Mathematicians write like novelists

As I attempt to teach myself something about stochastic calculus, I have been reading a great many articles, and several textbooks, on the subject. It has left me with the distinct notion that mathematicians hate to spill the plot too early in the story. Each proof builds upon clues that have been scattered throughout the text, just as a writer might have the butler passing down a hallway in the second chapter for no apparent reason. Lemmas and sub-theorems that, to all outward appearances, are wholly unrelated to the general theme begin to appear. Then slowly, sometimes painfully, these logical manipulations are pulled together as the proof draws to a conclusion. But even so, the mathematician is reluctant to come out and say, “the butler did it.” Rather, phrases like “it is clearly obvious,” and “it is easily proven” are used to inform the reader that the point of mathematics is the mental challenge of figuring out how the pieces fit together.

There is no enjoyment in reading a novel that lays out the entire plot in the first paragraph. So a plot is simply a literary construct used by authors to evoke emotion, just as sculptors do with statues, and dancers with physical movement. Likewise, spelling out every mathematical truism and trick would make a proof lengthy and boring, with no pleasure left to be savored, right? So I think that mathematicians must write proofs (whether intentionally or not) in a manner intended to allow other mathematicians to sense the thrill of unraveling a logistic knot.

Sometimes I enjoy this artistry. Today I do not. πŸ™

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Career, Teaming, and Entrepreneurial Methods

Well, now that the fall semester has started at Purdue, I’m going to make an effort to post more regularly. Of course, just one post a month would be more regular than my recent performance. πŸ™‚ Anyhow, today’s e-mail brought a brochure for a course (IE 590 & ChE 597) titled, “Career, Teaming & Entrepreneurial Methods for Engineers.” It promises to teach engineers how to:

  • Be proactive in career development
  • Develop interpresonal and professional communications skills
  • Understand large technical project management
  • Develop an awareness of the technical marketplace and the supply of technology
  • Learn entrepreneurial methods

It sounds interesting. Why isn’t this type of class required for all engineering students?

I might be tempted to take it if I could afford more time away from my dissertation. However, I can’t graduate without defending, and I can’t defend without a dissertation. And it’s difficult to teach at a leading engineering school without a PhD. So more of my future blog entries may be more related to my research (and less devoted to engineering education in general). At least until I clear the hurdle of completing my final defense.

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The Marshmallow Challenge

Ever hear of the “Marshmallow Challenge?” Small teams of individuals are given the following assignment: use twenty sticks of uncooked spaghetti, one yard of masking tape, and one yard of string to construct the tallest possible free-standing structure that supports the weight of a marshmallow. Most people assume that since a marshmallow doesn’t weigh much, it shouldn’t significantly affect the support structure. Of course, even a small mass can produce structural failure when placed atop a long unsupported column.

So what profession does best at this task? According to Tom Wujec, a Fellow at software company Autodesk, the tallest structures are built by engineers and architects. They consistently outperform similar teams of lawyers, business school students, or corporate managers. This is not an unanticipated result, as we expect our engineers to know something about static structures. However, it is rather surprising to learn that youngsters, even kindergarten students, do far better than most adults—kids are simply not afraid to repeatedly fail as they search for an approach that works. (You may discover more about this learning exercise at MarshmallowChallenge.com).

Two insights come from this anecdotal report of group behavior. First, that engineers have been trained to think in a manner that is distinctly different from those in other professions. Second, that repeated rounds of prototyping and evaluation may be an effective means for dealing with the messy, unstructured, uncertain problems that engineers frequently encounter.

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Secret Tools of the Engineering Grad Student, Part 4: BibConverter

As you begin writing academic papers, you will need to cite the work of other researchers. From the prior two posts in this series, you know about using LaTeX to typeset your paper, and using JabRef to store your bibliographic references. However, typing in all the citation information by hand is rather tedious. Your ability to search the research literature is greatly enhanced if your academic institution provides you with access to databases of the engineering literature (Engineering Village, IEEE Xplore, Web of Science). All of these services allow you to export bibliographic data to a file on your computer, which you can then import into JabRef. However, there is a better way: BibConverter.

