I mentioned in my twitter that I’ve got about 5 different blog posts in my head that I haven’t managed to put to “paper” yet – and the one I’m taking the time to jot down tonight isn’t even one of them! (I’ve also got like two or three different drafts saved on here of ones I began but haven’t finished!).

Anyway, the one that I’m thinking about right now is about the constant battle that I find myself in: do I ensure that my students have a deep rich understanding of the topics at hand or do I make sure I cover every topic before the state exam? Without even the slightest hesitation, I tell you that I always opt for depth of understanding and mastery of the skills over coverage of content.

One thing I’m enjoying about my new school is that the other 6th grade teacher (there are only two of us) has been teaching the same content for many years (I actually observed her teaching in the same classroom during my graduate school placement back in 2009!), and she’s admitted she feels somewhat burned out and is excited to be working with me for all of the new ideas she gets from me. I receive some of her resources (and her advice/notes about the topics/order of the units/skills), and in turn, share with her my ideas for revisions and for next year. Some things she’s taken to more quickly than others, but I’m enjoying collaborating with her a lot, especially because she’s a mostly enthusiastic recipient of my ideas.

Our current topic is order of operations. She shared with me a bunch of her resources of things she’s done – some I really like/already knew (like Four 4s and Bowling), some I think are useful so I don’t have to make ten thousand expressions of my own (i.e. practice handouts), but some I think are repetitive and dull and not really helping students to grow their brains. I’ve been researching the problem with PEMDAS. I told my students it was banned, and if they HAD to use something, they should use GERMDAS, which I wrote as follows:

- Grouping Symbols (and gave examples, like the parentheses, brackets, braces, and absolute value bars, as well as the vinculum or the fraction bar)
- Exponents and Radicals
- Multiplication and Division
- Addition and Subtraction

>>>>>>>>>>>>>>>>>>>>>>>>>>> (from left to right)

We discussed that the paired operations were inverse relationships, and should be done from left to right. I thought I had done a pretty decent job of training my students to go beyond the basics that trip up high school and college students (stuff like 10 – 2 + 5, where they do 10 – 7 = 3 instead of 8 + 5 = 13). And then I kept reading and digging, and realized there are a couple more things I want to get my students to understand.

I want them to understand that the order of operations isn’t truly arbitrary! And that we can rewrite expressions to be more clear and less ambiguous by eliminating subtraction and division and rewriting them as addition of the additive inverse and multiplication by the reciprocal (also called the multiplicative inverse – in fact, I’m not honestly certain what the difference is between a reciprocal and the MI). So I decided tomorrow’s lesson is going to give the students some expressions that are intentionally ambiguous to uncover some of the potential problems they might still face AND also some problems where we can discover/use the “Boss Triangle” to talk about the hierarchy of operations and why it goes in that order.

I’m taking ideas from a bunch of different places:

http://mathforgrownups.com/the-problems-with-pemdas-and-a-solution/

http://www.mathematicalgemstones.com/gemstones/amber/pemdas-is-broken-and-how-we-can-fix-it/

I would way rather my students understand when and how we can ignore the “left to right” rules and the arbitrary order (and in fact that we can ignore the order of operations in certain cases and get the same answer because different parts of the expression don’t affect each other!), than for them to robotically repeatedly simplify expression after expression after expression…