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.
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