Origins

The International Mathematical Modeling Challenge (IM²C) began in 2015. Its purpose is to promote mathematical modelling and applications at all levels for all students. To achieve this, the IM²C gives an incentive in support of teachers’ efforts to include effective modelling instruction and an approach to mathematics that incorporates modelling as a central conceptual and operational feature.

The IM²C is a team-based competition. Teams comprise up to four students from the same school, and a team advisor (usually one of their mathematics teachers). They have up to five days to work on a centrally set modelling task, write a report, and submit it for evaluation. Team members collaborate with each other to develop a solution and report, using any inanimate resources they find.

Each participating country selects up to two team reports for inclusion in the international round of judging.

Australian engagement

Australia’s participation in the IM²C has been led and overseen by the Australian Council for Educational Research (ACER). ACER drew together a national advisory group of experts in mathematics, mathematical modelling, and leaders in mathematics education to oversee, guide and promote the introduction of the IM²C. ACER developers built a website to house information, materials and online processes to allow individuals to engage with the IM²C (www.immchallenge.org.au). Members of the advisory group developed a range of support materials. ACER prepared the materials for publication and posted these on the Australian IM²C website.

Australia first participated in the IM²C in 2016. About 100 schools around Australia expressed interest, which resulted in 25 teams being registered and 15 team reports submitted for evaluation. Australia’s two international entries were highly successful; both were awarded ‘meritorious achievement’, which is the second- highest international award. Teams were composed largely of students from Years 10, 11 and 12, with a small number of students from earlier year levels.

By the time of the second year of Australian participation, in 2017, interest and engagement in the IM²C had expanded three-fold. For the 2017 competition, more than 320 schools made direct contact and expressed interest via the Australian website. Registrations increased to 82 teams from 37 schools. Of those, 45 teams from 27 different schools submitted entries.

Table 1 Age and gender of 2017 team registrations and submissions

The age and gender of members both of registered teams and of teams that submitted reports was particularly notable, as shown in Table 1. Around 40 per cent of team members were girls. Teams were composed of students from Years 7 to 12.

From the spread across year levels, it is clear that several schools have seen the IM²C not only as an opportunity to win a competition, but, significantly, also as an opportunity to inject something new and of enduring benefit into their mathematics program. It is interesting to observe that a number of schools registered multiple teams at particular year levels, including in the lower secondary school year levels.

The distribution of teams’ schools is spread across Australia, and across school sector types. Expressions of interest so far have generated team entries from the Australian Capital Territory, Victoria, New South Wales, Queensland, and Western Australia. Government schools have generated more than half of all team registrations and team submissions.

Table 2 State and school sector (government, independent, Catholic) of 2017 team registrations and submissions

Competition outcomes

2016: Insuring against risk

In 2016, the IM²C problem was based on the notion of insuring against risk within the context of an organising committee planning an athletics competition. Teams had to consider the best way for the committee to cover potential liabilities if an athlete competed and broke a world record and thus earned a promised financial performance incentive. Teams were asked to consider various aspects of this potential risk, and to formulate advice to the committee as to whether they should purchase insurance against the risk of making payouts to record breakers.

Six teams’ reports were judged as national finalists and from these, the Australian judging process conferred the following awards:

• ‘Outstanding’ – Trinity College, Perth and Perth Modern School
• ‘Highly Commended’ – Mildura Secondary College, Victoria and Somerville House, Queensland
• ‘Merit’– Manea Senior College, Western Australia and Glen Waverley Secondary College, Victoria.

The reports of the ‘outstanding achievement’ awards were subsequently judged as ‘meritorious’ in the international judging, which is the second-highest category.

2017: Jet lag

The modelling problem used in 2017 was about the phenomenon known as jet lag. Teams were asked to develop an algorithm that meeting organisers could use to decide which location in the world would be suitable to hold a three-day meeting of participants from around the world, in order to minimise jet lag and maximise the productivity of the participants.

