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.
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 answer that question 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 with a calculator, for sure. But is the ability to figure 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 problems like this. 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?