As with most things, the farther you go in mathematics the sharper your sense of ignorance becomes.  There is just too much math out there, and too little time to follow every thread that crosses your path.  I’d like to share a few of the math facts that I’ve accepted for for much of my mathematical life, but which I (lamentably) have never seen proven.  I choose these (out of the domain that our former Secretary of Defense Donald Rumsfeld would describe as my “known unknowns”) because they are common facts, they are interesting and accessible even at the undergraduate level (or earlier),  and they seem hard (in the sense that their proofs are not easily communicable in a few sentences, or twenty minutes at a chalkboard).  Here goes:

1.  That is transcendental (or, indeed, irrational)(same for ).

This is a biggie, since (let’s face it) most numbers are transcendental, but we only really have a couple of examples of them in common circulation.

PROOF:  This fact follows from the Lindemann-Weierstrass Theorem, a MUCH more general result, or the Gelfond-Schneider Theorem, a MUCH MUCH more general result (which also settles that pesky question about the rationality of ).  As you may have guessed, I do not know the proofs of these theorems…

2.  That there is no general formula for solving polynomial equations of degree 5.

Of course, we have algorithms for finding roots of quintic equations to arbitrary precision — we just can’t always express the roots exactly using arithmetic operations, radicals, and so on.

This is a great one to bring up when discussing the quadratic formula – it’s so natural to ask about equations with higher degree, and the expectation from students is that, while there probably are “quadratic-type formulas” for any polynomial, they are going to be complicated.  It’s a bit of a shock to find that degree 4 is as high as you can go!

PROOF:  For this one, you need a paradigm-shifting change in perspective on polynomial equations, Galois Theory (which I have never studied, blah blah blah…)

3.  That can’t be integrated. 

Of course, the function can be integrated — else our whole theory of continuous probability would crumble — but once again, it’s the existence of a nice expression for that integral, in terms of other elementary functions, that fails.

This is a great problem to give your Calculus II students, after they’ve spent a month or so mastering the different techniques of integration.  After struggling with the problem for a while, they will be ready (expectant!) for a “neat trick” that solves the problem — and the fact that NO such trick exists is a shocker!  This is also a nice lead-in to a discussion about the relative difficulty of differentiation vs. integration  (ironic, since many more functions are integrable than differentiable).

PROOF:  There is a nice (and perhaps unexpected — but perhaps not, see below) connection between this and the previous problem — the proof comes out of Differential Galois Theory, the analogue of Galois theory for differential fields.

Negativity is hard

The examples above are all negative results, in the sense they they show “something is NOT” — either not in some easily defined class or not expressible by some means (in example 1, “transcendental” is just a shorthand for “NOT algebraic”).  Somehow, showing something is NOT seems to be much harder than showing something IS.  Why so hard?  It must have something to do with the difference between producing an example illustrating an idea, and (on the other hand) clearly delineating the boundaries of the idea.  The former is specific, and the latter is global.  The former requires you to provide some kind of construction, the latter requires you to show that no such construction will work.  In many cases, the negative result requires some fundamentally new perspective — and the consequences of the resulting proof extend far beyond the motivating problem!

There are many many more examples of this kind of thing, across all different areas of mathematics:  in number theory, we have Wiles’ proof of Fermat’s Last Theorem (here’s a nice blog-in-progress detailing the proof for the interested amateur), and in set theory  the independence of the Continuum Hypothesis (and development of forcing).  Power to the negative!

Professor Esther Wilder, Associate Professor of Sociology at Lehman College, recently received a $600,000 NSF grant for her Quantitative Reasoning (QR) research. Her bio on Lehman’s website is utterly impressive, and yes, of course, she is also a super mom with two young daughters! How does she do it? What’s her secret? I met her in the Carman Hall cafeteria on a beautiful day in March (the photo above is the magnificent Music Building), with tons of stars and question marks shining and twinkling in my eyes.

I’d love to first ask you “How do you do it?” but back up a little.  Let me start by asking about your childhood.

I grew up in Pelham, Massachusetts, in a very academically and artistically-oriented family.  My father was a professor of microbiology at the University of Massachusetts. My mother was a very talented artist and interior designer, though she had actually studied microbiology in college!  Perhaps owing to my mother’s influence, I loved art when I was a child and remember really enjoying pottery and stained glass. I enjoyed a lot of sports as well. One of my big claims to fame is when I won a basketball foul shooting contest at camp and succeeded in getting 5 out of 5 free throws!  I still have the certificate! I also loved animals and exploring nature as a child. Frogs and toads were my favorite animals!  I used to bring them home all the time and I still have some very vivid images of my mother squirming and telling me to “take it outside.”  Our family also had two dogs, both cairn terriers: our first one was “Jovy” and our second was “Brindle.”

What kind of student were you?

