陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「On multiple choice questions in mathematics」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

Now that the project to upgrade my old multiple choice applet to a more modern and collaborative format is underway (see this server-side demo and this javascript/wiki demo , as well as the discussion here ), 一个常见想法是 it would be a good time to collect my own personal opinions and thoughts regarding how multiple choice quizzes are currently used in teaching mathematics, and on the potential ways they could be used in the future. The short version of my opinions is that multipl

In principle, it would seem that the unambiguous and precise nature of mathematical statements would lend itself well to the multiple choice format; in contrast to some other disciplines of knowledge, many questions in mathematics do have a single and objective correct answer, with all other answers being agreed upon as being incorrect. With a multiple choice quiz, a student can be tested on such questions in an objective manner; indeed, the grading for such quizzes can even

已知结果和反例

On the other hand, the multiple choice format, as it is currently used in maths exams, has a number of serious weaknesses which, in my opinion, render it inferior to other examination options for most upper-division maths courses, although there are ways to remove the most glaring defects of the format. Perhaps the most obvious problem is the zero-tolerance approach to mistakes, which can distort the relationship between aptitude and credit: a student who had the right approa

A more insidious problem, however, is that these quizzes give a misleading impression of what mathematical problem solving is, and how one should go about it. In actual mathematical research, problems do not usually come with a list of five alternatives, one of which is correct; often, figuring out what the potential, plausible, or likely answers could be, or even what type of answers one should expect or whether one should ask the question at all, is as important as actually

证明或构造的主线

But perhaps most of all, multiple choice questions promote the idea that the answer to a mathematical question is more important than the process used to arrive at that answer (and the insights acquired during that process, and the art of communicating that process effectively to others). In truth, the process is far more important than the answer, particularly if the answer is to an artificial question, such as one designed specifically for examination purposes. Knowing the

I have discussed my reservations about the use of multiple choice quizzes in classroom examinations, particularly in upper-division mathematics courses. On the other hand, I do feel that such quizzes can play a very useful supporting role in self -examination for such courses, particularly with regards to foundational material (e.g. definitions or basic rules of calculation). I will illustrate this with a hypothetical course in high school algebra, though the point is certain

阅读时建议盯住的点

Suppose this algebra course is intended to teach students how to solve various algebraic equations. There are of course several standard pitfalls that the student encounters when actually trying to solve such equations; a common one is starting with an equation such as and concluding incorrectly that , when instead the best one can say is that or . Now, one can caution against this error in classes, and the student may even write down this warning when taking notes, but it st

This is where a self-administered multiple choice quiz (in particular, an online quiz) can help, with questions such as

值得单独记下的条目

• x is either equal to or .
• f is smooth and rapidly decreasing.
• f is absolutely integrable.
• f is continuous and compactly supported.
• f is a tempered distribution.
• Integration by parts, differentiating and integrating .
• Integration by parts, differentiating and integrating .
• Trial differentiation, using functions such as .

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源