How to deal with misapplying mathematical rules?

I have noticed that though I understand the mathematical material in my classes nigh perfectly, I still make frequent careless mistakes. Roughly 55% (95% CI: (37, 72) of these errors are due to me misapplying mathematical rules. For example, I may in a derivation go from $a = 4(c + b)$ to $a = 4c + b$, even though I am aware that $q(r + v) \neq qr + v$ in general.

How can I make this misapplication of rules less devastating on tests?

I am slow at math, and my tests are fast-paced, so I usually barely have enough time to finish tests when dong several steps at once and not checking my work. This prevents me from using many techniques. It has been suggested to solve problems multiple times in the same way, which I rarely have time to do. It has also been suggested to not do multiple steps in a single line, but again, I seldom have time to do this, and even when I do do it, it only slightly decreases the number of errors I make.

The scientific papers I’ve seen don’t help much.

I currently keep a journal of errors, but I have found few patterns in them, so I have a hard time determining what errors to check for.

It has been suggested to continue practicing, as this will make me eventually improve. However, despite practicing roughly forty hours per week, I have not noticed an improvement other than ones that resulting from me improving my techniques.

One way I have succeeded in improving my techniques is by first checking to see if the solution seems correct, for example by seeing if it seems reasonable or plugging the answer to an algebra problem into the original equation. Then, if the answer is incorrect, I go back to the first line in the derivation, cover up the next, determine what the next line should be, look to see if that is what was written, and then repeat this for all lines. The problem with this is that it if time-consuming, which prevents me from finishing the test, and it only prevents roughly half of errors.

So, how can I deal with misapplying mathematical rules?

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I taught gifted students at the elementary level and have had success helping students become less careless. I suspect some of my strategies might help you.

If you make mistakes, you are practicing making that mistake, so now you have to undo practicing getting them correct. I would suggest you isolate the kinds of mistakes you make (eg distributing incorrectly – a=4(c+b) becomes a=4c+b)

Once you have a list of such mistakes, go to Khan Academy and practice the specific skill that you are making careless mistake with. On Khan Academy you can isolate a skill and practice it. You will have to get 5 right in a row, before they consider that you have practiced it. After you have practiced whatever skills you have isolated, you can then take Khan Academy’s mastery challenges which you do over several days. They will give you different skills that you have practiced and give you a chance to move up levels until you have mastered the skill. This will ensure that you have really made these “math rules” second nature and will also help you speed up.

I have found in general that students who take tests slowly need more practice. With enough practice they will go faster.

I hope this strategy helps.

Here are some more aspects, which could be helpful:

  • Distinguish between understanding and realisation

It’s good and necessary to gain a proper understanding for mathematical rules. But to correctly apply them is a different story. To do calculations without mistakes you need a lot of routine. The more you perform these calculations, the more you apply mathematical rules, the better you will perform. So, the message is do it, do it again and again.

  • There’s more than one way to do it

Typically there is not only a single way to solve a problem. Think about different possibilities to correctly apply the mathematical rules you want to master. This will enhance your creativity and it also enables you to check the solution of your problem from a different point of view. It’s a smart way to find mistakes.

  • Test and training situations are usually different

When you do exercises at home, don’t think too much about timing constraints during tests. Use the private time to apply with care the calculation rules and don’t be hasty. This way you can better train doing correct calculations. But sometimes it could also be useful to do some kind of test simulation.

  • Have fun and don’t worry

It’s not helpful for your mathematical development to worry too much about mistakes. Better laugh about them and try to see them as opportunity for improvement. Nobody is perfect and the others also fail sometimes.

  • Be confident! 🙂

In the interest of not repeating things that have been told to you already, I’ll try to offer something else. Be it in mathematics, athletics, music, public speaking, or anything based on performance, you will never reach your best unless you practice under duress. From what I’ve understood, the issue here is test-taking skills, not math skills.

If you’re anything like me, during a test, you get nervous. Your heartrate increases, you start sweating a little bit, and your mind starts racing a bit more than usual. I’ve had the same experience in all of the above activities: mathematics, athletics, music,and public speaking. The only way you can conquer this is effectively simulating duress and practicing under it. You have to get used to it, or else you’ll always perform at $<50$% of your capacity.

A few good methods for practicing are as follows (in order of increasing efficacy):

1) Practice with a timer. That is, if you have an hour long test, time yourself for an hour uninterrupted. For me, this doesn’t work, because I know in my head that it doesn’t really count for anything.

