I am finally getting myself together here for Melanie Bettinelli’s linkup on “Learning Notes.” Melanie is a homeschooling mother of many who writes a very fine blog. Check out the series she’s doing on Shakespeare with kids, for example. There are innumerable ways and styles of homeschooling, and if you are curious about this growing phenomenon and want to understand its appeal to families, I think Melanie’s blog is a great place to start. Conversations, creating, exploring with people who love you? …the best kind of education, for sure.
My first foray into this linkup isn’t going to be a day-by-day account this time because I’ve got math on the mind these days.
For a non-mathematician, I think a lot about math, and this blog post has finally spurred me to put down some thoughts on it.
I come from an academic, humanties-centered household. There was no mathiness or science or business-type activity to speak of in my parents’ lives or in their parents’ lives. My mother joked about her wildly contrasting verbal and math scores on the GRE. I did fine in math in high school, took the minimum I needed to in college, and that was it. I had no opinion of it one way or the other. I certainly had to work hard and think things through in the higher math (the highest I got was what they called “Advanced Math” in the day – maybe there was a bit of Pre-Calculus in it, and a little trig, but I never even attempted calculus. I don’t think the school offered it, come to think of it.), but I often had the weird experience of hitting a wall at night when I was doing my homework, then waking up the next morning, saying “Ah-ha!” – my brain evidently having worked it all out when I was sleeping.
As a parent, I’ve had one older kid who needed help in math, but the other two breezed through on their own, doing very well. My second son never studied in high school and made straight A’s, even in Calculus. Daughter had to study, but still did well, and liked it – “Math is like a puzzle to me, and I love puzzles” is what she’s always said. And now she’s studying for the LSAT which she was emboldened to do, not just because she took a Civil Rights/Liberties class and really enjoyed doing case analyses, but also because she looked into what the LSAT is and joyfully discovered, “It’s LOGIC!” So.
And then, for the others…. it was time to homeschool.
Math is something that some non-mathy homeschooling parents dread, but I never have, mostly because I picked a program that I found easy and even interesting to work with – from The Art of Problem Solving. I’ve written about this program before, so I won’t repeat myself. I’ll just say that Joseph worked through the Pre-Algebra text last year and is making “A’s” in Algebra in 8th grade right now. He never minded it too much, and neither did I – in fact, in many ways, I found it illuminating. Plus I love the videos. There, I admitted it.
Now, I have a theory about teaching. I actually think that people who are a “natural” at a subject don’t necessarily make the best teachers of that subject. Think about it – if you have an intuitive grasp of a topic or skill, it might be a challenge for you to communicate the process to someone who doesn’t have a clue. On the other hand, if you’ve had to work through a process step-by-step and have actually struggled with various aspects of it…you might just be a really effective teacher to the equally clueless.
All that is to not to say that I’m a fabulous math teacher. But it is to say that I’m not a bad one – at least to my own children – and I think it’s because I understand their lack of understanding.
Anyway, math is not only on my mind these days, it’s on the mind of many because of Common Core-related issues. I’ll say straight up that I’m (not surprisingly) opposed to Common Core simply because I’m opposed to all federal standards in educational content, period, without exception and also because I believe that the push for Common Core is primarily profit-driven. As I’ve said before, no one makes money when teachers are using five-year old textbooks using methods they’re familiar with. People make money when new textbooks must be written and printed, when workshops on new pedagogies must be paid for, when consultants must be consulted and when – above all – children must be tested.
But what has gotten folks riled up above all is the content of the standards, especially in math. I saw a bit of this in the text Joseph was using in his old school, and which we used in the first year of homeschooling (because at that point we weren’t sure if he would be returning to school after our fall in Europe…just in case he was, he needed to be on track.) I rather liked the text because it invited the student to look at problems in a number of different ways and introduced various problem-solving strategies, but I could see how it could be confusing.
(My problem, though, with how this is shaking out in schools is this: I think the various strategies should be introduced. What I don’t think is right is then tying “success” of a child – and by extension, a teacher and a school – to that child’s mastery of all of the strategies. It’s terribly confusing and really confounds the purpose of introducing various strategies, doesn’t it?)
So now, to the present. With the Art of Problem Solving and the curriculum which my younger son is using from the same group, Beast Academy, we are encountering “new” strategies. That is, they are new to us, all of us having been taught more or less “traditional” math, even if it has been 40-45 years apart.
And here’s the thing.
They’re so much better.
They make sense. They are, as far as I can tell from my limited perspective, truer reflections of what is going on with the numbers with more explanatory power than anything I was taught, which was mostly about learning rules and formulas and plugging in the numbers and doing the computations, period.
I’m going to start with a simple example.
(Caveat – I’m only going to say this once, but it applies to every example. You may have learned this stuff during math. Maybe I was taught it, too. But I don’t think I was, and if I was, it didn’t stick.)
When my older son started PreAlgebra with AOPS, he re-learned a lot about basic arithmetic operations. It seemed, at first glance, kind of silly, but it wasn’t because, as we soon discovered, it really helps to understand exactly what these basic operations are. So take division. What is division? Well, division is a few things, I suppose, but one of the things division is is simply multiplying by the reciprocal of a number. So…20 divided by five is also 20 times one-fifth. Right? So there’s your definition of division: Multiplying by the reciprocal.