I was a perfectionist and a bit of geek.  I guess some things never change since I think I am still that way today! I think my parents’ love for learning and their high values of education strongly influenced my siblings as well as myself. All three of us went on to obtain our doctorates, my brother in mathematics and my sister in chemical engineering.

How about the influence you received from teachers? Who were the best teachers you had?

Over the years, I think I have had several wonderful teachers, but I think the ones that I would describe as “the best” were those that had an enthusiasm and passion for their subject and the students, and who really engaged students in the course materials.  Since I majored in journalism, I was doing a lot of writing as an undergraduate. I remember Professor Norman Sims at the University of Massachusetts, who taught a course entitled, “The History of Journalism.” His lectures were so enthralling and well-delivered that I would lose track of time while in his class and he could answer any question that students raised. And he also gave us an interesting assignment where we needed to write about a famous journalist, including an examination of the journalist’s writings. 

Another faculty member, Professor Albert Chevan at the University of Massachusetts, had a similar impact on me, but more through the way he structured his class as opposed to his lecture style. I took a course on “The Demography of Minority Groups” and he taught the course in such a way that we were actively engaged in working with census data to learn about the various groups.   There were so many courses that I took that focused on memorizing facts and ideas and reporting the information back on quizzes and exams.  But what I loved about Professor Chevan’s class is that he really taught the course as a social science course and we were actively engaged in the process of scientific inquiry using raw data. And I found that to be very exciting and ultimately, I went on to graduate school to study demography.

… and have become a professor of sociology! You mentioned that you did your undergraduate work at many different schools including Haverford College, Tel Aviv University, and UMass, and were exposed to many different ideas and perspectives in terms of culture, history, politics, etc. throughout your education. How would you compare such experience as a student and your experience as a professor/teacher at CUNY?

I feel so fortunate to be at CUNY where the students are so sincere and committed and really very loving in many different ways. I typically teach two classes at Lehman including Sociology of Healthcare, and Death, Dying and Bereavement, but I have recently developed several new courses on the sociology of disability and data analysis in the social sciences that I will be teaching in the years ahead.   In any case, I may be a teacher, but I have learned so much from CUNY students, who have really taught me so much and inspired me in countless ways.  I think the diversity of the student body and what they bring to the table, in terms of experiences, perspectives, etc., makes for such a dynamic and engaging classroom environment, especially when one is teaching courses in sociology. 

Now, congratulations on your NSF grant! After learning about your interests in a wide variety of things and subjects, your conducting this interdisciplinary project makes sense to me. Will you tell us a little bit about it?  

Thank you! This project is to develop a Numeracy Infusion Course for Higher Education (NICHE).  This course will teach faculty in various disciplines how to infuse Quantitative Reasoning (QR) throughout the curriculum. We have a team of more than a dozen faculty and staff throughout CUNY who are involved in this project, including two co-PIs, Professors Dene Hurley at Lehman and Frank Wang at LaGuardia, as well as a research advisor, Professor Elin Waring at Lehman. The NICHE project represents an extension of several other QR initiatives I have been involved in over the years. For example, I am currently the co-director of Lehman’s QR initiative. From 2004-2007, I served as the PI on a different NSF-funded project to improve sociology students’ scientific literacy by Integrating Data Analysis (IDA) throughout the curriculum. That project represented an adaptation of an initiative developed by researchers at the American Sociological Association (ASA) and the University of Michigan, for which Lehman had been involved.

As an aside, I don’t know if I would ever have embarked upon all these pedagogical if hadn’t come to CUNY. As you know, many CUNY students are first-generation college students who have experienced tremendous hardship, particularly in terms of economic conditions but also with regard to college preparedness.  I have an article that states that the quantitative literacy gap of minority students is a key factor in explaining their underemployment and unemployment and I have seen firsthand the challenges that our students encounter on the job market.  I feel like we owe it to them to better prepare them for this challenging world we live in and sincerely hope our project helps!

I sincerely hope so, too! It’s customary during my interview to ask the professor about her/his hobbies and interests outside their academic entity. What do you like to do in your free time?

Free time…

Um, in your non-academic time? If you have any?

(Laugh)  I love spending time with my family, especially my daughters, who are 2 and 5.  And I enjoy all kinds of activities ranging from reading to road trips to outdoor activities such as swimming, hiking, etc.  And I still enjoy arts and crafts… I’m really hoping to learn how to knit sometime in the next several years.

Well then! I’ll have to connect Professors Wilder and Dedlovskaya (my 2/6 post) and have them start a knitting club. Okay, it’s time for full disclosure. We DID have a face-to-face interview over a healthy lunch (a chicken caesar salad for her and spicy shrimp rolls for me). However, this post is mostly based on Professor Wilder’s WRITTEN responses that she sent me via E-mail. Her stories being so fascinating, it was not an easy task for me to create this shorter version (Please contact me if you’re interested in reading the full version, which I know you are!). At the end of her E-mail, Professor Wilder challenged me to go one-on-one in basketball with her, to which she added, “though I’m really out of practice these days and you might win!” This post may have an unprecedented follow-up entitled “One-On-One with Professor Esther Wilder.”

