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

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

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

### A2. A Trip Around Antarctica

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

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

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

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

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

### Math and Sex

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

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

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

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

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

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

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

### Opinion Piece on Math Education

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

### To Test or Not to Test

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

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

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

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

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

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

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

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

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

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

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

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

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

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

### The Math Wars

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

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

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

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

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

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

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

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

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

### Starting a Tutoring Service

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

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

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

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

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

### Beliefs about Teaching

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

### E8. Comparing triangles.

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

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

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

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

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

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

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

### E7. Non-standard dice

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

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

I really like the problem because:

(1) The result is quite surprising.

(2) It can be solved by an average middle-school student, requiring only some logic and persistence.

(3) There is an extremely neat advanced solution.

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

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

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

I really like the problem because:

(1) The result is quite surprising.

(2) It can be solved by an average middle-school student, requiring only some logic and persistence.

(3) There is an extremely neat advanced solution.

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

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

### A Surprising Probability Result

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

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

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

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

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

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

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

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

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

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

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

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

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

### E6. Lattice triangles and tetrahedrons

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

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

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

Click here to see the solution.

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

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

Click here to see the solution.

### John Donne and Mathematics

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

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

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

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

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

### You Haul 19 Pounds

(Title with apologies to Merle Travis.)

A couple of days ago I requested an examination copy of Single Variable Calculus by John Ragowski from W. H. Freeman. Today a 19-pound package arrived at my door, containing 5 books: The book I requested in hardback plus Volume II of the paperback version, plus both paperback volumes of the Early Transcendentals version, plus a 1425-page Instructor's Solution Manual (Early Transcendentals). In addition, there was an Instructor Resources CD, and a nice canvas bag with the publisher logo and the slogan "No Teacher Left Behind". To top it all off, the fulfillment service mistakenly slipped in a packet of signage for a Bruegger's Bagels franchise. (I wonder if Bruegger's Bagels got a packet of calculus materials, and if so what they made of it.)

While I appreciate W. H. Freeman sending this so me so promptly, I doubt that all this is necessary. One book would have been enough for me to make an adoption decision. Sending out all these books seems to be a very non-sustainable practice. Even if I adopt the book, I have at least 3 books that I will never use. I have to ask how much this practice contributes to deforestation, the burning of fossil fuels, and the high price of textbooks.

So, are the books any good? I don't know yet, but it looks pretty much like a dozen other calculus textbooks.

A couple of days ago I requested an examination copy of Single Variable Calculus by John Ragowski from W. H. Freeman. Today a 19-pound package arrived at my door, containing 5 books: The book I requested in hardback plus Volume II of the paperback version, plus both paperback volumes of the Early Transcendentals version, plus a 1425-page Instructor's Solution Manual (Early Transcendentals). In addition, there was an Instructor Resources CD, and a nice canvas bag with the publisher logo and the slogan "No Teacher Left Behind". To top it all off, the fulfillment service mistakenly slipped in a packet of signage for a Bruegger's Bagels franchise. (I wonder if Bruegger's Bagels got a packet of calculus materials, and if so what they made of it.)

While I appreciate W. H. Freeman sending this so me so promptly, I doubt that all this is necessary. One book would have been enough for me to make an adoption decision. Sending out all these books seems to be a very non-sustainable practice. Even if I adopt the book, I have at least 3 books that I will never use. I have to ask how much this practice contributes to deforestation, the burning of fossil fuels, and the high price of textbooks.

So, are the books any good? I don't know yet, but it looks pretty much like a dozen other calculus textbooks.

### Lockhart's Lament

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

For a copy of this paper, go to Keith Devlin's MAA Column

where you can read a short appreciation of Lockhart and link directly to the paper. It is impassioned, funny, and as as over-the-top as a good polemic should be.

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

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

For a copy of this paper, go to Keith Devlin's MAA Column

where you can read a short appreciation of Lockhart and link directly to the paper. It is impassioned, funny, and as as over-the-top as a good polemic should be.

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

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

### Why do students have such trouble with quantifers?

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

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

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

Figure 1:

Figure 2:

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

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

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

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

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

Figure 1:

Read this document on Scribd: LargeRegularHexagon

Figure 2:

Read this document on Scribd: LargeTriangleWithHexagon

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

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

### Update on Quadrilateral Paper

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

### NES/MAA Meeting

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

### Abstract versus concrete in mathematics education

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

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

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

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

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

“The motivation behind this research was to examine a very widespread belief about the teaching of mathematics, namely that teaching students multiple concrete examples will benefit learning,” said Jennifer A. Kaminski, a research scientist at the Center for Cognitive Science at Ohio State. “It was really just that, a belief.”

