This describes Moebius transformations, which are maps of the complex plane which have the form

for fixed . This video really emphasizes how anything having to do with the complex numbers is really about rotation.

I don’t think we have a plug-in for embedded video yet. Maybe I’ll work on that later. Until then, enjoy the link. [UPDATE: We now have embedded youtube functionality, as can be seen. -- Ed]

A significant theme at all levels is the promise of technology to flatten barriers and cut costs in education. At the same time, using these new methods to their fullest will require modifying fundamental attitudes about the role of educators and the purpose of institutions. There is no doubt that many of my prescriptions will sound counter-intuitive in a modern world where education seems so important. I urge you to resist the fallacy of considering proposed changes against what the current system promises, rather than what it delivers.

I believe that the crucial failing of our modern educational system is a failure to admit that elementary school has a two-fold purpose. In a modern economy, school’s function as a daycare is just as important as its role as an educational institution. If we take this notion seriously, the ramifications are far-reaching.

First and most importantly, there is no longer any call for forcing twelve-year-olds to sit at a desk seven hours a day. Much of my interest in educational practice comes from sheer indignation that mine was not carried out better, and I had an excellent education by most metrics. Once we accept that school is as much about minding kids as teaching them, it becomes acceptable to allow them to run around and play for a few hours every day. These pictures of a young test-taker speak more eloquently than I ever could:

Of course, school is also about education, but today’s curriculum is crammed full of information that the average student neither wants nor needs to know. In place of this, I believe we need a basic (very basic) universal curriculum, nationally defined, which is required of all students for “graduation” from elementary school (no yearly assessment). This basic curriculum would consist of only those skills which are actually necessary for day-to-day life: reading, writing, basic arithmetic, perhaps basic computer competency and a few others. Most importantly, this test is not intended to set a high standard, but rather to assess basic competency; this should be something that most 4th or 5th graders could pass handily.

What then of the other subjects: science, literature, history? Without a doubt these should still be offered, but they should no longer be required of all students. There is certainly the worry of producing students without even basic knowledge in these crucial subjects but, truth be told, the same is true today.

I don’t want to pass this off as an idle concern, but we should weigh it against the presumptive benefits for students who remain in classes (and teachers, as well). Without forcing unwilling students to sit through classes, we should expect that misbehavior and disruptions should drop significantly; this would also allow teachers to throw out students that are a perpetual disruption in their class.

In many cases, we can then aim for a higher standard for those students who remain. Students who are engaged and want to learn do so faster and easier; we should also be able to cut out much of the tedium and repetition which results from trying to force ideas into unlistening ears. I have little doubt that the top quarter of motivated students could learn a typical 5-6 year arithmetic curriculum in under two.

Nor does the curriculum need to be completely unguided. I would envision a system of dependencies. Knowledge builds on itself, and key components like arithmetic and reading are necessary for everything from science to music to woodshop. The pressure of peer groups will encourage adequacy in basic areas, as well. Once everyone your age has learned to read, there is a strong incentive for you to do so as well. This is especially true in an environment where knowledge is viewed as a commodity rather than a prescription.

Where does technology come in? The most immediate change which we can expect (I hope) is a relaxation in the crushing grip of textbook publishers like Houghton Mifflin and Prentice-Hall. We are very nearly at a point where the physical book becomes irrelevant, and at that point we no longer need to pay for historical ideas and methods which belong to all of us. More generally, as we allow students greater freedom in following their own interests at school, it is inevitable that those interests will outstrip the knowledge, the capabilities and (when different for every student) the time of their instructors. Digital texts will have to take up the slack. Fortunately, digital texts offer a much more powerful means of expression than static books; I can hardly guess at the diversity of these new methods, although I have some ideas about the presentation of mathematical ideas (that will wait for another day). To sum up the basic idea in a pithy statement:

Although a computer may never teach as well as a good teacher, we can certainly design them to teach better than a bad teacher, and there are plenty of those.

That describes the basis of my position. Most importantly, give kids more freedom to do and explore what they want while they’re at school. We don’t have to take them off the leash all together, but we need to let them run around a bit. While we’re at it, let’s only expect kids to know those things that we do; that doesn’t mean we have to give up on them learning anything more, it just means we shouldn’t tell them what that ‘anything’ has to be. From there, our biggest challenge is to make good on our promise to teach them ‘anything.’ Technology and free, open information is the best, most efficient way for us to do that.