 people

31. What is partitioning? • Partitioning involves the ability to divide an object or objects into a given number of non-overlapping parts.

32. Partitioning strategies • Simple partitions: • halves, quarters, eighths … (halving) • evenness/equal-sized parts • More complex partitions: • larger number of pieces (e.g., 6ths, 10ths, 12ths …) • odd number of pieces (e.g., partition into 3rds, 5ths …) • Partitioning complex shapes: e.g., two squares

33. Part-whole fractions

34. Part-whole fractions: example What fraction is shaded?

35. Part-whole fractions: example What fraction is shaded?

36. Part-whole fractions: example How much of the square is shaded?

37. Part-whole fractions: example How much of the shape is shaded?

38. Part-whole fractions: examples Given the part find the whole … or another part If is 1/4 what fraction is ? What would a whole look like? If is 2/3 of all the counters, show how many 1/3 is

39. Comparing fractions

40. Comparing fractions: examples Show or explain which fraction is larger. 1/9 or 1/8 2/3 or 3/5 7/6 or 8/9

41. Strategies for comparing fractions Less sophisticated More sophisticated Attempting to use whole number knowledge Drawing pictures Identifying fractions with the same denominator or numerator Benchmarking fractions to well known fractions Using equivalent fractions

42. Animations • Immediate response • Computer medium • High motivation to “do again” • Independent • Formative information for the teacher • Sets of objects: NM0129 (unit fractions) & NM0130 (non unit fractions)NM1231 (unit fractions)NM1232 (non unit fractions)

43. Animations – fractions of a set

44. Animations – immediate results

45. Exploring Fractional Thinking

46. Check it out on the website Research Concept maps Assessment strategies … and the Teacher information pages

47. Assessment Resource Banks www.nzcer.org.nz/arb Username: ARB Password: guide