Learning from the Australian Mathematics Competition (in 2 parts)

1. Learning from the Australian Mathematics Competition(in 2 parts) GilahLeder La TrobeUniversity <g.leder@latrobe.edu.au> and MonashUniversity <Gilah.leder@education.monash.edu.au>

2. Part 1 Implications for Instructionfrom large scale data – using the AMC Part 2Whatever happened to …?Medallists in the Australian Mathematics Competition have their say

3. The Australian Mathematics Competition [AMC] • Introduced in 1978 • Now: Australia + ~ 40 other countries • For: students of all standards (open competition) • Initially grades 7-12 (now also grades 3-6)

4. More details • 3 papers Junior [7-8], Intermediate [9-10], Senior [11-12] • 30 questions / 75 minutes • Multiple choice • Questions graded: easy  difficult “Students of all standards will make progress and find a point of challenge “ Visually Impaired Students

5. AMC AIMS • To highlight the importance of mathematics as a curriculum subject • To give students an opportunity to discover talent in mathematics • To provide resources for the classroom and general discussion

6. PART 1What can we Learn from Large Scale Testing? Using the AMC Implications for Instruction

7. Success rates (in percentages) on common items AMC data (item numbers refer to the Junior paper)

8. Success rates (in percentages) on common items

9. What can we conclude?

10. No change in performance of top 1% of students, but whole group improved from grade 7 to 10 [Q16] The digits 1, 2, 3 and 5 can be arranged to form 24 different four-digits numbers. The number of even numbers in this set is (A) 1 (B) 2 (C) 6 (D) 12 (E) 18 Answer correctly: ~2/3 grade 10 students; almost all top grade 7 students

11. Performance of top 1% of students, improved from grade 7 to 10 but no change for whole group [Q25] Four singers take part in a musical round of 4 equal lines, each finishing after singing the round through four times. The second singer begins when the first singer begins the second line, the third singer begins when the first singer begins the third line, the fourth singer begins when the first singer begins the fourth line. The fraction of the total singing time that all four are singing at the same time is (A) 3/4 (B)3/5 (C) 2/3 (D) 5/6 (E) 8/15 Answer correctly: ~ 15% grade 10 students / top students ~ 60% grade 7; ~85% grade 10

12. Implications • Routine, multi step exercises: best students → max performance in grade 7; whole group improves with grade level (though still performance below best in grade 7) • Non-routine problems requiring considerable synthesis of ideas: difficult for whole group, all grades, but suitably challenging for best students

13. PART 2Whatever happened to …?Medallists in the Australian Mathematics Competition have their say

14. M:F participation rates in the AMC2005 Total (N) > 250,000 (Australian entries)

15. Success rates (2005) (%M of N(category awarded)) Medallists ~ 1 in 10,000 Generally, few F [2005: 31 medallists (5 F) in grades 7 to 12 at Aust schools]

16. Aims Examine how exceptionally high achievers in mathematics perceive mathematics, and To gain insights into their background, motivations, work habits, and occupational choices. Important & Timely: • the drift away from demanding mathematics courses • the widespread concerns about the declining popularity of mathematics.

17. Selection ofPrevious research • SMPY – exceptionally high achievers at junior high school (Julian Stanley & colleagues; Lubinski & colleagues) • High achievers in mathematics (Csikszentmihalyi, Rathunde, and Whalen (1993); Gustin (1985); Wieczerkowski, Cropley, and Prado (2000). • Mature mathematicians (Burton, 2004) • Cross cultural comparisons (Andreescu et al. 2008)

18. Gender differences e.g., Secondary analysis of TIMSS data- maths(Robitaille & Beaton ,2002) • Males: over-represented among high performing students and • Gender differences particularly prevalent among high performing students • “Yet the results also indicate that females are capable of achieving at high levels in … advanced mathematics”(Mullis & Stemler ,2002, p. 289)

19. Method/Theoretical model WEB based survey (Likert Format & Open ended & using SurveyMonkey) To explore • personal qualities and characteristics (subject specific and broader attitudes and beliefs, expectations, motivations, self-perceptions, …) • Environmental factors (the cultural milieu, the home, peer and educational environments) + 2 x 2nd survey

20. Significant predictors of success – Rationale: Csikszentmihalyi, Rathunde, and Whalen’s (1993) study of talent development Eccles’ (1985) model of academic choice Mullis & Stemler (2002)

21. Eccles et al model

22. Sample Selection 1 • Purposeful Sampling • The AMC medallists • Between 1978 and 2006, 690 medals awarded to students at Australian schools (few females)

23. Sample Selection 2 • ~ 420 letters sent1 • 52 letters sent back • By cut off date 113 responses – 90 usable Response rate • 90 out of 368 ≥ 24% • 90 out of 113 → ~80% 1~40% (both M & F) were multiple medallists

24. Survey response rate 1 Sample used: purposive sample “The response rate varies significantly among methods of administration. Surveys printed in magazines may have a 1% or 2% response rate. Mail surveys often have return rates between 10% and 50%” (McBurney & White, 2004,p. 247) McBurney, DH, & White TL (2004) Research methods. Belmont CA: Wadsworth

25. Survey response rate 2 “The non-response rate wouldn’t matter if we could be certain that those that do not respond are very similar to respondents on all relevant variables” (Muijs, 2004, p. 43) Single & Multiple medallists; Good age range; N(F) = 10 (11%) Muijs, D (2004). Doing quantitative research in education with SPSS. London: Sage Publications

