Teacher to Adviser

1. CENTRE FOR EDUCATIONAL DEVELOPMENTStudents making the connections between algebra and word problemshttp://ced.massey.ac.nz

2. Teacher to Adviser Team Leader, Numeracy and MathematicsCentre for Educational DevelopmentMassey University College of EducationPalmerston North New Zealand a.lawrence@massey.ac.nz

3. Palmerston North (New Zealand)

4. NZAMT-11 conference

5. New Zealand schools Years 1- 6 Primary Years 7 & 8 Intermediate Years 9 -13 Secondary Full primary Year 7–13

6. Issues in education in New Zealand Numeracy and literacy Curriculum Assessment NCEA Technology National testing

7. You didn’t tell me it was a word problem ..\little league movie_WMV V9.wmv

8. Difficulties with word problems Educators frequently overlook the complexity of Mathematical English Vocabulary Connectives Word order Syntactic structure Punctuation Half of the sum of A and B, multiplied by three Half of the sum of A and B multiplied three

9. Context is complicated Contextualising maths creates another layer of difficulty – the difficulty of focusing on the maths problem when it is embedded in the ‘noise of everyday context’ (Cooper and Dunne, 2004, p 88) Placing mathematics in context tends to increase the linguistic demands of a task without extending the mathematics (Clarke, 1993)

10. The national standard in NZ “use algebraic strategies to investigate and solve problems… Problems will involve modelling by forming and solving appropriate equations, and interpretation in context” “must form equations…at least one equation” (assessment schedule, NZQA)

11. Algebra word problems in NAPLAN

12. Skills assessed in NAPLAN 2008 Identifies the pair of values that satisfy an algebraic expression. Solves a multi-step algebra problem. Solves algebraic equations with one variable and expressions involving multiple operations with negative values. Determines an algebraic expression to model a relationship.

13. Algebra word problems in NAPLAN

14. What is it about algebra word problems? What are algebra word problems? Why do students find them difficult? What can teachers do to help their students tackle them with more success?

15. Solve this word problem A rectangle has a perimeter of 15 m Its width is 2.2 m Calculate the length of this rectangle

16. It is a word problem… A rectangle has a perimeter of 15 m Its width is 2.2 m Form and solve an equation to calculate the length of this rectangle 2.2 + 2.2 = 4.4 15- 4.4 =10.6 10.6 / 2 =5.3

17. It is a word problem … but is it an algebra word problem? What makes an algebra word problem? What solution strategies are we expecting? Is this algebra? Is this an equation? 2.2 + 2.2 = 4.4 15- 4.4 =10.6 10.6 / 2 =5.3

18. Algebra word problems in NAPLAN

19. Methods of solving word problems Do you have a preferred way of solving word problems? What do you consider when you are deciding how you will tackle a word problem? What makes you decide to use algebra to solve a word problem? Can you write a word problem that all your students use algebra to solve?

20. Solving algebra word problems Experts tend to solve algebra word problems using a fully algebraic method. They translate into algebra and use algebra to find the answer. Students commonly use a variety of informal solution strategies. They work with known numbers to find the answer.

21. Informal methods Trial and error, guess and test, or guess, check and improve, involve testing numbers in the problem. These methods involve working with the forwards operations. Logical reasoning methods involve first analysing the problem to identify forwards operations, then unwinding using backwards operations.

22. Informal methods work well When 3 is added to 5 times a certain number, the sum is 48. Find the number. Forwards : multiply by 5, add 3 Backwards: subtract 3, divide by 5

23. Focus on translation Four problems

24. Focus on translation Four problems (cont) (Stacey & MacGregor, 2000)

25. Informal methods have limitations Informal methods can be effective for simple word problems. More complex problems such as those with ‘tricky’ numbers as solutions and those involving equations with the unknown on both sides are not readily solved by informal methods.

