Showing posts with label mathematics education. Show all posts
Showing posts with label mathematics education. Show all posts
Hippasus & The Discovery of Irrational Numbers
In November I gave a speculative talk to the New England Section of the Mathematical Association of America on the discovery of irrational numbers by the Pythagorean Hippasus through an examination of the mystic pentagram, the sacred symbol of the Pythagoreans. I have expanded it a bit, adding a method of recursively computing the golden ratio, phi. If you want to see how these topics are related, please check my paper on Scribd: https://www.scribd.com/doc/294006886/Irrational-Numbers-the-Mystic-Pentagram-and-Eigenvectors.
Multiplication & Division of Polynomials
When I was in high school I learned to multiply polynomials by using a technique that was reminiscent of the multiplication of multi-digit numbers that I had learned in earlier grades. That is, after filling in any "missing terms" with zeros the two polynomials were written one over the other, with the terms arranged by degree, with the constant term at the right. If the polynomials were of degrees m and n, with m <= n, the polynomial of degree m was written on the bottom. We would multiply the top polynomial by each of the m terms of the bottom polynomial, to produce m rows, each of which was indented one position to the right. Then all terms of the same degree were located above one another, and we would add each column to get the answer. Similarly, the algorithm for polynomial long division was based on the algorithm for multi-digit integer division. Polynomial division never really clicked for me. I learned how to do it, but somehow it felt like magic.
These days, most elementary and middle school students - even many students from better schools - are not being taught multiplication and division of multi-digit numbers, instead being told they can rely on the calculator app on their smartphone. This has caused much hand-wringing, particularly among traditionalists. How can we teach students to multiply and divide polynomials when they haven't learned to multiply and divide integers? Non-traditionalists might say that students can use CAS to multiply and divide polynomials and don't need to be able to do it by hand, but I am not willing to go that far.
In the course of tutoring a high school student, I recently was exposed to a method for multiplying polynomials that is being taught currently in some schools, called the galley method . I really like it, for several reasons:
(1) The method does not not depend on knowing the analogous arithmetical algorithm.
(2) The method is directly related to the definition of the multiplication of m and n as the area of an m x n rectangle (or as the cardinality of the Cartesian product of sets of cardinalities m and n, if you prefer). These definitions are, in the opinion of many educators, superior to regarding multiplication as repeated addition.
(3) The first time I used the galley method, I found it slightly easier to use than the traditional method. I would use it myself in cases where a CAS is not necessary or available.
(4) The galley method immediately leads to the reverse galley method for the division of polynomials.
Here's how it works: To multiply polynomials of degree m and n, create an (m+1) by (n+1) table, the galley. (The term galley apparently comes a French word that indicates an oblong tray for holding setup type.) Write the terms of the degree n polynomial above the n+1 columns, in order of decreasing degree as you move from left to right. Write the terms of the other polynomial along the right side of the galley, in order of decreasing degree. Then, fill in each cell with the product of the monomials above its column and to the right of its row. Finally, observe that the terms on any diagonal (of slope 1) have the same degree, and the degrees corresponding to different diagonals are different. Add along these diagonals to combine like terms, and write the result in the first empty space outside the galley. The result is now in front of you, in an L-shaped pattern along the left and bottom sides of the galley. If you find this difficult to visualize, you can find an excellent description with diagrams here.
The galley method can be used in reverse for polynomial division. Suppose you want to divide a polynomial of degree m by a polynomial of degree n, where n < m. Create a galley of n+1 rows and m-n+1 columns. Write the n+1 terms of the divisor along the right side of the galley and the m+1 terms of the dividend in an L-shaped band beginning in the area one unit to the left and one unit below the upper left-hand corner of the galley. Then fill in the terms in the remainder of the galley as well as the terms above the galley one by one, beginning with row 1 column 1 of the galley, then row 0 column 1 (just above row 1 column 1), then the rest of column 1. Next move to row 1 column 2, and so forth. The solution will appear above the galley. If there is a remainder, some of the last n-1 terms in the divisor (the L-shaped band) will not in fact be the sum of the terms on their diagonals, and the remainder is the difference between the sum of the terms on the diagonals and the corresponding terms in the divisor. Again, if you need further description, go here.
