Learning from the Australian Mathematics Competition in 2 parts

1
Learning from the Australian Mathematics
Competition(in 2 parts)

• Gilah Leder
• La Trobe University
• ltg.leder_at_latrobe.edu.augt
• and
• Monash University
• ltGilah.leder_at_education.monash.edu.augt

2
Part 1 Implications for Instruction from 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
MF participation rates in the AMC2005 Total
(N) gt 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 Whalens (1993)
  study of talent development
• Eccles (1985) model of academic choice
• Mullis Stemler (2002)

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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)

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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
• 140 (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 wouldnt 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
Place of Birth

• General Population born outside Australia 23
• Medallists 26 born outside Australia
• 23 of the males
• 40 of the females (cfAndreescu et al. 2008)
• (China, Malaysia, Russia, South Africa)

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 (lt10)
• home duties 10

29
Fathers occupation

• Common
• Engineer (almost 20)
• Mathematician/maths teacher /computing/IT/accounta
  nt (25)
• Doctor (10)
• Manager (10)
• Parents of MF 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
Favourite subject at school why nominated maths

• Good at it
• Logical
• Unambiguous
• Challenge of problem solving / like non-routine
  problems

• Intellectually stimulating
• Beauty
• Like extension work

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,
  didnt 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 gt90 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
Responses to 2 items

• Hadamard, J. (1945), The Psychology of Invention
  in the Mathematical Field, Princeton Princeton
  University Press.

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?

51

• Yes, I quite often get stuck while working
  consciously on a problem, and only after sleep,
  or a break, do I make further progress. But the
  conscious work is important too, in bringing to
  the front of my mind the ingredients of the
  solution, while a break or other activity helps
  to combine them in a more fluid way than can be
  done consciously.

52

• I'm not sure whether I've ever dreamed a
  solution, but I have thought of solutions to
  problems while lying in bed trying to fall
  asleep. Some times I've even jumped out of bed to
  write down the solution, only to find that it
  doesn't actually work as I expected.

53

• Worked in my sleep! Well, I've had some
  nightmares... I wish I could solve problems in my
  sleep. Like most people, I sometimes work for
  hours on a problem and get nowhere. Hours or days
  later, the solution will present itself pretty
  much straight away. Often the cause of the
  frustration is some trivial, stupid error or
  oversight, that I cannot see due to fatigue. It
  is quite normal for the solution to come when one
  is fresh. It's the same in any line of work.

54

• Have your main discoveries been the result of
  focused, conscious work, or have they come to
  you unexpectedly and spontaneously? If the
  latter, can you give a specific example

55

• I find it difficult to parcel a discovery into a
  simple little entity. I think most research
  involves trying to understand some broader
  picture. As such, it requires quite a bit of
  thought and organization of ideas. Certainly
  there may be some steps where the way to proceed
  is not clear, and the right approach may come
  unexpectedly and spontaneously, but these are
  generally just steps towards the main discovery.
  E.g. the final step towards one of the result in
  my thesis came to me while I was wandering around
  in Paris during a stopover on the way to a
  conference, but this result also relied on months
  of calculations which preceded this final
  discovery. So I wouldn't have been in the
  position to make this discovery without the
  previous months' work.

56

• I think it is often a mixture of both. My
  'discoveries' are not really of the form that you
  can pin down to a precise thought or idea.
  Rather, they are the culmination of many smaller
  steps, some of which come through focused work
  and others that come subconsciously. For
  example, I was trying to derive simple formulae
  for a particular statistical model that I was
  using. I spent many hours writing out equations
  and making a lot of progress, but it was all
  quite messy and hard to keep track of what I had
  done and what I still hadn't. The next day I had
  a much clearer mental picture of the task, and
  could easily structure the derivation in a more
  logically coherent manner. Both the focused and
  unfocused aspects of the work were important here
  -- the focused work was required to work out all
  the equations I needed, and the unfocused 'big
  picture' vision was required to assemble them all
  together well. The big picture here was inspired
  by the pieces, but I doubt the converse would
  have happened as easily.

57
The creative process has been described in
different ways

• 2 groups inspiration (best) logically
  mathematically evolved
• Long period of work / gestation / sudden
  solution
• Period of intensive activity no solution /
  switch off solution
• Activity / Incubation / Solution

58
How I work mathematicians

• One can spend a lot of time thinking about a
  problem without seeming to get anywhere. Yet this
  effort is crucial to making an eventual
  discovery, which may occur when one has turned
  one's mind away from the problem. But generally I
  approach my work systematically where possible
  one starts by writing down the steps to follow in
  a logical manner. When this systematic approach
  fails, then it is often best to leave the office
  and leave the pen and paper, and take a stroll to
  focus one's mind on the problem. When this
  doesn't work, ask everyone you know who you think
  might be able to help. Then give up and start
  something new, while hoping that inspiration will
  arrive at some unexpected moment. Eventually, one
  discovers the unfinished work under a pile of
  papers on one's desk, and starts to think about
  it anew. Then the cycle repeats. Is there any
  other way to do mathematics?

