How to not make basic math mistakes in complicated math classes
August 29, 2020 10:19 PM   Subscribe

I just finished taking calculus III, which while I'm sure is not super high level math to some, is the highest level I've achieved. I did not do as well on the midterm and final exams as I wanted, and it is at least partially because I made dumb mistakes on simple calculations that threw my answers off, or just didn't pay close enough attention on a few questions and came to the wrong conclusion when if I had paid closer attention I would have realized I was wrong. How to not do this?

I did really well on the written assignments - like almost 100 percent on everything - and got 100s on a majority of the weekly quizzes, so I feel like I have a good grasp on the material from the course. But somehow I got Cs or Bs on the major exams as a result in many cases of making simple mistakes. Certainly there were a few questions I just didn't do correctly on both exams because I failed to understand how to do them, but at least half the points I lost were basically just for mistakes/calculation errors. I assume I may get some amount of partial points anyway but as of now I just know that I probably got a solid B (or even a high C but hopefully not) on the final and it's immensely disappointing given the amount of time I spent doing practice problems and reviewing. I did take time to double check my answers at the end of the exam, but apparently I didn't do it thoroughly enough. Like, one of the two questions I know I got wrong, I double-checked all the calculations during the exam but didn't realize that I'd forgotten to take the square root of something.

Anyone else out there who has run into this issue in math coursework with some good tips? Most of my background is more social sciences than hard math or science and I am an adult student going back to school in a more math-heavy field but taking some preparatory courses at the community college. I've been doing well in all the other math courses I took until now (Calc I-II, Linear Algebra) but for some reason this class just got me. Maybe I need to learn how to study for math/science or take their subject exams better or something? Resources/ideas for that, or other analyses of the situation?
posted by knownfossils to Education (17 answers total) 9 users marked this as a favorite
This is similar to the problem of software quality control and debugging. You want to break out the larger problem into as many discreet steps as you can, so that you can re-check them individually.

1. Break up the problem in to individual steps.
2. Double check that the way you've broken down the steps makes sense, and that you haven't made any obvious errors that will change the outcome.
3. Solve each step in order.
4. Double check each step in order.

This is helpful because if it takes longer to do it this way, then you know that you needed extra time to understand the larger problem. ...and if you end up with the wrong answer, then you probably didn't understand the larger problem as well as you thought you did.
posted by Anoplura at 10:47 PM on August 29, 2020 [2 favorites]

The best way to check your own work is to come up with options that do not involve going through your own work line by line. There's a tendency to just end up confirming what you wrote when you do that, and still overlook that square root because you've spent all your brainpower on the steps you didn't get wrong.

One often taught trick is to work backwards. If you've just done an integral, take a derivative. Even just reworking a problem from scratch--if you have time--will often find things you would miss reviewing it.

Another common trick, and much faster and IMO more valuable, is to test your solutions with example numbers or simple cases. This might be things like plugging in zero or very large numbers into equations. Or if you are supposed to be at a local maximum, plug in a slightly larger number and a slightly smaller one, make sure they are both less. (I'm not sure what calc III focuses on, but hopefully you get the idea.) Do this enough and in some cases you'll have a strong intuition that something is wrong when you make a mistake.
posted by mark k at 10:54 PM on August 29, 2020 [29 favorites]

Carefully Do tons of textbook problems, way past the formal hw assignments, to develop “muscle memory” for not making silly mistakes.
posted by shaademaan at 1:42 AM on August 30, 2020 [2 favorites]

For me this was to do with exam stress. At some stage I went from being good at exams to underperforming in them and it was because I wasn’t clear-headed enough in the exam room. Unfortunately I kept thinking of myself as ‘good at exams’ past the point when it was actually true.

I don’t have any specific suggestions for you but looking back at the time of my life when I had to do exams, I wish I spent some of my exam prep time learning about techniques to calm myself down and reduce the adrenaline. I don’t know what that would have looked like, but there must be resources out there that could help.
posted by Bloxworth Snout at 2:52 AM on August 30, 2020 [1 favorite]

From tutoring experience: I noticed most of "silly" mistakes came from skipping steps in reasoning or calculations. I made them spell out each step and not calculate anything in their head. I mean, literally, spell out each step they were making out loud (or write it out). Never do two things in any step even if they are just simple multiplication.

