Given a square ABCD of side length a, let M be the midpoint of AB and N be the midpoint of BC. Draw AN and CM, and let their intersection be O. Find the area of AOCD.
I saw this problem on an Internet math forum, along with an solution involving Cartesian geometry. The solution was straightforward and not particularly difficult. However, I'm including the problem with a challenge to do it without introducing coordinates. I think the solution that I found is much prettier than the coordinate-based solution. What do you think?
Note: There is something wrong with scribd which is not allowing me to post my solution, which is a short pdf file. I will add a link later when I can upload my file.
E14. Two-block calendar
A calendar consists of two cubes of the same size, about 2 inches on a side. Each cube contains a single digit. When placed together, the front faces of the two cubes display the day of the month, from 01 to 31. Note that single digit days must be displayed as two digits, with a leading 0. Describe what digits to place on each face of the cubes for this to work.
Cambridge Math Learning, Inc start up
I've been away from the blog for a while now, doing lots of things necessary to get Cambridge Math Learning ready. We hope to begin teaching classes within 4 months. In designing lesson plans, I've been putting together a list of interesting yet fairly simple math problems suitable for warm up exercises, to be done in groups of 3 or 4 people. One of the problems in my list is the two-block calendar problem, given next.
Education, Training, and Instructional Design
Oh, you know all the words, and you sung all the notes,
But you never quite learned the song she sang. –Mike Heron
At Cambridge Math Learning we are developing a system for teaching mathematics to adult learners with math anxiety. We plan to develop it in the context of corporate training sessions, and eventually market it to individuals who need to learn mathematics for career advancement, helping their children with homework, or other reasons. Thus, we need to span main two learning environments: training and education.
We define training as the delivery of knowledge needed to enable an individual to perform a specific type of task, and education as knowledge delivered for its own sake. Training is often delivered in a "just in time" framework, where the knowledge that the worker learns will be used immediately. This is economically efficient because research shows that learning left unused is soon forgotten. This seems to be somewhat less true of education than of training, since the goal of education is to teach high-level principles, while training focuses on facts and procedures. If the reader thinks back to their school days, he or she will find that it is general methods of thought rather than specific facts and procedures that are most clearly remembered.
In recent years, both education and training have been increasingly dominated by the methodology of instructional design, a field which attempts to make instruction into a science. Although there are different theories of learning which can underpin instructional design, including cognitivism and constructivism, behaviorism is the oldest and most pervasive theory of learning used. The basic idea is to divide the material to be learned into small chunks. Each chunk is identified with one or more behavioral objectives, sometimes called performance objectives. These performance objectives must be testable. Once the student has achieved pre-defined "mastery" of these objectives, they are deemed to have learned the material.
It is easy to see why this methodology has become so popular. It promises to make learning efficient, quantifiable and replicable. We recognize that it has been effective in many spheres. And yet, something seems missing. By breaking up learning into bits, the participant may learn the words without learning the song. Creativity can be stifled, and material learned in this way is often soon forgotten.
As an example, suppose an adult with no musical training wants to learn to play an instrument, say a piano. In the traditional approach, the person is first taught how to play scales, which they practice over and over. Then they are shown how to play chords, which they practice over and over. Next they are shown how to play very simple tunes. This is a basic behaviorist approach. Many people have learned to play instruments this way, but many more have become bored and discouraged. We might even say they develop "music anxiety". However, holistic approaches to learning music do exist. For example, Paul Winter teaches a workshop in which non-musicians are put into small groups, each with a different instrument. The facilitators show people how to make noise come out of the instrument, and leave them on their own. After a few hours, as if by magic, the random noises begin to develop coherence. Through childlike play, the adults have tapped into their innate, long-buried musical talent. They are enjoying making music. At some point they can seek more formal instruction.
While you may agree that a behaviorist approach to learning an art is not ideal, you may say that mathematics is not an art, and that its mastery consists of learning step-by-step procedures, making it ideal for a behaviorist approach. This would be wrong. Almost all mathematicians (and almost no non-mathematicians) consider math to be an art. This view has been well advanced in G. H. Hardy's book A Mathematician's Apology. Even though our goal is not to make learners appreciate mathematics as an art, we think we can teach it as a skill that can be enjoyed rather than drudgery that must be endured. Also by focusing on main ideas of mathematics rather than minutia, we hope to provide learners with a foundation by which they can learn or relearn mathematics as they need it.
