GL(s,R)

January 8, 2012

4 oz. > 100 mL

Filed under: Uncategorized — Adam Glesser @ 5:19 pm
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Most everyone is familiar with Santayana’s admonition that, “Those who do not learn from history are doomed to repeat it.” What is the analagous statement for mathematics? Are those who fail to learn mathematics doomed to work in food services? Doomed to playing the lottery? Doomed to credit card debt and forclosures on the house they can’t afford?

Actually, the answer is usually none of the above. In fact, most of us probably know fairly successful people whose background and skill in mathematics was as minimal as they could get away with. I know several scientists (mostly in biology) who are extremely good at what they do, but who would be terrified to sit through even our freshman level math courses. No, it does not seem that an individual’s lack of mathematical background will necessarily cost that person anything substantial. Ah, but what of a nation or a world?

Suppose a society fails to learn mathematics? I don’t mean that all individuals fail to learn, just a number so overwhelming that they permeate the government, regulatory agencies, businesses, and schools. What if a sufficient fraction of the population learns to think intuitively, rather than critically? What if enough people agree that what feels right, is right? What if the system of checks and balances fails because the counterweights just can’t keep up? What if people start to believe that these rocks are what keep the tigers away?

I am writing this as I live the answer to these questions. To which circle of hell am I referring? Circle 9er: Airport security.

I know: It is such an easy target for scrutiny, and yet without a doubt the people who work in airport security are an honest group who are doing their job and who likely sympathize with many of the travelers inconvenienced by policies they never asked for. Am I annoyed that the security officer here at Heathrow just confiscated from me the half-full 4oz. bottle of contact lens solution—which passed through American security without notice—because 4oz. is 118mL and the British limit is 100mL?* No. I am annoyed that many people in power in both (all?) countries believe that such trivial differences matter.

*Technically, they should have confiscated it in the United States since they match Britain with a 3.4oz (roughly 100mL) limit. 

How should we decide the appropriate level of security at our airports? Should experts come up with a list of reasonable ways a terrorist might attempt to take over or destroy an airplane, and then enact sufficient security measures to make those avenues of destruction prohibitively difficult? It sounds pretty good. It feels like the right solution.

Of course, if you’re one of the those anal mathematicians, then you might start questioning the definition of `reasonable ways’ and `prohibitively difficult’. At that point, it might occur to you that probability and statistics are at play here, and that these are necessary to consider before deciding upon a course of action. But those of us without a Ph.D. in pointdextery know that probability and statistics is a just a smart person’s attempt to get around the immutable law that either the plane crashes or it doesn’t; you either stop the terrorist or you don’t. Never mind the regular reports of journalists sneaking weapons or TSA agents sneaking people through security. We’d all rather be alive with a little less liberty (and contact solution) than free and dead at the bottom of the Atlantic, right?

So, I guess this is my answer: The cost of a society failing to learn mathematics is giving up some of its liberty for, well, the appearance of security? But hey, at least those badges the TSA officers wear now are keeping the tigers away.

[Update: This post was retroactively inspired (that is, I read it after I wrote this) by an article of Keith Devlin.]

September 9, 2011

Hey, I wrote a book review!

The following is a book review written this summer for the Center for Teaching Excellence at Suffolk University. The shortness of the review is not a function of the content of the book, but rather the medium (a newsletter).

Creating Significant Learning Experiences:
An Integrated Approach to Designing College Courses
L. Dee Fink
Copyright © 2003 by John Wiley & Sons, Inc.

Reviewed by Adam Glesser

Fink argues that a new paradigm is emerging in college teaching, one that encourages a focus on activities that produce significant learning experiences, valuing the quality of learning over the quantity of content coverage. In order to frame the discussion, he defines a Taxonomy of Significant Learning consisting of three categories that essentially mimic Bloom’s taxonomy of educational objectives:

  • Foundational Knowledge
  • Application
  • Integration

and three categories that go beyond Bloom:

  • Human Dimension (“students learn something important about themselves or about others” (p. 31))
  • Caring (about the subject, phenomena, ideas, their own self, others, the process of learning, etc. (p. 49))
  • Learning How to Learn

Fairly little attention is given by faculty to the latter three in the course design process, although I suspect that when pressed, most professors would espouse these as goals of their courses. In the sciences, I see some of these categories as long-term goals, built up through the entire curriculum and difficult to foster in a single course. This suggests that we need a concerted effort to consider these values collectively, not merely in isolation.

