Maths Competitions

The following article outlines the main mathematics competitions that we do here in Queensland and the benefits to participating in and training for them.

Term 3 is a big term in the life of an enrichment program at a school; mainly due to the sheer amount of competitions that occur.  ICAS (International Competitions and Assessments for Schools) and the AMC (Australian Mathematics Competition) are the two competitions that most people have heard of, but there are many others of varying degrees of difficulty and it is very much up to individual schools (and teachers) to decide what competitions they are going to offer to their students.

It's worth pointing out that mathematics competitions are not the same IQ tests; they can be prepared for.  Below is the solution I did out for the first question on this year's AIMO paper. It should hopefully give a feel for the types of questions and thinking required to do well on maths competition questions.

Interestingly, in the past under the old OP system there was little motivation to offer genuine mathematics enrichment programs to extend our top students.  At the time, the main data point (for better or worse) that parents used to judge a school was the amount of OP 1's it received.  As there were around 600 OP 1's awarded every year, the bar to get an OP 1 was high but achievable for a smart, hard-working student that was in a school that knew how to optimise the OP system.  Administering a genuine mathematics enrichment program was generally not seen as a good allocation of limited school resources. 

However, now that we have transitioned to the ATAR system, the number that is used to judge a school is the ATAR 99.95.  There are only around 30 ATAR 99.95's awarded across the state, as a consequence they are much more difficult to achieve.  In addition, due to how subject scaling works, only students who do well in mathematics, and subjects requiring a high competency in mathematics (engineering, physics, chemistry etc.) have a chance at getting an ATAR 99.95. The net effect of this is that mathematics enrichment programs (i.e. participating in and training for mathematics competitions) have become much more important to schools wanting to attract and challenge students capable of achieving the top ATAR score.

For many of our highly capable students, there is little to challenge them in their junior years of schooling. Mathematics competitions are one of the few chances they have of being exposed to genuinely difficult questions; questions that they cannot do straight away, but still have the necessary background knowledge to be able to solve.  The skills they learn from being confronted with such questions prove highly valuable in their senior years when they have to do the complex unfamiliar style questions on their internal and external exams. I always encourage our top students to participate in all of the mathematics competitions that we offer at school.  Many of the students who "only" achieve credits or even lower on the harder competitions often go on to attain ATAR scores of 99.85 or higher.

Aside from helping attain a high ATAR score, there are also many other benefits to training for and sitting maths competitions. These main ones are listed below:

When I first started teaching at QASMT, one of the things that struck me most was how fragile a lot of the students were.  Many had gotten straight A's for all of their schooling and found getting anything less than that highly traumatizing.  Usually whenever I handed back exams I would always take a box of tissues along for the students that would cry.  I would also make them fill out a well-being survey that I would check before allowing them leave the classroom. One of my most poignant memories was of one of our top students who ended up leaving the school because wasn't able to accept getting less than straight A's all of the time.

When I transferred to Brisbane State High, due to a personal connection, I soon ended up administrating the Mathematics Olympiad program and running all of the competitions that we use to identify students to be invited into their training pathway. I was really impressed by the students in the training program; there was a resilience, teachability, even humility to them that I rarely encounter in students that smart. Being confronted with hard problems forces a student to acknowledge the limits of their own ability and become much more open to mentoring and learning from others.

The experience of high school mathematics for very intelligent students is one that is oddly divorced from reality. Most of the time they are simply required to select a procedure they already know and apply it to a contrite situation.  As a result, the experience of being confronted with a question that cannot immediately be solved can be foreign and incredibly uncomfortable for these students. Being able to find creative solutions to problems that do not already have pre-made answers is one of the greatest skills that can be learned from studying mathematics.  Mathematics competitions are extremely valuable in this regard, in that they will allow students to develop skills that will prove highly valuable in whatever future career path they choose to pursue.

Many of the scholarships offered by the big universities require more than just a high ATAR score.  Placing in a mathematics competition is a way to distinguish oneself from other scholarship applicants.  The Australian Maths Trust publishes the names of any students that do particularly well in any of their competitions.  This is something that students can use as evidence when they are applying for highly competitive places in prestigious universities. Already at our school we have two grade 12 students that have been accepted into the Australian National University before their ATAR scores have been published. This is in part due to their achievements in the various competitions and Olympiad programs they have taken part in.

There are a wide variety of mathematics competitions that are available for a school to offer their students.  It is very much up to individual schools what they choose to administer.  Below are the different competitions that I am most familiar with listed in order of difficulty.

The ICAS tests are offered for a broad range of subjects (Maths, English, Science etc) and the tests are standardized internationally.  The tests are measures of general academic performance and as a result, predominantly reward students who are able to work error-free as opposed to those who are good problem solvers.  I've had students in the past who have topped the state in year 10 ICAS mathematics but went on to underperform in year 12 when the content became genuinely difficult.

The AMC is probably the most well known maths competition in Australia.  There are five different divisions and each division caters for two year levels (i.e. Middle Primary division is for years 3 - 4 etc.). This means that students from years 3 - 12 are able to sit the AMC.  The AMC consists of 25 multiple choice questions and 5 short response questions. Primary divisions have 60 minutes to complete the paper, with secondary divisions having 75 minutes.  The questions are generally similar enough to what students are doing in class such that most high achieving students are able to do quite well on the AMC without any extra training. The AMC questions are a fantastic enrichment resources to use in a general class setting.

The QAMT problem solving competition is run by the Queensland Association of Mathematics Teachers and is a step up in difficulty from the AMC.  Similar to the AMC, each division caters for 2 year levels and there are three divisions (years 7-8, years 9-10, years 11-12). The paper consists of 5 questions, and students have 2 hours to complete the paper. The questions are difficult, and students that have not had any training in maths competition style questions can really struggle with the paper.

AIMO is the most difficult maths competition that is open for all students to participate in.  Only students in grades 7 - 10 can participate in the AIMO and they all sit the same paper.  An understanding of year 9 maths is necessary (i.e. Pythagoras' Theorem, Similar Triangles etc.) to be able to fully engage with all of the questions.  There are 10 questions, no calculator is allowed, and students have 4 hours to complete the paper. The questions are genuinely difficult, most require significant insight to be able to make progress on, and it is not uncommon for students who get straight A's in their mainstream maths classes to fail to get a single question out.

These competitions are invitation only to students who are already in training programs run by the Australian Maths Trust.  Students are selected into the training programs based on their performance predominantly in the AIMO, but also in the AMC and QAMT problem solving competitions.  Invitation to participate in any of these competitions is an achievement in itself, and something invited students should be incredibly proud of.