On the distant planet of Lanogy, all of the population centers are in sight of (that planet's version 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. Any time the topic comes up, it usually ends in something at about the level (intellectually, emotionally, and decibellically) of a shouting match between two principal camps. One camp basically believes that the problem is that the mandatory mountaineering classes have been dumbed down, or that kids these days are just not willing to do the work necessary to learn mountaineering, or some combination of both.
Another camp basically considers the problem to be with the classes themselves, and argue that it's the classes, not the kids, that need to change.
Rather than plunging into this debate blindly, let's first sneak into a mountaineering class and see what goes on there.
Mountaineering is among the least popular subjects in school. It doesn't actually start out that way, but it gets there quickly. Younger students start off learning to recognize a handhold in rock. Then they learn to navigate up a stretch of mountain by simply moving from one handhold to the next. This is more or less the last point where everyone in class is together in their understanding. The next part of the instruction introduces climbing tools.
The first tools that the children see are of ancient design. Learning to use them requires a bit of practice. Some people take to it right away, and others have varying degrees of success. Although the climbing tools allow one to make progress on a mountain far, far more quickly than climbing by hand, the children are not yet experienced enough in the world to understand the value of that. At this age, they have never had a need to obtain mountain-borne resources themselves, and tool practice, for those that don't take to it naturally, can quickly become repetitious and boring.
This, in fact, is where the aforementioned debate begins. The standard approach to teaching tool skills is repetitive drill. The advantage to this approach is that it can be administered almost effortlessly by an instructor. Even an instructor who has no deep understanding of the tools, who doesn't like to use the tools, and who would never venture onto a mountain voluntarily themselves can drill the students in tool use. Not only that, basic tool ability is is easy to measure--in fact, it can even be done by a machine! In fact, whenever the debate comes up, the traditionalists often put up a simple challenge--when a new teaching technique shows the students getting better results in the mechanical tool-ability checker, they will be happy to support that technique. It makes for quite the uphill battle for those who want to change things.
However, there are compelling reasons for change as well. Under traditional methods, even if people develop a mechanically demonstrable proficiency, they certainly don't like it. Furthermore, they neither understand how and why the tools work, nor do they have any idea how they might modify them or use them in a novel way given a never-before- encountered type of terrain. Also, recall that most people end up with a healthy dislike for mountaineering, and avoid doing it even to the point of altering their career paths if necessary. Naturally, this means that they don't use the skills, and so a great deal of it is largely forgotten. In addition, there is lots of data that seems to show that traditional instruction methods put certain socioeconomic groups at a strong disadvantage. That fact alone is a major driving force behind the opposition.
There are some things that the main body of the opposition and the traditionalists are in agreement on. Nobody is proposing that mountaineering be made an optional subject, for example. There is a little more contention on what aspects of mountaineering should be taught and when, but for the most part both sides agree on the general types of tools that should be introduced to kids. Where the opposition shows its clearest difference is in the subject of how the skills are taught.
Those who would reform the system essentially feel that repetitious drill and direct instruction are inefficient and counterproductive. The students quickly become bored or frustrated, and the dreaded question "why are we doing this?" surfaces both in class and, more importantly, in the children's heads, again and again and again. Reformers point to research that suggests that their might be a an even better reason than "kids hate this" to change. It seems, according to this research, that there might be fundamentally better ways to teach mountaineering, ways that work better than the standard "demonstrate, drill, repeat" cycle. And one of the advantages is that these techniques seem to do a better job of crossing socioeconomic boundaries.
Of course, whether any of this research is valid is itself hotly debated. But to many in the reform camp, this is almost a side issue--the new techniques just "feel right", and their own experience with them is justification enough.
