# Posts

## BSME vs BSM

I arrived in Hungary, and am getting to know the Budapest Semester in Math Education (BSME) program. This program, now in its 3rd year of operation, is for undergraduate (and recent graduates) who are interested in the teaching of mathematics. They currently offer 5 courses that touch on different aspects of math education, and BSME students can also take BSM courses.

Yes, there is BSME and there is BSM, which is the Budapest Semesters in Math program, now in its 33rd year. They are not the same program, yet they hold classes in the same building, are both coordinated out of St. Olaf College in the US, and are generally nicely coordinated.

The 5 courses are the following, and I’ll write about them each in future posts.

Practicum – Visit math classrooms in schools around Budapest, and debrief.

Discovery Learning: The Pósa Method – Learn about the math instructional method developed by Lajos Pósa.

Problem Solving in Secondary School Mathematics – Develop problem solving skills, engage in mathematical tasks to foster deep math learning.

Concept Building through Games and Manipulatives – Just that, developing mathematical thinking through informal and formal hands-on learning.

Directed Research: Gender Issues in Mathematics – Think of this as a math education REU.

Tune in later for more details about these courses, and other aspects of BSME!

## The Budapest Semester in Math Education

I’m traveling to Hungary next week to visit the new Budapest Semester in Math Education (BSME) program. This program is intended for undergraduate students and recent graduates who are broadly interested in teaching mathematics, and promises to share Hungarian insights into mathematics and math education. BSME is already in its 3rd year, and has just announced the addition of a summer program.

I do feel compelled to admit that I have a fascination with mathematics in other countries. When I taught in Bard College’s Master of Arts in Teaching (MAT) program, my students and I read and discussed The Teaching Gap by Hiebert and Stiegler, Knowing and Teaching Elementary Mathematics by Liping Ma, and we tried our hand at Japanese Lesson Study. Like many math circle leaders around the country, I’m curious about the content and pedagogy of math circles in Eastern European, especially Bulgaria, Hungary, Romania and Russia, where the US math circle movement traces its roots. I’ve also peeked into math textbooks like Singapore Math, and Russian textbooks.

I do also have a fascination with mathematics in this country. The history of math and math teaching in the US includes the math curricular reform movements like New Math, the Common Core State Standards, and everything in between, and before. It’s important to know about the Inquiry Based Learning (IBL) movement, the summer math “Epsilon Programs”, math contests and student math journals. The debate between “Back to the Basics” and “Conceptual Understanding” is just a small part of the required reading.

But what can we learn about math teaching from Hungary? I plan to write more about that during my visit to BSME.

## How to catch a cheater

I proctored the AIME II contest this week, and caught a cheater. Here are some details and thoughts about the occasion.

At about 4pm the day before the contest, I started getting emails and phone calls from parents, from tutors, some students, and even my math colleagues at Bard who had been contacted as well, in desperate attempts to contact me. Their children had all planned to take the contest at Kean University in NJ, but for logistic reasons, the contest manager had to cancel, less than 24 hours in advance.

Up until then, I only expected to proctor one student, since the other two AIME qualifiers from when we hosted the AMC 10B and AMC 12B contests had also qualified on the A schedule a few weeks earlier. But now I imagined a packed room with 20 or 30 AIME students! The AMC response (amcinfo@maa.org) was quick and responsive, and they granted permission to include as many students as I had AIME answer forms for (I counted, and had 20 forms), even though I had registered and paid for only 10 students.

I passed on the good news to everyone, and quickly drafted out a liability and photo release form for the newcomers, along with detailed instructions on how to drive up to Bard College (more than 2 hours away from New Jersey) the next morning.

But then, over the next few hours, all of them found closer places, and cancelled, thankful that the worst case scenario – a long drive up to the Hudson Valley – was there if needed, but the drive could be avoided.

All but one, that is: the cheater. He is a high school freshman at one of New Jersey’s most prestigious public high schools. He was driven up by his math competition tutor, who described himself as a friend of the family. He coaches the cheater, and four other NJ students in competition math. All the rest of the students had also planned to come up to take the AIME II at Bard, but had found more convenient locations overnight.

They arrived at Bard, and were welcomed in by the local math student (a middle school student) who had qualified, and his mom. When I arrived a little later, I set up the room, and welcomed the mom and the tutor to enjoy Bard College, a wonderfully secluded campus where they could stretch their legs, catch breathtaking views of the Catskill Mountains and the Hudson River, and explore eclectic architecture (including a Frank Gehry building in north campus). Or, they could set up their laptops with Bard’s wifi, and catch up on work. Bard is on spring break this week, so parking is plentiful, and so are study nooks.

