This week in screencasting: The polar express

It’s been a little quiet on the screencasting front lately, but in the next couple of weeks my colleague teaching Calculus III will be hitting material for which I volunteered to provide some content: namely, using MATLAB to visualize some of the surfaces and solids used in multiple integration. Yesterday, I finished two of these. The first on is on polar coordinates and polar function plotting in MATLAB:

And the second one is on cylindrical coordinates and plotting two-variable functions in cylindrical coordinates:

MATLAB doesn’t provide a built-in function for plotting in cylindrical coordinates. Instead — and this is either ingenious or annoying depending on how you look at it — to plot something in cylindrical coordinates, you generate all the points you need in cylindrical coordinates and then use the pol2cart function to convert them en masse to cartesian coordinates, then plot the whole thing as usual in cartesian coordinates.

I think this is smart, since by avoiding the use of a specialized function for cylindrical plots and sticking instead to a single command for 3D plotting, you learn one command for all 3D plots and you get to use all the extras available, such as adding a contour plot onto the cylindrical plot. Overloading the pol2cart function so that it can accept and produce the third coordinate makes this all work. Overall I like how MATLAB doesn’t try to make a function for everything but rather creates a well-featured set of relatively simple tools that will do lots of things.

But I can see where some people — especially MATLAB novices — would find all this annoying, since the entire process takes several steps. There’s a workflow diagram for doing this in the screencast, but a better way is to make an M-file that holds all the steps. Here’s the one I flashed briefly at the end of the screencast:

% Script for plotting a cylindrical function.
% Written by Robert Talbert, Ph.D., 10/20/2010

% Theta: Change t1 and t2 to set the starting and ending values for theta.
t1 = 0;
t2 = pi/2;
theta = linspace(t1, t2);

% r: : Change r1 and r2 to set the starting and ending values for theta.
r1 = -5;
r2 = 5;
r = linspace(r1, r2);

% Create meshgrid for inputs:
[theta, r] = meshgrid(theta, r);

% Apply the function to create a matrix of z-values. Change the function to
% match what you want to plot.

z = r*cos(theta);

% Convert to cartesian and plot using mesh:

[x,y,z] = pol2cart(theta, r, z);
mesh(x,y,z)

It would be simple enough to modify this so that it’s a function rather than a script, accepting the arrays theta and r and a function handle for z, and then producing the 3D plot. Or, one could even make an “ez” version where the user just enters a string containing the function s/he wants. If somebody wants to try that out, and you want to share your results, just put the source code in the comments.

The third one in this series will be up later this weekend. It’s on spherical coordinates and it’s pretty much the same process, only using sph2cart instead of pol2cart. There might be a fourth one as well, dealing with some special cases like constant cylindrical/spherical functions (you can’t just say “rho = 5”, because rho has to be a matrix) and how to plot not just the surfaces but the volumes underneath them.

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Why change how we teach?

Sometimes when I read or hear discussions of innovation or change in teaching mathematics or other STEM disciplines, whether it’s me or somebody else doing the discussing, inevitably there’s the following response:

What do we need all that change for? After all, calculus [or whatever] hasn’t changed that much in 400 years, has it?

I’m not a historian of mathematics, so I can’t say how much calculus has or hasn’t changed since the times of Newton and Leibniz or even Euler. But I can say that the context in which calculus is situated has changed — utterly. And it’s those changes that surround calculus that are forcing the teaching of calculus (any many other STEM subjects) to change –radically.

What are those changes?

Photo by http://www.flickr.com/photos/brostad/

First, the practical problems that need to be solved and the methods used to solve them have changed. Not too long ago, practical problems could be neatly compartmentalized and solved using a very small palette of methods. I know some things about those problems from my Dad, who was an electrical engineer for 40 years and was with NASA during the Gemini and Apollo projects. The kinds of problems he’d get were: Design a circuit board for use in the navigational system of the space capsule. While this was a difficult problem that needed trained specialists, it was unambiguous and could be solved with more or less a subset of the average undergraduate electrical engineering curriculum content, plus human ingenuity. And for the most part, the math was done by hand and on slide rules (with a smattering of newfangled mechanical calculators) and the design was done with stuff from a lab — in other words, standard methods and tools for engineers.