Several Australian schools evidently saw the IM²C as an opportunity to engage students from several different year levels in a challenging mathematical activity. The jet lag problem could be worked on with different levels of sophistication, and teams produced a wonderful variety of approaches and outcomes.

Eight teams’ reports were judged as national finalists and from these, three awards were conferred:

• ‘Outstanding’ – John Monash Science School, Victoria
• ‘Meritorious’ – Perth Modern School, Western Australia
• ‘Honourable Mention’ – Box Hill High School, Victoria.

The two top reports were forwarded to the international IM²C centre to be judged against the 49 entries from 27 countries that participated in the IM²C in 2017.

Feedback about participants’ experiences

We used surveys and a variety of other sources to gather information about how schools and their teams have engaged with the IM²C, and the results of how this engagement related to mathematics teaching and learning.

Team advisors were surveyed after each IM²C in 2016 and 2017 . Direct contact was also made with two groups of advisors whose experience was expected to be of particular interest: those who coordinated multiple teams in 2017, and those who had participated in both the 2016 and 2017 challenges.

Team advisors who have now been involved in multiple years of operation are a further source of information about the mathematical and other benefits of participation. Four of the advisors of teams that submitted reports in 2016 also advised 2017 teams that successfully submitted reports. Information from these advisors offers a longitudinal perspective, including the actions taken in the second year in response to experiences and lessons learned in the first.

2017 team statistics

Team advisors’ responses provide potentially useful information about different aspects of the operation of the IM²C, for example, the different ways teams were formed, the different possible modes of team operation and management, and the mathematical activities that were promoted as a result.

Seven schools registered three or more teams in the 2017 IM²C.This included five schools that entered virtually all students in a particular class. Those seven schools accounted for 43 of the registered teams, and 23 of the teams that went on to submit a report, which amounted to about half of the IM2C activity for 2017.

Reviewing cases where substantial parts of a class or of a year level cohort were involved in the IM²C may generate useful insights into the benefits and remaining challenges of activities like this. Other cases where teams from several different year levels registered to participate may also provide useful information. It seems that what the IM²C offers goes beyond the narrow attractions of a competition, and extends to a more strategic domain-wide focus for mathematics at schools.

Logistics and modes of operation

The 2017 IM²C took place in Australia between 14 March and 7 April. This timeframe was chosen in anticipation of a substantial increase in participation and coincided with various Term 1 dates in the different educational jurisdictions. This also ensured that reports could be received and judged in time to identify the Australian entries to the international competition. These were required by 8 May 2017.

The 14 March – 7 April timeframe meant that teams from schools in the Australian Capital Territory, New South Wales, the Northern Territory, South Australia and Tasmania had 19 school days to complete their teamwork; schools in Queensland and Victoria had 14 days. This was a slightly shorter window than was available in 2016. Teams could choose a working period of up to five days, which could include a weekend. The chosen dates imposed schools with severe constraints, particularly in their coordination of end-of-term activities (such as assessments and report writing).

The team advisor survey conducted after the 2016 challenge gathered information on how teams had chosen and used their available five days, and other matters related to their engagement. Seven out of 15 team advisors responded. The most common mode of operation was for team members to use mathematics class time and in some cases some other available time, during three school days, and two weekend days, to complete their work. Two teams restricted their efforts to mathematics class time during five school days (and some additional out-of-class times during that period). Advisors found that a key logistical challenge was to fit the work around other school classes and commitments.

The survey showed that most teams spent from 10 to 20 hours on the problem over the five-day period. Teams had typically not received mathematical modelling teaching, or to had worked intensively in a collaborative group. Team members used communications technology in innovative ways to collaborate and share tasks without necessarily being physically together.

The 2017 survey gathered further information about some aspects of team engagement. Currently, survey responses have been received from 10 of the schools from which teams submitted reports, and from four schools that registered teams but did not proceed to complete and submit reports.

Teams were fairly evenly divided between four common modes of operation in the way they used the five days:

• team time was restricted to mathematics class periods
• teams used three school days (with students attending other activities as required) and two weekend days

• teams used three whole school days and two weekend days
• teams used five school days with students attending other activities as required.