I helped the students bubble in the answer forms with their names and other information (kids these days don’t remember their street addresses), and made sure they understood the AIME contest rules. Then I started the exam by clicking on the countdown timer (Google search for “internet timer”) set to 3 hours.

Ten minutes later, the cheater pulled a cell phone out of his pocket and put in on his lap. Did he not understand that electronic devices were strictly not allowed on the AIME? I walked over to him and confiscated his phone immediately. I let him continue the exam, because I saw that he hadn’t received any information from the phone.

In the middle of the exam, he excused himself to go to the bathroom. He returned a few minutes later. Later, with half an hour to go, he excused himself again to go to the bathroom and left the room.

A minute later, the other student’s mom opened the door to ask me a question. In the hallway, she told me that the NJ student had gone outside. He told her that he needed to get his photo ID, and she wondered if it was okay that her son didn’t have an ID with him. She was worried about that, but I reassured her that since her son had been to the AMC contests at Bard for several years in a row, I recognized him, and no ID was needed.

Then the other student came back from outside, not from the bathroom. That was strange. Back in the testing room, the NJ student got back to work, and started to open up a folded piece of paper. I stood up and walked over to him, explaining that outside notes are not allowed on the exam, and asked him to hand it over. I unfolded the paper, and found a numbered list of 3 digit numbers. The answer to each AIME problem is a 3 digit number.

I was shocked at how blatant this student was in his cheating, and how easy he was to catch.

I decided to let him continue working, mostly to avoid distracting the other student, and also to maintain control over the situation.

When the time was up, I asked the students to sign the statement on their answer form that all the work on the contest was their own. Both students signed. I collected the answer forms and excused them from the testing room.

I told the NJ student that I wanted to speak with him and with his tutor, and he went outside to the parking lot. Then I took the opportunity to thank the local student and his mom, and to ask what problems he found interesting. They left soon after that.

But the NJ student and his tutor didn’t come back. I waited and waited, and then gathered my things and started leaving. Funny – I still had the student’s cell phone and his cheat sheet. Right before I left the building – it seemed about half an hour after the contest ended – the tutor appeared.

The tutor asked what had happened. We talked, and the tutor explained that he had given the student the answers on a sheet of paper, because he thought the exam was over (it was just after 12pm when I confiscated it).

He told me that one of his other NJ students had finished the exam early, about 10am, at some NJ location, and texted an image of the exam to him, which he worked on in his car.

The student showed up soon after that, and I talked with them both. The student admitted to cheating, and said that he wanted the opportunity to take the USAMO. He said that the tutor didn’t know about the cheating, but I don’t believe him. The tutor also claimed not to know about the cheating, even though he admitted to receiving an electronic copy of the contest around 10am, during the exam, working on it, and handing his student a neatly-written list of answers. He didn’t seem at all upset that his student had lied to him, and asked me to give his student a chance and to forgive the cheating.

It was fascinating to interview a cheater and his accomplice, and to feel in control of the situation. I asked them question after question to collect information to share with the AMC. I was curious if they would show any remorse for what they had done, and so I asked them questions to open opportunities for that. But there was no remorse at all, so I ended the conversation and left.

The cheater ran after me again to ask for his cell phone back, but I refused to, and told him to email me and I’d ship it back to him.

1. How important is it to protect the cheater’s identity? I shared his name and other information with the AMC, of course. But I could also easily contact his high school through personal contacts. I could publish his name on the web. I could contact the Kean University contest manager, and other university based contact managers in the region to let them know. Should I take any or all of those steps? Or leave it to the AMC to take action.
2. Is it important to protect the tutor’s identity? I have contact information for another of his students and that student’s father. What action on my part is appropriate? I was offended on many levels that this tutor placed no priority at all on honesty. In fact, he is a key figure in a cheating gang. If he were tutoring my child, I would want to know.
3. How welcoming should I be of AIME students from outside the Bard Math Circle community? I want to promote and develop a culture of math enrichment in the Mid-Hudson Valley. I want to open up opportunities for students whose schools don’t have a math club, a math team or a math circle. But this cheating incident was very, very disappointing.
4. What would you do with the cell phone? (I ended up shipping it to his parents by express mail, carefully bubble wrapped and insured. I included a brief letter explaining that the phone had been confiscated from their son during the AIME exam.)

