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.
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I vaguely recall (it's been almost forty years) that you're supposed to use parentheses. Is that no longer done, or was it always training wheels?
And from what I remember about order of precedence, exponents come before multiplicands.
Then again, computers make that relative null since if you enclose in () it gets evaluated first, before exponents and other items.
You're supposed to use parentheses to override the default order of precedence. That is, if you really wanted people to perform the addition in 3 + 5 * 7 before the multiplication, you could write (3 + 5)*7. Otherwise the multiplication comes first, the addition second.
We also use parentheses for clarity. If you want to underscore the precedence of the multiplication, you could write the expression as 3 + (5 * 7). That merely emphasizes the default order of operations and is therefore, strictly speaking, unnecessary, but it can head off quite a few calculation errors.
Interesting. I remember seeing a study many years ago by a linguist at U. Rochester in which reading scores went up by some small but reliable amount if one simply inserted an extra space between phrases within the sentence (presumably by helping readers parse it better). This seems to be a numerical equivalent of the same phenomenon.
Polish Notation--very often Reversed (i.e., RPN) as in: 3 5 7 Ã +
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.
I wonder if the PEMDAS (Parentheses, Exponents, Multiply, Divide, Add, Subtract) pneumonic would work as effectively as the visual groupings? That was the cue that told me to multiply 5*7 first and then add 3 next.
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.
That's a pretty neat effect. I wonder if the results reveal a left-to-right directional bias (i.e., faster problem solving when the movement is left-to-right vs right-to-left) since left-to-right is also the direction of reading (for English anyway). Also, I bet there would be something like a Simon Effect whereby a motion opposite to what is required to solve a problem will slow problem solving.
A few years ago I read of a draw for, IIRC, a car in which all the correct answers to a 'skill-testing' question were rejected because the person who set it performed the operations as they came rather than in the correct order. I never did find out how it was resolved.
PEMDAS
The order of operations.
Parentheses---> Exponents---> Multiplication---> Division---> Addition---> Subtraction.
"Please Excuse My Dear Aunt Sally"... ahhh... memories :-)
For the reasons above, regarding calculators and orders of operations, I always go parintheses-happy on my scientific calculator. I've had my equasions go bad one too many times to take chances.
I must say, I love the math-psychology post! I'd love to see more like it.
As a working programmer, I *always* use parentheses to make it crystal clear what I mean. I don't care (and don't even remember!) what the default precedence is supposed to be.
Pattern-chaser
"Who cares, wins"
BODMAS = Brackets, Division, Multiplication, Addition and Subtraction.
My answer to 3+5 * 7 = ?
is 38 because it really is 3 + (5 x 7).