# Tag Archives: order of operations

It’s summertime here at OMT. After an initial flurry of posting back at beginning of June, things have slowed down. But only on the blog. Behind the scenes, we’ve been busy as beavers. The interns have been trained in, and we’re rarin’ to go.

In particular, I’ve had a number of really interesting (to me) conversations by email. You’ve heard of email right? Kind of like Twitter, only no character limit, so most people use full sentences and spell words correctly.

Except one colleague of mine who sends the following sort of crap to all faculty (he is referring to Rate My Professor, and I wish I were making this up):

I dfer 2 yor sensibilities in this matter as I’v not gone there in yrs b/c I feel that things lyk worrying about how ‘hot’ students find us contribute 2 grade inflation—wich is a horse I’ve ben riding 4 sum tym, now

I digress.

Email conversations. Right.

I’d like to tell you about a few of these over the next week or so. Like the one I had with Justin Yantho as we hashed out whether the following video represents good teaching (with thanks to Frank Noschese for alerting us to it).

In a sign of summer torpor (and of the OMT interns’ inexperience), the following is copied and pasted mostly verbatim from one of my replies in the conversation.

That video is “training”, not “teaching” in my view. Effective training. But training nonetheless. I think of this analogously to training dolphins to jump through hoops. Stimulus (hoop), response (jump), reward (fish). Self-contained system, disconnected from other behaviors.

I’ve only watched four minutes or so of video. But it’s offered as “exemplary”-both in the sense of being an example of what’s being promoted, and in the sense of being very good. And as an example of what teaching should look like? I’m opposed.

To be sure, I’m opposed to a lot of what I’ve seen in other sorts of classrooms (including all too frequently in my own!)

What bothers me the most here is related to my reaction to another video Frank Noschese sent around recently in which “Integers are important because they’re a state standard.”

In the former, teacher says, “Tell your neighbor about the four operations”. In the latter, the teacher says, “In a pair/share, talk about why is it important for you to understand integers?”

In both instances, we’re setting the standard that “talking about math” equates with “repeating previously stated information” rather than with “exploring ideas, wondering or processing”.

Now, I get that I’m drawing gross generalizations based on small sample size (short video snippets). But each of the videos is purporting to demonstrate an aspect of good practice. These people want us to learn from the teaching we are seeing; they want teachers to emulate the model. That makes it fair to pick the examples apart, I think. And I’m totally ready to eat my words if you find videos in either one of these sites that pushes kids to really think about mathematics.

But I’ve been in a lot of math classrooms over the years. Lessons rarely move from this sort of rote opening into a mode involving rich thinking and dialogue. Not never, but it’s rare.

This puts me in the mind of The Teaching Gap (a book I cannot recommend highly enough). The authors of that book draw on evidence from a well-designed international video study to outline important differences in classroom practice in three countries: the US, Japan and Germany.

The connection here is that the teaching we see in these videos is an extreme example of how US teachers spend their class time-recitation and practice, in stark contrast to how Japanese teachers spend their class time-problem solving and discussing ideas.

And I haven’t even addressed the error(s), right? Why does the set of “order of operations” have six elements if there are only four operations? Is exponentiation not an operation? Why does the opening example involve an operation about which she does not speak?

Thanks to Justin for giving me permission to reference our conversation. As you can see, I didn’t really let him get a word in edgewise. He did a fabulous job of arguing back, though. If you’re not following him on Twitter, do so now.

## Pedmas questions revisited

Where does factorial fit in the order of operations?

Is $n+1!=(n+1)!$? I think not.

Is $3n!=(3n)!$? Now I’m not so sure.

So is it PEMDFAS?

Or PEFMDAS?

Or PFEMDAS?

## Calling all you PEMDAS (or BEDMAS, or PERMDAS) fans…

Consider the expression:

$256^{(\frac{1}{2})^{3}}$

Is its value 2 or 4096?

And can we all agree that this question doesn’t matter? Can we agree that it’s like debating the meaning of a poorly written sentence, when we should really be reprimanding the person who wrote it and imploring them to be more clear next time?

In short, can we allow Vi Hart to lead us into the light on this one?