## Mathematical Writing

Forgive me, but I’m often overjoyed to provide feedback on student writing as a math professor. Who knew?! Below are some of my favorite pieces of advice about writing mathematics, and I hope you will find them useful.

Paul Halmos wrote How to Write Mathematics (original available here), and these are his key points:

1. Say something. To have something to say is by far the most important ingredient of good exposition.
2. Speak to someone. Ask yourself who it is that you want to reach.
3. Organize. Arrange the material so as to minimize the resistance and maximize the insight of the reader.
4. Use consistent notation. The letters (or symbols) that you use to denote the concepts that you’ll discuss are worthy of thought and careful design.
5. Write in spirals. Write the first section, write the second section, rewrite the first section, rewrite the second section, write the third section, rewrite the first section, rewrite the second section, rewrite the third section, write the fourth section, and so on.
6. Watch your language. Good English style implies correct grammar, correct choice of words, correct punctuation, and common sense.
7. Be honest. Smooth the reader’s way, anticipating difficulties and forestalling them. Aim for clarity, not pedantry; understanding, not fuss.
8. Remove the irrelevant. Irrelevant assumptions, incorrect emphasis, or even the absence of correct emphasis can wreak havoc.
9. Use words correctly. Think about and use with care the small words of common sense and intuitive logic, and the specifically mathematical words (technical terms) that can have a profound effect on mathematical meaning.
10. Resist symbols. The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it.

The Math Association of America is now hosting an entire webpage devoted to Mathematical Communication. This looks like it will become a fantastic resource going forward.

Over at Harvey Mudd College, Francis Su has posted his Guidelines for Good Mathematical Writing. This seems to be part of a wider campus-wide effort in communication. In this post, Rachel Levy explains why Every Math Major Should Take a Public-Speaking Course.

Francis Su also has an article Some Guidelines for Good Mathematical Writing in the August/September 2015 MAA Focus Newsmagazine (a subscription or MAA membership might be required).

Annalisa Crannell at Franklin & Marshall College offers A Guide to Writing in Mathematics Classes for undergraduate math students, but the advice is useful for a much wider audience.

1. Why Should You Have To Write Papers In A Math Class?
2. How is Mathematical Writing Different?
3. Following the Checklist
4. Good Phrases to Use in Math Papers
5. Helpful Hints for the Computer
6. Other Sources of Help

I’ve also been told from several people to consider Serre’s How to Write Mathematics Badly. I found the video, but had a tough time watching it through to the end. Maybe Serre should have taken the public speaking course at Harvey Mudd College?

Dave Richeson from Dickinson College writes about The Nuts and Bolts of Writing Mathematics at his Division by Zero blog. He also has a very nice Checklist for Editing Mathematical Writing, and an example of a poorly written proof for a class exercise.

## Uniqueness of Factorization

A few days ago I came across a proof of the Fundamental Theorem of Arithmetic (aka Unique Factorization) in Courant and Robbin’s What is Mathematics that I hadn’t seen it before. I liked it enough to learn it.

Then another surprise – I saw it again yesterday in Primes and Programming by Peter Giblin, a book that Larry Zimmerman had recommended to a student from the summer high school program.

The usual proof that I know is based on Euclid, and basically is a proof by Strong Induction. This new proof is by the Principle of Least Element. So the key is to suppose that unique factorization fails, and to reason about the least positive integer $$N$$ that has more than one factorization into primes. Even though we’ll show this number doesn’t exist, we can deduce lots of information about it!

First, some notation. Let’s say that two distinct prime factorizations of $$N$$ are

$\text{(1)}\qquad N = p_1 p_2 \dots p_r \text{ and } N = q_1 q_2 \dots q_s$

Of course, we’ll arrange the primes in non-decreasing order, so that in particular, $$p_1$$ and $$q_1$$ are the smallest primes in those factorizations.

The other primes don’t take a big role in what comes next, so let’s write $$P = p_2 \dots p_r$$ and $$Q = q_2 \dots q_s$$, so that we have

$N = p_1 P = q_1 Q.$

The first observation is that $$p_1$$ and $$q_1$$ are different primes, otherwise if they were equal, we could factor them off and then $$N/p_1 = P$$ would be a smaller positive integer with two distinct prime factorizations.

