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

Confirming the Overlapping Waves theory in children learning single-digit multiplication. Sanne van der Ven University of Amsterdam. Thanks to : Dr. Jan Boom Dr. Evelyn Kroesbergen Prof. dr. Paul Leseman. How to measure how children learn mathematics ?. Developing math knowledge.

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

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  1. Confirming the Overlapping Waves theory in children learning single-digit multiplication Sanne van der Ven University of Amsterdam Thanksto: Dr. Jan Boom Dr. Evelyn Kroesbergen Prof. dr. Paul Leseman

  2. How tomeasurehowchildrenlearnmathematics?

  3. Developingmathknowledge Mathematics is more thanaccuracy on a test • Childrendevelop: how do theylearn? • Performance is more thanaccuracy:howdid a childreach the answer? • Why are somechildrenfasterlearnersthanothers?

  4. Strategies

  5. An example: strategies in addition ‘What is 4 + 5?’ • No understanding/guessing • Countingall • Counting on • Counting on fromlarger (min procedure) • Decomposition • Retrieval

  6. Strategies - Beware!

  7. Development: Overlapping Waves Siegler (1996)

  8. Aims of the study Aim 1 • Test overlapping waves model statistically Four steps: • Choosemathability • Measurethisabilitylongitudinally • Identifyandcategorizestrategies • Buildand test statistical model Aim 2 • Explainindividualdifferences in development

  9. Mathematics in the Netherlands • Social constructivism: ‘realistischrekenen’ (realistic mathematics) • Children construct theirownknowledge • Focused on understanding: math talk based on real world examples rather than drill and practice • Not evidence-based: Heavily debated!

  10. Example page workbook (grade 2)

  11. My study

  12. Assignment: identifystrategies In groups of 3, devise a meaningful way tocategorizechildren’sstrategiesfor single digit multiplication. Make sureyou span the entirelearningperiodfrombeginningto end, but alsotryto limit the number of categories!

  13. Apply your categorization – does it work? • Categorize the verbalandvisualexamples • Adaptyourcategoriesifnecessary. • You have a small selection: in totaltherewere 98 children * 8 weeks * 15 problems = 11,760 responses

  14. My own solution • Start broad, thennarrow down

  15. Initialcodingscheme:single strategies • Don’tknow • Guessing • Addition (8 x 6 = 14) • Repetition (7 x 4 = 7) • Other wrong strategies • Strategyunknown • Drawingandcounting • Fingercounting • Counting out loud (or silently) • Drawing a number line • Repeatedaddition • Repeatedaddition in smaller steps • Repeatedaddition in larger steps • Doubling • Using neighbours: 9x = 10x – x • Using neighbours: 6x = 5x + x • Using neighboursotherwise • Retrieval

  16. Initialcodingscheme:hybridstrategies • First repeatedaddition, continue on fingers • First repeatedaddition, thendoubling • First repeatedaddition, thencounting out loud • Reverse andrepeatedaddition • Reverse and double • Reverse and retrieval • Double, thenrepeatedaddition • Using a neighbour, thencounting

  17. Thenreduce the number of categories • Don’tknow • Guessing • Addition (8 x 6 = 14) • Repetition (7 x 4 = 7) • Other wrong strategies • Strategyunknown • Drawingandcounting • Fingercounting • Counting out loud (or silently) • Drawing a number line • Repeatedaddition • Repeatedaddition in smaller steps • Repeatedaddition in larger steps • Doubling • Using neighbours: 9x = 10x – x • Using neighbours: 6x = 5x + x • Using neighboursotherwise • Retrieval RepeatedAddition Wrong DerivedFacts Counting Retrieval

  18. Results - Descriptives Retrieval Derived Facts Repeated Addition Counting Wrong

  19. Results - Descriptives

  20. Retrieval Derived Facts Repeated Addition Counting Wrong

  21. So, howto model? • Combination of twotechniques in one model: • IRT (graded response model)  createscontinuousvariable (latent trait) • Latent growth curve modeling growth of this latent trait

  22. Graded response model: assumptions • Onestrategyused at a time • Strategies are ordered • Underlyingdimension (“mathematicalmaturity”) • Non-linearlyrelatedtostrategyuse

  23. Categorical Growth Intercept: - M = 0 - sd = 1.02 Slope: - M = 0.97 - sd = 0.90 Retrieval Derived Facts Repeated Addition Counting Wrong χ2(2151) = 2937.42, p < .001, NC = 1.37, CFI = .90, RMSEA = .06

  24. Contextualandplainproblems Retrieval Derived Facts Repeated Addition Counting Wrong

  25. Easy anddifficultproblems Retrieval Derived Facts Repeated Addition Counting Wrong

  26. Accuracy Intercept: - M = 0 - sd = 0.44 Slope: - M = 0.28 - sd = 0.45 χ2(1998) = 2071.75, p = .12, NC = 1.04, CFI = .96, RMSEA = .02

  27. Interrelations

  28. Differences in development • Working memory – relation has been shown in many studies • Anyideaswhy?

  29. Connectionisttheory

  30. strategy choice working memory accuracy Hypotheses

  31. Working memory tasks • Digit Span Backwards • Odd One Out • Keep Track 3 1 6 5  5 6 1 3 ?

  32. Questionsforfuture research • How general is the model? • Different mathabilities • Different ages • Different countries • Are therechildrenthatdeviatefrom the model, andwhy? • Why was there no relationshipbetweenworking memory and the twoslopes (development)? • Measureearlierduringdevelopment? • Should we promotesmarterstrategies, betterexecution, both, neither? • Perhapstailortoworking memory profiles?

  33. Questions? Retrieval Derived Facts Repeated Addition Counting Wrong

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