On Mistakes

It’s too much for me to lost it. I’ll copy and paste it here. This is a reply to a Piazza question on specific suggestions for avoiding math mistakes (that the author has listed).


Some practical advice on the list of mistakes you humbly provided. I can see two major type of mistakes.

Type I: misread the question (1b, 5b) / glossed over essential details or distinctions (2) / jump to conclusions prematurely (1c, 6)

Let me summarize how other students tackle this from what I saw marking the midterm

-          Circle / underline / highlight details as you read (and I suppose it should be done SLOWLY, I feel that you really rushed too much)

Reason: Be careful in the beginning but getting slack later IS BETTER than being “a little bit more” careful all the time.

Even if you are not the “careful” type but the “impatient type”, make sure you stay patient in reading the question AND PROVIDE HINTS for yourself when you dive into the panic of relating everything from the mind to the paper.  The advanced student actually ADD relevant equations, simplifications, naming variables NEAR THE highlighted details.

Good habits of marking details DECREASE your time to find them, and there is a higher chance that you get the correct SOLUTION STRATEGY at the beginning. It is WELL WORTH the time.

My elementary school math teacher FORCES us to underline the SHORTEST part of a question that is the SUBJECT of the question. I cannot thank her more for this training. (You won’t give an answer in thetas when it’s an indefinite integral on a function of x, for instance)

-          Quick checking steps in crossed-out scribble / near the margin in a box / on the back on the previous page.

Reason: From my impression, almost all super-high accuracy people have this habit, and checking IS NOT just repeating / eyeballing the steps you did, that’s too infrequently useful. While checking, you ask questions like:

  1. Does the value make sense?
  2. Does it feel too complicated / simple?
  3. Could I reverse the steps to the first line? (Very very important in integral calculus)
  4. Am I answering the subject of the question?

However, checking techniques are specific to question types. You should BUILD UP YOUR CHECKING HABIT during the time you do the suggested problems. I’m afraid webwork only wouldn’t work that well because you can just try again. Deliberately train yourself to apply checking habit while you do written work.

-          Quick sketches are always your friend. It seems that the number of students who get the right overall strategy if the sketch is correct, is higher than the number of those who don’t sketch at all.

Reason: Whenever applicable, visualizing is a powerful skill to hint at the solution strategy AND PREVENT silly conclusion-jumping. It is simply harder to jump to the conclusion if you’re shown the correct picture. Like Q6, if you try to sketch the area profile x e^{-x}, you probably can see why neither x =  0 nor x = 2 could possibly be the solution. The critical point of this function is easy to find (answer x_c = 1), and you could imagine the maximum volume is obtained by choosing an x somewhere before x_c=1. (Come on, don’t just read what I say, you won’t understand what I say in any REAL SENSE, if you don’t do the sketch right now)

I was really curious how you guys performed on Q6 (I’m marking another question), and I found many who found some in-roads and got partial marks have a pretty good sketch and label the unknown they define on their sketch. Usually, they also write down the correct integral but may not succeed in the algebra to optimize the integral.

Type II: Technical mistakes in problem solving

This is way harder to avoid. Hopefully, rectifying Type I would give you a grade above 70s already. Below are my personal advice, feel free to comment on.

-          Make sure you zoom in and zoom out.

Reason: A good problem solving process nearly always involving “zooming in” to some details, then “zooming out” to make sense of it, then “zooming in” again to other details and so forth. One seasoned problem solver usually has a sense of the structure of your solution, and you know how much you should zoom out to make sure you’re still good. The most important time to “zoom out” is when you feel you are writing down the solution / important key interim steps.

-          Defeat the wishful thinking of a “1-step solution” in the first second.

Reason: Though 1a) was really easy, if you didn’t think twice before going to other questions, you might not be careful enough. Sometimes, it’s not your mathematical thinking that went wrong, but your wishful thinking (that can happen especially when you’re emotionally stressed and panicking) is HINDERING you to see the question as it is. Be skeptic whenever you think I’m going to “see the solution”, like right now. Then ask yourself subconsciously “what’s that trick that will get me there immediately?” Sorry, math just doesn’t work that way, and I suppose many more things in life neither. A scholar usually has the character to practise “useful skepticism” and has developed a personal art of it.

I was thinking many less people would not have lost the 3 painful marks on 1c) (Q6 is only  4 marks!!!), if they were just a little bit more skeptical of their “one-step solution”.

-          Practise structured writing, and use symbols / special marks to HIGHLIGHT your key steps

Reason: I have seen too many students’ work in the midterm which I believe is unintelligible even to the authors, and I strongly suspect if these people will be able to tell me what they did on that question. That ought not to be so. Having a poor presentation doesn’t really harm the marker but yourself, because you don’t stand a second chance of correcting most of your mistakes. Unless you’re pretty sure you WILL get it right in the first trial (or you’re sure you won’t have time to go over your solution again…), it doesn’t make much sense to obstruct yourself from checking it.

Nobody deliberately hinder him/herself, but without a clear structure (headings, sections, paragraphs, key questions in a new paragraph, boxed substitution step etc), the chance that you got a perfect solution goes down exponentially with the length of your writings. That’s probably why many math / computer science people are fans to write out their work as “elegant” and “clear” as possible, simply because it makes the THINKING inherent in your writings transparent.

ABOVE ALL, DELIBERATELY PRACTICE THESE ADVICE (AND ANYTHING ELSE YOU GOT ELSEWHERE), WHILE YOU DO YOUR ROUTINE PRACTICE (e.g. make your webwork written work, do suggested problems or other problem sets)

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