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).

Unitizing

Rational Numbers

Quantities and Change

Ratio Sense

Relative Thinking

Partitioning

M1

3

9

M2

12

36

Proportional Reasoning

x 3

x 4

Singer’s ExperimentWhich 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?

$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

What is 15% of 40 kg?

% Kg

0 0

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

Student Explanations

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?

Josef says: “I have about 40% left in my glass.” What do you think?

Reflective Journals: Explanations

Reflective Journals: Conversations

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

Kaleidoscope of Experiences