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The 'algebra as object' analogy: a view from school

The 'algebra as object' analogy: a view from school. Dr Geoff Tennant and Kate Colloff g.d.tennant@reading.ac.uk. Plan of session. Background to the issue; Original research ; Some reflections on the process; Questions?. The issue.

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The 'algebra as object' analogy: a view from school

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  1. The 'algebra as object' analogy: a view from school Dr Geoff Tennant and Kate Colloff g.d.tennant@reading.ac.uk

  2. Plan of session • Background to the issue; • Original research; • Some reflections on the process; • Questions?

  3. The issue Teaching of early simplification of algebra, need to communicate that: 3a + 2a = 5a Algebra as object: ‘a’ is the object ‘apple’, so can read this as ‘3 apples plus 2 apples is 5 apples’

  4. From: CRANK, V et al. (1998). Key Maths 7. Cheltenham: Stanley Thornes.

  5. From: CRANK, V et al. (1998). Key Maths 7. Cheltenham: Stanley Thornes.

  6. From: CRANK, V et al. (1998). Key Maths 7. Cheltenham: Stanley Thornes.

  7. Number: Ian, Chris and Tim Statistics: Helen, Fiona, Chris, Mairead, and Tim. Number: Ian, Chris and Tim Statistics: Helen, Fiona, Chris, Mairead, and Tim. Questions from Oxford GCSE Maths for Edexcel, 2010, p41.

  8. Three objections (Tennant 2009) 1. It’s wrong: this is abbreviation, not algebra! Tennant, G. D. (2009). 2a+3a=5a without apples: the algebra as object analogy. Mathematics in School, 38(1), pp. 4-6.

  9. Three objections (Tennant 2009) 2. It ‘works’ only in a very narrow band of examples eg. 3a – 2a = a would work but -2a + 3a doesn’t, nor does 3a – 4a …..

  10. Three objections (Tennant 2009) And what about ab or a2 ?

  11. Three objections (Tennant 2009) 3. It leads to problems further on (Clement 1982); Eg 1: in a University there are six students to each professor. If S is the number of students and P is the number of professors, write down a formula connecting S and P; For one professor there are 6 students: P = 6S Clement, J. (1982). Algebra word problem solutions: thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13(1), pp. 16-30.

  12. Three objections (Tennant 2009) Eg 2: 5 miles is (approximately) the same as 8 kilometres. If m is the number of miles and k is the number of kilometres, write down a formula connecting m and k. 5 miles = 8 kilometres 5m = 8k

  13. Textbook approaches to an introduction to simplification • ‘Algebra as object’ already noted; • Ambiguous; • ‘Algebra as object’ merges into algebra as variable; • Algebra as variable.

  14. Textbook approach: ambiguous a + a + a + a + a is 5 x a. In algebra it is written as 5a (ie. with the multiplication sign left out). 5a is called an algebraic term. In the same way 3y = 3 x y = y + y + y Exercise: write down each of the following as an algebraic term: 1. m + m + m …. From GREER, A (1993). A complete GCSE mathematics basic course: third edition. Cheltenham: Stanley Thornes.

  15. Textbook approach: algebra as object merges into variable From MILLER et al. (1992). MSM Mathematics 2w. Walton-on-Thames: Nelson.

  16. Textbook approach (3): algebra as object merges into variable From MILLER et al. (1992). MSM Mathematics 2w. Walton-on-Thames: Nelson.

  17. Textbook approach: simplification with algebra as variable From VICKERS, K M et al. (1995). National Curriculum Mathematics 5. Cheltenham: Stanley Thornes (Publishers) Ltd. DISCUSSION EXERICSE Is a + b the same as b + a? Is 5a + 3a = 8a? Is 5a – 3a = 2a? Is 3a + 4b = 7ab? Is 5a + 4b + 3a + 7b = 8a + 11b? Is 5a + 4b – 3a – 7b = 2a -3b? Discuss. As part of your discussion replace a and b with many different numbers. Continues with ‘ambiguous’ approach.

