What do you do when they don’t need you?

I’ve tried an experiment over the last two days – my advanced algebra students and geometry students each had some challenging tasks that I sort of left to them to figure out. Last year, I taught them very explicitly how to do the tasks at hand, modeled some examples along side their own work, and then gave them time in class to practice. For homework, I gave them more problems that were similar to those we did in class, giving them more chances to practice what I had assigned them.

This year I turned it around. In geometry, we are starting proofs. I gave them a couple relatively simple ones, and asked them in groups of two to construct some sort of logical reasoning to go from a starting point to proving the statement I had given them. There was a lot of struggling, difficulty stating using facts why one logical statement led to another. Over time, they did start communicating with each other and sharing what they were thinking. I did occasionally poke one group in a certain direction, but didn’t lead the whole group in that way. Eventually they were all thinking along the lines that I envisioned at the beginning. I could have modeled for them what I did last year, but I saw a lot of really good conversations along the way. By the end, they were much closer to making their own proofs than they had in the beginning. By the end, they were clearly seeing the connections between thoughts. This was only the second class period during which we had talked about proofs. While I don’t think any of them would wager large sums of money over constructing geometric proofs, I think they at least see how the system can be used to make logical statements that are irrefutable.

I did something similar with the advanced algebra group which was to figure out graphing absolute value functions during our lesson last Friday. I gave them an exploration that was, in hindsight, confusing and didn’t do much aside from frustrate them with Geogebra commands. I told them that I wanted them to use Geogebra, the textbook, Wolfram Alpha, and any other resources available to learn how to graph any arbitrary absolute value function by hand. At the end of the class I broke down and apologized for giving a poorly designed exploration. I told them I would put together a video on graphing functions, and I did – posted it on the wiki.

When we had class on Tuesday, I found out that none of the students had actually taken advantage of the video. They had looked in the textbook. They had graphed functions over and over. When I asked students to share what they had figured out, one student used a table of values and a piecewise function based on the sign of the argument of the absolute value function. Another student had graphed both the argument of the absolute value function and its opposite since that was what this student had observed, and then erased either the top half or bottom half of the graph. Another student broke down and did what the book said to do. By the end, all of them were graphing absolute value functions using their own method. I wasn’t sure about understanding, but in the end, I admit that I didn’t quite mind. They all had their own models for what was going on, and they were confident that they could use the technology to confirm whether what they were doing was right or not.

I have always wondered about what I would do in the situation where it was clear a group of students didn’t need me to be there. In a way, this is part of my fear of tools like Khan academy – if there are others out there that are more engaging, better at explaining ideas, or better at coming up with really interesting questions that got students thinking about what they were doing, and these people happened to make videos: what would I do in my classroom if students got hold of these videos? Would I be mad? Or would I figure out a way to take advantage of the fact that students had figured something out on their own and use it to do something even more interesting or impressive?

I think I do a good job of engaging students – today we were talking about differentiability and we were joking like crazy about whether functions would be differentiable at a given point or not. Really? Were we really joking about this? It seemed like everyone was minimally entertained, but based on the questions I was bouncing around from person to person, it seemed like they also understood the concept. I know teachers that are more effective though at making kids understand and be entertained and do problem after problem until concepts are so painfully clear that they become automatic. These teachers could easily make a career in stand-up or television based on their comedic brilliance and presence – what if they decide to make videos?

What then? What do I do? If I get to that point, is it the end of my usefulness as a teacher?

Or is that just the beginning? Maybe that’s the point where I can assume my students have a certain level of basic knowledge and I can then build off of that level to do even cooler stuff. Maybe that’s where I can assume, for once, that my students have a base level of skills, and can then rise above to analyze bridges or patterns in nature or create a mathematical model that inspires a student to choose to be a doctor or engineer where he or she never would have if I hadn’t done the right project or group activity or lesson. Maybe the fact that I entrusted my students to try to figure out something on their own is enough that they feel empowered to try things even if failing again and again is a possibility. Make mistakes and come back asking questions about why their theories were incorrect. I remember worrying at one point in my career what horrible things would happen if I somehow introduced students to a concept in a way that it caused them get a question wrong and cause them just one more failure in a line of failures. If I could teach in a way that makes students feel OK coming in the next day saying “I didn’t get it, but this is what I tried” I’d feel pretty good about myself. That is, ultimately, the sort of resilience that a person needs to survive in this world.

If the students show us that they don’t need us to show them how to solve a specific problem, then we as teachers should honestly accept this fact. Our goals, if they can do the problem in a way that is mathematically correct, should shift to applying that ability to doing something more profound and relevant, be it communicating that solution or applying the solution to a new situation that is different but connected in a subtle way. Our job puts us in contact with some really amazing minds that are eager to do what we say in some circumstances. In the cases that they demonstrate that they don’t need us – that is when we must apply our professional judgment and teach them to expand their knowledge to something bigger than themselves.

Not sure why I’m waxing so philosophical today, but I’ve been really impressed with my students this week, and it’s only Wednesday! After these few days, it feels like what I’m doing with this group is like using a super computer to do word processing. I only hope they are enjoying the process as much as I am.

Nice work when you can get it….

I graded my first set of physics tests today. With the group deciding not to take the class at the Advanced Placement level, we’ve been able to slow down and spend time experimenting and really engaging with kinematics and projectile motion. I assigned them problems and helped them learn, but I was more impressed with the experiments they worked on and their engagement level during those activities. I was concerned about what would happen when we returned to solving problems, but I was very pleasantly surprised.

I’m interested in sharing student work, so this is the first time I will be doing it. When I asked the student if it would be OK to share, the student agreed and was really excited that I would want to show the work to other teachers.

This student started out solving problems in a very scattered way: calculations here, sketches there, units nowhere to be found. When I showed the structure I wanted students to use to solve problems, it was initially a burden. The student didn’t like doing it. Upon grading, I was very happy to see this:

The degrees vs. radians issue is one that I always battle, made especially difficult this year because students have me for physics (when I insist on degree mode almost exclusively) and then calculus (when I change my insistence to radian mode) right in a row. Yes, the student should have noticed that multiplying by the ratio should not have resulted in a negative sign for initial y-velocity. Yes, it should have again become obvious that something was up when he found he needed to ‘add’ a negative sign in the answer at the end to make the sign of the answer make sense according to his own sign convention.

The fact that the student can notice these things (and that I can see where the errors are) would not even be something I could discuss if it wasn’t for the structure I put in place. By learning to use the structure to organize thoughts, this student became able to solve problems in a logical manner rather than with calculations all over the page. I don’t like teaching procedures, but this is an example of where it pays off.

We like students exploring and experimenting and constructing their own knowledge. These are really good ways for them to spend time in our classroom. I include using correct mathematical notation, showing steps, labeling axes, learning terminology, and other things of that nature as part of my class expectations – at times a battle that seems unimportant in the context of what I really want students to know how to do in five or ten years. There is room for procedural knowledge, however, and this student’s success is evidence of why we do it.

The main point (and the thing I’ve been working to change compared with how I did things for a while) is that these procedures should not be the meat of a lesson or the main focus of instruction all the time. These things CAN be taught by computers or videos and don’t necessarily need a human in the room. It is important for students to have skills and have access to resources that help them develop those skills.

But getting answers is not the point – this is the tricky part that we have to do a better job of selling to students, at least I do. Clear communication of reasoning and sharing the logic of our ideas are some of those “21st century skills” that students should have when they leave our classrooms. If a student needs to learn a structure to help them with this process, it is worth the time needed to help them learn it.