Math in the Media: Gender Bias in Teachers?

Yes, it does take a village to raise a child....  (H/T to The Honorable Hillary Rodham Clinton.)  But it seems it also takes a village to discourage girls from achieving their full potential in math and math-related fields....  (Sigh!)

In a new study, headed by Edith Sand, an economist at the Bank of Israel and an instructor at the Tel-Aviv University’s Berglas School of Economics, teachers themselves contribute to the problem of too many female students shying away from higher-level math courses as they progress in their education.  The study found a gender bias in performance evaluation;  Teachers who knew the identities of their students tended to grade girls more harshly and boys less so on exams than teachers who did not have any information about the students.   Unconscious or not, our influence as teachers on students always goes far beyond the content of our lectures and exercises.  But this influence may not always be constructive.  What care we need to always take....

The article, reported by Linda Carroll for Today, is  here:

Teacher Bias May Help Discourage Girls from Math, Study Finds

The study is published by the National Bureau of Economic Research in Cambridge, Mass.

Depressing...?  Yes.  Hopeful?  Also.  Knowing of an unconscious bias can contribute to its cure, eh? 

Remember School House Rock?  "Knowledge is Power"

Math in the Media: Homer vs. Pierre?

I just had the pleasure of watching a neat 8 minute video detailing some of the mathematics injected in to the Simpsons animations.  Apparently, there are mathematicians among the creative staff who cannot help themselves throwing in a little math humor into the background every so often. 

The video, listed here on YouTube by Numberphile is titled

Homer Simpson vs Pierre de Fermat

 Do give it a watch.  It is always good to know where the subliminal messages about how cool math really is are lurking, no?

What is Mathematics? Passages....

So I was asked during the break between the semesters to take part in an Intersession course designed to bridge the void between the sciences and the humanities (mind the gap!) by having professors from all stripes discuss common topics to a diverse audience.  The idea is that each professor sees the topic from the perspective of their chosen field and the same topic often looks quite different to different people.  It is a wonderful idea hatched and developed by Dr. Kristin Cook-Gailloud, the Director of the Program in French Language and Culture here at Hopkins.  This is the second year running this course and I thoroughly endorse it.  Alas, due to scheduling issues, I did not participate this year.  But a topic in this year's course, Passages, stuck in my head.  The idea of movement from one state to another is something innate to a mathematician, if regarded as movement from a state of ignorance and confusion to clarity and enlightenment. 

So I wrote an essay to clarify my idea of passage in mathematics.  It is here:

Passages in Mathematics

Enjoy and do let me know what you think....

Math in the Media: Prime Gaps....

I am always amazed at how some of the most vexing, curious and fascinating puzzles in mathematics can be stated so simply, even as they evade solution or even complete understanding for centuries.  It is one of the more alluring aspects of this trade. 

Here's one:  Just how big can the gaps between consecutive pairs of prime numbers get as one traverses the natural numbers out toward infinity? 

One would expect the gaps to get larger and larger and also tend toward infinity in the long run, no?  But showing this, and providing some sort of measure of the growth of the size of the gaps as one goes "out there" has been remarkably elusive. 

I'll let you read this nice article by Erica Klarreich in Quanta Magazine, to "see" that progress has recently been made, and there is promise of more progress coming. 

Mathematicians Make a Major Discovery About Prime Numbers

Will math ever cease to amaze....

Playfully Serious Math: A glimpse at Vi Hart and Fibonacci

I am often struck by just how repulsive mathematics is to some people when, in my eyes, it is all an absolute kaleidoscope of color, art and logical splendor.  But it is not always easy to get someone else to see what you see.  This is what education is all about, I guess.  One step at a time....

I was recently turned on to an absolutely wonderful math and science educator whose videos would do well to provide the backbone of the next generation of the Common Core, at least in math education.  Vi Hart is a videographer (is that what one would call someone who makes videos) who specializes in a playful, though very serious approach to expose and illustrate complicated science concepts and techniques.  One of her series, entitled Doodling in Math Class, exposes the rich, playful and beautiful structure inherent in every math class but lost in the tedium of sterile, and solely utile function.  Below is a three part video explaining why and how the Fibonacci Sequence (not a series, really) shows up so often in nature.  It is mind-boggingly well done, IMHO:

Doodling in Math: Spirals, Fibonacci, and Being a Plant

The other two parts follow immediately from this one.  Give them a look!