Let's consider a mathematical example. An even number, as you know, is a whole number that is divisible by 2. Suppose you are given a large number and asked whether it was even. You could use a calculator to divide the number by 2, of course, but there is a much faster way, which most everyone knows: just look at the last digit of the number. It is even if the last digit is even (0, 2, 4, 6, or 8); otherwise it is odd. So you can tell that 254953321650 is even faster than I can enter it into my calculator, even if my calculator will hold such a large number. There are similar, very quick rules that will determine whether a number is divisible by 3, 4, 5, 6, 8, 9, or 10.
The rule for determining whether a number is divisible by 3 is to add the digits. If the sum of the digits is divisible by 3, so is the number; otherwise not. For example, if you are asked whether 237910068 is divisible by 3, you could say 2 + 3 + 7 + 9 + 1 + 0 + 0 + 6 + 8 = 36. Since 36 is evenly divisible by 3, so is 237910068.
One could establish behavioral objectives to test whether the student has learned these divisibility rules, but that would be missing the point. The divisibility rules in themselves are of small value, other than that students find them interesting. What is important is that the student understands why these rules are true. To do this they must develop mathematical styles of thinking. Some of the mathematical ideas include the ability to search for patterns, a fairly deep understanding of the meaning of a the digits in a multi-digit numeral, prime numbers and their importance in factoring, and the distributive law. If students want to go on and look for a divisibility rule for 7, and understanding of modular arithmetic and algebraic notation will come into the mix.
At Cambridge Math Learning we recognize that many students come to us with very limited goals. Perhaps they need to be able to use basic statistical formulas, for example to determine the mean and variance of a distribution. We will teach them what they want, but we will teach them more; we plan to teach the song as well as the words. In this way they will gain long-term retention of the information, or the ability to reconstruct it.
(c) Peter Ash, Cambridge Math Learning
But you never quite learned the song she sang. –Mike Heron
At Cambridge Math Learning we are developing a system for teaching mathematics to adult learners with math anxiety. We plan to develop it in the context of corporate training sessions, and eventually market it to individuals who need to learn mathematics for career advancement, helping their children with homework, or other reasons. Thus, we need to span main two learning environments: training and education.
We define training as the delivery of knowledge needed to enable an individual to perform a specific type of task, and education as knowledge delivered for its own sake. Training is often delivered in a "just in time" framework, where the knowledge that the worker learns will be used immediately. This is economically efficient because research shows that learning left unused is soon forgotten. This seems to be somewhat less true of education than of training, since the goal of education is to teach high-level principles, while training focuses on facts and procedures. If the reader thinks back to their school days, he or she will find that it is general methods of thought rather than specific facts and procedures that are most clearly remembered.
In recent years, both education and training have been increasingly dominated by the methodology of instructional design, a field which attempts to make instruction into a science. Although there are different theories of learning which can underpin instructional design, including cognitivism and constructivism, behaviorism is the oldest and most pervasive theory of learning used. The basic idea is to divide the material to be learned into small chunks. Each chunk is identified with one or more behavioral objectives, sometimes called performance objectives. These performance objectives must be testable. Once the student has achieved pre-defined "mastery" of these objectives, they are deemed to have learned the material.
It is easy to see why this methodology has become so popular. It promises to make learning efficient, quantifiable and replicable. We recognize that it has been effective in many spheres. And yet, something seems missing. By breaking up learning into bits, the participant may learn the words without learning the song. Creativity can be stifled, and material learned in this way is often soon forgotten.
As an example, suppose an adult with no musical training wants to learn to play an instrument, say a piano. In the traditional approach, the person is first taught how to play scales, which they practice over and over. Then they are shown how to play chords, which they practice over and over. Next they are shown how to play very simple tunes. This is a basic behaviorist approach. Many people have learned to play instruments this way, but many more have become bored and discouraged. We might even say they develop "music anxiety". However, holistic approaches to learning music do exist. For example, Paul Winter teaches a workshop in which non-musicians are put into small groups, each with a different instrument. The facilitators show people how to make noise come out of the instrument, and leave them on their own. After a few hours, as if by magic, the random noises begin to develop coherence. Through childlike play, the adults have tapped into their innate, long-buried musical talent. They are enjoying making music. At some point they can seek more formal instruction.