These days, most elementary and middle school students - even many students from better schools - are not being taught multiplication and division of multi-digit numbers, instead being told they can rely on the calculator app on their smartphone. This has caused much hand-wringing, particularly among traditionalists. How can we teach students to multiply and divide polynomials when they haven't learned to multiply and divide integers? Non-traditionalists might say that students can use CAS to multiply and divide polynomials and don't need to be able to do it by hand, but I am not willing to go that far.
In the course of tutoring a high school student, I recently was exposed to a method for multiplying polynomials that is being taught currently in some schools, called the galley method . I really like it, for several reasons:
(1) The method does not not depend on knowing the analogous arithmetical algorithm.
(2) The method is directly related to the definition of the multiplication of m and n as the area of an m x n rectangle (or as the cardinality of the Cartesian product of sets of cardinalities m and n, if you prefer). These definitions are, in the opinion of many educators, superior to regarding multiplication as repeated addition.
(3) The first time I used the galley method, I found it slightly easier to use than the traditional method. I would use it myself in cases where a CAS is not necessary or available.
(4) The galley method immediately leads to the reverse galley method for the division of polynomials.
Here's how it works: To multiply polynomials of degree m and n, create an (m+1) by (n+1) table, the galley. (The term galley apparently comes a French word that indicates an oblong tray for holding setup type.) Write the terms of the degree n polynomial above the n+1 columns, in order of decreasing degree as you move from left to right. Write the terms of the other polynomial along the right side of the galley, in order of decreasing degree. Then, fill in each cell with the product of the monomials above its column and to the right of its row. Finally, observe that the terms on any diagonal (of slope 1) have the same degree, and the degrees corresponding to different diagonals are different. Add along these diagonals to combine like terms, and write the result in the first empty space outside the galley. The result is now in front of you, in an L-shaped pattern along the left and bottom sides of the galley. If you find this difficult to visualize, you can find an excellent description with diagrams here.
The galley method can be used in reverse for polynomial division. Suppose you want to divide a polynomial of degree m by a polynomial of degree n, where n < m. Create a galley of n+1 rows and m-n+1 columns. Write the n+1 terms of the divisor along the right side of the galley and the m+1 terms of the dividend in an L-shaped band beginning in the area one unit to the left and one unit below the upper left-hand corner of the galley. Then fill in the terms in the remainder of the galley as well as the terms above the galley one by one, beginning with row 1 column 1 of the galley, then row 0 column 1 (just above row 1 column 1), then the rest of column 1. Next move to row 1 column 2, and so forth. The solution will appear above the galley. If there is a remainder, some of the last n-1 terms in the divisor (the L-shaped band) will not in fact be the sum of the terms on their diagonals, and the remainder is the difference between the sum of the terms on the diagonals and the corresponding terms in the divisor. Again, if you need further description, go here.
Don't Be a Math Teacher
I have enjoyed teaching math for several decades and
currently teach K–12
math teachers about education and mathematics. So you'd think that I would
recommend that a mathematically-inclined person should give strong
consideration to becoming a math teacher. You'd be wrong.
If when I began my career the state of the mathematics
teaching profession, both at the school and university levels, was what it is
today, I don't think I would have gone into it. Math teaching now has the
following difficulties.
First, digital technologies have resulted in students with
shorter attention spans and more dependence on instant gratification. You would
be surprised how many high school students are unable to add or multiply two
two-digit numbers without a calculator, and in many cases unable to add or
multiply even one-digit numbers. This matters, in much the same way that it
would matter if students couldn't read words of more than one or two syllables,
and depended on print-to-voice readers to “read” literature. Also, most students faced with a problem
that requires more than five minutes of work, most will say that they can't do
it. A Korean student that I had in a Calculus course in an elite U. S. woman's
college earned an A+ told me privately that she felt she was not good at
mathematics but was best in the class because the rest of the girls were lazy.
I think we will only see more and more students who are unwilling to do the
hard work required to become good at math.
Second,
in the United States, virtually all K–12 students must pass tests in order to
advance in grade or to graduate. Teachers must teach to the test. While
teaching to the test is, in my opinion, a good thing for formative assessments,
teaching to the test for these high-stakes one-size-fits-all tests at best
limits the ability of the teacher to approach the curriculum in innovative ways
and at worst results in much time wasted with students taking practice tests
and learning test-taking tricks that have nothing to do with mathematics. And
of course, if the students do not do well on these tests, the teacher's job is
on the line.