  18. Textbook approach (4d): simplification with algebra as variable From LEDSHAM, A & WARDLE, M E (1996). Foundation GCSE mathematics: revision and practice. Oxford: Oxford University Press. The perimeter of an equilateral triangle is given by the formula p = a + a + a or p = 3a Find the perimeter of an equilateral triangle whose side length a is 5cm. The perimeter, p = (3 x 5) cm = 15 cm. a a a

  19. Textbook approach (4d): simplification with algebra as variable From LEDSHAM, A & WARDLE, M E (1996). Foundation GCSE mathematics: revision and practice. Oxford: Oxford University Press. The perimeter of an isosceles triangle is given by the formula p = a + a + b or p = 2a + b Find the perimeter of an isosceles triangle with a = 3cm and b = 5cm. The perimeter, p = (2 x 3 + 5) cm = (6 + 5) cm = 11 cm. a a b

  20. Textbook approach (4d): simplification with algebra as variable From LEDSHAM, A & WARDLE, M E (1996). Foundation GCSE mathematics: revision and practice. Oxford: Oxford University Press. The perimeter of a square is given by the formula p = l + l + l + l or p = 4l Find the perimeter of a square whose side length is 7cm. The perimeter, p = (4 x 7) cm = 28 cm. l l l l

  21. Textbook approach (4d): simplification with algebra as variable From LEDSHAM, A & WARDLE, M E (1996). Foundation GCSE mathematics: revision and practice. Oxford: Oxford University Press. The perimeter of a rectangle is given by the formula p = l + w + l + w or p = 2l + 2w or p = 2(l + w) Find the perimeter of a rectangle with l = 5 cm and w = 3 cm. The perimeter, p = 2(5 + 3) cm = (2 x 8) cm = 16 cm. l w w l

  22. Textbook approach (4e): simplification with algebra as variable From ATKINSON, C et al. (1998). Nelson GCSE mathematics: intermediate 1. Walton-on-Thames: Thomas Nelson & Sons Ltd. The function is n → 3n + 1. Copy and complete this box. n → 3n + 1 6 → ? 23 → ? ? → 100 ? → 1

  23. Textbook approach (4e): simplification with algebra as variable • The function is n → 3n + 2 + 4. • Copy and complete this box. • What is a simpler way of writing the function? From ATKINSON, C et al. (1998). Nelson GCSE mathematics: intermediate 1. Walton-on-Thames: Thomas Nelson & Sons Ltd. n → 3n + 2 + 4 8 → ? 23 → ? 0 → ? n→ ?

  24. Textbook approach (4e): simplification with algebra as variable From ATKINSON, C et al. (1998). Nelson GCSE mathematics: intermediate 1. Walton-on-Thames: Thomas Nelson & Sons Ltd. Copy and complete these boxes, and write each function in a simpler way: n → 5n - 3n 6 → ? 10 → ? 13 → ? n→ ?

  25. Further suggestions (1):consecutive numbers • Ask volunteer to give a number (eg. 23), offer a prize for any pupil who can get add up that number and the next 4 consecutive numbers (ie. 23+24+25+26+27) faster than you. • Repeat until it becomes clear to the pupils that you have a method which enables you to get the answer quickly.

  26. Further suggestions (1):consecutive numbers Suggest that they gather information systematically: First no. Sum Total 0 0+1+2+3+4 10 1 1+2+3+4+5 15 2 2+3+4+5+6 20 3 3+4+5+6+7 25 10 10+11+12+13+14 60 Formula 5x0 + 10 5x1 + 10 5x2 + 10 5x3 + 10 5x10 + 10 n n+(n+1)+(n+2)+(n+3)+(n+4) 5n+10 5n+10

  27. 48 20 28 8 12 16 3 5 7 9 1 2 3 4 5 Further suggestions (2):number pyramids Volunteer gives 5 numbers (eg. 1,2,3,4,5) which go into the bottom of the pyramid. Then each box in the rows above is the sum of the two boxes below.