Do tons of textbook problems, way past the formal hw assignments, to develop “muscle memory” for not making silly mistakes.

This. Knowing your material cold and doing A LOT of sample problems reduces the likelihood of mistakes. Just "getting it", ie getting the reasoning right is not the same as mastering the material. It's just the first step.
posted by M. at 3:28 AM on August 30, 2020 [3 favorites]

Seconding M, don't skip steps.

Also, knowing where common mistakes are likely and paying extra attention to those spots.
posted by rakaidan at 5:07 AM on August 30, 2020 [1 favorite]

- ez win, especially don't choke on the arithmetic.
- there's only so many 'flavors' of problems in undergrad calc. As mentioned above, do a ridiculous number of practice problems. Back in my day, when we wore onions on our belt, Schaum's Guides were the gold standard. Dunno what the web equivalent of that is.
posted by j_curiouser at 5:22 AM on August 30, 2020 [1 favorite]

I'd say that getting the calc right and the arithmetic wrong suggests you're well on your way to becoming a real mathematician.

My thought is you have to slow your brain down when you get to the simple steps. Don't do a subtraction while half your brain is thinking about the Hamiltonian or Legrange multiplier in the step. Don't worry your instructor will think you a dolt if you write out 999-997 = 2 in the margin.
posted by SemiSalt at 6:08 AM on August 30, 2020 [8 favorites]

I used to TA calculus in college, and this to me sounds like a test problem, not a math problem. (ho, ho.) In other words, you know what to do, but test stress is giving you brain farts. Here are some methods I found useful for my students:

— Try to study or do practice tests in circumstances as similar as possible to the test. For example, can you do your problem sets in the classroom where the exam will be held? (Maybe not, but if you can, do.) Can you use the same pen/pencil you will use for the test? If you usually work in your PJs but get dressed for class, practice your problems in real clothes. Etc. Basically you want your body to be like “ah, yes, the circumstances under which I perform my ordinary math excellence” and not “my goodness, where are we?”

— If your class has a TA who has been grading your problem sets, they might know your math habits and be able to tell you like “you tend to skip the second-to-last steps” or something.

— Don’t do anything in your head. EVERY thought gets written down. Partial credit adds up, but also, it’s just good math hygiene.

— Instead of just checking line by line, pretend you are telling a story. “What I’m looking for is an area under a curve, so that means I need two things…” or “It is I, the math god, here to flatten you a dimension by taking a derivative! Spheres become circles! Fear me! First, I will..." It’s very corny, but it works really well. Depending on the kinds of errors your making, it can also help you spot a mistake of scale, so something way too big or way too small will be more obviously wrong.

— Eat breakfast!
posted by Charity Garfein at 6:28 AM on August 30, 2020 [10 favorites]

I was a math major and had this problem (for me it's definitely being lazy about and kind of bad at arithmetic, not exam stress, though ymmv). Once trick is that most college math instructors at American colleges will make the arithmetic for test question nice and pretty when the actual math is difficult, so you can often tell you've taken a wrong turn on the arithmetic when you wind up with weird fractions or having to do nasty multiplication (this is extra true once you get to linear algebra).

Also, n'thing write down every step. It'll often get you more partial credit if the TA/professor can see you understood the math but temporarily forgot how to add than if you just come out with the wrong solution at the end. Plus, you're more likely to catch your own mistakes that way.

I'm still terrible at arithmetic, so I've got no long term answer for you. But, double checking work, being skeptical of involved arithmetic, and writing every step down got me a math degree.
posted by snaw at 7:16 AM on August 30, 2020 [5 favorites]

"The best way to check your own work is to come up with options that do not involve going through your own work line by line."

Yes, I'm a big fan of this! And make it part of your practice--your work on a problem isn't done until you've found a second independent way of getting the same answer, or some other way of checking your work. If that means you spend more of your practice time on fewer problems, that's OK.

Sometimes there are also ways you can get a rough estimate before you start--do you expect the answer to be positive or negative? Can you estimate an upper bound? Etc.

In my brief experience as a calculus teacher, I heard "I understood this problem but I just made a dumb mistake" a *lot*. Usually I took it as a sign that their understanding was more fragile than they thought--they'd learned one way to do that sort of problem, but only one, and didn't have the intuition to tell them when they were way off.

But, yes to being methodical and writing everything down as well.

