On Friday AJ and I were in Vicksburg, MS looking for a scenic grassy area, possibly with an interesting history, as a venue for our picnic. She had a vague memory of the battlefield (National Military Park) from an elementary school field trip, so we decided to check it out. We found the entrance, paid the $8 entrance fee, and followed the sign that said, "picnicking only at Tour Stop #12 and Cairo Museum." About four miles later, we realized we were stuck in a nightmare of '50s car-culture-meets-'00s -SUV-culture: a drive-through national park being appreciated by aging baby boomers who remembered the park from their childhood and suburban parents trying to introduce their kids to the nation's (south's) history without ever leaving the comfort of their car.

I'll spare you the agony of being stuck behind 2 mph crawlers reading about the exploits of the 108th Infantry Division and Logan's Farthest Advance. In my ire (magnified by hunger as our picnic sat untouched in our trunk), I began thinking of cars as an interface between humans and the built environment. They were invented as a means of transportation, which became a lifestyle, and an environment was built to accomodate this lifestyle. The environment was built with the expectation that its users would n0t only have cars, but prefer to stay in their cars as much as possible. What started as an empowering tool soon became a necessity, as car-culture and auto-centric design created a positive feedback loop.

I'm not the first person to notice this by any means. But I'm not trying to publish original research, I'm just writing a blog. My question is this: if this is a desirable architecture (relying on cars as an interface), is the interface designed properly for human interaction? The ipod taught us the value of ergonomic interfaces. Do cars meet the requirements of human-centered design? How about this: are you comfortable interfacing with the world through a car? In practice, most of us are because it's what we know. But in theory, do you like that idea? I don't.

Americans aren't going to stop using cars in the next 10 years. They will continue to interface with their surroundings through them, but I propose that the built environment should offer its users a choice of interfaces. Make them car-, bike-, and pedestrian-friendly (not to mention wheelchairs) . It's been done before, including by Randy Brown at Village Point East, where he put a parking lot in front and a sidewalk leading to the back (or perhaps the other front) from a nearby residential neighborhood. It can be done, and it has been done, and it is being done. I just wish it was a lot more common.

## Sunday, May 31, 2009

## Tuesday, May 19, 2009

### Conceptual Calculus

When I was in high school, roughly a senior, it came to my attention that there was a set of ideas known as calculus that was somehow related to change over time (until then, I had always thought of calculus as an intimidating and far-off college class that made sure only intelligent, hardworking people became engineers and scientists). While reading Feynman's Six Not-So-Easy Pieces I encountered dy/dx notation for the first time and was thoroughly stumped in my attempt to follow his derivations. I made it my goal to learn calculus by whatever means possible, and set off to the library. I read through a variety of textbooks, David Berlinski's A Tour of the Calculus, and part of an OCW calculus course, but in the end my attempts to teach myself calculus were stymied. I learned the power rule, encountered the quotient rule, learned more than I ever wanted to know about the inner personalities of functions (thank you, Mr. Berlinski), and learned that derivatives could be used to describe how things change, but I was still no closer to understanding what a derivative was.

This breakthrough wouldn't come until I was a freshman in college, taking Calculus I: Differential Calculus with a math teacher who had retired several years before. He spent the first lecture explaining how far the Greeks had gone without ever figuring out how to a) find the slope of a curve at any point and b)find the area inside any shape. He then explained how the Cartesian coordinate system enabled mathematical representations of the shapes with which the Greeks had been fascinated, and then digressed to the topic of limits. Within three or four lectures, we had a secant line of a curve described mathematically, then found the limit as the section of the curve marked off by the secant line approached zero. Before I knew what was happening, we had found what the Greeks in all of the glory only dreamed about. We had taken a derivative! I became so obsessed with the limit definition of a derivative that I wrote it constantly on chalkboards in random classrooms, on every page of my notebook, and possibly on a bathroom stall. . .

The rules that applied to derivatives and made them easy to compute didn't interest me much (except for the chain rule - I was fascinated by the chain rule), but the limit definition opened up the whole world to me. I could now understand Feynman, and the rest of the world. Anything that changed, really. Of course, it takes the fundamental theorem to really put it all together, but that came soon enough.

