I am starting work on a project to develop self-study materials to teach mathematics to adults with math anxiety, phobias, or just plain stress. I've recently been looking into methods that involve getting the student into a state of relaxed awareness prior to a session of math awareness. These use music (at about 1 beat/second, such as Baroque music), yogic breathing techniques, or other methods to teach the student to synchronize body and mind and facilitate communication between the two brain hemispheres. Claims for these techniques are astounding. They have mostly been used in teaching language, or other subjects where memory is paramount.
These approaches have been dismissed as pseudoscience by some, but there seems to be quite a lot of evidence that they work. See the "suggestopedia" method of Georgi Lozanov (http://en.wikipedia.org/wiki/Suggestopedia) or the Institute of HeartMath (http://www.heartmath.org/education/overview.html).
I would love to hear from anyone who has experience in these methods, or in related ones, particularly as applied to mathematics learning.
E13. The Very Bad Key Drive
The following problem was a joint effort. I don’t know the original author. The original version, involving a poisoned keg of wine, was passed on to me by Arshag Hashian of Northeastern via Sandy Blank. My nephew, Michael Ash, modified the statement of the problem to make it politically correct. My sister, Arlene Ash, improved the exposition of my solution.
Here it is:
You have five expendable computers and 240 key drives. Exactly one of the drives has a very bad problem. Any computer that has mounted the bad drive during the previous day will be destroyed when the cron system maintenance runs on the computer at midnight.
You need to use the data on the 239 good drives on a nonexpendable computer in 48 hours and so you can have only two rounds of testing. How can you determine which is the bad drive?
For the solution, click here.
Here it is:
You have five expendable computers and 240 key drives. Exactly one of the drives has a very bad problem. Any computer that has mounted the bad drive during the previous day will be destroyed when the cron system maintenance runs on the computer at midnight.
You need to use the data on the 239 good drives on a nonexpendable computer in 48 hours and so you can have only two rounds of testing. How can you determine which is the bad drive?
For the solution, click here.
E12. Ladder against a wall (Part II)
Here is another ladder against a wall problem, from Coxeter's classic Introduction to Geometry. It is somewhat atypical of the book in that the interesting part seems to be the algebra, rather than the geometry. To make this more of a challenge, try to do it without using a CAS.
A 24-foot long ladder rests against the horizontal ground and a vertical wall in such a way that it touches a cube. The cube is 7 feet on a side and is placed flat on the ground, touching the wall. Find the height of the top of the ladder.
This is a two-dimensional problem which could be stated a little less colorfully in terms of a square and a line segment. It's easy to see that there must be (at least) two answers because of the symmetry of the problem; if a line segment of length 24 passes through (7,7) with endpoints on the positive coordinate axes, the reflection of that line segment in the line y = x will satisfy the same conditions. I found several ways of setting up the problem, all of which result in a 4th degree equation. However, the solutions are quadratic irrationals. In one approach, the biquadratic factors into two quadratics with integer coefficients. In another, a somewhat obvious substitution does the trick.
A 24-foot long ladder rests against the horizontal ground and a vertical wall in such a way that it touches a cube. The cube is 7 feet on a side and is placed flat on the ground, touching the wall. Find the height of the top of the ladder.
This is a two-dimensional problem which could be stated a little less colorfully in terms of a square and a line segment. It's easy to see that there must be (at least) two answers because of the symmetry of the problem; if a line segment of length 24 passes through (7,7) with endpoints on the positive coordinate axes, the reflection of that line segment in the line y = x will satisfy the same conditions. I found several ways of setting up the problem, all of which result in a 4th degree equation. However, the solutions are quadratic irrationals. In one approach, the biquadratic factors into two quadratics with integer coefficients. In another, a somewhat obvious substitution does the trick.
E11. Ladder against a wall
A ladder is placed against a (vertical) wall and the bottom of the ladder is moved away along the (horizontal) ground. What is the shape of the curve traced by the midpoint of the ladder?
