Confirming the Overlapping Waves theory in children learning single-digit multiplication

Sanne van der VenUniversity of Amsterdam

Thanks to:

Dr. Jan Boom Dr. Evelyn Kroesbergen Prof. dr. Paul Leseman

How to measure how children learn mathematics?

Developing math knowledge

Mathematics is more than accuracy on a test

1. Children develop: how do they learn?

2. Performance is more than accuracy: how did a child reach the answer?

3. Why are some children faster learners than others?

Strategies

An example: strategies in addition

‘What is 4 + 5?’

• No understanding/guessing• Counting all• Counting on• Counting on from larger

(min procedure)• Decomposition• Retrieval

Strategies - Beware!

Development: Overlapping Waves

Siegler (1996)

Aims of the study

Aim 1• Test overlapping waves model statistically

Four steps:– Choose math ability– Measure this ability longitudinally– Identify and categorize strategies– Build and test statistical model

Aim 2• Explain individual differences in development

Mathematics in the Netherlands

• Social constructivism: ‘realistisch rekenen’ (realistic mathematics)

• Children construct their own knowledge• Focused on understanding: math talk based on real

world examples rather than drill and practice• Not evidence-based: Heavily debated!

Example page workbook (grade 2)

My study

Assignment: identify strategies

In groups of 3, devise a meaningful way to categorize children’s strategies for single digit multiplication.

Make sure you span the entire learning period from beginning to end, but also try to limit the number of categories!

Apply your categorization – does it work?• Categorize the verbal and visual examples• Adapt your categories if necessary.

• You have a small selection: in total there were 98 children * 8 weeks * 15 problems = 11,760 responses

My own solution

• Start broad, then narrow down

Initial coding scheme:single strategies• Don’t know• Guessing• Addition (8 x 6 = 14)• Repetition (7 x 4 = 7)• Other wrong strategies• Strategy unknown• Drawing and counting• Finger counting• Counting out loud (or

silently)• Drawing a number line

• Repeated addition• Repeated addition in smaller

steps• Repeated addition in larger

steps• Doubling• Using neighbours: 9x = 10x – x• Using neighbours: 6x = 5x + x• Using neighbours otherwise• Retrieval

Initial coding scheme:hybrid strategies• First repeated addition,

continue on fingers• First repeated addition,

then doubling• First repeated addition,

then counting out loud• Reverse and repeated

addition• Reverse and double• Reverse and retrieval

• Double, then repeated addition

• Using a neighbour, then counting

Then reduce the number of categories• Don’t know• Guessing• Addition (8 x 6 = 14)• Repetition (7 x 4 = 7)• Other wrong strategies• Strategy unknown• Drawing and counting• Finger counting• Counting out loud (or

silently)• Drawing a number line

• Repeated addition• Repeated addition in smaller

steps• Repeated addition in larger

steps• Doubling• Using neighbours: 9x = 10x – x• Using neighbours: 6x = 5x + x• Using neighbours otherwise• Retrieval

Wrong

Counting

Repeated Addition

Derived Facts

Retrieval

Results - Descriptives

Retrieval

Derived Facts

Repeated Addition

Counting

Wrong

week 1 week 2 week 3 week 4 week 5 week 6 week 7 week 80

2

4

6

8

10

12

14

Number Correct

Results - Descriptives

week 1 week 2 week 3 week 4 week 5 week 6 week 7 week 80

2

4

6

8

10

12

14

Number Correct

strongmiddleweak

1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

Easy7 x 2 and 5 x 3

1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

Intermediate I7 x 3

1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

Hard8 x 6 and 6 x 9

1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

Intermediate II3 x 4 and 6 x 5

Retrieval

Derived Facts

Repeated Addition

Counting

Wrong

So, how to model?

• Combination of two techniques in one model:

– IRT (graded response model) creates continuous

variable (latent trait)

– Latent growth curve modeling growth of this latent trait

Graded response model: assumptions• One strategy used at a time

• Strategies are ordered

• Underlying dimension (“mathematical maturity”)– Non-linearly related to

strategy use

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1.0

4 x 6

Categorical Growth

χ2(2151) = 2937.42,p < .001, NC = 1.37, CFI = .90, RMSEA = .06

Retrieval

Derived Facts

Repeated Addition

Counting

Wrong

Intercept:- M = 0- sd = 1.02

Slope:- M = 0.97- sd = 0.90

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1.09 x 4

(plain)

Strategy ability

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1.04 x 6

(context)

Strategy ability

Contextual and plain problems

Retrieval

Derived Facts

Repeated Addition

Counting

Wrong

Easy and difficult problems

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1.08 x 5

(plain)

Strategy ability

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1.09 x 4

(plain)

Strategy abilityRetrieval

Derived Facts

Repeated Addition

Counting

Wrong

Accuracy

χ2(1998) = 2071.75, p = .12, NC = 1.04, CFI = .96, RMSEA = .02

Intercept:- M = 0- sd = 0.44

Slope:- M = 0.28- sd = 0.45

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1.0

4 x 49 x 36 x 8

Interrelations

Differences in development

• Working memory– relation has been shown in many studies

• Any ideas why?

Connectionist theory

Hypotheses

working memory

accuracy

strategy choice

Working memory tasks

• Digit Span Backwards

• Odd One Out

• Keep Track

3 1 6 5 5 6 1 3

?

Questions for future research

• How general is the model?– Different math abilities– Different ages– Different countries

• Are there children that deviate from the model, and why?• Why was there no relationship between working memory

and the two slopes (development)?– Measure earlier during development?

• Should we promote smarter strategies, better execution, both, neither?– Perhaps tailor to working memory profiles?

Questions? Retrieval

Derived Facts

Repeated Addition

Counting

Wrong