(The interview and E-mail correspondence took place on March 7 and 9, 2012)

There are certain arguments that are beautiful and convincing but not admissible as proofs. They give the right answer, and show in an important way why something should be true, but they don’t pass muster as formal mathematics. Frequently they are shunned as “intuitive” or “heuristic” arguments that aren’t sufficiently “rigorous.” Certainly there is something about them that seems to lack deductive strength. But I am sometimes saddened by the injustice of relegating these stunning ideas to the apocrypha, especially when they served as prologue to a famous result.  It can be interesting to summon up the feeling that these arguments are “right” in a way that we perhaps do not understand.

Here I want to look at a few of these mysterious quasi proofs.

The first and probably most famous is Archimedes’ argument of the lever, which he used to find the quadrature of a parabola. Quadrature refers to the problem of finding a region bounded by straight lines which has the same area as a given curved region.  In the problem I will present, Archimedes is trying to find the quadrature of a parabolic segment. There is a very nice discussion of the problem on Wikipedia.

A picture of a parabolic segment is given here:

A parabolic segment

As Wikipedia reveals, Archimedes formally established his result using the method of exhaustion, and described his proof in a letter to Dositheus.

But in his work The Method, Archimedes establishes the result by recourse to a magical argument.

Consider the following picture.

The argument of the lever

We will go into the picture in detail, but the main idea is that the parabolic segment defined by the parabola and the line , when dangled from , balances the triangle which is suspended on the lever from its center of gravity.  Archimedes then concludes, from the placement of the fulcrum, that the triangle is exactly three times heavier than the parabolic segment, and therefore must have three times the area.

For those who are curious about the construction, the line containing the segment is drawn to be tangent to the parabola at .  We then draw to intersect the tangent at , with parallel to the axis of symmetry of the parabola.  This yields the triangle .

Now draw to bisect and then produce the line to so that .  Now consider an infinitely thin “slice” of which I have drawn in red and green.  Archimedes shows, using mechanical principles, that the whole slice (both red and green parts) balances at the fulcrum with just the green part suspended from .  Since this slice was arbitrarily located, the argument is completed by “summing” over all of the infinitely many slices of the triangle.

Interestingly Archimedes remarks in his argument that the slices of the triangle and the slices of the parabolic segment are equal in number. As Katz says in his book A History of Mathematics, this makes one wonder if Archimedes had some notion that there are different infinite cardinalities.

Before going on to the next magical argument, I want to remind the reader about a theorem from geometry. The idea of a “dynamic triangle” is the key ingredient in Euclid’s proof of the Pythagorean Theorem, and is given in Book I of The Elements.  Consider the following picture.

ABC has the same area as BDC

To explain the picture, suppose that a triangle is given of a certain height .  Draw a line through parallel to and on that line select a point .  Then the areas of the triangles and are the same.  It is hard not to see the triangle as a “moved” version of . This dynamic property is a powerful source of geometric intuition.

The following argument for the area of a circle was given by Kepler.

Kepler's circle

The idea is to divide the circle into infinitely many slices of pie, each one so thin that the circular segment at its base is virtually a straight line.  One then “peels” the circle as shown in the picture to form a triangle of height , where is the radius of the circle.  By the argument on dynamic triangles, each pie slice has the same area both in the circle and in the peeled version. Therefore the area of the circle and the triangle are the same.  But the triangle has height and base (because the base is the circumference).  By the familiar formula for the area of a triangle, the area of a circle must be .

Kepler’s argument naturally makes one wonder if something analogous can be done for the sphere.  Recall the following theorem on the “dynamic” properties of a cone.

A volume preserving movement of a cone

Though we usually only learn the formula for the volume of a “right circular” cone, the formula is much more general.  A cone of any base and height has volume where is the area of the base.  This gives a “dynamic” picture of cone volume similar to the one for Euclid’s triangle — movements of the “tip” of the cone in the plane parallel to the plane of the base do not affect the volume.

Now (I will not attempt a graphic for this!) decompose a sphere into infinitely many cones, all meeting at the center.  To aid the imagination, suppose that the base of each cone is an infinitesimal hexagon, which is completely flat.  Now peel the sphere and lay it flat, while keeping the center unmoved.  The cones (whose volumes have been preserved) now unite to make one cumulative cone in a way similar to Kepler’s triangle.  The base of this cone will have area equal to the surface area of the sphere, and the height of the cone will be , the radius of the sphere.  This shows that the volume of a sphere should be one third its surface area times its radius, which is in fact the case, if you consider the traditional formulas:

   and  

Note that this uses a special property of the sphere:  all of the original cones have the same height .

