Peter Ash's Thoughts on Math and Education

E19. Cutting a square cake

2011-11-16T07:42:00.000-08:00

I found the following puzzle in Jacob's Geometry: Seeing, Doing, Understanding (3rd ed.). Given a square sheet cake, 9" on a side, divide it into 5 pieces that have the same amount of cake and the same amount of icing.

To be more mathematically precise, the problem is: Given a square, a units on a side. find a dissection of the square into 5 polygonal pieces, each with area a²/5 and each containing the same length of boundary of the original square, namely 4a/5. I think this is not too hard, but I will post my answer if anyone asks.

Extra credit: To generalize and sharpen this result, show that we can replace 5 by n, where n is any integer greater than or equal to 3. Also show that all polygons can be triangles or convex quadrilaterals, and that even so for any fixed n there are an infinite number of essentially different such dissections. (Two dissections are essentially different if one contains a polygon that is not congruent to any polygon in the other dissection.)

The Birthday Problem

2011-09-20T08:49:00.000-07:00

A well-known problem asks for the smallest number of people (N) who must be in a room before it is more likely than not that two share the same birthday. The answer, surprising to most people who have not heard the problem before, is N = 23.

I thought it would be interesting to modify the problem where we ask for people who share that same day of the month for their birthday. While the answer is not as surprising as the original problem, the computation is much easier. Direct computation for the first problem using factorials will result in overflow on scientific calculators such as the TI-83. Also, the answer to the day-of-month problem (N = 7) is more suitable for empirical testing in small classes. Simply ask each student for their birth day (1 - 31) and record on a large month calendar. For N = 11 the probability of a match increases to almost 88%.

The formula for the probability of one or more matches amongst a group of N people is
Prob = 1 - (31)(30)...(32 - N)/31^N
= 1 - 31! /[(31 - N)! * 31^N]

Mathematics and Humor

2011-09-12T08:35:00.000-07:00

"Time flies like an arrow. Fruit flies like a banana." -- The Flying Karamazov Brothers.

Have you ever told a joke to someone who "doesn't get it"? If you patiently explain the referents you may get them to "understand" the joke, but they will probably respond something like "So, why is that funny?"

In the simple example above, you probably found this funny if (1) you are familiar with the maxim "Time flies like an arrow", (2) your knowledge of the English language allows you to understand that "flies" can be a verb meaning "passes swiftly" or a plural noun referring to a type of insect and that "like" can mean both "as" and "enjoy" (3) your knowledge of writing style leads you to expect that when the same word appears in two successive short sentences, it will usually have the same meaning in both sentences.

I think we face the same problem when we try to teach mathematical understanding. A proof is most memorable to us when, like in getting a pun, we make a connection between two or more apparently unconnected thoughts, what is often called an "Aha!" moment. Without previous deep knowledge of the constituent thoughts, the student may be able to follow the step-by-step logic, and may be able to remember the proof for tomorrow's test, but the proof will not be memorable, and both the theorem and the proof will soon be forgotten. One implication for pedagogy is that the curriculum must be carefully planned so that, when a mathematical topic is introduced, the students will understand the constituent parts and be able to appreciate their connection. Otherwise, we are mostly wasting our time.

I recently came across a proof of the Pythagorean Theorem that was new to me that gave me an aha! moment. This was given in Sanjay Gulati's excellent "Mathematics Academy" blog as a Geogebra demonstration. He does not indicate the original source of the proof. The aha! moment comes for the connection between the Pythagorean Theorem and an apparently unrelated theorem that I always teach in my elementary geometry class, the "crossed chords" theorem. The aha! moment occurs from looking at the following picture.

Then the crossed-chords theorem tells us that (c + a)(c - a) = b², or c² - a² = b².

Twitter Recommendations

2011-09-08T04:34:00.000-07:00

Check out
http://www.bestcollegesonline.com/blog/2011/09/07/the-50-best-twitter-feeds-for-math-geeks/

Harold Jacobs' Geometry

2011-08-12T10:00:00.000-07:00

I've been considering a new text for a course in Euclidean Geometry that I teach for middle school teachers. I've been using Essentials of Geometry for College Students by Lial et al. The students seem OK with it, but I find it very boring. I supplement it with lots of my own exercises using Geometer's Sketchpad, paper folding, MIRA(tm), etc. to keep things interesting.

In looking for a replacement, the best book I have found so far is Geometry: Seeing, Doing, Understanding by Harold R. Jacobs. The latest (3rd) edition was published in 2003. Although I will probably use this book, I will transform many of the problems I assign from pencil, paper, ruler, and protractor to Geometer's Sketchpad. I would love it if the publisher W. H. Freeman would commission an update.

This is a high school text, but it is more challenging than Lial. The applications to "real life" are the most realistic and compelling that I have seen anywhere. I keep finding things that I didn't know, and ways of looking at geometry problems that I hadn't considered.

In one example on page 503 Jacobs shows a closed smooth curve bounding a convex region and consisting of circular arcs. One student said that the sum of the arc measurements must be 360 degrees, and the other doubts it because the curve is not a circle. From the nature of Jacobs' construction, it is easy to show that the sum of the arc measures is indeed 360 degrees. A good teacher could connect this with the fact that the sum of the exterior angles of a convex polygon is 360 degrees.

In another example, Jacobs gives an "Area Puzzle" where he guides students to prove a curious fact about triangle areas. If each vertex of a triangle (ABC in the figure below) is connected to a point 1/3 of the way from the next vertex (in CCW order, say) to the following vertex, and the intersections of these 3 segments (Cevians) are connected, an inner triangle (DEF) is formed. The area of DEF turns out to be 1/7 of the area of ABC. I have known this for some years, and even published a paper (with my brother Marshall and my nephew Michael) generalizing it to quadrilaterals and to ratios other than 1/3. The proof I used involved using analytic geometry to establish the result for a right triangle with vertices (0, 0) (1, 0), and (0, 1) and then arguing that the area ratio is preserved by affine transformations, so the result holds for all triangles.

Jacobs presents a neat synthetic proof that clearly shows where the strange ratio 1:7 comes from. He constructs 6 more triangles, each a translate of the central triangle, and then guides the student to show that the triangles can be dissected and reassembled to fill the original triangle. See the diagram below.

An Application of Triangle Geometry

2011-03-01T09:36:00.000-08:00

I am always on the lookout for attractive, simple, and real applications of synthetic Euclidean geometry to share with my students. That's why I enjoyed an article in the March 2011 issue of The College Mathematics Journal by Robert K. Smither, "The Symmedian Point: Constructed and Applied". The symmedian point of a triangle is the intersection of the three symmedians, where a symmedian is the reflection of a median of the triangle in the angle bisector (at the vertex through which the median passes). See the picture below, where the dashed lines are the medians, the green lines are the angle bisectors, the red lines are the symmedians, and P is the symmedian point of triangle ABC.

Smither worked for the Navy in the post-WWII era, well before computers were widely available. His job was to design a system for locating mines that might be dropped by aircraft on a harbor. As a mine hit the water, it would be observed from three stations. On a map, rays would be drawn from each station location in the direction in which the mine had been sighted. In theory, the three rays would be concurrent at the location of the mine. In practice, due to measurement error, the three rays would not be concurrent, but would form a triangle ABC. The most likely location of the mine was assumed to be the point for which the sum of the squares of the distances to the sides AB, BC, AC was minimum.

The hand calculations required to locate the point in question using analytic geometry were horrendous and error-prone. By examining the results, Smither was led to rediscover the symmedian point, which turns out to be the point that minimizes the sum of the squares of the distances to the sides of the triangle. He also discovered a neat method of constructing this point, which is easier than using the definition and appears to be original.

E18. Alphabetical ordering of whole numbers

2011-02-25T07:35:00.000-08:00

Suppose the integers from 1 to 10,000,000,000 are ordered alphabetically as spelled out in American English, according to the rules below. Find the first odd number in this ordering.

(1) Numbers are spelled out formally, using neither "and" nor common shortcuts. For example, 2400 is two thousand four hundred, not twenty-four hundred.
(2) To alphabetize, ignore any spaces or hyphens. For example, sixteen comes before six thousand.

This problem is one of the easier ones in Peter Winkler's excellent collection, Mathematical Puzzles: A Connoisseur's Collection. It is an excellent problem to give to students working together in small groups. Usually students will have to make many attempts along the way to finding the solution, giving everyone a chance to participate. Note the specification of American English. Many people don't realize that there is a difference between American English and British English in some number names. For example, "billion" has different meanings in the two forms of English.

It might make an interesting exercise to write an algorithm for converting an integer (say in the range specified in this problem) to the spelled out form.

Curriculum for Overcoming Math Anxiety Course

2011-02-14T06:09:00.000-08:00

(c) Peter. F. Ash, Ph.D. 2011
The following is the Curriculum for my Overcoming Math Anxiety course offered at Cambridge Center for Adult Education February 23 – March 9, 2011 over three two-hour classes:

1.Let's get personal
What brings you here? Why do you need to overcome math anxiety? When did your dislike or fear of mathematics first develop? Start keeping a "math journal".

2.Everyone can learn math
Is there something in your brain that means you can't learn math? What is dyscalculia? Overcoming handicaps.

3.Math phobia?
A serious fear of math may be a phobia, and may require treatment. A treatment you can do yourself, called TAT (Tapas Acupressure Technique) can help you. Our special guest lecturer shows you how.

4.One size doesn't fit all (even if your teacher thought it did)
Different people have different learning styles. If you know your preferred learning style you can learn math better. Are you a quantitative or a qualitative learner? Learning through different modalities: visual, aural, or tactile/kinesthetic.

5.Math myths
If you were taught with traditional methods, you probably learned that being good at math required prodigious memory and the ability to regurgitate what the teacher told you. You may believe that there is one way to solve a math problem, and that math must be done while sitting still and keeping quiet. Wrong!

6.The new way to learn math
Modern reform mathematics suggests that math instruction be focused on solving interesting complex problems which can be solved in different ways, that students work in groups and communicate their ideas to one another, and that students learn to do mathematics with deep understanding, not by rote.

7.A sound mind in a sound body
Research shows that regular aerobic exercise helps you to beat stress, improve memory, and sharpen your thinking. Schedule your exercise before doing your math and watch what happens.

8.Learning is not all in your head
Learning cannot be separated from movement. The fact that proper movement leads to optimal learning underlies Brain Gym®, We'll practice basic Brain Gym exercises to help reduce stress and make learning easier.

9.The mind-body connection
Learn to reduce stress and improve focus with meditation-based techniques. Use Zen meditation, yoga, TM, the relaxation response, or simple diaphragmatic breathing to reduce stress and empty your mind of chatter so you can learn better.

10.Music hath charms…
Playing certain classical music in the background can help you energize and focus. I'll play the CD and you'll hear if it helps.

11.Manipulate and understand
Learn what a mathematical manipulative is and how it helps visual and tactile/kinesthetic learners understand math concepts. Experience the power of multiple representations in math.

12.So, can I really do math?
Sure you can! You'll investigate a few math problems working in a group. Try out your new-found math anxiety reduction skills and enjoy some interesting open-ended problems.

13.Help! I need to take a math class
How to tell if you have a good teacher. What to do if you don't. Important study skills
.
14."Teach your children well"
What you can do so your children learn to like math, not to fear it.

15.Where do you go from here?
I'm here to help. Send me an email if you'd like a bibliography on math anxiety and math learning. Contact me if you are interested in math tutoring or math classes.

Car Talk Math Puzzler Solution Wrong!