Now, however, problems are completely different and cannot be so easily encapsulated. I can again pull an example from my Dad’s work history. During the last decade of his career, the Houston Oilers NFL franchise moved to Tennessee. Dad was employed by the Nashville Electric Service and the problem he was handed was: Design the power grid for the new Oilers stadium. This problem has some similarities with designing the navigational circuitry for a space capsule, but there are major differences as well because this was a civic project as well as a technical one. How do we make the power supply lines work with the existing road and building configurations? What about surrounding businesses and the impact that the design will have on them? How do we make Bud Adams happy with what we’ve done? The problem quickly overruns any simple categorization, and it required that Dad not only use skills other than those he learned in his (very rigorous!) EE curriculum at Texas Tech University, but also to learn new skills on the fly and to work with other non-engineers who have more in the way of those skills than he had. Also, the methods use to solve the problem were radically different. You can’t design a power grid that large using hand tools; you have to use computers, and computers need alternative representations of the models underlying the design. And the methods themselves lead to new problems.

So it is with calculus or almost any STEM discipline these days. Students today will not go on to work with simple, cleanly-defined, well-posed problems that fit neatly into a box. Nor will they be always doing things by hand; they will be using technology to solve problems, and this requires both a different way of representing the models (for calculus, think “functions”) they use and the flexibility to anticipate the problems that the methods themselves create. This is not what Newton or Leibniz had in mind, but it is the way things are. Our teaching must therefore change to give students a fighting chance at solving these problems, by emphasizing multiple representations of functions, multiple methods for solution of problems, and attention to the problems created by the methods. And of course, we also must focus on teaching problem-solving itself and on the ability to acquire new skills and information independently, because if so much has changed between 1965 and 1995, we can expect about the same amount of change in progressively shorter time spans in the future.

Photo by http://www.flickr.com/photos/wonderlane/

Also, the people who solve these problems, and what we know about how those people learn, have changed. It seems undeniable that college students are different than they were even 20 years ago, much less 200 years ago. Although they may not be natively fluent in the use of technology, they are certainly steeped in technology, and technology is a primary means for how they interact with the rest of the world. Students in college today bring a different set of values, a different cultural context, and a different outlook to their lives and how they learn. This executive summary of research done by the Pew Research Foundation goes into detail on the characteristics of the Millenial generation, and the full report (PDF, 1.3 Mb) — in addition to our own experiences — highlights the differences in this generation versus previous ones. These folks are not the same people we taught in 1995; we therefore cannot expect to teach them in the same way and expect equal or better results.

We also know a lot more now about how people in general, and Millenials in particular, learn things than we did just a few years ago. We are gradually, but also rapidly, realizing through rigorous education research that there are other methods of teaching out there besides lecture and that these methods work better than lecture does in many situations. Instructors are honing the research findings into usable tools through innovative classroom practices that yield statistically verifiable improvements over more traditional ways of teaching. Even traditional modes of teaching are finding willing and helpful partners in various technological tools that lend themselves well to classroom use and student learning. And that technology is improving in cost, accessibility, and performance at an exponential pace, to the point where it just doesn’t make sense not to use it or think about ways teaching can be improved through its use.

Photo by http://www.flickr.com/photos/softpixtechie/

Finally, and perhaps at the root of the first two, the culture in which these problems, methods, people, and even the mathematics itself is situated has changed. Technology drives much of this culture. Millenials are highly connected to each other and the world around them and have little patience — for better or worse — for the usual linear, abstracted, and (let’s face it) slow ways in which calculus and other STEM subjects are usually presented. The countercultural force that tends to discourage kids from getting into STEM disciplines early on is probably stronger today than it has ever been, and it seems foolish to try to fight that force with the way STEM disciplines have been presented to students in the past.