The heart of this book is the two chapters on course design. My teaching mimics that of my own teachers and so, like them, I am a member of content-aholics anonymous: the group of professors ashamed that their courses are creatively designed to include as much content as possible. Conveniently, Fink offers up a 12-step plan for designing a course. Although few of his suggestions are innovative, many of them will make you say, “That makes so much sense! Why haven’t I been doing that?”

My only complaint is the relative lack of attention given to the the grading system and its role in fostering significant learning. While the author accepts the need for “the development of a feedback and assessment system that goes beyond just grading and contributes to the learning process” (p. 142), he gives an example of a grading system that is “fair and educationally valid”, but which reduces the course, for many students, to the calculus of point grubbing.

The title sets the bar: the book is a failure if reading it is not itself a significant learning experience. Fortunately, the author succeeds in the ultimate accomplishment in pedagogical writing: he made me put the book down at times, frantic to work on designing one of my courses.

March 22, 2011

The title of this post was already taken

Filed under: Uncategorized — Adam Glesser @ 12:10 am

Tricks of the Trade

(with Professor Glesser)

My all-time favorite differentiation technique is logarithmic differentiation. The implementation is right there in the name: take a logarithm and then differentiate. If you are a pro with your log rules, you will understand why this would be useful.
There are two canonical types of functions where this technique is often used in standard calculus courses. The first is where you have a product and/or quotient of functions, potentially raised to a rational power. For example:
y = \sqrt[3]{\dfrac{(3x-2)^2\sqrt{2x^3+1}}{x^4(x-1)}}
If you apply the natural log to both sides—we choose base e so as to avoid unnecessary constants when differentiating (recall that when u is a function of x that \log_a(u)' = \dfrac{u'}{u \ln(a)}—then we can deconstruct the right hand side using the log rules to get:
\ln(y) = \dfrac{1}{3}\left[2\ln(3x-2) + \frac{1}{2}\ln(2x^3 + 1) - 4\ln(x) - \ln(x-1)\right].
Differentiating both sides is now a snap:
\dfrac{1}{y}\dfrac{dy}{dx} = \dfrac{1}{3}\left[\dfrac{2\cdot 3}{3x-2} + \dfrac{6x^2}{2(2x^3+1)} - \dfrac{4}{x} - \dfrac{1}{x-1}\right].
Multiplying both sides by y—which is the orignal function—gives us the derivative.
The second example is one that gives students no trouble at all, but gives teachers fits. The simplest such example is
y = x^x.
Three-quarters of the class knows that you use the power rule to get
\dfrac{dy}{dx} = x\cdot x^{x-1} = x^x.
The remaining group of students will point out that the power rule only works with a constant exponent, so instead you need to use the exponential rule which gives
\dfrac{dy}{dx} = x^x \ln(x).
Of course, the teacher is squirming right now because they know the exponential rule only works you have a constant base. In fact, neither rule is correct! However, in a way, they are both half-right. Applying the natural log to our orginal equation gives
\ln(y) = \ln(x^x) = x\ln(x).
Differentiating—using the product rule on the right—gives
\dfrac{1}{y}\dfrac{dy}{dx} = x \cdot \dfrac{1}{x} + \ln(x) = 1 + \ln(x).
Multiplying both sides by y now gives
\dfrac{dy}{dx} = y(1 + \ln(x)) = x^x + x^x\ln(x), the sum of the two incorrect answers.
SWEET
Using logarithmic differentiation on functions of the form f(x)^{g(x)}, we can get a general rule which is not particularly well-known:

SPEC (Super Power Exponential Chain) Rule

If y = f(x)^{g(x)} is differentiable, then \dfrac{dy}{dx} = g(x)\cdot f(x)^{g(x) -1} \cdot f'(x) + f(x)^{g(x)}\ln(f(x))\cdot g'(x), i.e., evaluate the derivative using the power and chain rules, then with the exponential and chain rules, and finally add the two incorrect answers together.
In short,
\begin{array}{cl} variable^{constant} & \longrightarrow \text{ Power rule (+ Chain rule)} \\ constant^{variable} & \longrightarrow \text{ Exponential rule (+ Chain rule)}\\ variable^{variable} & \longrightarrow \text{ SPEC rule: Add power rule and exponential rule answers together} \end{array}
An Example
On my midterm, I asked the students to compute the derivative of y = [\sin(x)]^x. The SPEC rule makes this a piece of cake: the power rule gives x \sin(x)^{x-1}\cos(x) and the exponential rule gives \sin(x)^x\ln(\sin(x)). Adding them together gives:
\dfrac{dy}{dx} = x\sin(x)^{x-1}\cos(x) + \sin(x)^x\ln(\sin(x)).
The title
When coming up with this, I thought that the perfect title would be “Two Wrongs Make a Right.” Unfortunately, this was already taken by the authors of the paper of nearly the same name (which has to be the most obvious title for a paper—ever).  They don’t give a name to this rule, so in honor of my anime loving friends, I stick with SPEC rule for the moniker.
What about logarithmic differentiation?
One of the faculty asked me if students will avoid logarithmic differentiation now. For the first type of problem: absolutely not—the SPEC rule doesn’t really apply. For the second type: I hope so—logarithmic differentiation is useful because it simplifies calculations; why not use a trick to simplify it even more?