Third,
computers are in the process of making math teaching obsolete. If a student can
learn a subject online, where they can have their instruction completely
personalized and can have their homework graded instantly, why should a school
system invest in expensive and inefficient human beings to do the teaching?
Most teachers will be replaced by low-paid “facilitators” of online courses who
do not need to even be trained in mathematics. Instead of there being thousands
of college instructors nationwide teaching Calculus I, a dozen or so master
teachers will make videos and consult with courseware designers to put together
online courses. Math teaching, as we know it, will exist only at the graduate
school level where economies of scale do not apply, or at pricey private
schools.
So my career
advice is only to become a math teacher if it is really a calling for you. Good
teaching will be harder than it has ever been, with fewer rewards and little in
the way of job security.
Open Letter to Harper's
Harper's Magazine published a long essay by Nicholson Baker in the current (September 2013) issue, entitled: "Wrong Answer: the Case against Algebra II". Baker is taking up the same anti-algebra banner that was waved by Andrew Hacker in last year's NY Times. Baker rightly points out many problems with the current math curriculum in the schools, but I think he makes many errors of logic and interpretation.
Baker starts off by establishing the difficulty of algebra by quoting a solution of Cardano to the partial cubic equation from 1545, stating that "the algebraic rules that Cardano described and codified are variants of the techniques that students are taught, with varying degrees of success, today". It is as if I argued that the reading of English is too difficult for students to learn by giving a quotation from Chaucer, whose language is a "variant" of what students are taught today. In point of fact, (1) the notation that we use today has been refined over the last 5 centuries or so, precisely to make the techniques of algebra as easy as possible, (2) Cardano did not "codify" algebra in any reasonable meaning of the term, and (3) the solution of the cubic equation in radicals was no longer being taught in Algebra II when I took it 50 years ago and is, practically speaking, a fairly useless algorithm today.
It is no secret that large numbers of students dislike high school mathematics. Baker seems to feel that this proves that high school mathematics is too difficult for most children to master. It never seems to have crossed his mind that perhaps it is the way that algebra is taught is the problem, not the fact that it is taught at all. In fact, advances in understanding how people learn make it possible to teach high school mathematics to anyone except the most learning disabled. The problem is that these methods are not being used because teaching mathematics well is not an easy task. It is one that we entrust to teachers who are overworked, underpaid, and inadequately trained. In the elementary grades, where the foundations need to be set that will allow the child to master algebra, teachers often themselves have a misunderstanding and dislike of mathematics, which is all too easily passed on to children.
All this might not be so bad if we subscribe to Baker's theory that most students only need a one-year survey course of mathematics in high school: they need to learn "courtesy and kindness, the times tables, fractions, percentages, ratios, reading, writing, some history -- the rest is gravy, really". He cites the fact that American technology ruled the world in the 1950s when only a fourth of students took high school algebra, and seems to think this proves that, as a country, it is fine if most students don't learn algebra. I find it impossible to follow this logic. Haven't a few things changed since the 1950s that might make American technological supremacy less sure and teaching our children mathematics a tad more important?
One of the things that has changed is that we now at least pay lip service to the ideal of equity in education. The idea that it is fine that 3/4 of students (mostly female and people of color) will never be able to enter technical fields that require college mathematics is no longer acceptable to most of us. When algebra, geometry, and trigonometry were not required subjects, the lower classes went to the lower classes where they learned a little shop math or "general math". I don't believe we want to go back to those days. I don't think that students should be allowed to have their future position in society cemented in ninth grade because they could opt out of a subject they might not like. As a nation, we need to teach mathematics better and make it more relevant, not make it optional. Anything less is a cop-out.
Baker starts off by establishing the difficulty of algebra by quoting a solution of Cardano to the partial cubic equation from 1545, stating that "the algebraic rules that Cardano described and codified are variants of the techniques that students are taught, with varying degrees of success, today". It is as if I argued that the reading of English is too difficult for students to learn by giving a quotation from Chaucer, whose language is a "variant" of what students are taught today. In point of fact, (1) the notation that we use today has been refined over the last 5 centuries or so, precisely to make the techniques of algebra as easy as possible, (2) Cardano did not "codify" algebra in any reasonable meaning of the term, and (3) the solution of the cubic equation in radicals was no longer being taught in Algebra II when I took it 50 years ago and is, practically speaking, a fairly useless algorithm today.
