*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:

- Instructor tells them about something, includes worked examples.
- 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.
- 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…

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