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Modifying arithmetic practice to promote understanding of mathematical equivalence

Modifying arithmetic practice to promote understanding of mathematical equivalence. Nicole M. McNeil University of Notre Dame. Seemingly straightforward math problem. 3 + 5 = 4 + __ 3 + 5 = __ + 2 3 + 5 + 6 = 3 + __. Mathematical equivalence problems. Why we care about these problems.

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Modifying arithmetic practice to promote understanding of mathematical equivalence

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  1. Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame

  2. Seemingly straightforward math problem 3 + 5 = 4 + __ 3 + 5 = __ + 2 3 + 5 + 6 = 3 + __ Mathematical equivalence problems

  3. Why we care about these problems • Theoretical reasons • Good tools for testing general hypotheses about the nature of cognitive development • E.g., transitional knowledge states, self-explanation, etc. • Practical reasons • Mathematical equivalence is a fundamental concept in algebra • Algebra has been identified as a “gatekeeper”

  4. Most children in U.S. do not solve them correctly 16% % of children who solved problems correctly Study

  5. Why don’t children solve them correctly? • Some theories focus on what children lack • Domain-general logical structures • Mature working memory system • Proficiency with “basic” arithmetic facts • Other theories focus on what children have • Mental set, strong representation, deep attractor state, entrenched knowledge, etc. • Knowledge constructed from early school experience w/ arithmetic operations

  6. But isn’t arithmetic a building block? • Knowledge of arithmetic should help, right? • Children’s experience is too narrow • Procedures stressed w/ no reference to = • Limited range of math problem instances • Children learn the regularities • Domain-general statistical learning mechanisms that pick up on consistent patterns in the environment 12 + 8 2 + 2 = __

  7. Overly narrow patterns • Perceptual pattern • “Operations on left side” problem format • Concept of equal sign • An operator (like + or -) that means “calculate the total” • Strategy • Perform all given operations on all given numbers 3 + 4 + 5 = __

  8. Overly narrow patterns • Perceptual pattern • “Operations on left side” problem format • Concept of equal sign • An operator (like + or -) that means “calculate the total” • Strategy • Perform all given operations on all given numbers

  9. Overly narrow patterns • Perceptual pattern • “Operations on left side” problem format • Concept of equal sign • An operator (like + or -) that means “calculate the total” • Strategy • Perform all given operations on all given numbers 3 + 4 = 5 + __

  10. “Operations on left side” problem format

  11. “Operations on left side” problem format

  12. “Operations on left side” problem format

  13. Equal sign as operator Child participant video will be shown

  14. Add all the numbers Child participant video will be shown

  15. Recap 12 + 8 2 + 2 = __ 12 + 8 2 + 2 = __ 3 + 4 + 5 = 3 + __

  16. Recap 12 + 8 2 + 2 = __ Internalize narrow patterns 12 + 8 2 + 2 = __

  17. Recap 12 + 8 ops go on left side 2 + 2 = __ = means “get the total” add all the numbers Internalize narrow patterns 12 + 8 2 + 2 = __ 2 + 7 = 6 + __

  18. The account makes specific predictions • Performance should decline between ages 7 and 9 • Traditional practice with arithmetic hinders performance • Modified arithmetic practice will help

  19. The account makes specific predictions • Performance should decline between ages 7 and 9 • Traditional practice with arithmetic hinders performance • Modified arithmetic practice will help

  20. Performance should get worse from 7 to 9 • Why? • Continue gaining narrow practice w/ arithmetic • Strengthening representations that hinder performance • But… • Constructing increasingly sophisticated logical structures • General improvements in working memory • Proficiency with basic arithmetic facts increases

  21. Performance as a function of age Percentage of children who solved correctly Age (years;months)

  22. The account makes specific predictions • Performance should decline between ages 7 and 9 • Traditional practice with arithmetic hinders performance • Modified arithmetic practice will help

  23. The account makes specific predictions • Performance should decline between ages 7 and 9 • Traditional practice with arithmetic hinders performance • Modified arithmetic practice will help

  24. Traditional practice with arithmetic should hurt • Why? • Activates representations of operational patterns • But… • Decomposition Thesis • “Back to basics” movement • Practice should “free up” cognitive resources for higher-order problem solving

  25. Ready Solve Set 3 + 4 + 5 = 3 + __

  26. Performance by practice condition Percentage of undergrads who solved correctly Practice condition

  27. The account makes specific predictions • Performance should decline between ages 7 and 9 • Traditional practice with arithmetic hinders performance • Modified arithmetic practice will help

  28. The account makes specific predictions • Performance should decline between ages 7 and 9 • Traditional practice with arithmetic hinders performance • Modified arithmetic practice will help

  29. Performance by elementary math country Percentage of undergrads who solved correctly Elementary math country

  30. Interview data • Experience in the United States • Experience in high-achieving countries

  31. Effect of problem format • Participants • 7- and 8-year-old children (M age = 8 yrs, 0 mos; N = 90) • Design • Posttest-only randomized experiment (plus follow up) • Basic procedure • Practice arithmetic in one-on-one sessions with “tutor” • Complete assessments (math equivalence and computation)

  32. Smack it (traditional format) 9 + 4 = __ 7 + 8 = __ 2 + 2 = __ 4 + 3 = __

  33. Smack it (traditional format) 9 + 4 = __ 7 + 8 = __ 7 2 + 2 = __ 4 + 3 = __

  34. Smack it (nontraditional format) __ = 9 + 4 __ = 7 + 8 7 __ = 2 + 2 __ = 4 + 3

  35. Snakey Math (traditional format)

  36. Snakey Math (nontraditional format)

  37. Assessments • Understanding of mathematical equivalence • Reconstruct math equivalence problems after viewing (5 sec) • Define the equal sign • Solve and explain math equivalence problems • Computational fluency • Math computation section of ITBS • Single-digit addition facts (reaction time and strategy) • Follow up • Solve and explain math equivalence problems (with tutelage)

  38. Summary of sessions homework homework homework homework

  39. Understanding of math equivalence by condition Arithmetic practice condition

  40. Follow-up performance by condition Arithmetic practice condition

  41. Computational fluency by condition

  42. Computational fluency by condition

  43. Interview data • Experience in the United States • Experience in high-achieving countries

  44. Effect of problem grouping/sequence • Participants • 7- and 8-year-old children (N = 104) • Design • Posttest-only randomized experiment (plus follow up) • Basic procedure • Practice arithmetic in one-on-one sessions with “tutor” • Complete assessments (math equivalence and computation)

  45. Traditional grouping 4 + 6 = __ 4 + 5 = __ 4 + 4 = __ 4 + 3 = __ In this example: 4 + n

  46. Nontraditional grouping 6 + 4 = __ 5 + 5 = __ 4 + 6 = __ 3 + 7 = __ In this example: sum is equal to 10

  47. Understanding of math equivalence by condition Arithmetic practice condition

  48. Follow-up performance by condition Arithmetic practice condition

  49. Computational fluency by condition

  50. Computational fluency by condition

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