Solving Equations: Students’ Algebraic Thinking

1. Solving Equations: Students’ Algebraic Thinking Chris Linsell Supported by TLRI

2. Outline • Introduction • Conceptual Difficulties • The PhD Study • The TLRI project • Some early conclusions • Where to next?

3. Introduction • Differences in perception between maths teachers and students • One. ..fact must astonish us, or rather would astonish us if we were not too much accustomed to it. How does it happen that there are people who do not understand mathematics? If the science invokes only the rules of logic, those accepted by all well-formed minds. ..how does it happen that there are so many people who are entirely impervious to it? (Poincare, 1908)

4. Hewitt 2001 • A girl in the class was looking at the following: • Example l: x+3=9, x=6 • Example 2: x+7=10, x=3 • I asked her whether she understood what was written on the board and she replied “I don't understand, I just copy it down.” • I then asked “What is the 'ex' on the board?” • To which she replied “What 'ex'? That's a times.”

5. Algebraic Thinking? • Is it appropriate to talk about algebraic thinking? • We no longer talk about simply arithmetic thinking, but specify the strategies that students use in a number of domains. • Suggested domains of algebra - Generalised number • Patterns and relationships • Specific unknowns

6. Specific Unknowns • How do students solve equations? • What are the prerequisite skills and knowledge? • How can we find out? • Observational studies • Diagnostic interviews

7. Some Conceptual Difficulties • Arithmetic structure • Inverse operations • Notation and convention • Unknown / generalised number / variable / parameter • Operating on unknowns • Binary / unary operations • Integers

8. Process / object duality and procepts • Counting→natural→subtracting→negative→roots→imaginary and complex • Adding→sums • Evaluating expressions→equations • Procepts, pivotal symbolism

9. The = sign • 3+5= • compute now • is equivalent to • Process/object again • 3+5=8+2=10 • lack of appreciation of the structure of mathematical statements • 3x+154=475 and 3x=475+154

10. Numeracy • x+3=7 can be solved by advanced counters by guess and check, but can be solved much more easily by part/whole thinkers able to visualise 7 as 3+4 • 2x+3=11 requires an understanding of numbers beyond simple additive part/whole or multiplicative part/whole thinking • 2(x+3)+2x=22 requires an understanding of the use of brackets, the hierarchy of operations, and an understanding of the commutative, associative and distributive laws

11. The PhD Study • Qualitative • 4 Year 9 students studied for 27 lessons • Activity based approach • Data – videotapes of lessons, SRIs, student’s work, field notes • Transcription and analysis

12. Process/object and procedure • Superficial similarity of process and procedure • Analogy of two-digit subtraction • Getting the right answer • Clues from errors • Clues from notation • Learning – process 2/4 object 0/4

13. Prerequisite Numeracy • Numeracy as a predictor of learning • Arithmetic structure • Inverse operations • Operating on integers • Binary/ unary view

14. Strategic Thinking • Multiplicative part / whole • Expressions as objects • Value of guess and check • Constraints on choice of parts • Unknown parts • Algebraic part / whole thinking • Equations as objects • Time spent on stages

15. The TLRI Project • How do students solve equations? • What diagnostic questioning is appropriate for eliciting the knowledge and strategies used by students? • Action Research Paradigm

16. Structure of Diagnostic Assessment • Separated into knowledge and strategy components • Strategy assessed by interview, need for supplementary questions • Knowledge assessed by written test, guided by literature and our own experience

17. Strategy • A focus on HOW students solve equations • A series of increasingly complex questions, students solve and explain strategies • Teachers classify the strategies used by students • Parallel questions – in context and abstract

18. Strategy – Sample Questions • 3n – 8 = 19 • Here are 3 packets of beans, that started with the same number in each packet, but a mouse ate 5 of the beans. There are now 16 beans left in the packets, like this pile over here. How many were in each packet at the beginning?

19. Strategy Classification • 0.         Unable to answer question • 1a.       Known basic facts • 1b.       Counting techniques • 1c.        Inverse operation • 2.         Guess and check • 3a.       Cover up • 3b.       Working backwards • 3c. Working backwards then known facts • 3d. Working backwards then guess and check • 4.         Formal operations / Equation as Object • 5. Use a Diagram

20. Knowledge – Areas Investigated • Using symbols and letters to represent an unknown • Manipulating symbols/unknowns (lack of closure) • Forming expressions with unknowns/symbols in them • Understanding of the equals sign • Operations on integers • Understanding of arithmetic structure • Understanding of inverse operations

21. Knowledge – Sample Questions • What answer do you get for each of the following? • 5 + 6 × 10 • 8 × ( 7 – 5 + 3 ) • 6+12 3 • 18 -12 ÷ 6

22. Applying the Strategy Analysis • Watch the videos, try to classify the kind of strategies used by the students in each case. • In groups discuss the strategies used. Although we do not have a hierarchy yet, what would be the next teaching step for these students?

23. 3) n+46 = 113 Andrew(n=67) Video removed

24. 3) n+46 = 113 Jack(n=67) Video removed

25. 3) n+46 = 113 Oliver(n=67) Video removed

26. 3) n+46 = 113 Oscar(n=67) Video removed

27. 3) n+46 = 113 Sean(n=67) Video removed

28. 5) 4n+9=37 Andrew(n=7) Video removed

29. 6) 3n-8=19 Andrew(n=9) Video removed

30. 5) 4n+9=37 Nicola(n=7) <20 seconds to answer 7 How did you work that out? I timesed 4 by 7 and added 9 Did you guess seven? I took a close estimate So you tried seven and it was right? Yes

31. 6) 3n-8=19 Nicola(n=9) Takes longer than previous question, but still less than <30 sec 9 How did you work that out? I just took a close estimate. Was 9 the first number you tried? No I tried 8. I went 24 – 8 So you went 3x8 is 24 minus 8 is … Not 19! So then you tried 9? And 9 was right

32. 5) 4n+9=37 Susan(n=7) Approx 40 sec to answer 7 How did you work that out? Because 9 minus 37 is 28 and 28 divided by 4 is 7

33. 6) 3n-8=19 Susan(n=9) After 4 minutes of trying prompted and replies: Something like 3 times 4 remainder 1 or something. Because 3x4 is 12 but 8-19 equals 11

34. Some early conclusions • Clear parallels with NDP approach • Numeracy teaching is having an impact • Instead of looking at how hard equations are to solve and whether students get them right, it is more useful to look at what strategies the students use • The strategies we have identified are consistent with the research literature but extend it

35. More early conclusions • Contexts are a help, not a hindrance! • At some point contexts stop being helpful • “Working backwards” is less homogeneous than has been previously suggested • Formal operations, which treat equations as objects, are clearly difficult for students

36. What next? • Guess and Check and Cover Up are worth teaching! • We want to develop an Algebra Framework similar to the Number Framework • The next step is to gather lots of data, using the diagnostic interview we have developed

37. 2007 • Find interested teachers • Teachers must have numeracy data on their students • Train teachers to administer diagnostic tool • Teachers and research team to gather data on 800 – 1000 students • Enter results into database

38. 2008 – Data Analysis • Factor Analysis, Rasch Analysis • What is relative difficulty of test items? • What is relative difficulty of strategies? • What prerequisite knowledge is required for each strategy? • What stage of numeracy is required for each strategy? • What is the impact of context?