Now. Flash back to..I don’t know. Fourth, fifth grade math. When you were taught how to multiply and divide fractions. Multiplying: easy. Just multiply straight across. But dividing? Ooooh…tricky. You had to remember that weird thing you had to do – you had to flip the divisor and multiply by the resulting reciprocal. I don’t know about you, but I never understood why you did that. Why do we have to do that? Who knows? It’s a rule!
But hey….isn’t that what division is? Isn’t that the definition?
So when you have to divide fractions, you multiply by the reciprocal…because that’s what division is.
My point is – this was taught to me as a rule with no theoretical foundation. I probably would have had an easier time remembering it if I’d been taught the reasoning behind it in this really very simple way.
Properties were another thing. Every year we’d be taught those blasted properties, and never did any of them except the Commutative (because that’s easy) make sense to me. I had to relearn it every year, and barely did so, because the properties were presented as one little section in one chapter and then essentially neglected, probably until Algebra.
In these AOPS books, students are taught the properties early on, and they use them..constantly. Multiplication and Division are taught within the framework of the Distributive property – basically, they are taught to break the numbers apart in order to both more easily mentally compute, but also to understand, once again, the operations from the ground up. And really, this is something a lot of us do anyway, right? I know I do, and always have – if I have to compute, say, 78 times 6 in my head, I do so by breaking it up into 70 times 6 plus 8 times 6. It’s just that I never knew what I was doing.
SO. Finally. Back to this blog post – in which the author says that the way kids are taught to do multiplication – the algorithm (or system) – undercuts their understanding of place value.
If you want kids who get right answers without thinking, then go ahead and keep focusing on those steps. Griffin gets right answer with the lattice algorithm, and I have every confidence that I can train him to get right answers with the standard algorithm too.
But we should not kid ourselves that we are teaching mathematical thinking along the way. Griffin turned off part of his brain (the part that gets 37 times 2 quickly) in order to follow a set of steps that didn’t make sense to him.
Ding-ding-ding. As a non-mathematician, I am in total agreement.
So now back to my issue. Have you ever tried to explain 2 and 3 (and more) digit multiplication to a kid? And what “carrying the ones” means? And keep any sense of place value? I mean..try it. Right now. Explain why you do those things to an imaginary (or real) nine year old.
It works, sure. You get the right answer. And there’s a reason it works. But now….let me tell you about how Michael is learning multiplication of 2, 3 and more digit numbers.. It may not be new or radical to you…perhaps it’s being incorporated in some of the other new math materials out there. But it’s new to me, and I’ll admit when he first started, I got nervous. I was thinking, “Wait. This isn’t the way I was taught. I mean, I don’t really understand the way I was taught..but this is different! I don’t think it’s what they’re doing in regular school. WILL HE BE AN OUTCAST?”
Well, not really on that last part. So let’s go to the photos:
That’s how I learned it. You, too, probably. Again, imagine explaining to a kid why you carry the 1 and then the 3 and why you put a zero in the units place on that second line. Try.
Now here how Michael’s learning.
Do you see? It’s the Distributive property, in action.
In case you don’t – it’s (3X5) + (3X40) + (70 X 5) + (70 X 40). It’s an accurate, clearly laid-out expression of what is happening in the act of “multiplying” these numbers.
The beauty of it is that if you can do part of it your head – if you know that 45 X 3 is 135 right off the bat – feel free to just put it down that way. Doesn’t mess anything up.
This makes so much more sense. To me, a non-math person. Yes, it takes up a bit more space on the paper, but it preserves the sense of what the numbers are and what is going on in the act of multiplication. In other words, it’s not just a “rule” but a clear process.
Has this been the dullest blog post ever on my blog(s)? Probably. But at least I got it out of my system.
The examples of “Common Core” math that I have seen do, indeed seem unnecessarily complicated and frankly convoluted. I think the intention is to encourage a deeper “number sense,” but they end up confusing instead. My point is that what I have encountered in the AOPS programs has certainly been new to me, but as not-mathy person I haven’t found them confusing at all,but rather illuminating and quite interesting. There is a way of teaching a way of doing math that is a more accurate expression of what is going on and which doesn’t seem so random, especially to the non-mathy person. The tragedy is that a worthy end is being massively screwed up and, as a consequence, raising suspicions against any attempt to develop better ways to teach our children math, better ways that are out there and that are not crazy or needlessly confusing – in fact, are the opposite.
Below are some of 9-year old Michael’s math pages from last week. You’ll probably have to click on them to get a better view.
This was the first workbook page on which he had to work with this new algorithm. Robots optional.
On this page, he was given just a few numbers of each problem and had to work out the rest. So, for example in #144, he would have to work out what do you multiply 6 by that gives you a number with 8 in the units digit..well, it could be 3 or it could be 8..so you have to go from there and figure it out. We left the last one to do as review later.
He started exponents late last week. On this page he had to work out where to put parentheses so the equation would work. If it worked without parentheses, circle it. (Obviously it was also an exercise in understanding Order of Operations.)
The way that Beast Academy is planned (they haven’t finished all the books yet…) the student will be ready for Pre-Algebra after competing level 5 (this is 4C, with one more to go in the 4th level) – I had my doubts when I heard that, but as we go on…I can see it. Michael is going to have a completely different, deeper understanding of math than any of his siblings..and it will be better, I have no doubt.