Now that that’s done, let’s assume without loss of generality that $$p_1 < q_1$$, and we’ll form a new number:

$\text{(2)}\qquad M = (q_1 – p_1) Q$

By equation (2), it’s clear that $$M$$ is a positive integer that is less than $$N$$, and therefore does factor uniquely into primes. Now we rewrite $$M$$ as follows:

$M = (q_1 – p_1) Q = q_1 Q – p_1 Q = N – p_1 Q = p_1 P – p_1 Q = p_1 (P – Q)$

That is, $\text{(3)}\qquad M = p_1 (P – Q)$

We’re almost there. Note that because of equation (3), the prime $$p_1$$ is a prime factor of $$M$$. Now consider the factorization of $$M$$ given in equation (2). Since we were careful to list the primes in non-decreasing order, $$p_1$$ can’t be any of the primes in $$Q = q_2 \dots q_s$$, and so it must be a factor of $$(q_1 – p_1)$$. Suppose that $$(q_1 – p_1) = p_1 t$$. Then solving for $$q_1$$, we find that $$q_1 = p_1 (t+1)$$. And this is a contradiction, since then $$q_1$$ would not be a prime number!

## Typesetting synthetic division

I’m teaching an Algebra course that highlights the Fundamental Theorem of Algebra. So of course we’re looking closely at polynomial division, and in particular at synthetic division. My students are preparing their homework assignments using LaTeX, so this begs the question about how to typeset their computations.

One of my students found the LaTeX package polynom, which can automatically compute and typeset polynomial long division and synthetic division. The examples in the polynom manual are:

\polylongdiv{X^3+X^2-1}{X-1}

which yields

and for synthetic division, the polynom command

\polyhornerscheme[x=1]{x^3+x^2-1}

which yields

But what if you want to show division by x - m using synthetic division, rather than x - 1? These commands no longer work.

Instead, you can make fancy use of \cline and \multicolumn in the array environment:

\begin{array}{r|rrrr} & 1 & 1 & 0 & -1 \\ m & & m & (m^2 + m) & (m^3 + m^2) \\ \cline{2-5} \multicolumn{2}{r}{1} & (m + 1) & (m^2 + m) & (m^3 + m^2 - 1) \end{array}

which yields

Interesting! If you divide a polynomial by x - m, the remainder is just the value of that polynomial at m!

## How to Give a Good Math Talk

There are a lot of sites with useful advice out there. This post is intended to collect several links for future reference.

Technically Speaking – Videos of math research students presenting their findings. I love the examples of bad style along with good style. This comes out of an NSF-funded project.

Joe Gallian’s Advice on Giving a Good PowerPoint Presentation from his article in Math Horizons.

Giving a Conference Talk by Mike Dahlin.

Oral Presentation Advice by Mark D. Hill, includes “How to Give a Bad Talk” by David A. Patterson.

Giving an Academic Talk by Jonathan Shewchuk.

I also have several resources on good mathematical writing that I’ll share later.

## A brief 4,000 year history of Diophantine Equations

I filled in for a NY Math Circle class over the weekend. Since the topic was Primitive Pythagorean Triples, I had a blast. I also shared the following outline with the students. Each item is full of wonderful mathematics and anecdotes!

Plimpton 322, a Babylonian cuneiform tablet @ Columbia University. From 1900BCE – 1600BCE, and allegedly includes the Pythagorean triple (12709, 13500, 18541).

Pythagoras was born ca 580BCE on the island of Samos. Famous quote: “All is Number”. His proof of the theorem that bears his name involved cutting up a square of side a+b and rearranging the pieces.

Proclus (5th century CE) credits Pythagoras with the formula (2n+1, 2n^2 + 2n, 2n^2 + 2n + 1) of (necessarily) primitive pythagorean triples where the hypotenuse is 1 more than one of the sides. Proclus also credits Plato with the formula (2n, n^2 – 1, n^2 + 1) where the hypotenuse is 2 more than one of the sides. (For what n is this primitive?)

Euclid of Alexandria (born 365 BCE) is famous for the 13-book Elements. The Theorem of Pythagoras is Book I, Proposition 47. Analysis on Primitive Pythagorean Triples appears as Lemma 1 “To find two square numbers such that their sum is also a square” in Book X, just before Proposition 29.

By the way, Elisha Scott Loomis, an early 20th century mathematician, published The Pythagorean Proposition, which has 370 proofs of the theorem, and not a single one used trigonometry. This was republished in 1968 by the NCTM.

Diophantus of Alexandria (200CE – 298CE) is famous for his book Arithmetica. He sought integer (and perhaps rational too?) solutions to algebraic equations. The term Diophantine Equation typically refers to equations where we seek positive integer solutions.

