1. Self-Regulated Learning and Proportional Reasoning: Charles Darr and Jonathan Fisher Explorations into SRL in the Mathematics Classroom

2. Applying Self-Regulated Learning to Mathematics Instruction “… a major objective of mathematics education, on the one hand, and … a crucial characteristic of effective mathematics learning on the other” (De Corte et al, 2000). Self-Regulated Learning is...

3. What is Self Regulated Learning? Theories on self-regulated learning (SRL) describe how students become: “ … masters of their own learning processes” (Zimmerman, 1998). Forethought Performance control Self-reflection According to Zimmerman, SRL involves cyclical processes of forethought, performance control and self-reflection

4. Students get there by passing through stages of observation, emulation, self- control and self-regulation. In Mathematics Education, SRL is particularly relevant to problem solving. Becoming a Self-Regulated Learner Observation Emulation Self Control Self- Regulation

5. Expert Problem Solvers Fully regulated. Analyse Plan Explore Verify Naive Problem Solvers Haphazard Use Direct Translation Methods Problem of Inert (non- transferable) Knowledge Self-Regulation and Problem Solving?

6. Inert Knowledge? Knowledge that is in the student’s mind, but which can not be applied in new situations.

7. What does SRL Look Like in the Mathematics Classroom? realistic and challenging tasks; variation in teaching methods including teacher modelling, guided practice, small group work and whole class instruction; classrooms that foster positive dispositions towards learning mathematics. In a review of research into SRL in mathematics, De Corte et al (2000, p.196), list three components of instruction that appear to foster self-regulation:

8. Our study explored how components of SRL might be integrated into classroom teaching and learning in the area of proportional reasoning. Taking a lead from Moss and Case (1999) we designed a series of interactive lessons that began with instruction on percentages. We hoped to: Our Study Appeal to students intuitive sense of proportionality Develop opportunities for classroom discourse that modelled and supported self-regulation. Motivate them to engage in problem-solving behaviours

9. According to Piaget it is: …. a capability which ushers in a significant conceptual shift from concrete operational levels of thought to formal operational levels of thought (Piaget & Beth, 1966). What is Proportional Reasoning?

10. Proportional Reasoning is in essence a process of comparing one relative amount with another (Sophian and Wood, 1997, p.309). When two quantities vary in such a way that one of them is a constant multiple of the other, the two quantities are proportional (Stanley et al, 2003, p.2).

11. Unitizing Rational Numbers Quantities and Change Ratio Sense Relative Thinking Partitioning

12. M1 3 9 M2 12 36 Proportional Reasoning x 3 x 4

13. Singer’s Experiment Which box is more crowded?

14. Percentages as a Site for Proportional Reasoning What is 15% of 40? Not long ago $100 in $NZ was worth about $40 in $US. How much would have $15 in $NZ been worth in $US? When my scale is 1:100 the length is 15. How long will it be when the scale is 1:40? A stack of 40 books is 100 cm high, how high will a stack of 15 books be? If I can buy 40 ice-blocks for $100, how many can I buy for $15?

15. $NZ 100 15 $US 40 ? x 0.15 x 0.4 Not long ago $100 in $NZ was worth about $40 in $US. How much would have $15 in $NZ been worth in $US?

16. Data Sources Pre and post interviews Pre and post test Written journal responses Classroom video Context and Data Sources Context 12 lessons in a Year 7 class Mid-decile school Class of 32 students

17. We found two elements of Maths instruction that enhanced opportunities for students to practices or observe self- regulating behaviour. These were, the use of: Rich representations (or models) of problem situations; and... Reflective journalling. Enhancing SRL

18. We used... Cuisenaire rods Geometric shapes Cardboard strips and Double-number lines. Models of Proportional Problem Situations Models allow students to develop rich representations of problem situations. They can involve concrete materials, graphic designs or abstract ideas.

19. Using a double number line enables learners to represent proportional situations graphically. Models of Proportional Problem Situations: The Double Number line

20. 100 40 104 52 156 What is 15% of 40 kg? % Kg 00

21. 100 40 104 52 156 Not long ago $100 in $NZ was worth about $40 in $US. How much would have $15 in $NZ been worth in $US? $NZ $US 00

22. Models of Proportional Problem Situations: The Double Number line The double number line was introduced through a series of ‘concrete’ activities centred on 2-litre milk containers. For example: Drawing/creating scales showing % and capacity Identifying faulty scales Verifying scales Estimating how full a number of bottles were

23. Student Explanations

24. Models of Proportional Problem Situations: The Double Number line

25. Rich discourse Students comparing methods Students recognising patterns and strategies from analogous problems. Students verifying answers. Models of Proportional Problem Situations: The Double Number line When the double number line was established we observed: … all important components of SRL

30. Reflective Journals: Explanations

31. Reflective Journals: Conversations

32. If students in mathematics are going to become self-regulated learners, they need to be confronted with opportunities that allow them to reveal their thinking and to observe and emulate the thinking of others. Self Regulated Learners in Mathematics

33. Kaleidoscope of Experiences