Incredible frequency of careless mistakes

Ever since high school, I've had a serious problem with math classes. Be it discrete math, algebra, calculus or linear algebra, I seldom have trouble understanding the texts or lectures, but when it comes to executing calculations on homework and exams, I can never do it: I keep making mistakes. It's been a constant source of frustration, and right now, I'm on the verge of failing linear algebra and second year calculus (which I've been avoiding for as long as I could).

I'm currently going through all the exercises and videos on Khan Academy, beginning with one-digit addition, working my way back up to integrals. I was making a lot of mistakes with multiple digit subtraction and multiplication, so initially, I was hopeful that maybe with enough practice in those two things, I'd be good to go again. But it turns out that my mistakes aren't confined to any one task, and the number of mistakes I make per exercise just accumulates as the tasks become more and more computationally involved:

• Thinking that 7 + 5 = 13
• Neglecting negative signs when transcribing
• Carrying in multiple-digit addition when a digit hasn't surpassed 10
• Using the wrong exponent laws, log laws, limit laws etc.
• Mixing up trigonometric function properties
• Forgetting to reverse signs when expanding negated expressions of multiple terms

A typical page from my notebook:

I've tried everything: going for walks, ensuring that I have good sleep, writing out every step with almost hilarious verbosity, changing rooms, starting with fresh notebooks, reviewing everything I write, trying different notations, heck I even spent a month refining my handwriting and posture for good measure.

I'm certain I spend about twice as much time as other people practicing too. I avoid taking other courses in the same semesters that I take math courses.

As far as personality goes, in general, I'm not exactly OCD, but I'm not careless either. I probably have more patience than most people -- a bit too much, as my friends would tell me. I'm sure you can also tell from my adequate grammar, punctuation and spelling that I probably don't have any kind of learning disability that would prevent me from following symbolic rules. I've gotten near 100% in a first order logic course too.

Yet when I do math, I always screw up. It's been this way for over a decade, maybe more.

I don't know what I'm doing wrong anymore, and I'm losing hope.

What could I do?

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Have you gotten a full physical, including a neurological workup? – marty cohen Sep 28 '11 at 18:45
Dyscalculia sounds like a possibility; it is not incompatible with excellent writing skills. It might be worth your while to get an evaluation from an expert in learning disorders. – Brian M. Scott Sep 28 '11 at 18:54
@Brian Some of those symptoms are familiar, though probably not to as severe a degree as Wiki seems to suggest it should be. If I do have dyscalculia, it's probably not a result of working memory problems: I scored off the charts in numerical memory when I had my IQ tested in elementary and middle school. Though I'm certain my IQ's dropped quite a bit since then, I think arithmetic was frustrating for me even back then. – Rei Miyasaka Sep 28 '11 at 19:48
I’d guess that if dyscalculia is the problem, it’s in a relatively mild form, but I really know very little about it; I’ve just had students who had been diagnosed with it. – Brian M. Scott Sep 28 '11 at 19:59
Should this be CW? – JavaMan Sep 28 '11 at 21:06

@Rei, I've gone through pretty much the same problems as you in school, and also made the same sort of mistakes! And I'll assume yours isn't a medical problem. You mentioned a big list of mistakes as being your problem. You probably need to ask yourself first: is there something fundamentally wrong in your understanding, or is it that you just make mistakes when pressed for time, etc.? If it's the former, get the concept right first, saving the calculations for later. If it's the latter, it's a much more common problem that we all get into. I can't give you a more specific answer in that case without knowing more about when you make these careless mistakes.

If that seems too abstract, here's some concrete help about your $x=480\times 72$ problem. Don't do it multiple times in the same way - instead check whether the answer makes sense in different ways! (This is an extremely important step that I've seen many people neglect, because they get a correct answer. That's really not understanding maths.) First off , $x = (480 \times 72) > (y=480 \times 70)$, where $y$ should be very easy to do :$y= 48 \times 7 \times 100 = 33600$. This alone should tell you that , if you got the correct answer in your worksheet, it must be 34560!

Next step: (I assume you know basic algebra) We know that $x = 480 \times 70 + 480 \times 2 = 480 \times 70 + 960$. If that seems too difficult to compute, just take $x \approx 480 \times 70 + 1000 = 34600$. Now, we finally get your answer of $34560$, by subtracting 40!

Ok, suppose you're still not sure. That's fine, we'll do another quick check. It's easier $z= 500 \times 70$ which is just $5\times 7$ with three zeros to follow, so $z=35000$. And $x$ should be a little less; if you're interested you can find the difference. Otherwise, you can check your computations by finding which answer you got was closest to $35000$, and discarding the rest.

