My wife works at Penguin Group, where naturally they have an incredible “take pile” of books they publish. That’s where I found Number: The Language of Science, a fascinating and completely accessible history of the human traditions that give us today’s number systems. Apart from the amazing stuff about numbers, my favorite thing so far is the variety of ways in which mathematicians have been wrong.

Too often our culture treats “real mathematicians” like infallable geniuses, but in fact, deep-seated misconceptions were central to the development of mathematics. This got me thinking about my students.

* * *

My first year at Saint Ann’s, I had a rather heated exchange with a seventh grader over whether or not fractions fit infinitely in between each other. The class and I came up with all kinds of arguments to convince her otherwise, but she was sure that a stick could only be chopped so fine before it was just “in pieces.” At the time, this seemed like a major gap in her her understanding. I mean how were we supposed to get to _____ if she wasn’t “getting” *this*? I’m sad to say, though I never told her, I thought this was really bad.

Last year, my fifth graders were talking about stars and polygons, using protractors to make their own, when a student had a brilliant realization. After making a twelve pointed star from a dodecagon, she explained that a circle was just a polygon with 360 sides! This isn’t really true, so we had a conversation about what a 720-gon might be like. She got that, but “make enough sides,” she said, “and that’s a circle.” She was definitely onto something and really enjoying her work, so I let it slide.

I knew enough of Archimedes’ method of exhaustion to know that the fifth grader was thinking like a “real mathematician,” but even then I think I looked back in awe at the misconceptions of that seventh grader.

* * *

Then I read the chapter, called “This Flowing World,” about the centuries of debate on these exact issues. The seventh graders concerns were legitimate issues articulated by Zeno and the greeks, and their resolution was of fundamental importance in the development of the differential calculus! Of course I want my students to see the poofs and results that will bring them “up to snuff,” but when they pause to wrestle with a classical paradox, why should I object?

When it comes to mathematics, this kind of wrongness is essential. As in the Monty Hall problem, our intuition often leads us astray, and it’s the act of proof that forms the heart of mathematical reasoning. Wrong notions represent a successful model of thought that reflects a current state of understanding. If a model is weak it will demand attention sooner or later. If not, then it’s not weak.

The genius view of mathematicians has meant, among other things, that most students leave school thinking they’re not one. Amazingly, mathematicians are people that have good ideas and bad ones – make mistakes and have breakthroughs. This year, I want to share in the growth and mathematical discovery of my students, without needing them to get it all right.

I can still be better about this.

I think your student is right! If you cut my pizza into 1000 parts and give them all to me you haven’t given me a pizza! The problem is the math is a model of the real world, not the world — and we have to be careful in interpreting what the math tells us. I claim this is actually more sophisticated than “just doing the math” — and a lot of the college juniors in my physics class have a lot of trouble getting this point.

The Unabashed Academic ;-)

Great point, Joe. There is certainly something to her argument. I mean, if it’s good enough for Zeno! I think what I found toughest in that moment was that everyone in the class was convinced and generating good mathematical arguments to convince her, but she was obstinate. (Isn’t every seventh grader a little bit?) That was a tough class for me, and I could have done a much better job of it. Thankfully teaching is a regenerative career, and I get a brand new start this year!

Great thoughts here! I love the story about the 5th Grader and the circle. You’re right! Being wrong is just fine sometimes. I particularly liked this phrase: “Wrong notions represent a successful model of thought that reflects a current state of understanding.”

In science it’s all about the process. If students can go through the steps of the scientific method by forming a hypothesis, conducting the experiment, gathering data and then summing it all up, it doesn’t matter how off base their ideas were when they started. It’s the thought processes they went through along the way I care about most. We’re training how to think, not what to think, right?

Absolutely. Mathematical thinking is the whole thing. The circle one was particularly interesting, because she had gotten very excited like she had had a real breakthrough in understanding shapes. To shut her down would have been almost cruel!