While you may agree that a behaviorist approach to learning an art is not ideal, you may say that mathematics is not an art, and that its mastery consists of learning step-by-step procedures, making it ideal for a behaviorist approach. This would be wrong. Almost all mathematicians (and almost no non-mathematicians) consider math to be an art. This view has been well advanced in G. H. Hardy's book A Mathematician's Apology. Even though our goal is not to make learners appreciate mathematics as an art, we think we can teach it as a skill that can be enjoyed rather than drudgery that must be endured. Also by focusing on main ideas of mathematics rather than minutia, we hope to provide learners with a foundation by which they can learn or relearn mathematics as they need it.
Let's consider a mathematical example. An even number, as you know, is a whole number that is divisible by 2. Suppose you are given a large number and asked whether it was even. You could use a calculator to divide the number by 2, of course, but there is a much faster way, which most everyone knows: just look at the last digit of the number. It is even if the last digit is even (0, 2, 4, 6, or 8); otherwise it is odd. So you can tell that 254953321650 is even faster than I can enter it into my calculator, even if my calculator will hold such a large number. There are similar, very quick rules that will determine whether a number is divisible by 3, 4, 5, 6, 8, 9, or 10.
The rule for determining whether a number is divisible by 3 is to add the digits. If the sum of the digits is divisible by 3, so is the number; otherwise not. For example, if you are asked whether 237910068 is divisible by 3, you could say 2 + 3 + 7 + 9 + 1 + 0 + 0 + 6 + 8 = 36. Since 36 is evenly divisible by 3, so is 237910068.
One could establish behavioral objectives to test whether the student has learned these divisibility rules, but that would be missing the point. The divisibility rules in themselves are of small value, other than that students find them interesting. What is important is that the student understands why these rules are true. To do this they must develop mathematical styles of thinking. Some of the mathematical ideas include the ability to search for patterns, a fairly deep understanding of the meaning of a the digits in a multi-digit numeral, prime numbers and their importance in factoring, and the distributive law. If students want to go on and look for a divisibility rule for 7, and understanding of modular arithmetic and algebraic notation will come into the mix.
At Cambridge Math Learning we recognize that many students come to us with very limited goals. Perhaps they need to be able to use basic statistical formulas, for example to determine the mean and variance of a distribution. We will teach them what they want, but we will teach them more; we plan to teach the song as well as the words. In this way they will gain long-term retention of the information, or the ability to reconstruct it.
(c) Peter Ash, Cambridge Math Learning
Algorithms Meet Art Puzzles & Magic
This was the title of a talk given by Erik Demaine at MIT two days ago, at Frank Geahry's Stata center, itself an example of art and puzzle. It was a lovely talk. Erik has done much of his work (starting at age 6!) with his dad, artist Martin Demaine. Martin was in the audience and even participated, and it was great to see the warm relationship between them. The theme of the talk was the way mathematics and art can inspire one another. The mathematics was broadly sketched, which enabled the talk to be much more accessible and personal than the typical math presentation. Erik's has an deep interest in origami, and he had many neat examples to show, including large pleated pieces. He concluded the talk (show?) with a number of magic tricks some of which were very funny and included members of the audience. In one trick he described a method of hanging a picture using two nails in a way that the picture would fall if either nail was removed. The mathematics involved was fairly simple, involving the commutator of a group. In his demonstration he had a volunteer hold out two arms to simulate the nails, and draped a large rope around his arms. It was quite effective, especially when he brought up a second volunteer to demonstrate that the method could be extended to any number of nails (4, in this case). All in all, this was an incredibly enjoyable presentation.
Poincare's Prize
I recently read Poincaré's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles by George C. Szpiro. I recommend it highly. Some time back I recommended another book on the same topic, The Poincaré Conjecture: In Search of the Shape of the Universe by Donal O'Shea. If you can only read one book on the topic, I recommend the Szpiro book.
