Gowers on Elsevier

Fields medalist and Polymath founder Timothy Gowers has some bellicose thoughts on the state of the journal industry.

 

Tao and Gowers

Terence Tao and Timothy Gowers

As we all know, mathematics journals tend to be written, edited, and typeset by volunteers.   This can make their exponentially increasing price something of a mystery (see figure below, borrowed from an article on the Crisis in Scientific Publishing.)  In addition to mundane economic reasons for the climb in journal value, some allege there to be questionable marketing practices at play.  As Gowers notes, Elsevier (a publisher with many desirable titles) insists on bundled sales of journals to libraries, making it impossible to select really essential pieces a la cart.  This has the obvious effect of raising the cost of maintaining a well-equipped academic library.

Though much less visible than high textbook prices, students stand to be affected by these policies through tuition and fee increases.

The Increasing Price of Academic Publications

The Increasing Price of Academic Publications

 

Fellow Fields medalist Terence Tao has seconded some of Gowers’s opinions.  While Gowers has resolved neither to publish nor to referee for Elsevier, Tao will only commit to not submitting to their journals in the future.

It is interesting to note in relation to these issues that Princeton has recently stated that its faculty will publish only in journals which allow the author to retain copyright.

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Disturbing violations – mixed numbers, PEMDAS and more

1. Mixed numbers, the spittle on the front steps of our oh-so-coherent-and-sensible mathematical edifice

I learned to loathe mixed numbers, such as 3 \frac{1}{2}, after dealing amicably with them for thirty years or so.  It was on the Appalachian Trail, I was a new-ish grad student and adjunct at CUNY, and I was hiking and talking about teaching math with friend and mentor Roman Kossak.  Perhaps you have some inkling why I feel this way — maybe you’ve suffered the same realization — but if not, beware: the following thoughts, once thought, cannot be un-thought.  In our discussion, Roman indicated his dislike of mixed numbers and I expressed my surprise, so he asked the following question:

In the number 3 \frac{1}{2}, how are the quantities 3 and \frac{1}{2} connected?

In particular, what operation sits between them?  From early in our mathematical lives we are drilled on the following convention: when two quantities (terms, expressions, etc.) are written next to each other, with no operation in between, then they are connected by multiplication.  But this is clearly not the case with mixed numbers — if we multiply 3 and \frac{1}{2}, we obtain 1.5, but 3 \frac{1}{2} is equal to 3.5.  They are connected by addition. What? What the what?  I’ve hated mixed numbers ever since (not the numbers themselves, which are fine and even useful — splitting up the whole number part and fractional part of a number can be quite handy! — but the abomination that is our current notation).

2. PEMDAS and dividing rational functions

I could go on an on about the order of operations and the accompanying mnemonic PEMDAS (in reality, something like PE(MD)(AS)), but for now I want to focus on the MD.  The rule here indicates that Multiplication and Division have the same precedence, and when a number of multiplications and divisions occur in a row they are to be evaluated from left to right.  This means that when confronted with the expression:

12 \div 2 \times 3
we should first divide 12 by 2, and then multiply the result by 3, yielding and answer of 18.  This is tricky enough for someone wrestling with these ideas, since the “general notion” is that we carry out the operations according to the order they appear in PEMDAS, which would seem to indicate that we do the Multiplication first. However, this kind of discrepancy can be singled out and emphasized to students when they first encounter it, without (too much) harm.  My problem comes a little later in the curriculum, when we confront division of rational functions — and, in particular, when the second function consists simply of a term.  An example of such a problem might appear like this:

\frac{12x^3}{y} \div 2x
How are we to interpret it?  The intended meaning, and the way the problem is inevitably presented, is that the first fraction is divided by the second expression, like so:

\frac{12x^3}{y} \cdot \frac{1}{2x}
This interpretation is quite pervasive — an informal poll of colleagues and graduate students shows just-about-universal agreement.  But following PEMDAS and taking the previous example as a guide, the correct simplification is:

\frac{12x^3}{y} \cdot \frac{1}{2} \cdot x
Once again, WHAT?