…

Though the experiment tested college students, the researchers suggested that their findings might also be true for math education in elementary through high school, the subject of decades of debates about the best teaching methods.

In the experiment, the college students learned a simple but unfamiliar mathematical system, essentially a set of rules. Some learned the system through purely abstract symbols, and others learned it through concrete examples like combining liquids in measuring cups and tennis balls in a container.

Then the students were tested on a different situation — what they were told was a children’s game — that used the same math. “We told students you can use the knowledge you just acquired to figure out these rules of the game,” Dr. Kaminski said.

The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing. Students who were presented the abstract symbols after the concrete examples did better than those who learned only through cups or balls, but not as well as those who learned only the abstract symbols.

The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems.

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

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

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

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

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

This biography of one of the greatest geometers of the twentieth century was a mixed bag for me. Donald Coxeter was a fascinating and brilliant man, and a list of his correspondents and admirers is a veritable who's who of twentieth century mathematicians, artists, and scientists. Donald was long-lived, even for a mathematician. He lived from 1907 – 2003, and was mathematically creative until nearly the end. A man of this rich creativity presents a formidable challenge to a biographer; how to sketch the trajectory of his life while neither getting bogged down in details on the one hand, or being superficial on the other.

A search on Amazon reveals that this book is the first published by Ms. Roberts, and the book definitely has some rough edges. I feel she needed more editorial guidance than she received. Ms. Roberts, who is not a mathematician, has apparently interviewed a number of Coxeter's colleagues at great length, and she has let them do the talking. As a result, I feel the absence of a strong author's voice, and a resulting fragmented picture of Coxeter. I found the book hard to plow through in places, which really surprised me given the strong interest that I have in Coxeter's work.

The flip side of this is that it is wonderful to read accounts from such luminaries as John Conway, Walter Whitely, Freeman Dyson, M. C. Escher (through his son, George), Buckminster Fuller and Douglas Hofstadter (who wrote the forward).

I found the glimpses of Coxeter's life outside of mathematics to be quite fascinating. He was a pacifist and a vegetarian, and seems to have been highly regarded as a person by most who knew him. His early life included problems with a withdrawn mother and an overbearing father, and a brief psychoanalysis by Freud's student, Stekel. I would have liked to have been told more about Coxeter the man.

Perhaps the best compliment I can pay the book is that I came away from it wishing that I had known Coxeter, and determined to read more of his work.

A search on Amazon reveals that this book is the first published by Ms. Roberts, and the book definitely has some rough edges. I feel she needed more editorial guidance than she received. Ms. Roberts, who is not a mathematician, has apparently interviewed a number of Coxeter's colleagues at great length, and she has let them do the talking. As a result, I feel the absence of a strong author's voice, and a resulting fragmented picture of Coxeter. I found the book hard to plow through in places, which really surprised me given the strong interest that I have in Coxeter's work.

The flip side of this is that it is wonderful to read accounts from such luminaries as John Conway, Walter Whitely, Freeman Dyson, M. C. Escher (through his son, George), Buckminster Fuller and Douglas Hofstadter (who wrote the forward).

I found the glimpses of Coxeter's life outside of mathematics to be quite fascinating. He was a pacifist and a vegetarian, and seems to have been highly regarded as a person by most who knew him. His early life included problems with a withdrawn mother and an overbearing father, and a brief psychoanalysis by Freud's student, Stekel. I would have liked to have been told more about Coxeter the man.

Perhaps the best compliment I can pay the book is that I came away from it wishing that I had known Coxeter, and determined to read more of his work.

