数学建模是一项具有挑战性的活动,不但新手有这样的感觉,已经从事数学建模多年的专家也会有这样的感觉。积累建模经验无疑能提升解题的效率,但绝不可指望照搬前一个问题的解决方法到新问题的求解上。时间和环境的改变会影响模型参数和假设条件的设立,某些参数或许需要放大,某些假设或许需要更改,某些参数或假设或者需要舍弃。已经轻车熟路的问题,即使参数和假设都不变,仍有可能仅仅因为需要回答新的问题而变得难解。

数学建模的这些特点使参加MCM竞赛变得意义非凡。无论竞赛结果如何,同学们愿意付出一大块自己的时间去思考和解决一个不熟悉的问题,仅此一点就足以体现同学们的强烈求知欲望和进取精神。而参加建模竞赛所体验到的成就感以及所获得的结果对知识的贡献,无疑会吸引更多的同学参加数学建模活动,好比打高尔夫球,击出的好球能改变原本乏善可陈的局面,并希望继续打下一场。

一位作家朋友说过,小说能获得心灵感应。作者用文字向读者描绘一个场景、表达一个想法、或叙述脑海中的一个情节,如果写得好,就能使世界另一边的读者“看”到作者想表达的意图和景象。数学教学、特别是数学建模的教学,在许多方面是一个类似的过程。

作为教育工作者,我们使用各种符号、图表、文本和图像向学生们解释难懂的概念,希望同学们能从其中一些表述方式中对这些概念获得更深刻的认识,并以此为立足点建立自己的符合逻辑的理解。这种理解能帮助同学们在数学、科学、及工程领域内获得创意,而这些创意是不能通过其他手段获得的。模式匹配和模仿,虽然表面上也许能产生类似的结果,但却不能从根本上激发创新。

自MCM竞赛开办以来,评委们对如何将数学建模的各个部分恰如其分地融入竞赛论文有许多认识,这套系列丛书陆续发表了许多这方面的建议和评论,其主要目的是帮助新参加数学建模的团队和机构了解MCM竞赛的目的及细节、指导教师在帮助团队准备参赛中的重要性、以及如何在有限的时间内完成MCM竞赛所要求的各项任务。

建议参加MCM竞赛的同学,即使已经受过良好的数学建模训练,在递交论文前也应按这些忠告详细检查论文内容的完整性。这些忠告不是给同学们提供按部就班的取胜秘诀,而是提供一个框架,帮助团队启动建模步骤并明确应该朝什么方向努力,写出满意的解答。

生活中许多事情不可急于求成,没有捷径可走,数学建模就是这样的事情。学习建模技巧和使用建模步骤的模板能对建模过程有所帮助,但理解需要时间。正如此,每个团队的解答都是特殊的,我们必须牢记这一点。



Patrick J. Driscoll 博士

美国西点军校运筹学教授

MCM竞赛主席

2016年4月

Foreword by Patrick J. Driscoll

Mathematical modeling is a challenging endeavor whether you are new to the task or have been professionally engaged in it for many years. Efficiencies are gained with experience, but never at the cost of assuming a new problem can be solved in the same way with exactly the same ingredients as a previous one. Time and circumstances affect modeling parameters and assumptions, amplifying some and negating others. Simply asking a different question about the exact same situation can introduce intractable elements into a previously well-defined problem.

These facts, among others, mark the participation of MCM teams as extraordinary, no matter what level of award is achieved. The mere fact that undergraduates are willing to commit a solid block of their own time to tackle an unfamiliar problem speaks highly of their motivation to learn and develop. Ah, but the satisfaction of experiencing a model function as intended, and yield significant results pertaining to a line of inquiry – that is what will bring someone back for more. Like the one beautiful golf shot on an otherwise unremarkable round.

A writer friend of mine once said that a novel is a bit like achieving mental telepathy in that an author conveys a scene, an idea, or a plot existing in her mind into the mind of a reader using text or illustrations. If successful, someone on the other side of the world could read the text and ‘see’ the imagery that the author intended to convey. In many ways, teaching mathematics, and mathematical modeling in particular, is a similar endeavor.

As educators, we use various symbols, diagrams, text, and graphs to present our understanding, our imagery of challenging concepts to students in the hope that an insight gained from one or more of these approaches will become a foothold upon which students can construct their own logical understanding. When students attempt to do this, active learning occurs. When students attain this ownership, a level of understanding occurs that leads to innovation and creativity in mathematics, science, and engineering that is not possible to achieve through other means. Pattern matching and mimicry, while appearing to produce similar results, are deceptively inadequate to the task.

Since the beginning of MCM, we’ve published a good deal of recommendations and critique concerning what judges consider to be elements of a proper engagement with mathematical modeling. This advice was primarily intended for teams and institutions new to mathematical modeling and its nuances, recognizing both the important role that a faculty advisor plays in preparing teams to compete and the limited time within which the tasks required by MCM must take place.

Teams participating from institutions with an existing strong commitment to mathematical modeling might use such advice as simply an editorial check prior to submitting their paper. Regardless, the advice was never intended to prescribe a lockstep adherence to a process as a recipe for success, but rather to provide a framework within which MCM teams might initiate their efforts and bring about a respectable level of completion.

There are some things in life that cannot be rushed, no expedients exist that can accelerate them, and while technologies and templates exist to augment and enhance processes associated with these activities, understanding demands time. Mathematical modeling is one of these. And this, we must remember, is what makes the accomplishments of every MCM team so special.



Patrick J. Driscoll, Ph.D

United States Military Academy at West Point

Professor of Operations Research

Director of the Mathematical Contest in Modeling

April, 2016.