Both authors are fine writers. The books are of similar length. O'Shea's book is 200 pages followed by 72 pages of supplementary material: endnotes, two glossaries, a timeline, and an 11-page bibliography. Szpiro's book is 262 pages followed by 32 pages of endnotes and bibliography. Each book provides a different interesting aspect of Poincarés life: Szpiro's book relates Poincarés career as a mining engineer, in the course of which he exhibited great personal courage and deductive ability worthy of Sherlock Holmes to investigate a mining disaster. O'Shea spends a fairly lengthy chapter on the Klein-Poincaré correspondence which has been put forth as an example of the way academics can cooperate even when their countries are mortal enemies. O'Shea's careful reading shows the antagonism simmering beneath the surface of their "polite" academic discussion.
Szpiro introduces a great deal of the mathematics that led to the proof of the conjecture by Grigori Perelman in 2002, almost 100 years after Poincaré made the conjecture. He illustrates the math by very clever analogies, avoiding any attempt to go to deeply into the mathematics, which it seems to me is the only way to present material of such awesome complexity and abstraction to a lay audience (in which group I include myself.)
Like O'Shea, Szpiro shows mathematicians warts and all, as he discusses priority disputes such as the Smale-Stallings-Zeeman controversy of the proof of the Poincaré conjecture in higher dimensions (which preceded the proof of the original three-dimensional conjecure). He is not afraid of picking sides: He argues that Smale deserves credit for the proof, but that his abrasive personality made it difficult for him to get help in establishing priority.
Nor is Szpiro shy in assigning full credit for the final proof to Perelman, though standing on the shoulders of many giants, especially William Thurston and Richard Hamilton. Perelman has been pictured as an eccentric loner, refusing the Fields Medal and the $1,000,000 Millennium prize for no good reason. Szpiro sees him as a man of utmost integrity and great friendliness to those who share his seriousness. It is not surprising, then, that Szpiro takes the great Chinese mathematician Yau Shing-Tung to task for pushing the claims of his students Cao and Zhu, who wrote a paper in which they claimed to given the first real proof of the conjecture, based on Perelman's "outline".
If you are interested in mathematics, you owe it to yourself to read either Szpiro's or O'Shea's book on the Poincaré conjecture.
Both authors are fine writers. The books are of similar length. O'Shea's book is 200 pages followed by 72 pages of supplementary material: endnotes, two glossaries, a timeline, and an 11-page bibliography. Szpiro's book is 262 pages followed by 32 pages of endnotes and bibliography. Each book provides a different interesting aspect of Poincarés life: Szpiro's book relates Poincarés career as a mining engineer, in the course of which he exhibited great personal courage and deductive ability worthy of Sherlock Holmes to investigate a mining disaster. O'Shea spends a fairly lengthy chapter on the Klein-Poincaré correspondence which has been put forth as an example of the way academics can cooperate even when their countries are mortal enemies. O'Shea's careful reading shows the antagonism simmering beneath the surface of their "polite" academic discussion.
Szpiro introduces a great deal of the mathematics that led to the proof of the conjecture by Grigori Perelman in 2002, almost 100 years after Poincaré made the conjecture. He illustrates the math by very clever analogies, avoiding any attempt to go to deeply into the mathematics, which it seems to me is the only way to present material of such awesome complexity and abstraction to a lay audience (in which group I include myself.)
Like O'Shea, Szpiro shows mathematicians warts and all, as he discusses priority disputes such as the Smale-Stallings-Zeeman controversy of the proof of the Poincaré conjecture in higher dimensions (which preceded the proof of the original three-dimensional conjecure). He is not afraid of picking sides: He argues that Smale deserves credit for the proof, but that his abrasive personality made it difficult for him to get help in establishing priority.
Nor is Szpiro shy in assigning full credit for the final proof to Perelman, though standing on the shoulders of many giants, especially William Thurston and Richard Hamilton. Perelman has been pictured as an eccentric loner, refusing the Fields Medal and the $1,000,000 Millennium prize for no good reason. Szpiro sees him as a man of utmost integrity and great friendliness to those who share his seriousness. It is not surprising, then, that Szpiro takes the great Chinese mathematician Yau Shing-Tung to task for pushing the claims of his students Cao and Zhu, who wrote a paper in which they claimed to given the first real proof of the conjecture, based on Perelman's "outline".
If you are interested in mathematics, you owe it to yourself to read either Szpiro's or O'Shea's book on the Poincaré conjecture.