2011-01-23T20:05:00.001-08:00

On the NPR program Car Talk, hosts Tom and Ray presented a puzzler in which a truck has a broken fuel gauge. The trucker wishes to determine when his fuel tank is one-quarter full. The tank is in the shape of a circular cylinder on its side. The trucker can put a stick in the filling opening at the top of the tank, and use it as a dipstick to mark the height of the fuel. Obviously, the half-full mark is the radius of the circle from the end of the stick. Where is the quarter-full mark? A solution was requested that would use a minimum of advanced math. Solvers could use a pizza box, string, a pencil and a knife.

I loved the problem, and their method of solution which involved using the string and pencil to draw a circle the size of the tank cross section on the top of the pizza box, then using the bottom of the pizza box as a to draw a diameter of the box, and using the box as a square to draw a radius perpendicular to the diameter. Finally, they cut out the resulting semi-circular region, and balanced the semicircle on the pencilpoint to determine the center of mass of the region. They claimed that the quarter-full horizontal line would pass through the center of mass.

Not so! Look at the fuel tank "head on", so you see a circular region. Imagine drawing 3 horizontal lines on the region. The first line is the diameter of the circle, or the half-full line. The second line is the one that passes through the centroid of the lower semicircular region and is parallel to the first line. Let's call this the TARL (Tom and Ray line). If Tom and Ray are right, the TARL is the quarter-full line. However, they're wrong so there is a third line, the BL (bisector line) which is parallel to the previous two lines and divides the semicircular region into two pieces of equal area. If the circle involved has a radius of 10 inches, the TARL is approximately 4.24 inches from the half-full line, while the BL is approximately 4.04 inches from the half-full line.

The mathematical details are interesting, and can be shown without any explicit reference to integral calculus. See them at Scribd.

Why do clocks move clockwise?

2011-01-03T18:05:00.000-08:00

Here's a question that can be answered with a little knowledge of mathematics and history, and some common sense.

Why do "normal" clocks always turn in the direction we are used to, and not in the opposite direction?

Twenty Incredible Math Talks

2010-12-09T08:04:00.000-08:00

Florine Church of Bachelorsdegree.org sent me a link to http://www.bachelorsdegree.org/2010/12/08/20-incredible-ted-talks-for-math-geeks/. I had known that TED.org has some of the most wonderful and thought-provoking lectures that I have heard online (or anywhere else) but I was not aware that they had many talks on mathematics, including applications and education. I'm looking forward to listening to these talks, and suggest that my readers see them as well.

Thanks, Florine

E18. Spider and Bug

2010-11-24T14:42:00.000-08:00

A room is in the shape of a rectangular prism, 12 feet high, 12 feet wide, and 30 feet long. A spider is in the center of one of the 12 x 12 walls, one foot from the ceiling. A bug is in the center of the opposite 12 x 12 wall, one foot from the floor. The spider wants to reach the bug by the shortest possible route, and can only travel on the surface. What is the shortest distance, and what is the route? (Hint. The shortest route is NOT the obvious one of going straight up to the ceiling, straight across the middle of the ceiling, and straight down the opposite wall for a total of 42 feet.)

I remember this problem from my school days, and managed to find it again in the Math Forum archives (1995). I would appreciate hearing from anyone who knows the original source. I expect it may be due to Ernest Dudeney.

Teaching Partial Fractions

2010-11-22T14:50:00.000-08:00

Students commonly encounter the method of partial fractions for the first time (without proofs) in Calculus II, as a method to aid in integrating rational functions. These days, partial fractions are sometimes not taught at all, since students can determine most any common indefinite integral by using a CAS. Without taking sides in the debate over how much methods of integration should be taught, I would like to make a case that partial fractions should be taught in high school or below.

Of course, partial fractions are a technique that comes up when discussing the algebra of rational functions. However, they also come up very naturally in arithmetic. I propose introducing them in the context of solving a problem that students might find interesting. I call this the problem of the base-p rulers.

The smallest distance measurable by an ordinary English-units ruler is 1/2^n inch, where n is typically 5 (32nds) or 6 (64ths). Define a base-2 ruler to be an idealized version of this ruler, where all coordinates of the form a/2^n are marked, where a and n are non-negative integers. It's clear that not all rational distances are measurable with such a ruler, for example 1/3 is not. To measure all rational distances, we can create an infinite number of base-p rulers, where p varies over the prime numbers. A base-p ruler has all co-ordinates of the form a/p^n, where a and n are non-negative integers. A length of length a/b can be laid with base-p rulers, provided a/b can be expressed as a sum of signed base-p numbers a/p^n. For example, the length 1/6 can be laid out by measuring 1/2, and then backing up 1/3: 1/6 = 1/2 - 1/3.

We want to have students discover that every rational number length a/b (in lowest terms) can be expressed using base-p rulers, where p varies over the primes that divide b.

Providing a proof requires some number theory. Clearly, it is necessary and sufficient to show that every number of the form 1/b can be represented in the required form, and the number theory involves finding a generalization of the fact that if (a, b) = 1 there is a solution in integers to ax + by = 1.

The mystic pentagram and the discovery of irrationals

2010-11-10T16:53:00.000-08:00

According to one legend, the Pythagorean Hippasus of Metapontum first discovered that not all numbers are rational by proving that the square root of two is irrational, and he was murdered by other Pythagoreans who believed that all numbers are rational.

However, some people believe that a Pythagorean, possibly Hippasus, discovered the existence of irrational numbers in a different way, by considering the mystic pentagram. Since this figure was sacred to the Pythagoreans, they must have been curious about determining its dimensions. And it is not too difficult to imagine that one of them was led to the discovery of irrationals this way. Indeed, if the five diagonals of a regular pentagon are drawn, forming the mystic pentagram, the ratio of the length of a diagonal of the pentagon to the length of a side is the irrational number phi, the golden ratio.

It makes a great exercise for good beginning geometry students to prove, as some early Greek geometer must have, that the ratio mentioned above cannot be a rational number, using what is essentially Fermat's method of infinite descent. I'll sketch an outline of the proof below.

Consider a regular pentagon of side length s and let the length of each diagonal be d. A regular pentagon is formed inside the original one. Let its length be s', and the length of its diagonals be d'. After drawing the diagram, the student needs to make repeated use of the following elementary facts:
(1) The interior angle in a regular pentagon is 3 * 180 / 5 = 108 degrees.
(2) The sum of the angles in a triangle is 180 degrees.
(3) Two sides of a triangle are equal iff the two angles opposite the sides are equal.
Using these facts it becomes apparent that the diagram has 36 degree angles all over the place [ 36 = (180 - 108)/2] and lots of isosceles triangles. Using this information, the following simple relations can be determined:

(4) s' = 2s - d
(5) d' = d -s

If the ratio d:s is rational, then (by scaling) we may assume that d and s are positive integers. But according to formulas (4) and (5), this means that d' and s' are integers too, and obviously from the diagram d' < d and s' < s. Now, we can imagine continuing the process of drawing diagonals and producing smaller and smaller nested pentagons over and over. Each time the length of the side and the length of the diagonal is a smaller positive integer. But after (much) less than s iterations, the length of a side will be less than 1, not an integer. So d and s can not both be integers.

To see where phi arises, use (4) and (5) to write

(6) d'/s' = (d - s)/(2s - d)

Since the original pentagon and the nested one are similar, we can replace the left hand side by d/s, and then by dividing the numerator and denominator of the right hand side by s we obtain a quadratic equation for (d/s). The positive solution is phi = (1 + sqrt(5))/2.

Is the NCTM Opposed to Mathematics Education?

2010-09-29T19:01:00.000-07:00

The National Council of Teachers of Mathematics (NCTM) is the nation's largest professional organization for K-12 teachers of mathematics. The idea that the agenda of the NCTM would in fact be opposed to teaching mathematics seems, on the face of it, absurd. And yet that is the thesis of David Kline's paper, "A Brief History of K-12 Mathematics Education in the 20th Century". If Kline were an isolated crank, this idea would not matter much. But he is a professor of mathematics at California State University Northridge and according to Google Scholar his paper has been cited by 51 researchers since its publication as a chapter of Mathematical Cognition in 2003. Furthermore, the paper was described as must reading for all interested in mathematics education by John Mighton, author of the influential education best-seller, The End of Ignorance: Multiplying Our Human Potential, which I reviewed earlier.

Klein is firmly in the traditionalist camp of mathematics education, and in fact presents himself as a member of Mathematically Correct, the most famous of the traditionalist groups. To boil down a detailed argument of over 40 pages to a few sentences, Klein feels that the mathematical reform movement which came to prominence in the 1990s under the auspices of the NCTM and the National Science Foundation (NSF) disenfranchised students by offering mathematics instruction grounded in constructivist theory and based on "textbooks with radically diminished content and a dearth of basic skills". He traces the reform movement to the progressive education movement beginning in the early 20th century, whose leaders had a documented hostility to mathematics. For one example of many, he quotes the influential progressive educator William Heard Kilpatrick as saying that mathematics is "harmful rather than helpful to the kind of thinking necessary for ordinary living". He states that the NCTM was created by the MAA (Mathematical Association of America) in part to counter these progressivist ideas, though later the NCTM embraced these same ideas.

As is well-known, the reform movement in mathematics is characterized by educational constructivism, the theory that "only constructed knowledge – knowledge that one finds out for oneself – is truly integrated and understood". Constructivism is originally a psychological term, and Klein quotes several psychologists who claim that educators misapplied the concept, so that claims by constructivists they support "brain-based learning" ring hollow. In any case, educational constructivism is now connected with child-centered, cooperative, self-paced, problem-based discovery learning.

Education can be viewed as a wedding of pedagogy and content. In theory, the two are separate. However, constructivist learning takes longer since the student spends time exploring blind alleys on the way to getting a correct answer. For example, if the student is discovering their own algorithm for multi-digit addition it will take longer than if they are simply told how to do it, and moreover the algorithm they derive may be inefficient, making subsequent work take longer. So when traditionalists say they are designing a pedagogy-neutral curriculum, they are being somewhat disingenuous. By proposing a lengthy list of content to be covered, they insure that strict constructivist approaches will not work.

In essence, Klein feels that the NCTM has been taken over by professional educators, not teachers, and that these educators are pursuing a progressivist agenda with little regard for actually teaching mathematics.

While this critique seems to make sense, it does not fit with my personal observations of many practitioners of constructivist mathematics education. Most of these people have a deep love of mathematics and of teaching. After all, anyone who has done original mathematics has practiced discovery learning. Indeed, the "Moore Method" pioneered by topologist R. L. Moore at the University of Texas was an extreme form of discovery learning for graduate mathematics majors, Moore is regarded as one of the most successful teachers of graduate mathematics in American history, based on the number and quality of his Ph.D. students.

I think most mathematics educators would agree that students should receive some direct instruction in standard algorithms and basic theory and some opportunity to explore mathematics on their own. As always, balance is important.

I also think that most mathematics educators would agree that being able to teach well using a constructivist approach is more difficult than using a traditional approach. Part of the reason why constructivist approaches have not been more successful therefore has to do with inadequate training of teachers, and of a failure to recruit the best students into a difficult and underpaid profession.

For some eloquent defense of constructivist, problem-based learning from people who clearly love mathematics see:
A Mathematicians Lament by Paul Lockhart
In Math You Have to Remember, In Other Subjects You Can Think About It by Keith Devlin

Fractal Video from Teamfresh

2010-09-12T19:20:00.000-07:00

I found this on Steven Strogatz's NY Times Math Blog

Classic newton fractal from teamfresh on Vimeo.

