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Self-Regulated Learning and Proportional Reasoning: PowerPoint PPT Presentation


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 .

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Self-Regulated Learning and Proportional Reasoning:

Explorations into SRL in the Mathematics Classroom

Charles Darr and Jonathan Fisher


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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).


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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


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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


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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


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Inert Knowledge?

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


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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.


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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.


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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).


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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).


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Partitioning

Unitizing

Relative Thinking

Proportional Reasoning

Quantities and Change

Rational Numbers

Ratio Sense


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Proportional Reasoning

M1

3

9

M2

12

36

x 3

x 4


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Singer’s Experiment

Which box is more crowded?


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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?


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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


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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


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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.


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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.


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Models of Proportional Problem Situations: The Double Number line

Using a double number line enables learners to represent proportional situations graphically.


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%

Kg

100

40

10

4

15

6

5

0

2

0

What is 15% of 40 kg?


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$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?


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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


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Student Explanations


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Models of Proportional Problem Situations: The Double Number line


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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


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Percentage

Cuisenaire 1


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Reflective Journals:

Explanations


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Reflective Journals:

Conversations


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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.


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Kaleidoscope of Experiences


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