September 29, 2010

An exam by any other name

Filed under: Uncategorized — Adam Glesser @ 7:10 pm

As I’ve mentioned in the past, I don’t have full control of my finite mathematics course. I asked for certain leeway in my grade breakdown, but had to settle on leaving the four class tests at 60% of the overall grade (in contrast to my precalculus where there are two tests making up 30% of the grade). Consequently, the students still treat my class as if nothing counts until exam week. Sure standards comprehension make up 30% of the overall grade, but they aren’t EXAMS!!!! How do I convince these kids that the typical college attitude will not cut it? They have an opportunity to spend four years living the life of the mind. Instead, they spend four years taking their mind off life. The night before a quiz, I stayed up late making screencasts of linear programming problems. One student told me how much he appreciated it, only he hadn’t had time to watch it because, “you know, the Yankees were on.” Guess how he did on the quiz?

Maybe I have to follow the rules and set my percentages like everyone else, but is there a chance that I could change the name (retention assesments?) and de-emphasize the importance of exams so that the day-to-day stuff gained relevance? Or are students socially constructed to work this way: slack, slack, slack, cram, rinse, repeat?

August 6, 2010

A geometry question

Filed under: Uncategorized — Adam Glesser @ 8:59 am
Tags: , ,

One of the things most pleasing to me about this whole inter-web-o-blog-o-sphere is that I can ask so many people for help in such a short amount of time. I am going to put the over-under on an answer at three hours (I would put it lower if I was on the West Coast) [Update: It turns out the winning time is 33 minutes. Thanks, Justin!].

In a 2008 MAA Mathematics Magazine “Proof Without Words” article, Sidney H. Kung gave the following proof of the tangent of the sum formula:  The proof is transparent to me in all but one step. Why does \angle BDF =\beta? I guess it must have something to do with the line segments being chords in the circle, since I haven’t used the circle anywhere in the proof, but I’m just missing something obvious or forgetting something from geometry. Help!

July 4, 2010

Travel Log

Filed under: Uncategorized — Adam Glesser @ 9:52 pm
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Hey everyone,

I’m back from Europe and simultaneously elated, bummed, excited, nervous, stressed and a little sleepy. I had a productive trip and I can’t wait to tell you all about it. First, though, I need to turn the title into a pun.

Just kidding, the real pun is what follows:

A Question

Let me fast-forward to the second half of my trip. My roommate in Lausanne, the Korean dynamo Sejong Park, gave me the following problem (suitable for high school math competitions) on the first day. It took me a mere 10 days to solve it (although, to be fair, I only worked on it about 3 hours per day).

Decide which of the following quantities is greater:

\dfrac{\log(5)}{\log(3)} or \dfrac{\log(3)}{\log(2)}

where the base of the \log is fixed, but arbitrary. Try to work on it before you read the answer. If you don’t get it in the first 10 minutes, don’t feel bad, neither did any of the mathematics professors I asked (however, Romanian super-stud Radu Stancu apparently solved this in two minutes).

Oxford

Starting from the moment I walked through passport control in Heathrow airport and saw some bizarre parabolic light fixtures, I wished I had my camera. One of the highlights was getting to eat dinner at the Christ Church High Table (those of you into obscure literature and cult films might recognize the room as being where a certain H. Potter sups. Note that, unlike in the film, the room does have a ceiling). A combination of jet lag and hay fever made the first few days a bit of a blur, but eventually my host, David Craven, and I got down to work and watched Team America: World Police. After that, we finished the paper that just wouldn’t die and then I worked a bit on editing his new book on fusion systems.