The above arguments are similar to the following magical method of finding the volume of a torus.

A straightened torus

Finding the volume of a torus is a fairly challenging exercise for students in Calc II.  The exercise is considerably simplified if you snip the torus along the green circle and then pull it straight.  The volume of the resulting cylinder is the same as the torus. One could alternatively think of this as a “restacking” of the infinitely many discs of which the torus is composed.

The last example of a magical argument I borrow from Jerome Keisler‘s freely downloadable Elementary Calculus: An Infinitesimal Approach.

It is an example of something I find especially intriguing:  an infinitesimal diagram.

A version of this diagram is used in Chapter 7 to support the argument that

Here it is:

An infinitestimal diagram showing a differential triangle for sine and cosine

As students of calculus will know, is a small perturbation of the angle .  By the definition of radian measure (the diagram shows a unit circle) the red line in the diagram is also equal to . By imagining that is infinitely small, we can treat the red segment of the circle as if it were straight.  From basic trigonometry, it then follows that

or in other words

which is the familiar formula for the derivative of the sine function.

What do all of these arguments have in common? All of them are highly visual, and involve the infinitely small.  None of them are deductive.  There are no established axiomatic systems currently in existence that justify them.

And yet each of them is, in its own way, evidence for the truth.  From a modern viewpoint it is easy to say that these crude ideas have been replaced by the more precise idea of the limit.  But that is too dismissive of the heuristic power the arguments have to point the way towards plausible theorems.

I think something even stronger could be true, which is that the arguments are valid in the context of some logical superstructure which has not yet been discovered.  Our enormous store of mathematical knowledge has the potential to give us a kind of epistemological chauvinism.  To combat this, we should try to see ourselves as we see the mathematical figures of the past: in pursuit of something powerful, but not yet grasping its full nature.  I find it particularly humbling to consider the Egyptian and Babylonian mathematicians who could not or did not conceive of fractions with non-unit numerators, which could have greatly simplified their calculations.  There is no reason to be confident that we are not still similarly benighted.

There is a mysterious relationship between the curved and the straight, between things with magnitude and things without.  We have found deductive work-arounds for dealing with some phenomena related to these dualities, but they are still out there, and we still do not understand them.

A 30-clue Sudoku puzzle. Image from WikimMedia Commons

Sudoku (rules can be found here) has always had a problematic association with mathematics.  A first reaction by the ‘man-on-the-street’ to all those numbers is that it’s “too math-y” —  ironic, since the numbers are just employed as placeholders and could easily be replaced by letters, pictures, or other symbols.  A Sudoku fan would counter with the (equally ironic?) retort “it’s not math at all — it’s just logic.”  In any case, the pattern-searching and deductive reasoning employed in teasing out the unique solution are skills any mathematicians would find familiar.

However, the real math behind Sudoku starts to rear its head when we turn the problem around, and try to build Sudoku puzzles.  We have to fill in enough boxes so that there is only one correct solution, but few enough that determining that solution presents a challenge.  Generally speaking, the more clues we fill in the easier the puzzle becomes to complete.

Project idea:  Work with a student (perhaps a motivated student studying probability or discrete math — anywhere that you are discussing combinations and permutations) to describe all possible completed Sudoku puzzles.  Even this may be quite hard, in the end, but the process would be illuminating — just finding different ways of describing the problem would be a good challenge.  How many ways of filling in nine digits in a 9×9 grid?  How do we begin to eliminate “repetitions” in rows, columns, and regions?  Etcetera.

An obvious question presents itself – what’s the “hardest” puzzle we can make?  How few of the boxes can be filled in, and still give a unique solution?  Examples have been found with as few as 17 clues.  On the other hand, it is not hard to see that very few clues will not work — if we try to build a puzzle with only 7 clues, the solution cannot be unique (since we have only used 7 of nine digits, any solution we obtain can be transformed into a different solution by swapping all occurrences of the remaining two digits).  But the gap between 7 clues and 17 is quite large — yes, it has been reduced further, but not by as much as you might expect (until this year, see below).

It is interesting to reflect on the status of this work in the relatively recent past — this post by Gordon Rolye on his blog SymOmega is about 15 months old, and asks (but doesn’t answer) the question “Is there a 16-clue Sudoku puzzle?”  In one of the top comments, he provides a reasonable argument why he expects the answer is no:

Well, it is still open, but my hunch is that there is NO 16-clue Sudoku puzzle. A single 16-clue Sudoku puzzle would give us a whole bunch of 17-clue puzzles and given how hard it now seems to eke out a single extra 17-clue puzzle, I would guess that it just doesn’t happen.