What have we gained in the process?

• We admitted that certain computations are hard to do by hand.
• Hence, we looked for concepts to simplify those computations, and looked for approximate answers.
• We checked our answers several times, in multiple ways.

Here're some thumb rules to keep in mind:

• Human beings make errors in a variety of situations, so you're no exception!
• Since you mentioned errors in multi-digit addition, subtraction , etc. you may want to try those first,ignoring everything beyond.It's not important that you compute very fast, but it's very important that you get the algorithms very clearly.
• When you get an answer, just stop and ask, "how can I check this"? Check your answer any way you want - use a calculator, computer, etc.

Here're some simple exercises to try out; you may want to try finding an approximate answer first, without doing any pen-and-paper calculation:

• $99 \times 99$
• $3.14 \times 2.99$
• $-1 - (-1)$

As an example, I'll give some hints for the first one. Let $a=99^2$. I find it hard to square 99, so I'll just square 100 and say $a$ is slightly less than 10000. Next, I'll maybe use an identity like $a=(100-1)^2$ and zoom in on how much $a$ is less than 10000. I'll perform a crude check by stating that $a>90 \times 90 = 8100$. So $a$ lies between $8100$ and $10000$.

Hope I've taken the keen edge off your despair!

Does that help?

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Try working out problems with someone else who can watch over your shoulder and point out mistakes in real time as you make them. Then you can correct the mistake right away, before it propagates into rest of your work. This will allow you to notice and appreciate incremental improvements where you need fewer corrections, as opposed to having one less reason your final answer was wrong.

It may also help to reverse the process, to look for mistakes others make as they work, to train your ability to notice mistakes, which you can then apply to your own work.

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That might help. Thanks. – Rei Miyasaka Sep 29 '11 at 1:42
"To look for mistakes others make as they work, to train your ability to notice mistakes." How would spotting mistakes make me better at spotting mistakes? What would I learn from it? – Kelmikra Oct 26 '15 at 8:12

If you know you are having certain problems, try focus on the specific problem. Also, don't work on the problem in isolation -- work on it in context too.

Also, work out your own way of doing things. From looking at your notebook paper, my first instinct is that part of your difficulty is "skipping steps" -- trying to do several steps in your head when you still have difficulty with the individual steps is a surefire recipe for disaster.

Look at your work there -- all of your errors occur in the same spot, where it looks like you are doing two things:

• You are multiplying $4 \times 7$ in your head
• You are adding $5$ to the result (because it carries over from $8 \times 7$)

You may have better luck working out the problem in a way that doesn't require you to do two things at once. For example, you might compute $480 \times 7$ separately in a separate diagram: $$\begin{matrix} & & 4 & 8 & 0 \\ \times & & & & 7 \\ \hline & & & & 0 \\ & & ? & ? & \\ + & ? & ? \\ \hline & ? & ? & ? & ? & \end{matrix}$$

Or you might try "lattice multiplication" which lets you do the individual single-digit products independently of the addition steps.

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Skipping steps is definitely a habit that I have. I try my best to shake it. Sometimes I have a better success rate when I try very hard not to skip steps; sometimes it doesn't make much of a difference. – Rei Miyasaka Sep 28 '11 at 19:56
"trying to do several steps in your head when you still have difficulty with the individual steps is a surefire recipe for disaster." I make an incredibly large number of errors, but I haven't noticed that doing multiple steps at once increases the rate of errors. It does, however, make me much faster, so I can't call it a "surefire recipe for disaster." – Kelmikra Oct 26 '15 at 8:15

Personally I spent a couple of weeks improving my fundamental math skills on Khan Academy.

I practiced mentally doing 1-digit addition, then 1-digit subtraction, then multiplication, then continued with 2-digit calculations.

It's like any other skill. If your fundament is bad you will not be able to excell. Define your challenges - on paper - and work through the most fundamental problems until it sits real tight. Hehe..

And don't do too many steps in your head at once (unless your fundament is beyond SOLID)

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Come to think about it I have a neat little theory. It is that we are not limited in the amount of information we can process at a given moment, but that it deterioates fast. Think about how much information you process subconsciously at any given moment.

Combine that with the fact that we myelinate our brains when we practice something, which in turn increases the speed and precision at which the neurons are able to fire signals and you have a REALLY strong argument for improving the fundamental mathematical skills and practicing them regularly!

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