## Friday, May 13, 2011

### Listing Math Preconceptions: Where to go next?

On twitter, @dcox21 shares a list he was making of math preconceptions. In the list, various preconceptions have been organized by math topics, for example, by fractions, integers, absolute value, rational functions, geometry, etc.

Organizing by math topic has its advantages, but it masks important things about students, because it's not organized by either the nature of the mistakes or by the student thinking that generates those mistakes.

Consider these three items that were listed as preconceptions.

To me, each of these is strongly related, despite the fact that they are listed in the table under the different topics of fractions, distributions, and non-linear functions.

In the first example, the student multiplies by two everywhere they can, which is top and bottom. In the second example, the student multiplies in one place when need to multiply in everywhere. In the third example, they square both terms, but in doing so, they miss the operations of x*y and y*x. Although the topics and grade levels are different, something about nature of the mistake is similar. Building a table of preconceptions solely around topics masks those similarities.

Next, consider these examples:

“You can’t divide smaller numbers by larger numbers.”

“Division always makes a number smaller.”

-10 > -6

1/3 > 1/2 (because 3 > 2)

These preconceptions were listed under the different categories of "integers", "exponents", "fractions" and "inequalities". But to me, all of these are related by the fact that they seem to stem from ideas that students are have around whole numbers and whole number operations. Children build up a lot of intuitions, ideas, reasoning, and procedures around whole numbers, and we are seeing here that students try on and rely on those ideas in a variety of ways as they learn about fractions and integers and exponents.

Placing the difficulties in different categories masks the fact that they seem to have a common origin-students' prior understandings and thinking around whole numbers.

What's the point?

I think there are lots and lots of potential connections and stories to tell by looking across the difficulties and ideas students have. I believe that a list of student difficulties is only useful if we do the work of trying hard to make sense of it by looking for connections and telling stories that help us see links between students' thinking and mathematical thinking.

Often it begins by asking, "Why would a student do this?" or "What does this mistake imply?" I often ask, "What good ideas do they have that would lead them to do this?" or "What could a student be trying to figure out what to do?" This often leads me to ask, "What ideas seem to be in place?" or "What ideas are they missing?" In the process, I often come to see my students and the discipline in a different way.

I'm curious to hear from other, what connections or stories they can see from the list?

1. I like the groupings you have here. Teaching at the college level, I have noticed that I have little patience with students who make these kinds of mistakes. What I like now is that we have a math center with a similar focus to our writing center so we can encourage students who have these sorts of misconceptions to get the help they need (rather than from crabby me).

Here's a quick tip on how to get bad evals: one year I wrote f=n (v/2L) on the right side of the board. Moving back to the left side and erasing I said "let's write that again and move on." I then wrote f=(nv/2L). Suddenly I heard lots of murmurs so I asked what was up. "We have NO idea how you got that." Long pause here. Me: "You should be ashamed of yourselves."

2. I like the idea of a math center. Is it well advertised and used? Yeah I can imagine why that would lead to bad evals... ;)

I often think of the evolution of physics education researchers in a similar way: We often start being shocked /disgusted by what students do wrong. We then often transition to being intellectually curious about what students do wrong and why. Many then transition to becoming empathetic--actually being able to see, understand, and relate to what students are doing and why. Lastly, there can be a transition toward participation--actively seeking opportunities to participate in the and with the thinking of students.

That trajectory: "Shock" -> "intellectual curiosity" -> "empathy" -> "participation" is really not at all different from what happens when individual encounter new cultures or unknown lands. For most of us, students' minds are anthropologically new territory.

3. Brian, how did you get inside my head?? Andy, I'm glad to know that I'm not the only one having a secret reaction of revulsion (or not so secret). That's not the only reaction, obviously. I think that the "shock->curiosity->etc." trajectory is more of an alloy, slowly undergoing oxidation in contact with students.

Ok, gonna tackle your actual question now, in a separate comment.

4. Tried to post this last night, but blogger still seems to be having trouble with comments.

First group: I came up with 8 ideas about the first example. Ideas in place, ideas missing, etc. But when I compared with the rest of the group, the link that I saw was "gathering like terms."

It's a sensible thing to do. If it doesn't make sense to compare apples and oranges, why does it make sense to multiply them? My students really, really want it to be kilowatts per hour, not KilowattHours. And what on earth does a FootPound look like?

I saw the "whole number" theme you mentioned for group 2, but only for some of the examples. The third, fourth, and sixth example (I'll call them 2c, 2d, and 2f) reminded me of problems like "a bus holds 20 people and your group has 67. How many buses do you rent?" Sensible (but not listed in the multiple choice) answers include things like "3 -- then I save money by getting a minivan."

For 2a, I'm imagining the thought process "4 multiplied by itself no times is, well, nothing!" I don't see that as being connected to notions about whole numbers -- it looks more like a misplaced application of the correct ideas that exponents are about multiplication, and multiplication by zero yields zero.

2b looks more like 1a, as far as I can tell.

For -10 > -6... To me, this is the easiest one to understand. In any practical sense, -10 really is more of something. A deeper hole. A bigger debt. The mathematical statement is wrong if you're measuring hills or assets, but the problem is in defining what you're measuring, and then resisting the temptation to *change* the terms of measurement when you are measuring down instead of up.

Have you come across the book Twice As Less? The author explores how the everyday meaning of words like "fraction" or "divide" or "less" can interfere with students' ability to make sense of the mathematical meaning. She prescribes Euclidean proofs for all.

I appreciate your point that we can structure our topics around the students' approaches -- so that things that are likely to get confounded get addressed together, where you can see their similarities and differences. This does require a lot of prior knowledge by the instructor about what those approaches will be.