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Learning at its Best

Letting students be dead wrong

via Lost In Recursion

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.

About Paul Salomon

Life is not about discovering yourself. It's about creating yourself.


5 thoughts on “Letting students be dead wrong

  1. Paul,
    Thank you so much for this post. This is an eye opening understanding for me, about what is missing in traditional math in Western Culture. It seems as though Math has become is dogmatic for so long, before it finally becomes questionable and discuss oriented. I never really understood why 1+1 = 2, I just was always told that was it was and that I needed to know that, in order to move on to the next assignment or “learning.”

    Moments like the one you’ve described, points out why it is so important to still learn through critically questioning, even in a proven subject such as math.

    Paul, are you familiar with ethno mathematics? Its the theory regarding that math needs to be directly connected to our lives — as a Dewey educator, this fascinated me, and while its implementation into Western Society, on any type of grand scale, seems impossible, small ways through “story problems” already does this. It is an interesting take on a hard science subject.

    Cheers and thanks for the post,

    Posted by caseykcaronna | August 17, 2011, 7:30 pm
  2. This is a very thought-provoking posting. It is all too easy to correct a misconception, but a much higher level teaching skill to really learn from it, and to lead learners to a deeper understanding. I have also seen teachers ‘correct’ misconceptions which turn out to be equally valid as the teachers own ideas, just because the teacher is trying to stick with what is on the curriculum.

    By the way, Alex Bellos addresses the question of infinite fractions in a really engaging way in ‘Alex’s Adventures in Numberland’ (‘Here’s Looking at Euclid’ in the US).

    Posted by Paul Richardson | August 18, 2011, 8:06 am
  3. These are important anecdotes to share, Paul – before we pounce on misconceptions (and kids), we should work to recognize critical thinking and hypothesizing and wondering and imagining for the crucial, human skills that they are. Heck, I remember that when I was a kid I thought that people traded houses when they moved. When my family moved, I learned differently, but it felt great not knowing and trying to figure out how people moved. I really like the cognitive dissonance and ambiguity I felt in the middle of my head – it was like stereo music.

    Anyway, yes: we need to ask kids what they think and to listen to kids’ explanations and to go from there with our teaching in a way that celebrates what our students accomplish for themselves in wondering about their worlds and gives them new directions to explore on the way to deeper solutions.

    What other stories do we have on the Coöp of this kind of critical thinking from students questing after something?

    With thanks,

    Posted by Chad Sansing | August 19, 2011, 9:31 am
  4. Hi Paul, One of the things I have been doing this weekend, instead of the things I should have been doing, is reading this book: Little Bets: How Breakthrough Ideas Emerge from Small Discoveries

    In it, Paul Sims describes the power of “small bets” to reposition big enterprises (maybe like education?), and also as a way of thinking about experimentation in learning. Check this out:

    (by Peter Sims)

    AT the recent Aspen Ideas Festival, the New York Times columnist Thomas L. Friedman said that when he graduated from college, he was able to go find a job, but that our children were going to have to invent a job.
    Enlarge This Image

    Jobs, careers, valued skills and industries are transforming at an unheard-of rate. And all of the change and uncertainty can make us risk-averse and prone to getting stuck.

    Despite these realities, our education system emphasizes teaching and testing us about facts that are already known. There is much less focus on our ability to discover, create and reinvent.

    The same often holds true in the workplace. Perfection is rewarded, while making mistakes is penalized. It’s no wonder that “failure” has taken on a deeply personal meaning, something to be avoided at nearly all cost.

    The skills we’re taught work well for familiar situations, yet we’re trained to perfect our ideas and use the past to predict the future with linear plans in a nonlinear world. As such, we need a completely new mind-set. Linear thinking is a death knell for creativity.

    When I worked as a venture capital investor, I found that most successful entrepreneurs don’t begin with perfected ideas or plans — they discover them. Entrepreneurs think of learning the way most people think of failure.

    A prime example is Howard Schultz, one of the most successful entrepreneurs of our time. When he started what would become Starbucks, he modeled the first stores after coffeehouses in Milan, a new concept for the United States in the 1980s. He was clearly onto something, but the baristas wore bow ties — which they found uncomfortable — and customers complained about the nonstop opera music and menus that were written primarily in Italian. And the early stores had no chairs. Mr. Schultz routinely acknowledges that he and his team made a lot of mistakes. But they learned from them, as they did from countless other experiments.

    Consider another example — what it takes to create great comedy. Editors at The Onion, the humor publication, estimate that they try out hundreds of headlines each week before they finally decide to use only a small percentage of them.

    Even the most successful stand-up comedians, like Chris Rock, try thousands of new ideas in front of small club audiences in order to develop a one-hour act. Some jokes fail, but Mr. Rock is willing to be imperfect; he persists night after night because every small bet takes him closer to a brilliant act on the big stage.

    This is how comedians and entrepreneurs must work — by making countless small bets to discover what works. The real genius is in the approach.

    The same holds true for leaders, managers and collaborators. They must to be willing to learn from mistakes. Affordable risks should be encouraged, and small failures celebrated — these are the mark of learning organizations. Otherwise, risk aversion will lead to stagnation and decline.

    In a time when valued skills and occupations shift constantly, we must be able to discover interests, opportunities and careers by experimenting. Or by reinventing ourselves altogether.

    The architect Frank Gehry, for instance, designed relatively conventional buildings for much of his early career. But inspired by how contemporary painters and sculptors worked, Mr. Gehry performed a series of experiments on his own house in Santa Monica, Calif., during the late 1970s .

    Working with plywood, corrugated metal and chain-link fencing, he built a new exterior around his original house.

    His experiments were the precursor to what would become his distinctive style, evident in the Guggenheim Museum Bilbao in Spain and the Walt Disney Concert Hall in Los Angeles. The money was good in conventional architecture, yet he decided to start anew, using his own style and voice.

    INVENTION and discovery emanate from the ability to try seemingly wild possibilities; to feel comfortable being wrong before being right; to live in the world as a careful observer, open to different experiences; to play with ideas without prematurely judging oneself or others; to persist through difficulties; and to have a willingness to be misunderstood, sometimes for long periods, despite the conventional wisdom.

    All these abilities can be learned and developed, but doing so requires us to unlearn many of our tendencies toward linear planning and perfectionism.

    As the technology pioneer Alan Kay put it: “The best way to predict the future is to invent it.” It begins with a little bet. What will yours be?


    Seems like this is all about what you are saying in your post?


    Posted by Kirsten Olson | August 21, 2011, 3:48 pm
  5. Great examples, Kristen! (Thank you to everyone else for your comments.)

    You’ve got it dead on. Experimentation, trial, failure, retrial are huge pieces of a process that is learning. The entire notion of standardization is built on training our students, and equipping them with a standard set of tools. In fact, however, these are not essential tools for the world in which they will work or even live now. If we teach the learning and creative processes in individual ways, allowing for all sorts of wrongness, we will almost certainly better prepare a generation of thinkers and creators, better suited to regenerate our world.


    Posted by Paul Salomon | August 21, 2011, 4:11 pm

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