Metaphors, misconceptions, and doing mathematics

In our brains, there are some hardwired, a priori ideas. For lack of a better word, they are innate. Built-in. I've been reading 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?

1. 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.
2. Failure of wrong picture: They think they have the right picture, but the analogy is wrong.
3. 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.
4. 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:

1. 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.
2. 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.
3. 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.

review strats.docx

A problem-centered math curriculum

I like the gardening metaphor for education. The children are the plants; the teachers are the gardeners; and the teachers' job is to create good conditions for the plants to grow. The metaphor reminds me that the learning is a product of my students' own efforts to understand the concepts and do the problems; however much I might do, I can't grow for my plants. Er, students. My number one rule as a teacher is, "The children are supposed to be doing the thinking." Watching me do math on the board is like a pep talk to my tomato plants. I'm not opposed to telling them how to do problems; it's just that I recognize how much practice they need on their own. So, I water my plants with carefully curated, appropriately challenging problems. I weed, watching my students work and correcting their errors. I give my plants fertilizer: ... hmm. Well. I guess this is like formal definitions and concepts -- a little bit can help the students' grow very rapidly, but too much can burn the soil.

The problem with the metaphor is that it sounds so peaceful; it is nothing like the lived experience of teaching this way. I give a micro-lecture on a concept. I am very conscious of how much I talk. As a former debater, I could prattle on endlessly, but I try to keep it very brief. Then I pick questions I think are appropriately challenging -- ones they can do but that stretch them a bit -- but sometimes I pick wrong. It's tricky: if the problem is too complex, then too-many details will overwhelm the students; if it is too easy, the students won't have to do any mathematical thinking, just recall of previously-learned facts. I write up a problem and am met by blank stares. Time ticks by in fractions of hundredths of a second.

Do I give a hint to give them an in-road, or is that cutting them too much slack? By definition, appropriately challenging questions are at the periphery of their knowledge, so the first steps are from safe territory into unknown lands; they are tentative and do not need to be rushed. The creeping roar of silence in my ears is loud. I am thinking back to graduate school, where we watched videos of U.S. math teachers and videos of other nations' math teachers. The U.S. teachers asked easy "yes-no" or other simple questions; they rattled the questions of rapid-fire, giving students only a few seconds to answer. We don't play that game in my classroom. They've been thinking for a while now, and the wheels are now beginning to turn. Many students are scratching diagrams or ideas down. I sweep the room for the stragglers who are truly stuck, and then we're off and running, and the real work begins in earnest. The complete silence lasted for no more than a minute.

I walk around the classroom and am a triage nurse and ER doctor, rolled into one. I try my best to diagnose problems at a glance. Wrong concept. Oops, an algebra mistake here. Simple addition error. Solid work, you. And I must decide who's going to be able to catch their own error and self-correct. It depends on the kid, on the concept, and on the type of error. Simple addition errors and algebra mistakes are rarely noticed. Those need to be pointed out, but some kids require a very delicate correction because they'll erase all of their work -- and I do mean all of it -- if they don't realize it's only a very minor error. Some kids will see a conceptual error quickly if I pose a counterexample, but other kids need a thorough explanation to see the error, and even then, they're sometimes just humoring me. Generally speaking, I try to offer the lightest help I can to get a student back on track, and if I'm unsure, I follow the doctor's rule to "do no harm." Don't freak out a kid who's just starting to get it. If he is just starting to see A, don't point out B, C, and D. This process is fast and messy, but the room is not chaotic. Not quiet, but focused.

After about 10 minutes of this, the kids are in thoroughly different places. The stronger kids are on the next, more difficult set of questions, but the weaker kids are still struggling on the first set. These are not cookie cutter-style problem sets: "Here are 50 right triangles. Use the Pythagorean theorem. Go, go, go!" For some kids, my class is the first time they are asked to determine which math tool is appropriate for a problem. ("Is this actually a right triangle? Why do you think you can assume that?") One kid noted with surprise, "Hey, this problem is actually last week's concept." Yes, things keep coming back! It should not be unusual in a math problem set, yet I know it is too rare. After 10 minutes of solo or small group work, we discuss. The stronger students contribute more, of course, but because the weaker students wrestled with the questions, they get a lot out of the discussion. More than if every single problem was explicitly modeled for them first.