It is no secret that large numbers of students dislike high school mathematics. Baker seems to feel that this proves that high school mathematics is too difficult for most children to master. It never seems to have crossed his mind that perhaps it is the way that algebra is taught is the problem, not the fact that it is taught at all. In fact, advances in understanding how people learn make it possible to teach high school mathematics to anyone except the most learning disabled. The problem is that these methods are not being used because teaching mathematics well is not an easy task. It is one that we entrust to teachers who are overworked, underpaid, and inadequately trained. In the elementary grades, where the foundations need to be set that will allow the child to master algebra, teachers often themselves have a misunderstanding and dislike of mathematics, which is all too easily passed on to children.
All this might not be so bad if we subscribe to Baker's theory that most students only need a one-year survey course of mathematics in high school: they need to learn "courtesy and kindness, the times tables, fractions, percentages, ratios, reading, writing, some history -- the rest is gravy, really". He cites the fact that American technology ruled the world in the 1950s when only a fourth of students took high school algebra, and seems to think this proves that, as a country, it is fine if most students don't learn algebra. I find it impossible to follow this logic. Haven't a few things changed since the 1950s that might make American technological supremacy less sure and teaching our children mathematics a tad more important?
One of the things that has changed is that we now at least pay lip service to the ideal of equity in education. The idea that it is fine that 3/4 of students (mostly female and people of color) will never be able to enter technical fields that require college mathematics is no longer acceptable to most of us. When algebra, geometry, and trigonometry were not required subjects, the lower classes went to the lower classes where they learned a little shop math or "general math". I don't believe we want to go back to those days. I don't think that students should be allowed to have their future position in society cemented in ninth grade because they could opt out of a subject they might not like. As a nation, we need to teach mathematics better and make it more relevant, not make it optional. Anything less is a cop-out.
Sketchpad - Jacobs problems now available
I have finished writing 23 pages of problems for Jacobs' high school geometry book that use GSP. It is posted on Scribd as a pdf.
http://www.scribd.com/doc/146440260/Extra-Problems-for-Jacobs-Using-GSP
http://www.scribd.com/doc/146440260/Extra-Problems-for-Jacobs-Using-GSP
My Jacobs - GSP Project
I am using the textbook Geometry: Seeing, Doing, Understanding (Third Edition) by Harold R. Jacobs in my course for prospective middle school math teachers. I love many things about this book, but I also am a strong believer in having my students use Geometer's Sketchpad (GSP) software. Jacobs has lots of hands-on work, but he made a decision not to use geometry software, so I have been supplementing his book with Sketchpad work. Mostly, I have taken a number of exercises from Jacobs and made very similar Sketchpad exercises out of them.
So far, I have written out seven pages of exercises, and I imagine I will end up with 20 - 30 pages. I key each of my exercise sets to section of Jacobs. I intend to class-test my exercises, and once they are complete to post them on Scribd. In the meantime, I am happy to send my work-in-progress to anyone who wants to send me an email for it. My address is peter.ash@MathForTheRestOfUs.com, and I appreciate feedback.
So far, I have written out seven pages of exercises, and I imagine I will end up with 20 - 30 pages. I key each of my exercise sets to section of Jacobs. I intend to class-test my exercises, and once they are complete to post them on Scribd. In the meantime, I am happy to send my work-in-progress to anyone who wants to send me an email for it. My address is peter.ash@MathForTheRestOfUs.com, and I appreciate feedback.
The Power of Cryptograms
When I was a boy, a relative bought me and my siblings a copy of a book about solving cryptograms, which described techniques for solving substitution cyphers and gave a lot of cryptograms of varying difficulty to play with. I really enjoyed the book. At the time, it seems like many newspapers would publish a daily cryptogram, though I don't see them very much any more. My local newspaper, the Boston Globe, publishes a daily crossword, a Sudoku, a Kenken, and a few other puzzles, but no cryptograms. I think that's a loss. The kind of thinking used to solve cryptograms is very similar to what is used in solving mathematical problems of all kinds, and cryptograms is something that can be enjoyed both by mathematically-oriented people and literature-oriented ones, since the quotations that are encrypted can be quite memorable. Also, an interest in simple cryptograms could lead to a long-term interest in cryptography, the importance of which in today's Internet-driven world can scarcely be overestimated.