26. The Medallists • Place of birth • Parents’ occupations • Favourite subject at school • About mathematics • Careers • Leisure Occupations • Winning a medal • Working preferences & motivations • Mathematicians - Self descriptions • Females …

27. General Population born outside Australia: ~23% Medallists: 26% born outside Australia: 23% of the males 40% of the females (cf:Andreescu et al. 2008) (China, Malaysia, Russia, South Africa) Place of Birth

28. Mothers occupation – many with tertiary qualifications Common professions: • Teachers: primary/secondary/tertiary • Nurses Other • Doctors/dentists/pharmacist/dietician/ speech pathologist • Accountant/engineer/computer/IT • Secretarial duties/in sales (<10%) home duties ~10%

29. Father’s occupation Common: • Engineer (almost 20%) • Mathematician/maths teacher /computing/IT/accountant (~25%) • Doctor (~10%) • Manager (~10%) Parents of M&F medallists: similar

30. Favourite subject at school Most common • Mathematics: (~60%) • Another science subject (~25%) Other subjects • English (~10%) • History (~5%) [F: Maths ~20%; science subject ~40%; English ~30%]

31. Good at it Logical Unambiguous Challenge of problem solving / like non-routine problems Intellectually stimulating Beauty Like extension work Favourite subject at school~ why nominated maths

32. To me mathematics is … • An incredibly stimulating and fascinating world of order, logic, beauty and power. At the same time it is nothing - it exists purely in the minds of man, and were we to disappear, it would go too.(medical doctor) • A beautiful construction by the human intellect. It also happens to be useful for understanding the world.(software engineer)

33. The only pursuit which both allows and requires pure brilliance - it provides the worst trade-off between long hours/hard work and ability, and mathematical achievement is therefore as little mired in circumstance as any measure of a person I have encountered.(completing PhD in astrophysics)

34. a language - the language of absolute truth. If you want to understand the universe when it speaks, then you must learn mathematics. I don't want you to think I'm an extremist - there are many important and fundamental human truths about which maths says nothing.(completing PhD in pure mathematics)

35. About the importance of rules and precision (legal academic) A fascinating subject… certainly in teaching and music making. I believe I think quite mathematically often Instrumental music teacher (strings) a stepping stone to career opportunities and a good way to exercise the mind (senior executive manager in a large firm)

36. No obvious link (yet)between mathematics analogy and occupational choice

37. (Intended) Occupation (M) • Mathematician/statistician/computing ~20% • Engineer ~15% • Doctor ~15% • Actuary ~10% • Manager ~10% • Economist/financial analyst/hedge fund/venture capital ~10% • Other ~20%

38. (Intended) Occupation (F) • Doctor (4) • Freelance orchestral musician • Medical scientist • Meteorologist • Physicist/statistician • Artificial intelligence researcher /software engineer • Unsure –just completed PhD(cross discipline: English / human nutrition)

39. Leisure occupation Eclectic and wide ranging.They included: • sport (particularly football, golf, hiking, rock climbing, running, soccer, squash, swimming, tennis, volleyball), • music (including guitar, piano, singing, violin and writing music), • card games, playing chess, photography, • reading, • writing • socializing/spending time with family

40. Benefits of winning a medal - • None mentioned negative aspects. Many: • great satisfaction and pride in having their mathematics achievement recognized; • valued the actual award giving ceremony; • valued the opportunities to attend special courses & do advanced mathematical work with others who liked mathematics and were good at it; • talked of longer term benefits and/or specific doors being opened.

41. A source of pride - we were immensely competitive in a good-natured way at school and there were 3 or 4 students in my year who won AMC medals in various years. We still get together every year to do the Westpac / AMC competition paper over dinner (our 15th year this year) (M – surgeon)

42. Selection into the Mathematical Olympiad training program, with many flow on benefits, including: learn much more mathematics and at a higher level, meet like-minded people many of whom are now good friends, encouragement to continue with mathematics. (completing a PhD in statistics at Oxford university)

43. The AMT sent some extra challenging problems, but it wasn't really followed up. I did do some of them. If my school had given me any encouragement or some time off the incredibly boring school maths classes to do them, I would probably have done a lot more. So actual benefits - negligible. (F - musician) ctd

44. … I had much better and more encouraging teachers in music. In maths, the teachers treated it more like an embarrassment that I was good at it, didn’t really know what to do with me and certainly gave me no extra stimulation or room to expand my talents. The fact that I won an AMC medal is due entirely to two years of my schooling. The first was in Japan at age 8; when I came back I was two years ahead of my classmates in maths. The second important year was when I was 10, in fifth grade primary school. ctd

45. Working preferences in %(N(respondents >90% of sample)

46. Working preferences in %

47. Medallists • Thrive on doing difficult, challenging, and highly skilled work • Persist with a task • High motivation and task commitment • Some like working cooperatively; others competitively • Want to do well, irrespective of peers’ reactions Much overlap between M & F responses

48. Factors important in choice of career • Makes best use of my talents • Provides freedom from close supervision • Leaves room for other things in my life • Financial reward – somewhat important • Prestige of career – Not important

49. Hadamard, J. (1945), The Psychology of Invention in the Mathematical Field, Princeton: Princeton University Press. Responses to 2 items

50. “Have you ever worked in your sleep or have you found in dreams the answers to your problems? Or, when you waken in the morning, do solutions which you had vainly sought the night before, or even days before, or quite unexpected discoveries, present themselves ready-made to your mind?”