Quick, solve this problem
3 + 5 * 7 = ?
If you still recall high school algebra, you’ll remember that you should be doing the multiplication problem first. So the answer would be 35 + 3, or 38. But if you just punch the numbers into your calculator (or if you haven’t had occasion to do algebra since the ninth grade), you might do the addition problem first and come up with a different answer.
But even when people are reminded about the algebraic solution to the problem, when the numbers are grouped together a little differently, it has a significant impact on whether they’re able to solve the problem correctly. For example, if you see the problem grouped like this:
3+5 * 7 = ?
Even though you’ve now been reminded about the order of operations, the problem becomes a little more difficult, and if you’re doing a large number of problems, you will probably make more mistakes than if the numbers are spaced evenly.
You’ll do even better if the problem is written like this:
3 + 5*7 = ?
This visual grouping was the subject of David Landy’s talk, “Real Physical Symbol Systems for Mathematical Reasoning.” Landy and his co-author Robert Goldstone also asked volunteers to actually write out equations, and they found that people spaced out addition and subtraction more than multiplication and division, matching with their writing the method that makes solving the equation more efficient.
And of course, if you use algebraic shorthand for writing equations, you might omit the multiplication symbol entirely:
a + bc = d
So the shorthand that’s been developed over the years by mathematicians actually reflects the order of operations itself.
Landy and Goldstone have another interesting experiment which I’ll discuss below.
How would you solve this equation?
y – 4 = 6
You might go through the formal steps of adding 4 to both sides of the equation, then canceling out the 4 on the left to come up with the answer. But you might take the shortcut of simply moving the 4 to the right side and switching its sign.
Since this is a motion, from left to right, the researchers wondered if seeing a similar motion would help people the problem faster. So they created an experiment where little dots moved from left to right (or right to left) in the background while people solved similar equations.
Indeed, people were better at solving the equation when the dots were moving the same direction that the number needed to be moved. Pretty cool stuff.