Aren’t we creating models of the world? At least that’s what your students are doing (as opposed to staying on the Platonic plane). All models are wrong, some models are helpful. The 360-sided polygon being a circle may be a “good enough” model for some situations. I think this realization for a student is key, and pointing out the 720-gon is a great way to go about it. Brian Frank at http://teachbrianteach.blogspot.com/ talks about the value of misconceptions, and how we might want to spend more time finding what’s right in student misconceptions rather than just trying to destroy them.

In the same way, the inability of your seventh grader to divide a stick up into more and more pieces is, in fact, pointing towards a rather sophisticated physical understanding. At some point that stick is chunky, not continuous. It’s a real thing, made of atoms (another model that’s useful… until you go farther and break it, too!). Water may seem continuous, but we know it’s actually lumpy. In fact, many theoretical physicists think space may, in fact, be lumpy, the shortest possible displacement being the “Planck length.” Zeno’s paradox resolved!

Yeah, completely fascinating stuff. So when our standards demand that students understand some concept, we force some kids to leave behind their own legitimate and powerful thinking. Thanks for the comment.

I should also add that the seventh grader said, “maybe someday a mathematician will come along and prove I’m right.” I was certain this wouldn’t come from a mathematician.

This was as if, in say a Chemistry class, you were discussing various atomic models, speaking historically, but at the end one student said, “no way. It’s plum pudding!” Then you went over all of the terrific experimental set ups and results that lead to an alternative understanding, with every other student buying it all the way, and she said, “nope! It’s plum pudding.” There was something extremely frustrating in it, because there was an element of unthinking in what she was saying, but in hindsight I’ve come to grips with it.

I should also add that this student told her teacher the following year that she changed her mind. All weak models get bashed up at some point.

I think it was great.

Thanks a lot, Christina.

Often the experience we get as students in a classroom is that what is written in the text and taught by the teacher is absolute, has always been known, forever, and is written in stone. I still get that feeling as I upgrade my skills and knowledge, even though intellectually I know it is not true.

Mistakes are necessary to progress. They prevent stagnation in static knowledge and are the means, when explored, to education over indoctrination. They also signal where and when learning, or at least consideration and thinking, are taking place. There is real truth to the saying, “We learn from our mistakes”. I like the way Mark worded it: “we might want to spend more time finding what’s right in student misconceptions rather than just trying to destroy them”.

I also agree with Kate. Our job is to train students how to think, not what to think. The topics we teach to do this are just cover stories for the real, hidden teaching and learning. If the cover stories are useful or authentic, so much the better.

This “shift” in perception of what education entails might help those who think standardized tests are informative change their minds. (The tests are testing the wrong thing: content over thinking. When I entered university, I discovered that you can aim for the grade (what to think) or for the learning (how to think), but usually these were two very different goals.)

Teaching students how to think, instead of what to think, modelling and exploring incorrect thinking in the historically God-like experts, and publicly examining what is right in misconception, or why a misconception points out several scales of looking at a thing, allows our students to grow dynamically, curiously and thoughtfully into ever more productive citizens rather than stagnate in static acceptance of what is taught and written in stone.

Great article as always.

Shawn

Terrific comment! Very well articulated thoughts, and I’m write there with you. Thanks for your thoughts!

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It is tempting to view the truth as a binary value. I am reminded of Physicists’ models, classical physics did not become untrue the day Einstein finished his paper. Newton was onto something. If a model reflects a perception of reality, it has value at the time of the perception. Older models will continue to have value if reality is still perceived with respect to them. As perception deepens, so the model will be asked to deepen. Why take away the joy of discovery just because the discovery isn’t the latest version?

This is clearly the difference between “education” and “indoctrination”. The latter being the transmission of societies’ mental models, and the former being teaching the ability to create and manipulate one’s own mental models. While both are the purpose of our system, it appears the latter is the most often served.

Thank you!

John

Great comment. This point about modeling is very much a scientific one, but were only recently coming out of a time when mathematician, scientist, and philosopher were synonyms. It is an essential part of the mathematical process, even if you’re only modeling imagined objects and scenarios.

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