Photographs of mathematicians
In his review of Mariana Cook’s new book, Mathematicians: An Outer View of the Inner World, Boston Globe writer Mark Feeney writes "There has yet to be a mathematician maudit, or a Byronic mathematician (other, that is, than Byron’s daughter, Ada)." To which I reply, "What about Evariste Galois?"
The book is 92 black-and-white portraits of mathematicians, and looks quite interesting.
The book is 92 black-and-white portraits of mathematicians, and looks quite interesting.
My Want Ad
I've decided to move ahead with some ideas I've been developing on teaching mathematics to individuals with mathematics anxiety. The object is to develop and sell a software-based product that can be used by adults in their own home. I believe that much mathematics anxiety in adults is a form of PTSD (post traumatic stress disorder) and needs to be addressed before mathematics content can be mastered. My program would teach students relaxation techniques they need to use before attempting mathematics lessons. The lessons themselves would be tailored for adults likely to experience stress in learning mathematics.
I have placed an advertisement looking for help (no pay yet) with a start-up business to bring this all about. So far, this advertisement has been sent out to Acton Networkers, a local group of mostly technically savvy job seekers. For a few more details, see http://www.scribd.com/doc/17717501/Advertisement-for-StartUp-Workers.
I have placed an advertisement looking for help (no pay yet) with a start-up business to bring this all about. So far, this advertisement has been sent out to Acton Networkers, a local group of mostly technically savvy job seekers. For a few more details, see http://www.scribd.com/doc/17717501/Advertisement-for-StartUp-Workers.
Learning Theory and Mathematics
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.
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.
E10. A formula for weighted totals
Professor Blank wants to use a spreadsheet to compute for each student a weighted total of four quiz grades, where the students highest grade is multiplied by four, the next highest grade by three, the next highest by two, and the lowest by one. He requests a formula that will calculate this weighted total using only addition, subtraction, multiplication by an integer, and the min and max functions. Find such a formula.
The problem is a little harder than it appears at first. For a solution, go here.
The problem is a little harder than it appears at first. For a solution, go here.
Freeman Dyson's Problem
My friend John Lamperti turned my attention to a number theory problem in an article on Freeman Dyson in the March 29 New York Times Magazine Section. It is an excellent article, which I recommend. The section with the problem is the following:
The discovery (or proof) of the smallest such number, 105263157894736842, makes a good problem for an elementary number theory course or a bright high school student.
The first time John told me the problem, he had heard it second-hand, and it was backwards: Is there an integer which if you take its first digit and move it to the back you can exactly double the value? In this case the answer is no, and the proof is simpler than the solution for Dyson’s problem.
To see my solutions, go here.
[T]aking problems to Dyson is something of a parlor trick. A group of scientists will be sitting around the cafeteria, and one will idly wonder if there is an integer where, if you take its last digit and move it to the front, turning, say, 112 to 211, it’s possible to exactly double the value. Dyson will immediately say, “Oh, that’s not difficult,” allow two short beats to pass and then add, “but of course the smallest such number is 18 digits long.” When this happened one day at lunch, William Press remembers, “the table fell silent; nobody had the slightest idea how Freeman could have known such a fact or, even more terrifying, could have derived it in his head in about two seconds.” The meal then ended with men who tend to be described with words like “brilliant,” “Nobel” and “MacArthur” quietly retreating to their offices to work out what Dyson just knew.
The discovery (or proof) of the smallest such number, 105263157894736842, makes a good problem for an elementary number theory course or a bright high school student.
The first time John told me the problem, he had heard it second-hand, and it was backwards: Is there an integer which if you take its first digit and move it to the back you can exactly double the value? In this case the answer is no, and the proof is simpler than the solution for Dyson’s problem.
To see my solutions, go here.
The Measure of Reality
At an alumni event I encountered a fellow member of my class of Reed College ‘06, Steven Shapin, who is now Franklin L. Ford Professor of the History of Science at Harvard University. After mentioning some books about the history of that had deeply affected me, I asked him to recommend some books on the subject that I might enjoy. He gave me a short reading list, and I will be reviewing these books as I get to them.
I find books about the history of science and particularly about the history of mathematics to be helpful to my teaching. The nineteenth century philosopher Herbert Spencer claimed that “If there be an order in which the human race has mastered its various kinds of knowledge, there will arise in every child an aptitude to acquire these kinds of knowledge in the same order.... Education is a repetition of civilization in little.” (Wikipedia) While I doubt that this is literally true, I have found that examining the long halting development of mathematical ideas by different cultures helps me understand the difficulty that students have in mastering them.