2) Make bets with friends. If you have friends in your classes, and good material to use as sample tests, put money (or some other bet) on your performance on a practice test. Maybe this isn’t ideal, but you’ll likely feel some sort of anxiety, and most people like to compete to beat their friends.

3) Practice before you sleep and when you aren’t in the state of mind for focusing. If you’ve been reading silently for a few hours, no doubt you are pretty relaxed and able to focus. If you just get in the door and sit down to practice, you don’t get time to collect yourself- just like on a test.

4) Practice beyond what you think is necessary. Many times I have felt ready, but in hindsight realized I wasn’t. Many other times, I have spent in excess of $25$ hours studying for an exam I felt I was ready for initially. The end result for me has been that the latter case always results in success- no matter how hard the test or how much I panicked.

5) Learn everything. Provided you have time, don’t dismiss things as “probably not on the test.” If you don’t have adequate time, prioritize.

A few good test-taking tips (which you may/may not have heard) are as follows:

1) Easy questions first. This speaks for itself.

2) If time-pressure is your problem, make your goal to solve all but one problem. Often-times the lack of time pressure makes you work better, and you end up having time for the test after-all. Not to mention, $9/10$ is better than $5/10$ from making mistakes everywhere.

3) Eat some chocolate or something with mild sugar content beforehand. This will make you a little bit more attentive. No coffee immediately before.

4) Sleep enough the night before, always. How much is enough is in the eye of the beholder, for me its $6$ hours, but if you need $8$, sleep for $8$.

5) Time out the problems, allot a roughly even amount of time for each one. If you indeed do the easiest questions first, you’ll be ahead of schedule.

Maybe this is nothing new, but the way I read your question, this sounded more important than advice on mathematical mechanics.

This is not a question about mathematics; it’s a more general question about learning to be consistent under pressure. How does a musician learn to perform a difficulty phrasing? How does a tennis player development an accurate backhand?

The answer is the same in both instances. You practise again and again to be perfect under relaxed conditions. Only when you can do it flawlessly without any pressure do you increase tempo or the level of difficulty, and then only slowly.

It’s the same with mathematics. When you practise, you should be much more concerned with accuracy than speed. You should make sure that you have a really complete understanding of all the nuances of the problems you look at. When you are given enough time, you should not make a single mistake. Working tirelessly at eradicate mistakes at a low tempo will breed good habits; in particular, the habit of not making mistakes.

Will this solve your problems? Not necessarily. Not everyone can learn to be as quick as von Neumann. Not everyone can play the violin like Jascha Heifetz, or tennis like Björn Borg. But everyone can improve, and I don’t think there is a better way to do so.

I used to have this problem quite a bit, and it still regularly relapses. It tends to happen the most when I am unintentionally thinking about previous steps or future steps in the solution, rather than the one I’m actually, currently doing. I also have a tendency to just “jump right in” and start solving before I have thoroughly reviewed the question, and it often happens then.

The method that helps me most is to be thorough in the solution, mainly by recording all the steps and avoiding mental calculations. Consider the example:

\int \frac{1}{x+1}dx = \int \frac{1}{u} du = \ln |u| + C = \ln|x+1| + C

I know the answer from the start, but I still record every possible step. It takes more time, but it means that I can very quicly find a careless mistake. It also helps to find easier ways to check your work. If I had a very complex definite integral, it might be easier to estimate the area using geometry than by re-analyzing all of my steps. Also use your calculator to check arithmetic.

If all else fails, temporarily skip the problem and come back after successfully solving two other problems. You will likely forget about the original problem and give your brain a chance to refresh.

I realize I haven’t provided a method to avoid the mistakes altogether. I’m not sure there is one. Hopefully this advice will at least help you manage the problems better; it definitely helps me.

May I ask you to scan a piece of your calculations? I know it seems strange but I’m saying that because I’ve been noticing that many times the reason for miscalculation in highschool/undergraduate students tests is a calligraphic problem.

Most of the times students write numbers and long calculation in a way which very very often lead to miscalculation. Even if the writing itself is fine, but many times is not suitable for long mathematical-analytical calculations.

For example letters too big, to circular, too separated, while perfectly fine in an english test or even in higher mathematics, many times in straight and force brute calculations can lead to errors.