Many thanks to my colleague and friend Thomas Johnstone for pointing this out and discussing it with me ad nauseum.

3. More order of operations nonsense

Consider at the following two examples:

  1. \sin 2x
  2. \sin x \cos x

Most mathematicians would agree that, in the first expression, we are meant to first multiply 2x, and then evaluate \sin of the result.  In the second expression, we first evaluate \sin x and \cos x, and then multiply the results.  Onscreen, the typsetting power of LaTeX gives us a visual clue, by grouping the 2x in the first example but separating the two trig functions with a bit of space in the second — but this is subtle, and disappears entirely on the chalkboard.  What kind of rule regarding order of operations can apply here?  Do we evaluate first, or multiply first?  Either way we have it, one of them is plain wrong.

Rant follows

TL;DR:  This situation is crazy!

We teach the general rules, and we give examples of how they work in practice.  We hope the rules make an impression on malleable brain matter, but the examples are the heart of how our students learn (perhaps this changes as they move into the upper reaches of our mathematics curriculum, but perhaps not).  Enough examples, and they start to internalize them and extract some version of the general rule of their own accord.  If the examples are good enough, varied enough, and consistent enough then hopefully “their version” of the rule will more-or-less agree with “the rule.”  If the examples don’t follow the rules, what the heck are our students supposed to do, anyway?

Surprisingly, the discrepancies between these examples and the rules they purport to follow never bothered me before someone pointed them out — I would solve these problems according to the examples I had been given, without questioning the discrepancies between the examples and the rules.  Some of these examples are completely inculturated, and have been integrated into our mathematical canon (mixed numbers, for example).  Some exist more-or-less in “blackboard world,” a special realm in which the typesetting conventions of textbooks don’t always apply.   Some are simply accepted without a thought.  In some cases, there is disagreement (even among major software implementations of mathematics, such as Excel vs. Mathematica).  Despite the common argument that mathematics is a language, perhaps the one pure language — rational, rule-based and unambiguous above all — it seems it seems it is infected with the same kinds of exceptions, special cases, and nonstandard usages that plague our students in their English classes.  What’s to be done?  And more to the point, what are our students meant to do with this (aside from carrying on, in much the same way as always)?  Your suggestions are most welcome…

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Chicken and Rice with Professor Marina Dedlovskaya

Always soft-spoken, warm, and welcoming, Professor Marina Dedlovskaya is Associate Professor of Mathematics at LaGuardia Community College. She was a PI of a CUNY Improving Mathematics Learning grant winning project and that is how I met her more than 2 years ago. She and I had a little conversation at a cafeteria in LaGuardia’s E-Building.

You told me that many women knit in your hometown. Where is it?

I’m originally from Orenburg, Russia. It’s right on the border between Europe and Asia, on the Ural River. I was a math teacher there, working for a high school and a college. I then became interested in pure math and got a doctoral degree from Moscow University. 

Did you like math as a young student? Tell us a little bit about your childhood and your family.

Yes, when I was a child, I liked math and physics. I decided I liked math better because my math teacher was better than my physics teacher. My mother is a medical doctor, but her parents were both teachers: My grandmother was an elementary school teacher and my grandfather was a PE teacher who became a principal. My uncle, my mother’s brother, is a literature teacher.

Was becoming a teacher a natural path for you because there are many teachers in your family?

I would say so. I chose to go to a university in my hometown, Orenburg State Pedagogical University, because my parents didn’t want me to move out.

You got married in Russia and came to the US in 2001… and have an 8-month-old daughter now! How’s your family life in New York?

My husband and daughter are both good. He used to be a jazz guitarist and is enjoying jazz in New York. We have a lot of guitars and CDs… tons of them in our house! He and I watch a lot of movies, too. Of the ones we saw recently, I liked “Red State.” I was a little disappointed by “Midnight in Paris” though.