BTW, THIS is what mathematics is really about.  Vi's money quote (at the end of the third part):

This is why science and math are so much fun.  You discover things that seem impossible to be true, and then you get to figure out why it is impossible for them not to be true.

Math in the Media: TEDx and me....

Late last year, I was asked to give a TEDx talk (the 'x' means locally organized) for the inaugural TEDx event here at Hopkins (called TEDxJohnsHopkinsUniversity).  I gladly accepted, seeing it as a chance to say something I've been wanting to say for a while:  I wanted to give a talk on what mathematics means to me and why I chose it as a lifestyle.  On February 22, 2014, here on campus, I gave the talk, entitled "Why Mathematics?".

Here it is in full:

Why Mathematics?

It was a great experience, and the organizers did an excellent job.  I hope you find the talk interesting.

Math in the Media - Perhaps the Matrix is....

Here is an article filed under the category "Thoughts to Ponder":  Edward Frenkel, a mathematician from Berkeley, posits that the university perhaps is just a giant simulation and we are simply participants.  How would we know?  Can we detect if we were?  The article is an OpEd in the New York Times, and can be found here:

Is the Universe a Simulation?

Frenkel gives some sense to this idea by differentiating (no pun intended) mathematical ideas (manuscripts, really) from literary ones in the following way:  Mathematical ideas are somewhat universal.  The laws and constructions of Pythagorus, Euclid, Newton, etc., would surely have been created (discovered?) even if these greats had never existed.  It may have taken longer for someone else to develop them.  But the structure of mathematics (its logical framework) exists as it is whether we discover it or not.  Try that with a sonnet sans Shakespeare....

It is a very nice read, this article, and again, gives a sense for how mathematics seems different from other disciplines of study.  Frenkel mentions that many mathematicians consider themselves Platonists, believers that everything exists in the ideal, and what we perceive in this world is simply real versions of that ideal.  It works for me.  I would believe that it would work for most all mathematicians, really.

Frenkel even goes so far as to say that the giant computer simulation that we exist in is, like all computer simulations, not entirely without anomalies, coding inaccuracies that render the coding conspicuous.  Perhaps all of our logic in mathematics is simply facets of the coding that can be detected "from within"?

Certainly a "thought to ponder"....     

Beauty in Math

Mathematicians often talk about their craft in emotional terms.  We get excited by elegant, beautiful, clever constructions and the hidden insight in the logical relationships we uncover.  We can easily be stunned into awe when our intuition leads us astray, and something we did not expect pops out of our reasoning.  And when we see a formula or other type of mathematical construction that not only looks aesthetically pleasing, but contains meaning far beyond its simple symbolic patterns, we treat it as something that should be hanging in the Louvre....

Beauty has profound meaning in mathematics, at least to us.

Not sure you believe me?  Well, a paper just published in the journal Frontiers in Human Neuroscience may just change your mind.  Researchers use Functional Magnetic Resonance Imaging (fMRI) to map the brain activity of mathematicians as they viewed various mathematical formulae and constructions that they have rated on an ugly-beautiful scale.  That part of the brain that is activated when people see beautiful art, or hear beautiful music (yes, there is a specific place)?  Evidently, that place lights up when we view math that we see as beautiful.  At least to us, it is real. 

Give it a read (H/T to CG!!): 

Mathematical beauty activates same brain region as great art or music

BTW, the most beautiful formula of the study:  Why Euler's Identity, of course!  Can you see the beauty?

The importance of mathematical "error"?

I recently participated in an intersession course here at Hopkins (a short, three week course offerd between our fall and spring semesters) called "Thinking though the Fields, A Round Table on Bridging Science and the Humanities at Hopkins"  (sound like a title in need of an acronym, eh?) The course was run by Dr. Kristin Cook-Gailloud.  It was a wonderful experience, with a topic of the day presented in a short talk by a few academics in diverse fields, and then a general discussion about the different interpretations and experiences.  I talked about the importance of puzzles and game playing in mathematics and mathematical research.  Neat.