I know seems strange and maybe off-topic but since nobody pointed that out until now I wanted to ask …

Be CLEAR. Write every step out and be clear with what you’re doing. Also MOST classes will have answers that look somewhat reasonable. The answer isn’t going to be some awful fraction most of the time. Your solution shouldn’t be more than a certain length depending on the class. If most problems similar to the one you’re doing is a half page, then you should make sure you have around a half page of work.

Also on tests, things that you know “pretty well” turn into things that you “kind of know” and things you “kind of know” turn into things you don’t know.
Mistakes are just a part of math. You can try to be clear and study and practice, etc. but mistakes are a part of life. You just try to learn from them.

Never trust anyone who doesn’t make mistakes. 🙂

When the examiners mark papers they are more interested in correct methodology instead of algebraic accuracy, so if you make an ‘algebraic slip’ but your logic is correct you will only lose at most one or two marks. This is known as “error carried forward” and to illustrate this; imagine you were asked to evaluate $$I=\int_{x=1}^{\sqrt{3}}\frac{1-x^2}{1+x^2}\mathrm{d}x$$
and you recognise that an appropriate substitution would be $x=\tan\theta\implies\mathrm{d}x=\sec^2\theta \mathrm{d}\theta$ then you change the limits $x=1\implies \theta = \tan^{-1}(1)=\frac{\pi}{4}$ and you accidentally write $x=\sqrt{3}\implies \theta = \tan^{-1}(\sqrt{3})=\color{red}{\frac{\pi}{6}}$ (instead of the correct $\color{green}{\frac{\pi}{3}}$). So you continue and deduce that

$$\int_{x=1}^{\sqrt{3}}\frac{1-x^2}{1+x^2}\mathrm{d}x=\int_{\theta=\frac{\pi}{4}}^{\frac{\pi}{6}}\frac{1-\tan^2\theta}{1+\tan^2\theta}\sec^2\theta \mathrm{d}\theta$$ and you successfully recall the identity $1+\tan^2\theta \equiv \sec^2\theta$ then $$I=\int_{\theta=\frac{\pi}{4}}^{\frac{\pi}{6}}\frac{1-\tan^2\theta}{\sec^2\theta}\sec^2\theta \mathrm{d}\theta=\int_{\theta=\frac{\pi}{4}}^{\frac{\pi}{6}}({1-\tan^2\theta}) \mathrm{d}\theta=\int_{\theta=\frac{\pi}{4}}^{\frac{\pi}{6}}({1-(\sec^2\theta-1)}) \mathrm{d}\theta$$$$=\int_{\theta=\frac{\pi}{4}}^{\frac{\pi}{6}}({2-\sec^2\theta}) \mathrm{d}\theta=\left(2\theta-\tan\theta \right)\bigg|_{\theta=\frac{\pi}{4}}^{\frac{\pi}{6}}=\left(\frac{2\pi}{6}-\tan\frac{\pi}{6}\right)-\left(\frac{2\pi}{4}-\tan\frac{\pi}{4}\right)=\left(\frac{\pi}{3}-\frac{1}{\sqrt{3}}\right)-\left(\frac{\pi}{2}-1\right)=\color{red}{1-\frac{\pi}{6}-\frac{1}{\sqrt{3}}}$$ instead of the correct answer of $\color{green}{1+\frac{\pi}{6}-\sqrt{3}}$. If this question was worth $10$ marks in an exam you would get at least $8$ or maybe $9$ marks even though the end answer (marked $\color{red}{\mathrm{red}}$) is incorrect.

So, since time is an issue the best advice I can give you is to check your working once but no more. Do this to save time for showing the examiners correct logical reasoning (which is the most important part); as the objective is to score as many marks as possible your priority should be to move on as quickly as possible and answer as many questions as fully as possible.

Also, when I took my math exams, to acquire some confidence early on in the paper I used to underline the questions that I felt I could handle; and answer those first. Since you are not obligated to answer the questions in order.

You are experiencing a paradoxical problem.
You understand the concepts, but you misapply them.
You barely finish tests in time, but you have had great confidence that
you did them perfectly or nearly perfectly.

Many people find they “freeze up” on tests due to stress,
and they then forget how to do things.
Possibly you have the opposite problem.
You may be too relaxed.

In my experience, to do well on a timed exam I need to be in an unusual
psychological state. I did not quite realize this until I had been
out in the workforce for a few years, where I had days, weeks, or even months
to work out the solutions to problems, and then decided to to back to school.
I found out I had forgotten how to take timed exams, and had to relearn it.