It is very easy to work out the answer to this problem, and I won't bother to do that here. If you haven't seen the problem before, test your intuition. Try to sketch what you think curve looks like before solving the problem. (In particular, is the curve concave up or concave down?) The first time I saw this, my intuition was wrong.
It is very easy to work out the answer to this problem, and I won't bother to do that here. If you haven't seen the problem before, test your intuition. Try to sketch what you think curve looks like before solving the problem. (In particular, is the curve concave up or concave down?) The first time I saw this, my intuition was wrong.
Blogger etiquette
I was recently going through my old blog postings, and I found a thoughtful and positive comment by "Sarah" dated September 16, 2008. The original post was about research that purports to show that students learn mathematics better from abstract models rather than concrete ones. In the comment, Sarah apologizes for responding so late. (I'm not sure when the original post was; sometime in Summer 2008.) This is probably why I missed it.
Although I didn't have too much to add to what she said, I wanted to at least acknowledge her comment. I was able to visit her Blogger profile and her two blogs, but neither blog is terribly current and I would feel odd leaving my comment attached to a totally unrelated topic. However, if I were to leave the comment where logic dictates--on this blog, next to the original comment--it seems clear she would never see it.
I'd appreciate suggestions as to how to deal with this type of situation.
Although I didn't have too much to add to what she said, I wanted to at least acknowledge her comment. I was able to visit her Blogger profile and her two blogs, but neither blog is terribly current and I would feel odd leaving my comment attached to a totally unrelated topic. However, if I were to leave the comment where logic dictates--on this blog, next to the original comment--it seems clear she would never see it.
I'd appreciate suggestions as to how to deal with this type of situation.
Lure of the Labyrinth
A couple of weeks back, I went to a meeting of the Association of Teachers of Mathematics in Massachusetts. The keynote speech was by Scot Osterweil of MIT's Educational Arcade. The goal is to produce games that teach mathematics in a way that is engaging to students and has true educational value. He showed us a game, Lure of the Labyrinth, that embodies these principles. It teaches mathematical topics, such as proportions, at the middle school level, and is quite engaging, even to adults. It reminded me a bit of Myst, although there is definitely a more kid-friendly feel.
Try it yourself by going to http://labyrinth.thinkport.org. You must register, but I don't think that there is any downside to that. You can choose either Game or Puzzles. I'd suggest trying Puzzles first, and selecting the first puzzle.
I really liked this and I agree with Scot that properly designed puzzles and games are one of the best ways to teach mathematics.
Try it yourself by going to http://labyrinth.thinkport.org. You must register, but I don't think that there is any downside to that. You can choose either Game or Puzzles. I'd suggest trying Puzzles first, and selecting the first puzzle.
I really liked this and I agree with Scot that properly designed puzzles and games are one of the best ways to teach mathematics.
A New Business Card
I recently designed a new business card, using an interesting geometrical structure as a design element. The design is based on a circular Dirichlet tessellation, also known as a Voronoi diagram with multiplicative weights. The design seemed appropriate because I have done research on these structures in the past, in a paper I wrote with Ethan Bolker in the eighties.
In the multplicative Voronoi diagram, we start with a finite number of sources (points) in the plane, each assigned a positive weight w. The diagram consists of the circles or circular arcs that divide the plane into regions, where the region corresponding to point P consists of all points X such that
|P-X|/w(P) is less than or equal to |Q-X|/w(Q) for every other source Q.
You can think of the sources as being restaurant locations and the weights as being a desirability rating, so if w(P) is r times w(Q), a customer is willing to travel r times as far to go to P as to go to Q. For the case of two sources, the boundary is the circle of Apollonius of ratio r. The case where all weights are equal reduces to the classical Voronoi diagram, where the circular arcs degenerate into straight lines.
If you would like to play around with these diagrams, you can use the applet written by Gabi Knuppertz at http://www.pi6.fernuni-hagen.de/GeomLab/VoroMult/. I was not able to find the needed plugin for Firefox, but got it to work fine in Internet Explorer.
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