A short time afterwards, I received a request from some of the students in the class.  They were doing a class project to compile a booklet on topics of a similar capacity.  They asked me to answer a few questions about how mathematicians use error in their work.  My answers are below.  Enjoy:


In relation to your field, how do you define error?

• In Mathematics, we define or use error in many ways.  Perhaps two important ones are:  (1) As a means to study levels of inaccuracy in estimation and approximation, and (2) as a means to address falsity in claims of truth, like proofs.  For (1), mathematics is the study of the logical structure of complicated things.  Many times, these complicated things are systems defined by equations involving numbers.  When used to model something physical, we must accept that our model might not be completely accurate, due to the fact that we cannot properly account for some influential effects in our model.  Think how a model of a pendulum may take into air resistance when predicting its position at some future time, but possibly not that the humidity of the air may affect the constant that we use for air resistance.  Instead of trying to account for everything, we make an approximation and hope that we are fairly accurate in the end, accepting the errors that will accrue, but hoping that they are small.  Also, when modeling mathematics on a computer, another kind of error we see is the fact that computers cannot be precise in the way that we are when doing arithmetic.  For example, there is really no such thing as zero on a computer.  When defining arithmetic on a computer, and assigning numbers to variables we must determine a level of precision (number of bits to devote to a number.) This works well for normal calculations, but when doing high precision work, if one were to multiply an extremely small number to a very very large one, the result may be inaccurate, since the very small number may only be accurate to a finite degree and the multiplication may bring some of the inaccuracies up to the range of what we consider normal numbers.  There is a field of mathematics that studies errors in calculation like these, call numerical analysis.  For (2) , any new mathematical structure or concept or theorem is an abstract idea that must be proven to be consistent with all other mathematical ideas.  Many times, a new idea is claimed to be proven, but under scrutiny by other mathematicians, it is shown to not be proven completely.  There is an error in the proof.  Either the claim is wrong, or the claim is not fully justified as proven.  At this point, the idea is NOT a fact, and dangerous to try to use to help prove other possible facts.  All mathematical ideas claimed to be proven are scrutinized extremely carefully by the mathematical community, either in research paper review, or by other independent verification.  It is a strength of the field that nothing is really proven until verified fully.       

How do you deal with and interpret error in your field of work

• Mostly, the above answer works here also.  For (1), we deal with errors in accuracy by trying desperately to manage it and/or minimize it.  Typically, on a computer, decreasing error means increasing computational time and effort.  Hence there is often a trade off between how accurate you want your answer to be and how long you want the computer (or you) to spend trying to compute the answer.  For (2), when a new idea seems to be proven, a mathematician will immediately go to colleagues and collaborators to have them assess the value, correctness and completeness of the proof.  Errors are often found and arguments (statements of the proof) are changed to address the criticisms.  Once a research paper with some new result (proof) is submitted, there is a formal review process where independent mathematicians with knowledge in a particular field assess the correctness of the proof.  Papers are deemed unacceptable for publication when not correct, and must be reworked or abandoned, depending on the nature of the errors.  Sometimes, when a paper is published with an error, the error must be fixed either with an addendum to the original paper, or with withdrawal of the paper from the journal.  There are no instances where errors are tolerated in mathematical proof.

During your career has erroneous findings led to any key or luminous findings?

• In my work personally, no.  Although some work has not been published due to errors unseen in the original drafts.  However, so much beautiful, amazing mathematical ideas have come from initial errors.  The famous Fermat's Last Theorem, proved only recently but stated 300 years ago, was a simply stated idea that was claimed to have a simple proof by Pierre de Fermat.  Alas, he never wrote down his proof, and the community has been trying to find it for 3 centuries.  The idea is now a fact (theorem), but the recent proof is not simple at all.  However, two things come out of this:  (a) So much beautiful math has been developed in the search of this proof, and (b), it is now basically universally believed in the mathematics community that if Fermat indeed had an idea for a simple proof, it had an error.  We will never know, but....  And in the 70's, Stephen Smale claimed to prove that Chaos (the theory of unpredictability in deterministic mathematical models) does not exist in mathematics.  His proof was in error, and this was shown by another mathematician who produced a counterexample (a single example of something that shows that a supposed fact is incorrect.)  Smale set out to prove he was indeed correct, and in doing so, developed a new branch of mathematics called hyperbolic dynamics, centered around his famous "Smale Horseshoe".  Alas, he only really proved he was originally mistaken, but the error is considered a beautiful one due to what came out of it!

