When I learned arithmetic in school, it was heavy on lumbering processes, and light on theory.
When summing or subtracting two numbers, there was a lot of "carrying the one," but not a lot of explanation as to why. The whole goal was to go through the a set of repeatable steps to arrive at a final answer. It's not that satisfying.
But today I discovered there's a totally new way to do math that makes much more sense, and is way more educational.
On Facebook, John Carney has a nice little post explaining how subtraction used to be taught vs. how it's taught today.
First, here's the old way subtraction was taught:
Take this: 474-195.
Old way: Try 4-5. Nope. So cross out 7, carry the 1. Add 1 to 4. Now subtract 6 5 from 5. Write down 0.
Wait. That's wrong. It's not add 1+4. It's 10+4. So cross out my 1. 10+4=14. Minus 5. Write down 9.
Next subtract 9 from 7. Caring again. But remember it's 9 from 6. Dammit. Cross out 4. Add a one...wait, a 10 to 7...err, rather 6. 16 minus nine is 7.
The four is crossed out. So it's a three. Minus one
My answer is: 279.
To get that I had to add and subtract a lot. You can actually count the operations.
(1) 4-5.
(2) 7-1
(3) 10+4
(4) 14-5
=9
(5) 6-9
(6) 4-1
(7) 10+6
(8) 16-9
=7
(9) 3-1
=2
= 279.
Notice how many occasions for error and how much switching between addition and subtraction is required. This is a system built to fail.
Now here's the new way subtraction is taught:
They key to (new way) is realizing that this subtraction problem is asking you to measure the distance between 474 and 195. You do that, in turn, by measuring the distance between landmarks (easy, round numbers). It's turning math into a road map.
So 474-195.
Starting point is 195. How do we get to 474? Well, first we'll drive to 200.
(1) 200 is 5 from 195
(2) 400 is 200 from 200
(3) 474 is 74 from 400
(5) 74+200 = 274.
(6) 274 + 5 = 279.
Not only are there fewer steps, the steps are far less complex. You aren't carrying, or worrying about adding ten then subtracting the other thing, then remembering to subtract one from the other column. It's much straighter.
As Carney puts it, there are far fewer steps, and it's much harder to make mistakes. More crucially, solving the problem this way gives the learner an intuitive sense of what subtraction is: A way for measuring the distance between two numbers. This approach is satisfying, because as Carney says puts it, math is turned into a road map.
A lot of us ultimately taught ourselves to do math this way in our heads (similar techniques work for multiplication and so on) but it obviously makes so much more sense to just start this way, and learn a simple approach that's also theoretically elegant.
Incidentally, the new way of doing math is part of the controversial "Common Core" approach to math, which aims to establish a set of common approaches to doing math all around the country. Conservatives aren't happy about the one-size fits all approach to national