Fermat (1608CE – 1665CE) wrote in the margin of his copy of Arithmetica that there are no integral values of x, y, z so that x^n + y^n = z^n if n > 2.

Andrew Wiles of Princeton University announced a proof in 1993 of Fermat’s Last Theorem, after working in secret for 7 years. An error was found in his proof, which was salvaged in 1994. Wiles’ proof was published in 1995.

Jumping back to 1900, David Hilbert asked mathematicians at the International Congress of Mathematicians to devise a method to determine whether a Diophantine equation has solutions. This is known as Hilbert’s 10th Problem.

Julia Robinson (1919-1985) was a Californian mathematician who worked on Hilbert’s 10th problem for decades, from the 1940s until final achieving a solution (in joint work with Martin Davis, Hilary Putnam, Yuri Matiyasevich and others) in 1970. In general, there is no such algorithm!

Mirroring the Hilbert Problems of 1900, the Clay Institute of Mathematics issued the Millennium Problems in 2000.

## Kevin’s Books

One of the great friends that I made in my time at Vanderbilt University was Kevin Blount. Kevin knew all the graduate students and professors, and often hosted dinners and movies at his nearby apartment.
Kevin ended up writing his Ph.D. dissertation On the Structure of Residuated Lattices with Constantine Tsinakis, and moved on to an academic position at Sacred Heart University in Connecticut.

Kevin passed away on May 30, 2006, which surprised all of us. I had just moved back East, and so one of the first trips was attending his memorial service at Sacred Heart.

At the end of 2008, Kevin’s wife, Xiaoyu, gave me Kevin’s math books. After some brief discussions with some mathematical colleagues, the books ended up being stored in my attic. I’m now offering these books to those who knew Kevin. I am sure that Kevin would have been happy to have his math books shared among his friends and colleagues, and I hope that this will help keep Kevin’s memory alive in the mathematical community.

If you are interested in some of the books below, leave a comment or contact me privately. I’ll send them your way. (Pardon the typos – I’ll correct those as they are noticed).