 

Math in the Media - A matter of motivation, not IQ?

I am and have always been convinced that ANYONE can do mathematics if they have the proper motivation, interest, and access to good mentors and training material, at least to the levels found up to and including the first year in college.  I believe that ineffective education and cultural and societal biases are reasons why mathematics education has a mystique about is as something less than the primer coat of all higher level thinking.

So articles on studies like that mentioned here in a Scientific American article

Like Math?  Thank Your Motivation, Not IQ, 

by Tia Ghose and Live Science, are quite refreshing to read.   

To me, the findings are not surprising.  It just seems perfectly natural that a person's motivation to learn a skill is extremely important to their ability to master it.  More important than intelligence?  Ahhh....  Read the article.    

A Find: In Praise of Lectures

Currently vogue in internal university discussions involving education is the idea that the standard lecture format for a course is not the most effective means to educate students.  We here at Hopkins are quite interested in understanding better how serve our entering students in the large-lecture courses we call Gateway Science courses (Our study of this issue here at Hopkins is appropriately called the Gateway Science Initiative.  Also, you can read a JHU-centric white paper on this issue).  I am on the Steering Committee studying this issue.  There is a lot of talk about active learning, and other alternatives to inspire students who do not benefit from the simple instructor-led lectures.

I definitely agree with the idea that the classroom experience could benefit from a purposeful study of how our students acquire knowledge and an active design approach to how we teach.  However, I was always a bit troubled by some of the criticism leveled at the standard lecture format.  I love lecturing, feel comfortable in leading a classroom this way, and see great value in the experience.

It turns out I am not alone.  Thomas Korner, a mathematician in Trinity College at Cambridge University, has written a defense of the lecture format:

In Praise of Lectures

 I find this essay particularly inspiring.  Give it a read.  It definitely says things that I strongly agree with.

Math in the Media: To save this math class, we must destroy it!

Words failed me (mostly) when I read this article today in the Washington Post:

At Virginia Tech,  Computers Help Solve a Math Class Problem

It is a common problem that the transition from high school mathematics courses to those at university can be quite difficult to make.  Even in courses whose content is basically the same, like Calculus AB in the AP system and what most universities call Calculus I, the treatment of that content is much different here at the university.  Of course, students sometimes do not do well.  And I am sure that sub-standard teaching from some of us up here may be a part of it.  We need very much to analyze how we teach and learn to do a better job!  And many of us are.  IN fact, here at Hopkins, we are devoting a LOT of resources precisely to this problem of how to better and more comprehensively educate our incoming students.

But to help cure the "problem" of not-high-enough passing rates by essentially removing instructor face-time from teaching!?!  That is patently absurd in my book.

Mathematics is absolutely NOT about learning a few techniques to apply to standard problems set up to test those techniques, which is exactly what many unit-based, worksheet driven, math courses seem to be like pre-college level.  Porting that type of course to the university level may in fact raise passing rates.  But without the ability to study nuanced mathematical ideas and relationships via discussion and debate (think Socrates), one never learns how to THINK mathematically.  Only to calculate.

Maybe that is what VTech is looking for.  I, for one, am not.  

Math in the Media: Flipping a Lecture???

Here at Hopkins, we are engaging in a project to better understand the general purpose and success of how the large-lecture, so-called, Gateway Science classes (the calculus, chemistry, physics, biology, etc.) in preparing students for the higher-level, specialized study of their future majors. It is a huge affair, and taken as a holistic, university-wide endeavor, has the potential to transform the general curriculum here at JHU in far reaching ways.

But more on that later. There is an interesting article in the Chronicle of Higher Education on turning the standard lecture-type model for university instruction into a much more interactive and enriching experience. The article

How 'Flipping' the Classroom Can Improve the Traditional Lecture

is a very good read. Eric Mazur, a physicist from Harvard, gave a talk here at Hopkins recently on his efforts to enliven the classroom experience. Engaging, he was, and his notion of peer-instruction, whereby students learn by active discussion with their peers while under the direction of the instructor, is just one aspect of the search for new models to engage students and promote a better, deeper sense of learning.