No team used all of the five consecutive school days for work on the Challenge problem. Factors that affected team decisions about time use related largely to the need to fit the work around other scheduled classes or other school activities and to minimise disruption to other important commitments.

For the 2017 survey, the most common response to the question about the average number of hours each team member spent was 10 hours or fewer. The second most common response was more than 10, but less than 20 hours. A small number of teams reported they spent more than 20 but less than 40 hours. No team reported spending more than 40 hours on the task.

The perspectives of advisors who were involved in both 2016 and 2017 further enrich the picture. Differing views were expressed about preferred logistic and timing arrangements; however, these views illustrate different legitimate and viable ways to organise participation. In one case, in 2016, a weekend was incorporated in the team’s working period, and in 2017 only school days were used. This latter approach allowed for more collaboration among team members and better monitoring and support of the team, and will be used in future. In a second case, the team advisor concluded that the preferred approach would be to include a weekend within the five day timeframe in order to minimise the conflict with other school work. In a third case, inclusion of a weekend was definitely the preferred option since it provided far greater flexibility and freedom for team members to collaborate. Decisions about preferred timing were also significantly affected by the IM²C dates in a particular year in relation to other school demands. A key conclusion was that the IM²C dates should be included in schools’ yearly schedule as early as possible.

Helpful practical support made a difference to one team’s use of time and motivation. One team advisor booked a meeting room that the students could use for extended periods during the school days they worked on the challenge, and provided coffee, chocolate milk and biscuits for the team in the board room, and noted that ‘this seemed to help with their motivation too’.

Team advisors promoted the IM²C in different ways. One school posted photographs of the previous year’s team on the wall of the maths classroom. This worked to provoke awareness and incentive for students to consider it the next year. Students expressed interest, and a team was nominated from a Year 11 Specialist Mathematics class that worked on the problem during their maths class time, and used some additional time in school. The team used Google Docs as a collaboration tool. A similar tool was also used by another school that participated in the IM²C for a second time. The usefulness of technological collaboration tools is clear.

In another school, the team advisor set up a ‘senior school IM²C mathematics club’. It met half a dozen times for half an hour on Friday lunchtimes. A displayed photograph of the successful team from the previous year also provided motivation. Students were directed to sample material on the Australian IM²C website, including competition material from previous years, and these were referred to in the preparation stages prior to the 2017 IM²C period.

In a third school that had been involved over two years, team members were also encouraged to refer to the material provided on the IM²C website, which was seen as important and worthwhile.

Additional survey information from team advisors

Participant selection methods

Different schools formed teams using individual selection processes.

The intention for next year is to start preparation earlier and include students in Years 7–10.

Factors that limited teams’ ability to complete and submit reports

Most teams used mathematics class time and some other available time during the week chosen for the challenge. Some incorporated a weekend into their working time. A key to success was building strong and practical approaches to collaboration, which in many cases involved using suitable collaboration tools.

Teams experienced difficulties in working within the timeframe and managing their time. Survey feedback revealed:

• students (and advisors) misjudged the amount of time required, and ran out of time – six hours of school time was insufficient

• a Year 12 advanced maths team underestimated the time required and failed to submit anything
• teams that failed to submit tended to be disorganised and did insufficient work to make progress
• a project management platform that allowed students to collaborate easily outside class hours, ask questions, and make their teamwork visible was really useful – teams used a combination of ‘Slack’ for discussion and questions, and ‘Trello’ for project management, which worked well
• one team used the school vacation to minimise conflicting school demands, but this meant some students were unwilling to devote the time (hence did not submit), and it was more difficult to monitor team progress; as a result the team will do things differently next time
• team members worked together at the back of their regular maths classes for the week, plus some time out of class; they identified tasks for each other, worked on them overnight, and then shared and discussed results the next day.

Team advisors who responded to the 2017 survey for teams that did not proceed to complete and submit a team report identified these main factors:

• in-school arrangements did not facilitate effective team function
• team members were overwhelmed by other school-related activities at the same time
• a lack of awareness or understanding of modelling processes.