Computer-aided proof, and No 16-Clue Sudoku

On New Year’s Day 2012, a team of researchers posted this article on the arXiv: There is no 16-Clue Sudoku: Solving the Sudoku Minimum Number of Clues Problem.  Their proof is an interesting combination of subtle mathematics and brute-force computing.  One might ask why we can’t just check all the possible 16-clue puzzles, and here we run into trouble with exponential growth — there are simply too many (waaaaaay beyond the combined computing power of every processor ever built by man).  The mathematical subtlety arises in reducing the number of possible 16-clue puzzles to check (this is not,  strictly speaking, the method, but it captures the essence).  Even so, the number of possibilities is enormous, requiring about  7 million CPU hours to check them all.  In the end, no such puzzle was found having a unique solution.  Conclusion?  The minimum number of clues required for a Sudoku puzzle must be 17.  Here’s a nice video on the subject from the numberphile blog.

This amazing, and unsatisfying.

Amazing, because we have leveraged our  hands-dirty, real-world, physics-harnessing technological abilities to extend our purely mathematical knowledge (where usually the information flows the other way, with mathematics offering assistance to the more-applied disciplines).

Unsatisfying for at least two reasons:

First, how do we know it’s right?  No human has actually checked all the possibilities (nor will they!).  Here, we rely on the correctness of the hardware and software that ran the computation — and, given our daily difficulties with computers, this is a leap of faith on a pretty high level.  However, this is a tractable problem.  For example, it is much easier to check the algorithm used than to check every aspect of the hardware and software that implements it — and if the algorithm is sound, and produces consistent results regardless of implementation, we can have some confidence that it really does what we want it to do.

Second, and more troubling from a mathematical perspective, is that this proof doesn’t offer any kind of insight into the problem.  As mathematicians, we have a motivation to ask “why?” and we take great satisfaction when we finish a proof and finally understand not just the result but the reason behind it.  Indeed, when a mathematician shares a proof with a colleague, the essence of what they are trying to communicate is why the result holds.  So we ask: Why is 17 the minimum number of clues?  How can we generalize this to higher-order puzzles?  What’s the logical reason that no 16-clue puzzle will have a unique solution?  The result is true, but the reason is still beyond us.

Below I will give what I think are controversial opinions on mathematics education. It isn’t my intention to offend anyone, only to motivate discussion. I also want to say in advance that when I talk about higher math skills I am not talking about problems with basic numeracy.

I subscribe to a heresy. The heresy is this.

The idea that many people need to know a lot of higher mathematics for utilitarian purposes is patently not correct. The mathematics that the vast majority of people need for work is summarized in tables, or done by algorithms prepared by others. Certainly, it is beneficial for a small number of self-selecting and creative people to know mathematics. But there is generally no materialistic benefit to society for teaching most college students one mathematical fact rather than another – to factor quadratics rather than compute toy probabilities, for example.

Of course, materialistic results are arguably not the principal value to society of mathematics education, though we often present them as such.

Humanity has long believed that something important happens when a person learns to perform abstract reasoning. Plato’s inscription above the Academy stated that only those who had studied mathematics could enter. The Rhind Papyrus, written 4000 years ago, asserts that accurate reckoning is the entrance into the knowledge of all existing things, and all obscure secrets. There are many other authorities who say something equally grand and mystical.

Somerset Maugham observed that intensive training in any subject changes one’s view of the world. To justify mathematical education, after abandoning claims for its utility, the change that attends mathematics must not only really exist, but also be unique and desirable. This presents the question: what is the effect of a mathematical education?

The effect of a mathematical education is the ability to process abstraction–to solve problems which are not specified all the way down to the sense-particulars. This is the ability to handle things which have, rather than sensory qualities, only some kind of blank existence: objects which are ontological (and perhaps literal) ciphers.

The idea of a placeholder for a real-world entity is very powerful and mysterious. Some people seem able to handle this with amazing facility and others not at all. Those who can do it are generally those who are able to solve problems.

What is a problem? A problem is an emergence, an exception, something for which there is no known protocol, at least not known to us. It is not always clear how the entities in a problem should be understood. It is fortunate then that there are those who are able to understand things without understanding them– that there are those who are able to move inchoate entities with numb hands and blind eyes into felicitous configurations that are intuited by a seemingly miraculous 6th sense. And then, when correctly arranged, the problematical objects tend to leap into real existence, to develop sounds and colors and textures. They have been turned and displayed artfully, and sanded so that our intuitions can gain traction. At this stage, a problem has been solved.

There is a notable dissimilarity between what I describe above and what takes place in a typical mathematics class, particularly at the low level. I am not sure that Plato would have much use for a graduate of a college algebra course.

We substitute algorithms for thought. A glance at a typical pre-calculus syllabus shows a numbing litany of processes and procedures that lay so thick on the ground that a view of anything interesting is completely impossible. And this characterizes all the mathematical education that most people ever see, with the possible exception of high-school geometry (which is perennially under threat).

Why?