The first book from Steven Shapin’s list is The Measure of Reality: Quantification and Western Society, 1250 – 1600 by Alfred W. Crosby. This is one of those “big picture” books, like Guns, Germs and Steel that attempts (rather successfully) to explain the success of an entire civilization over a period of centuries.
The books’ argument is well described on the first page:
The thesis of the book is contained in the word “thus”. That is, Crosby believes that the success of Western imperialism is due to the development of quantification. It would be pointless to argue whether this is true or not. Clearly, evidence can be found to either support or refute a thesis that is this broad. But what is fascinating to me are the examples that Crosby introduces, including a description of some of the high points of late medieval and renaissance culture: polyphony in music, perspective in art, the beginnings of modern physics and mathematics. I even gained an appreciation of the role of double-entry bookkeeping in enabling complex business arrangements.
What I found most interesting is trying to imagine the mentalité, or mind-set, of pre-quantitative people. How does one experience time, when one has never seen a clock? How does one picture a scene when viewing a picture of it that does not obey modern rules of perspective?
Despite its big picture, I found the most endearing feature of this book some of the details. For example, I have never realized that the invention of the staff to write music in the fourteenth century prefigured the Cartesian coordinate system. Note that the staff is a graph in which time is the horizontal axis and pitch the vertical. One wonders why it took centuries for the mathematicians to catch up.
I find books about the history of science and particularly about the history of mathematics to be helpful to my teaching. The nineteenth century philosopher Herbert Spencer claimed that “If there be an order in which the human race has mastered its various kinds of knowledge, there will arise in every child an aptitude to acquire these kinds of knowledge in the same order.... Education is a repetition of civilization in little.” (Wikipedia) While I doubt that this is literally true, I have found that examining the long halting development of mathematical ideas by different cultures helps me understand the difficulty that students have in mastering them.
The first book from Steven Shapin’s list is The Measure of Reality: Quantification and Western Society, 1250 – 1600 by Alfred W. Crosby. This is one of those “big picture” books, like Guns, Germs and Steel that attempts (rather successfully) to explain the success of an entire civilization over a period of centuries.
The books’ argument is well described on the first page:
Western Europeans were among the first, if not the first, to invent mechanical clocks, geometrically precise maps, double-entry bookkeeping, exact algebraic and musical notations, and perspective painting. By the sixteenth century more people were thinking quantitatively in Western Europe than in any other part of the world. Thus, they became world leaders in science, technology, armaments, navigation, business practice, and bureaucracy, and created many of the greatest masterpieces of Western music and painting. [Emphasis added.]
The thesis of the book is contained in the word “thus”. That is, Crosby believes that the success of Western imperialism is due to the development of quantification. It would be pointless to argue whether this is true or not. Clearly, evidence can be found to either support or refute a thesis that is this broad. But what is fascinating to me are the examples that Crosby introduces, including a description of some of the high points of late medieval and renaissance culture: polyphony in music, perspective in art, the beginnings of modern physics and mathematics. I even gained an appreciation of the role of double-entry bookkeeping in enabling complex business arrangements.
What I found most interesting is trying to imagine the mentalité, or mind-set, of pre-quantitative people. How does one experience time, when one has never seen a clock? How does one picture a scene when viewing a picture of it that does not obey modern rules of perspective?
Despite its big picture, I found the most endearing feature of this book some of the details. For example, I have never realized that the invention of the staff to write music in the fourteenth century prefigured the Cartesian coordinate system. Note that the staff is a graph in which time is the horizontal axis and pitch the vertical. One wonders why it took centuries for the mathematicians to catch up.
Persistence in Solving Math Problems
I’m currently teaching a course to a middle-school math teacher in Teaching Mathematics through Problem Solving. I’ve taught this course a number of times before. One of the things I do is to ask students to work on problems which I think they will find difficult, but doable. My students – all teachers themselves – become frustrated if they encounter a problem that takes them more than a few minutes. They are so used to routine problems that will yield to a known method of attack, that they don’t know what they are capable of.