From a Spreadsheet Problem to the Umbral Calculus: A Mathematical Odyssey

2010-09-11T15:42:00.000-07:00

I'm planning to write a paper where I describe how a colleague's challenge to come up with an Excel formula to compute a weighted average of grades led me to make a couple of mathematical conjectures, and how I was able to prove the conjectures and solve the problem. Along the way, I got a lot of help from many people and I discovered a lot of combinatorial mathematics that I had not known, including the Binomial Inversion Formula and the Umbral calculus. In describing this odyssey I will explore the social nature of mathematics and the different ways that people from different disciplines approach mathematical problems. Also, I hope to show that experiences of this sort can be replicated in the classroom through a problem-based method of learning.

My Problem Published

2010-08-19T17:21:00.000-07:00

My problem about finding the kth largest element of a set has been published (with a very slight misprint) in the American Mathematical Monthly:
Monthly Problem 11520

E18. A locus related to a rectangle

2010-07-04T11:06:00.000-07:00

This problem is related to my earlier E16.

Let ABCD be a rectangle. Find the locus of all points P such at PA + PC = PB + PD.

The End of Ignorance: Multiplying our Human Potential

2010-06-29T19:29:00.000-07:00

I've just finished reading The End of Ignorance: Multiplying our Human Potential, by John Mighton who has developed a mathematics education program called JUMP (Junior Unrecognized Mathematics Prodigies). His system has met with amazing success with a very wide range of elementary school students and considerable hostility from the Mathematics Education establishment in his native Ontario.

He challenges the NCTM orthodoxy, and the tenets of constructivist math education. I feared that this might be another "Mathematically Correct" screed, but it is far from that. Mighton has an enviable record of success in reaching the most "hopeless" students, and an admirable humility in recognizing that his system is not the only way to improve math education.

Mighton has a Ph.D. in mathematics, a career as a playwright, and a firm grasp of philosophy. He and a large cadre of volunteers have developed the program over a number of years, and refined it by trial-and-error. The major ideas are:

Learning takes place with a balance of concrete and symbolic, guided and independent, and procedural and conceptual.
Compared with constructivist methods, the teacher is expected to be a very active guide. Concepts are broken into small units, gaps in student understanding are detected and filled, lessons are carefully designed, sequential, and scaffolded. Weaker students are motivated by carefully graduated challenges, and stronger students are given extra challenges.
Whole-class lessons allow students to experience the thrill of discovery collectively.
Teachers give frequent and specific encouragement to all students.
Formative assessments are given continuously, and used to modify instruction. Students who don't know the material necessary to begin the lesson are given additional instruction before learning the new topic.
There is a strong emphasis of the development of procedural knowledge through use of workbooks and individual work.

I think any educator who reads Mighton's calmly recollected stories of the hostility and closed-mindedness that his ideas have generated among certain math curriculum consultants (some of whom are subject to conflicts of interests due to their relationships with textbook publishers) is bound to feel a sense of embarrassment for our profession.

Mighton's book has caused me to rethink some of my pro-constructivist positions. I also will be following up on reading some of the work on cognitive psychology that he cites as having been seriously misinterpreted by the mathematics education establishment as supporting constructivist and situated learning approaches.

NES/MAA Meeting

2010-06-21T07:57:00.000-07:00

The NES/MAA meeting I mentioned in my last post was held at Salve Regina University, which is located on the grounds of an opulent mansion on the ocean at Newport Rhode Island. There were a number of interesting invited presentations. I particularly enjoyed the talk by David Abrahamson and Rebecca Sparks on Baseball Statistics and Keith Conrad's talk on Check Digits (in credit card numbers, etc.). Both dealt with fairly elementary mathematics, but related the results to the everyday world in a compelling way.

Ed Burger's Battles Lecture on p-adic norms was a bit more technical, but Ed's high-energy and humorous style of presentation made the medicine go down very well.

My presentation on The Pythagorean Theorem (Revisited) was well received. I was a bit surprised and gratified that none of the mathematics professors or students previously knew the main theorem I was presenting, the Pythagorean Theorem for right tetrahedrons.

Presentation at NES/MAA Meeting, June 12

2010-06-04T11:10:00.000-07:00

I will be attending the New England Section of the Mathematical Association of America meeting in Newport Rhode Island, June 11-12. I have had a paper accepted. It is an expository paper presenting some simple and interesting facts relating to the Pythagorean Theorem that many professional mathematicians do not know. To view my notes for the presentation, go to http://www.scribd.com/doc/32536190/Pythagorean-Theorem-Notes.

Comments on the paper are welcome.

Coded arithmetic puzzles

2010-04-24T02:13:00.000-07:00

In a course I am developing, I want to give out some math problems for people to work on that should be in the grasp of adults without much math background at all. One such problem is what I call "coded arithmetic puzzles". The commonest example I know is "Send More Money": Solve S E N D + M O R E = M O N E Y, where each of the 8 different letters in the equation represents a different digit.

I would like to find a collection of these types of puzzles that would enable me to give different classes different puzzles. The puzzles should not be too tedious and should not require too much cleverness; in other words of the difficulty of Send More Money, or easier. Perhaps someone has written a program that would generate puzzles of this sort.

Does anyone know if there is a formal name for this type of problem? "Coded arithmetic puzzles" is not a very helpful Google search.

Name Change

2010-04-10T19:31:00.000-07:00

Cambridge Math Learning, Inc. is now doing business as Math for the Rest of Us. I think this emphasizes the mission of the company, which is to teach mathematics to the "bottom 80%" of adult math learners; those who have been poorly served by mathematics instruction in the past and most of whom now have anxiety when facing mathematics that they must learn.

More on Ordering a Multiset

2010-04-10T19:13:00.000-07:00

I posed problem A3, to find a formula for the k-th largest element of an n-element multiset A. I found a very interesting formula that is unknown to several famous combinatorists, including Donald Knuth, and I have submitted a problem to the MAA Monthly Problems section which asks for the solution that I found, a linear combination of certain symmetric functions. However, Knuth told me that there is a simpler known formula of a different type. Knuth's formula is

min(max_k)

where (max_k) is a set of C(n,k) numbers, each of which is the maximum of a different subset of A of size k.

Pretty cute!

Mandelbrot Set

2010-03-11T17:05:00.000-08:00

A friend just sent me a link to a fantastic video: Mandelbrot Fractal Set Trip to e214 by teamfresh. The video runs about 9 minutes and zooms in on the Mandelbrot set to a magnification of 10^214. Wow!

It feels like there must have been some pretty clever programming and lots of computer time used to produce this video. The idea of using video to zoom in on the Mandelbrot set is so powerful that it seems to make the beautiful still pictures that I am familiar with, obsolete. To teamfresh, I say Bravo!

I am amazed and humbled by the incredible complexity that can be contained in the simplest mathematical formulas, as shown in this video. Truly our own inventions can take a life of their own.

E17. A 1-2-3 counting problem

2010-02-15T07:35:00.000-08:00

The following problem seems at first to be quite difficult, but if you look at it the right way it isn't.

How many n-digit integers are there that contain no digits other than 1, 2, or 3, subject to the condition that any two consecutive digits differ by exactly 1.

This problem (for the n = 10 case) appeared in the ATMIM newsletter, Winter 2002, where it is credited to http://www.mathkangaroo.org, an interesting math enrichment and contest Web site.

I think this problem is too easy for me to post an answer, but if anyone asks for one, I will.

The buckling train track

2010-02-10T08:50:00.000-08:00

Here's another gem from the ATMIM conference last month, suitable for any class where the students know the Pythagorean Theorem. Imagine a length of train track, two miles = 2 x 5280 ft long. To accommodate expansion of the track on hot days, the track is hinged at both ends and at the middle. If the track expands slightly, the middle of the track will rise, forming a shallow isosceles triangle. Supposed the track expands 2 feet. How high with the track be in the middle?

The teacher asks the students for guesses, which tend to be around 1 foot. Then he leads the students through the calculation of the answer. The height is the length of a leg of a right triangle where the hypotenuse is 5281 feet and the other leg is 5280 feet. This works out to be about 102.8 feet!

A3. Ordering a multiset

2010-01-15T16:48:00.000-08:00

Given a multiset of real numbers {a₁, ...,a_n} find expressions e₁, ...,e_n such that {a₁, ...,a_n} = {e₁, ...,e_n}, and {e_i} is a non-increasing sequence, where each expression is formed from the a_i, the max function, and elementary arithmetic operations.
I will not post the answer to this problem, because I plan to publish it if it is not already known. If it is a known result, I would appreciate a reference.

E16. A Property of rectangles

2010-01-15T16:38:00.000-08:00

Someone showed me the following problem at the meeting of ATMIM (Association of Teachers of Mathematics in Massachusetts.) It's proof is a great non-routine use of the Pythagorean theorem. I think the proof is too easy to post, but if anyone can't figure it out, post a comment and I will make the solution available.

Let ABCD be a rectangle, and P any point in the interior. Prove that AP² + PC² = BP² + PD².

E15. Area in a square

2009-12-11T01:29:00.000-08:00

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

2009-11-02T06:11:00.000-08:00

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

2009-11-02T06:02:00.000-08:00

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

2009-10-05T18:44:00.000-07:00

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

Algorithms Meet Art Puzzles & Magic

2009-09-17T04:24:00.000-07:00

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

2009-08-26T15:09:00.000-07:00

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.

Photographs of mathematicians

2009-08-24T13:42:00.000-07:00

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.

My Want Ad

2009-07-27T09:35:00.000-07:00

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.

Learning Theory and Mathematics

2009-06-24T07:48:00.000-07:00

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

2009-05-19T16:19:00.000-07:00

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.

E12. Ladder against a wall (Part II)

2009-04-23T07:26:00.000-07:00

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.

E11. Ladder against a wall

2009-04-19T17:59:00.000-07:00

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.

Blogger etiquette

2009-04-09T12:09:00.000-07:00

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.

Lure of the Labyrinth

2009-04-08T08:19:00.000-07:00

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.

A New Business Card

2009-04-07T08:15:00.000-07:00

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

2009-04-05T17:00:00.000-07:00

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.

Freeman Dyson's Problem

2009-04-01T06:42:00.000-07:00

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:

[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

2009-03-28T11:29:00.000-07:00

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:

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

2009-03-23T07:52:00.000-07:00

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.

Math Humor

2009-03-20T04:37:00.000-07:00

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?”

Follow up to problem E9 - Composite values of integer polynomials

2009-03-17T07:39:00.000-07:00

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.

Both a Leader and a Follower Be

2009-03-16T10:56:00.000-07:00

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Thoughts from Oliver Selfridge Memorial

2009-03-16T06:26:00.000-07:00

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.

Two Geometric Mathematical Induction Proofs

2009-02-11T12:26:00.000-08:00

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.

E9. A prime generating function - not

2009-01-04T15:04:00.000-08:00

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

Oliver Selfridge 1926 - 2008

2008-12-09T12:17:00.000-08:00

On Wednesday December 4 I had an appointment to have lunch with Oliver Selfridge. I had met Oliver a few years ago when I was visiting a mutual friend, Wally Feuzeig, at BBN Educational Technologies, and had recently written him reminding him of our mutual interest in mathematics education. Oliver was enthusiastic, and send me some of his work: A Math Quiz that offers some very challenging problems for older children, and a list of Abstracts of 23 booklets that he had written (or was writing) to help interest children in mathematics. I had hoped to talk with Oliver about these projects as well as to mention to him some initial thoughts I had about writing a geometry book, and to find out if he might be interested in some sort of collaboration.