Let's look at how the instruction might proceed in a reformist class. The first part, identifying handholds and learning to move from one to the next up a mountain, might look pretty much the same. But the first time a tool is to be introduced, you will notice a significant difference. Whereas the traditional teacher would simply pull out the tool, state its name, demonstrate its use, and start the students practicing, the reformer would attempt to present students with a climbing situation that would be hard to do without tools. Then she might ask them to come up with ways to solve the problem, possibly through some sort of group activity, or with some sort of physical objects that could be used to model the climb. At some point the concept of the tool begins to emerge through the activity and discussion, and the students work toward understanding what the tool is for and how it is used.
A teacher using the reformist approach well can really work wonders with it. For one thing, the students in the mountaineering classes will be fully engaged in what is going on. That's much harder to do with the introduce, demonstrate, drill method. Furthermore, the reformist approach encourages students to really think and talk about the tools they are learning to use, which seems a huge advantage over the traditionalist approach. Also, there seems to be a significant impact on leveling the socioeconomic differences of the other method (although, as stated above, that research itself is hotly contested).
But of course the picture is not all roses. It's possible that this approach could lead to lower scores on the mechanical tests because more time spent trying to understand means less time memorizing and practicing, and less ability to simply rattle off the procedure. There is another difficulty which is probably more fundamental than this, however--the fact that many of the teachers who are doing the mountaineering instruction are just like the rest of the population. They dislike the topic, avoid it when possible, and never really got a fundamental understanding of the principles underlying the tools involved. Since the reformist approach comes precisely from a standpoint of understanding the underlying principles of the tools, this puts many teachers at a strong disadvantage in delivering by the reformist method. Without a strong facilitator, the activities can become just activities, and very little gets learned by anyone.
Much could be said about the arguments involved, but we are about to divert our attention from them entirely. As is occasionally the case when a battle is raging furiously between two large entities, things happening on the side get little notice. But there is a third group, not very well organized and often acting individually, with a very different vision. They come, mostly, from that tiny segment of the population that is almost completely shrouded in mystery--the mountaineers. Now, it's not at all surprising that professional mountaineers would have a viewpoint that's radically different from the rest of the public. After all, not only to they apparently like mountaineering, but, for some completely unfathomable reason, they do just mountaineering (whatever that might mean). They don't do it in support of commerce, building, or any of the many other fields that use mountain resources. They simply mountaineer. Bizarre. So, as we have said, we're not surprised that they have a different viewpoint, but it turns out the the viewpoint itself is very surprising indeed. And what is that viewpoint? That mountaineering is, in fact, fun. Fascinating. Intriguing. Interesting. But, and this is the shocking part, not just to mountaineers.
Of course, it would seem that there is a huge body of evidence contradicting this view. Why do these mountaineers see it differently?
We have not touched yet much on the public's view of the mountains. They mostly see a relatively sheer rock face, some of which can be scaled by hand, but with increasing difficulty and quickly requiring the use of various tools. We talked a little about their first introduction to tools. From there, the tools more or less rapidly increase in complexity. All further instruction and exercise of tools uses the previous tools, and anyone that hangs on long enough will eventually be instructed in the use of tremendously powerful mountain navigating tools that allow them access to a huge variety of mountain-borne resources. However, the attrition rate is high, and although most people have heard of these tools, most people are out of the subject entirely long before seeing them.
But the mountaineers themselves see mountaineering radically differently precisely because of something no one else has any reason to even suspect. Mountaineers, you see, have seen radically different parts of the mountains.
This study led to some mountaineers exploring parts of the mountain that no one had ever thought of looking at before. What they found was an entirely different world. They soon realized that they would need completely new tools for navigating those parts of mountains.
Not only that, but other mountaineers started looking at existing tools and finding common principles that manifest themselves in seemingly disparate tools. They were able to use these observations to build tools that allowed them to get to still other parts of the mountain that had been glimpsed but as yet unreachable. But when they got here, they realized that they, too, had an entire new world to explore.