26. The expert model

27. The expert model When 3 is added to 5 times a certain number, the sum is 48. Find the number. Comprehension - Read and understand problem Translation - Write as an algebraic equation 5 x +3 = 50 Solution - Manipulate equation to find x

28. Comprehension When 3 is added to 5 times a certain number, the sum is 48. Find the number. Comprehension - Read and understand problem Translation - Write as an algebraic equation 5 x +3 = 50 Solution - Manipulate equation to find x

29. Translation When 3 is added to 5 times a certain number, the sum is 48. Find the number. Comprehension - Read and understand problem Translation - Write as an algebraic equation 5 x +3 = 50 Solution - Manipulate equation to find x

30. Translation When 3 is added to 5 times a certain number, the sum is 48. Find the number. Comprehension - Read and understand problem Translation - Write as an algebraic equation5 x +3 = 48 Solution - Manipulate equation to find x

31. Solution When 3 is added to 5 times a certain number, the sum is 48. Find the number. Comprehension - Read and understand problem Translation - Write as an algebraic equation 5x+ 3= 48 Solution - Manipulate equation to find x x = 9

32. In the expert model “Equation solving is a sub-problem of story problem solving, and thus story problems will be harder to the extent that students have difficulty translating stories to equations” (Koedinger & Nathan, 1999, p. 8)

33. Few students use the expert model Even after a year or more of formal algebraic instruction, many students find word problems easier than algebraic problems (van Amerom, 2003)

34. Students use informal methods Many students rely on informal, non-algebraic methods even in problems where they are specifically encouraged to use algebraic methods (Stacey & MacGregor, 1999)

35. Difficulties with translation and solution Students who do try to follow the expert model may have difficulties at any of the three stages… BUT the major stumbling blocks for secondary students are the translation and solution phases. (Koedinger & Nathan, 2004)

36. Focus on translation Expert blind spot is the tendency to overestimate the ease of acquiring formal representations languages, and to underestimate students’ informal understandings and strategies (Koedinger & Nathan, 2004, p. 163)

37. Symbolic precedence view Secondary pre-service teachers prefer to use an algebraic method regardless of the nature of any given word problem. They tend to use formal methods regardless of the problem and view the algebraic method as “the one and only ‘truly mathematical’ solution method for such application problems” (Van Dooren, Verschaffel, & Onghena, 2002, p. 343)

38. Mismatch between approaches The mismatch between teachers’ and students’ approaches is reinforced by textbooks which commonly portray methods that do not align with typical students’ algebraic reasonings. Teachers need to critically view tasks and create or select activities and problems that are appropriate.

39. Teachers lack explicit strategies I am not even sure I know how I tackle word problems. I have never been taught how to go about problems myself.I just seem to know what to do, so when it comes to teaching kids, well, I don’t know what to say…

40. Key words Key words are something I do use… but I am not sure how well they work

41. Problems with the key word strategy Keyword focus tends to bypass understanding completely so when it doesn’t work students are at a total loss. Key words are only able to be identified in simple word problems. Key words can be misleading with more complex problems.

42. So what strategies are effective? Explicit expectations

43. The algebraic problem solving cycle

44. Effective strategies Explicit expectations the problem solving cycle Focus on translation from English to algebra (encoding) from algebra to English (decoding)

45. Focusing on translation both ways I liked how we learnt from both views - putting it into word problems and taking a word problem and putting it into algebraic. I understand it much better now.

46. Effective strategies Explicit expectations Focus on translation from English to algebra (encoding) from algebra to English (decoding) Create the ‘press for algebra’

47. Tasks encourage informal strategies Teachers commonly start with problems that are easy for students to do in their head in order to demonstrate the “rules of algebra”…. BUT Most students only see a need to use algebra when they are given problems that they cannot easily solve with informal methods.

48. A common problem A rectangle is 4 cm longer than it is wide. If its area is 21 cm2, what is the width of the rectangle? This one is not hard. You know that 21 is 7 times 3 so it’s got to be 3.

49. It’s obvious Once you see it, it’s obvious… Why would a student use algebra? But algebra is what I would always do first. At least now I know I will have to be so careful with the problems I use.

50. Effective strategies Explicit about expectations Focus on translation Create the ‘press for algebra’ problems with ‘tricky’ numbers problems that don’t ‘unwind’ Focus on the whole problem the complete problem solving cycle