When Oliver did not show up for lunch, I went to his nearby office, and met Wally who told me that Oliver had been badly injured in a fall at his home the night before, and it was not known whether he would survive. In fact, he had just died.

If the name Oliver Selfridge is familiar to you, it is probably because of his pioneering work in Artificial Intelligence at MIT. You can read about it, and some other aspects of his fascinating life in the NY Times obituary. I would like to add that he was extremely generous, devoted to the education of children, and had kept a very child-like sense of wonder. I will miss him.

A2. A Trip Around Antarctica

2008-11-21T09:25:00.000-08:00

I found this neat problem in Peter Winkler's excellent book, Mathematical Puzzles: A Connoisseur's Collection. I've dressed it up a little.

You have planned an expedition to travel in a 8000 mile loop around Antarctica. Your advance team has set up 20 fuel caches along the route, and has distributed 8000 miles worth of fuel among the caches. You know the amount of fuel at each cache, and the amount of fuel required to travel between any two consecutive caches. Prove that, regardless of the spacing of the caches or the amounts of fuel in each cache, you can complete the trip, assuming that you have an infinitely large fuel tank. Determine how to pick a cache you can start from.

(This might be an elementary problem, depending on how you look at it.)

Math and Sex

2008-11-18T04:14:00.000-08:00

The late physicist Richard Feynman once said "Physics is like sex. Sure, it's useful but that's not why we do it." I think anyone who has been seduced by mathematics can appreciate that this applies to mathematics as well.

I was thinking that the analogy can be pushed a little further into mathematics education. I have been somewhat disheartened by the hostility that many in the mathematical research community have shown to the reform movement in K-12 mathematics education, particularly as regards discovery learning. As an example of this hostility, see the open letter to Secretary of Education Richard W. Riley that was signed by approximately 200 research mathematicians and scientists in 1999.

And yet, there is an example of the successful discovery learning technique in higher mathematics that whose success is, so far as I know, uncontroversial in higher mathematics circles. I refer to what is called the Moore Method, used by topologist R. L. Moore at the University of Texas. Mathematics graduate students were forbidden from reading about the subject and instead were led to rediscover the theory by themselves, given a set of axioms and very unobtrusive guidance. See here for an account by one of Moore's students. Based on the number and productivity of his Ph.D. students, Moore was one of the most successful mathematics thesis advisors of the first half of the 20th Century.

Granted, pedagogy that works for mathematics graduate students might not work for K-12 students. But, given the success of the Moore method, I wonder if the hostility to even considering the benefit of asking students to develop their own algorithms for arithmetic (for example) is akin to a parent's telling a teenager that sex is great, but only for when they are older. While such prudence may be sensible (if futile) with regards to advice regarding sexual experimentation, I doubt that it is helpful in developing a student's mathematical passion, or even ability.

Opinion Piece on Math Education

2008-10-10T09:32:00.000-07:00

I have written an opinion piece in math education that was published in the Lexington Minuteman on October 9, 2008 under the (slightly inaccurate) title "Math is key to success in the world economy". It seems to be no longer available on the Lexington Minuteman, so I've posted the original on scribed.

To Test or Not to Test

2008-10-08T06:30:00.000-07:00

There has been a lot of debate on the issue of high-stakes testing that determines whether a student will be able to graduate high school and whether schools will be taken over by the state. In Massachusetts, this centers on MCAS (Massachusetts Comprehensive Assessment System), which was set up in response to the Massachusetts Education Reform Act of 1993. Since the adoption of the federal No Child Left Behind Act (NCLB) of 2001, similar assessment systems have been established nationwide.

The goals of the assessment system are laudable. Most observers agree that the public schools, particularly those in poor urban areas, have a history of failing their students. And it seems obvious to me that if a student cannot pass a fair test of basic skills, there is something wrong. I do not support granting diplomas to students who lack the most basic skills.

But as currently implemented there are big problems with MCAS. The importance of this test to schools has severely distorted priorities.

If a school is facing penalties unless they raise their MCAS scores, there is a tremendous incentive to transfer resources to those students who score in the lower ranks. There is no incentive to help student who are already scoring excellent.

Benefits for those students who MCAS and NCLB were designed to help are not clear either. High school drop out rates are up and anecdotal evidence as well as logic indicates that students who cannot pass MCAS after several tries are more likely to drop out. And since average scores will go up as more of these students drop out, schools have a disincentive to retain these students.

A personal communication by a community college science teacher reveals that most graduates of the Boston public schools who attend this community college place into a basic math course which begins at the third grade level. I asked how this could be possible, since these students have all passed the 10th grade Mathematics MCAS, which I regard as a reasonable test of high school math knowledge. The teacher replied,

... students who fail the MCAS tests are put into intensive "MCAS prep" programs. These are designed for one purpose only -- to get them past the test. I. e. it is "teaching to the test" in its purest form. Many students are indeed then able to pass the MCAS math test, and still be grossly deficient in math skills.

I think the problem is analogous to that of car manufacturers that seek to improve the quality of their product. One way to improve quality is to devote more resources to inspecting the final product. A better way is to do what the Japanese have done and improve the process of car production. There will still be final inspections, but less defects will be found. In the same way, improvement in education must precede the institution of high-stakes testing.

The Math Wars

2008-10-04T08:53:00.000-07:00

The "Math Wars" have been framed as a debate between “traditionalists” and “reformers”. I don't take a side in this debate, but rather I think the debate is unproductive.

The reformers actually represent the educational establishment, and their position has been the official position of the National Council of Teachers of Mathematics (NCTM) for over 20 years. They believe in learning by discovery, cooperative work in small groups, and an emphasis on communicating one's thinking. Their philosophy of education is constructivism, and although the word constructivism does not appear in the Principles and Standards for School Mathematics, it is clearly a constuctivist document. The traditionalists, who include many parents and a large number of university mathematicians and scientists, have developed as a reaction to what they see as the excesses of the reformers and a perceived decline in the abilities of college students. A more informal movement, their position seems to be well stated by the Mathematically Correct movement. Traditionalists stress the importance of individual competence, ability to instantly recall number facts, and the ability to perform important algorithms. Although I have not seen them espouse a theory of education, from their prescriptions they implicitly adopt behaviorism and cognitivism.

There is a hidden but clear political dimension to the math wars. The NCTM Principles and Standards states "All students should have the opportunity and support necessary to learn significant mathematics with depth and understanding. There is no conflict between equity and excellence." I see no evidence that the traditionalists agree with this, and my conversations with traditionalists indicate that most believe that mathematics teachers need to put forth a challenging curriculum, and essentially serve those students that are able to rise to the challenge. They also contend that the reform agenda has put equity far ahead of excellence.

My belief is that traditionalism and reformism are not as diametrically opposed as they seem, and that the future will see a convergence in these movements. I think that most reformers now see that students must spend a substantial amount of time on rote learning. For example, the number of high-school graduates who struggle to make change without electronic assistance is disturbing. Without the instant recall of basic number facts, such as the single-digit multiplication table, students are severely handicapped in trying to solve more complex problems. And I think that most traditionalists see value in the goals of the reform movement. I think we can both do a better job of education the top 20% of students who we need as a technological elite and the bottom 80% who will have to find jobs that are increasingly more quantitative and will also need to be informed citizens in a world that depends more and more on numerical analyses.

Those who care about mathematics education need to move beyond the math wars and work together to improve the quality of the teachers and schools that provide this education.

Starting a Tutoring Service

2008-09-29T06:42:00.000-07:00

I am starting a mathematics tutoring service for high school students. I plan to take students in the Massachusetts towns near my home such as Bedford, Lexington, Concord, Lincoln, Carlisle, and Arlington. I will be specializing in geometry and calculus, and can also tutor college students. I can help students with their homework problems, and also help them to succeed in high-stakes tests such as SAT, ACT, Advanced Placement, and MCAS.

For a flyer describing what I offer, see http://www.scribd.com/doc/6297124/Tutoring-Flyer.

I offer a free evaluation. After that, my fee is $100 per hour.

Beliefs about Teaching

2008-09-21T19:33:00.000-07:00

As someone who has taught university mathematics and also taught mathematics education, I've noticed the huge disconnect between views of teaching held by research mathematicians and by K-12 teachers (and the mathematics education professors who teach the teachers.) Below I list some beliefs that I have found to be widespread in the university community (in italics) followed by contradictory beliefs that are widespread in the mathematics education world (in bold). I think the best thing that could happen to mathematics education in this country would be to open up a dialog between these two groups, since each has information and skills that are critical to improving mathematics education.

Any good research mathematician who is interested in teaching can do a better job teaching mathematics than most public school teachers, at least from grades 5 up. The research mathematician can present mathematics as an exciting intellectual endeavor, and the teacher cannot.

Virtually no one can successfully teach K-12 who does not understand the basic techniques of teaching, including classroom management and performance objectives. A K-12 teacher who understands teaching but whose knowledge of mathematics is limited to a basic understanding of the mathematics to be taught can be an excellent mathematics teacher.

Good teaching is a matter of laying out the material in a clear and elegant manner, and answering student questions when needed.

Good teaching requires the instructor to develop a list of performance objectives for every class, and let the students know what these objectives are. The objectives must be specific and testable: for example, "The student should be able to solve a quadratic equation in standard form with real roots, where the coefficients are integers of absolute value less than 20, within 30 seconds, using the quadratic formula. The lesson is not successfully completed until all students can meet the objectives.

"Teaching to the test" is bad. Every test should contain at least some problems that are non-routine and require the student to synthesize knowledge. These problems are the only way to ensure that students have truly learned the material. A student who cannot solve problems that are somewhat different from what they have seen before does not deserve an A.

Tests should determine whether the student has met the performance objectives. In other words, teaching to the test is the essence of good educational practice.

There are two acceptable ways of assigning grades in a course. The first is to "grade on the curve", that is to use grade cutoff points that make the distribution of letter grades follow a normal distribution, with approximately the same number of F grades as A grades, the same number of D grades as B grades, and C grades the most common. This is the safest way to grade. The other way is to build the grade cutoffs into the test based on the professor's subjective opinion. For example, one says that a student must score 90% on a given test to deserve an A. This method may result in unacceptably high levels of failure.

There is no scientific basis to support the idea that student grades ought to form a normal distribution. In fact, one popular theory states that any student who works at it ought to be able to meet the course objectives and get an A; the smarter students will simply reach that point sooner than the not-so-smart.

I could go on, but I hope the reader gets the point. I've tried to present actual ideas that I've heard expressed. To some extent these dichotomies represent a clash of values, and so may be impossible to resolve completely. But I still think they are ideas that we must talk about if mathematics education is to improve.

E8. Comparing triangles.

2008-09-14T12:04:00.000-07:00

I saw the following problem on the Internet a few years ago.

Let T be a triangle with side lengths a, b, and c. Let T' be a triangle with side lengths a', b', and c'. Suppose a < a', b < b', and c < c'. Must it follow that Area(T) < Area(T')?

The answer is quite simple, but surprising to most people. It makes a good question to put to a beginning geometry class.

If you need to, you can find the answer here.

E7. Non-standard dice

2008-09-01T07:11:00.001-07:00

A standard pair of dice consists of two identical cubes, each with the integers from 1 to 6 occurring once each. When the dice are thrown, the total on the faces can be any integer from 2 to 12; where the frequency of occurrence are 1 for 2 or 12, 2 for 3 or 11, and so forth up to a frequency of 6 for the total 7. A non-standard pair of dice has a positive integer on each face, the totals on the faces can be any integer from 2 to 12, and the frequencies of occurrence are the same as on a standard dice, yet the numbering is not identical to a standard pair of dice. Show that a non-standard pair of dice exists, and it is unique.