The brainchild of Kjell Magne Fauske, BibConverter is a free online service that converts a web page of information into a BibTex reference. While you are browsing through one of the article databases, you will likely find a paper you want to reference. Instead of downloading a citation file, simply copy the entire web page to your desktop, go to BibConverter, and paste the clipboard contents into the provided box. Click on the “Convert” button, and now you have a valid BibTex reference, that will look something like this:

@ARTICLE{Kalman1960,
  title = {New approach to linear filtering and prediction problems},
  author = {Kalman, R. E.},
  year = {1960},
  volume = {82},
  number = {1},
  pages = {35--45},
  month = mar,
  abstract = {Classical Wiener problem (filtering and prediction) is re-examined in
       discrete case using author's new ("state transition") method of
       analysis of dynamic systems; general solution is developed in
       terms of conditional expectations; this gives result of
       greatest possible generality when only first and second-order
       statistical averages are used; basic concepts of theory of
       random processes reviewed.},
}

Copy this data to the clipboard, and go to JabRef. Create a new BibTex entry with “Ctrl+N”, select the “Article” entry type, and then replace the “BibTex Source” entry with your clipboard contents. Once you’ve done this once or twice, it will seem quite natural, and it saves you the time and mess of having to clean up all the citation files that will start to litter your system. You can even download a bookmarklet from the BibConverter site to save you the trouble of surfing to the BibConverter site and selecting the proper database format. This service has saved me hours of time over the past several years.

[Note: IEEE Xplore recently changed it’s online format, and it looks like BibConverter is currently unable to process data from that service.]

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Secret Tools of the Engineering Grad Student, Part 3: JabRef

If you write academic papers, then you need to maintain a database of the references you cite. Assuming that you use LaTeX, this is typically accomplished by creating a BibTeX file that contains the needed bibliographical data. (Note that, in addition to identifying this particular data format, you may also see the term “BibTeX” being used to reference the software program that pulls information out of the BibTeX file and integrates it into a compiled LaTeX document.)

While it is possible to create a BibTeX file using nothing more than a text editor, it saves time to import such information into a program designed for this purpose. Always pleased to uncover free software, I’ve found the open source program JabRef to be a powerful citation manager. Many of the engineering article databases (Compendex, IEEEXplore, Web of Science, etc.) allow you to export bibliographical data for the papers they store. It turns out that there are many formats for exporting such data, but JabRef allows most of these formats to be imported directly into its BibTeX database.

Since JabRef runs on the Java Virtual Machine, it works with Windows, Mac OS X, and Linux. In fact, if you don’t want to be bothered downloading JabRef, you can launch the software over the web.

Alternatives to JabRef include Firefox extension Zotero, Mac program BibDesk, and commercial program Endnote.

[Update: I’ve been made aware of a program, SciPlore MindMapping, that takes a new approach to reference management. You can view a YouTube video about this application to learn about its features.]

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Secret Tools of the Engineering Grad Student, Part 2: LaTeX

If you are going to write a dissertation (or any other paper for that matter) with significant mathematical content, you will discover that the typesetting of your equations proceeds much better if you use LaTeX. While there is a steep learning curve, you will save a good bit of time down the road if you get comfortable with LaTeX (pronounced “Lay-tech”) early on. Here’s an equation for the Fourier transform rendered with this typesetting system:
Fourier Transform

This equation is created with the following code:

 F(f) = \int_{-\infty}^\infty f(t) e^{-j2\pi ft} dt 

As you can probably figure out, mathematical symbols are created in LaTeX with text keywords preceded by a backslash. In addition to the improved typesetting, this means that you can quickly update many equations at once by simply searching for, and replacing, text strings. Thus, if you wanted to convert the above equation to be a function of g, instead of f, a simple text replacement would update all the equations in one fell swoop. Contrast this to the equation-by-equation corrections required if one is using MathType to typeset mathematics inside a Microsoft Word document.

While there are many benefits to using LaTeX, it does take a little getting used to. In particular, you may find yourself trying to control a lot of factors (margins, paragraph spacing, etc.) that are easy to modify in a word processor, but difficult to adjust in LaTeX. In the beginning, don’t worry about trying to control the output; focus instead on getting your equations to typeset correctly. Also, expect to spend some time searching for documentation. While most everything you will want to do has been done already, it sometimes takes a while to hunt down the correct command. (Hint: If you absolutely must play with the margins, use the geometry package.)

