Dealing with frustations at tests

Question

I'd like to talk about some things that are quite often for me when dealing with math tests... I don't know if this is the right place to talk about it, so if it's not, please comment and I'll delete this post.

I've always been very good at maths, but terrible at calculations and algorithms. The thing is that, recently, I'm getting really frustrated in almost every test because of my lack of attention in some details during math tests... I'm able to prove things without worries, but when it comes to applying theorems in numerical examples, I'm a disaster... I fully understand how they work, what they tell and how to prove them, but I don't know what happens that, in almost every test that I take, I fail at the questions involving calculations.

For example, today I was applying the Euclidean Algorithm for polynomials and wrote $-x^3-x^3=0$ at the first line of the algorithm. The value of the test was $30$ points and this question was valued with $6$ points. I lost $6$ points because of that stupid calculation.

I studied for 2 years as an undergrad in Physics and changed to mathematics, so I'm relatively new in maths. I don't know if this is lack of attention, lack of maturity, or what is it, but that is really making me frustrated, and I'd like to know if some of you guys went through some similar things, and If so, what have you done to overcome.

Thank you, I needed to let off some steam!

Answer 1

Not sure if this is the best place for this sort of question, but I'm going to give some modicum of response anyhow because I can relate to you a lot here.

My personal thought is that some people are just more inclined towards the "higher" thoughts - proofs, ideas, theorems - and others are more inclined towards computations. Sometimes you're just good at noticing the underlying details in calculations, and sometimes you're just good at noticing the nuances involved in proof-writing. Being good at math doesn't necessarily mean you're amazing at both.

I'm personally inclined towards the same things as a math major. The computational details often screw me over, more than I'd like to admit. (The fact that I have poor handwriting makes it worse - I've literally missed whole problems on tests because I misread my own handwriting!) My proofs and such are usually pretty good and I have an understanding of the ideas, but sometimes I get too ahead of myself in the calculations. Hell, not 5 minutes ago, I had to correct a small error I made in helping someone on here on MSE (and sadly it's not my first time doing that).

Anecdotally, I feel like this is a remnant of my middle/high school math team competition stuff, where speed was emphasized. There was a "speed math" sort of thing where we had to solve problems very fast in order to get points. I was the good one on the team early on but at some point - despite remaining the good one for other reasons - the speed lent itself to errors.

What has helped me curb this somewhat and maintain my grades was double-checking my work, going line by line, trying to justify each and every bit - both for calculations as well as proofs. It's not perfect and you have to consciously ensure that you don't get hasty again, but it has helped me quite a bit. Heck, reworking it all over again - but carefully - helps, as would checking your answer. (Checking being, for example, trying a few values for your solution, differentiating an integral, trying to "break" your proof with counterexamples, whatever the context would demand. And in reworking a problem, trying to not "bias" yourself by basically copying down your previous solution - reworking it a different way if at all possible, even!)

It's not much and it might not be even possible if you don't have time - it depends on the testing environment. I've had exams where I've had literally hours to spare, and others where I was working right to the very end, so it all depends. But that's the main thing that I've been trying and it has helped somewhat. Just slowing down, checking your work, reworking if possible ... it's small and too often said, but it really helps if you can maintain the slowing-down bit.

I don't think it's so much a matter of maturity or intelligence or skill or anything like that. Personally, I'd say skill is only in part the computational abilities, and moreso are the higher things like proof-writing. (Thus why I also think it somewhat absurd that you lost all of the points on that question for a simple mistake.) I feel like if you understand the ideas and proofs and theorems and whatever, that's like 80% of the battle. Anyone can plug-and-chug - I've done it without even understanding the material or where the formula comes from - but understanding the material is truly the important thing. At least to me.

Answer 2

I absolutely love mathematics, and I have been teaching myself calculus ever since I took high school geometry last year. It may be hard for some to believe, but I never memorized my multiplication tables. As a consequence of my ineptitude with arithmetic, I hate and am horrible at number-crunching. I prefer math more when it uses more symbols and less numbers.

In any case, I have found that using a high-tech calculator like the TI-Inspire has saved me a lot of points on homework assignments.

If it's a test or something that doesn't allow calculators, I usually walk myself through the really simple calculations. Example: what the hell is $6950-339$? Solution: $$6950-339\\=6000+900+50-300-30-9\\=6000+900-300+50-30-9\\=6000+600+20-9\\=6600+11\\=6611$$ High accuracy, minimal effort.

Other times, it can be really helpful to write down formulas you are about to apply, even if you know them by heart. This helps me when I am sort of stressed and likely to have a brain fart and make a stupid error in calculations.

Trying to plug numbers into an un-simplified formula then simplifying the numbers is a waste of time and likely to cause errors. Always simplify your formulas as much as possible before plugging in anything.

As far as I can tell you don't have any trouble remembering formulas, but in the event that you do, it could help to learn how to derive them. One fully understands a formula once one can prove it, and once one understands it, it is hard for one to forget it. In most cases learning the proof often requires more effort, but if you're like me, that shouldn't be any problem because proofs are all worth learning for their own sake.

And of course your intelligence is not quantified by the speed of your mental calculations or by how many digits of $\pi$ you've memorized. What matters is the fact that you are willing to learn mathematics.

Hang in there :)

Inclusion:

If time on tests is the problem, try talking to your prof. so you can get extra time on tests. I have a really slow processing speed, and it's not that things are conceptually difficult, I just need more time. I've often been the last student to finish, but one of the highest scores in the class.

Answer 3

One thing I've noticed happening when working on a piece of maths is that sometimes, even after I've checked mentally that what I'm about to write down is correct, my fingers have other ideas and what actually appears on the paper is something else. For example I might intend to write $a_n$, and think that I've written $a_n$, but what's actually there is $a^n$.

I'm not sure why this happens. I think it's either because I'm already thinking about the next step, or because my hand has the "muscle memory" needed to write the wrong expression and simply goes ahead and performs the movements. (This kind of automaticity is a goal when learning a musical instrument.)

I don't know whether this happens to other people too. But in my case, it means it's not enough to carefully check what I'm about to write—I have also to check that I did actually write it.

Just another thing to consider in trying to identify where the tendency to make errors is coming from.

Edit: I posted some related thoughts on error causes in my answer here: https://matheducators.stackexchange.com/questions/14853/how-to-make-a-student-not-overlook-easy-mistakes-such-as-the-wrong-sign/14911#14911