Kevin’s Math Books

1. Albert, Introduction to Algebraic Theories, Matt Insall
2. Artin, Geometric Algebra, Tracts in Mathematics Number 3 – John Raymonda
3. Baumslag and Chandler, Group Theory
4. Bennet, Affine and Projective Geometry
5. Bjarni Jonsson, Topics in Universal Algebra, Lecture Notes, Vanderbilt University, 1969-70 – John Snow
6. Bollobas, Graph Theory, GTM 63, Matt Insall
7. Bond/Keane, An Introduction to Abstract Mathematics, Instructors
8. Bondy and Murty, Graph Theory with Applications, Matt Insall
9. Bonele, Non-Euclidean Geometry, Dover
10. Bourbaki, Elements of Mathematics, General Topology, Part 1, Addison Wesley
11. Burnside, Theory of Groups of Finite Order, Princeton
12. Curry, Foundations of Mathematical Logic, Dover
13. Dieudonne, Introduction to the Theory of Formal Groups, Dekker
14. Gruenberg and Weir, Linear Geometry, Van Nostrand
15. Hodel, An Introduction to Mathematical Logic – Jan Gałuszka
16. Hall, The Theory of Groups, Macmillan
17. Springer, Geometry and Analysis of Projective Spaces
18. Sternberg, Lectures on Differential Geometry, Prentice Hall
19. Hennie, Introduction to Computability
20. Lang, Introduction to Algebraic Geometry, Tracts in Mathematics Number 5 – John Raymonda
21. Hochschild, Introduction to Affine Algebraic Groups
22. Nagaia, Local Rings, Tracts in Mathematics Number 13
23. Spanier, Algebraic Topology
24. Lipschitz, Discrete Mathematics, Schaum McGraw Hill
25. Solutuion Manual for, Brooks/Cole
26. Curtis Clark, An Approach to Graph Achievement Games: Ultimately Economical Graphs
27. Hungerford, Algebra, Springer 73 – John Snow
28. Mendelson, Introduction to Mathematical Logic, 3e, Wadsworth & Brooks/Cole Mathematics Series
29. Moore, Elementary General Topology, Prentice-Hall
30. Mitchell, Theory of Categories, Academic Press, Matheamtics XVII
31. Herken, The Universal Turing Machine, Oxford – John Snow
32. ??, Fundamental Concepts of Algebra, Addison Wesley
33. Kelley, General Topology, Van NostrandO??, Theory of LIE Groups, Princeton
34. Munkres, Elements of Algebraic Topology
35. Sawyer, A Geometric Approach to Abstract Algebra, Freeman
36. McCoy, Rings and Ideals – John Raymonda
37. Smullyan, First-Order Logic, Dover
38. Veblen and Young, Projective Geometry, Volume 1
39. Gelfond, Transcendental & Algebraic Numbers, Dover – John Snow
40. Freese & McKenzie, Commutator Theory for Congruence Modular Varieties – Jonathan Farley
41. Davey & Priestley, Introduction to Lattice and Order, Cambridge – John Snow
42. An Algebraic Introduction to Matheamtical Logic, GTM 52 – John Snow
43. Categories for the Working Mathematician, GTM 5 – John Snow
44. Manin, A Course in Mathematical Logic, GTM 53
45. Burris/Sankappanavar, A Course in Universal Algebra, GTM 78 – John Snow
46. Hirsch, Differential Topology, GTM 33
47. Gaum, Elements of Point Set Topology, Prentice-Hall
48. Bourbaki, Elements of Mathematics, General Topology Part 2
49. Eisenberg, Topology, Holt Rinehart Winston
50. Pareigis, Categories and Functors
51. Kaplansky, Set Theory and Metric Spaces – John Raymonda
52. McKenzie, McNulty, Taylor, Algebras, Lattices, Varieties, Volume I, Wadsworth & Brooks/Cole Matheamtics Series, – Jonathan Farley
53. Monk, Mathematical Logic, GTM 37 – John Snow
54. Lightstone, Symbolic Logic and the REAL NUMBER SYSTEM – John Snow
55. The Essentials of Logic
56. Halmos & Givant, Logic as Algebra – Jan Gałuszka
57. Jacobson, Lectures in Abstract Algebra, I Basic Concepts, Van Nostrand – John Raymonda
58. Husain, Introduction to Topological Groups, Matt Insall
59. Tarski, Introduction to Logic and to the methodology of deductive sciences, Dover
60. Church, Introduction to Mathematical Logic, Princeton – John Snow
61. Rosenbloom, an introduction to Symbolic Logic, Dover
62. Krantz, How to Teach Mathematics: a personal perspective
63. Thomas Rishel, Teaching First – A Guide for New Mathematicians, MAA
64. Hobby and McKenzie, The Structure of Finite Algebras, AMS Comm 76, – Jonathan Farley

## AP Classes Are a Scam

I heard about the following Atlantic article from @stevenstrogatz : October 13th, AP Classes Are a Scam which I found quite interesting.

I thought much the same in those years when I taught a lot of freshman Calculus. My main observations were that

1. Most students who had taken AP Calculus in High School had to take the Calculus sequence anyway, and resented that they had to essentially repeat a course.
2. Students who had not taken AP Calculus in High School felt intimidated that they were in class with students who had, and felt completely inadequate.
3. As a result, the AP students barely worked at all, since they had a superficial knowledge of Calculus, while the non-AP students worked very hard.
4. Since AP Calculus is not college level Calculus, the effects were clear by midterms: the AP students had fallen too far behind, and the non-AP students were learning the material, and starting to enjoy it.
5. I suspected that there were other, successful AP students, who weren’t in my class, and never took another math class in their lives. Thus, some of the most enthusiastic math students at the high school level were diverted out of the math major, since they saw Calculus as the final math class.

So why did I come away with the impression that AP Calculus, presented as the highest level math class one could take in college, was essentially a terminal math class, serving to prevent bright and hard-working high school math kids from continuing in mathematics?

I’ve come to understand this more in the contrast between acceleration and enrichment. Our educational system emphasizes acceleration, and works hard to move kids rapidly through material. There are a lot of incentives for this, like granting college credit. An alternative is to enrich the curriculum, and allow students to go deeper into the material.

When I was in high school, I used math team to enrich my studies, as well as my own mathematical reading of fantastic authors like Martin Gardner. I don’t think I ever earned an academic credit for this enrichment, but it was profoundly enjoyable, and directed me into mathematics. I did benefit from acceleration as well, but my most memorable mathematical moments were from some inspirational math enrichment.

So, why do we bother with AP courses? I think our students would benefit greatly if a similar amount of resources were invested into academic enrichment. I’d love to see after-school math circles, math clubs and math teams in every school, and I can imagine similar enrichment in other subjects.