I'd suggest anyone teaching math in the middle grades think about challenging their students with cryptograms. As a starting point, I found Simon Singh's web page at http://simonsingh.net/cryptography/cryptograms/ to be a nice introduction.
I'd suggest anyone teaching math in the middle grades think about challenging their students with cryptograms. As a starting point, I found Simon Singh's web page at http://simonsingh.net/cryptography/cryptograms/ to be a nice introduction.
E20. The 5-5-9-9 Problem
Introduction: A problem that has been well publicized on the Internet is the "Four Fours" problem. It asks for a representation of all integers from 1 to N (where N is as large as possible) using the digit 4 exactly four times, in addition to basic arithmetic symbols as needed. See, for example, http://www.wheels.org/math/44s.html. I have found this problem to be excellent for group work for students at the middle school or high school level, because it allows students of many different ability levels to participate. However, when I gave this problem to a group of teachers in an online course, one teacher responded by listing the answers available from an online source, verbatim, even though it was clear she did not even understand some of the symbols used in "her" solution. I thought this was a clear case of cheating, but the student felt that using the Internet was a legitimate way to solve a mathematics problem. Maybe she had a point. In any case, it became clear to me that it was time to modify the problem to make it one whose solutions would not be available on line. See the following.
Problem: Show how to represent each of the integers from 1 to 50 using the digit 5 exactly twice and the digit 9 exactly twice, in addition to basic mathematics symbols as needed. Allowed symbols are plus, minus, times, divides (including fraction bar), parentheses, square root, exponentiation, decimal point, factorial, floor function, and ceiling function.
Comments:
Problem: Show how to represent each of the integers from 1 to 50 using the digit 5 exactly twice and the digit 9 exactly twice, in addition to basic mathematics symbols as needed. Allowed symbols are plus, minus, times, divides (including fraction bar), parentheses, square root, exponentiation, decimal point, factorial, floor function, and ceiling function.
Comments:
- The numbers 5, 9, and 50 in the problem are all arbitrary, and the teacher is free to change them to customize the problem.
- The definition of "basic mathematics symbols" is crucial to the problem. For example, if you allow the successor function (++ suffix in the C programming language), the problem is trivial, and uninteresting. On the other hand, disallowing the decimal point, floor function, and ceiling function would make the problem much more difficult.
- There are some ambiguities that will come up that must be resolved, such as whether (9-5)(9-5) may be used to represent 44. A question like this would make for good class discussion.
- The following lemma makes the problem easier to solve: If a number can be represented using a subset of the 4 allowed digits, it can be represented using all 4 allowed digits. Proofs involve showing how to use any number of the allowed digits to produce either a factor with value 1, or a term with value 0. I think most students at the high school level could arrive at this lemma without being told about it. In any case, this lemma might make the idea of using a simplifying lemma, so important in mathematics, something students would remember.
Simple but remarkable math facts
I'm putting together a list of simple but remarkable math facts that could be used to spice up a math class, more-or-less at the level of high school math. These should be presentable in the form of questions that could be asked of a class, using some showmanship to draw students into them. I'll give a starter list of questions. Most are from geometry, a couple from probability. Happy to hear from anyone of similar questions.
(1) A length of railroad track is two miles long. To account for expansion in hot weather, the track has been joined in the middle. In hot weather the track expands, with each one-mile long piece expanding by one foot. It will form a shallow isosceles triangle, with base exactly two miles long, and sides 2 miles + 1 foot. How high will the track be at the middle? Less than 1 inch? More than one inch but less than one foot? More than one foot but less than 10 feet? More than 10 feet but less than 20 ft? More than 20 ft?
(2) Tennis balls are frequently sold in packages of 3, with the 3 balls packed into a cylinder. Which is greater, the height of the can or the circumference of the can? By how much? The answer to this question will be much more surprising if the teacher displays such a can. Most students will guess from looking that the height of the can is considerably greater than the circumference.