Lausanne

A week in Oxford and then we took off for Switzerland. Thanks to a clever disguise and some misdirection, I was able to get into the country despite asking earlier in the week if Switzerland had been named after the font, Helvetica. Anyhow, the conference had everything: talks, night club dance-offs, barbecues, workshops, water closets, fencing, fighting, torture, revenge, giants, monsters, chases, escapes, true love, miracles… Simply put, it rocked. I got to hear about biset functors from Serge Bouc (and who better to hear them from, the guy wrote a Bouc on the subject) and learned the correct technique for pouring beer from the tap (as a lifelong teetotaler, this wasn’t a skill I had picked up before). Good things came to an end, eventually. I started missing my family and was worried about my growing (more beautiful by the day) wife  who is now 23 days and counting until the due date of the yet-to-be-named baby number 3 (I already suggested Glesser Escher Bach and my wife said, “no”). When I started thinking about how I could use \{\infty, -\infty \} \cup \mathbb{Z} to demonstrate a non-associative binary operation to my older son, I knew it was time to come home.

Back home

So, I finished a paper and have between one and three new collaborations started. This might be a great year for research. However, my homecoming was not all roses. For starters, I see that while I was gone, Mr. Cornally set up and closed up shop on the beta version of his Standard Based Gradebook. Major bummer that I missed this. Eighty-seven blogs posts to go and I’m caught up on the math edublog section of my reader. What else did I miss (is that a cool new background over at f(t))?

Coming soon

I wrote down several posts (or at least the outline of several posts) while on my trip. Some are pedagogical, others are more of those page view killers where I pretend to know something about something. In any case, they should be out reasonably soon, unless, that is, I preempt them with…

Coming sooner

I start teaching a summer precalculus course this week if enrollment is high enough. This is going to be the official start of my SBG experience. I finished crafting the syllabus today and tomorrow will finalize the course standards; by finalize, of course I mean that they are still subject to change as soon as I figure out what I’m doing. I’ve decided to split the grade up as 80% for standards and 20% for the final. As someone who always wanted to keep the final worth at least 40%, I’m still a bit wary of taking it down to Cornally levels. I also decided to toss out the midterm since the course is only 6 weeks long. As soon as I finish my standards list, I’ll post them here for your convenience and, hopefully, your criticism.

The picture below is only inserted to stop you from seeing the answer to the Travel Log problem

Okay, here it comes…

An Answer

If you’ve struggled long enough and would like to see the answer, here it is:  \dfrac{\log(3)}{\log(2)} > \dfrac{\log(5)}{\log(3)}.

What? You say you had figured that out already by plugging it into a calculator? Well that isn’t very sporting, is it. Anyway, one can deduce the above inequality without resorting to newfangled technology. Let’s make a couple of observations. First, using the change of base formula, we can rewrite the two fractions:

\dfrac{\log(5)}{\log(3)} = \log_3(5) and \dfrac{\log(3)}{\log(2)} = \log_2(3).

Clearly, both of these are greater than 1 and less that 2. Let’s make some room by multiplying them all by 2. We then get 2\log_3(5) = \log_3(25) < 3 and 2\log_2(3) = \log_2(9) > 3 and we’re done. The general idea here is that if we can bound two logs by the same integers, then multiplying by a suitable positive number, we can tell the logs apart. Simple when you see it!

June 14, 2010

Time to go to work

Filed under: Uncategorized — Adam Glesser @ 8:00 am

Although teaching has been my main focus the last two years, I try not to lose sight of what got me into this mess in the first place: I love doing mathematics. As such, I will make my annual sojourn to Europe to hobnob with people much smarter than me in the hope that they will drop a crumb off of their plate that I might legitimately turn into research. I’ll spend a week at Oxford working with über-genius David Craven and then make my way up to Lausanne, Switzerland for the conference celebrating the 60th birthday of algebraist Jacques Thévenaz, author of one of the five books I’d take to a deserted island, G-algebras and Modular Representation Theory.

Anyway, this little trip won’t be about teaching or blogging, but actually doing mathematics. Sweeet. So I’ll be back at the beginning of July and ready to tell you all about the class (or possibly classes) I’ll be teaching (one is actually for high school students!).

Rest well blog-o-sphere,

Adam

P.S. In case you’re wondering, the other four books would be Gorenstein’s Finite Groups, Ayn Rand’s Atlas Shrugged, Heller’s Catch-22 and the U.S. Army Survival Manual.

May 31, 2010

When I’m Not Teaching

Filed under: Uncategorized — Adam Glesser @ 9:02 pm
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Actually, it is a trick title: I’m always teaching. However, when I’m not in front of a formal classroom or working on material for said classroom, I spend most of my time with my family. But, when I’m not trying to corrupt youth (at school or at home) I read a lot of blogs, play board games, learn Spanish (or, more recently, French) and write a very popular blog. 🙂

When, though, I finish doing all the above things, I try to find time to do some mathematics. In case you’re interested in what I do or you’re a family member looking to see what I’m up to recently (hi, Mom and Dad), here is a paper accepted last week into the Proceedings of the Edinburgh Mathematical Society.