I originally saw this problem in an old Martin Gardner Scientific American column, and I posted it to a Problem of the Month column that I used to run on the Cambridge College Web site. Since that column no longer exists, I thought I would reprint it here.

I really like the problem because:
(1) The result is quite surprising.
(2) It can be solved by an average middle-school student, requiring only some logic and persistence.
(3) There is an extremely neat advanced solution.

For the elementary solution, one way to start is to determine the largest number that can occur on any face.

The advanced solution was suggested to me by the probabilist and friend John Lamperti, and uses generating functions. To see that solution, click here.

A Surprising Probability Result

2008-08-30T08:26:00.001-07:00

At dinner last night, several of us were discussing the Chinese one-child per family policy, when Sandy Blank posed the following question.

Suppose the probability that a child is male is exactly 1/2, and that each couple continues to have children until a male is produced, and then stops. What fraction of the new generation will be male?

Upon hearing this question, most everyone will guess that the number of girls will be considerably greater than the number of boys.

I reasoned that each completed family will have one boy, and that I could compute the expected value of the number of girls by summing n x p(n) for all positive integers n, where p(n) is the probability that the couple will have n consecutive girls before having a boy. See here for the details of the computation, which are relatively simple.

I was shocked to find that the expected value of the number of girls was also one, so the new generation will be 1/2 male.

My sister Arlene, a statistician, was well aware of this problem, and she presented an incredibly simple solution. Since each birth has a probability of 1/2 of being male, the new generation will be approximately 1/2 male. It doesn't matter when families decide to stop having children.

I think that (with either solution) this is a neat problem. It might also make a good bar bet.

E6. Lattice triangles and tetrahedrons

2008-08-17T11:21:00.000-07:00

In two dimensions, a lattice polygon is a polygon in a Cartesian coordinate plane such that the two coordinates of each vertex are integers. In three dimensions, a lattice polyhedron is a polyhedron such that the three coordinates of each vertex are integers.

(a) Prove that a lattice triangle cannot be an equilateral triangle.

(b) Is it possible for a lattice tetrahedron to be a regular tetrahedron?

Click here to see the solution.

John Donne and Mathematics

2008-08-15T16:12:00.000-07:00

I recently found myself thinking about the play Wit by Margaret Edson. I saw a production a couple of years ago, and was extremely moved by it. The play is about a professor of literature who is in the hospital dying of ovarian cancer. The play has been made into a movie, which I haven't seen.

The literature professor is an expert in the poetry of John Donne, and a major motif in the play is a teacher's insistence on the correct punctuation in one of Donne's sonnets. She complains about an edition in which a semicolon has been replaced by a comma. I suppose this could have been an excuse for a put-down of pedantry, but on the contrary the playwright made me believe that the correct punctuation was important, even vital.

In the same way, most students must regard the distinctions that mathematicians make as mere pedantry. Why make a big deal over the difference between rational and irrational numbers? According to the calculator, sqrt(2) = 1.414213562, and if you use that value for any practical application it won't matter that it is not exact. But it does matter. I wish I had the skill of Ms. Edson to make my students understand that this is not an unimportant detail, but rather is the heart of mathematics itself.

You Haul 19 Pounds

2008-08-14T06:34:00.000-07:00

(Title with apologies to Merle Travis.)
A couple of days ago I requested an examination copy of Single Variable Calculus by John Ragowski from W. H. Freeman. Today a 19-pound package arrived at my door, containing 5 books: The book I requested in hardback plus Volume II of the paperback version, plus both paperback volumes of the Early Transcendentals version, plus a 1425-page Instructor's Solution Manual (Early Transcendentals). In addition, there was an Instructor Resources CD, and a nice canvas bag with the publisher logo and the slogan "No Teacher Left Behind". To top it all off, the fulfillment service mistakenly slipped in a packet of signage for a Bruegger's Bagels franchise. (I wonder if Bruegger's Bagels got a packet of calculus materials, and if so what they made of it.)
While I appreciate W. H. Freeman sending this so me so promptly, I doubt that all this is necessary. One book would have been enough for me to make an adoption decision. Sending out all these books seems to be a very non-sustainable practice. Even if I adopt the book, I have at least 3 books that I will never use. I have to ask how much this practice contributes to deforestation, the burning of fossil fuels, and the high price of textbooks.
So, are the books any good? I don't know yet, but it looks pretty much like a dozen other calculus textbooks.

Lockhart's Lament

2008-08-13T07:04:00.000-07:00

Paul Lockhart, a research mathematician and K-12 math teacher, has written a scathing critique of the way that mathematics is taught in our schools. His point of view is that mathematics is an art and needs to be taught as something that is as inherently enjoyable as music or painting, rather than as a subject that must be endured so that the students can pass their exams and the country can become more competitive. Unfortunately, most teachers have never done any real mathematics and have never learned to appreciate mathematics as an art.

For a copy of this paper, go to Keith Devlin's MAA Column
where you can read a short appreciation of Lockhart and link directly to the paper. It is impassioned, funny, and as as over-the-top as a good polemic should be.

I suggest this paper to my mathematics education students at Cambridge College just to shake things up a bit.

For one book in the spirit of Lockhart's ideas, see Trimathalon: A Workout Beyond the School Curriculum by Judith and Paul Sally.

Why do students have such trouble with quantifers?

2008-08-11T10:41:00.000-07:00

During a course I taught this summer in Non-Euclidean Geometry for middle school and high school teachers I used Joel Castellanos' excellent NonEuclid program. I assigned as homework a few problems from the list of activities that accompanies the program. Only one of the 13 students in the class answered the following question correctly:

In Euclidean geometry, any polygon can be completely enclosed in some sufficiently large triangle. This is so obvious a statement that I have never even seen it written as a theorem. In, hyperbolic geometry, this is not an obvious statement. Is it a true statement?

.

The correct answer is that the statement is not true. Counterexamples are easy to come by. For example, consider a regular hexagon, whose center coincides with the center of the Poincare Disk. If the vertices of the hexagon lie on a sufficiently large circle, as in Figure 1, a little experimentation should convince the student that it will be impossible to enclose the hexagon in a triangle. A simple proof (not required for the homework) is based on the fact that with proper normalization the area of a hyperbolic triangle is equal to the defect = (pi - sum of the angles). Thus, no triangle can have an area greater than pi. However, the hexagon can be decomposed into six triangles, each of which has defect = 2*pi/3 - eps, where eps > 0 can be made as small as desired by increasing the radius of the circle. Thus the area of the hexagon = 4*pi - 6*eps can be made significantly larger than pi. Since the part cannot be greater than the whole, no triangle can enclose such a hexagon.

Twelve of 13 students showed, in effect, that given a triangle, they could find a regular hexagon inside the triangle. One student even wrote that her initial attempt didn't work because her hexagon was too big, so she had to use a smaller hexagon. Clearly, her problem was in the interpretation of the question. I am convinced that that was the problem of the other students as well, since almost all of them had previously constructed "large" regular hexagons. See Figure 2 for a typical student production.

Figure 1:

LargeRegularHexagon - Upload a Document to Scribd

Figure 2:

LargeTriangleWithHexagon - Upload a Document to Scribd

This can be viewed as a problem with understanding the difference between existential and universal quantifiers that bedevils college students, as anyone who has taught beginning calculus knows. I think that there is a psychological component as well. Most students originally go into mathematics because they are good at following directions. For example, they are asked to multiply two polynomials, and they are rewarded when they can do so. The idea of discovering that something is impossible rubs the wrong way. It is satisfying to be able to create a regular triangle inside a given triangle. It is disturbing to have to conclude that there is a hexagon which cannot be enclosed in any triangle.

If our teachers think that solving a problem in mathematics consists of following some procedure to produce a positive result, how are students going to view mathematics as a search for truth, whether the result be positive or negative?

Update on Quadrilateral Paper

2008-08-11T10:37:00.000-07:00

Great news! Our paper, "Constructing a Quadrilateral Inside Another One" was accepted without further changes by The Mathematical Gazette, and should appear in the November 2009 Issue. This is the same version that I have posted on scribd.

NES/MAA Meeting

2008-06-04T06:00:00.000-07:00

Last weekend I attended the annual meeting of the Northeastern Section of the Mathematical Association of America (NES/MAA) held this year at St. Michael's College, near Burlington Vermont.

While on-campus housing was available, Leslie and I chose to stay at a Days Inn across the street from the campus, which worked fine, except for their WiFi, which unusably slow. Weekend weather was a mix of rain, clouds, and sun. The locals said that the rain was needed, and it was easy to put up with. The meeting was Friday afternoon through early Saturday afternoon, and we stayed on through Sunday morning, spending Saturday night doing the tourist thing in Burlington. Burlington is a beautiful small city, with lots of nightlife, particularly since there was a Jazz Fest going on. We had a tasty meal outdoors at the Irish Pub. Just after we ordered a downpour began, our waiter came out and reduced the height of our cafe umbrella, and we had a great time eating as the temperature cooled down and rain poured down inches from us.

I gave a 15-minute talk on my paper (described elsewhere on this blog), "Constructing a Quadrilateral Inside Another One". The talk was one of seven "Contributed Papers", which were scheduled in three rooms between 8 AM and 9 AM on Saturday morning. My talk was the first one, and not surprisingly the audience was small, but the talk was well received. My presentation used both PowerPoint and Geometer's Sketchpad and (since the local PC did not have Sketchpad) I had hook up my laptop. Fortunately, everything worked fine.

The talks were quite interesting, and the organizing topic seemed to be mathematical modeling in biology and environmental science. Even though Leslie is very much a non-mathematician, she is quite interested in the application areas, and she attended and enjoyed a couple of the talks. The most interesting talks for me were those by George Pinder, Christopher Danforth, and Charles Hadlock.

George Pinder of the University Vermont described a simulation of alcohol-assisted bioremediation of superfund sites.

Chis Danforth, also of UVM, gave the after-dinner Battles lecture entitled "Chaos and the Mathematics of Prediction: Hurricane Katrina, Harry Potter, and Happiness." The reference to Hurricane Katrina had to do with the difficulty of predicting weather, the Harry Potter reference is about the difficulty of predicting which children's book out of hundreds published annually might be the next blockbuster hit, and Happiness refers to trying to determine the emotional well-being of large populations over time by an analysis of the numbers of positive and negative words published online or in song lyrics.

Charles Hadlock of Bentley College spoke on Agent-Based Modeling in Teaching and Research. Agents are individuals who are part of a large population and whose behavior is directed (either probabilistically or deterministically) by nearby individuals. Think of Conway's Game of Life. Agent-based models are easily programmed, making them a good choice for student learning, and resulting animations, shown by Charles, have a mysterious beauty that appears very much like natural phenomena.

I found Mohammed Salmassi's short talk on using Spherical Easel for geometry education to be the most immediately useful, since he convinced me to use that software as part of my course on non-Euclidean geometry that I am teaching this summer.

Abstract versus concrete in mathematics education

2008-05-13T17:16:00.000-07:00

In the New York Times, April 25 2008 Kenneth Chang wrote an article reporting on research done by Jennifer Kaminski, Vladimir Sloutsky, and Andrew Heckler at Ohio State University and reported in Science. I've quoted the core of that article below.

