The Arc Cotangent Controversy

I love this discussion at squareCircleZ. All my readers should check it out. Which is the graph of arccot(x)?

from squarecircleZ

from squarecircleZ

I especially like this controversy because some big players have weighed in on each side. Mathcad and Maple prefer the first interpretation, Mathematica and Matlab prefer the second.

For a more thorough treatment, check out the original post here. Three cheers for great math blogging! đŸ™‚

Could your math teacher be replaced by video?

Before I get to the titular topic, let me share some links. I’ve been meaning to post links to a couple of online resources that are astonishingly thorough. I strongly encourage you to check all these out.

  • Drexel Math Forum — This site has been around for years, I’m just getting around to posting about it now. But if you’ve never been there, I highly recommend it. Almost any math question high school students could asked has been answered and cataloged on this site (including misconceptions about asymptotes like I posted about the other day).
  • Interact  Math — When you first link to this page you’ll be unimpressed. But select a book from the drop down menu and then pick a chapter and set of exercises. Then, click on an exercise and prepare to take an interactive tour of that problem. The site let’s you graph lines, type math equations, do multiple choice problems, and more. If you have trouble with the problem, it will interactively walk you through each step, asking you simpler questions along the way. What a fantastic resource! Unfortunately, almost none of our books are on the drop down list. That doesn’t keep it from being useful. Just find problems similar to what you’re struggling with and try those.
  • Khan Academy — A nonprofit organization started by Sal Khan, this site has 1800+ youtube instructional videos, nicely organized by course and topic. You can go learn everything from basic arithmetic to college level Calculus (and Differential Equations, Linear Algebra, Statistics, Biology, Chemistry, Physics, Economics…). Sal’s mission is to provide a world class education to anyone in the world for free. It’s very exciting to see how this site will grow, and possibly change how we do education.

Math Teaching by Video

Some of these sites, especially the Khan Academy, make me wonder how long our modern American school system will remain in its present form.  Will we always have a teacher in the front of the math classroom delivering instruction?

I’m not afraid of the idea that we (teachers) could be partially replaced by video lessons. It’s actually a pretty good idea. The very best instructional practices could be incorporated into a flawlessly edited video. Teachers wouldn’t make frustrating, careless mistakes, students could replay the videos at any time, and substitute teachers could easily run the class. Every school, even the poorest and most marginalized would be able to deliver top-notch, world class instruction.

And what would teachers do, then? Qualified teachers could turn their efforts toward more of “coaching” and “discussion leading” role, concentrating on one-on-one sessions, remediation, reteaching, providing feedback, grading, seminars, open forums, field trips, and inquiry-based instruction that supplements the more formal video presentations. And don’t forget blogging! đŸ™‚ So much of a teacher’s time is currently spent preparing lessons and teaching them that they have very little time for all those other (more?) important aspects of teaching. All this time devoted to preparation is being spent by teachers everywhere. Imagine the possibilities if we devoted the bulk of our time to these other aspects instead of preparing instruction. Sounds really great to me.

15 degree triangle

Check out this fun problem at the math challenges blog. I picked it in honor of what we’re doing in Precalculus right now, since it has a trigonometric flavor. Don’t look at the answer until you’ve tried it! In fact, after looking at the answer, I found I had done it a different way. So let your creativity run wild. I’d love to see your proof, if you figure it out. And if you want a hint, come see me or email me.