If you are here at Hopkins, you WILL see more of this in the years ahead. For now, give the article a good read.

NPR on the JMM

Well, here is something you do not hear every day: A human interest story on a national radio news program focusing on the joys and wonders of a national meetings of 6000+ mathematicians!

Go figure!!!!

National Public Radio decided to attend the Joint Mathematics Meetings of the American Mathematical Society and the Mathematical Association of America, the national gathering place for the year of all stripes of mathematicians, to see just what was happening there. The result was a report by Ari Daniel Shapiro entitled

A Unique Expression Of Love For Math

detailing the huge diversity of expression and study, both in the art and the science of mathematics, that mathematicians bring to their profession. The transcript and the audio of the piece is at the link.

What a nice way to view the world of mathematics that we see every day, but which most people never get a glimpse of.

Thank you, Ari and NPR!!

I'll be on Cogito.org next week....

I will be hosting a discussion forum on the website www.cogito.org for the next couple of weeks. Cogito is a math and science website and online community for talented youth, and part of the Center for Talented Youth (CTY) family here at Hopkins. I'll be taking questions and offering advice on whatever I can (involving mathematics, I suspect).

It sounds like it will be a lot of fun. I'll post my thoughts here in the interim.

My response to the NYT Op-Ed on Math Ed

Well, it has been a while since I commented here on the New York Times Op-Ed on Math Education and its ills. My rebuttal here in this blog caught the interest of a talk show in California, though I did not reply in time to attend the discussion. I did write a rebuttal to the article and submitted it to the NYT. Alas, it was ignored. Oh, well.

Here it is, though. At least I accept my own submission. Enjoy:

Why Not Teach Math for Math’s Sake?

It is quite conventional wisdom, with lots of supporting evidence, to believe that the way we teach primary and secondary mathematics here in the US is generally failing our young students. This was detailed in the recent article in this forum “How to Fix our Math Education”, by Sol Garfunkel and David Mumford, and I see the general effects of pre-university education in students daily from my perspective as the director of an undergraduate program in mathematics at an American university. I believe the problems discussed in that article are real and demand action, and I applaud the authors for writing the piece. I disagree, however, with the conclusions of Professors Garfunkel and Mumford.

From my perspective, students come to university with a view of mathematics as a giant tool box they carry around with them, the tools being techniques useful to solve many kinds of diverse math-based problems. Pre-university education seems to be filled with disparate situations where a new concept is introduced (abstract or applied) to solve a certain kind of numerical problem, a technique is drawn up and the student receives a worksheet containing 40 or so variations of the same type of problem. Once completed, the class moves on to the next idea. While this assessment of pre-university education is simplistic, the outcome is that students never really learn how to think analytically, reason deductively, understand why these tools exist in the first place, or see just how each idea fits into the whole. Context via applications to real world phenomena (the kernel of the above authors’ proposed solution) may help in this regard, but there is a deeper problem with simply embedding math into applications to prove its usefulness.

The idea that mathematics lives only to serve its applications and functions only as the language of the sciences is absurd on many levels. More like music and poetry than physics or engineering, mathematics is an art, the art of pure reason. One could say that mathematics is the distillation of pure rational thought. When we teach math, we are not teaching how to solve problems. Instead, we are teaching how to think analytically; how to analyze any given complex situation, discover and understand its underlying logical structure, and figure out how to abuse that underlying logical structure to say something useful or conclusive about that situation. Numerical problem solving is solely one manifestation of this process. In the general sense math has very little to do with actual numbers at all. It is just that the use of a number system as one of our basic building blocks allows for a natural logically consistent system. Instead of giving students tools for solving problems, we should be teaching them the very nature of how and why these tools exist and were developed. Questions like why there is a quadratic formula, and why does the sine function exist at all are much more thought provoking and fundamental than how they work. We should be engaging students to actually design and redesign the tools themselves, a process of self-discovery which enables them to own the math they create. We should be giving them the confidence and experience to be able to see a problem as an opportunity for creativity and ingenuity, rather than an obstacle to overcome. And we should be teaching them that mathematical constructions have an innate aesthetic quality. They exist simply for what they are: beautiful constructions, often useful, whose existence lies entirely in the imagination, but whose manifestations in the real world are everywhere.

