# Category Archives: Reflection

## Where does math come from?

As math teachers, we need to stay vigilant about how we represent our subject to our students.

At her recent workshop for teaching artists here in the Twin Cities, Malke Rosenfeld said to the gathered group,

It would be fair to say that most of think about math inside a textbook context.

She paused.

Heads nodded, eyes wide with recognition. Malke prepared to demonstrate that this is not the full story.

Math comes from, and lives within, textbooks. I am not OK with this.

So what can we do in every lesson every day to represent mathematics as a subject that comes from, and lives within, the minds (and bodies) of our students?

## Reflections on teaching

I am working on a ton of interesting projects right now. Not least of these is my classroom teaching at the community college. My fingertips are sore from typing.

And yet there is always more to say. More to think about. More conversations to have. Here is a peek into one that is ongoing.

Malke Rosenfeld and I have been going back and forth about math, dance, Papert and learning for a few months now. I am learning a lot from the conversation. She asked some questions this morning.

Malke: A thought just entered my head — why are you offering TDI? Is it based on a question you are unsure of and want to see what others think? Or are you seeing a deficit in math teachers’ thinking that you want to shore up?

Me: When ranting on Twitter, I could see that some of my assumptions about baseline teacher knowledge about fraction/decimal relationships as they pertain to developing children’s thinking were unfounded. That is, I was assuming teachers knew a lot more than they seemed to. Which has implications for my Khan Academy critiques, and lots of other writing on my blog. Yet people were also curious. So I wanted to say more in a way that would draw from and build on a larger collective knowledge, so it’s not just my spouting off.

Malke: Is there a reason you offered it specifically as a course, and not a moderated discussion (which it sort of seems like right now)?

Me: When you view learning as a social process, you tend to think of courses AS moderated discussions. I mean this quite seriously. I know that it goes against the grain of online (and face-to-face) course design. But that’s not because I think of online instruction differently from others; it’s because I have a particular view of learning that runs much deeper than that. If I tell and quiz, you’re not learning very much. If I propose a set of ideas, listen to what you have to say, encourage you to interact with others and move the conversation in directions that seem useful based on those interactions, you’re probably going to learn a lot.

As long as I can keep you engaged in that process. Which is a different challenge online than in the classroom.

Malke: Is there a place you specifically want your students to get to by the end of the seven weeks?  Or are you just curious to see what develops?

Me:  I am curious to learn what I can about teaching at every opportunity. I want to produce “students” who can articulate important questions (see? learning as having new questions to ask?) about curricular approaches to decimals. Ideally, I would help them to develop a critical voice that speaks to/through them when they work with individual students, when they plan lessons and when they talk with their colleagues in a variety of settings. In short, I want to change the way teachers view the territory of decimals, fractions and children’s minds. Strange mix of lofty and specific there, huh?

## Another great question from College Algebra

Here is something cool that happened in College Algebra today. We were doing a short thing to summarize our domain and range work before moving on.

A student asked, Is the only way to find range to make a graph?

This stopped me in my tracks. I had not really thought about the knowledge I draw on when identifying the range of a function, and the question cut to the heart of the matter.

My gut instinct answer was yes. But I wanted to explore that a little. I concocted a silly function to do so. $\sqrt[3]{x^{5}+x^{2}}+x^{2}-sin(x)$. I wanted to say that I would need to graph that to know its range.

But the longer I looked at it, the more clear it was that I knew a lot about this silly thing without graphing it. The $x^{2}$ term dominates, for instance, in the long run, so I know it goes to infinity on both sides of the y-axis. I could see that 0 is in both the domain and the range.

But I wasn’t 100% sure whether there were any negative values for the function.

Later in the day, this got me thinking about end behavior. This is why we teach that end behavior silliness, right? It’s not about end behavior, it’s about knowing what values can come out of a function, and having a basis for knowing this.

I am brainstorming here. The point is that the student question showed a sign of her learning, and it pushed me to rethink something too. Win-win.

Another cool thing happened, too. We were comparing $y= x^{2}$ and $y=2^{x}$, looking for sameness and difference. I had to push to get domain and range on the table.

We agreed that the two functions have the same domain—all real numbers. We were split on whether they have the same range.

But not for the reason I expected. Not at all.

A student argued that The only time when they are the same is when x=2. Therefore they do not have the same range.

My students found this argument compelling.

Ignore the second intersection point in the left half-plane. Focus on the essence of the argument.

Do these functions have the same range? is interpreted as Do these functions intersect?

That seems like a useful insight into the mind of a College Algebra student.

## I get that there is no perfect lesson

I get that there is no perfect lesson. Really I do.

And I get that students leave my classes with wrong ideas. But the thing is, when I come across these wrong ideas, I try to do something about it.

A couple of tweets from the field last week (sender’s tweets are locked, sorry).

S came in today claiming to have used Khan academy last night to learn about decimal place value. Was adamant 0.63 > 0.7 cc @Trianglemancsd

Was also adamant that 0.4 < 0.40. Not feeling overly confident about Mr. Khan and his explaining abilities…

If I don’t like the videos, I am told it’s not about the videos; it’s about the exercises.

If I don’t like the exercises, I am told there are new ones in the queue.

If I don’t like the trial versions of the new ones in the queue, I am told that the particular exercises don’t matter; it’s about the knowledge map.

When I say that the knowledge map is flawed, I am told that it doesn’t matter because students can move around in Khan Academy in any way that they like.

And then every day kids are going to Khan Academy for help with decimals. Some of these kids, such as the one in the tweets above, are going there independently. And some of them are going there because their entire state is piloting it as a primary instructional resource!

Whoa there! they say. Khan Academy isn’t meant to be a primary instructional resource.

But then here is a video that Khan Academy produced…

At 20 seconds in, a student teacher in mathematics says this:

When I first [learned] about Khan Academy, it was mostly “my teacher said this, but I can’t remember what he said, so I’m going to go check it out on Khan Academy. So it was more of a personal resource.

That’s kind of where I was thinking it would be in my classroom down the line. “If you’re struggling with this, go check out Khan Academy.”

But now, after coming to this, it can be that first step. It can be the go-to. “Hey, go learn this. Go learn the foundations, and then we can take it to the next level in our classroom, and put in those hands-on activities.”

Just to be clear, Khan Academy produced this video. I am not misrepresenting KA here. They are proud to share that a math teacher at a training views Khan Academy as a good primary instructional resource.

Now, I have long been critical of textbooks that introduce decimals as though they were a logical extension of the whole number place value system (just ask my students!) I am no fan of what Hung-Hsi Wu calls Textbook School Mathematics.

But if you are going to get introduced by the publisher of The New York Times  [at about 3:00 in the linked video] as  ”a true pioneer” who is “breaking down barriers” with “heart”, “guts” and “innovation”, I think you have a responsibility at least as great as that of the average textbook author. You have to strive to do better and you have to pay attention to what people already know.

If you are going to repeat that your mission is “a free world-class education for anyone anywhere,” you need to spend some time concentrating on the meaning of world-class, rather than imitating the bad textbooks that presently exist.

I have taught many crappy lessons, and I surely have many ahead of me. I do not fault Khan Academy for having a few crappy lessons.

But I seek feedback from my students on what they are learning.

I consult research on learning for the topics they are struggling with. I collaborate with colleagues near and far to improve my lessons.

I do not defend my crappy lessons by calling them unimportant. I own their crappiness.

And I strive to do better next time.