Friday, June 26, 2009

Math is Like Mountain Climbing

It must be true; I've read it in 3 different places! :^)

I like the analogy: they're both very hard work, and both give their enthusiasts lots of pleasure in what they achieve, along with a view of the world that most people don't get.


In Out of the Labyrinth: Setting Mathematics Free, a book full of wisdom gleaned from their experiences conducting math circles, Robert and Ellen Kaplan write:
Those who love to climb mountains have a very different view of them, and it may be no accident that so many mathematicians are also mountain walkers and climbers. It isn't just the exhilaration of solving the rock face, but the fresher air along the way and the long views from the top that draw them on. ...

We aim to take acrophobia away by having our students do the climbing however they will, with us as their Sherpas. We bring up the supplies and peg down the base camp; we point out an attractive col or a dangerous crevasse; but they do the exploring on a terrain we've brought them to. (page 11)
In an earlier description about why math might be particularly difficult to teach well, they write:
Fall from a ledge and the odds are slim that you'll climb back up to and past it. (page 8)

The handholds seem to grow fewer the higher you climb. Mathematics is all ledges. You no sooner acclimate yourself to breathing the thin air at this new height than the way opens up to one still higher... (page 10)


In The Art and Craft of Problem-Solving, one of the first treatments of problem-solving I've found that doesn't just regurgitate Polya's lovely four step process*, Paul Zeitz writes:
You are standing at the base of a mountain, hoping to climb to the summit. You first strategy may be to take several small trips to various easier peaks nearby, so as to observe the target mountain from different angles After this, you may consider a more focused strategy, perhaps to try climbing the mountain via a particular ridge. Now the tactical considerations begin: how to actually achieve the chosen strategy. For example, suppose that strategy suggests climbing the south ridge of the peak, but there are snowfields and rivers in our path. Different tactics are needed to negotiate each of these obstacles. For the snowfield, our tactic may be to travel early in the morning, while the snow is hard. For the river, our tactic may be scouting the banks for the safest crossing. Finally, we move onto the most tightly focused level, that of tools: specific techniques to accomplish specialized tasks. For example, to cross the snowfield we may set up a particular system of ropes for safety and walk with ice axes. The river crossing may require the party to strip from the waist down and hold hands for balance. There are all tools. They are very specific. ... (page 3)

As we climb a mountain, we may encounter obstacles. Some of these obstacles are easy to negotiate, for they are mere exercises (of course this depends on the climber's ability and experience). But one obstacle may present a difficult miniature problem, whose solution clears the way for the entire climb.For example, the path to the summit may be easy walking, except for one 10-foot section of steep ice. Climbers call negotiating the key obstacle the crux move. We shall use this term for mathematical problems as well. A crux move may take place at the strategic, tactical, or tool level; some problems have several crux moves; many have none. (page 4)

After so richly developing his metaphor, Zeitz uses it to explain mathematical problem-solving. For example:
Let us look back and analyze this problem in terms of the three levels. Our first strategy was orientation, reading the problem carefully and classifying it in a preliminary way.Then we decided on a strategy to look at the penultimate step that did not work at first, but the strategy of numerical experimentation led to a conjecture. Successfully proving this involved the tactic of factoring, coupled with a use of symmetry and the tool of recognizing a common factorization. (page 6)
I haven't finished this book. It's full of hard mathematical problems that I can come back to over and over - a whole mountain range I can carry with me!


Each of these 3 authors has used the metaphor to a slightly different purpose. Mike South's piece (here) reminds me of Lockhart's Lament. They're both about how destructive 'teaching' can be in math. As a teacher, I continue to struggle with this conundrum.

I'll attempt to tantalize you with the beginning of his essay:
On the distant planet of Lanogy, all of the population centers are in sight of ... mountains. It's not just because there are lots of mountains on the planet, although that is also true. There are certain resources which are only available in the mountains. You need them to build cities, hence the proximity. But not only that, the mountains are the source of resources needed to facilitate trade in Lanogian economies, so all trade that takes place is near mountainous areas out of convenience. In addition to that, many other technologies turn out (some times unexpectedly) to benefit dramatically from the resources the mountains have to offer.
Now, interestingly enough, despite how useful the mountains are, almost no one on Lanogy likes them. Spending time in the mountains voluntarily is, to almost anyone you talk to, such a laughably improbable concept that it would only occur to them in jest. Everyone knows that builders and commerce agents have to do their share of mountaineering as part of their jobs (in fact, that very fact encourages a lot of people to eschew those professions), but the only people that would ever spend most of their time there would be the mountaineers. These very rare and very peculiar people (those whose only job is to climb mountains) might, possibly, do it voluntarily. But what they would do, why they would do it, and, indeed, what they do professionally is a complete mystery to the rest of the Lanogians.
Now, the reason for this general dislike of climbing in and retrieving resources from the mountains could be due to the simple fact that most people are really not very good at it. Now why, in a society that can obviously see the value of the resources obtained from the mountains, people still aren't good at climbing them, is widely disputed.
I am not into mountain climbing myself. Too scary. But I can wish I were braver, and I can understand better so many people's fears of math by reading these pieces. I can also get better at math myself by using Zeitz's strategies, tactics, and tools.

Polya's work, written back in 1944, was so helpful most of us mere mortals haven't discovered anything much more to say. I wrote a problem-solving handout for my classes, in which I updated Polya's language, and added a few ideas too basic for him to have included, like: Write 'Let x =' the quantity you're trying to find. Maybe I can write a separate post on Polya, and include it there.


  1. I think the fabulous ideas of Polya can be made more effective by combining them with techniques for note-making that are adapted to math. Here are 2 approaches:
    a) I've collected a number of ideas on using mind maps in math:
    b) The other approach is to use two separate columns on your notes sheets: The main column for computations, diagrams etc., and a smaller right column for reflection on your problem solving. More ideas in

    (I apologise for including links, but as you will see, the material is difficult to condense into a blog commentary.)

  2. I remember two other metaphors for math problem solving.

    The first is "Polya's mouse", which is mentioned in Paul Zeitz' book in chapter 2.1: A mouse tries to escape from an oldfashioned trap, a cage with a trapdoor. After several futile attempts, the mouse finally finds one place where the cage bars are slightly wider apart, and escapes. Polya's conclusion: "to try, try gain and vary the trials so we do not miss the few favorable possibilities [...]".

    The second is Andrew Wiles' now well-known metaphor:
    "Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room, and it's dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and finally, after six months or so, you find the light switch. You turn it on, and suddenly, it's all illuminated. You can see exactly where you were." (From:

    I find both metaphors interesting, the first one being rather accessible as a guideline for own problem solving situations.

  3. Thanks, Thomas. I like the second, because it explains how things can seem so clear after you've 'turned on the lights'. I might try that one the next time a student seems to be feeling bad about how long it took them to 'get it'.

  4. Thomas, your first comment was stuck in google's spam filter, and I hadn't known to check that. Here it is. (I'm marking it to look later. Traveling today...)


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