59

• The descriptions given are all to do with the
  unpredictable nature of research progress, which
  I have already commented on in my responses to
  Hadamard's questions. I certainly identify with
  them, and would like to add that this
  unpredictability is what makes research both
  frustrating and rewarding. It is perhaps more of
  an emotional roller coaster ride than many other
  types of work, so a decent amount of emotional
  stamina / discipline is required to handle it
  well.

60
Reality check!

• Most of my work is done as a part of large
  projects, so waiting for several weeks or months
  for inspiration to strike isn't really an option.

61
Mathematicians students views

• Picker, S. Berry, J. (2000). Investigating
  pupils images of mathematicians. Educational
  Studies in Mathematics, 43 (1), 65-94.

62
Self descriptions (mathematicians)

• Curious, active, optimistic, opinionated, aspire
  to be rational (but know enough not to claim to
  always be), aspire to be empathetic (all the time
  rather than just some of the time).
• Important influences Parents who are caring and
  not dogmatic, friends with diverse backgrounds
  and interests, and the opportunity to engage in a
  variety of activities, hobbies and sports.
• (PhD student in mathematics)

63

• Irritable, passive aggressive, a little obsessive
  compulsive, a bit reserved, mercurial, a little
  anxious, a little paranoid, worry a little too
  much, a bit of a hypochondriac, complain a lot,
  not easily offended or shocked, vulgar at times,
  easy going, depressive at times, irreverent, not
  too serious, good sense of humour...
• Likes Keeping in touch with friends and family
  via email or phone working (ie studying
  mathematics, research, teaching, etc) leisure -
  music, swimming, walking, reading (news,
  literature, etc - not mathematics) walking
  cooking socialising (restaurants, bars, cinema,
  picnics, etc)
• (PhD student in mathematics)

64

• Apparently aloof but secretly quite perceptive
  (math professor at university in USA)
• Very easy going like spending time with my wife
  I get paid to do Math what's not to love!
  (math professor at university in USA)
• Open-minded though skeptical. Talkative though
  self-reflective Likes Talking to family.
  Talking to friends. Teaching. Eating and drinking
  well and cooking. Small amounts of regular
  exercise. (mathematician at Australian
  university)

65
F why not mathematics?

• Probably doing maths at high school and feeling
  like I had to compete with people who were not
  only very good at it, but also tremendously
  enthusiastic and energetic about pursuing it.
• Probably because it was badly taught and there
  was almost no encouragement.
• I discovered in Year 12 that I enjoyed computer
  science even more than mathematics, and this
  continued throughout my undergraduate degree.
• I didnt make a decision not to become a
  mathematician I made a series of decisions to
  study other things and work in other fields.

66

• At uni I was interested in a lot of things, but
  eventually narrowed it down to applied maths and
  later meteorology
• A perception that research mathematics wasnt as
  interesting or enjoyable as the problem-solving,
  competition maths I was heavily involved in and
  enjoyed. A perception that it would be difficult
  to find an interesting and rewarding job as a
  career mathematician. Being female my father
  believed it was disadvantageous to be female in
  the science/engineering fields, and so dissuaded
  me.

67
Conclusions

• Many findings mirror those of previous studies
  thrive on challenge, persist sensibly, high
  motivation, like to do well, like competition
• Many likeable, well-rounded individuals
• Variety of occupational choices
• Insights into mathematicians (working) lives

68

• Note recurring theme
• At school The most exciting and fulfilling
  mathematics came from opportunities to do
  advanced mathematical work with mathematically
  talented peers outside the regular school
  curriculum.

69
SMPY gender difference summary after 35 years

• in the SMPY cohorts, although more mathematically
  precocious males than females entered
  math-science careers, this does not necessarily
  imply a loss of talent because the women secured
  similar proportions of advanced degrees and
  high-level careers in areas more correspondent
  with the multidimensionality of their
  ability-preference pattern (e.g., administration,
  law, medicine, and the social sciences). By their
  mid-30s, the men and women appeared to be happy
  with their life choices and viewed themselves as
  equally successful. (Lubinski Benbow, 2006, p.
  316)