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Travelling the road to expertise: A longitudinal study of learning

Travelling the road to expertise: A longitudinal study of learning. Kaye Stacey University of Melbourne Australia. A journey. 2.71828 0.6 0.3 repeating 4,08. with 3204 students. as they learn to understand decimal notation. over seven years (Grades 4 – 10). Grades 4 – 6 “primary”;

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Travelling the road to expertise: A longitudinal study of learning

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  1. Travelling the road to expertise: A longitudinal study of learning Kaye Stacey University of Melbourne Australia

  2. A journey 2.71828 0.6 0.3repeating 4,08 with 3204 students as they learn to understand decimal notation over seven years (Grades 4 – 10) Grades 4 – 6 “primary”; Grades 7 – 10 “secondary”

  3. Understanding how students think about decimals Tracing students’ progress in the longitudinal study Looking at teaching interventions Creating computer games using intelligent tutoring and AI (Bayesian nets) Developing CD and website for teachers Thanks to Vicki Steinle Liz Sonenberg Ann Nicholson Tali Boneh Sue Helme Nick Scott Australian Research Council many U of M honours students Dianne Chambers teachers and children providing data Our decimals work in summary

  4. Why decimals? * place value, fractions, density of real numbers etc • Practical Importance • Is my blood alcohol over 0.05% or not? • Is my p value over 0.05? • Links to metric measuring • Fundamental role of number in mathematics (e.g. understanding 0) • Known to be complex* with poor learning • A case study of students’ growth of understanding, which was able to start from a good research base (incl M.Swan)

  5. In preparation for the journey: • Who? 3204 students(12 schools, all SES, volunteer teachers) • Transport: ordinary teaching • Territory and map – see later • The destination – “understanding decimals”

  6. The destination: understanding decimal notation Why is such a simple rule as rounding hard to remember? A convention carrying distributed intelligence 27.483 3.145

  7. Sample • Cohort study • aimed to follow as many students as possible for as long as possible • 1079 students first tested in primary – nearly 60% followed to secondary school • over 600 students completed 5,6 or 7 tests (i.e. followed into a third or fourth year) • Quantitative analysis of longitudinal data conducted by Dr Vicki Steinle (PhD thesis)

  8. Sample data • Two tests per year – one per “semester” • 9862 tests completed • Students tracked for up to 7 semesters • Tests average 8.3 months apart. Absentees not chased.

  9. Interconnections between the map and the mapping tool “It is only by asking the right, probing questions that we discover deep misconceptions, and only by knowing which misconceptions are likely do we know which questions are worth asking”, (Swan, 1983, p65). The longitudinal study uses one type of question: which of two decimals is larger? e.g. 0. 8 or 0.75 (CSMS item, 1981) 4.8 or 4.63 (Resnick et al, 1989)

  10. Similar items – different success • Select the largest number from0.625, 0.5, 0.375, 0.25, 0.125Correct: 61% • Select the smallest number from0.625, 0.5, 0.375, 0.25, 0.125Correct: 37% • Why such a large difference? Foxman et al (1985) Results of large scale “APU” monitoring UK. All sets given here as largest to smallest; not as presented.

  11. Common patterns in answers 0.625 0.5 0.375 0.25 0.125 Largest Smallest • 0.625 0.125 correct • 0.625 0.5 well known error “longer-is-larger” • 0.5 0.625 identified 1980s “shorter-is-larger”

  12. Persistent patterns Select the smallest number from 0.625, 0.5, 0.375, 0.25, 0.125 predicted from other responses by our sample Our sample is typical

  13. Early explorers expert Map prior to 1980 4.8 4.63 Longer-is-larger Comparison used by Brueckner 1928

  14. Shorter-is-larger Early explorers expert 5.736 5.62 Map by 1990 4.8 4.63 Use patterns of responses to sets of comparison items – Sackur, Resnick, Peled and others Longer-is-larger

  15. Shorter-is-larger (S) expert (A1) Map for this longitudinal study other A U Longer-is-larger (L)

  16. Decimal Comparison Test (DCT2) We now have better version set of similar items set of similar items

  17. Decimal Comparison Test (DCT2) We now have better version set of similar items set of similar items

  18. Complex test, easy to complete • Several items of each type • Codes require consistent responses • Items within types VERY carefully matched

  19. Longer-is-larger additional information – zero makes small 4.71 4.082 whole number analogy 4.8, 4.63 “fat columns” 4.71 has 71 tenths, 4.082 has 82 hundredths