“The motivation behind this research was to examine a very widespread belief about the teaching of mathematics, namely that teaching students multiple concrete examples will benefit learning,” said Jennifer A. Kaminski, a research scientist at the Center for Cognitive Science at Ohio State. “It was really just that, a belief.”
…
Though the experiment tested college students, the researchers suggested that their findings might also be true for math education in elementary through high school, the subject of decades of debates about the best teaching methods.
In the experiment, the college students learned a simple but unfamiliar mathematical system, essentially a set of rules. Some learned the system through purely abstract symbols, and others learned it through concrete examples like combining liquids in measuring cups and tennis balls in a container.
Then the students were tested on a different situation — what they were told was a children’s game — that used the same math. “We told students you can use the knowledge you just acquired to figure out these rules of the game,” Dr. Kaminski said.
The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing. Students who were presented the abstract symbols after the concrete examples did better than those who learned only through cups or balls, but not as well as those who learned only the abstract symbols.
The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems.

I find the results of this study partly obvious and partly misleading. It is very obvious that when trying to teach a concept, giving examples with too many extraneous details will confuse the student. Imagine trying to devise checkers strategy when each of your checkers was a different shape. It would be very hard to avoid coming up with strategies that had nothing to do with the rules of the game, like: "a square checker should always jump over a triangular checker". Some degree of abstraction is necessary in order to allow the transfer of knowledge from one domain to another.

The study was done with undergraduate college students. I find it hard to believe that the researchers will replicate their results with children in grades K – 12, given what is known about the differences in learning with age. I also question the usefulness to practice of studies that report the statistical distribution of learning results from two teaching methods (say an "abstract" method and a "concrete" method). It is well established that different students have different learning styles, and that these styles may be determined, either by testing or informally by trained teachers. A teaching method that works well for a student with a strong verbal-procedural learning style may work poorly for one who has a visual-kinesthetic learning style. If one style is predominant in a population and the abstract style works better for that style, the results will be better on average, but it might be a great mistake to teach all children using that method, in effect writing off the students with the minority style.

The system the researchers used is isomorphic to the cyclic group with 3 elements, or addition modulo 3. There was a concrete model where the elements were measuring cups filled 1/3 full, 2/3 full, and 3/3 full, which I found the easiest to understand. The abstract symbolic model where the elements were circles, diamonds, and an irregular figure seemed much more mysterious and indeed Richard Weiss pointed out to me that it is possible to construct two non-isomorphic addition tables that fit the abstract model. See the New York Times article for more detail.

One reason given by proponents of multiple concrete representations is motivation, related to the idea that students must see mathematics as relevant to their lives before they will invest effort in learning it. I've always felt this is dubious, because in most textbooks or classrooms the supposedly real-life scenarios seem cooked up. I feel there are two ways to enhance motivation through curriculum design: (1) Go all out. Get students interested in some activity that excites them and requires real mathematics, such as the design of computer games, building a robot or a racecar, etc. The problem with this approach is that it fails to address the testing mania which grips our country. Students who learn mathematics this way are going to acquire skills in a non-standard order, and will not know some topics that they need to pass high-stakes tests. (2) Teach mathematics through problem solving. Eliminate, as much as possible, phony word problems. Instead tap children's innate curiosity and competitiveness with questions like: "Can 2467432 be a perfect square (no calculators allowed)?"

The King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts, Walker & Co, 2006

2008-04-01T16:46:00.000-07:00

This biography of one of the greatest geometers of the twentieth century was a mixed bag for me. Donald Coxeter was a fascinating and brilliant man, and a list of his correspondents and admirers is a veritable who's who of twentieth century mathematicians, artists, and scientists. Donald was long-lived, even for a mathematician. He lived from 1907 – 2003, and was mathematically creative until nearly the end. A man of this rich creativity presents a formidable challenge to a biographer; how to sketch the trajectory of his life while neither getting bogged down in details on the one hand, or being superficial on the other.
A search on Amazon reveals that this book is the first published by Ms. Roberts, and the book definitely has some rough edges. I feel she needed more editorial guidance than she received. Ms. Roberts, who is not a mathematician, has apparently interviewed a number of Coxeter's colleagues at great length, and she has let them do the talking. As a result, I feel the absence of a strong author's voice, and a resulting fragmented picture of Coxeter. I found the book hard to plow through in places, which really surprised me given the strong interest that I have in Coxeter's work.
The flip side of this is that it is wonderful to read accounts from such luminaries as John Conway, Walter Whitely, Freeman Dyson, M. C. Escher (through his son, George), Buckminster Fuller and Douglas Hofstadter (who wrote the forward).
I found the glimpses of Coxeter's life outside of mathematics to be quite fascinating. He was a pacifist and a vegetarian, and seems to have been highly regarded as a person by most who knew him. His early life included problems with a withdrawn mother and an overbearing father, and a brief psychoanalysis by Freud's student, Stekel. I would have liked to have been told more about Coxeter the man.
Perhaps the best compliment I can pay the book is that I came away from it wishing that I had known Coxeter, and determined to read more of his work.

The Poincaré Conjecture: In Search of the Shape of the Universe, Donal O’Shea, Walker & Company, 2007

2008-03-05T07:50:00.000-08:00

I recommend this book highly. It explains the history, context, and importance of the Poincaré conjecture, as well as many of the attempts to solve the problem, culminating in the successful solution by Grigory Perelman. The writing style is lively, and the explanations seem to be about as clear as possible, given the complexity of the mathematics, the amount of material covered, and the relatively short length of the book (200 pages + 75 pages of footnotes and appendices). The book is semi-popular; while the target audience is probably people with an undergraduate degree in mathematics or the equivalent, there is enough historical and personal narrative to appeal, I would think, to the non-mathematical reader. The book ends with a page-long appreciation of the civilizations and people who erected an edifice of thought that culminated in the statement and proof of this conjecture, which makes a major contribution to the understanding of the universe in which we live.

O’Shea shows how the history Poincaré conjecture is the history of much great mathematics, extending from ancient to modern times, and including such intellectual giants as Euclid, Gauss, Riemann, and Poincaré, and such stellar contemporary mathematicians as Milnor, Smale, Freedman, Donaldson, Thurston, Yau, and Hamilton.

For me, a major side benefit of reading this book was its recommendation of Jeffery Weeks’ book, The Shape of Space, which explains many of the big ideas of modern topology and geometry while demanding little mathematical background, only a willingness to think hard and work at visualization.

My Philosophy of Mathematics

2008-02-12T07:58:00.000-08:00

Inspired by Where Mathematics Comes From by Lakoff and Nunez, reviewed below, I am setting forth my own philosophy of mathematics. It is probably not original; in fact it seems like common sense. It also seems a bit like Kant’s philosophy of ontology, if I understand Kant. Before presenting my own ideas, I’ll summarize the major philosophies of mathematics prevalent today.

Lakoff and Nunez discuss the three major philosophies of mathematics prevalent today and put forth their own, called the philosophy of embodied mathematics. To me, these philosophies are rather like the proverbial blind men trying to describe an elephant. The man touching the trunk says an elephant is like a fire hose. The man touching the tail says the elephant is like a rope. The one touching a leg says an elephant is like a tree. Each blind man correctly describes a part of the whole.

The philosophy of formalism states that mathematics is the production of valid formulas that follow from a small number of axioms. A proponent, Bertrand Russell, defined mathematics as “the subject where we never know what we are talking about, nor whether what we are saying is true.” For formalists, mathematics exists independent of meaning. This seems preposterous on the face of it, and I doubt that many working mathematicians hold it to be true. However, it contains a grain of truth in that I (or any mathematician that I can think of) would be disinclined to accept as true something that could not, in principle, be given a formal proof. That Godel has shown that there are propositions about arithmetic that cannot be proven true or false does not invalidate this.

The Platonic philosophy of mathematics, or Platonism, states that mathematics is concerned with the discovery of truths about a realm of abstract mathematical ideas, and that this realm has an objective reality outside of any human minds. This is the philosophy that most working mathematicians intuitively have. The deeper one enters into a mathematical subject matter, the more it seems to have an objective reality. However, this view has been pretty well torn apart by the philosophers. Many of the arguments against Platonism are well summarized in Where Mathematics Comes From and elsewhere, and the gist of them seems to be the impossibility of a physical being knowing a non-physical reality. My philosophy of mathematics, given below, contains a modified form of Platonism, which I think meets the objections.

The post-modernist explanation of mathematics states that mathematics, like all systems of ideas, is purely a product of the culture in which it arises. In its extreme form, this philosophy is not only wrong, it is pernicious. If there is no objective reality and truth is what society wishes it to be, then we are in the world of Orwell’s 1984, where Winston Smith’s reeducation by the state is complete when he is willing to believe that two plus two is five. Extreme postmodernism is the philosophy of totalitarianism, and it seems to me that a professor who believes this theory is in the wrong line of business. That said, a moderate form of postmodernism does illuminate mathematical thought. Mathematicians, like everyone else, are products of their culture. The areas of mathematics that are deemed important and the methods of proof that are accepted are determined, in large part, by culture.

The philosophy of embodied mathematics has its own problems, and also supplies valid insights. This philosophy states that mathematics does not have an objective reality but is not totally culturally determined either. Instead, “Mathematics is the product of human beings. It uses the very limited and constrained resources of human biology and is shaped by the nature of our brains, our bodies, or conceptual systems, and the concerns of human societies and cultures.” Lakoff and Nunez take special aim at Platonism, which is what they see at the center of an elitist “romance of mathematics”.

I think that mathematics is more than a product of human beings. It can only be understood as a blend of internal and external reality. Let me offer an analogy. Imagine we are in a forest, and I (pointing) say “this is a tree” and “that is a tree”. Most people would agree that the trees are actually there, but I am saying much more. I am saying that both objects I am pointing to are instances of the same kind of entity, namely “tree”. So the concept of tree is rooted both in the external entities (which are apparent to any sentient beings, not just humans) and internal concepts and abilities, including language and the idea of a category of “tree”. In the same way, mathematics is implicit in the regularities of the universe but must be made explicit by human thought. It is both a feature of the universe and a product of our minds.

There is no way to prove this, but it would be hard to imagine that an alien that was intelligent in a way that would enable it to talk with us would not believe that 2 + 2 = 4. On the other hand, imagine that this alien, like Saint-Exupery’s Little Prince, lived on a very small spherical planet. Such an alien might have no idea of Euclidean geometry, because that would be irrelevant to its environment. The ratio of the circumference to the diameter of a circle (C/d) would not be pi, but rather it would be variable. A circle drawn around the equator would have C/d = 2, and progressively smaller concentric circles in the northern hemisphere would have ratios closer and closer to pi. Whether the alien would regard pi as an important number or not is unclear. Given the ubiquity of pi in our mathematics, it is hard to believe that it would not appear in alien mathematics as well. Our ideas about mathematics are not just about us, and not just about the external world, they are about the complex and interpenetrating interaction between the two.

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, George Lakoff and Rafael E. Nunez, Basic Books, 2000.

2008-02-11T18:49:00.000-08:00

Recently a student of mine – a middle school teacher – asserted that he did not believe that 0.999… = 1. I tried several methods of convincing him that he was wrong; for example calculating 1 = 3(1/3) = 3(0.333…) = 0.999…, and showing that 0.999 … < 1 led to a contradiction. I browbeat him into submission, but I don’t think I convinced him. After reading the book under review, I see I could have handled things better. I learned that in non-standard analysis, 0.999 … < 1 is true, and more importantly I learned that understanding even the simplest infinite processes involves mastering some tricky metaphors.