This is a difficult question to answer. My conviction is that teaching algorithms is easier. It is easier for the educational agencies because it creates an objective measure of success, and enforces a minimum amount of achievement. It is easier for the teacher, who may be unsuited for adventuring. It is certainly easier for the students, though they trade exhilaration and difficulty for agonizing monotony and boredom.

The ability to solve problems varies wildly; requiring genuine problem solving in a class can instantly give rise to a satisfied elite of competent students, and a miserable complement of failures. This situation is such an anathema that we prefer the frigid status quo, which at least puts a balm on the unpleasant and difficult realities which result from compulsory mathematics classes. The effect is that even though we receive more training in mathematics than any society in history, it is sapped of its real worth.

To the extent that students are allowed to think, however, they certainly intuit the difference between the sunshine of open-ended investigation and the pabulum of algorithmic methods. At certain times we can, despite everything, find our way through the endless jungle of canonical tautologies in introductory courses, and introduce a little light.

There is a general feeling that the internet and our new “connectedness,” have tremendous power to foster new communities.  However, sometimes the internet can feel like a very lonely place (in the same way, I suppose, that New York City can be the loneliest crowded place on earth).  I wanted to point the way towards two amazing communities centered around mathematics.  They are both long-running and large enough to have developed a “community feeling,” and have enough dedicated members to keep that feeling positive and continuous.  They are both “question and answer” type sites, but the questions and answers often spark additional commentary and sometimes lively debate.  The target audiences are quite different.

The first, math.stackexchange.com, is aimed at people studying mathematics at any level, from high school to graduate work.  Can’t figure out how to factor that polynomial?  Want some broad context on the challenges of integration?  Can’t prove that theorem in group theory?  Post your question here.  Homework questions are welcome, if asked honestly and explained properly, but don’t expect to post your assignment and return to find all the answers — for the most part, there is a nice culture of explanation, encouragement, and providing hints.  Some sample questions:

• Factor the binomial 
• Rotating two vectors to point in the same direction
• Prove: If and and are nilpotent, then is nilpotent.

The second, mathoverflow.net,  is aimed purely at research-level mathematicians.  A number of world-class mathematicians, including several fields medalists, post regularly on this site. Working through a famous paper, and don’t understand a key step in a proof?  Want to know if the hypotheses of a given theorem are really necessary?  Have a question, and not sure if it’s been studied before?  Studying something out of your area of focus and want some information on background, relevance, or sources?  This is a great place to ask.  Beware, the “research-level” condition is strictly enforced, and askers are expected to put a little effort into seeing if an answer exists on the internet already.  Violators are gently directed to math.stackexchange.com (referred to as “math.SE” on the site).

• Is there a natural random process that is rigorously known to produce Zipf’s law?
• How to memorize (and understand) Nakayama’s lemma and its corollaries?
• There are few “big-list” type questions (as opposed to focussed questions with a single answer) — these questions are not really encouraged, but past instances have provided some very rich reading, including:
  A single paper everyone should read

Sometimes I see something in the news that just makes me excited about math.  This is the kind of development I would love to explore with a class if I had a few days or weeks to spare!  On the other hand, it might form the basis of a great project for a motivated student or group of students.

The company Lytro has introduced the first commercially available light field camera.   A traditional camera captures the intensity of light (including relative color intensity) arriving at every point on a film or, in a digital camera, a sensor.  A light field camera captures both the intensity and direction of light as it strikes the sensor — it record vectors, rather than scalars (and, in particular, in place of a single scalar it must record multiple vectors, since many different light rays will be striking in the same place).  What is the practical upshot?  Among other things, you can change the focus after the picture is taken!  Follow this link to give it a try (click on the picture to change the focus).

Project ideas

1. Learn about focus, and how traditional cameras record a picture.  Discuss limitations.
2. Learn about vectors and light fields.  There is a range of mathematics to explore that could be adapted to different skill levels, from an introduction to vectors up through the algorithms used by the camera.
3. Explore the practical and mathematical possibilities of a light field camera.  If you can get your hands on one (they’re about $399), take some photos and do a demo!
4. How does this “lack of focus on focus” affect the process of photography?  Does it affect the way you frame your shots, the kinds of subjects you choose, and so on?

Resources

An overview for a general audience can be found on the Lytro “Science Inside” page.  For more details, check out Lytro CEO Ren Ng’s surprisingly readable Stanford dissertation on the subject.  This is only the latest work to come out of Stanford’s Computer Graphics Laboratory on light fields and related technologies.

He is a doer. You can easily see that about Professor G. Michael Guy, Assistant Professor at Queensborough Community College, just by taking a quick glance at his website. Being a CUNY Improving Mathematics Learning grant winner and a 2011 Chancellor’s Mathematics Instruction Award winner, he never seems to be fully content with his present self and is always striving for a better and improved self. Professor Guy picked one of his favorite burger joints on the Upper East Side for our interview.

