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Algebra: Directions for the Future.

Algebra: Directions for the Future. Marj Horne, Australian Catholic University. m.horne@patrick.acu.edu.au. Think of a number. Multiply your number by 4 Add 12 Divide the number you now have by 2 Add four Halve the number Subtract the number you first thought of

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Algebra: Directions for the Future.

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  1. Algebra: Directions for the Future. Marj Horne, Australian Catholic University m.horne@patrick.acu.edu.au

  2. Think of a number • Multiply your number by 4 • Add 12 • Divide the number you now have by 2 • Add four • Halve the number • Subtract the number you first thought of • Find the letter in the alphabet that occupies that position • Think of an animal that starts with that letter

  3. Was your animal Grey? (or pink?)

  4. I’m thinking of a number. Four times the number plus three is the same as three times the number plus nine. What number am I thinking of? Murray Britt - NZ

  5. Change in pedagogy • Students must take control of their own learning (teachers must be willing to give up control) • Recognition of the importance of discussion in the classroom both in groups and whole class • The use of language • Problem solving and investigations • Formative assessment – feedback which supports learning • Evidence based approaches

  6. Major movements • Impact of technology - CAS • Algebra from early years on

  7. Impact of technology • When ordinary calculators arrived teachers ignored them or actively argued against them. • Calculators can be used as “answer getting” machines or as tools, particularly for learning • Now they are accepted by many as useful learning tools • Change in emphasis – move towards building number sense • The “thinking” curriculum

  8. New calculator technology • Introduction of graphic calculators more planned • Need for professional development to use creatively as learning tool rather than just answer getting tool • We need to build “algebra sense” just as we build “number sense” • CAS changes the range of problems and applications possible and opens up investigations and problem solving as well as providing a tool for learning

  9. CAS • Focus on understanding and making connections rather than routine skills • Applications to “real” problems and to investigations opens up • Classroom changes – students more in control – more group work • Students still develop skills

  10. Algebra in the early years • New Zealand has acknowledged this for longer than most other countries • Structure of systems • Connections between arithmetic and algebra • Patterns • Generalisations

  11. How many legs are there? 2 lions, their 4 cubs and 4 storks

  12. How many legs are there? 2 lions, their 4 cubs and 4 storks 2  4 = 8

  13. How many legs are there? 2 lions, their 4 cubs and 4 storks 2  4 = 8 + 4  4

  14. How many legs are there? 2 lions, their 4 cubs and 4 storks 2  4 = 8 + 4  4 = 8 + 16 = 24

  15. How many legs are there? 2 lions, their 4 cubs and 4 storks 2  4 = 8 + 4  4 = 8 + 16 = 24 + 4  2

  16. How many legs are there? 2 lions, their 4 cubs and 4 storks 2  4 = 8 + 4  4 = 8 + 16 = 24 + 4  2 = 24 + 8 = 32

  17. Standing 40 m away from a flagpole on level ground a man used a theodolite to find the angle of elevation of the top of the flagpole as 60o. Find the height of top of pole from ground if the angle was sighted from 2 metres above the ground.

  18. h 2m 60o 40m h = 40.tan60o = 40  1.732 = 69.28

  19. h 2m 60o 40m h = 40.tan60o = 40  1.732 = 69.28 + 2 = 71.28

  20. h 2m 60o 40m h = 40.tan60o = 40  1.732 = 69.28 + 2 = 71.28 m.

  21. Cos A = 0.5 = 60o • 2x – 5 = 9 = 2x = 14 = x = 7

  22. Understandings of “=” • 7 + 8 + 9 = ‘makes’ or ‘now work it out’ also ‘now do the next step’ hence misuses shown earlier • x = 3 assigning a value

  23. Understandings of “=” • 27 = 5 × 6 – 3 ‘wrong way around – the single number is always on the right’ (reinforced by classroom rules such as ‘x terms on the left and numbers on the right’ when solving equations) • 20 + 4 = 6 × 6 – 3 × 4 ‘if you work out each side you get the same answer’ (quantitative sameness)

  24. Understandings of “=” • 3x + 2 = 5 ‘both sides are the same only when x is 1’ • 3 + 2 = 5 identical • 3x + 2x = 5x ‘identical, equivalent – true for all values of x (identity) and in fact also true for x as any object thus fruit salad algebra ‘bothsides are the same when x is 1’ • 2x + 4 = 6(3x – 2) ‘the = means the two sides balance’