Also, this may seem like small consolation, but: you will get there eventually. If you keep at this, you'll start to pick up additional ways of approaching each problem, you'll develop more facility at the calculations, you'll get more intuition for what to expect. The fact that you're doing well on the homework, and that you're able to do most of the work on the exams correctly, shows that you've made a lot of progress. That means you're capable of making more. Please don't assume you're broken in some fundamental way--you're not bad at arithmetic, or unable to concentrate on details, or whatever--you're just not quite there yet. Sometimes that's not only a matter of studying hard, but also a question of *time*, to reflect and let things sink in. You'll get good at this stuff, even if it's not till halfway through the next course....
posted by floppyroofing at 7:34 AM on August 30, 2020 [5 favorites]

I majored in mathematics in university and I had a similar experience. For me, it was connected to the time pressure of the exam, which made me feel like I had to rush.

Because I was rushing through the exam, I often found myself with lots of time left in the exam period after I had finished my work. The trick I learned to catch some of my errors was to physically put the exam paper aside and think about something else for 5-10 minutes - enough time to “forget” the details of your answers. Then when I went through the exam paper again, I didn’t automatically gloss over some things because I remembered what I wrote.

It also helped to go through the questions again in a different order.
posted by bkpiano at 7:59 AM on August 30, 2020 [2 favorites]

"Silly mistakes" (my second-least favourite term, after "lazy") in mathematics exams were the first clue that I had ADHD. Managing attention during the mechanical arithmetic part of exams has always been very difficult for me.
posted by scruss at 11:46 AM on August 30, 2020

I had my first case of test stress when I went back to school as an adult, too. My extremely old school prof verified that I was actually doing the homework I was turning in and then recommended having a beer or two before the exams. I hardly drink, so I adapted the advice and had a really calming diner breakfast instead - grease! Protein! Carbs! A bit of jam at the end! Sweet milky tea! Worked a treat - my body was pretty sure everything was A OK.

Plus doing so many practice problems that the relevant arithmetic was beyond familiar. Sometimes I needed to back up and do practice from an earlier class in the sequence, but this often clarified and locked in the earlier material now that I knew where it was going.
posted by clew at 11:53 AM on August 30, 2020 [2 favorites]

This is a very common problem. I agree with the idea to come up with cross checks, and I would suggest asking your instructors for ideas about how to do them too.

Also, to make sure you get the partial credit and to make your own re-checking easier, keep your work organized and write down steps. It’s very hard to recheck three steps at once, and even harder for your grader to figure out what you did if you made some mistake in a hidden step.

Last, when you are doing practice problems, make sure some of them are like test problems. Many homework problems involve just one concept, while test problems may involve a few concepts together. If your instructors give exams like that, ask for practice problems that also combine a few concepts together.
posted by nat at 1:00 PM on August 30, 2020 [1 favorite]

When I have TAed and seen mistakes in homework and tests that were "just an calculation error", often the resulting answer "obviously" (to my eyes) didn't make any sense: for example, a formula for a probability that could easily be more than 1, or a formula for an area that clearly had units of distance. I don't know how many of your errors are catchable with common sense like that, but I bet more than zero are. Depending on what kind of math you're doing, some of the following standard checks may make more or less sense than others:

- Is your answer always logically possible no matter what the variables are? (e.g., if your answer is a probability, is there no way to make it negative?)
- Do your units make sense? (e.g., if you're calculating a volume, it would be really weird to see r ² in your answer rather than r³ or dr ²)
- Does your answer make sense if various variables go to 0 or infinity?
- Does it make sense if you plug in numbers that are really easy to work with, like 1 everywhere?

This checklist often doesn't help if you just forgot to divide by 2 somewhere or thought that 6 x 9 = 48, but it does catch a lot of my own calculation mistakes.
posted by dfan at 1:59 PM on August 30, 2020 [1 favorite]

Thanks everyone for your thoughts! Certainly it's a good point that if I had a 100% deep understanding of the material it might have been obvious to me that the answer I got was wrong, and I feel like that sort of logical check is something that didn't really occur to me to do and could have given me a faster inkling into an error if I didn't have time to go back line by line through the entire problem. All of your suggestions and various bits of encouragement are very helpful and exactly what I was looking for, and hopefully will contribute to additional math successes in the future.
posted by knownfossils at 5:28 PM on August 30, 2020 [1 favorite]

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