(3) This is a very common optical illusion. Draw two lines exactly the same length. At the end of one place arrowheads, and at the end of the other reverse arrowheads, something like this:
<--------------------> >--------------------<
Which line is shorter? By what percentage? While not strictly a math problem, it does show that we cannot always depend on the evidence of our eyes, and that in this case measurement with a ruler is more reliable than vision.
(4) Display two statues. The statues are similar (in the sense of geometry, same shape but different size). Tell the class that both statues are made of the steel (or gold?) that the smaller statue weights 10 pounds, and the larger statue is exactly twice the height of the other. How much would they guess it would weigh? You could tie this into the use of a scale model of the Titanic in James Cameron's movie. This is also related to the misuse of 3-d figures in statistical graphs, where typically oil consumption might be indicated by the size of an oil barrel, with the height of the barrel proportional to a country's consumption, greatly exaggerating the difference in consumption between different countries.
(5) A rope has been stretched around the Equator of the Earth (25,000 miles, approximately). How much longer does the rope need to be if it is to be raised two feet off the ground and remain a closed loop?
(6) How many people, picked at random, must be in a room before the probability that two or more people share the same birthday is greater than 1/2?
(7) The Monty Hall problem. You are a contestant in Monty's TV show. You are presented with three doors. There is a prize behind just one of them. You choose one door. Without opening that door, Monty opens one of the other two doors that does not hide the prize. (If your first pick was correct so both unpicked doors have no prize, Monty chooses randomly which of those to open.) He says you may stay with your original pick, or switch to the other unopened door. Should you switch or not? How would your choice affect the probability of your winning?
(1) A length of railroad track is two miles long. To account for expansion in hot weather, the track has been joined in the middle. In hot weather the track expands, with each one-mile long piece expanding by one foot. It will form a shallow isosceles triangle, with base exactly two miles long, and sides 2 miles + 1 foot. How high will the track be at the middle? Less than 1 inch? More than one inch but less than one foot? More than one foot but less than 10 feet? More than 10 feet but less than 20 ft? More than 20 ft?
(2) Tennis balls are frequently sold in packages of 3, with the 3 balls packed into a cylinder. Which is greater, the height of the can or the circumference of the can? By how much? The answer to this question will be much more surprising if the teacher displays such a can. Most students will guess from looking that the height of the can is considerably greater than the circumference.
(3) This is a very common optical illusion. Draw two lines exactly the same length. At the end of one place arrowheads, and at the end of the other reverse arrowheads, something like this:
<--------------------> >--------------------<
Which line is shorter? By what percentage? While not strictly a math problem, it does show that we cannot always depend on the evidence of our eyes, and that in this case measurement with a ruler is more reliable than vision.
(4) Display two statues. The statues are similar (in the sense of geometry, same shape but different size). Tell the class that both statues are made of the steel (or gold?) that the smaller statue weights 10 pounds, and the larger statue is exactly twice the height of the other. How much would they guess it would weigh? You could tie this into the use of a scale model of the Titanic in James Cameron's movie. This is also related to the misuse of 3-d figures in statistical graphs, where typically oil consumption might be indicated by the size of an oil barrel, with the height of the barrel proportional to a country's consumption, greatly exaggerating the difference in consumption between different countries.
(5) A rope has been stretched around the Equator of the Earth (25,000 miles, approximately). How much longer does the rope need to be if it is to be raised two feet off the ground and remain a closed loop?
(6) How many people, picked at random, must be in a room before the probability that two or more people share the same birthday is greater than 1/2?
(7) The Monty Hall problem. You are a contestant in Monty's TV show. You are presented with three doors. There is a prize behind just one of them. You choose one door. Without opening that door, Monty opens one of the other two doors that does not hide the prize. (If your first pick was correct so both unpicked doors have no prize, Monty chooses randomly which of those to open.) He says you may stay with your original pick, or switch to the other unopened door. Should you switch or not? How would your choice affect the probability of your winning?
Changes in Mathematics Research Methods
The following math problem appeared in a math forum that I
follow:
You have a deck of 56 cards, labeled 1 through 56. All the
cards are drawn randomly one at a time. What is the probability that for
exactly one card, the face value of the card is one less than the face value of
the card before it.