Math, like music and poetry, has a few constituent parts (notes and keys in music, words with contextual meanings and rhyming schemes in poetry) and a few logical rules which they follow. But with these few rules and parts, no one questions the infinite beauty and variance of musical creations or the fact that a few well-placed and possibly rhyming words can draw such emotion (think Shakespeare). And no one questions the value of teaching primary and secondary students music for music’s sake. Why not teach math for math’s sake? View it as a ground up endeavor where applications can serve as motivations for new mathematical ideas, but where the math lives outside of any application; where the beauty of self-exploration and discovery of fascinating concepts arises simply out of the aesthetic appeal of the constructions; and where the process of developing the skills of analysis and deduction in abstract logical systems becomes the goal of mathematics at the primary and secondary level. The application-based problem-solving skills could come along for the ride, and be reinforced in the other science-based classes. But the math would exist on its own.

Someone once said to me, “When are we going to stop getting students to solve problems and start getting them to POSE problems?” At the research level in math, we design and use the tools we need to pose and solve questions and problems as we need them. Teaching children the rudimentary process of doing this would go a long way to curing our math education woes.

What is that old saying “Teach a student a technique, and she will be able to solve some problems. Teach a student how to develop techniques, and she will be able to solve any problems.” I just made that up. But if we can teach our children to think analytically (read mathematically) before they reach university, imagine what we can do with them in university and beyond.

Math Play: Every Natural Number is Interesting.

I just read this in the first chapter of the text we use for our senior class Advanced Algebra. The book is Introduction to Advanced Algebra by W. Keith Nicholson. I like it, so I will pass it on.

If you study math, you have probably heard of the Well-Ordering Axiom, a property of the integers which is equivalent to the Principle of Induction. The Axiom states: Every non-empty set of non-negative integers has a smallest member.

Intuitively clear, no? Here's how it works in practice:

I claim that every positive integer is interesting. To show this, let's assume it is false (this is a proof technique known as proof by contradiction, in which one assumes the claim is false, then works by deduction until something absurd follows, a clear contradiction. If the logic is solid, only the assumption can be flawed. Thus your original statement must be true. )

Since the statement is assumed false, there must be a non-empty set of uninteresting positive numbers. But by the Well-Ordering Principle, there then must be a smallest uninteresting number. But an extreme element of any ordered set is automatically interesting (in that it is special)! Hence we arrive at the contradiction, making our assumption false and the original statement true.

Silly, yes?

How to (what?) Our Math Education?!!?

Having concern for the state of mathematics education here in America is such a common thing among the population that there are probably almost as many ideas for a solution as there are people with concern. And while the issue is unsettled, it is great to have voices loud enough to keep the discussion lively and vogue. A recent (August 25) addition to the discussion is the New York Times Op-Ed piece

How to Fix Our Math Education

by Sol Garfunkel and David Mumford. I guess it is a hit piece on the No Child Left Behind initiative, but more it is an indictment on the standard idea of teaching math for math's sake at the elementary and secondary level. Their view seems to be that since mathematics was developed in tandem with science and applications, it should be taught that way. Bringing in the deep conceptual beauty of mathematical relationships in a class focusing on engineering or finance would better serve the students' educational needs rather than teaching the pure elements of, say, algebra in their own right.

I will let you read the article and form your own conclusion. My take? Little can be farther from the truth! The applications of mathematics are many, varied and beautiful. But the essence of mathematics is the study and development of pure rational thought. It is precisely the abstract nature of pure mathematics that should be taught to young students in our schools. And it should be taught as an art at every level, with applications only to serve as neat ways to display its innate beauty. The lack of a cohesive story about abstract mathematical relationships and patterns in our math class sequences is what fails our educational systems today. And not the fact that we do not apply math correctly. This is just my opinion.

I found the letters in rebuttal to this article of most interest to me: Read here for some of them:

Math = The Practical and the Beautiful

One real money quote that gives away my take on this whole business? From the Computer Scientist Jonathan David Farley's letter at the end:

You do not study mathematics because it helps you build a bridge. You study mathematics because it is the poetry of the universe. Its beauty transcends mere things.

Pure candy, that quote is!!