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