The reason for considering the number 56 was not specified,
but that doesn't matter because it is clear that to solve the problem in any
kind of satisfying way we have to solve it in a more general case, where 56 is
replaced by n. In this case, the probability is P(n)/n!, where P(n) is the
number of permutations p of (1, … ,n) for which p(k – 1) – p(k) = 1 for exactly
one k. The problem reduces to finding P(n).
The answer is rather neat, and (SPOILER ALERT) I give it below.
However what I find most interesting is the way my finding the solution to this
problem depended so much on technology. To solve it I did not need to know much
mathematics at all. If I had had to solve the problem 30 years ago, it would
have taken much more knowledge and/or much more time.
The first thing I did was to generate some numerical
evidence. By writing out the acceptable permutations and counting them, I found
that P(2) = 1, P(3) = 2, P(4) = 9, and P(5) = 43. I sent the problem to a
friend of mine, Ken Cutter, who is a MATLAB guru, and he wrote a short program
in MATLAB, using perms(1 … n) to list all permutations and the diff function to
compute the differences p(k) – p(k-1). He discovered that I had missed one
acceptable permutation of (1 … 5) so that actually P(5) = 44. He also gave me
the list of a few more values of P, including P(6) = 265.
I now knew I was looking for a sequence 1 , 2, 9, 44, 265, …
, so I went to the Online Encyclopedia of Integer Sequences (OEIS, https://oeis.org/)
and entered 1 , 2, 9, 44, 265 into the search field, and out came two known
sequences. Only the first of these two sequences seemed to relate to
permutations, and sure enough, one the descriptions of this sequence identified
it as what I was looking for.
The answer turns out to be that P(n) number of derangements
of P(n), that is, the number of permutations with no fixed points. The formula
for P(n) is given on OEIS as P(n) = n!*Sum((-1)^k/k!, k=0..n). On the Wikipedia page for derangements,
it is noted that a common notation for P(n) is !n, so the desired probability
is !n/n! = Sum((-1)^k/k!,
k=0..n) which, as
Wikipedia notes, converges rapidly to 1/e. In fact, it is the first n terms of
the Taylor expansion for exp(x) about 0, evaluated at x = -1. For the original problem with n = 56, the
answer for all practical purposes is 1/e.
Looking
back, the most crucial tool in my finding the solution was The Online
Encyclopedia of Integer Sequences, started by Neil J. A. Sloane, and a
wonderful resource for anyone involved in mathematical work. MATLAB was also
very helpful. Without it, I might have still gotten the answer if I had gone
over my work and discovered my mistake which made me get P(5) off by one. If I
had entered the correct first 4 terms, Online Encyclopedia of Integer Sequences
would have returned 17 sequences, but it would still have been pretty easy to
find the correct sequence from amongst these 17. But MATLAB (and Ken Cutter) saved
a lot of time. Lastly, Wikipedia was a help in explaining the result.
When I was
in school, none of these resources were available. If I knew more about
combinatorial mathematics, I might have known the answer immediately, or at
least recognized it once I generated the numerical examples. Otherwise, I would
have had to look at my lists of acceptable permutations more carefully and perhaps
derived a recursion relation or induction step that would enable me to find the
solution. I would be interested in hearing from anyone would like to send me a
solution to this problem from scratch, imagining that they do not know the
answer already.
This
experiences brings home to me how much the methods of mathematics research have
changed since I was in school. I wonder what changes we should be making in
K-12 math education that take into account these changes.
Reed Magazine Profile
My undergraduate Alma Mater, Reed College, just published a profile of me focusing on Math for the Rest of Us at http://www.reed.edu/reed_magazine/march2012/articles/alumni_profiles/ash.html. I feel particularly honored because not only do I have great fondness for Reed College, but Reed Magazine is one publication I make a point of reading. The writing is superior to most college alumni magazines I have seen, and the articles describe varied, idiosyncratic work by fascinating people in many intellectual areas.
The Birthday Problem
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)/31N
= 1 - 31! /[(31 - N)! * 31N]
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)/31N
= 1 - 31! /[(31 - N)! * 31N]
Mathematics and Humor
"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.
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.
Harold Jacobs' Geometry
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.
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.
Curriculum for Overcoming Math Anxiety Course
(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.
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.
Twenty Incredible Math Talks
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
Thanks, Florine
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
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.
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:
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.
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.
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.
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