*Where Mathematics Comes From*, by George Lakoff (yes, the Lakoff who analyzes political metaphors), and they try to root out some of the basic images, ideas, or metaphors that we layer upon to build mathematical thinking. I have not finished reading the book, but there is obviously a qualitative shift that happens for every student from innate to learned. While greater than-less than is probably an innate idea when we talk about the size of physical objects, all the ideas about greater than-less than with numbers or variables is learned. Clearly, the ideas about comparing the size of numbers or variables layer upon the innate understanding -- at first we reason by analogy to physical size, i.e., a group of six things is bigger than a group of four things. If, by chance, we met someone who could not grasp the idea of comparing physical sizes, we would think it quite odd and have a hard time helping them learn math. Yet despite the fundamental nature of the innate ideas, we know that most ideas in mathematics have to be learned. Plato gets himself into phenomenal trouble (pun deeply intended) by thinking that all mathematical ideas are a priori -- essentially, memories -- and not learned.

What is a mathematical concept or idea? It is actually really just a set of relationships (built up from some very basic, innate idea, perhaps). For example, "equilateral triangle" is a set of relationships: what a side is, what a polygon is, what equal lengths are -- and that is just the definition. Then all sorts of other properties get tied in: that its angles are equal, how to find the length of its altitude, how to find its area, that its altitude is its median is its angle bisector, that all the properties true for isosceles triangles are also true for equilateral triangles, the relationship of its inscribed circle's area to its circumscribing circle's area, how to tessellate it, etc., etc. To take another dig at Plato, he might say that this complexity shows equilateral triangles are the basis of all reality! (Yes, he really thought this.) Sorry, Plato, a lot of concepts are this interconnected.

What goes wrong when a student does not understand a concept?

- Failure of imagination: They are having trouble understanding the basic idea, connecting it or analogizing it to innate ideas and/or ideas they already understand.
- Failure of wrong picture: They think they have the right picture, but the analogy is wrong.
- Failure of oversimplified heuristic: The basic idea is right, but the student has a shortcut that is insufficiently complex to solve problems. In other words, the foundational understanding is there, but the mathematical thinking that has been layered on top is sketchy.
- Failure of overgeneralizing: They understand an idea and know the mathematical strategies, too, but they apply this idea inappropriately. Connections to other ideas are supposed to enrich an idea AND LIMIT it. For example, students should use isosceles triangle properties on equilateral triangles -- but not in the other direction! Yet many see the ideas map in one direction, and assume both directions of idea-mapping are acceptable.

One particular example of overgeneralizing that drives me nuts: CPCTC. For those who are not geometry teachers, this means "corresponding parts of congruent triangles are congruent." A rather obvious property, once you think about it, which should not really merit a name. Yet students overapply this like crazy. At one point, I had to scold my students: "CPCTC is not magic pixie-dust that you sprinkle on your work whenever you hope a miracle will happen!"

We teachers can feed into this, of course, by overgeneralizing. My worst example is the prohibition against side-side-angle congruence. We tell students it does not work, but that is an overgeneralization. Side-side-angle WILL work if the angle given is right or obtuse. And it even works for acute angles if the opposite side is longer than or equal to the adjacent side. It really only fails to prove congruence in one instance: given an acute angle and the adjacent side longer than the opposite side. So why do we tell our students to avoid this method of proving congruence? It actually makes a fun problem to foray into. Here is a fun little GeoGebra to begin exploring the ideas.

Aside from failure (1), the others are accurately called "misconceptions." The students think they have the idea, which makes it especially challenging to correct. I am going to quote Shawn Cornally at length here about how to root out misconceptions:

**You have to tee up the misconceptions**. This does*not*look like simply saying the misconception followed by the “correct” model. This does look like getting the students to display their dizzying array of pre(mis)conceptions. Use whiteboards, video tape each other and compile it, and come to a consensus–*even if it’s the wrong consensus.***You have to provide an experience that frustrates and confuses.**This is based on some awesome neuroscience; basically, the human brain is not like a computer file system. When you edit the file on a computer, the magnetic dipole or electronic domain are flipped with an application of current. Human memories are repetitive, sensory-correlated, and plastic. You have to have the file open, and you have to chisel at it repetitively from different angles, and the misconceptions will often spend some time living in bizarre discord with the more appropriate model. This is ok, if not somewhat drawn out.**You have to measure the extensibility of the new model**. Can the student take the new model on the road? Is their understanding of rates limited to iterations of meters and seconds, or can they take that and apply it to dollars per volume? Have you measured their abstraction level?

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Ben Orlin has a great post where he compares doing mathematics to rock-climbing. It is worth reading in its entirety, but here is the part that struck me.

Too many students think math ability lies along a single axis. But soon they discover the sprawling taxonomy of mathematical skills. There’s speed--in computing, in connecting, in spotting errors. There’s organization--of arithmetic, of arguments, of ideas. There’s understanding--of concepts, of technicalities, of deep truths. And so on. When you’re doing math, you’re not just a single mechanism. You’re a whole brain...

Lots of students try to do math like a man. They stare intently at an example, and then tackle a practice problem, diligently striving to reproduce the method step for step. Sweat gathers on their brow. It’s a draining, painful process, and not at all how math is meant to be done. Like rock-climbing, math rewards the nimble. People who employ their full toolbox fare better than ones who rely exclusively on memorization and formalisms.

This strikes me as just right. Mathematics is supposed to be elegant, flexible, graceful, and subtle. The attitude of a student who does math "like a man" is abetted by a curriculum that allows that approach to be somewhat successful. I think a good curriculum ought to convince students fairly quickly that this is a dead-end approach; I have written about it here, here, and here. To summarize, a student who does math "like a man" probably never had a teacher "tee up" to a misconception.

One last thought: I have a meta-study guide I give my students at several points during the year. I want to emphasize to them that studying -- good, effective studying -- is about thinking conceptually, rooting out misconceptions, and working to understand things. My guide gives them some concrete suggestions to do so on their own.

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