Embed Size (px)
Citation preview
Learning from the Learning from the Australian Mathematics Australian Mathematics
CompetitionCompetition
(in 2 parts)(in 2 parts)
Gilah LederGilah LederLa Trobe University La Trobe University
<[email protected]><[email protected]> andand
Monash UniversityMonash University<[email protected]><[email protected]>
Part 1 Part 1 Implications for Implications for
InstructionInstruction from large scale from large scale data – data – using the AMCusing the AMC
Part 2Whatever happened to …?
Medallists in the Australian Mathematics Competition have
their say
TheThe Australian Mathematics Australian Mathematics Competition [AMC]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)
More details 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
AMC AIMSAMC 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
PART 1PART 1
What can we Learn from What can we Learn from Large Scale Testing?Large Scale Testing?
Using the AMC
Implications for Instruction
Success rates (in percentages) on common items
AMC data (item numbers refer to the Junior paper)
Grade/Item
Total Top 1 %9 10 7 8
2 98.2 98.4 100 99.8 4 78.0 81.9 97.8 99.110 58.5 61.9 94.9 96.612 34.0 42.8 89.7 94.613 22.3 31.0 77.3 90.816 53.6 59.1 94.4 96.917 11.0 12.0 38.1 47.4
Success rates (in percentages) on common items
Grade/Item
Total Top 1 %9 10 7 8
18 46.0 54.1 90.2 92.419 58.8 67.1 94.6 96.420 35.4 42.1 92.4 97.125 12.0 15.2 58.4 75.327 12.5 14.2 62.1 72.828 14.0 14.4 28.4 44.529 9.6 9.3 27.9 33.930 8.7 8.6 25.9 33.3
What can we conclude?What can we conclude?
No change in performance of top 1% No change in performance of top 1% of students, but whole group improved of students, but whole group improved
from grade 7 to 10 from grade 7 to 10 [[Q16]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
Performance of top 1% of Performance of top 1% of students, improved from students, improved from
grade 7 to 10 but no change grade 7 to 10 but no change for whole group for whole group [Q25][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
ImplicationsImplications• 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
PART 2PART 2
Whatever happened to Whatever happened to …?…?
Medallists in the Australian Medallists in the Australian Mathematics Competition have Mathematics Competition have
their saytheir say
M:F participation rates in M:F participation rates in the AMCthe AMC
2005 Total (N) > 250,000 2005 Total (N) > 250,000 (Australian entries)(Australian entries)
GRADE % MALES
7 51
8 51
9 50
10 51
11 54
12 56
Success rates Success rates (2005) (2005) (%M of N(category (%M of N(category awarded))awarded))
Grade 7 %M 12 %M
Prize (<1 in 300) 68 76
High Distinction (top 2% [5% Senior since 2006] & no higher award)
68 75
Distinction (top 15% [25%])
58 66
Credit (top 50% [60%]) 53 59
Participation (rest) 48 50
Medallists ~ 1 in 10,000 Generally, few F
[2005: 31 medallists (5 F) in grades 7 to 12 at Aust schools]
AimsAimsExamine 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.
Selection ofSelection of
Previous researchPrevious 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)
Gender differences 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)
Method/Theoretical Method/Theoretical modelmodel
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
Significant predictors of Significant predictors of successsuccess – –
Rationale:
Csikszentmihalyi, Rathunde, and Whalen’s (1993) study of talent development
Eccles’ (1985) model of academic choice Mullis & Stemler (2002)
Eccles et al model
Sample Selection Sample Selection 11
• Purposeful Sampling• The AMC medallists• Between 1978 and 2006, 690 medals awarded to students
at Australian schools (few females)
Sample Selection Sample Selection 22
• ~ 420 letters sent1
• 52 letters sent back• By cut off date 113 responses – 90
usableResponse rate1. 90 out of 368 ≥ 24%
2. 90 out of 113 → ~80%
1~40% (both M & F) were multiple medallists
Survey response rate Survey response rate 11
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
Survey response rate Survey response rate 22
“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
The MedallistsThe 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 …
Place of BirthPlace of BirthGeneral 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)
Mothers occupationMothers occupation – many – many with tertiary qualificationswith tertiary qualifications
Common professions:• Teachers: primary/secondary/tertiary• NursesOther• Doctors/dentists/pharmacist/dietician/speech pathologist• Accountant/engineer/computer/IT• Secretarial duties/in sales (<10%)home duties ~10%
Father’s occupationFather’s occupationCommon:• Engineer (almost 20%)• Mathematician/maths teacher
/computing/IT/accountant (~25%)• Doctor (~10%)• Manager (~10%)Parents of M&F medallists: similar
Favourite subject at schoolFavourite subject at schoolMost common• Mathematics: (~60%)• Another science subject (~25%)Other subjects• English (~10%)• History (~5%)[F: Maths ~20%; science subject ~40%;
English ~30%]
Favourite subject at schoolFavourite subject at school
~ why nominated maths~ why nominated maths
• Good at it • Logical• Unambiguous• Challenge of
problem solving / like non-routine problems
• Intellectually stimulating
• Beauty• Like extension
work
To me mathematics is …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)
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)
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)
• 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)
No obvious link No obvious link (yet)between mathematics (yet)between mathematics analogy and occupational analogy and occupational
choicechoice
(Intended)(Intended) Occupation (M) Occupation (M)• Mathematician/statistician/computing
~20%• Engineer ~15%• Doctor ~15%• Actuary ~10%• Manager ~10%• Economist/financial analyst/hedge
fund/venture capital ~10%• Other ~20%
(Intended) Occupation (F)(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)
Leisure occupationLeisure occupationEclectic 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
Benefits of winning a medal - 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.
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)
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)
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
… 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
Working preferences Working preferences in %in %(N(respondents >90% of sample)(N(respondents >90% of sample)
SA/A N D/SD
Enjoy working competitively 67 14 19
I thrive when work is challenging and difficult
96 4 -
Prefer being in charge when working in a group
48 45 7
Relations with fellow workers more important then task performance
35 41 25
I worry that my success will make others dislike me
27 19 54
Working preferences Working preferences in %in %
SA/A N D/SD
It is important for me to perform better than others on a task
51 31 18
I try harder when cooperating 58 25 17
Sometimes I work at less than my best so others won’t resent me
6 8 86
I prefer working alone 39 27 34
Once I start a task, I persist 79 17 5
It annoys me when others do better 41 26 33
It important to me to have a job that pays well
61 27 12
I prefer working on tasks requiring a high level of skill
94 5 1
MedallistsMedallists• 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’ reactionsMuch overlap between M & F responses
Factors important in choice Factors important in choice of careerof 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
RESPONSES TO 2 ITEMSRESPONSES TO 2 ITEMS
Hadamard, J. (1945), The Psychology of Invention in the Mathematical Field, Princeton: Princeton University Press.
“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?”
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.
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.
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.
“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”
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.
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.
The creative process has The creative process has been described in different been described in different
waysways
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
How I work: mathematiciansHow 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?
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.
Reality check!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.
Mathematicians – students’ viewsMathematicians – students’ views
Picker, S. & Berry, J. (2000). Investigating pupils’ images of mathematicians. Educational Studies in Mathematics, 43 (1), 65-94.
Self descriptions Self descriptions ((mathematicians)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)
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)
• 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)
F – why not mathematics?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 didn’t make a decision ‘not to become a mathematician’; I made a series of decisions to study other things and work in other fields.
• 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 wasn’t 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.
ConclusionsConclusions• 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
• 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.
SMPY gender difference SMPY gender difference summarysummary “after 35 years” “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)