I think it may be sometimes better to assign one difficult problem rather than 10 routine ones. And we need to get students to commit to trying to solve a problem even if they have to put it away for a while, let it percolate in their unconscious, and come back to it later. At the risk of sounding like an old fogy, students today are very much used to expecting instant gratification. We need to teach them the rewards of persistence.
I think it may be sometimes better to assign one difficult problem rather than 10 routine ones. And we need to get students to commit to trying to solve a problem even if they have to put it away for a while, let it percolate in their unconscious, and come back to it later. At the risk of sounding like an old fogy, students today are very much used to expecting instant gratification. We need to teach them the rewards of persistence.
Math Humor
Here are a few math education jokes I heard recently. Two of them come from my Tai Chi teacher. I think they all contain a bit of wisdom as well as humor.
The teacher draws a right triangle on the board, labels the legs 3 and 4 and the hypotenuse x, and asks the student, “Can you find x?” The student rushes up to the board and pointing, says “Here it is!”
The teacher writes (a + b)2 on the board, and asks if anyone can expand it. A student comes up and writes
( a____+____b )2.
The teacher writes an equation on the board, and says “Suppose x is the solution to this equation” and starts to do some algebraic manipulation. A student waves his hand wildly. When the teacher calls on him, he says “But sir, suppose it isn’t?”
The teacher draws a right triangle on the board, labels the legs 3 and 4 and the hypotenuse x, and asks the student, “Can you find x?” The student rushes up to the board and pointing, says “Here it is!”
The teacher writes (a + b)2 on the board, and asks if anyone can expand it. A student comes up and writes
( a____+____b )2.
The teacher writes an equation on the board, and says “Suppose x is the solution to this equation” and starts to do some algebraic manipulation. A student waves his hand wildly. When the teacher calls on him, he says “But sir, suppose it isn’t?”
Follow up to problem E9 - Composite values of integer polynomials
I’ve been doing some more thinking about Problem E9, which asks the reader to prove that the range of a quadratic polynomial evaluated on the integers must include a composite number. In addition to the algebraic proof I linked to the original problem, my brother provided a proof based on congruences. I think both are interesting.
Each proof applies to all non-trivial polynomials, not just quadratics, and each actually shows (or can be extended to show) that the range of the polynomial must include an infinite number of composite numbers.
One question to which I do not know the answer is whether the range of a polynomial must contain an infinite number of prime numbers. Of course, the answer is “no” if the polynomial factors over the integers. If the polynomial is prime, however, I suspect the answer is “yes”.
For linear polynomials, the prime or composite nature of the ranges have been well studied. In 1837, Dirichlet proved that every sequence (an + b) contains an infinite number of prime numbers, iff (a, b) = 1, and moreover the fraction of all primes ≤ x that are in such a sequence approaches 1/φ(a) as x approaches infinity, where φ(a) is the number of natural numbers less than a that are relatively prime to a. As a corollary, this also shows that there are an infinite number of composite numbers in the range.
Dirichlet’s Theorem is famous for being the first that used complex analysis to solve a major problem in number theory. Recently (2004), Green and Tao (in an important and difficult paper) proved that there exist arithmetic sequences of arbitrary length that are all prime. The proof is non-constructive.
Each proof applies to all non-trivial polynomials, not just quadratics, and each actually shows (or can be extended to show) that the range of the polynomial must include an infinite number of composite numbers.
One question to which I do not know the answer is whether the range of a polynomial must contain an infinite number of prime numbers. Of course, the answer is “no” if the polynomial factors over the integers. If the polynomial is prime, however, I suspect the answer is “yes”.
For linear polynomials, the prime or composite nature of the ranges have been well studied. In 1837, Dirichlet proved that every sequence (an + b) contains an infinite number of prime numbers, iff (a, b) = 1, and moreover the fraction of all primes ≤ x that are in such a sequence approaches 1/φ(a) as x approaches infinity, where φ(a) is the number of natural numbers less than a that are relatively prime to a. As a corollary, this also shows that there are an infinite number of composite numbers in the range.
Dirichlet’s Theorem is famous for being the first that used complex analysis to solve a major problem in number theory. Recently (2004), Green and Tao (in an important and difficult paper) proved that there exist arithmetic sequences of arbitrary length that are all prime. The proof is non-constructive.
