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Self-Regulated Learning and Proportional Reasoning:
Charles Darr and Jonathan Fisher
Explorations into SRL in the Mathematics Classroom
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 ...
What is Self Regulated Learning?Theories on self-regulated learning (SRL) describe how students become: “ … masters of their own learning processes” (Zimmerman, 1998).
ForethoughtPerformance control
Self-reflection
According to Zimmerman, SRL involves cyclical processes of forethought, performance control and self-reflection
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
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?
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?
• 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:
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
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?
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).
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?
$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?
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
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
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.
Using a double number line enables learners to represent proportional situations graphically.
Models of Proportional Problem Situations:
The Double Number line
100 40
10 45 2
15 6
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
0 0
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
You have been employed by the milk factory to help them redesign their container.
They would like a scale on the side to show how much milk is left. They want you to design the scale and then test how ac-curate it is.
1. Design a scale using a double number line that shows both the percentage left and the amount in millilitres. It should increase in 10% amounts.
2. Test how accurately your scale meas-ures the following amounts: 400 ml, and 700 ml. Record your results in a table like the one below.
% of 2000 ml Our Scale (ml) Actual Measure-ment (ml)
Difference (ml)
200
700
Models of Proportional Problem Situations: The Double Number line
• 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
If the big shape is 100%, what percentage is the triangle?
If the parallelogram is 100%, what percentage is the triangle?
If the shape on the left is 100%, what percentage is the triangle?
If the big shape is 100% what percentage is the ?
If the shape on the left is 100%, what percentage is the ?
If the shape on the left is 100%, what per-centage is the ?
?
What am I ?
1. I am 60% as long as the orange rod. 2. I am 50% as long as the pink rod. 3. I am 50% shorter than the dark green rod. 4. I am 10% shorter than the orange rod. 5. I am 25% shorter than the brown rod. 6. I am 20% longer than the yellow rod. 7. I am 25% shorter than the pink rod. 8. The black rod is 75% longer than I am. 9. The red rod is 100% longer than I am. 10. The blue rod is 120% longer than I am.
1. I f pink represents 100%, what do the f ollowing colours represent?
Light Green Red White 2. I f yellow represents 100%, what do the f ollowing colours represent?
Light Green Pink Red White 3. I f white represents 25%, what do the f ollowing colours represent?
Pink Light Green Red White 4. I f pink represents 50%, what do the f ollowing colours
represent?
Brown Light Green Dark Green Yellow Red White 5. I f dark green is 60%, what do all the other colours represent?
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