  20. everything wrong – quite smart! “unclassified”: we don’t know

  21. 0.6 0 0.0 0.00 expert on comparison task Only known to be OK on easy items on DCT2 – maybe not OK on harder items 4.77777 vs 4.7 truncation – no meaning for later dec pl; failed algorithms – e.g. comparison of space and 0 4.4502 4.45

  22. Shorter–is-larger some place value considerations – all thousandths smaller than all hundredths e.g. 5.62 greater than 5.736 analogies with fractions or negatives (e.g. 0.4 < 0.3) Distinguish behaviour and way of thinking

  23. Can subdivide these with improved DCT Shorter–is-larger some place value considerations – all thousandths smaller than all hundredths e.g. 5.62 greater than 5.736 analogies with fractions or negatives (e.g. 0.4 < 0.3) Distinguish behaviour and way of thinking

  24. Courtney - a “text book” case derived from research interviews – not too obvious for our student teachers Hidden Numbers Making the biggest and smallest numbers Number Between Example of S thinking

  25. S behaviour: “through the looking glass” • 5.736 < 5.62 (because larger whole number, so reverse) • The mirror is a powerful metaphor underpinning some everyday and mathematical concepts (Lakoff & Johnson) • Fractions (and hence decimals) and negative numbers as “mirror images” of whole numbers, so everything is reversed (Stacey et al, PME, 2001) • negative is an additive inverse • reciprocal is a multiplicative inverse • Can even lead to getting all comparisons wrong (U2)

  26. S behaviour: false number line analogies Th H T U t h th -3 -2 -1 0 1 2 3 • Further right means smaller • 5.736 “further right” than 5.62 • Consequences of 0 and Units as “mirror position” 0 vs 0.6 • Further confusion with 1 as “mirror position” for fractions Some S-like students get classified into A by DCT2

  27. Characteristics of test • Reliable (56% of students in same code after one semester!) • Generally agrees with interviews • Weakness is in diagnosing expertise with consequence that all our estimates of expertise are overestimates • some “experts” follow rules without understanding • some “experts” cant do other tasks (e.g. could reshelve books in library, but don’t know metric properties of decimals) • Test has been improved over life of study – now have improved versions – cycle of improvement

  28. Examples of student journeys

  29. 210403026

  30. 310401041

  31. 400704005

  32. 410401088

  33. 500703030

  34. 500703030

  35. 600703029

  36. 390704012

  37. Where are the students in each grade?

  38. Test-focussed prevalence

  39. Prevalence of L codes • L drops exponentially (L = 440exp(-0.45*grade)) • L2 about 5% in Grades 5-8: some just accumulating facts, not changing concepts “just a few little things still to learn”

  40. Prevalence of A codes by grade A1 = expert • One quarter expert at Grade 5, one half in next 4 years, one quarter never • An issue for adult education e.g. “death by decimal” • Remember these are overestimates of expertise!

  41. Prevalence of A codes by grade A1 = expert Note: 10% in the non-expert A category 4.4502 / 4.45 is a difficult item – more about these later

  42. 10-15% • Peaks in early secondary - probably curriculum effect Prevalence of S codes * This graph was incorrect in the original, and is corrected here.

  43. Within the S region • Around 5% in S1 in all grades • 0.6 < 0.7  • 0.5 <0.125 x • Around 10% in S3 in all grades (more Grade 8) • 0.6 < 0.7 x • 0.5 <0.125 x • Early studies did not ask the S3 question! • S3 Possibilities: • analogy with fractions (one sixth, one seventh) • analogy with negative numbers (-6, -7) • doesn’t include any place value considerations

  44. Another look at prevalenceWhich towns are most visited?

  45. Another look at prevalence • Previously “test-focussedprevalence” - how many are at each town at eachtime? • Also “student-focussed prevalence” – how many students visit each town sometime on their journey? • S.F.P. > T.F.P. • S.F.P. measured over time – in primary school (Gr 4 - 6) and in secondary school (Gr 7 – 10) • S. F. P. increases if you make more frequent observations, so reports are under-estimates.

  46. Student-focussed prevalence SFP of A1 is 80% SFP of non-expert codes 15% - 30+% SFP primary different to SFP secondary

  47. Differences between TFP & SFP • TFP S < 25% • SFP S = 35% • TFP S1 = 6% (most grades) • SFP S1 (pri) = 17% • SFP S1 (sec) = 10% • TFP estimates, SFP under-estimates

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