The Origin and Development of the College Student Mathematical Modeling Contest (2)
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Due to the aforementioned shortcomings of the Putnam Mathematical Competition, especially because of the rapid development of computers, computational skills and abilities, as well as network technology, the scope of application of mathematics has been expanding day by day. More and more people recognize the importance of mathematics, especially mathematical modeling. There is a demand for mathematics education (including mathematics competitions) to make corresponding transformations. Since the United States leads in scientific development, it is understandable that the college student mathematical modeling contest first appeared in the United States. Let's still look at what Ben A. Fusaro, the founder of the American College Student Mathematical Modeling Contest, has to say. In his article summarizing the first American College Student Mathematical Modeling Contest (MCM-1985) (B.A. Fusaro, Mathematical Competition in Modeling, Mathematical Modelling, v. 6(1985), 473 - 485), he said: "I conceived the idea of a national college student applied mathematics competition in October 1983. This was due to the difficulties we encountered while organizing our school students to participate. Salisbury College has a large percentage of first-generation college students (Translator's note: i.e., no one in their family has attended college before). They often view facing such a terrifying exam (Translator's note: i.e., the Putnam Mathematics Competition) simply as an ordeal. The experience of participating in the competition and receiving extremely low scores further amplifies this disheartening effect. Finally, the lack of applied content in the Putnam Mathematics Competition cannot excite the interest of students who are interested in practical applications. However, the concept of an applied mathematics competition is much richer than just modifying the current Putnam competition.
This competition concept is based on my foundational philosophy developed over more than ten years. I am dissatisfied with the excessive emphasis on the purity, formalism, and almost impractical application content of well-established mathematics. Many campuses have no detectable application or structural (mathematics) existence. In my mind, (classical) applied mathematics, computational mathematics, statistics, and pure mathematics are all important components of mathematics science teaching activities and courses...
The problems involved in the Putnam competition reflecting the Bourbaki tendency after 1945 only touch on a very small area of the top part of mathematics... It is difficult to tell from the Putnam competition that computers are playing a role."
Fusaro firmly believed that his idea was correct. He sought opinions from many renowned applied mathematicians, experts in the Putnam Mathematics Competition, and the head of the non-profit organization "Consortium for Mathematics and Its Applications" (COMAP) in the United States. The responses he received were almost all supportive views and good suggestions. He also cooperated with others to apply for related topics and funds. The competition was finally held in 1985. The name of the competition was then changed to (American) College Student Mathematical Modeling Contest (The Mathematical Competition in Modeling, abbreviated as MCM), later renamed The Mathematical Contest in Modeling, with its abbreviation remaining unchanged. In 1999, the Interdisciplinary Modeling Contest (The Interdisciplinary Contest in Modeling, abbreviated as ICM) was added.
If the Putnam Mathematics Competition is a completely closed-book examination, then the College Student Mathematical Modeling Contest is a completely open examination. In recent years, the competition starts at 8:00 pm Eastern Time on Thursday of the second week of February each year, and ends at 8:00 pm Eastern Time on the following Monday, lasting for four days. Currently, MCM and ICM are organized by COMAP, and the competition is sponsored by the National Security Agency (NSA), the Operations Research and Management Science Society (INFORMS), the Industrial and Applied Mathematics Society (SIAM), and the Mathematical Association of America (MAA). Teachers wishing to register students for participation in ICM should do so through the website . Teams participating in ICM can only choose problem C, meaning they can only work on problem C during the competition, whereas teams registered for MCM can choose any one of problems A or B. These are annual major correspondence competitions mainly aimed at college students, with three college students forming a team to participate in the competition. Each team chooses one of the questions proposed by the competition organizers and uses mathematical modeling methods to solve or partially solve the practical problem, writing it into a paper and submitting it to the competition organizers within the specified time. Then, the MCM Committee and the ICM Committee respectively invite experts to review and award prizes. The award levels are divided into: Outstanding, Meritorious, Honorable Mention, and Successful Participant. All awardees and their instructors will receive a certificate. Outstanding papers will be published in the famous American university mathematics journal Journal of Undergraduate Mathematics and Its Applications (UMAP). Outstanding winners will receive cash rewards or be invited to present reports at professional society annual meetings.
Why are MCM and ICM completely open competitions?
In COMAP's explanation of the competition, it points out: "This competition consists of three optional problems A, B, and C. Importantly: teams participating in MCM can choose to work on either problem A or B but not problem C, and when submitting their solution papers, they can only choose one of either problem A or B. Teams registered for ICM can only work on problem C and cannot choose problems A or B.
What teams can and cannot do in preparing their solution papers:
Teams can use lifeless data resources or materials: computers, software, reference books, online materials, books, etc., but all materials used must cite their sources. Failure to do so will result in the revocation of the team's eligibility.
Except for discussions and research with team members, assistance or discussion from instructors or anyone outside the team is prohibited. Any input from outside the team is strictly forbidden. This includes emails, phone calls, personal conversations, online chats, or other question-answer system exchanges, or any other form of communication.
