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