It seems clear that participation in the IM²C is severely hampered if facilitating conditions are not provided. Proper advance planning has to have taken place, other staff need to be aware of students’ commitment to the challenge, and school processes need to be established to provide at least some level of support.

Mathematics education outcomes

An overarching objective of the IM²C is to encourage and facilitate change in some of the classroom practices of mathematics teachers and learners. Mathematical modelling is increasingly recognised as an important way to view mathematical activity. It provides a clear imperative to explore and exploit connections between the opportunities and challenges that arise in the world as experienced by individuals, and the mathematical knowledge those individuals have developed, or could develop in response to a challenge. Strengthening the connection between mathematical knowledge and skills, and the ways in which they can be used, is a central objective of many approaches to mathematics teaching and learning; the IM²C provides a very direct opportunity to develop the relevant capabilities.

Resources provided

The IM²C in Australia has been supported by resource materials that have been prepared for teachers in order to develop students’ knowledge about modelling and to foster specific modelling skills. They are freely available on the website (https://www.immchallenge.org.au/supporting-resources) and it is expected that additional materials will be developed.

About half of the respondent team advisors in the 2017 participant survey only gave specific guidance and teaching on mathematical modelling because of the Challenge. The other half of respondents was equally split between advisors who reported mathematical modelling as an element of previous mathematics instruction, and those who reported that no previous instruction on modelling had been given.

Teams were typically referred by their advisor to the resources available on the IM²C website. If these were used by teams, it was typically outside of normal class time; a small proportion of advisors reported systematic use of the material with classes that contained team members.

Team advisors observed the potential usefulness of the resources and their intention to make greater use of them in the future. Team advisors reported in a number of cases that the resources had been distributed to other mathematics teachers in the school to support mathematics teaching at different levels and, in one case, in a problem-solving elective.

Survey responses to whether additional resources that would facilitate future participation were split fairly evenly between those that said nothing, the existing resource is adequate, and those who asked for more of the same.

Increasing engagement and reach

The dramatic expansion of engagement with IM²C is a clear indicator that schools and teachers see it as a worthwhile area for greater focus of mathematics teaching and learning. The great majority of responses to the 2017 survey of team advisors indicated an intention to participate in 2018. Any hesitation about 2018 participation was because advisors did not know how the IM²C window would align with the Term 1 school program.

Reactions from 2016 participants

From the 2016 participant survey, team advisors observed that the activity generated a very positive response from students. They saw it as an enjoyable and valuable enrichment opportunity, an opportunity to develop collaboration skills important for this century, and a valuable opportunity to work on an extended mathematical exploration and understand the thought processes of other team members.

Team advisors surveyed after the 2017 IM²C registered similar comments about the benefits of participation. Several participants specifically mentioned the benefits of the team collaboration involved, the intellectual challenge involved in working on the problem, and in showing the connections between mathematics and ‘real-world problems’.

‘The students really enjoyed participating in the real-world problem solving challenge. They enjoyed the pressure it created and the creativity that came about from working together in a team. The school will benefit as we take on board some of the strategies required to solve this International Mathematical Modelling Challenge.’

‘The students learnt to be resourceful, to be self-reliant, to work cooperatively and independently, as well as how to problem solve when the question is open ended. I am hoping that the benefit to the school will increased interest in this type of activity and more participation in the future. This group of girls is already looking forward to participating next year.’

‘The students enjoyed working on a different type of maths question and they were able to see how it could be applied to real world problems.’

Reactions from 2017 participants

The 2017 survey of selected team advisors revealed similar feedback, summarised in the extracted comments following.

Mathematical input and outcomes

• Different teams used very different approaches. Some tried to use an algebraic approach, others were very visual and used a spatial approach.