Incomplete solution papers are acceptable. MCM/ICM reviews submitted papers without passing or failing criteria. No numerical scores are given. Reviews primarily examine the steps and methods of each team.
Abstract page (Note: MCM/ICM stipulates that solution papers must include a one-page abstract.) The abstract is an extremely important part of your submitted MCM paper. Reviewers give significant weight to the abstract, sometimes distinguishing between winning and non-winning papers based on the quality of the abstract. To write a good abstract, try to put yourself in the reader's shoes, thinking that readers may decide whether to read the entire paper based on your abstract. Therefore, the abstract should clearly describe your approach to solving the problem, most importantly stating your most important conclusions. Your concise abstract should inspire readers to want to understand the details of your work. Simply restating the competition problem or cutting and pasting clichés from the introduction of your paper is generally considered a poor abstract.
The conciseness and organization of your submitted paper are extremely important. Necessary statements should present the main ideas and results. Your paper should include the following: restating and explaining the competition problem in your own words. Necessary statements should present the main ideas and results. Logical assumptions and their reasons, emphasizing assumptions relevant to the problem; clearly listing all variables in your model; the design and rationality of the model you adopt/develop; testing and sensitivity analysis of the model, including error and stability analysis, etc.; discussing the advantages and disadvantages of your model and methods; providing algorithms represented by text, graphics, or flowcharts for the computer code you develop (as step-by-step algorithm steps)."
Since 1992, China has held an annual Chinese College Student Mathematical Modeling Contest, modeled after the American version. However, we have fully considered China's national conditions, achieving great success under the leadership and guidance of the Ministry of Education, cultivating a large number of outstanding students and teachers (see §4.2 for details).
3 Why Participate in the College Student Mathematical Modeling Contest
If we could answer in one sentence, it would be: because the College Student Mathematical Modeling Contest is an excellent, specific carrier for nurturing students' innovation and competitive abilities. Below, we discuss why to participate in the College Student Mathematical Modeling Contest from the perspectives of students, teachers, and leaders.
Actually, in our book "Tutoring Materials for College Student Mathematical Modeling Competitions (Volume Four)," Hunan Education Press, 2001, "Chapter One: How to Participate in College Student Mathematical Modeling Competitions," pages 1–22, we already had a relatively detailed explanation. Here, we mainly offer some insights for leaders and faculty and students at higher vocational colleges.
1. For the school's leadership (principals, deans of academic affairs, etc.), wholeheartedly improving the school (high-quality teaching, high employment rates, high-level teaching staff, and increasing reputation, etc.) is undoubtedly their pursued educational goal, and they will take various measures. However, regarding selecting and sending students to participate in the College Student Mathematical Modeling Contest, many leaders (even math teachers) will hesitate considerably: our math class hours are few, the teaching tasks are heavy, even if we participate, failing to win awards will not only fail to enhance the school's reputation but may even lead to some negative comments, etc. Actually, leaders have three issues they haven't considered enough: (1) having a full awareness of the extreme importance of mathematics. Students' future development and achievements are closely related to their solid foundation in mathematics. However, the current mathematics teaching indeed has many areas needing improvement, especially how to teach knowledge and also how to use this knowledge to solve practical problems, which needs strengthening. Encouraging some faculty and students to participate in mathematical modeling activities, especially the College Student Mathematical Modeling Contest, is certainly beneficial for promoting teaching reforms. (2) One key to running a good school is enhancing teachers' teaching levels. How to improve? Encouraging teachers to organize students to participate in mathematical modeling contests and activities like the College Student Mathematical Modeling Contest can help teachers better understand how to use mathematics to solve practical problems and assist math teachers in understanding what kind of mathematics other departments need and how they use these mathematics. Through mutual learning and discussions, they can generate concrete ideas on how to improve their teaching levels, how mathematics teaching can better serve other majors' subsequent courses, and even professional research projects, proposing feasible solutions, ultimately enhancing teachers' professional levels and teaching abilities, thereby raising the school's level. (3) Students' enthusiasm to participate in the College Student Mathematical Modeling Contest is very high, the key lies in how to organize and train them well. Actually, even in higher vocational colleges, there are definitely some students with a fairly solid foundation in mathematics, among whom there are those with strong interests in mathematics, especially using mathematics to solve practical problems. Why not organize them to participate? Cultivating some higher vocational college students with a good mathematical foundation and application capabilities will likely increase their chances of making good achievements in their future work. Having more graduates achieve success in their careers also signifies that the school is doing well and has a high level. Additionally, this will foster some good ideas about differentiated instruction.