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

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

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

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

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

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

### My Philosophy of Mathematics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

This is an important and flawed book. It has generated much commentary, pro and con, and if your curiosity is whetted by my review I suggest you check the reviews of the book in Amazon.com. Like

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

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

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

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

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

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

This is an important and flawed book. It has generated much commentary, pro and con, and if your curiosity is whetted by my review I suggest you check the reviews of the book in Amazon.com. Like

*The Number Sense*by Stanislaus Dehaene (reviewed below), this book applies cognitive science to an analysis of mathematical thought. But unlike that book,*Where Mathematics Comes From*goes far beyond an analysis of arithmetic skill, and analyzes some very sophisticated concepts of higher mathematics. Also, compared with Dehaene’s book, this book does not depend very much on laboratory science, but depends almost entirely on theoretical cognitive science.This is a long book, and a somewhat difficult read, unless you happen to be well versed in the jargon of both mathematics and cognitive science. Even though the authors cover a lot of territory in 450 pages, and the quality of the writing is generally good, I had the feeling that I was reading the same thing over and over. If you want an “executive summary” to get the gist of what the authors believe they have accomplished, I suggest looking at the section “A Portrait of Mathematics” on pages 377 – 379.

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

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

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

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

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

### E5. Duck, Duck, Goose

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

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

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

### Review of the Number Sense

The Number Sense: How the Mind Creates Mathematics by Stanislaus Dehaene, Oxford University Press 1997

This book, written by a noted neuropsychologist, explores the new field of mathematical cognition. That is, it attempts to root our understanding of the development of mathematics in the biology of the brain. It is one of those rare books written by a pioneering researcher in a scientific field who is also an excellent writer – in English as well as presumably in his native French. I think it is particularly valuable for those of us in education, because in order to teach mathematics we must understand how children actually acquire mathematics. While there is much to learn here, I also found much to disagree with, and I will deal with these points below. Perhaps the major drawback to the book may be its date of publication, since Dehaene indicates that the ten years following the writing of the book promise to be a time of unparalleled scientific advance in the field.

The book is organized into nine chapters:

Chapter 1, “Talented and Gifted Animals”, discusses scientific research that shows that many animals have innate primitive arithmetic skills, which enable them to add, subtract, and compare small integers. Calculations and comparisons of numbers become less accurate as the numbers involved increase beyond three.

Chapter 2, “Babies Who Count”, sets forth the contention, supported by ingenious research, that shows that, similar to animals, human babies as young as a few days old also have innate arithmetic skills, enabling them to understand and manipulate small integers.

Chapter 3, “The Adult Number Line”, discusses the conception that human adults have of number. Much of this chapter has to do with discovering the extent to which we can manipulate numbers very quickly, that is, without visible thought.

Chapter 4, “The Language of Number”, discusses the ways different cultures name numbers, and the effect this has on calculating abilities.

Chapter 5, “Small Heads for Big Calculations”, applies the results covered in the previous chapters to the difficulties of teaching arithmetic to children.

Chapter 6, “Geniuses and Prodigies”, presents case studies of a number calculating prodigies and mathematical geniuses, and attempts to show that their abilities are not different in kind from that available to any intelligent adult.

Chapter 7, “Losing Number Sense”, discusses the relationship between brain function and number sense as revealed by studying people who have lost various parts of their number sense due to lesions in particular parts of their brains, or to other brain injury.

Chapter 8, “The Computing Brain”, shows how modern advances in brain research cast light on relationship between calculation and the brain. The tools of positron emission tomography (PET) and electro- and magnetoencephalograpy are described, and some results obtained by applying these tools to mathematical cognition are discussed.

Chapter 9, “What Is a Number?” moves into the philosophy of mathematics. Dehaene tackles questions such as the merits of the formalist, Platonist, and intuitionism theories of mathematics, and the relationship between mathematical truth and reality.

A book of this wide coverage is bound to be controversial. I recommend reading it yourself and making up your mind about some of the controversial issues, but I’d like to bring up a few places where I disagree with the author.

It seems to me that one of the dangers of neuropsychology is that of reductionism, and although Dehaene is a sophisticated thinker I don’t think he escapes this.

I take issue with his apparent assumption, which seems unsupported by data, that ability to perform arithmetic calculations is strongly correlated with the ability to do higher mathematics. Among mathematicians I have known, some excel at arithmetic, some are poor, and many are in between. The type of thinking that is involved in geometry, for example, seems to have little to do with arithmetic ability.

I find particularly problematic his discussion of mathematical geniuses, for several reasons. First, he lumps together the self-taught Indian mathematical genius Ramanujan with autistic super-calculators and idiot savants. To me, this is as if one compared Shakespeare with a pre-typewriter clerk who filled thousands of pages of commercial transactions. Both men may have had unusual ability to produce fast legible handwriting, but we would only call one a genius. Second, Dehaene makes clear that he believes that anyone could be a super-achiever in mathematics or arithmetic if they devoted enough time and effort to the enterprise; that there is nothing special about the brain (or mode of thinking) of the genius. This is speculation, and I prefer the opposite speculation of Oliver Sachs, whose prime-number generating autistic twins seem not to calculate but rather to see the integers “directly, as a vast natural scene” or Ramanujan, who described his own mathematical discoveries as being handed to him by a Hindu god while he slept. Non-believers can imagine that Ramanujan’s unconscious mind allowed him to make his discoveries operating in a way that might be totally different from his conscious mind.