Teaching this way can be overwhelming (so many mistakes at once, sometimes). It can be bewildering when I can't figure out what a student has done wrong (but I know the answer is wrong). But the upside is huge: I uncover more mistakes, more bad assumptions, more misunderstandings in one class than I would in a month if all I did was lecture and give cookie cutter-style problem sets. I want to see which students don't know that the Pythagorean theorem applies only to right triangles, which students make a lot of algebra mistakes, and which students are skimming by on a thin surface over a lake of incomprehension. This way, I have a chance to help them before the test. Although some students are hesitant to dive in at first, there's been little resistance to this problem-centered approach. They recognize they'll get lots of help if they stumble, and they feel proud when they can do it themselves. The real question is what homework looks like. Since I'm not there to help, I keep homework straightforward; if kids get stuck on the homework, they get frustrated fast.

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What makes a good math student? As a colleague recently put it, good math students see a rule once and do not make a mistake again. Perhaps the rule makes intuitive sense to them, but a good memory is a better bet. The biggest help, though, is what I would call an "object-oriented" mentality (to borrow a term from CS). Successful math students easily associate rules to their respective objects: these are properties of polygons, so therefore these properties apply to triangles; algebra has its rules; here is how you do a combinatorics problem. As the teacher shows them more, these "objects" and rules can get very fine-grained: the properties of all quadrilaterals, then parallelograms, then rhombuses, etc.

The weaker math students tend to approach it as a random grab-bag of properties, without organizing it schematically in their minds. These kids tend to look for analogies, and often, the analogies are superficial. The most heart-breaking thing I read in graduate school was an article where the researcher interviewed elementary school students who were working on arithmetic word problems. The researcher asked, "Why do you divide in this problem?" expecting to hear some reasoning about the context in the word problem -- Bill and Jane are splitting up the apples, so we need fair shares, or something to that effect. The kids' answer? "Well, there's a big number and a small number, so whenever we get those, it's a division problem."

I remember reading in a book by Stephen Jay Gould (can't remember which) about the early geologist who thought landmasses uplift. But he had come to this conclusion he thought the earth was like the human body. According to Gould, this was also the way most of his contemporaries thought: rivers were like veins, rocks were like bones, clouds were like lungs, etc. This idea seems strange to us, so pre-scientific, because there is something very literary/artistic about analogical thinking. Today, we tend to believe we should put analogies aside when we do the actual science. Yet many students still reason through analogy. They tend to over-generalize. They forget rules. They make the same mistakes over and over again.

One of my goals is to help these kids develop an intuitive sense about why a mistake is wrong. For example, take the common mistake $\sqrt{a^2+b^2}=a+b$. How do I work to unseat this error, permanently? First, we spend time looking at the triangle inequality, which states that the length of the longest side must be less than the sum of two shorter sides of a triangle: a + b > c. We do this several different ways until we have a solid grasp of the idea. Now we turn to right triangles. How long could the hypotenuse c possibly be? Well, it must be less than the two legs added together: in other words, $a+b>\sqrt{a^2+b^2}$. I want the students to understand that their mistake is like saying "This is a right triangle" and "This is a straight line and does not make a triangle" at the same time.

You may think that this approach bores the students who don't make this kind of mistake in the first place. I have found, however, that there's no correlation between a student having an object-oriented mentality and understanding of why a rule works. These object-oriented students benefit from our class foray into the triangle inequality. Putting everything in its own box can limit insight; working to build connections helps these students be more insightful, as well as helping analogy-based thinkers avoid mistakes. Before, I wrote that good math students are object-oriented. But I mean good at school math. The best mathematicians, I'm willing to bet, use both kinds of reasoning. Object-oriented to systematize their existing knowledge, but also analogy-oriented to look for new ideas, to deepen an understanding of an old idea, or to explore connections. New fields of math come from those analogy-oriented thinkers.

