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## When to introduce the matrix exponential?

In the study of linear systems, a familiar relationship is the homogeneous state-space equation \dot{\mathbf{x}}=A(t)\mathbf{x}(t), where \mathbf{x}(t) is an n-vector, and A is an n \times n matrix. The time-invariant solution, (i.e., when A is a constant matrix), is \mathbf{x}(t) = e^{At} \mathbf{x}_0. When this subject is first introduced, the solution is often assumed, rather than derived.

The thinking is that since the solution to the homogeneous scalar equation is x(t) = e^{at} x(0), then students will willingly accept a matrix-friendly equivalent that solves the state-space differential equation. So the definition for the exponential matrix is given, and is shown to work for the homogeneous case:

\begin{aligned} \dot{\mathbf{x}}(t) & = \frac{d}{dt} \left( e^{At} \mathbf{x}_0 \right) \\ & = \frac{d}{dt} \left( e^{At} \right) \mathbf{x}_0 + e^{At} \frac{d}{dt} \left( \mathbf{x}_0 \right) \\ & = A e^{At} \mathbf{x}_0 + e^{At} \left( 0 \right) \\ & = A e^{At} \mathbf{x}_0 \\ & = A \mathbf{x}(t) \end{aligned}

It seems to me that this presentation sequence, however, masks what is really going on with the system; that there is an infinite recursion on the initial state, \mathbf{x}_0, that converges to a value for \mathbf{x}(t):

\begin{aligned} \mathbf{x}(t) & = \mathbf{x}_0 + A \int_0^t \mathbf{x}(\tau)\, d\tau \\ & = \mathbf{x}_0 + A \int_0^t \left[ \mathbf{x}_0 + A \int_0^t\mathbf{x}(\tau)\, d\tau \right]\,d\tau \\ & = \mathbf{x}_0 + A \int_0^t \left[ \mathbf{x}_0 + A \int_0^t \left[ \mathbf{x}_0 + A \int_0^t\mathbf{x}(\tau)\, d\tau \right] d\tau \right]\,d\tau \end{aligned}

This recursion obviously repeats ad infinitum. However, the matrix exponential can now be defined by collecting terms on the right hand side, leading to:

\begin{aligned} \mathbf{x}(t) & = \left[ \mathbf{I}_n + At + \frac{1}{2!} \left( At \right)^2 + \dots \right] \mathbf{x}_0 \\ & = e^{At} \mathbf{x}_0 \end{aligned}

Presented in this order, the exponential matrix is developed based on system response, rather than the other way around. This strikes me as being easier to comprehend than “guessing” that some seemingly arbitrary function might solve the problem. Is this conceptually easier for anyone else?

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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 ‘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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## Sorting Out the Lines of Thought

It is my hope that, by making frequent blog entries, I will slowly sort out the tangle of thoughts that go through my head each day. These ideas and notions are often related to the engineering profession or engineering curriculum—and they all seem tangentially related to one another in some way as they pass through my consciousness. Without stopping to write them down, however, all I retain is an emotional agitation that comes from knowing that things are changing, but not being sure what to do about it. It is somewhat akin, I must confess, to the way that I felt about my stock investments throughout most of last spring.

So, as a first pass, I see these issues as needing resolution to put my tiny brain at ease:

Role: Are engineers to continue as problem solvers, or should they (could they?) become advisers to society? In a Talk of the Nation interview on NPR, former marine biologist Randy Olson talks about why scientists need to involved in presenting their findings to the general public, and how they might do so effectively. It seems to me that as the world becomes more complex, we need engineers to speak up about the inevitable compromises that are part of any sufficiently robust system. The concept of relying on facts, rather than anecdotes, is only now starting to get due attention in management circles. Courtesy of Stanford professors Bob Sutton and Jeffrey Pfeffer, the notion of evidence-based management reached the readers of the Harvard Business Review in 2006. If not evidence, on just what have managers been basing their decisions up to now? Could engineers really do any better, or are they so lacking in charisma and social skills that they could barely stay afloat in the choppy waters of corporate politics?