You are a self-claimed food-lover, but are you a vegetable-hater? I noticed your burger doesn’t have lettuce or tomato on it.

I’m not a vegetable-hater, but tomatoes are one of the very few things I don’t really like. I’m still a food-lover.

Okay. It’s always fun to talk about good restaurants and different kinds of food, especially in New York. What’s your favorite place to eat?

I like to try new things and have spent lots of time in Flushing in the last few years. There’s a food court at the bottom of the New World Mall on Main Street, and it’s a fun place to have a food adventure. You can try out all kinds of Asian dishes.

What else do you enjoy in your free time other than eating?

When I’m looking for a change of pace from doing math and work, I often spend time programming. I have two fun websites I launched over the last year focused on strategy for two popular games Hanging with Friends and Scramble with Friends. I programmed solvers and analyzed the outcome to help people improve their game play. Some may think using my websites are cheating, but I think this is just a highly informed strategy.

Do you like music?

I don’t listen to music. There’s no music on my iPod.

(Gasp!) I’m shocked. I assumed EVERYONE likes some type of music.

Not me. It mostly sounds like noise to me.

Well then. Let’s talk about math education. One of the projects you’ve been conducting with Professors Cornick, Holt, and Russell at QCC is Arithmetic WARM UPS. It’s a modularized and accelerated program for students in the basic arithmetic course, which has been producing positive outcomes.

Yes, this is the second semester to scale the project to about 1,000 students and 20 instructors. It’s a lot of work to track the data of this many students and instructors, but the results seem to be equally good.

According to your website, you’re working on some other projects as well. Tell us a little about them.

Some of my colleagues and I are currently working on a small, locally-funded study. We’re focusing on helping students in remedial math courses develop good study skills and study habits. Our department and our college as a whole are very supportive for trying new ideas. I’ve been learning a lot through these projects, becoming a part of research communities and implementing interventions based on data and outcomes, not just on our intuition.

It’s evident that you and your colleagues care so much about your students. Did you have a good model? Who was the teacher who influenced you the most when you were a student?

I have been influenced by many teachers, including the ones who taught me things like how you shouldn’t treat your students. Of course, I have learned from many great teachers and professors also. As far as the best and most influential teacher goes, it was Dr. Pat Rickels at the University of Louisiana. She was the director of their honors program when I was an undergrad. She was one of the people who helped open my eyes to the world beyond myself, and taught me to appreciate it.

The topics of our conversation ranged widely from some stereotypical questions we have gotten based on where we’re from (e.g., “Do you have alligators in your yard?” “Do you eat sushi every day?”) to our pets (his cat Clara and my dog Basil; we had to show their photos to each other, of course). Or at least we seem to have talked about them. The burger and fries, with chocolate chip custard in addition to them, made the accuracy of my memories highly questionable. Thank goodness I recorded the entire session.

(This interview was conducted on February 10, 2012)

I’m nostalgic for an era that never included me–the days of the log table.

Eli Maor’s book e: The story of a number, turned my attention to the history of logs.  He tells the story of the Scottish wizard Napier with easy but brainy discursions into the nature and imperfections of the prototypical logarithm idea.  Here is a summary, cribbed from Smith’s History of Mathematics, of early suggestions for a logarithm function (expressed in terms of the modern definition).

As a sample of human flavor, relating to Napier’s relationship with Henry Briggs:  Upon their first meeting, these two men regarded each other in silent admiration for fifteen minutes before speaking.

I did not learn from Maor (but rather from Wikipedia) that Napier was also thought to carry a black spider in a small box, and that the black rooster was his familiar spirit.

But there are a few things I’ve come across that I find equally strange.  One is a quote from Hume:

“[Napier is] the person to whom the title of ‘great man’ is more justly due than to any other whom his country has ever produced.”

This is strong praise from a great historian living over a century after Napier died.

There is the well-known comment of Laplace, describing the logarithm as:

“[a]n admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations.”

Both of these remarks make clear that the advent of logarithms were something of considerable significance in the history of science, if not civilization (the logarithm idea rapidly spread around the globe).

I have been guilty of presenting logs to my students in their now watery vestigial form as just one more purposeless graph to learn (or not) and manipulate (or not).  But they have a character of their own and a history that makes them into something more than just an interesting curve with curious properties.

The logarithm is (or was) useful. Napier was responding to a particular problem: multiplication is hard.

How hard is it?

If you multiply two six digit numbers, such as 329874 and 718765, you can see that the traditional method requires you to do 6×6 = 36 single digit multiplications, and then roughly another 36 single digit additions.

So in short, multiplying digit numbers requires about operations.  Anyone who has ever graphed a parabola knows that this is going to present problems as becomes large (as it very well may in problems based in the sciences).