Now I'm coming to my point. You've already impressed me with your patience, so hold on a little longer. It is my opinion, based on my experience with Feynman, that a truly well-educated person needs an understanding of calculus, at least of differentiation and integration and how they relate to each other. Conventional wisdom says, "most college students can't even pass algebra; if you add calculus to the general education requirements no one will ever graduate." The real difficulty in learning calculus, though, is in remembering the rules and knowing how and when to apply them. Differentiating isn't too bad, but when you get to integration and all of the glorious guesswork involved there, it becomes taxing on even the most intrepid math student. My rebuttal is this: there is no reason for the average well-educated person to be proficient at integrating and differentiating (what one of my math teachers referred to as "computational ability"). In fact, due to the prevalence of computer algebra systems, I would argue it's hardly necessary for engineers to be proficient at any of this, including solving DEs. So, what's left? Merely the concepts.

Conventional wisdom would then say, "you can't learn math without solving problems." Can't you? How many engineering students truly come to an understanding of the Wronskian after finding a dozen pairs of linearly independent solutions to a DE? They merely learn to apply algorithms efficiently, something a computer will always do far better than them. I'm not arguing that engineers shouldn't solve problems, but I'm arguing that business majors, social scientists and the like should be introduced to the concepts of calculus without being expected to solve many problems.

My proposal is this: develop a "Conceptual Calculus" course that covers integration, differentiation and the Fundamental Theorem in a non-computationally-intensive way, to be taught in one semester to non-science majors who have a basic understanding of algebra. This course could (should) then be introduced as part of the core curriculum at universities (or liberal arts colleges, more likely) dedicated to producing well-rounded individuals capable of understanding topics across disciplines.

This breakthrough wouldn't come until I was a freshman in college, taking Calculus I: Differential Calculus with a math teacher who had retired several years before. He spent the first lecture explaining how far the Greeks had gone without ever figuring out how to a) find the slope of a curve at any point and b)find the area inside any shape. He then explained how the Cartesian coordinate system enabled mathematical representations of the shapes with which the Greeks had been fascinated, and then digressed to the topic of limits. Within three or four lectures, we had a secant line of a curve described mathematically, then found the limit as the section of the curve marked off by the secant line approached zero. Before I knew what was happening, we had found what the Greeks in all of the glory only dreamed about. We had taken a derivative! I became so obsessed with the limit definition of a derivative that I wrote it constantly on chalkboards in random classrooms, on every page of my notebook, and possibly on a bathroom stall. . .

The rules that applied to derivatives and made them easy to compute didn't interest me much (except for the chain rule - I was fascinated by the chain rule), but the limit definition opened up the whole world to me. I could now understand Feynman, and the rest of the world. Anything that changed, really. Of course, it takes the fundamental theorem to really put it all together, but that came soon enough.

Now I'm coming to my point. You've already impressed me with your patience, so hold on a little longer. It is my opinion, based on my experience with Feynman, that a truly well-educated person needs an understanding of calculus, at least of differentiation and integration and how they relate to each other. Conventional wisdom says, "most college students can't even pass algebra; if you add calculus to the general education requirements no one will ever graduate." The real difficulty in learning calculus, though, is in remembering the rules and knowing how and when to apply them. Differentiating isn't too bad, but when you get to integration and all of the glorious guesswork involved there, it becomes taxing on even the most intrepid math student. My rebuttal is this: there is no reason for the average well-educated person to be proficient at integrating and differentiating (what one of my math teachers referred to as "computational ability"). In fact, due to the prevalence of computer algebra systems, I would argue it's hardly necessary for engineers to be proficient at any of this, including solving DEs. So, what's left? Merely the concepts.

Conventional wisdom would then say, "you can't learn math without solving problems." Can't you? How many engineering students truly come to an understanding of the Wronskian after finding a dozen pairs of linearly independent solutions to a DE? They merely learn to apply algorithms efficiently, something a computer will always do far better than them. I'm not arguing that engineers shouldn't solve problems, but I'm arguing that business majors, social scientists and the like should be introduced to the concepts of calculus without being expected to solve many problems.

My proposal is this: develop a "Conceptual Calculus" course that covers integration, differentiation and the Fundamental Theorem in a non-computationally-intensive way, to be taught in one semester to non-science majors who have a basic understanding of algebra. This course could (should) then be introduced as part of the core curriculum at universities (or liberal arts colleges, more likely) dedicated to producing well-rounded individuals capable of understanding topics across disciplines.

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