Math Tutoring Service

See my Mathematics Tutoring Service on Thumbtack

Twenty Incredible Math Talks

Florine Church of sent me a link to I had known that 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

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

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

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?

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

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

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

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

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

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:
  1. Learning takes place with a balance of concrete and symbolic, guided and independent, and procedural and conceptual.
  2. 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.
  3. Whole-class lessons allow students to experience the thrill of discovery collectively.
  4. Teachers give frequent and specific encouragement to all students.
  5. 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.
  6. 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

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

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

Comments on the paper are welcome.

Coded arithmetic puzzles

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

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

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


where (maxk) 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

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

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, 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

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

Given a multiset of real numbers {a1, ...,an} find expressions e1, ...,en such that {a1, ...,an} = {e1, ...,en}, and {ei} is a non-increasing sequence, where each expression is formed from the ai, 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

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.