  25. Understanding of equals and the language used from early schooling on. • Concentration on calculation outcomes • Restricted understanding of the arithmetic operations – seeing them as combining only rather than also in terms of change (or relational). (Elizabeth Warren)

  26. 2 + 4 = 5 + 1

  27. 2 + 3 ? 5 + 1

  28. Relational thinking 78 + 34 = 112 78 + 35 = ? 69 + 57 = + 58 367 + = 562 + 364

  29. If I know that 78 + 34 = 122 what else do I know? If I know that 23  16 = 368 what else do I know?

  30. Some reasons for difficulties in algebra • obstructions caused by different understandings of the symbols between children’s arithmetic understanding and algebra. • inappropriate generalisations and interpretations • alternative approaches to semantics deduced from the “concrete” situation

  31. Understanding of operation symbols • + a sign meaning to combine two numbers and accompanied by action • 5 + 7 is 12 • Once it is 12 the parts are no longer visible • In a + 7 the + sign does not mean actively combine the two parts as it did in 5 + 7 • While a + 7 can be seen as a single object, the components maintain their identity

  32. Understanding of operation symbols Seeing the operations as combining leads to the incorrect 4x + 3 = 7x A critical part of algebraic development is “acceptance of lack of closure” (Collis, 1975)

  33. Understanding of operation symbols Another way of seeing 5 + 7 is in a relational way as 5 more than 7. a + 7 then becomes 7 more than a number a The use of this type of language rather than translating it into words as a plus 7 or a and 7 is one that seems to be of great assistance in making sense of algebra.

  34. Inappropriate generalisations • x is any number • Guess and check being reinforced in spite of teachers’ approaches • Backtracking leading to inappropriate recording and limiting development Equation to solve for x: (x – 8)/2 = 3 Student response: x = 39

  35. 41 students, three schools, six weeks after completed a unit on algebra including equation solving If c = 5b + 2, and c = 27, what is b?

  36. 41 students, three schools, six weeks after completed a unit on algebra including equation solving If c = 5b + 2, and c = 27, what is b? 38 gave correct answer but

  37. 41 students, three schools, six weeks after completed a unit on algebra including equation solving If c = 5b + 2, and c = 27, what is b? 38 gave correct answer But 3 said thought at first question must be wrong – should have been c = b5 + 2 then b = 2

  38. Those who were correct were asked If g = 4f + 3, and g = 12, what is f ?

  39. Those who were correct were asked If g = 4f + 3, and g = 12, what is f ? Initially only two were correct. Only one used a teacher taught method. A third changed answer when asked to explain

  40. Those who were correct were asked If g = 4f + 3, and g = 12, what is f ? Initially only two were correct. Only one used a teacher taught method. A third changed answer when asked to explain The rest said it was impossible.

  41. Inappropriate generalisations • x is any number • Guess and check being reinforced in spite of teachers’ approaches • Backtracking leading to inappropriate recording and limiting development Left to their own devices students are unlikely to develop the semantics of algebra as we know them because the experiences they have are limited and often lead to alternative representations which are situation specific.

  42. Developing rules from patterns • The sequential patterns focus the children’s attention on an aspect which actually limits their understanding of function.

  43. xy 1 4 2 7 3 10 4 13 5 … 10 … 100 …

  44. x y 3 7 9 1 6 4 5 8 20 Ryan & Williams

  45. Mary has the following problem to solve “Find the value(s) for x in the following expression: x + x + x = 12 ” She answered in the following manner A. 2, 5, 5 B. 10, 1, 1 C. 4, 4, 4 Which of her answer(s) is (are) correct? Circle the letter(s) for each correct answer.

  46. Would your answer have changed if the question was + + = 12 A. 2, 5, 5 B. 10, 1, 1 C. 4, 4, 4 Which of her answer(s) is (are) correct? Circle the letter(s) for each correct answer.

  47. Jon has the following problem to solve “Find the value(s) for x and y in the following expression: x + y = 16 ” He answered in the following manner A. 6, 10 B. 9, 7 C. 8, 8 Which of his answer(s) is (are) correct? Circle the letter(s) for each correct answer.

  48. Again would you have answered differently if it had been + = 16 He answered in the following manner A. 6, 10 B. 9, 7 C. 8, 8 Which of his answer(s) is (are) correct? Circle the letter(s) for each correct answer. ?

  49. One question that arises in looking at early algebra is the use of symbols When should letters be introduced?

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