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I have a problem where it is often very difficult for me to solve simple math problems without thinking about them visually, but at the same time, I have poor spatial reasoning and so I'm unable to keep clear diagrams in my head for long. Therefore, this combination makes solving simple problems take much longer than needed. It may be due to the fact that since I was young, I always thought thinking visually (whenever it's possible) is the "most comprehensive" way to understand a problem, and so I've trained my brain to always think visually, even if it makes me very inefficient.

I'd like to know how to train my brain to not think visually anymore when it's not needed, particularly for simple problems that do not need to be thought about visually.

Here's an example:

Bob and Jake both started off their schooling by going to kindergarten. The cutoff for entering kindergarten in their area is that you had to be 2 years old by September of that year (otherwise you would have to wait till the next year to join).

By years (ignoring the month they were born and just looking at their birth year), Bob is 4 years older than Jake. But Bob was born in October, and Jake was born in June. How many grade levels apart are they (assuming once they entered kindergarten, they didn't skip or fail any grades)?

The way I would think about this is by drawing out a picture in my head like below and reasoning based on the picture (even though it's time consuming and difficult to do):

Picture One (plotting their birth years on a timeline): The picture in my head I use to think about the problem

However, coming up with the right picture in my head and then reasoning visually takes a while (like 5-7 minutes).

I asked my friend about this, and he just thinks this way:

Normally Bob would be 4 grades ahead of Jake. But since Bob missed the cutoff and had to wait another grade, he's now only 3 grades ahead of Jake.

That's a much simpler thought process that took my friend substantially less time, but even to try to understand that statement, I tend to draw out this type of picture in my head, but then I get stuck with a question:

Picture Two (plotting their grade levels on a timeline): My picture in my head to understand my friend's answer, and then getting stuck with a question

From some type of vague sense of logic, I could guess that since Bob missed the cutoff, he'd be one grade level less than he would normally be. But I'd have a good chance of getting this wrong and it's very difficult to make this make sense to me, unless I go back to thinking about their birth years on a timeline as in Picture One, because that makes it crystal clear that Bob "loses" a grade level and that their grade level difference shrinks by one.

My question is: how can I ditch the need to think as in Picture One/Picture Two, and just think verbally/logically like my friend did, so that I can speed up my thought process?

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  • $\begingroup$ I realize the school-age problem is only an example, but FYI I would just pick a specific pair of consistent birth years and assume without loss of generality that school begins in September, say Bob was born Oct 1950 and Jake was born Jun 1954. Now consider Sept months 2 calendar years after their birth years. Two calendar years after Bob was born is 1952, and in Sep 1952 Bob has not yet turned 2, but will turn 2 in Oct 1952, so Bob begins school Sep 1953. Two calendar years after Jake was born is 1956, and in Sep 1956 Jake will be 2 (having turned in June), so Jake begins school Sep 1956. $\endgroup$
    – Dave L. Renfro
    Commented Feb 10 at 23:56
  • $\begingroup$ @DaveL.Renfro Yeah, this might help, thanks. In the end though, you are reverting back to Picture One in the sense that you are considering their birth years. My friend solved the problem without considering their birth years at all, and just working with their grade level difference. However, if I were to think about their birth years, it's probably better to choose specific numbers and working with those (as you did), rather than visually trying to diagram the situation for arbitrary numbers. $\endgroup$
    – asdfasdf
    Commented Feb 11 at 1:48
  • $\begingroup$ Perhaps worth pointing out for its relevance to the types of consideration one deals with in more sophisticated mathematical situations is that picking a specific pair of birth years and then looking at a specific pair of subsequent grade levels doesn't actually show that the difference in grade level is independent of both the specific choice of a pair of birth years and the specific choice of a pair of subsequent grade levels. Although this more general result is fairly obvious here, there are situations in math in which what seems to be fairly obviously true is in fact not true. $\endgroup$
    – Dave L. Renfro
    Commented Feb 11 at 9:22
  • $\begingroup$ Taking a short break from some day-job work stuff, here are a couple of things where superficial intuitive reasoning leads you astray. [1] For each value of $x$ and each value of $r,$ the derivative of $x^r$ is $rx^{r-1}.$ So in the special case where $x=r,$ the derivative of $x^x$ is $xx^{x-1}=x^x.$ [2] If $x = 1+2+4+8+\cdots,$ then $2x=2+4+8+16+\cdots$ and hence $x=2x-x=(2+4+8+16+\cdots)-(1+2+4+8+\cdots)=-1.$ Therefore, we have $1+2+4+8+\cdots = -1.$ $\endgroup$
    – Dave L. Renfro
    Commented Feb 11 at 20:32

2 Answers 2

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You probably can't make large changes in the way you naturally approach problems. I suspect that when you say you "trained your brain" to think visually you were just reinforcing a particular skill you happen to have.

So use your visual strength. You will do better on problems where your approach works best. Seek out those problems. Resign yourself to the fact that sometimes other people's strategies work better or faster than yours.

On a school test or a math competition there's an advantage if you can solve some problems quickly. In real mathematical research insight counts for much more than speed.

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  • $\begingroup$ Thanks for the answer. It's possible that this may be a natural proclivity (since I like drawing), but I also have OCD and thus from a young age I was very rigid in the way I thought I "understood" things. In any case, the issue is that: 1. I'm don't have a high spatial intelligence, so it's difficult for me to keep these diagrams in my head or to reason with them, thus making my thinking process painful and inefficient. 2. I'm not interested in math research or gaining deep insight -- this is more of a daily life problem where it takes me longer than normal to understand simple things. $\endgroup$
    – asdfasdf
    Commented Feb 10 at 22:52
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This is a shot in the dark, but it may help you to see if you can find an inductive approach to a problem. This often involves generalising the problem and gives you more examples to think about. In your example, you could consider the case where Bob is $n$ years older than Jake for some $n = 0, 1, 2, \ldots$. The case when $n=0$ is then a very specific example that should give you a clue to the general case (they will be in the same grade when $n = 0$). Then you can think about what happens when $n = 1$ and so on.

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  • $\begingroup$ Thanks. Are you suggesting that I first consider the case where they were born on the same year (Bob in October and Jake in June)? Then Bob has to wait an extra year before starting school, so he'll be a grade level lower than Jake. I can think about the cases where Bob is born 1+ years before Jake and conclude that their grade difference = their age difference - 1. That makes sense, but I'm still relying on thinking about their birth years, similar to "Picture One" but maybe less visual. My friend did it without considering their birth years and just working with their grade level difference. $\endgroup$
    – asdfasdf
    Commented Feb 11 at 1:58
  • $\begingroup$ Yes, I am suggesting you generalise the problem and think about edge cases. $\endgroup$
    – Rob Arthan
    Commented Feb 11 at 20:56

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