Millenials are interested to a (perhaps) surprising degree in making the world a better place, which means they are a lot more interested in solving problems and helping people than they are with epsilon-delta definitions and deriving integrals from summation rules. The globalized economy and highly-connected world in which we all live has made almost every problem worth solving multidisciplinary. There is a much higher premium now placed on getting a list of viable solutions to a problem within a brief time span, as opposed to a single, perfectly right answer within an unlimited time span (or in the time span of a timed exam).

Even mathematics itself has a different sort of culture now than it did even just ten years ago. We are seeing the emergence of massively collaborative mathematical research via social media, the rise of computational proofs from controversy to standard practice, and computational science taking a central role among the important scientific questions of our time. Calculus may not have changed much but its role in the larger mathematical enterprise has evolved, just in the last 10-15 years.

In short, everything that lends itself to the creation of meaning in the world today — that is, today’s culture — has changed from what it used to be. Even the things that remain essentially unchanged from their previous states, like calculus, must fit into a context that has changed.

All this change presents challenges and opportunities for STEM educators. It’s challenging to go back to calculus, and other STEM disciplines, and think about things like: What are the essential elements of this subject that really need to be taught, as opposed to just the topics we really like? What new facets or topics need to be factored in? What’s the best way to factor those in, so that students are really prepared to function in the world past college? And, maybe most importantly, How do we know our students are really prepared? There’s a temptation to burrow back in to what worked for us, when faced with such daunting challenges, but that really doesn’t help students much — nor does it tap into the possibilities of making our subjects, and our students, richer.

Just so you’d know

…I’m all done with the ASEE and headed off on vacation in nearby Holiday World. Whatever it is I’ve left out about ASEE, I hope to fill in once I’m back home. If there’s something specific you’d like to know about what I’ve seen here, let me know in the comments.

Funniest remark of the ASEE so far

…goes to Robert Grondin of Arizona State University Polytechnic Campus, who made this remark in his talk in the Liberal Education for 21st Century Engineering session:

We do projects at the beginning of the course, because projects are fun, and we want to fool students into thinking that engineering is fun.

This was apropos of how engineering curricula usually incorporate projects — either at the beginning of the curricula via a freshman design course, or at the end via a senior design course, or both. But you can pretty much substitute any discipline and get the way we often think about how projects fit into the curriculum, right?

Prof. Grondin, on the other hand, has designed a generic Engineering degree — not Mechanical Engineering, Electrical Engineering, or whatnot… just Engineering — for ASU Polytechnic that requires only 20 hours of engineering coursework beyond the freshman core and in which there’s a design project course in every semester. That’s what you call taking project-based learning seriously, and I’d daresay that these general Engineering students are better prepared for real engineering work than many students with specialized engineering degrees.

What (some) engineers think about liberal education

I’m currently at the American Society for Engineering Education conference and symposium in Louisville. There is a lot to process as I attend sessions on student learning, technological literacy, liberal education, and so on, all from the perspective of engineers and engineering educators. There is an entire division (a sort of special interest group) within the ASEE for Liberal Education, and I attended one of their paper sessions this afternoon.

Engineers have a quite different perspective on liberal education than those in “liberal arts” disciplines (by which we usually mean social sciences, arts, humanities) and those of us math/science people working in liberal arts colleges, but surprisingly — at least for the engineers I hung out with in the session — the two conceptions largely agree. We all conceive of liberal education as education that integrates multiple perspectives into understanding what we study and do. We believe in the importance and relevance of disciplines other than our own and seek to learn about other disciplines, connect with practitioners and colleagues in other disciplines, and incorporate other disciplines in meaningful ways into our courses. We believe in teaching students metacognitive skills and preparing them to be human beings, not just workers.

Of course there are engineers who don’t think this way and in fact look down on other disciplines in direct proportion to their methodological distance from engineering (the less data and design involved, the greater the disdain). But consider too that there are also poets, philosophers, historians, mathematicians, sociologists, and so on who feel the same way about their own disciplines. The departmental silos exist all over campus.