2. For math teachers, organizing and guiding students to participate in the College Student Mathematical Modeling Contest will bring great benefits to themselves. (1) Requiring oneself to further study and diligently research the mathematics and methods of mathematical modeling, considering how to help students before the competition, motivating students to actively learn, and summarizing improvements with students after the competition will surely greatly enhance the teacher's teaching and professional levels. (2) Formulating specific ideas on how to conduct teaching reforms, possibly making one's teaching style and teaching more popular with students. (3) Because of a deep understanding, and even personal practice and experience in using mathematics to solve practical problems, it lays a good foundation for exchanging and discussing with teachers from other professional departments, and even collaborating, possibly greatly enhancing one's applied mathematics research level. Of course, young math teachers generally have many class hours, plus family burdens, so it is somewhat challenging to set aside considerable time to organize and guide students to participate in the College Student Mathematical Modeling Contest. The key issue is the school's policy and the teacher's own initiative and dedication. With these, time can certainly be "squeezed" out.
3. For students, participating in the College Student Mathematical Modeling Contest might be a big event in one's lifetime, likely becoming a "once participated, lifelong benefited" good thing. It might even influence some students' entire lives. Because this competition helps students enhance their innovation ability, competitiveness, and some excellent qualities, in a certain sense, it provides an early understanding of the abilities and virtues needed after entering the workforce in the future. (See §4.2 for details.)
4 How to Participate in the College Student Mathematical Modeling Contest
Because there are already detailed and specific recommendations in Chapter One of "Tutoring Materials for College Student Mathematical Modeling Competitions (Volume Four)," here we combine our understanding of the experiences and lessons learned by key higher vocational colleges at various stages of the competition to offer several points of advice for faculty and students participating in the College Student Mathematical Modeling Contest at higher vocational colleges, particularly emphasizing that the post-competition continuation stage is extremely important and providing specific examples.
4.1 Training Stage
1. Combine continuous efforts with concentrated training. Ideally, starting from the second year, there should be an extracurricular activity group or elective course, which can be scheduled for half a day on weekends.
2. The training content can involve teachers启发式地讲解微积分、线性代数、概率统计初步以及数学软件等方面的裁减常识。重点是要提高学生自学的能力。更重要的是以讨论班的形式让学生详细了解竞赛要完成的任务，给学生过去优秀的论文，让他们自己去读、消化和发现问题，然后在讨论班上报告（自认为读懂了不等于真正懂。如果能够讲得让讨论班其他人都懂了，特别是能够在别人的质疑中清晰、准确地回答问题，才是真正的懂，是能力的提升）。在讨论班上老师的作用既要尽可能起到启发和答疑的主导作用，更要用心观察学生学习过程中的问题与困难之所在。
3. Conduct 2-3 mock exams to let students adapt to real combat situations. Guide students on how to write papers, carefully read their papers, and specifically point out strengths, weaknesses, and suggestions for improvement. Identify three students who can collaborate well and possess different abilities (such as mathematics, computer programming, and writing) to form a team. Enhance students' expression and writing skills.
Students' practical abilities are improved during this stage.
4.2 Three Days of Struggle
This is where students independently face the challenge of the competition, akin to the final kick in soccer, reflecting the results of the training while also showcasing students' adaptability. Sometimes luck plays a role. The most crucial tasks to accomplish are:
(1) Allow ample time for careful reading of the problem, conducting thorough discussions, and documenting all assumptions discussed, imagining various approaches, serving as references for necessary model modifications.
(2) Begin mathematical modeling according to the requirements of the problem, answering the questions posed, followed by elaboration.
(3) From the start, assign one team member to draft the initial version of the paper. Pay special attention to writing a good abstract. Ensure sentences are coherent, avoid exaggerations, and always cite others' results when referenced, clearly indicating them in the references section. Maintain a spirit of constant improvement by carefully reviewing, checking, and revising your own paper repeatedly.
(4) As it is impossible to implement all three members' ideas within three days, record any unfulfilled ideas, solutions, and references consulted prior to submission for potential use in the post-competition continuation phase.
4.3 Post-Competition Continuation Phase
The end of the competition does not signify the end of the challenges for participating students; in a sense, it marks the beginning of true gains. There are two reasons for this.
Firstly, most students have numerous ideas during the three-day competition period, but due to time constraints, they are unable to try them out, nor do they know if these ideas are better. Moreover, they need to consider and deeply study whether there are flaws in the results they have already achieved, properly summarize, and discover their own strengths and weaknesses. Only those who are good at summarizing and identifying their pros and cons can continuously improve and achieve greater success.
Secondly, the sole prohibition of the competition—no discussion of the problem with anyone outside the team in any way—no longer exists. Teachers and students can now freely discuss, deliberate, and further research related issues. For instructors or other young teachers, delving deeper into the competition problems often provides excellent research topics, even for Group B problems. Indeed, this has proven true, as evidenced by the numerous papers published by teachers or students in various journals (including foreign journals such as the Journal of Mathematical and Computer Modeling) related to the competition problems. These works have enhanced the academic levels and research capabilities of teachers or students, reciprocally improving the teaching levels of teachers. Many teachers have thus earned the title of "Excellent Teacher" and various rewards. Some profound results have also generated significant social and economic benefits.
(Written by Professor Ye Qixiao, Beijing Institute of Technology)