As eclectic homeschoolers (and just a step away from unschooling) we aren’t terribly affected by changes in school policies and standards, including Common Core. But I hear complaints from friends, follow news on the subject and see “examples” in my social media feeds.
So I read James Tanton’s latest essay, titled “Renewed Math Wars”, primarily out of general interest. I should’ve known better! After all, James’s other essays were so inspirational, that I ended up co-authoring a book with him and Maria Droujkova of Natural Math (hopefully, it will be published in early 2016).
Back to the “Math Wars”. Reading it helped me re-think some of the experiences I’ve had in the last few years homeschooling Rocket Boy. What follows are my thoughts and observations in the context of an elementary school-aged child (yes, just one).
The difference between familiarity and understanding.
James uses an example of subtraction with borrowing. Before I get to that, I’m going to give an example of addition with carry.
Simple enough, right? Applying the standard algorithm, a child would re-write this, placing 56 under 237, then add starting with units, then tens, then hundreds, arriving at the answer = 293.
That’s how I was taught. And that’s how I’ve been doing it for many years except… Ok, I’ll return to the “except” part in a second.
For Rocket Boy, however, “standard algorithm” is the least effective. Because of his neurological condition, writing is hard for him and writing neatly is impossible. And writing one number under another, aligning units, tens, hundreds falls into the impossible category. So he figured out his own way of doing addition, relying on mental arithmetic and working his way left to right.
… except… I also add left to right whenever I don’t have pen and paper in front of me. Which is most of the time.
Moving on to subtraction. Once again, the standard algorithm of writing the second number neatly under the first one, aligning units, tens, etc. doesn’t work for Rocket Boy who, once again, relies on doing all the calculations in his head.
And interesting problem arises when he deals with subtraction with borrowing. Doing it in your head following the standard right-to-left approach is hard because so much effort goes into memorizing the number “backwards”. Writing down, in Rocket Boy’s case, is just as taxing. So, as long as I insist on “doing it the right way”, he resorts to approximations instead of calculations, such as
237-56 = 180 (approximately)
Granted that Rocket Boy tends to interpret things literally. “What are you doing?” gets me the answer “I’m talking to you”. Since I did not ask for “exactly how much is 237-56”, an approximation can be used. Especially since in so many other situations it works out great.
What happens when I don’t insist on using the “standard algorithm”? Lots of interesting things happen. Ok, interesting is a word I use hours later, after I cool down and re-group and think about things in a calm way. Or sometimes they appear interesting right away if I remember to ask the “ok, I see, can you show me how it works” instead of jumping in with “no, let me show you how it’s done”.
Interesting example #1
64-28 = 0 because it’s the same as 28-64 and 64 is greater than 28 so 28 is completely “expanded”.
Why it is interesting – Firstly, it raises the idea of negative numbers. Can something be less than 0? How so? Secondly, and of more interest, is 64-28 the same as 28-64? It works for addition and multiplication (and Rocket Boy is big on symmetry). Will it work for subtraction? Will it work for some numbers, but not others?
Interesting example #2
64-28 = 28-64 = 40 and 4 because 8-4=4 and 60-20=40.
Why it is interesting – this goes back to modeling addition and subtraction as distances between numbers. If numbers are made up of tens and units, can we just find distances between units and, separately, between tens? Why certain ways of doing things work with addition or multiplication, but not with subtraction and division?
Interesting example #3
64-28 = 2 and 30 and 4 because we add 2 to get 30, then 30 to get 60, then 4 to get to 64.
Why it is interesting – because that’s a much easier way to figure out subtraction and carry it out without a pen and paper. Since we go to the flea market and the farmers’ market a lot, Rocket Boy gets to see and hear how sellers count out the change using this method.
So what’s the point here? The point is that although Rocket Boy is familiar with the standard textbook approach, he does not understand why it works. No matter how many examples I give him to work through, he will not understand the why, only the how. Fortunately, in his case, because the textbook approach happens to be the most difficult and least productive for him. Which means he has to look for other ways of doing things, test them, and try to understand whether they work or not and why.
To me, our experience is a good example of familiarity NOT being equal with understanding. This does not mean I don’t think fluency with math facts is unimportant. But I think it’s over-valued and over-emphasized the same way written output is.
The idea of “once she gets lots of practice adding, subtracting or doing whatever else, then she will be ready to understand deeper truths”… In my personal experience, it just doesn’t work. If it worked, given the volume of drills kids go through in school, the rates of math illiteracy wouldn’t be as high as they are.