What is it like to solve a math problem?

I take the numbers apart into more manageable elements and recombine them. For example, 14×36 = (10×36) + (4×36), and I rebreak (4×36) into 2(2×36). Then I back up and gather up the pieces and assemble them, breaking down the addition into chunks as needed. Some addition is expressed as little subtraction chunks as well; for instance, I might turn a 7 into a 10 – 3 or I might fill up the 7 to the next 10 by deducting 3 from one of the other components.

I see the parts as if they were little movable wooden blocks with finished or unfinished computations on them. The numbers have color values for me because of grapheme-color synesthesia, but the blocks appear to be colored randomly. I kind of slide them around with the pieces on them. They are always right side up, and they have the thickness of Scrabble tiles..

No, wait, actually the blocks seem to be either red or green. I have no idea why.

47 – 23 = 47 – 20 = 27 – 3 = .24. I think I tend to do things like that from left to right instead of ones first, then tens, etc.

617 + 3194 = 3194 + 617 = 3794 + 17 = 3790 + 4 + 10 + 7 = 3800 + 11 = 3811

I’m a word person, not a numbers person, so this process is slow for me—up to several minutes. I forgot to actually notice the time because the process blotted out other things.

504, 24, 3811

It took about a two minutes to do the multiplication one, 2 seconds to do the subtraction one, and about 30 seconds to do the sum.

The images that came to mind were the sort of things I would write on a piece of paper. The classic way of representing calculations that we’re taught in school – I place one number over the other, and draw a line below everything, and write my answer below the line.

Yes, when doing basic number manipulation, I do see these images. Of course, calculus and probability are an entirely different ballgame…

I would describe the experience as… similar to crossing the road. You have to have the confidence that you can make it, but you’ve done it so many times before that of course it has to succeed. And when you reach your destination you feel… not satisfaction per se, more like your expectations for yourself have been met.

It took me about 20 seconds.
36×14 I said was 360 + (72+72= 144) = 50 4= 504
47–23 = Insantly 23+1 = 24
617+3194 = 3700 + (94+17= 9+1—> 3800 + 4+7=11 —>3811

It is different for each problem. I do it left to right (the wrong way) and work with 10s.

617+3194 = 611+3200 = 3811.

47–23 = (40–20)+(7–3)= 24.

14×36 = 15×36–36 = 10×36+5×36–36 =10×36+(10×36)/2–36=
360+180–36= 360+(180–40)+4=504.

It took me a long time because I’m one slow son of a bitch.
I don’t see images; rather I talk to myself.
Keeping more terms in mind requires more short term memory effort, but I only remember the products of small numbers. The numbers 360 and 180 are special; 360 is the number of degrees in a circle and 180 is the number of degrees in a semi-circle—if one remembers this, there is no need to calculate the quotient (10×36)/2.

I am not very good at this. I visualise numbers as thin whitish lines on a grey background. As the numbers tend to fade if I am not concentrating on them I sometimes have to repeat steps and the first multiplication took at least a minute. The other two sums only took a few seconds. I ‘write’ the sums out in my head as I would write them out on a sheet of paper.

It looked like a brick wall.
I am borderline incapable of doing problems like this in my head. My brain just doesn’t function in that way. Once there are more than a couple of numbers in there, it’s like the whole program crashes. ;)

All three took me less than 10 seconds.

It took a few seconds, the addition and subtraction happened almost immediately, the multiplication took a little longer because I had to watch the numbers move as I carried the carried the 2.

Guess what? I completed 2 out of 3 and did not use a calculator. Not bad for a guy 67 years old.(it did take me 20 minutes, but I am not complaining).

20 sec total.
No images really. But I did use a process. For the first one, I used 36 * 10 is 360. Add half on to that to get 540. Remove 36 to get 504.

Similar type of stuff for the other ones.

As the math gets more abstract I start using images instead of tricks.

47 – 23 = 50 – 26 = 24. (the time it took to type)

14×36 = 10×36 = 360;
4×36 = 4×30 = 120;
4×6 = 24

Add them up; 504

617 + 3194
620 + 3200 = 3820
3820 – 9 = 3811

@john65pennington I don’t think that your math ability should diminish by age. I watched my father add up the grocery bill at the check-out counter. He not only added up his order but found a problem when the cashier gave him the total. I asked him about that, he said it is just something he always did. I, of course, never inherited that ability but I find that my basic math skills seem to work better in my head than they did when I was in my earlier years. It is possible that my ability to concentrate improved. I lost physical speed as I entered my 60’s but gained (I think) better and faster mental abilities. Of course that is just my opinion, when I’m sober. I also found it is a little easier to talk a cop out of a speeding ticket.

Wow, you guys are pretty amazing. I’m another one who’s really bad at this stuff. I still haven’t figured the answers out yet due to fatigue.

@flutherother, I find I have troubles holding numbers in memory. I can figure out half an answer, then I start the next half hoping to put the two half answers together but then I forget what answers I came to and I have to redo them.

@Jeruba I always wondered if different people with Grapheme synesthesia all see the same color for the various digits. Where do you see the color? Right on the number or is it kind of like a background color in your mind?

@gailcalled what is that trick you used for subtraction? Looks like you round the minuend to the nearest (comfortable) 10 and add the rounding-difference to the subtrahend. Does that always work?

Trying it:
76 – 75 = 80 – 79 = 1;... nice
902 – 36 = 910 – 44 = 866;... hot!