So what did Napier propose to do about it?  Roughly, he hoped that each number could be somehow be encoded into a number in such a way that .  If this were true, and is not much bigger than , then you can compute by decoding .  If the encoding/decoding can be done quickly, then the benefit is that the arduous process of multiplication has been transformed into the relatively easy process of addition.

Napier is said to have been inspired by the trigonometric identity

which has a suggestive similarity.

Something that strikes one when reading Napier is the hard-as-nails functionality of what he was constructing.  Messy (and large) constants abound.  There is no easy theoretical elegance in his writing, but rather the feeling that something is being made to work.

The reduction in computational complexity accomplished by logarithms is of course not free.  Much of it is hidden in the construction of enormous lookup tables, which Napier had to compute by hand (and with the help of his other aid-to-computation, his eponymous rods or, corruptly,  bones.)

But the computational price must only be paid once, and literally for all, by the willing sacrifice of those such as Napier, Briggs, and many others who constructed the tables that were in use by scientists and engineers for centuries.

The complexity of multiplication is reduced to the linear complexity of addition and the (ironically) logarithmic complexity of looking up an element in a sorted list.

The original A Modest Proposal, written in 1729 by Jonathan Swift, was one of the most scathing and viciously ironic pieces of political satire ever (if you are unfamiliar with it, he proposes the Irish poor sell their 1-year-old children to the wealthy for consumption, thus “solving” both the food shortage and the prevalence of poverty in one dire swoop) .  So when I opened the February issue of the Notices of the American Mathematical Society and found an article with the same title, on math education reform no less, I put on my hard hat and prepared to have my ears burnt.  Here’s what I found:

First, it’s worth a read.  The author, Alan Schoenfield, professor of mathematics and education at UC Berkeley, makes an argument that is familiar but valid: that students are not being taught to solve problems but simply to carry out proscribed algorithms in a mindless fashion.  He offers some great (terrible!) illustrations of this trend, including this one (the article referenced is K. Reusser, Problem solving beyond the logic of things, Instructional Science 17 (1988), 309–338):

Reusser [2] asked ninety-seven first- and second-grade students the following question:

There are 26 sheep and 10 goats
on a ship. How old is the captain?

More than 3/4 of the students “solved” the problem,
obtaining their answers by combining the integers
26 and 10.

He gives more examples along these lines, contrasts them with some of his own more positive learning experiences, and final gets down to business:  What do we do about it?  This is the part where I retreat, turtle-like, into my protective shell and prepare for apocalypse.  Here it comes:

I truly believe that all of the mathematics in the K–16 curriculum—or at least all of the mathematics that should be in the K–16 curriculum!—can be seen as a set of sensible answers to a set of reasonable questions. My immodest proposal is that we revise the entire curriculum so that all students experience it as such, so that they come to see mathematics as a domain that not only makes sense, but as one that they can make sense of. On a more truly modest scale, I propose that we all, each time we teach, stop to think about how and why the mathematics fits together the way it does and how we can help our students to see it that way. We owe our students no less.

Does it make sense? Sure! Do I agree? You betcha!  It’s a well-reasoned, well-intentioned seed of an idea.  If this kind of revision were carried out carefully, it would do a great deal to help the abysmal situation of math education in America.  But where oh where are the dead Irish children?  Where is the shocking idea, the part that makes you snap your head and say “Wha? I can’t believe he just said that!”  The proposal is great — it’s just too modest.

My disappoint me stirred a faint memory, and (with some digging around in my files) led me to revisit this less-than-modest proposal made by Joseph Malkevitch in his 1997 article Discrete Mathematics and Public Perceptions of Mathematics.  Here goes:

Let’s get rid of Algebra.

Algebra is the mainstay of our national math education industry at the high school, and in many cases even at the college, level.   A typical student entering CityTech, where I teach, might take two full semesters of credit-bearing algebra prior to Precalculus.  Malkevitch argues that, for non-STEM majors, the emphasis on Algebra is just not worth the struggle.   If a student is going to take one college math class, let it be something relevant, something that builds problem-solving abilities and gives them the experience of wrestling with practical problems and perhaps extracting some general principles therefrom.  He makes a strong case for Discrete Mathematics as an alternative “College Math” experience, contrasting typical algebra examples, such as:

Simplify:  

With alternative questions from discrete mathematics:

Design an efficient route for a pot-hole inspection truck, which must inspect every stretch of street in the street network in Figure 1 at least once, and which starts and ends its tour at the location marked A. (You may assume that the streets are two-way.)

He challenges the reader to consider these (and other) examples in several ways: What learning opportunities does the problem present? How accessible is it to a non-technical audience? What skills are being tested? How relevant are those skills to other areas of your life?

I highly recommend reading the first two pages of Malkevitch’s article — if they catch your attention, you’ll find the rest is worth a look as well.  I’m not ready to start crusading for Discrete Mathematics, but I’m willing to consider demoting algebra from it’s pre-eminent position.  Let’s talk about alternatives!