Exceptional, or ordinary with practice?

How do you solve math problems in your head? Perhaps a better question is, do you solve math problems in your head? With the availability of electronic devices to do it for us, I would not be surprised to learn that many people never try. This is a fascinating question, one I also consider briefly in my new book, Be Different.
I was reading Darold Treffert’s book on savants, and I was intrigued by a few examples of savant thinking. I tried solving some of the problems in his book to get a feel for how “comprehensible” they might be to me, with no recent practice calculating. Here is a simple example:
You have a carriage with a wheel that’s six yards in circumference. How many revolutions will the wheel make while traveling two hundred twenty miles?
This is how I get the answer in my head. I’d be interested in how you might do it:
Six yards is eighteen feet. I see that as a short line.
So one hundred revolutions of a six yard wheel would take me 1,800 feet. That’s a much longer line in my head, one that curves.
Three hundred revolutions would take me 5,400 feet – more than a mile. Now the line has curved back unto itself, making a circle.
How many rotations are there to a mile? Less than three hundred. A mile is a smaller circle. I can see those circles, on inside the other. They do not quite match.
I adjust the length of the longer line that forms the big circle. Try 290 . . . that’s 5,400 less 180, or 5,220. A mile is 5,280. Now I see the line laid flat, like a straight stretch of highway. Two hundred ninety revolutions leaves us sixty feet short of a mile marker. So what’s the fraction?
Three eighteens go into that sixty-foot remainder with the same six remainder. Adding that to the 290, I see the answer is 293 and a third. The six-yard wheel does not fit a one mile line, but it fits perfectly into a three-mile ring. If you put a mark on the wagon wheel, and mark any point where it touches the big circle, those points will touch every time the wheel rolls past. I like that.
If you roll the same wheel around a one-mile ring the points will only touch every third trip around, which is unsettling to me. I like smooth fits, so I will solve the next step using three-mile units.
I can now see the answer: 880 revolutions. A perfect fit. Six yards, three miles, and eight hundred eighty turns.
How many three-mile eight-hundred-eighty revolution units are there in 220 miles? My mind visualizes stacks or piles for this next step. Seventy units reach two hundred ten miles. I quickly see how seventy-three and a third are needed to reach the two-twenty goal.
Stacking seventy-three piles of 880 in my mind takes a little time. Eventually, the stacks add up and I see the result is 64,240. Now I just have to add the third (of 880) and I’m done. To do that, I add three hundred to the pile, making 64,540, and then take back six and two-thirds.
64,533 and 1/3 is the answer to the question.
As a further experiment, I scaled up the distance, to 2450 miles and then 20,315 miles to see if I could keep scaling up the numbers. There must be some limit to that, and it certainly took me longer, but I solved those bigger problems in a few more minutes. Solving the longer distance problems involved one and then two more levels of “stacking” in my mind.
It does not seem that hard to me. I often did similar calculations as a kid, for fun. I’m sure I could do it again, pretty quickly, with some practice.
I test my answer with a calculator. The process to do that is considerably simpler.
I multiply 220 (miles) by 5,280 (feet per mile) to get 1,161,600 – the total distance in feet.
I divide that by 18 (the wheel circumference) to get 64,533.333 – the revolutions turned.
It’s a lot faster to get this answer from a computer, for sure. But is the ability to figure problems like this out in one’s head really exceptional? In today’s world, I would not be surprised if kids never develop these skills. When I grew up, though, pocket calculators did not yet exist and I had to know how solve math problems on my own. Given my own ability, I suspect many people of my generation could solve a problem like this in their heads, but perhaps I am wrong. What do you say?


Lynette said…
I could solve it without a calculator, but not in my head! lol
jm2jmm said…
Dear Mr R- my husband can do this kind of stuff. I cannot. Probably not even w a calculator. It's fascinating to watch him mentally figure out calculations, and quite frankly, kind of an attractive side of him! :)
Janet Ha said…
Being an older person too, and one who likes math to boot, I could do this in my head (or more accurately, with paper.) My method would be somewhat different from yours, closer to how you did it with the calculator. I would have figured out how many revolutions make 1 mile, then multiply that by 220. But I work in a high school and see every day a divide between students who could do this and those who couldn't, even with a calculator. At the heart of math ability lies number sense, a general idea of how numbers work and what they represent. Schools must decide which they will focus on teaching to students who lack math skills: calulator or calculation? Many choose calculator. I can't blame them. By the way, some kids who lack math skills have strong skills in other areas.
But some of these kids really have no math base at all. When given a much simpler logic problem than yours- How many times would a painter paint the number 4 if he's painting the numbers 400 to 499?- these kids just hope they will randomly guess the correct number. 41? 57? They cannot see that, at minimum it must be 100. And these are neurotypical kids. Actually the ASD kids usually, but not always, have pretty good number sense.
Unknown said…
I work with children with Autism. I also train other education providers on how to work with children with Autism. I recommend your book. And this article gave me one more insight
into my students world. THANK YOU, Mark Warner.
Unknown said…
I calculated the number of rotations per mile, then multiplied by 220, coming up with 64,430? I was surprised it was that far off. My family and colleagues view my "mind-math" somewhere between parlor trick and frightening, often ending up in the smart or weird camps. My twelve-year-old daughter exhibits similar abilities, yet attempts are being made to assimilate her into "everyday" math in a highly-recognized school system. Often, it's the system that needs help! This is the first time I've found your blog, and I look forward to your future writing.
Unknown said…
I'm one of the youths you talked about who potentially are not able to solve this problem in their heads. Maybe this is just my education, but I have not been spoiled with being able to use a calculator. Unfortunately, I have executive functioning issues, so, although I was able to come up with the steps of how to solve this problem in my head, I needed to use pen & paper to do the actual arithmetic that you need to get the answer (and I still ended up making a mistake). I know many kids whho are the complete opposite of me; they can do calculations in their head very quickly, but cannot necessarily figure out how to actually do the problem. I think savants who can solve problems like this in their heads are extraordinary because they have a mind that can conceptualize the problem, develop a multi-step method to solve the problem and then do extensive arithemetic very quickly without using paper to provide organization or visuals.

So, to answer your question--yes, being able to solve a problem like this in your head is extraordinary, but not because kids nowadays rely too heavily on calculators.
Unknown said…
Will you be touring this new book in Northern California? Please, please do!
Unknown said…
Well, I did it by leaving all the zeros out and solved as if the "miles" were just feet. Then it was just a scaling problem, but since I hate doing long division in my head I stacked the 10's powers as I went forward and then kept going. Being an engineer since childhood, I didn't worry about the last two sigfig, they are noise.
Steve Borgman said…
John, that's an example of a mind wired for math. Mine's much more verbally wired, which may explain why I needed to take remedial math in college ;-)
Anonymous said…
Mmm, I just brute-forced it and came up with the answer in about a minute.

There are always other fun ways to do an arithmetic problem. But what's more fun is a book iike Gradshteyn and Ryzhik's.

Actually I just started reading about Asperger's and it's bothering me a bit, maybe I have it. It's a sort of sinking feeling.

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