• Task was very relevant, with students able to link it to their own experiences.
• Prior to the challenge, used ‘Supersize Me’ problem (on the IM²C website) with the class, and spent several lessons looking at previous year’s IM²C problems and solutions.
• Used the IM²C 2017 as part of formal assessment.
• Goal was to expose students to an open, challenging and intense team-based task over a short period – it was very successful.
• Students developed their resilience, modelling and computational skills as well as their report-writing skills.
• Students worked under different pressures – for self-regulation, preparation, team work – and used their mathematical thinking in a different way from normal.
• Opportunities for students to take a leadership role, take initiative, and collaborate with peers.
• The challenge gave students the opportunity to think mathematically, and to see real connections between what they learn in the classroom used outside.

Reactions of team members

• Students were very enthusiastic and tried very hard.
• It was a great experience, and my intention is to expand it next year and engage staff from other (non-maths) parts of the Gifted program.
• Team of strongest Year 10 and 11 students ‘found it incredibly difficult to agree on anything, but did some good work’.
• I think the competition is a fantastic thing, and is something we are keen to continue in years to come.
• The competition is excellent, and the problem itself was of a very high quality.
• Students indicated that they really valued the opportunity to work on a real- world problem that had meaning. They loved the collaboration, and learned important lessons about working in teams.

Reactions of others

• Other maths teachers very supportive and assisted where possible.
• Posters used to advertise the activity within the school; and expectation that the school community will be interested in recognising achievements and participation.

Part 2: Feedback on 2017 IM²C reports

The 2017 IM²C problem is reproduced here.

Problem: Jet lag

Organizing international meetings is not easy in many ways, including the problem that some of the participants may experience the effects of jet lag after recent travel from their home country to the meeting location which may be in a different time zone, or in a different climate and time of year, and so on. All these things may dramatically affect the productivity of the meeting.

The International Meeting Management Corporation (IMMC) has asked your expert group (your team) to help solve the problem by creating an algorithm that suggests the best place(s) to hold a meeting given the number of participants, their home cities, approximate dates of the meeting and other information that the meeting management company may request from its clients.

The participants are usually from all corners of the Earth, and the business or scientific meeting implies doing hard intellectual team work for three intensive days, with the participants contributing approximately equally to the end result. Assume that there are no visa problems or political limitations, and so any country or city can be a potential meeting location.

The output of the algorithm should be a list of recommended places (regions, zones, or specific cities) that maximize the overall productivity of the meeting. The questions of costs are not of primary importance, but the IMMC, just as any other company, has a limited budget. So the costs may be considered as a secondary criterion. And the IMMC definitely cannot afford bringing the participants in a week before the meeting to acclimatize or give them the time to rest after a long exhausting journey.

Test your algorithm at least on the two following datasets:

Your submission should consist of a one-page summary sheet. The solution cannot exceed 20 pages for a maximum of 21 pages. (The appendices and references should appear at the end of the paper and are not included in the 20-page limit.)

Key assessment criteria

The following criteria were used to evaluate the Australian team reports.

1. Problem definition

• Identify a real-world problem to be solved, and specify precise mathematical questions from the general problem statement

2. Model formulation

• Identify assumptions with justification
• Choice of variables
• Identify and gather relevant (needed) data
• Choice and justification of parameter values
• Development of mathematical representations

3. Mathematical processing

• Application of relevant mathematics
• Invocation and use of appropriate technology
• Checking of mathematical outcomes for procedural accuracy
• Interpretation of outcomes in terms of the problem situation

4. Model evaluation

• Explore adequacy and relevance of findings in relation to problem situation
• Provide further elaboration or refinement of problem
• Relevance of revised solution(s) following revisiting and further work within earlier criteria
• Evaluation of sensitivity of solution to changed assumptions or conditions
• Quality of answers to specific questions posed in problem statement

5. Report quality

• Summary page quality: succinctness, power to attract reader
• Overall organisation of the report: fitness for purpose, and logical presentation that includes (as appropriate):
  • description of the real-world problem being addressed.
  • specification of the mathematical questions
  • listing of all assumptions wherever they are made
  • indication of how numerical parameter values used in calculations were decided on
  • setting out of all mathematical working: graphs, tables, technology output and so on
  • interpreting the meaning of mathematical results in terms of the real world problem
  • evaluating the solution(s) in terms of the problem requirements.