Significant time savings occur with LaTeX because templates for most publications types have already been defined. Thus, if I want to publish an IEEE paper, I simply drop my document into an IEEE template. Same paper in ASME format? Simply change to the appropriate ASME template. Need advanced math formatting commands? Use the AMS package. While similar templates are typically available for Microsoft Word as well, I often find myself hunting from paragraph to paragraph in Word, trying to discover why the formating has gone askew midway through the document. This is rarely a problem in LaTeX. And to produce my dissertation? Simply use the appropriate thesis style (your university may have its own format).

On my XP system (yes, I’m a dinosaur), I’ve had good luck using MikTex as my LaTeX implementation. One of the nicest features about the MikTex software (other than it being free) is that, when it encounters a package name it does not have, it goes out on the net and attempts to find the package for you. This has frequently saved me from having to install such code manually. While any text editor will work to generate LaTeX documents, I’ve always used WinEdt. Although WinEdt is not free, I’ve not regretted the $30 it cost me for a student license, as it integrates quite nicely with MikTeX.

If you are interested in learning more about using LaTeX, there is some decent documentation on getting started available from the LaTeX project site, as well as the WikiBooks site. When you see references to “LaTeX2e,” this simply indicates the current version of the LaTeX program. Similarly, “LaTeX3” refers to the next generation of the LaTeX software. Learning LaTex is initially frustrating, but you’re an engineering grad student. You’re not the type to choose the easy path. So download the software and give LaTeX a try. I suggest starting with a study sheet of equations for an upcoming exam. You’ll learn how to construct equations without needing to worry about paragraph formatting.

[Hint: Find an equation you like in Wikipedia? Right click on the equation and access the image properties. The associated text will be the LaTeX code used to generate the equation.]

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Secret Tools of the Engineering Grad Student, Part 1: Desktop Search

Each engineering specialty makes use of certain software packages. For instance, in my area of automatic controls, just about everyone uses Matlab; those studying other disciplines make use of other topic-specific packages. However, certain tools (software and otherwise) will prove beneficial to just about any engineering grad student who must carry out research and produce a dissertation at the conclusion of their studies. However, these tools are rarely mentioned as key technologies for surviving as an engineering grad student. Over the next several posts, I will identify some tools that I had to discover on my own as I marched toward a PhD degree. Today’s category is desktop search.

A stack of papers

You will undoubtedly collect a lot of information on your computer as a grad student. This will include papers, notes, programs, and presentations. Regardless of the software you use to produce or store such information, a good search program will help you quickly access data when you need it. During the course of my research, I have stored thousands of documents in various file formats. Often times I can remember an author’s name, or a keyword, but cannot recall which document contains a particular quote or data item. A search program allows you to quickly identify the information you need, regardless of where it is located on your computer’s hard drive.

An admitted Luddite, I still use Windows XP as my operating system, so I can’t speak to whether Linux, Mac, Vista, or Windows7 users require an external search program. I’ve had good luck with X1 Search, which I started using back in 2004, when this program was named Yahoo! Desktop Search. Then, in 2006, the association with Yahoo! was terminated, but a free version of X1 remained available as X1 Client. Sadly, the last free version (5.6.3, Build 3453) is becoming difficult to find on the web anymore. Although X1 is no longer free, there are other free options in the desktop search category that should work just as well. Regardless of which desktop search program works best for your situation, you will save significant amounts of time by being able to rapidly search through your research documents and retrieve key bits of data.

Although I wasn’t surprised at my need to search journal articles, I have been amazed at how often I need to go back to find a section of software code that I had previously written. Over the course of the past several years, I’ve created thousands of lines of Matlab code, and I occasionally realize that I’m starting to rewrite an algorithm that I’ve already sorted out. So I open up my desktop search program, limit my search to Matlab files, and start typing in relevant keywords. I’m almost always guaranteed to find the needed code within a few minutes. If you’re an engineering grad student, you’ll likely get good use out of a desktop search client.

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Bidirectional Translation

Happened to stumble across Mango Languages last night. It seems to be a nicely constructed site for language instruction. Once upon a time I worked for a firm headquartered in Germany, and had taken some company-sponsored language lessons, so I poked around in the first-level German module. One of the things I noticed right away was that translation was required in both directions. First, I was asked to translate from English to German. Then, after practicing a phrase, I was asked to translate from German back to English. This was easy for the first couple of phrases, but became increasingly difficult as I had to juggle more and more words in my head. I gave up halfway through the lesson as it was getting late, and my head was starting to hurt. However, someone fluent in both languages has obviously mastered the translation going in either direction.