Particularly enlightening was a parallel given in a talk by Cherrice Traver and Doug Klein of Union College (a liberal arts college known for its strong and historically-rooted engineering programs) between the criteria for ABET accreditation of engineering programs on the one hand, and the learning outcomes of Liberal Education and America’s Promise (or LEAP; a prospectus from the American Association of Colleges and Universities) on the other. Here are ABET’s Program Outcomes and Assessment criteria:

Engineering programs must demonstrate that their students attain the following outcomes:
(a) an ability to apply knowledge of mathematics, science, and engineering
(b) an ability to design and conduct experiments, as well as to analyze and interpret data
(c) an ability to design a system, component, or process to meet desired needs within realistic
constraints such as economic, environmental, social, political, ethical, health and safety,
manufacturability, and sustainability
(d) an ability to function on multidisciplinary teams
(e) an ability to identify, formulate, and solve engineering problems
(f) an understanding of professional and ethical responsibility
(g) an ability to communicate effectively
(h) the broad education necessary to understand the impact of engineering solutions in a global,
economic, environmental, and societal context
(i) a recognition of the need for, and an ability to engage in life-long learning
(j) a knowledge of contemporary issues
(k) an ability to use the techniques, skills, and modern engineering tools necessary for
engineering practice.

The entire accreditation document is here (PDF).

Compare those with the LEAP outcomes:

Beginning in school, and continuing at successively higher levels across their college studies, students should prepare for twenty-first-century challenges by gaining:

Knowledge of Human Cultures and the Physical and Natural World

Through study in the sciences and mathematics, social sciences, humanities, histories, languages, and the arts

Focused by engagement with big questions, both contemporary and enduring

Intellectual and Practical Skills, Including

Inquiry and analysis
Critical and creative thinking
Written and oral communication
Quantitative literacy
Information literacy
Teamwork and problem solving
Practiced extensively, across the curriculum, in the context of progressively more challenging problems, projects, and standards for performance

Personal and Social Responsibility, Including

Civic knowledge and engagement—local and global
Intercultural knowledge and competence
Ethical reasoning and action
Foundations and skills for lifelong learning
Anchored through active involvement with diverse communities and real-world challenges

Integrative and Applied Learning, Including

Synthesis and advanced accomplishment across general and specialized studies
Demonstrated through the application of knowledge, skills, and responsibilities to new settings and complex problems

As the presenters mentioned, you can make an exercise of lining these two lists of learning outcomes side by side (in fact, they gave us a handout where this was done) and draw lines connecting learning outcomes in LEAP with corresponding, or even identical, criteria from ABET’s list.

What this means, I think, is that there is a strong base of support for liberal education among engineers. One might even say that those in charge of accrediting engineering programs want engineers to be liberally educated. Some engineers, like the ones in the session I attended, will even say that themselves.

What nobody seems able to explain just yet is the implicit and sometimes explicit resistance to liberal education among many engineers and engineering programs. For example, why do most engineering programs require monumental depth in a single engineering discipline — as undergraduates — with only token amounts of university-required coursework outside of engineering? The electrical engineering degree at one university, for example, requires 68 credit hours just in freshman and electrical engineering courses. Then 33 hours in math and science, and a 3-hour mechanical engineering course. Eighteen hours total are left over for electives outside math, science, or engineering — and six of those are prescribed courses (composition and communication), leaving just 12 hours to be chosen from General Education elective blocks.

That’s just four courses the student gets to pick out of sheer curiosity and personal interest for his or her entire college education! Can that possibly be in line with what ABET — or for that matter, the engineering community and its clients — really want?

Active learning is essential, not optional, for STEM students

This article (1.2 MB, PDF)  by three computer science professors at Miami University (Ohio) is an excellent overview of the concept of the inverted classroom and why it could be the future of all classrooms given the techno-centric nature of Millenials. (I will not say “digital natives”.) The article focuses on using inverted classroom models in software engineering courses. This quote seemed particularly important:

Software engineering is, at its essence, an applied discipline that involves interaction with customers, collaboration with globally distributed developers, and hands-on production of software artifacts. The education of future software engineers is, by necessity, an endeavor that requires students to be active learners. That is, students must gain experience, not in isolation, but in the presence of other learners and under the mentorship of instructors and practitioners.  [my emphasis]

That is, in the case of training future software engineers, active learning is not an option or a fad; it is essential, and failure to train software engineers in an active learning setting is withholding from them the essential mindset they will need for survival in their careers. If a software engineer isn’t an active learner, they won’t make it — the field is too fast-moving, too global, too collaborative in its nature to support those who can only learn passively. Lectures and other passive teaching techniques may not be obsolete, but to center students’ education around this kind of teaching sets the students up for failure later on.

