As teachers, we face many tensions about how to conduct our classes every time we teach! This has been even more prevalent than ever to me this term. Every time I have made a lesson plan this term, it has never gone as planned . . . actually, I don’t think it ever does, but it’s just been way more obvious to me this term than in the past.

First day, I walk in, and although I wanted to start them off with a problem, I felt the anxiety in the room. So I asked them what questions they had about the course, and we were able to cover many of the key aspects of the course outline without even looking at it! Then, I was about to give them a problem, but I thought, hmm, this is the first class, let’s see what their perceptions about math are . . .

So I asked, “What comes to mind when you hear the word ‘math’?”

Here’s what came out of this:

(I provided them with the ‘fake world math’ terminology after they described it to me!)

This was the best start I ever had, and it was because I created an opportunity to listen to them and to listen to what I sensed the room needed at the time in that context.

I have a very lively group this term, and am very happy with the energy in the room every time we have class. From the moment I entered the room on the first day, there was a classroom norm of asking questions set right from the start. I was blown away by the questions they were asking, and how collaborative this group was. It may be something to do with the size of the class. I haven’t taught a class of 30 in a while! Last term, my class was around 10 at one point, making it difficult to get random groups and vertical non-permanent surfaces work very well. But with 30, there seems to be enough density of ideas in the room to foster a ‘keep thinking’ atmosphere!

Now my greatest fear is – how do I keep this fire alive . . . all term???

So far, my favorite lesson was when we were reviewing order of operations. After getting them to remind me about what order of operations they know about, I asked them to (in random groups on whiteboards around the room) find as many ways as they possibly can to make the digits 1 through 9 to be 100. Digits need to be in order, any operations may be used, and numbers can be clustered if needed. (Source Martin Garnder’s “Further Mathematical Diversions” – see http://www.worldofnumbers.com/ninedig1.htm).

The room was buzzing!

Some groups had answers right away, but most didn’t. Everyone struggled, but most persevered. This group worked very hard, and by the end of it, they were SO PROUD of it!

This individual asked, “can it be super complicated?”

And this group, found that by identifying a way to make 50, they were able to get to this:

For those groups who after a while of persevering started to evidence a feeling of struggle, I told them to try some of the similar problems from Fawn Nguyen’s Foxy Fives. I came back, and they told me that was too easy, and they wanted to continue trying the 100 problem!!! I was in awe of their perseverance, and told them that this was what mathematics was all about!

We then could have a discussion about what they noticed themselves doing in this problem solving exercise, and how the process was more valuable than the product because in the process of trying to find a solution, they were using their order of operations abilities inadvertently.

Now, my tension as a teacher is that I don’t have such magical entry points to every topic in my course, and some topics are so dry that really the only way to ‘problematize’ it is to give them the tasks to figure out in their groups without me introducing any particular ideal way to approach the problem.

I’ve been playing around with this by giving them a more difficult problem from the section to solve before they have built the tools for that problem.

For instance, I asked them to find possible solutions to:

You have won the lottery and are about to invest $20,000, part at 3% and the rest at 4%. What is the most that you can invest at 3% and still be guaranteed at least $650 in interest per year?

Because they hadn’t yet been jaded by textbook approaches to this sort of problem, many of them began with trial and error, which is what I expected. This was great! During our reflective discussion, I guided them to seeing that if you create a trial and error process, you can eventually develop an algebraic statement that encapsulates that process. My aim was to show them that algebra is a generalization of a process.

Now, where my tension came in, was during the next class, when I was trying to introduce solving inequalities and absolute value inequalities. I started by asking them what the possible solutions were to:

However, I feel that we moved too quickly into questions that were too complex, and in this particular case, I think it would have maybe been better to start them off with simpler inequality problems, and have them work towards more complicated absolute value problems within their groups.

A lot of learning took place, I’m sure, but I left the class feeling as though there was still confusion about this whole topic, and when students were trying to solve the problems, they were making many mistakes. Mistakes are a place for learning, but I feel that this is a topic that I should have found a more interesting approach for.

So to identify the tensions that I have gone through these past couple weeks:

• Deciding which problem to give students next based on my interpretation of what the class needs at a given moment, regardless of any preexisting plan I have made.
• Deciding when to stop the problem solving process and move into a discussion.
• Deciding from which end of a topic to begin (complex to simple, or simple to complex)
• Feeling as though a topic may have been possible to make more interesting with all the aspects I theoretically know about such as creating a need, opening the middle, low floor/high ceiling, keeping students in flow, etc.

I have been learning about what being an “effective” mathematics teacher entails lately as part of my graduate work. Interestingly, there is no “theory of teaching,” there are only “theories of learning.” Not a big surprise given that teaching IS learning! Research also points to the fact that being an effective mathematics teacher is MUCH more than being good at mathematics. There are different explanations that we as mathematics teachers must develop and rework constantly. We also need to adapt continuously to ever changing conditions. In this sense, I’ve been drawn to theories of enactivism, which are based on the notion that we co-evolve with our environment in a continuous, complex, and adaptive way. We restructure ourselves to be able to remain viable. In a classroom, this means that we adapt to what students need in the moment. In many ways, we are always doing this. However, we also seem to be tied to our “lesson plans” and external curricular structures. These constraints can sometimes make us forget what teaching is really about.

Since learning more about this ecological perspective, and being inspired in particular by Brent Davis’ work, I’ve made a slight shift in mindset when teaching. This past term, I had quite possibly the best conditions for teaching in this manner. My room had two whiteboards on each of three walls, meaning I could finally try out a full version of Peter Liljedahl’s Vertical Non-Permanent Surfaces. I wasn’t able to blog about it regularly because of an extremely heavy term (teaching 4 courses, when 3 is a full time load, taking one graduate course, volunteering as part of a visioning committee for my institution that met weekly and had many assigned readings, and trying to work on my dissertation work). However, I resorted to a different form of tracking what I did in my classes. The purpose was more for my students, but it ended up being very helpful for a colleague, who used it to teach the course this term.

A typical class would consist of giving students a problem (by the end of it, I could really choose any problem because they were already in the ‘problematizing’ mindset). Students would form random groups, and would work on the vertical whiteboards. The benefit I experienced of vertical whiteboards vs horizontal ones, was that the ideas could spread across the room. I could also call a class meeting around a particular whiteboard to emphasize something someone had done. Here’s sort of what a typical class looked like for much of the period:

During each class, I would take pictures of every group’s work, and then plunk them into a document that I would share with the class. This way, students could take a look back through what all the other groups were doing during the class. The double edge of this is that now I have these “photo lesson plans” that my colleague is using to guide her problem based version of the course! This practice has become generative because she plays off my ideas, and now she’s doing the same thing, so I’ll be able to play off her ideas! For example, she came up with the idea to bring in a kids bike to class to pose the problem: How far will this bike travel with a quarter turn of the pedal gear?

(this is sort of what the bike looked like . . . I didn’t take a picture, I just saw my colleague walking around with a kids bike, and had to ask – just imagine a pretty lady in a dress with flats, walking around with a little kids bike! Ah, math teachers)

BRILLIANT! I hear it kept students engaged all period!

Notice how no one is using books – they are talking, and thinking. There were so many times last term that I found myself at the back of the class, watching everyone, in awe. However, what this meant, was that I had to be EXTRA observant. AT any moment, I needed to be ready to take them somewhere further, to a different realization. The vertical whiteboards helped me adapt to what they needed.

I only hope to further enhance my adaptive abilities next term.

Maybe I’ll learn something really neat at TMC15! . . . heck, I’m sure I will learn a lot of cool things that I will be able to adaptively incorporate into my lessons next term!