All engineers deal at some level with the language of their particular engineering specialty, as well as the language of mathematics. Fluency in math provides magnificent tools for creating and analyzing new engineering methods. However, my recent experience as a graduate student (as well as two decades as a design engineer) leads me to believe that the emphasis is too heavily focused on the making the translation from physical reality into the language of mathematics. Once the math domain has been entered, there is little effort to move back into the physical domain. This seems to me a great oversight, as I consider engineers to be those who function primarily in the physical realm, rather than the mathematical domain. I love the beauty of mathematics, but I fear that the current generation of engineers may lack much substantive understanding of how to convert mathematical results into an enhanced understanding of physical realities. Many of my classmates fail to “see” how a problem solution relates to any real-world situation.

On the other hand, I frequently find myself struggling for hours trying to make the connection between a solution and its physical meaning. Often times the available textbooks and reference materials made it sound as though this connection should be immediately clear to the reader. It drives me crazy when technical material makes no concession for those of us who are not yet be completely fluent in the “language.” Since learning is a lifelong process, all of us should be constantly entering domains to which we have not been previously exposed. While the use of a particular method or technique may be plain-as-day to its creator, it’s often not nearly so obvious to those of us struggling to acquire an understanding of the new concept. So I suppose this post should serve as a reminder for myself to keep looking for better ways to describe technical concepts in a manner that novices can comprehend, and that experts will still find insightful. I’m not sure that it can be done at the same time, or via the same communication method, but surely there has to be a better way.

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Narrative and Storytelling

While skimming through a year-old post over at edtechpost, I noticed the following reflection:

So I will forgive you if you ignore me from here on out as a perennial dimwit when I tell you that it took me this long to β€˜get’ how crucial narrative and storytelling are to everything we are doing, be it learning online, connecting, weaving one’s online presence, blogging…

What really caught my eye was the phrase “narrative and storytelling.” Why are these factors not more frequently incorporated into the teaching of technical issues? While sitting through long lectures that cover intricate mathematical development, I often long to hear more about the context in which the methodology was developed.

  • What problem drove the development of a new approach? These equations don’t just drop out of the heavens! Aspiring engineers need to understand that effective problem solving is within their grasp; that “correct” solutions are not just found in dusty old reference texts. Novel methods are driven by persistence and hard work—this reality is rarely emphasized.
  • How long did it take to create, prove, and document the approach? It’s easy to get frustrated when the development of a new method hits repeated roadblocks. There needs to be some understanding of the hundreds (or thousands) of hours that are often spent in developing a new solution. Even though a proof can be sketched out in two minutes, the path from problem statement to solution is usually not intuitively obvious.
  • Finally, a pet peeve of mine: All contributions are made by real people with real lives, not mystical figures existing beyond the earthly realm. Even if there is no time for biographical sketches of these individuals, what is the correct pronunciation of their names? Most engineers finally figure out that “Euler” is “oy-ler,” not “you-ler.” However, I’ve sat through many lectures where a theorem author is identified on the overhead slide, but their name is never mentioned aloud. And rarely have I heard any emphasis on correct pronunciation. This small detail seems central to allowing engineers to properly communicate with others in the language of mathematics, as well as providing some sense of human involvement. By the way, I often refer to the Mathematics Pronunciation Guide. How else would I learn that “Stieltjes” is pronounced “steel-tyuhs?”

Having taught college courses many moons ago, I am well aware that trying to incorporate contextual material into lectures means even more work for already overstretched professors or lecturers. However, I’ve come to the decision that it’s better to master a few topics than to be aware of many. A good story makes any topic easier to remember, and also promotes a richer understanding of the material.

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A Mother Lode of Engineering Education Information

Happened across Dr. Richard Felder’s website today. Wow! What a treasure trove of information. It’s going to take me a while to digest this information, but I’m thrilled to see that the research exists. You can view an hour-long presentation Dr. Felder recently made at Penn State University, titled “Engineering Education in Five Years (or sooner).” Too bad we can’t see his slides most of the time, but an interesting talk nonetheless. My take away quote: “The power of the interactive tutorial is huge.”

Thanks to Teaching College Math for leading me to Prof. Felder’s site!

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