One could argue the same thing for any kind of engineer, or any of the STEM disciplines in general, since careers in those disciplines tend to adhere to the same description as software engineering — a tendency toward applications (many of which don’t even exist yet), centered on interaction and collaboration with people, and focused on the production of usable products.

Links for Tuesday

• What’s that smell? It could be the latest in biometrics.
• At Slashdot, a discussion on combining computer science and philosophy. I think that, in general, there is a lot of really interesting yet uncharted territory in the liberal arts arising from combining computing with [fill in humanities subject here].
• Circuit City hits Chapter 11. The only reason I’m sorry to hear about this is because I know people who work for Circuit City who might lose their jobs. But that’s the only reason. There used to be a time, when I was a teenager, when going to Circuit City to paw over all the tech stuff was fun and exciting. Now when I go, it’s a game of “dodge the irritating service rep”.
• Some nice tips on getting the most out of Google Scholar. Especially useful if, like me, you’re in a place that doesn’t have access to a lot of technical journals.
• Mike at Walking Randomly is finding symbolic integrals that the new version of MATLAB can’t do. This is a really important series he’s doing, and his articles are a great resource for MATLAB users.
• Speaking of math, here’s Carnival of Mathematics #43.
• The University of Cincinnati is trying out a market-based approach to its various schools that might levy budget cuts on programs that don’t produce. What a concept! Of course the anti-free market people are running wild in the comments.
• Finally, make sure you thank an engineer today.

The suckage of being an engineering student

A blog post at Wired claims to give the Top 5 Reasons It Sucks to be an Engineering Student. Discussion is in the comments there and at this lively thread at Slashdot. The reasons given at the Wired blog are (in reverse order):

1. Awful textbooks
2. Professors are rarely encouraging
3. Dearth of quality counseling
4. Other disciplines have inflated grades
5. Every assignment feels the same

It sounds to me like the blogger at Wired is stereotyping, based on what goes on at large research universities. A student could avoid #2, #3, and maybe #5 just by doing a 3+2 program where the first three years are done at a liberal arts college (…shameless plug alert…).

As for the grade inflation, I admit there’s no solution to this short of doing the right thing and forcing real academic standards on some of the touchiest-feeliest portions of the liberal arts world. But I think that would lead to mass chaos, as the stability of many liberal arts college depends on having some department on campus to be the “good cop” which offers refuge to students who just aren’t that interested in getting good at something difficult. All I can offer is some sympathy, that math and science professors are often eviscerated on course evaluations by those very students, who are shocked — SHOCKED — that deadlines would be enforced, hard material would be on tests, and so forth.

So to all engineering students out there, keep on keepin’ on. It might suck a little in the short term, but when it’s over you get to run our entire society!

Waving the waiver wand

Senator Max Baucus (D-Montana) is planning to present legislation that would provide free tuition for math and science majors, provided that they work or teach in a related field for at least four years after graduation. The full legislation involves $25 billion in spending on education and includes additional spending on supplementing teacher wages in rural areas. (I’m sure it’s purely coincidence that 99% of the state of Montana is uninhabited rural.)

The goal, according to Sen. Baucus, is “…to better prepare children for school and get more of them into college to make the United States more globally competitive, particularly with countries like China and India”.

Waving the magic money wand at this problem is a typically ineffective political response, and it misunderstands the problem. College students who stay away from math and science majors typically do so for a combination of two reasons: (1) they are no good at math or science because they weren’t taught the subjects properly in K-12, and (2) they are culturally indoctrinated to having negative viewpoints on math and science.