The best problem sets:
- are manageably complex and appropriately difficult; this usually means any one question probably only uses one concept, but perhaps in a novel way;
- review concepts periodically;
- require students to think about which math tool is appropriate, so multiple concepts are used in a problem set;
- help students organize ideas schematically, so some questions ask students to summarize whether a concept applies in different contexts;
- help students make connections, which requires violating the first rule by tenderly bringing two concepts to bear and asking them to reflect on how one is related to another;
- and should be accomplish-able in 60 minutes (or whatever class time is)!

Ideally, students look at a problem, interpret it, recall some basic facts, think a bit more, and reflect on what they learned.

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I was a good math student in high school but not an especially thoughtful math learner. The concepts we learned were easy enough for my object-oriented mind to learn that I did not struggle to remember rules. But I had a superficial understanding of why most things worked; I did not make deep connections; and I didn't know how to do much mathematical investigation beyond what we had been shown. One particular memory: on my own, I tried to investigate how to use the diagonal lengths of a quadrilateral to calculate its area. The problem is rather simple, but I fumbled around at it. I had absolutely no idea how to process the pattern, what connections to make, etc. Yet I did well in high school math.

By the time I got to college, there was a pretty serious gap between what I had seen and what I actually understood. I struggled in some of my college math courses, but it never once occurred to me that my background knowledge was a key problem. Why would it have occurred to me? I'd always earned good grades. But looking back, I am aware that I had skated over lots of concepts for years -- but the math curriculum I went through never "caught" me. In retrospect, I wish it had! For a description of what it feels like to finally get caught, Ben Orlin describes it here. Math failure feels even worse if you've been told for years you're getting it and doing well.

I came to teach math reluctantly, because my college experience. But as I started teaching and dove in to different ways to present the ideas, I realized there were glorious, fantastic connections to be made. For me, it all started with writing bonus problems. I usually had a few neat ideas in a chapter for a bonus problem, and with a lot of follow-through, I figured out which ones were do-able and why. That lead me to more investigations, to more puzzles, to writing more creative problems than ones in the textbook. It is fair to say that I didn't start actually doing math until I started teaching.

I feel rather cheated that my math curriculum did not require me to do much mathematical thinking. This is not something I want to happen to my students. Rather than present finished formulas, I present problems. My job is to carefully select the problems, so the context of a problem and its place in the sequence help guide their thinking. But it's their job to do the thinking. If I ask, "How are these ideas connected?" and they don't get it right away, that is alright. If they make a not-quite correct connection, then I'll help them see the error and steer them in the correct direction. But my message is: "It's your job to understand this and make the connection. There are no mysteries in this course that you can't understand."

But it is wrong to set up a game where a thin understanding of basic facts and formulas is sufficient for success. I don't want my trajectory to be my students' trajectories.

Post-script

I ran across this really excellent paper on the importance of a problem-centered curriculum in creating intellectual need for students: http://math.ucsd.edu/~jrabin/publications/ProblemFreeActivity.pdf.

The highlight:

Finally, teachers should learn to value and pay attention to student thinking. Teachers and textbooks need not always be the sources of solution procedures; student ideas can drive much of the learning that occurs. To achieve these results, it is necessary to provide teachers with explicit examples of carefully-selected problem tasks that fit into a coherent unit. However, prevailing attitudes may need to change as well. Primary and secondary school teachers rarely encourage significant investment in understanding a particular mathematics problem. Instead, there is typically a strong expectation that students will quickly produce answers (though not always correct ones) to the problems they are given, leading to problem-free activity. In order to teach with intellectual need, teachers must set up a classroom environment in which making sense of a problem is more important than producing an answer. When students understand a problem thoroughly, the answers they offer are more likely to contain mathematical insight, even when those answers are not complete and correct. In addition, many student errors can be traced to the way students interpret problems, rather than simply their level of knowledge.

Can I put this above the door to my classroom?