This post is in response to this repudiation of old math. This has been building for a while, and I've been thinking about what exactly it is I disagree with. So, this isn't just in response to that article, but with a more general view of math and what it is we are trying to teach children.
So, to start off with the article at hand, the author claims that "the top doesn’t make sense, the bottom does, and the connection to Common Core is completely misunderstood. (Says this math teacher.)" I'm just going to call the top the 'old' way, and the bottom the 'new', but I hope by the end of this post you'll get why that doesn't matter. They further explain that the old way is "... just an algorithm. You can do it without thinking".
Ok, here lies the essence of my entire issue with this view. To be blunt, and to the probable chagrin of many other teacher: math is nothing but an algorithm. The problem for many is not understanding this because there is not just an algorithm. There are many, in fact infinity many, possible algorithms to subtract two numbers (just as an example). The 'new' way is also just an algorithm.
Ok, so what is the rational for the 'new' way that makes it better. The author claims that students can do the old way just fine, but don't know why it works. And that somehow the new math teaches what they call “number sense”, which is useful for "other math concepts". Also the new way is easier for making change out of $20.
I will start with a story from my childhood. I remember when I first learned subtraction of simple numbers, like say 13 - 7, the way I did it was I started at the lower number and counted up adding 1 each time until I got to the top number. I kept count on my fingers, so however many fingers I had was the answer. This is the essence of the new math system, although generalized to work with differences greater than 10. However, I remember the other students didn't get how I could be adding to compute the subtraction: that's crazy! I get that this concept is totally understandable to that age group because it was obvious to me at the time, at least.
However, this method is a terrible general algorithm for subtraction. It's fine for 20 - 4.30. But what if you had 154442132 - 498484, as an extreme example. One argument is that, well, students would never need to do that because we have calculators. Ok, that's fine. Now to my next point: the new way is even more arbitrary than the old way, and doesn't add any new intuition about what is happening.
The argument made was that the new way is useful to other concepts: what, exactly? Let's first actually define what the new algorithm actually is:
Subtract: 32 - 12
Do you know the answer already?
yes: give the answer.
no: pick a number between 32 and 12, call it x, such that you know the answer to x - 12
Do you know that answer to 32 - x?
yes: give the answer as (32 - x) + (x-12).
no: pick number between 32 and x, call it y, such that you know the answer to y - x.
Do you know the answer to 32 - y?
yes: give the answer as (32 - y) + (y - x) + (x - 12)
no: pick a number between 32 and y, call it z, such that you know the answer to z - y.
Do you know the answer to 32 - z?
yes: give answer as (32 - z) + (z - y) + (y - x) + (x - 12)
no: keep repeating etc, etc, etc
You will see this is exactly the algorithm being taught if I plug in z=30, y=20, x=15:
32 - 12 = (32 - 30) + (30 - 20) + (20 - 15) + (15 - 12) = 2 + 10 + 5 + 3 = 20
Ok, I made this look even more complicated because I put in all those variables. Why did I do that? Because they are totally and completely arbitrary. Why pick 15, 20, and 30? The argument is that they are "easier". But what that really means is that we already memorized the results of those subtractions. We aren't actually learning anything new at all that we didn't before. If instead I only knew how to subtract 1, like I did when I first learned subtraction, then the algorithm would be x = 13, y = 14, z = 15, and I would have to add a bunch more. Other students could might see other combinations, such as x=22, or just see the answer right away.
So, why does the new way seems to be better to some teachers? I have some speculations. It very well may be building some intuition. And it is true, it is easy to understand why it works, but actually harder to understand how it works, which are separate issues. We can teach why very easily without pushing one particular algorithm. And the algorithm we pushed should be based on utility, generality, and understanding of the algorithm itself. Math is not just intuition.
Now comes to a more recent story about my experience trying to teach computer science to a group of early level high school students.
I gave an assignment to write a program that would take a starting time for a clock, and a after a certain number of ticks of the second hand it would report the new time on the clock. And it had to work for any number of ticks and times. So if I said it started at 10:01:56, it ticked 8 times, it should read 10:02:04. Now, if I asked a student to do this in their head, they had absolutely no problem with the additions and carry overs needed to do that. But ask them to write a program? They couldn't decide exactly what it was they were asking the computer to do. They had the intuition, or number sense, to solve the problem, but they couldn't describe the algorithm they were using to accomplish it. And without that, they couldn't tell the computer how to do it.
The reason this method is so good at making change is because you are almost always subtracting from an even number. That's because we don't print bills in odd amounts. The quantities and differences are also usually small. Every example I see about how easy this new method is to use is based on these problems that are easy no matter how you go about it. Again, completely arbitrary and a special case.
I think what we should be doing for this type of problem is exactly teaching algorithms. If you don't think they understand the algorithm, then work on that. But, exploring the algorithms are the important part, not just the answer, which is I think the end point here anyway.