Since I’m a non-math person, I have to ask you more non-math question. What’s your favorite word?

It’s hard to say. Since I’m a math person, I never thought languages were something to think about. I used to think they were only tools. Now, because my native language is Russian and I had to learn English, I appreciate languages more and understand how complex and sophisticated they are.

Okay, now, back to math. Tell us about your love for math.

Math is a special language. When you learn a new language, you start with alphabet, then grammar, which is boring and doesn’t show you much. But later, when you have a certain foundation, you can enjoy literature and you can write and express yourself. Math is the same. In the beginning, you have to learn rules and other basics and you may think: Why do I have to do this? But later, you can enjoy different mathematical structures and become able to answer more questions. They’re like puzzles you have to solve to open a big picture. They’re beautiful.

In math, there isn’t much room for subjectivity. If two people understand one thing, they’re on the same page. I believe that’s part of the beauty of math.

I agree.  In Russia, during the Perestroika time, my friends and I always said, “2 plus 2 is always 4 in decimal system, regardless of the political climate of the country.”

We went on to talk more about math, mostly about the “Mathematics Across the Curriculum (MATC)” project conducted at Dartmouth College. Math was taught in the context of humanistic disciplines such as Art, Music, Philosophy, History, and Literature, by collaborative teaching teams. These MATC courses made math more “relevant” for students, and had positive effects on the students’ learning, motivation, and confidence. Professor Dedlovskaya, along with English Professor Karlyn Koh at LaGuardia, conducted a similar project called “Humanistic Mathematics: A Scholarly and Pedagogical Exploration” as a part of their CUNY Faculty Development Program.

(The interview was conducted on January 25, 2012)

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About the header image – the Cairo Tessellation

The header image was created by Richard Sheinaus, Director of Graphic Design at CUNY Central.  It depicts a tiling of the plane known as the Cairo Tessellation, so-called because it appears in the paving of several Cairo streets.  It has many notable properties, not least of which is that it may be our best approximation of a mathematical impossibility: a regular pentagonal tiling (in the case of the Cairo Tessellation, the pentagonal shape is not regular — not all sides and angles are equal — which saves us from a universe-destroying contradiction).

For more information, check out Wikipedia and the somewhat more exhaustive Dave Bailey’s World of Escher-like Tessellations.

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City of Solitude

Many people have noticed and  commented on recent article in the New York Times on the value of solitude in creative work.

In the article, Susan Cain remarks;

… the most spectacularly creative people in many fields are often introverted, according to studies by the psychologists Mihaly Csikszentmihalyi and Gregory Feist. They’re extroverted enough to exchange and advance ideas, but see themselves as independent and individualistic. They’re not joiners by nature.

She goes on to compare productivity in workplaces with and without private space, and presents some evidence that privacy improves the quality of work.

Susan Cain’s thoughts merit consideration here at CUNY, particularly in those fields, like math, where lonely cogitation is the norm.  After all, we find ourselves embedded in a seething metropolis, where it isn’t always easy to be alone.

The organized and the disciplined, those who already know how to study, will no doubt seek out private warrens and study carrels in the seemingly dwindling corners that harbor them.  But we have a few wayward souls who wander our libraries and hallways, overwhelmed by information and possibly not even aware of the fact.  Achieving a productive psychic state may be a skill which needs to be taught, particularly in a landscape in which it is so difficult.

I disagree with Ms Cain’s suggestion that group-work violates the principle of ‘solitude is best’ — indeed class time is social time, and constitutes those hours, referred to in her quote, when students should “exchange and advance ideas.”  It’s all the other hours that they should be spending like urban anchorites, chewing on pen tips while their mind tries to penetrate a problem.  But suppose there were an abundance of students with the urge to engage in this solitary thinking– where would they do it?

