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## Self-Regulated Learning and Proportional Reasoning:

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**Self-Regulated Learning and Proportional Reasoning:**Explorations into SRL in the Mathematics Classroom Charles Darr and Jonathan Fisher**Applying Self-Regulated Learning to Mathematics Instruction**Self-Regulated Learning is ... “… 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).**Forethought**Performancecontrol Self-reflection What is Self Regulated Learning? Theories on self-regulated learning (SRL) describe how students become: “ … masters of their own learning processes” (Zimmerman, 1998). According to Zimmerman, SRL involves cyclical processes of forethought, performance control and self-reflection**Becoming a Self-Regulated Learner**Observation 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. Emulation Self Control Self-Regulation**Self-Regulation and Problem Solving?**Expert Problem Solvers Fully regulated. Analyse Plan Explore Verify Naive Problem Solvers Haphazard Use Direct Translation Methods Problem of Inert (non- transferable) Knowledge**Inert Knowledge?**Knowledge that is in the student’s mind, but which can not be applied in new situations.**What does SRL Look Like in the Mathematics Classroom?**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: • 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.**Our Study**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: Appeal to students intuitive sense of proportionality Motivate them to engage in problem-solving behaviours Develop opportunities for classroom discourse that modelled and supported self-regulation.**What is Proportional Reasoning?**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?**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).**Partitioning**Unitizing Relative Thinking Proportional Reasoning Quantities and Change Rational Numbers Ratio Sense**Proportional Reasoning**M1 3 9 M2 12 36 x 3 x 4**Singer’s Experiment**Which box is more crowded?**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?**Not long ago $100 in $NZ was worth about $40 in $US. How**much would have $15 in $NZ been worth in $US? $NZ 100 15 $US 40 ? x 0.15 x 0.4**Context and Data Sources**• Context • 12 lessons in a Year 7 class • Mid-decile school • Class of 32 students • Data Sources • Pre and post interviews • Pre and post test • Written journal responses • Classroom video**Enhancing SRL**• We found two elements of Maths instruction that enhanced opportunities for students to practices or observe self-regulating behaviour. These were, the use of: • Rich representations (or models) of problem situations; • and ... • Reflective journalling.**Models of Proportional Problem Situations**Models allow students to develop rich representations of problem situations. They can involve concrete materials, graphic designs or abstract ideas. • We used ... • Cuisenaire rods • Geometric shapes • Cardboard strips and • Double-number lines.**Models of Proportional Problem Situations: The Double Number**line Using a double number line enables learners to represent proportional situations graphically.**%**Kg 100 40 10 4 15 6 5 0 2 0 What is 15% of 40 kg?**$NZ**$US 100 40 10 4 15 6 5 0 2 0 Not long ago $100 in $NZ was worth about $40 in $US. How much would have $15 in $NZ been worth in $US?**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**Models of Proportional Problem Situations: The Double Number**line**Models of Proportional Problem Situations: The Double Number**line When the double number line was established we observed: • Rich discourse • Students comparing methods • Students recognising patterns and strategies from analogous problems. • Students verifying answers. … all important components of SRL**Percentage**Cuisenaire 1**Reflective Journals:**Explanations**Reflective Journals:**Conversations**Self Regulated Learners in Mathematics**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.