This is an important and flawed book. It has generated much commentary, pro and con, and if your curiosity is whetted by my review I suggest you check the reviews of the book in Amazon.com. Like The Number Sense by Stanislaus Dehaene (reviewed below), this book applies cognitive science to an analysis of mathematical thought. But unlike that book, Where Mathematics Comes From goes far beyond an analysis of arithmetic skill, and analyzes some very sophisticated concepts of higher mathematics. Also, compared with Dehaene’s book, this book does not depend very much on laboratory science, but depends almost entirely on theoretical cognitive science.

This is a long book, and a somewhat difficult read, unless you happen to be well versed in the jargon of both mathematics and cognitive science. Even though the authors cover a lot of territory in 450 pages, and the quality of the writing is generally good, I had the feeling that I was reading the same thing over and over. If you want an “executive summary” to get the gist of what the authors believe they have accomplished, I suggest looking at the section “A Portrait of Mathematics” on pages 377 – 379.

In the first four chapters the authors describe the brain’s innate arithmetic, which is quite rudimentary and similar to the innate arithmetic of many other species, and then details how humans have learned to extend these basic concepts to an arithmetic which enables efficient calculation and obeys certain “laws”, such as the commutative property for addition. This section sets the tone for the rest of the book, by introducing basic cognitive mechanisms that the authors believe explain how mathematics is invented and understood. These include grounding metaphors, which yield basic, directly grounded ideas, and linking metaphors, which yield abstract ideas.

Chapters 5 – 7 study the linking metaphors that determine algebra, logic, and sets, and chapters 8 – 11 deal with “The Embodiment of Infinity”. The concept of infinity underlies most of modern mathematics including various number systems (integers, rationals, real numbers, and complex numbers). It appears in many different guises from points at infinity in projective geometry to cardinal and ordinal infinities in Cantor’s theory. The authors introduce a “basic metaphor of infinity” (BMI) that is supposed to account for our understanding of all these concepts. Chapter 11, “Infinitesimals”, is perhaps the most mathematically interesting part of the book. The authors present the hyperreal numbers of Robinson and Keisler, which include infinitesimal and huge quantities and provide be an intuitive and direct way of dealing with calculus. They also introduce a system of their own invention, the granular numbers, which is a subset of the hyperreal numbers that seems to be easier to use.

Chapters 12 – 14 continue the discussion of the infinite by critiquing the program of modern analysis that was pioneered by Dedekind and Weierstrass and continues to this day. While the authors profess the highest admiration for the intellectual achievements of these men and their followers, the subtext here is that their (Dedekind’s and Weierstrass’) purpose in separating analysis from its roots based in an intuitive geometric understanding was to obfuscate the subject and make mathematics the preserve of a specially trained elite. I disagree. My understanding is that the development of technology was beginning to make the traditional conceptions of mathematics inadequate. For example, on page 307 the authors approvingly cite James Pierpont’s (1899) list of “prototypical properties of a curve” including that it is continuous and has a tangent. With this definition, it would be difficult to have a consistent theory of Fourier series necessary for the analysis of radio waves or to solve differential equation with a driving function given by a step function in electrical engineering. The Mandelbrot set and related constructions are rooted in the real world (Mandelbrot’s prototypical example is the coastline of England.) and these “monstrous” sets seem to describe nature better than classical curves and regions.

Chapters 15 and 16 contrast the author’s implications for a philosophy of mathematics that is grounded on the human mind –“embodied mathematics”– with other philosophies that either posit mathematics existing outside the real world (Platonism and “the romance of mathematics”) or see mathematics the manipulation of essentially meaningless strings according to given rules (formalism) or as a cultural construct. I will describe my own view in a separate posting.

The last section of the book is a “case study” in four parts shows that the methods developed in the book can be used successfully to teach mathematics in a way that focuses on meaning. The subject is Euler’s famous equation e^(i*pi) = -1. This is quite good, though I might do a few things differently

E5. Duck, Duck, Goose

2008-02-08T06:51:00.000-08:00

This problem was sent to me (as is) by Walter Carter of Seattle.

Some children have made up a simple version of the game “Duck, Duck, Goose”. In this game a group of people stands in a circle, and the person who is “it” taps the first person on the shoulder and says “duck”. The next person is tapped and called “duck” The next person is tapped and called “goose”, and the process is repeated. Every person who is called “goose” must sit down when they are tapped.

If there are a million people in a circle, and they are labeled sequentially from 1 to 1,000,000, and the tapper starts at person 1 going around and around until only one person is left standing, then what is that last person’s number?

Review of the Number Sense

2008-02-01T19:39:00.000-08:00

The Number Sense: How the Mind Creates Mathematics by Stanislaus Dehaene, Oxford University Press 1997
This book, written by a noted neuropsychologist, explores the new field of mathematical cognition. That is, it attempts to root our understanding of the development of mathematics in the biology of the brain. It is one of those rare books written by a pioneering researcher in a scientific field who is also an excellent writer – in English as well as presumably in his native French. I think it is particularly valuable for those of us in education, because in order to teach mathematics we must understand how children actually acquire mathematics. While there is much to learn here, I also found much to disagree with, and I will deal with these points below. Perhaps the major drawback to the book may be its date of publication, since Dehaene indicates that the ten years following the writing of the book promise to be a time of unparalleled scientific advance in the field.
The book is organized into nine chapters:
Chapter 1, “Talented and Gifted Animals”, discusses scientific research that shows that many animals have innate primitive arithmetic skills, which enable them to add, subtract, and compare small integers. Calculations and comparisons of numbers become less accurate as the numbers involved increase beyond three.
Chapter 2, “Babies Who Count”, sets forth the contention, supported by ingenious research, that shows that, similar to animals, human babies as young as a few days old also have innate arithmetic skills, enabling them to understand and manipulate small integers.
Chapter 3, “The Adult Number Line”, discusses the conception that human adults have of number. Much of this chapter has to do with discovering the extent to which we can manipulate numbers very quickly, that is, without visible thought.
Chapter 4, “The Language of Number”, discusses the ways different cultures name numbers, and the effect this has on calculating abilities.
Chapter 5, “Small Heads for Big Calculations”, applies the results covered in the previous chapters to the difficulties of teaching arithmetic to children.
Chapter 6, “Geniuses and Prodigies”, presents case studies of a number calculating prodigies and mathematical geniuses, and attempts to show that their abilities are not different in kind from that available to any intelligent adult.
Chapter 7, “Losing Number Sense”, discusses the relationship between brain function and number sense as revealed by studying people who have lost various parts of their number sense due to lesions in particular parts of their brains, or to other brain injury.
Chapter 8, “The Computing Brain”, shows how modern advances in brain research cast light on relationship between calculation and the brain. The tools of positron emission tomography (PET) and electro- and magnetoencephalograpy are described, and some results obtained by applying these tools to mathematical cognition are discussed.
Chapter 9, “What Is a Number?” moves into the philosophy of mathematics. Dehaene tackles questions such as the merits of the formalist, Platonist, and intuitionism theories of mathematics, and the relationship between mathematical truth and reality.
A book of this wide coverage is bound to be controversial. I recommend reading it yourself and making up your mind about some of the controversial issues, but I’d like to bring up a few places where I disagree with the author.
It seems to me that one of the dangers of neuropsychology is that of reductionism, and although Dehaene is a sophisticated thinker I don’t think he escapes this.
I take issue with his apparent assumption, which seems unsupported by data, that ability to perform arithmetic calculations is strongly correlated with the ability to do higher mathematics. Among mathematicians I have known, some excel at arithmetic, some are poor, and many are in between. The type of thinking that is involved in geometry, for example, seems to have little to do with arithmetic ability.
I find particularly problematic his discussion of mathematical geniuses, for several reasons. First, he lumps together the self-taught Indian mathematical genius Ramanujan with autistic super-calculators and idiot savants. To me, this is as if one compared Shakespeare with a pre-typewriter clerk who filled thousands of pages of commercial transactions. Both men may have had unusual ability to produce fast legible handwriting, but we would only call one a genius. Second, Dehaene makes clear that he believes that anyone could be a super-achiever in mathematics or arithmetic if they devoted enough time and effort to the enterprise; that there is nothing special about the brain (or mode of thinking) of the genius. This is speculation, and I prefer the opposite speculation of Oliver Sachs, whose prime-number generating autistic twins seem not to calculate but rather to see the integers “directly, as a vast natural scene” or Ramanujan, who described his own mathematical discoveries as being handed to him by a Hindu god while he slept. Non-believers can imagine that Ramanujan’s unconscious mind allowed him to make his discoveries operating in a way that might be totally different from his conscious mind.
Another oversimplification is Dehaene’s belief that young Oriental students do better than Western students at learning mathematics because the Eastern languages have shorter more user-friendly names for the digits. He seems to not consider the cultural differences that lead Oriental families to value hard academic work more than Occidental families do, which by itself is enough to explain differences in achievement.
In terms of pedagogic implications, Dehaene’s research has led him to the belief that the human brain is not well designed for calculation: “Ultimately, [innumeracy] reflects the human brain’s struggle for storing arithmetical knowledge”. He therefore feels that “by releasing children from the tedious and mechanical constraints of calculation, the calculator can help them to concentrate on meaning.” This is a position I have long shared; however I am now teaching middle-school mathematics teachers, and they mostly report that their students, who have grown up using calculators, are grossly innumerate. Since many algorithms of elementary algebra have counterparts in arithmetic algorithms, these students are not able to progress in algebra. I now advocate getting children to a state of competence in calculation before letting them use the calculator freely. However, I agree with Dehaene on the usefulness of concrete computational representations (manipulatives) in the classroom.
Dehaene gives a good description of the basic theories of mathematical epistemology: Platonism (mathematical objects have a reality, and the mathematician discovers this reality rather than inventing it), formalism (mathematics is about the formal manipulation of strings of symbols following basic laws of logic), and intuitionism (mathematics is a construction of the human mind, so that alien intelligences would create different a mathematics different from the human one.) He comes down for intuitionism, but it seems to me that his dismissal of Platonism is entirely too glib. He asks, rhetorically, “If these [mathematical] objects are real but immaterial, in what extrasensory way does a mathematician perceive them?” I would argue that they are perceived in the same way that we perceive a coherent world from the streams of sense data that enter our brains. We create our mental worlds, and this seems to be true whether or not the basis of the world is “material” or whether is grounded in ideas. Both the material and mathematical mental worlds are subject to laws of internal consistency, and both are subject to judgment by members of a community.

Physical Models for Non-Euclidean Geometry

2008-01-29T08:57:00.000-08:00

I strongly believe in the use of physical models, whenever possible, to introduce mathematical concepts. For example, when teaching non-Euclidean geometry to high school teachers, I like to have them create triangles on actual physical spheres, using rubber balls, push pins, and rubber bands to create geodesics. It is easy to “discover” that the sum of the angles in spherical triangles is greater than 180 degrees, and that the excess of a triangle (the sum of the angles minus 180 degrees) is additive and hence proportional to the area of the triangle. These demonstrations can be easily done for specialized triangles, so the student becomes familiar with the geometric fact before thinking about how it might be proven.

What I would like to do next is to show that the sum of the angles in a triangle on a surface of negative curvature is less than 180 degrees. This leaves me with two problems.

(1) How can I make (or obtain) a physical model of a simple saddle surface for experimentation by students. Ideally, models should be cheap enough so that I can supply each pair of students with a model to work with.