Overview of approach

An essential starting point is to clarify exactly what the IM²C problem requires. The problem statement asks teams to develop an algorithm, and to test that algorithm on at least two given scenarios. The question does not ask where those two meetings should be held, but asks for a systematic process (an algorithm) that could be followed to determine a location for a meeting of participants from different home locations that maximises productivity of the participants, especially in relation to the effects of jet lag.

Another essential step in an effective modelling activity is to transform the statement of what is required into a mathematical objective. As well as developing a clear understanding of what would constitute an answer to the question in its context, the goals of the exercise must also be expressed in clear mathematical terms. For example, the mathematical objective could be to minimise total distance travelled by the meeting participants, or time spent travelling, and so on.

The end-point of the modelling process is to communicate the results in a form that can be understood and used by its audience. The report required for the IM²C comprises three parts: a one-page summary sheet, a report of the solution, and appendices, which includes references. However, the exact way a report is constructed should be determined in light of its purpose and audience. A modelling report is not the same as a school mathematics assignment. It might take the form of a recommendation or set of recommendations to a committee, together with an explanation and justification of the recommendations. In between the beginning process of defining the goals of the task and defining that in mathematical terms, and the end process of writing a report, the processes of model formulation, mathematical processing, and model evaluation take place. Those processes would often occur multiple times, since the first attempt to solve the problem might expose other issues to be taken more into account to provide the best possible solution. A better mathematical formulation might be needed, which might require an adjusted model that a different kind of mathematical processing could assist, and an updated interpretation of the results and evaluation of the outcomes would then be needed.

An example taken from the 2017 IM²C reports was from a team that did an excellent job of defining the cumulative effects of jet lag across the meeting participants as the total number of time zones needed to be crossed to get to the meeting’s location. After this team established that mathematical objective, team members then systematically tested each potential location time zone to determine the total number of time zone changes that would occur. This enabled them to narrow their solution to the time zone that created the fewest time zone changes for all participants, which is where their investigation stopped. Another iteration of the modelling process for that team might have been to then apply other factors, such as existing flight routes, or climate factors, to further narrow their recommendation.

Identify relevant variables

This section describes some of the features of different approaches taken by 2017 IM²C Australian teams. It is structured around particular issues and factors that helped distinguish better approaches to the problem from less effective ones. The submitted reports were of extremely varied quality, and more importantly, this variety reflected the possibility that the problem could be tackled at different of levels of sophistication. Teams with younger students could gain from their work, recognising it might be conducted at a lower level mathematically compared to students with more mathematical knowledge and experience.

The extent to which teams identified relevant factors, and how those factors were then treated in the analysis and report was interesting. Some factors that were said to be important included time zone changes, climate, weather, characteristics of the destination city, hours of sunlight, optimal working conditions, activities outside of the meeting. A key issue then was the extent to which factors said to be important were treated effectively in the solution process.

A key feature of mathematical modelling is the need to identify assumptions that are made, and to explain why they are made; also to consider how these assumptions influence the solution found, and how changing assumptions might affect the solution.

Other factors that could be considered include:

• the difference in jet lag effect between travel towards the east and west
• the added impact of travel fatigue
• the possible addition of consideration of cost-related factors.

How ‘distance travelled’ was treated

Some of the considerations that were important to the distance travelled were:

• considering ‘as the crow flies’ versus plausible flight routes
• whether minimising distance travelled would alone solve the problem
• consideration of journey time (for example, whether direct or multiple flights might be needed)
• accuracy of complex distance calculations.

How time zones were treated

The relevant factors in the consideration of time zones were:

• an absolute reference system was essential– most teams used Coordinated Universal Time (UTC)
• the degree of overlap of ‘alert’ periods for participants
• accuracy in calculations of time zones for different locations and different times of the year
• the treatment of ‘recovery time’.