Reason (2), the cultural reason, is the more serious and prevalent. Our pop culture today abhors academic excellence and the intellect in general and math/science fields in particular. Students are steeped in this culture from infancy and grow up with fixed and regularly reinforced negative ideas about math and science. You are not going to make this problem go away by offering free tuition, which generally benefits only the students who would have majored in math or science anyway. If you want to prepare students for careers in math and science, and if you want to train up the next generation of workers who can compete on the global scale, this cultural issue has got to be addressed. (And I don’t see how the federal government is qualified or even sanctioned to do so. A cultural problem requires a cultural response, not a government response.)

In fact, waiving tuition would probably have the effect of decreasing the proportion of qualified graduates in math and science. Waiving tuition would (maybe) increase the total number of students in math and science, but the people who are enticed by the money are likely to be the ones who are not skilled or do not like math or science well enough to major in those fields without the enticement. So it will merely become harder to find the math and science graduates who know what they are doing. (Assuming they complete their degrees at all.)

[Hat tip: Slashdot]

Engineering a reality check

I’ve spent the better part of the last two weeks resurrecting the moribund 3+2 program in engineering that my college is proposing to the local Big University. My goal is to get the agreement with the BU completed and get the program on the books and advertised at our place by the end of the summer. It looked like things were on track to meet these goals this time last summer, but a bunch of things — heavy fall workload, the discovery of major scheduling issues with the proposed course plans, and so on — conspired to drive the whole project to “inactive” status. But I’ve revamped and streamlined the five-year plans, and it looks like we’ll be able to have the thing up and running soon, barring any other weirdness.

I’ve also been advising a student who came in as a freshman last fall wanting to major in Applied Math and then go to grad school to become an electrical engineer, who had never heard of our plans for a 3+2 program. This student was very interested and so, although we still have no official agreement with the BU, I put the student on the proposed five-year plan and had the student take courses on faith that the program would eventually be completed.

I’ve learned a few simple but important facts about engineering programs, and being a student in one of them, as a result of all this work.

1. You really have to be making A’s and B’s in pretty much everything at the freshman and sophomore level — including the year-long sequences in Calculus, physics, and chemistry — in order to have viability as an engineering student. Engineering is an intense discipline to study. It’s even moreso in a 3+2 program where students are working on two degrees — one in engineering from the BU, the other in Applied Mathematics from us, each of which would be intense enough on its own — in five years. Your level of skill in basic science and math has to be above suspicion. If your grades are merely lackluster in these basic areas, it’s unlikely you’ll improve over time as the courses get harder and harder.

2. Of all the courses in the program, it seems like Calculus II is the real make-or-break experience. Lots of students in this program will do fine in Calculus I because most of them had it in high school. But when you get to Calculus II, you typically reach the limit of the math you learned in high school. And it’s here that you have to move from remembering math to learning it. If a student can successfully pick up Calculus II and understand even just 3/4 of what goes on in the class, I’d give that student a good shot at doing well in the other math, science, and engineering courses as well. They are showing that for the most part, they can learn new tricks in a complicated technical subject. But if the student is making a C or below in Calculus II, I get very worried. If they make below a C-, I am really worried, because that’s not a passing grade in most departments. And if you need to take a year of Linear Circuit Analysis in your electrical engineering program, and the first of that requires Calculus III, and that requires Calculus II which you didn’t do very well in…

3. Your performance in a science, math, or engineering class should be determined only by your work if you are going to have a chance at making it in an engineering program. What I mean is that your academic skill and intellectual focus on these classes needs to be such that external factors don’t make much of a dent on your performance in those courses, even if the factor is serious. “External factors” include boy/girlfriend problems, family problems, part-time jobs, athletic or Greek involvement, and — this seems to be awfully common in engineering — an inability to understand a professor who is not a native English speaker. The students who go on to success in a program like this do sometimes have serious external problems in their lives, but they can focus past them and get the job done anyway.

Those three points fit not only my experiences from working on this agreement and the students in it, but also my experiences in going to an engineering school for undergrad and in having a dad, a sister, and two brothers-in-law who are professional engineers. Got anything to add to these?

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