Now, I have an ulterior motive for bringing this up. In preparation for my post on story telling in the classroom, I wanted to first run an idea by the blog-o-sphere and if the feedback is positive, I might try it. Stay tuned.

May 22, 2010

Here we go

Filed under: Uncategorized — Adam Glesser @ 11:46 am
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Inspired by Dan Meyer, Sam Shah and, most recently, Shawn Cornally (as well as Kate Nowak, Mr. Sweeney, David Cox, etc.) I’ve decided to start a blog, cleverly self-titled \mathrm{GL}(s,\mathbb{R}). No, wait. They didn’t inspire me to start a blog; they inspired me to radically question the way I approach teaching. If you ask my former students, I’m sure that they would tell you that I wasn’t such a bad teacher to begin with, but I know better. I’ve wasted opportunity after opportunity to enrich the lives of my students because of a silly need to cling to a system that I fondly remember rebelling against. What’s up with that?

A Story

I taught multivariable calculus in the fall of 2009 and had a student who, despite prior mathematical success, simply didn’t engage. Being a college teacher, I didn’t feel responsible for making her engage. After a horrendous midterm, she came to me to ask about what could be done to improve her lot in the class. I crunched the numbers and even with (unlikely) respectable numbers the rest of the way out, she had little hope of passing the class, much less getting the grade she actually wanted. Being rational, she decided to cut her losses: she shifted whatever attention she had wasted on my class to her other classes and simply conceded the F in my class.

It seems to me that if I teach for long enough to have 100 students face this same predicament, I will end up with 100 students choosing the same road. Please, help me save just a few!

Standard Based Grading (SBG)

Actually, this is telling the story in reverse order.  The changes to my teaching started in the classroom, not on the syllabus. However, there is not too much I can say about how to teach (generally, anyway) that isn’t already in the textbooks, blog-o-sphere, or whatever. Watch Dan Meyer’s TED talk for just a few minutes and you will learn more about pedagogy than I could ever teach. No, I want to start with Standard Based Grading because that is my big change for the fall and that is what the majority of the posts in this blog will be about (more on the minority later). In a nutshell, SBG takes the point of view that if the goal of a class is to learn (and to learn to apply) certain material, then your grade in the class should be a measure of how well you are meeting that goal. Important in the last part of the last sentence is the word are (present tense). If you were doing well at factoring, but are now unable to do so, your grade reflects that reality. If, however, you were unable to recognize a difference of squares, but now you can, your grade will reflect it. This is so intellectually satisfying to me that I can hardly believe I didn’t think of it before. What is the cost? Obviously, there is the cost of me having to pay attention more closely. Also, I have to give up the notion that my students need to learn how, when and where I tell them too.  For the former, well, it is what I should be doing anyway, isn’t it? For the latter, oh darn. I’ve argued for years that college is about learning how to learn. Yet, all I ever do is enforce arbitrary barriers to learning.

What will this blog be about?

Here is the plan. Probably, I will find a dozen other things to talk about, but my goal is to chronicle my journey from what I have been to what I will become. Along the way, as a way of breaking up the diary monotony as well as to create a convenient reference tool for myself, I will publish a few tricks of my trade. Some of these things will be shortcuts for computation, mnemonics and perhaps a bit of insight from the point of view of a practicing mathematician.

Why not just read Think Thank Thunk?

Well, that isn’t a very polite question to ask on the first real post. All the same, I’ll try to answer it. First, Cornally’s blog is designed to…um…actually be good. I have no delusions of grandeur here. My writing is terrible, but I will be brutally honest. You will read the good and the bad from my experiences. Second, the majority of excellent math edublog sites (at least of which I’m aware) are written by and for high school teachers. I am a college professor and so my perspective and challenge is slightly different. For instance, I get to teach cool advanced math courses (e.g., complex analysis and differential equations) and, at this level, the interaction with students is far from the psychological exercise of dealing with freshman algebra students. Having said that, this fall, I get to teach precalculus and finite mathematics. While science majors almost exclusively populate the first class, the second is full of students for whom mathematics is torture. By the way, if you see a post or two on here about probability and combinatorics, it is because of that class.

What’s coming next?

I’ve already started preparing my classes for the fall and so my next post will tell you about the process of selecting standards for my courses. I will also have my first Tricks of the Trade post entitled: Log rules!

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