Another oversimplification is Dehaene’s belief that young Oriental students do better than Western students at learning mathematics because the Eastern languages have shorter more user-friendly names for the digits. He seems to not consider the cultural differences that lead Oriental families to value hard academic work more than Occidental families do, which by itself is enough to explain differences in achievement.

In terms of pedagogic implications, Dehaene’s research has led him to the belief that the human brain is not well designed for calculation: “Ultimately, [innumeracy] reflects the human brain’s struggle for storing arithmetical knowledge”. He therefore feels that “by releasing children from the tedious and mechanical constraints of calculation, the calculator can help them to concentrate on meaning.” This is a position I have long shared; however I am now teaching middle-school mathematics teachers, and they mostly report that their students, who have grown up using calculators, are grossly innumerate. Since many algorithms of elementary algebra have counterparts in arithmetic algorithms, these students are not able to progress in algebra. I now advocate getting children to a state of competence in calculation before letting them use the calculator freely. However, I agree with Dehaene on the usefulness of concrete computational representations (manipulatives) in the classroom.

Dehaene gives a good description of the basic theories of mathematical epistemology: Platonism (mathematical objects have a reality, and the mathematician discovers this reality rather than inventing it), formalism (mathematics is about the formal manipulation of strings of symbols following basic laws of logic), and intuitionism (mathematics is a construction of the human mind, so that alien intelligences would create different a mathematics different from the human one.) He comes down for intuitionism, but it seems to me that his dismissal of Platonism is entirely too glib. He asks, rhetorically, “If these [mathematical] objects are real but immaterial, in what extrasensory way does a mathematician perceive them?” I would argue that they are perceived in the same way that we perceive a coherent world from the streams of sense data that enter our brains. We create our mental worlds, and this seems to be true whether or not the basis of the world is “material” or whether is grounded in ideas. Both the material and mathematical mental worlds are subject to laws of internal consistency, and both are subject to judgment by members of a community.

This book, written by a noted neuropsychologist, explores the new field of mathematical cognition. That is, it attempts to root our understanding of the development of mathematics in the biology of the brain. It is one of those rare books written by a pioneering researcher in a scientific field who is also an excellent writer – in English as well as presumably in his native French. I think it is particularly valuable for those of us in education, because in order to teach mathematics we must understand how children actually acquire mathematics. While there is much to learn here, I also found much to disagree with, and I will deal with these points below. Perhaps the major drawback to the book may be its date of publication, since Dehaene indicates that the ten years following the writing of the book promise to be a time of unparalleled scientific advance in the field.

The book is organized into nine chapters:

Chapter 1, “Talented and Gifted Animals”, discusses scientific research that shows that many animals have innate primitive arithmetic skills, which enable them to add, subtract, and compare small integers. Calculations and comparisons of numbers become less accurate as the numbers involved increase beyond three.

Chapter 2, “Babies Who Count”, sets forth the contention, supported by ingenious research, that shows that, similar to animals, human babies as young as a few days old also have innate arithmetic skills, enabling them to understand and manipulate small integers.

Chapter 3, “The Adult Number Line”, discusses the conception that human adults have of number. Much of this chapter has to do with discovering the extent to which we can manipulate numbers very quickly, that is, without visible thought.

Chapter 4, “The Language of Number”, discusses the ways different cultures name numbers, and the effect this has on calculating abilities.

Chapter 5, “Small Heads for Big Calculations”, applies the results covered in the previous chapters to the difficulties of teaching arithmetic to children.

Chapter 6, “Geniuses and Prodigies”, presents case studies of a number calculating prodigies and mathematical geniuses, and attempts to show that their abilities are not different in kind from that available to any intelligent adult.

Chapter 7, “Losing Number Sense”, discusses the relationship between brain function and number sense as revealed by studying people who have lost various parts of their number sense due to lesions in particular parts of their brains, or to other brain injury.