At John Jay our new building houses an area for student clubs, where students can use offices for quiet work.  Our library offers some solitude and quiet, but little privacy, with its large glass windows and open layout.  Our labs, often full, have workstations which do little to shield students from the eyes of their classmates.  In short, those who desire privacy can probably find it, but there is more we could consciously do to provide for a distraction-free student environment.  We could equip our campuses with high privacy zones, in a way analogous to the old smoke-free partitions in buildings.

Open spaces, full of light, are wonderful.  But we should equally value and cultivate the crannies and tiny redoubts that can store a student’s body while the mind flies away.  We should also help students learn to treat themselves with the psychic balm of aloneness in an environment where solitude is not the default condition.

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Doodling in Math Class

Five texting in the back.  Three bored, overconfident, ignoring you.  Seven who were placed incorrectly in your class and, after struggling through the first exam, will simply give up.  Four who answer all the questions (three consistently correct, who will pass with flying colors but will not learn much, and the last doggedly incorrect but undaunted).  Two quiet, occasionally channeling their growing frustration into emotional outbursts.  Maybe one class clown, looking for distractions.  And the other fifteen mostly quiet, mostly well-behaved, mostly hoping against hope they will dodge the college-career-killing bullet that is their math requirement.  What am I to do with you, o class of mine?

Yes, there’s the course content.  For non-STEM majors, it’s going to be algebra plus maybe some other stuff:  rationals, radicals, powers and pi, etcetera minus four a c.  Will they ever use it again?  Probably not.  Is it worth making them learn it?  Great question, but not for today.  I don’t want to talk about the content — I want to talk about the joy of math.  I mean noodling around with an idea, trying things out and discarding them, all process, no destination.  How the heck do we make that seemingly-incompatible-with-our-set-syllabi magic take place?  I don’t know — but instead of wrestling with the how, which can be depressing, I’d prefer to distract myself with the what.  Check out this video by Vi Hart, an awesome example of playing around math.

Vi Hart on snakes:

 

Vi Hart on Doodling in math class:

If you like it, there’s lots more (videos and many other things) at her blog:

http://vihart.com/everything/

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Welcome to the CUNYMath Blog

Hello and welcome!  The CUNYMath Blog is intended as a sounding board by and for the mathematics educators, researchers and graduate students at CUNY.  Ours is a large community, but it can sometimes feel small — it’s easy to be isolated in your research area or your department, with little sense of the variety of interesting work being carried out and the common (and uncommon) challenges being faced by your colleagues around the five boroughs. We are here to write about and discuss the things we find important — mathematics, education, CUNY,  New York City and so on.  We welcome you into the conversation — leave us a comment, tell us what you think!  Want to write something for us?  See the section on “Becoming a Featured Author” below.

About the Blog

This blog was formed out of a collaboration between the CUNYMath website (the blog’s home), and the Math Matters group here on the faculty commons.  Many thanks to the members of both of these groups for their efforts to bring this blog to life!  Special thanks to Mari Watanabe-Rose of Math Matters and Terrence Blackman of the CUNYMath committee for their support, to Matt Gold for welcoming us to the CUNY Academic Commons and to Scott Voth for technical assistance, to our inaugural authors and editorial team Hunter Johnson, Jonas Reitz and Mari Watanabe-Rose, and to Michael Carlisle for sharing his vision and setting the project in motion.

Becoming a Featured Author

Our Featured Authors contribute one post per week over the course of their tenure, provoking and participating in the discussion that drives our community.  Featured Authors rotate on a monthly basis, and there are a limited number of slots available.  If you are interested in contributing or would like additional information please contact Mari Watanabe-Rose ([email protected]), Hunter Johnson ([email protected]) or Jonas Reitz ([email protected]).

Typing math

This blog supports the use of mathematical symbols and expressions in both the blog posts and in the comments.  Simply enclose your LaTeX commands in $latex... $ tags.  For example, typing this into a comment:

$latex e^{i \pi} = -1$

should produce this output when the comment is published:

e^{i \pi} = -1

More information about LaTeX support on the Commons can be found here.

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