  28. Further suggestions (2):number pyramids Possible questions to ask: (a) How would you arrange the numbers in the bottom row to get the biggest total on the top row? (b) As (a), the lowest total? (c) What happens to the number in the box on top if you add 1 to each of the boxes in the bottom row, one at a time? Can you explainyour answer? …… (d) If you put a, b, c, d, e, in the bottom row, can you write a formula for the number in the top row? (e) Look at the answers different people have got. What do you notice?

  29. a+4b+6c+4d+e a+3b+3c+e b+3c+3d+e a+2b+c b+2c+d c+2d+e a+b b+c c+d d+e a b c d e Further suggestions (2):number pyramids

  30. Further suggestions (3):matching exercise Make up sets of cards as a matching activity. Have students explain why they think any set of cards do or don’t match. 5 lots of n 2n + 2m n + n + n + n 3n + 3 3n + 2n n + (n+1) + (n+2) 25 5n + 10 + 6 – 2n 3n + 16 4n 2(n+m) 2n + 9 + n + 7 n + m + n + m 3n + 8+ 2m + 12 3(n+1) 5n + 20 5m + 20 BEWARE: not all cards have partners!

  31. New fieldwork • Review of up to date textbooks; • Interviews with colleagues; • Focus group of year 7 pupils; • Focus group of year 10 pupils.

  32. Textbook analysis • 10 texts and 3 electronic resources in current usage within school. Of the textbooks: • 4 straight to algebra; • 3 generalised arithmetic; • 2 symbols (implying algebra as object); • 1 kinaesthetic object (length of rod). No clear patterns by publication date, with a range from 1955-2010.

  33. Teacher interviews (1) Aware that ‘algebra as object’ not recommended: • You cannot multiply apples. • The letter represents a number, not an object. • You cannot get cubic or squared bananas. • It means that students struggle to access other things later. • It is then difficult when you introduce a negative number of apples. One teacher knew ‘not allowed’ but couldn’t explain why.

  34. Teacher interviews (2) • Teachers using eg. faces of pupils to illustrate simplification, rationalising the approach as not being ‘algebra as object’ ; • Differences of opinion as to how to teach basic simplification; • Differences of opinion as to ‘good’ approaches for high and low attaining pupils – latter more prone to ‘algebra as object’ through frustration and pressure for exam passes.

  35. Year 7 interviews (1) • 35 pupils interviewed; • In general, pupils who have previously achieved higher at maths able to answer simplification questions more accurately; • When explaining simplification, answers included: • “2+3 then put 'a' back on” (74%); • “Collecting like terms” (9%); • Wrong, including 5, 5aa and 7 (9%); • “Guess / don’t know” (9%).

  36. Year 7 interviews (2) • No year 7 pupil explicitly appealed to an ‘algebra as object’ approach (eg. “3 apples plus 2 apples is 5 apples”).

  37. Year 10 interviews (1) 30 top set pupils in total of 8 groups, asked how they would explain introductory simplification of algebra to year 7 pupils. 6 of 8 groups appealed to ‘algebra as object’ Remaining 2 went straight to algebra and appeals to ‘rules’: “It’s just law”.

  38. Year 10 interviews (2) Answers confused when dealing with simplification not easily undertaken with ‘algebra as object’: “a is blue, b is red, so ab is purple” “a2 is two apples” “a x b = apple x banana = apple and banana.”

  39. Implications Need to ensure approaches to algebra, making links with number and generalisations in words, covered thoroughly in ITT and CPD courses; Ensure plenty of examples of other approaches readily available.

  40. Questions arising Where does the idea of ‘algebra as object’ arise from between year 7 and 10? Do teaching approaches which avoid ‘algebra as object’ work in practice? What happens in other countries?

  41. Reflections on the process Keep research interests close to professional interests; Explore something that is a burning interest; Can you get students to work with you? So their assignments feed into your larger project? General principle: data is all around you – even your annual reports!

  42. And finally…. Thank you! Dr Geoff Tennant g.d.tennant@reading.ac.uk Personal website: www.geofftennant.name

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