Chapter 8, “The Computing Brain”, shows how modern advances in brain research cast light on relationship between calculation and the brain. The tools of positron emission tomography (PET) and electro- and magnetoencephalograpy are described, and some results obtained by applying these tools to mathematical cognition are discussed.

Chapter 9, “What Is a Number?” moves into the philosophy of mathematics. Dehaene tackles questions such as the merits of the formalist, Platonist, and intuitionism theories of mathematics, and the relationship between mathematical truth and reality.

A book of this wide coverage is bound to be controversial. I recommend reading it yourself and making up your mind about some of the controversial issues, but I’d like to bring up a few places where I disagree with the author.

It seems to me that one of the dangers of neuropsychology is that of reductionism, and although Dehaene is a sophisticated thinker I don’t think he escapes this.

I take issue with his apparent assumption, which seems unsupported by data, that ability to perform arithmetic calculations is strongly correlated with the ability to do higher mathematics. Among mathematicians I have known, some excel at arithmetic, some are poor, and many are in between. The type of thinking that is involved in geometry, for example, seems to have little to do with arithmetic ability.

I find particularly problematic his discussion of mathematical geniuses, for several reasons. First, he lumps together the self-taught Indian mathematical genius Ramanujan with autistic super-calculators and idiot savants. To me, this is as if one compared Shakespeare with a pre-typewriter clerk who filled thousands of pages of commercial transactions. Both men may have had unusual ability to produce fast legible handwriting, but we would only call one a genius. Second, Dehaene makes clear that he believes that anyone could be a super-achiever in mathematics or arithmetic if they devoted enough time and effort to the enterprise; that there is nothing special about the brain (or mode of thinking) of the genius. This is speculation, and I prefer the opposite speculation of Oliver Sachs, whose prime-number generating autistic twins seem not to calculate but rather to see the integers “directly, as a vast natural scene” or Ramanujan, who described his own mathematical discoveries as being handed to him by a Hindu god while he slept. Non-believers can imagine that Ramanujan’s unconscious mind allowed him to make his discoveries operating in a way that might be totally different from his conscious mind.

Another oversimplification is Dehaene’s belief that young Oriental students do better than Western students at learning mathematics because the Eastern languages have shorter more user-friendly names for the digits. He seems to not consider the cultural differences that lead Oriental families to value hard academic work more than Occidental families do, which by itself is enough to explain differences in achievement.

In terms of pedagogic implications, Dehaene’s research has led him to the belief that the human brain is not well designed for calculation: “Ultimately, [innumeracy] reflects the human brain’s struggle for storing arithmetical knowledge”. He therefore feels that “by releasing children from the tedious and mechanical constraints of calculation, the calculator can help them to concentrate on meaning.” This is a position I have long shared; however I am now teaching middle-school mathematics teachers, and they mostly report that their students, who have grown up using calculators, are grossly innumerate. Since many algorithms of elementary algebra have counterparts in arithmetic algorithms, these students are not able to progress in algebra. I now advocate getting children to a state of competence in calculation before letting them use the calculator freely. However, I agree with Dehaene on the usefulness of concrete computational representations (manipulatives) in the classroom.

Dehaene gives a good description of the basic theories of mathematical epistemology: Platonism (mathematical objects have a reality, and the mathematician discovers this reality rather than inventing it), formalism (mathematics is about the formal manipulation of strings of symbols following basic laws of logic), and intuitionism (mathematics is a construction of the human mind, so that alien intelligences would create different a mathematics different from the human one.) He comes down for intuitionism, but it seems to me that his dismissal of Platonism is entirely too glib. He asks, rhetorically, “If these [mathematical] objects are real but immaterial, in what extrasensory way does a mathematician perceive them?” I would argue that they are perceived in the same way that we perceive a coherent world from the streams of sense data that enter our brains. We create our mental worlds, and this seems to be true whether or not the basis of the world is “material” or whether is grounded in ideas. Both the material and mathematical mental worlds are subject to laws of internal consistency, and both are subject to judgment by members of a community.

### Physical Models for Non-Euclidean Geometry

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

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

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

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

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

Anyone have any ideas?

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

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

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

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

Anyone have any ideas?

### Visual Calculus

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

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

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

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

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

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

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

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

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

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

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

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

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

### Tri-Color chessboards

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

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

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

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

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

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

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

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

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

### Three Books on Riemann Hypothesis

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

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

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

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

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

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

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

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

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

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

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

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

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

Subscribe to:
Posts (Atom)