While out of school I had not forgotten how to do math,
but I had come to take twice as long to work a problem in an exam
setting than I did when I was a student.

A little panic (but not too much) can keep you moving and
help you maintain a focus on doing what you need to do to answer a question.
A little fear (but not too much) can make you more alert to mistakes that
you may be making.

Clearly, whatever you’ve done during practice has not helped,
except for a some minor improvements in your technique.
Actually, improvements in your technique should be a major goal of practice,
not an incidental byproduct.
Forty hours a week is an awful long time to spend on practice;
somehow, you need to practice much more effectively in less time.

I have rarely if ever tried doing the same problem multiple ways on an exam,
but I find it is useful to do the same problem multiple ways
outside of an exam setting.
There are various benefits to this. One benefit, of course,
is that you get practice in each technique you use,
but a particular benefit is that you have a chance to compare the
effectiveness of the different techniques that can be applied to a
particular kind of problem: which ones are easier to apply,
faster, and less error-prone.
The best technique for one kind of problem
may not be the best for another kind of problem, even when it is
applicable to both kinds of problems.
In fact, the best technique for someone else to solve a particular
problem may not be the best for you to use on that same problem;
this is something that only you can discover for yourself.

In short, the point of practice is to solve problems faster and with
fewer errors. If you see no improvement in speed and no improvement
in accuracy, you need to try different techniques or different variations
on your techniques.

The particular example you give, transforming $4(c+b)$ to $4c + b$,
is an all-to-likely mistake to make.
We write formulas out in a sequential fashion, left to right,
but the distributive law does not work in such a sequential fashion:
it’s more like filling in the cells of a rectangular grid.
If you are multiplying a multinomial of $m$ terms by a multinomial of $n$ terms,
you are (in effect) filling in the cells of an $m \times n$ grid.
Every cell needs to be filled with the product of some term from
the first multinomial with some term from the second multinomial.
Distributing a monomial (such as $4$) over a multinomial (such as $c + b$)
is really no different; the grid in this case is merely $1 \times 2$ cells.
If you remember that every term of the product of your
input expressions needs to be the product of two terms from the
input expressions, I think you’re a lot less likely to simply copy
the $b$ from one line to the next and forget to multiply by $4$.

There is even a technique in the U. S. Common Core math curricula that
has students literally drawing a grid in order to perform multiplication.
I have never done this myself, but I do find that a mental “grid” is
a useful visualization.
An easy and quick check of my accuracy is simply to count the number of
terms in the product: if I multiplied an $m$-term multinomial by an
$n$-term multinomial, there should be $mn$ terms in the product.
(For this reason, I will often delay combining terms, for example I may
write $+4xy$ and $+2xy$ in two places in my answer rather than just writing
$+6xy$ once, until after I’ve counted the terms.)
For a complicated product where there’s a chance I did one term twice and
forgot another, I might methodically step through each term of the first
multinomial and (for each term of the other multinomial) check that I have
written the product of the two terms; and when I find that product I literally
make a mark to check it off so I don’t count it again.

In some cases I have even written out products of multinomials in
“long multiplication” format (as if I were multiplying multiple-digit numbers)
on a piece of scrap paper.
Come to think of it, maybe that Common Core grid could be a useful
technique to use literally, sometimes.

You don’t necessarily want to use any of the techniques exactly as I have
described them, but you do want to find techniques that work for you to multiply multinomials by other multinomials or by monomials.
You need to find techniques that are fast and accurate when you use them.

You also need to find fast and accurate techniques for doing every other
thing on which you have made errors in the past.
There is unlikely to be a single “silver bullet” that will fix everything.
The patterns to look for probably aren’t going to be recurring throughout
all the mistakes you make; you might want to consider any mistake you have
made more than once to be a “pattern”.

After you have found fast and accurate techniques to do a few of the
operations that have been giving you trouble, try the
techniques again and make sure they are still fast and accurate.
It does no good to discover a technique if you later forget how to do it.

Until you can take a past exam paper, or a set of exercises from a textbook
comparable to one of your exams, and completely work them out under exam
conditions with plenty of time to spare (checking this with a clock),
don’t tell yourself you have mastered the techniques or even understood
the material.

And once you have mastered suitable techniques, and learned how to recognize
which techniques to apply to which problems, you should be finishing your
exams quickly enough that you do have time to apply some of the other
self-checking methods that have been suggested.