Archive for December, 2012
It has been 7 months since I last wrote because I have been super busy with big life changes. First of all, I got married. (I couldn’t possibly do justice to that here, but I had to mention it.) Then I moved from fifth grade to eighth grade. I have really loved the change both in spite of and because of the challenges that come with teaching a new curriculum. And perhaps most importantly, I’ve been busy with some pretty great math that has nothing to do with school:
But I’m back because I couldn’t think of a better place to ask for math help. And I really need it.
Let me digress – I never really appreciated how important it is to deeply understand the math you’re teaching until this year. Every day something happens that makes me feel like I don’t know everything I should, and I think, “Gosh, pedagogy and teacher education are frickin’ important.” I have always been really good at math but that doesn’t mean that I didn’t get by on a little bit of rote memorization here and there. That is really hurting me now. Kids can’t just memorize rules in math because there would be too many of them. It would literally be impossible. Understanding concepts and chunking them together lightens the cognitive load. But how can a teacher do this for her kids when she can’t do it for herself? Which brings me to my question…
I’m struggling to explain why we simplify expressions differently depending on whether we are dealing with factors or addends. I can’t seem to put my finger on the underlying math concept that distinguishes the two. For example:
The same concept pertains to simplifying radical expressions:
And it pertains to scientific notation:
I know that the ease of multiplication and division has to do with the commutative property. But how do you explain why it doesn’t work for addition and subtraction? I have used examples kids can solve with order of operations like (3×4)(2×5) and (3×4)+(2×5). But the main idea the students seem to come away with is “Something is wierd with addition and subtraction,” which is not really helpful.
I am writing because I am hoping that teachers who have taught this before or people who just simply enjoy thinking about math might be able to provide insight. I am especially interested in being able to explain the relationship between the examples I mentioned. I’d love an analogy or model that would allow me to really show the kids the basic concept that underlies it all. Frankly, I need an ah-ha moment.
I don’t know if this post will end up making me seem like an idiot or a smarty pants. But if nothing else, it highlights the depth to which teachers must understand their content and how difficult it is to get there on your own. So, come on, math geeks! I know there’s an explanation out there that my kids would love!