@ninjacolin “Does that always work?’ sure, it’s basic algebra, balancing an equaction but I never though of doing subtraction that way. good job @gailcalled

Epiphany: Math = Facts about number. The more you know, the more power you have over numbers. I always thought it was a matter of raw talent, although I’m sure there are handicaps for that too, but mostly it’s about just knowing/remembering useful facts about how numbers relate to one another. If you forget or if you never heard of a way, you have to do it a longer way.

I wanna try one more: 46 – 233 = 50 – 237 = -187… can’t believe this!

@ninjacolin Ummm…. not really. Lots of basic arithmetic can be done this way, but math in general is about concepts.

@roundsquare, what I mean is.. I never realized that in subtraction you could add an equal amount to the minuend and the subrahend and it would still give you the same difference… That’s what I would call a fact about subtracting numbers… that I was never alerted to. :(

I totally would’ve passed grade 4 if my teacher just told me that

@ninjacolin, they don’t. There are a few common perceptions—a lot of people see the letter A as red (as do I)—but in general, no.

The letter or number simply has the color. It’s in the seeing of it and not in the character itself: an attribute. The character has line, an external profile (the outside shape of it, as if you shrink-wrapped it and then traced the package), and color.

Subtraction took me three seconds, addition took ten seconds, multiplication took twenty seconds, and no images came to mind.

I consider myself to be very good in math, but I don’t really think time is of great importance. Doing a calculation in five seconds doesn’t make a big difference from ten seconds. Visual images may help, because there are actually people who can manipulate an imaginary abacus in their head.

I tend to find formulas easily.
EX: What is the sum of 1–2+3–4….-2010+2011? It took me less than twenty seconds to realize that for each positive term, the sum of it and everything before that is the positive term plus one, divided by two, and that the answer is 1006.

I generally break down and use a calculator for large number multiplication or division.

However, for subtraction problems I generally do this in my head:

47 – 23
47 – 3
47 – 2 = 45
45 – 1 = 44

44 – 20 = 24

It’s generally very painstaking for me to do very simple math like this because I tend to be very particular about how I break numbers apart. Sevens and nines are the worst, I always have to find a way to break them down into fives or make them into even tens before I can do anything more. This probably has to do with the following methods I use.

I developed visual ways to identify how many a number represented (I know this seems really weird but when I add or subtract I count the amount in a number like syllables in a word so I can add or subtract up or down from the initial amount by counting up or down). Most numbers 1 through 9 have an easily identifiable amount hidden in the shape of the Arabic number. Take 3 for example. It has three points. 4 is trickier, I always imagine fours with the simpler hand-drawn right angle on top of a vertical line. I see that more as a box with four corners. Hence, 4 represents four objects to me. 5 I see like the five on a die, with points on all the corners and a point in the middle. 6 I see, similarly, like it’s representation on a die. The reason I have such trouble with 7’s and 9’s is that there is no easy way for me to do this visualization and counting.

I’ve always been really slow at doing these simple calculations just because of my obsession with having to count up or down. Despite having such an annoying handicap in simple math, I graduated with a bachelor’s in Aerospace Engineering and have no problem in the higher kinds of math like Differential Equations.

I’m sure that if I used a method similar to @gailcalled I could get over a lot of the difficulties I’ve had. Making things nice even numbers appeals aesthetically to me, which is most of the problem I’ve always had.

@ninjacolin Ahh, I see. I was just pointing out that math is about more than “facts about numbers” if you get a little deeper into it.

Btw, if you want a way to think about it, imagine this:

23 – 17 = 6. That means if you have 23 apples and I take away 17, you have 6 left.
If I start you out with 7 more apples (so you start with 30) but than I take away the original 17 I took away and the extra 7 you started with, you’ll still have the same 6 apples.

There are a lot more “tricks” out there. My dad basically raised me on them :)

@ninjacolin, did you get what you were after here?

I got more out of this than intended, thanks for asking @Jeruba. I suppose there was a lot less imagery than I anticipated. Most comments mentioned that they see the numbers, some see the numbers as if they were on paper. For me, I often see a child sized plastic orange chair when I’m doing math equations. It’s just something I remember from my elementary school. If I’m not focused on the chair, I likely have some sort of a star scape as the background to the math work I do in my head.

So great to hear from @Shuttle128. That was a very rich response. I don’t have those kinds of images for numbers in my head but I do have a common workflow or two that I default to: Just now when adding 27 + 34 in my head, I realized that I often have to count distances to 10s. I know instantly that 2 + 3 is 5. So, the answer will be more than 50. The only thing I had to figure out was whether 7 + 4 was greater than 10. I don’t know 7 + 4 = 11 as readily as I know that 2 + 3 = 5. So, I had to count using the syllable-like method that @Shuttle128 uses. It’s also similar to counting on your fingers, except the sound itself counts as a finger somehow and I can count that way. So cool that he described that, I wouldn’t have even realized that it was relevant. (See, @Jeruba, more than I expected) Anyway, all together it went like this:

27 + 34 = X
2 + 3 = 5. So the answer will be at least 50.
4 counts from 7 goes like this: 8, 9, 10, 11.
So the answer is 50 + 11 = 61.

Or, 30 + 30 = 60, so your answer will really be at least 60.

Or, (27 + 3) + (34— 3) = 30 + 31 = 61.

@PhiNotPi, I’m curious what you might suggest as a mental tool, a trick, that allows you to identify patterns easier than most.

@gailcalled cool. I’m actually concerned that the syllable-like counting method is a slowdown point for me.
btw, I didn’t really fail grade 4, that was a joke