And these two efforts were only the beginning of a flood of new kinds of tools that allowed access to ever more unexpected parts of the mountains. You see, it turns out that the part of the mountain that the average person can see was really only a tiny fraction of what was there. Mountains on Lanogy don't form like any other mountains. Way up out of sight of anyone on the ground there is a tremendously intricate structure that turned out to be vastly larger than was once understood, and the indications are that there are still vast parts unexplored, even undreamed of.
It is these fascinating structures that attract mountaineers. They have no context from which to explain it to anyone else. The structures they are studying are much more intricate and bizarre than anything one encounters outside of the mountains. A mountaineer typically discovers (often by chance) some part of the mountains outside of the standard course of study, is immediately captivated by its exhilarating beauty, and spends the rest of his life exploring it.
Now, it might seem that, if these parts of the mountains were so wonderful, the mountaineers could just take some photographs and bring them back and show everyone else how wonderful it was. But there's a catch--you see, part of the atmosphere of Lanogy, and almost all of that which is around the mountains, is completely non transparent. This doesn't bother the mountaineers--the tools they use to get around the mountains are adapted to this, and when they get to a completely unexplored place there's the thrill of feeling it out and trying to figure out what sorts of tools will need to be created to explore it. But as far as getting the rest of the world to experience what it is that mountaineers find so captivating about the mountains, well, forget it. There's just no equivalent of a photograph that would work. Or so it seems.
One thing that you need to keep in mind is that the mountaineers almost all went through exactly the same kind of early mountain training that everyone else did. Only after they had survived all of the attrition did these really interesting parts of the mountains ever come into view (as you see now, I'm using the term "view" abstractly here, as I have been when I talk about "seeing" these other parts of the mountain). It's easy to think, when there is one road that almost everyone takes to get somewhere, that it's the only road. The average mountaineer is mostly thinking about the particular type of terrain they explore, and so one question, in particular, never occurs to them: is this the only way up?
The small group of mountaineers I was talking about before have begun to think that there might, in fact, be other ways up the mountain. They think that people might have only been assuming up until now that these fascinating parts of the mountain are inaccessible except through the long and often tedious path that is followed in the general school course. Might there be ways to show these parts of the mountain to the uninitiated? If so, would they be able to see the beauty that the mountaineers see, or would it just be received as a different kind of tedium?
In a way, these ideas don't seem so implausible. After all, mountaineers almost universally describe the mountains as being profoundly beautiful and fascinating. It seems quite possible that, if given a chance, many, maybe most other people would see it the same way. Not only that, it might provide a way to cut the Gordian knot representing the interrelated issues of the mountaineering teaching wars. For example, perhaps a curriculum could be designed around showing students parts of the mountain that were immediately interesting to them, and which would encourage the students to work on the problems of navigating these mountains with excitement and interest. Furthermore, it would be a wonderful thing if all of this beauty was accessible to the general public--not necessarily from any building or commerce related perspective, but just because it could enrich their lives the way it does the lives of the mountaineers. That seems like a worthwhile goal in and of itself.
And so the lone mountaineers work on their lonely quest--trying to develop tools that can get people to the fascinating "other" parts of the mountains, or find ways to those parts that don't require years of training to navigate. And little by little, progress is made. And one day, one of the Lanogians realizes that there might be a way to help the rest of the public at least understand the problem, even if the lure of the mountains and the value of the exotic types of mountaineering still didn't really resonate with them. And he does it by creating an analogy.
In it he imagines a planet where they have a great controversy over "mathematics education". Only on their planet, mathematics isn't just a recreation, it's integral to a large number of things in their economy, industry, and science. And not only that, there are tremendously interesting parts of mathematics that only mathematicians ever see. And he tells the story of how a few of these mathematicians start to wonder whether there might be some way that they can show non-mathematicians the beauty and power that the mathematicians are immersed in every day. And not only that, they wonder, might this be the key to getting kids hooked on math, interested in it for its own sake, and enjoyed for its own sake?
Maybe.