(2) How can students draw geodesics on such a surface? Rubber bands are not going to work here, because a band stretched between two points on the surface will not necessarily lie on the surface. This problem is sort of mathematical, because I think a good understanding of the nature of geodesics should lead to discovering a way of having students create them on a surface of negative curvature.

I am aware of some very good software that uses the Poincare disk model to do geometry on the hyperbolic plane, and I plan to use the software when I teach. But I want students to have real physical experience first.

Anyone have any ideas?

Visual Calculus

2008-01-08T13:53:00.000-08:00

I found a fascinating page: VisualCalc. This is a talk by Tom Apostol about Visual Calculus, a technique for finding the area bounded by curves without using traditional calculus developed by an Armenian mathematician living in California, Mamikon A. Mnatsakanian, which has been espoused by Apostol. Some of the results using this method would be very difficult if not impossible to uncover with traditional methods. The starting point is the following simple (but neat) problem, solved by Mamikon (as he calls himself) when he was 15:

Problem: A line segment is drawn tangent to the inner of two concentric circles, terminating at the outer circle. The length of the segment is 2a. What is the area of the annulus?

Answer: pi*a^2. It is rather counterintuitive that the result is independent of the radius of the inner circle.

Solution: Let the radius of the smaller and larger circles by r and R, respectively. The area of the annulus is pi*(R^2 - r^2). Draw the obvious right triangle with legs of length r and a, and hypotenuse of length R. Apply the Pythagorean Theorem.

Mamikon noted that if he knew in advance that the answer was independent of r, he could let r = 0, and the tangent segment would become a diameter of the larger circle, establishing the result another way. This led him to a rather breathtaking extension of the result.

Theorem 1. Let C be a smooth convex oval. Move a vector v (of fixed length) around the oval (with the tail on the curve) so that it is always tangent to the curve (at its tail). Then the area swept out by the vector is pi*|v|^2. [I'm not sure what the exact hypothesis is, but this is the basic idea.]

Proof idea: Let S be the set of translates of the vectors v(t), with a common tail formed as v(t) goes around the oval. Then S is a circle of radius |v|.

Tri-Color chessboards

2008-01-07T07:47:00.000-08:00

When coloring a checkerboard, the basic requirement is that squares that are full-neighbors (horizontally or vertically) have different colors. Clearly, there are exactly two ways of coloring an n x n checkerboard with two colors (black and red, say). Once a color has been selected for the lower left corner, all remaining square colors are forced. I wondered how many different ways one could color an n x n checkerboard with three colors. This led me to consider two problems:

(1) How many ways are there to color an n x n checkerboard, using at most 3 colors?

(2) How many ways are there to color a 3k x 3k checkerboard, using equal numbers of red, blue, and white squares?

Bridget Tenner of dePaul University immediately came up with the answer to problem (1) by searching in Neal Sloane's wonderful Online Encyclopedia of Integer Sequences, using "3-color" as a search string. The answer is given here as a special case of A078099 (for m x n checkerboards), which is defined recursively. The sequence grows very quickly: it is 3 times a sequence beginning 1, 6, 82, 2604, 193662, 33865632, 13956665236.

A sequence to answer question (2) does not seem to appear in OEIS, so this may be an open question.

Three Books on Riemann Hypothesis

2008-01-01T19:44:00.000-08:00

My first review will be of three semi-popular books about the Riemann hypothesis: Prime Obsession by John Derbyshire, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics by Marcus Du Sautoy, and The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics by Karl Sabbagh.

Recently there has been a spate of semi-popular books about the Riemann Hypothesis. This is doubtless due in part to the fact that several of the most famous problems of modern mathematics such as the Four-Color Map Theorem, Fermat's Last Theorem, and the Poincare Conjecture have now been solved, leaving the Riemann Hypothesis as the most famous problem standing. However, writing a semi-popular book about the Riemann Hypothesis is an intimidating mission. Unlike the Four-Color Map Theorem and Fermat's Last Theorem, it is difficult to explain to an educated layperson what the theorem states, or why it is important. Even the statement of Poincare Conjecture is easier to comprehend.

John Derbyshire's Book, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, is the best of the lot. He sets himself the daunting task of explaining virtually all of the major mathematical ideas needed to understand the statement of the RH, its relation to the distribution of prime numbers, and some of the major methods that have been used to attack the problem, in a book designed for an otherwise educated person who is ignorant of mathematics from high school algebra on. It sounds to me that this goal must have been imposed the publishers, because whatever the talents of the expositor, it is prima facie impossible to bring anyone but a latent mathematical genius on such a trip in the confines of a single 422-page book. However, what Derbyshire does, and does brilliantly, is to explain the RH to someone who has understood two years of college calculus, or the equivalent. The reader who has experience with integrals and infinite series should be able to follow the exposition.

The Reimann Hypothesis and its relation to the distribution of primes belongs to the branch of mathematics called analytic number theory. This subject is not easy to write about. I took a reading course in analytic number theory in graduate school. I was intrigued by the subject, but became discouraged when I found the text, by a famous researcher in the field (who shall remain nameless) riddled with errors. I ended up going into another specialty. Now that I have read Derbyshire's book, I'm tempted to read more. In addition to the mathematical exposition, Derbyshire quickly and deftly sketches the political and social milieu and the personalities involved in the development of the RH and the search for its solution.

I bring up one quibble because it relates to the first chapter, and might cause a reader to give up. Derbyshire introduces the harmonic series (and its divergence) by asking the reader to imagine constructing a bridge out of playing cards. It turns out that the n-th card from the top of this bridge can extend 1/(n – 1) card length from the card above it, so that the span of the entire n-card bridge is 1 + 1/2 + … + 1/n. I've seen this before, and it is cute, but it is not easy. There are easier ways to introduce the harmonic series.

Marcus du Sautoy's book, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, covers much of the same territory as Derbyshire's book, but goes into somewhat less mathematical detail. Du Sautoy is a professor of mathematics at Oxford, and an excellent writer. I recommend this book for the poetry of the language and the vividness of the stories of the mathematicians involved in the story. It is wonderful to read a book by a first-rate mathematician who is also a first-rate storyteller.

In Karl Sabbagh's book, The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics, the author, like John Derbyshire, attempts to explain the RH to the mathematically unsophisticated reader. In this case, the mathematical basics are covered in a series of appendices, called "Toolkits". This book was something of a disappointment. First, the author comes across as a journalist rather than as an advanced amateur mathematician (like Derbyshire) or a professional mathematician (like du Sautoy). The writing has more of a superficial feel to it, where more tends to be made of the physical appearance or personal idiosyncrasies of mathematicians rather than their ideas. In addition, Sabbagh spends much of the book conversing with and about Louis de Branges, who has claimed to have a proof of the Riemann Hypothesis. It is true that de Branges is a respected mathematician who solved an important long-standing problem, the Bieberbach Conjecture. However, very few mathematicians credit his claims to have, or be close to, a proof of the Riemann Hypothesis. Sabbagh is obviously charmed by de Branges, and spends, in my opinion, far too much time on this player who seems to deserve at most a short footnote in the story.

E4. Very Proper Fractions

2007-10-08T13:34:00.000-07:00

A proper fraction is one where the numerator is no larger than the denominator. We'll call a fraction with denominator 1 a VPF (very proper fraction). Imagine a civilization where the only way of representing fractions is as a VPF or a sum of (two or more) VPFs. To keep things interesting, they don't allow using the same denominator twice. So, for example, they can't write 2/5 as 1/5 + 1/5. However, 2/5 CAN be represented as a sum of distinct VPFs: 1/3 + 1/15. Since the denominator of these fractions are all 1, we can simplify the notation by simply writing the numerators. For example, we've shown that we can convert from our system to their system by 2/5 = (3,15).

Problem: Write 1143/1170 as it would be expressed in this civilization.

Extra challenge: Describe an algorithm for expressing any proper fraction as a sum of distinct VPFs, and prove that your algorithm works. (In particular, show that it terminates in a finite number of steps.) This challenge is an advanced problem.

E3. A Fraction Sequence

2007-10-08T12:38:00.000-07:00

Let S be the sequence of all proper fractions with denominator ≤ 20, arranged in order from smallest to largest, so the first fraction is 0/1 = 0 and the last one is 1/1 = 1. Represent each fraction in lowest terms. Define the gap between two consecutive fractions in this sequence to be the difference of the smaller from the larger.

Determine the smallest gap in the sequence, and find two fractions that have this gap between them.

The gap between the first two fractions is 1/20 – 0/1 = 1/20, as is the gap between the last two fractions. Not counting these fractions, find two fractions that have a gap that is as large as possible.

Personal Thoughts on Math Research

2007-09-21T06:22:00.000-07:00

I publish math research once in a blue moon, and always with co-authors. I've just finished some new research. There are a few things I like about it. (1) The result was surprising, and the work was challenging, but not too difficult. (2) It is a very concrete result in elementary geometry, that I can explain to most anyone. I think the result is nifty. (3) It was a family affair, as my co-authors are my brother Marshall and his son, Michael. It was a nice division of labor, with me doing the geometry part and Marshall and Michael doing the calculus part. Michael's TeX expertise came in handy, as well. (4) It is the first paper that I have contributed to that depended on mathematical software for its solution.

See http://arxiv.org/abs/0704.2716 for the paper, titled "Constructing a quadrilateral inside another one". Be sure to look at version 3, which is much improved over the earlier versions. It's only 9 pages, and a pretty easy read, as math papers go. We are at work on a new version, which should improve the exposition and simplify the proof of the main theorem a bit, but I think the current version is actually publishable quality. We'll be submitting the new version to The Mathematical Gazette.

This paper is an example of old-fashioned mathematics done with modern tools. There is nothing here that couldn't have been done in the 18th century, but without Geometer's Sketchpad I would never have come across the problem or have been able to obtain experimental evidence for our conjecture, now a theorem. And without Maple (or the reincarnation of Euler) we couldn't have done the calculations involved.

Technical production notes: Marshall wrote up the result using Scientific Workplace. Since I don't have a copy, I sent him my corrections as Word documents (using the MathType math editor), and produced wmf format graphics with Sketchpad. It would have really simplified our collaboration if we had the same software. What are other people using for collaborations?

E2. A Puzzling Fraction Representation

2007-08-22T15:42:00.000-07:00

Show that any positive rational number can be written in the form

where a, b, n, and k are non-negative integers and (nines) represents a string of one or more 9s.

A1. Relative and Absolute Extrema

2007-08-22T14:54:00.000-07:00

In finding minima and maxima, first-year calculus students often use the fact that, for a function differentiable on the entire real line, if the function has exactly one relative extremum, that extremum is an absolute extremum. The proof is simple: WLOG, say the function has a relative maximum at a. If a is not an absolute maximum, then there is a point b where f(b) > f(a). Then on the closed interval with endpoints a and b, f has a minimum value. Since the minimum is less than f(a), it is not attained at either a or b, so it is in the open interval, and thus is a relative minimum.

Does the above result generalize to R²? In particular, say f is differentiable on the entire plane, and has a relative maximum at (0,0), and no other relative extrema. Must the function have an absolute maximum at (0,0)? Why or why not?

E1. A Chemist's Question

2007-08-20T13:01:00.000-07:00

Years ago, a friend who was a Ph.D. chemist asked me the following question. I thought the question was very easy. If you find it too easy yourself, you might pose the question to a child of the appropriate age.

The chemist knew that certain fractions, such as 1/4, can be expressed as a terminating decimal, while other fractions, such as 1/3, can only be expressed as a non-terminating, repeating decimal. The chemist wanted to know how to tell whether a given fraction can be expressed as a terminating decimal.