Skills: Are the skills that students learn in college in any way related to the skills they need to be productive in society? It seems to me that engineering curriculum is too often subject to the tyranny of technique. Yes, students can calculate the maximum stress in a beam, but do they know what to do with the number they generate? They may be able to produce a Bode plot for a feedback system, but can they use that information to reduce system error? It is undoubtedly easier to teach and grade technique, but is this ultimately a disservice to students, and to society? Further, a majority of the engineers that I graduated with become project engineers, rather than designers or researchers. Would their classroom time not have been better spent learning more about project management, and less about the intricacies of partial differential equations? This is not to say that we could ever abandon mathematical rigor in the engineering sciences. However, with college costs climbing without bound, perhaps a more judicial use of students’ time and money is prudent; not every engineering student want to pursue an academic career. For those who want to proceed to grad school, the current arrangement may be fine. However, are the remaining students receiving an education that will allow them to acheive rapid proficiency throughout their working careers?

Education: Based on the roles and skills needed by engineers, it is possible to start addressing the education of engineering students. This topic is vast, and I might start by breaking it down into four subheadings:

• Topics: What skills should we be teaching? More software programming? More interpersonal skills? More hardcore engineering?
• Methods: By what method should we present these topics? Screencasts? Online lectures? One-on-one tutoring?
• Style: How might the material be best presented to allow students to quickly comprehend key concepts?
• Structure: What is the structure by which this education is best delivered? Are universities still the right venue for delivering an engineering education? Will new organizations, either ad-hoc or private enterprise, sprout up to deliver an education at a lower cost, and in less time?

I’ll try to work through these issues in future posts. If blogging fails to help me sort out these thoughts, then perhaps the “Preparing Future Faculty” program I enrolled in today will get me moving down the right path. By completing the course I am supposed to be able to:

• Explore and reflect on my assumptions about academic roles, positions, practices, missions, and institutions.
• Construct an institutional profile and relate my career goals and faculty skill sets with institutional missions and departmental goals.
• Construct a career strategic plan for enhancing and maintaining faculty skill sets and competencies.
• Develop a portfolio including curriculum vita, cover letter, research statement, and teaching philosophy.

Sounds like a good start to me!

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## Instructional Training

When I returned to school more than two decades after getting my master’s degree in mechanical engineering, it was with the intent of teaching at the university level. I had previously taught engineering technology classes, early in my career, at one of Purdue’s extension campuses, and I enjoyed the experience. But I had studied to be an practicing engineer, and I left academia to begin my industrial career soon after. Following twenty years in the private sector, though, I felt that I had accumulated enough useful insights to be of benefit to young engineering students. Teaching at the university level requires a doctorate, however; so, at an age that was twice the norm, I began work on my PhD.

As I wrap up my degree (hopefully defending yet this semester), I must admit to having nagging doubts about an academic career. Its not so much the hard work associated with beginning at the bottom of the academic ladder, or the drudgery of grading tests and homework, or even the murky waters of academic politics. Rather, as I will detail in future posts, I think that universities face some serious challenges in the near future. Further, I believe that much of the coming revolution in education will be driven by private enterprise, and not academic institutions. I would much prefer leading the change, rather than resisting it.

Granted, this may be making some unfair assumptions about today’s universities. I know that many schools are working hard to modernize their curriculum and instructional methods. And my experiences in higher education may not be typical. So I’ve decided to give the academic path one final look during the course of this semester. I’m signing up for the “Preparing Future Faculty” course being offered this semester. Perhaps, given further additional exposure to the academic system, I will feel more comfortable about making positive changes from within a university position. If not, well then I can say I gave it an honest evaluation.

In addition to the PFF course, I plan on attending a series of workshops given by Purdue’s Center for Instructional Excellence (CIE). I’m told by friends who have previously attended these seminars that the material is more suited for the humanities than for engineering coursework. However, I’ll try to report on the ideas and methods that seem the most appropriate for the engineering classroom.