Both a Leader and a Follower Be
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Thoughts from Oliver Selfridge Memorial
I attended the memorial service for Oliver Selfridge in Cambridge yesterday (March 15). There were lots of people there from the AI community, as well as friends who knew him from his interests in education, madrigal singing, gardening, skiing, sailing, poetry writing, and a few others. He was considerably more than a dilettante in all of these fields, and was know for his prodigious memory, his sense of wonder, and his desire to know what made the world work.
One of the speakers, Marvin Minsky, said that when meeting an incredible intellect like Selfridge, von Neumann, or Nash, he (Minsky) always concentrated not just on what the person said, but on trying to figure out how they were able to arrive at it. He mentioned that Oliver had an uncanny sense of direction, so that (for example) he was able to determine which way was North when emerging from underground after a complex subway trip. Minsky finally realized that Oliver was frequently checking the position of the sun in the sky, and the directions of the shadows. It turned out that he was doing this unconsciously.
Someone mentioned also Oliver's belief in the primacy of learning. He was quoted as having said (approximate quote): "A mind without learning is scarcely a mind at all", and it was this belief which informed his researches in AI.
One of the speakers, Marvin Minsky, said that when meeting an incredible intellect like Selfridge, von Neumann, or Nash, he (Minsky) always concentrated not just on what the person said, but on trying to figure out how they were able to arrive at it. He mentioned that Oliver had an uncanny sense of direction, so that (for example) he was able to determine which way was North when emerging from underground after a complex subway trip. Minsky finally realized that Oliver was frequently checking the position of the sun in the sky, and the directions of the shadows. It turned out that he was doing this unconsciously.
Someone mentioned also Oliver's belief in the primacy of learning. He was quoted as having said (approximate quote): "A mind without learning is scarcely a mind at all", and it was this belief which informed his researches in AI.
Two Geometric Mathematical Induction Proofs
Proof by mathematical induction is a powerful weapon in the mathematician's arsenal, but I confess that I don't care too much for this type of proof. Proofs by mathematical induction typically don't shed light on why the result is true or how it might have been discovered in the first place.
When I recently gave a two-hour workshop for high school teachers on using mathematical induction, I looked around for some easy but somewhat unusual examples. The two that I found that were proofs of geometrical results turned out to be the students' favorites. I present them here. Actually, I am stating the problems and will give a link to the proofs so that you can try to come up with the proof for yourself.
The two-color map theorem
A number of straight lines are drawn in the plane, dividing it into regions. Show that each region may be colored either red or black in such a way that no two neighboring regions have the same color. Solution is here.
Tiling with trominoes
Given an n x n chessboard where n is a positive power of two, with one corner square removed, prove that it can be tiled with trominoes. (A tromino is a figure that can exactly cover 3 contiguous squares, not all in in the same rank or file.) Solution is here.
When I recently gave a two-hour workshop for high school teachers on using mathematical induction, I looked around for some easy but somewhat unusual examples. The two that I found that were proofs of geometrical results turned out to be the students' favorites. I present them here. Actually, I am stating the problems and will give a link to the proofs so that you can try to come up with the proof for yourself.
The two-color map theorem
A number of straight lines are drawn in the plane, dividing it into regions. Show that each region may be colored either red or black in such a way that no two neighboring regions have the same color. Solution is here.
Tiling with trominoes
Given an n x n chessboard where n is a positive power of two, with one corner square removed, prove that it can be tiled with trominoes. (A tromino is a figure that can exactly cover 3 contiguous squares, not all in in the same rank or file.) Solution is here.
E9. A prime generating function - not
A student claims to have found a quadratic function f(n) = an^2 + bn + c, with a, b, and c integers, such that f(n) is prime for all positive integers n. Disprove the claim by showing that such a function always takes on a composite value. This is a special case of a well-known result for general polynomial functions. What is sought here is a proof that uses no more than high school algebra.
Hints. Without loss of generality, a > 0. If c = 0 the result is trivial and for |c| > 1, the result is clear, since c divides f(c), so it suffices to prove the result for |c| = 1.
To see my solution, click here
Hints. Without loss of generality, a > 0. If c = 0 the result is trivial and for |c| > 1, the result is clear, since c divides f(c), so it suffices to prove the result for |c| = 1.
To see my solution, click here
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