The Big Unteaching Experiment: Rebooting the Machine

OK. Sorry for the confusion earlier today folks. I accidentally posted things meant for my classroom blog to this one. And they got automatically tweeted out. Awesome. So, now you know that I keep a simple blog for my students each semester. That way we all have a running account of what happens each day. I find it especially useful for my Moore Method geometry course, but it serves well if a student might happen to miss a meeting.

We are now past the half-way point in the semester. In fact, spring break starts right about…now. I have been worried for some time about the state of my experimental differential geometry course, and today I discussed with my students how we will go forward in a manner different from the first half of the term.

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The Coming Disasters

Sometimes, teaching feels like this:

Students have this conception that the math will just happen to them. If they show up for every meeting they will somehow magically learn. I, the Great and Powerful OZ, will cause you to be enlightened! Yeah…no.

How do we tell them that they have to fight against their lack of understanding? How do we explain that only sustained and deliberate practice turns a novice into an expert?

There I am, waving at my differential geometry students. “Hey, do your project. Tell me what you need.”

There I am, waiving at my math in decision making students. “Really. You gotta just try it a bit. Don’t just sit there.”

And there I am, trying to nudge euclidean geometry students: “Which problem are you working on? Come see me. You can’t just watch all semester.”


Stuff I Say a Lot

Originally Posted 09-05-2012

Like any teacher, I find myself repeating a few items to students. Sometimes in class, sometimes out of class. Here is a list:

  • Half a proof is no proof, but progress is progress! You just need to reframe your results.

  • Mathematics is divided into two pieces for each person. There is the stuff you understand, which all seems easy. Then there is the stuff you don’t understand, which is all really hard. There seems to be a hard barrier between the two parts. Our job is to move the line a little bit every chance we get.

  • It is considered rude to use the word “obviously” when writing or speaking mathematics. It might seem obvious to you because you have been thinking about it for three days. It might look really hard to someone who hasn’t thought about it.

  • Mathematics is just as much a social construct as anything else. It somehow consists of the way in which we have learned to work effectively and to communicate to each other. We have our own standards for making arguments, and those change depending on your audience. When you are speaking or writing, you have to know who is in your audience.

  • Sometimes your brain does mathematics unconsciously when you are not paying attention. Your job is to seed your brain with as much useful data as you can when you are actively working. When you reach a frustration or exhaustion, let the problem go and see what your mind can come up with when you are doing something else.

  • That is awesome. What else can you do? What is next?

  • I intentionally put you in the position to get stuck on a hard problem. How can I help you find a way to get unstuck?

  • It only looks easy when a professor does it because they put in years of work that you didn’t see. Mathematics takes time.

  • Wait. This is [so-and-so]’s proof. We’ll talk about your idea next. First let’s evaluate this work.

Those are the ones I use the most. Teaching in an variety of IBL styles, I have to do a lot of managing expectations and psychology.

A Student Misconception: They know what Mathematics is

Originally Posted 09-03-2012

[Part Three of the Math Blogging Initiative.]

An Issue

I teach college students. It seems like the issue that holds my students back from progress in mathematics boils down to one simple thing.

They think they know what mathematics is. They are wrong.

Perhaps you can see the real problem. Students don’t understand what mathematics really is, so they have no chance of improving at it. They can’t even properly conceive of what it might mean to work for improvement. No matter how hardworking and dedicated they might be, they are stuck.

The Misconception

It is pretty clear that my students expect the following “plan of study” for mathematics classes:

  1. Instructor tells them about something, includes worked examples.
  2. Student memorizes as much as they can, reworks examples until they can reliably do the kinds that have shown up in lecture or in the textbooks they have read.
  3. Instructor asks students to repeat the standard style exercises under time pressure without notes.

There is so much wrong with that I don’t even know where to begin. Perhaps the most serious problem I have with that is that involves no actual thought.

What I think

Let’s try a couple of definitions.

Mathematics is the use of the axiomatic method (extremely rigorous logical reasoning) about the concepts of shape and number.

Mathematics is the use of abstraction, pattern and structure identification, and formal reasoning to solve interesting problems.

Mathematics is the smallest discipline including the study of the integers, planar geometry, and whatever related fields where mathematicians have learned to successfully apply their methods.

Mathematics is a socially constructed set of work habits and communication rules for dealing with problems using logical reasoning.

I’m sure that none of those is perfect, but each has an aspect of the truth. I am sure that I have stolen bits of them from others, but I have no idea where at the moment.

Back to the Students

A lot of my students are planning to be high school teachers, so I feel it is doubly important to introduce them to the true nature of mathematics. It is the only way to break the cycle of misinformation.

Extra Scary: my students think they are “good at math” when they likely have never seen it before. They are good at memorizing and following directions. Many of them succeeded in their algebra courses in high school because they made a leap up the ladder of abstraction more readily than their peers.

What is to be done?

Well, I wish to say that I knew exactly how to change all of this. I have some anecdotal evidence that teaching my class using Inquiry Based Methods is making a difference. I put real mathematics front and center–nothing else will do.

This flies in my Geometry class. Somehow, students show up expecting that things will be different from their high school algebra 2 class.

I’d say the same is true for upper level courses that are focused on clear argument making. Students are in for the long haul, and are willing to have their eyes opened. But exactly this mismatch between expectations (mine vs. theirs) makes it extra difficult to do an IBL setup in lower level courses.

For example, my Math in Decision Making Students constantly doubt that I am actually teaching them mathematics. I have been asked at least six times in the last two weeks when we are going to do some “real math”. But these studetns aren’t used to having success, as a rule. Even if they think I am insane, they are willing to try out something other than mindlessly plugging numbers into formulas and waiting for a bell to ring. Last semester they came around, and I have faith they will do so this term.

In linear algebra, though, I have a tougher job. These students expect the “standard” system. They had learned to navigate the old system. They are a bit upset that I have changed the “game” on them. But I don’t want a game. So I’ll have to keep preaching…