My qualifications


    1. Teaching
      1. UTeach
      2. Modern Physics lab
      3. The Princeton Review
      4. AVID volunteer
      5. Engineering physics (103n)
      6. Undergraduate teaching assistant for Digital Systems course at Cornell
      7. Volunteer with CTTC teaching web page design to adults
    2. Math & Science Training:

I will provide a transcript upon request. (I don't feel comfortable publishing the transcript online).


Ongoing training: Advances in Mathematics and Educational Practice

I subscribe to a variety of professional journals and educational magazines

* AAPT's American Journal of Physics
* NCTM's Mathematics Teacher

I plan to involve particularly enthusiastic students in independent research as early as possible in their education. My research experience in grad school gives me a grounding in scientific investigation, and I have previously supervised students in independent research resulting in published papers (the lock-in amplifier project and the Nonlinear pendulum projects).

I also incorporate goals-oriented feedback from students, teachers, adminstrators and experts: see the Evaluation strategy section.


Some Representative Mathematical Works

As a graduate student in physics, and in my mathematics education training, I've gotten to investigate many different interesting topics. I've chosen a few representative works to show how I personally approach questions of mathematical and scientific interest.

The Value of that 'New Car Smell'

(Computational Finance with Stathis Tompaidis)
[Mathematica Notebook, Presentation Notes, PDF]

My friend Chris and I came up with, and found some pretty strong insights into, the question "What is the value of 'New Car Smell'?" More prosaicly, it's well known that a car loses thousands of dollars in value once it is driven off the new-car lot (hence becoming a used car). There are a wide variety of reasons for this drop in value, yet a liquid new- and used-car market exists for every age of saleable automobile. Much of this discrepancy comes up because newer cars offer a variety of peripheral or intangible benefits -- MP3 players, traction control, 'coolness' factor, new-car smell -- whose value is difficult to capture precisely.

The idea, then, is to model the true costs of ownership, by age, for a class of cars. If our model correctly captures the observable factors (which it should), and the market for cars is both liquid and populated by rational buyers and sellers (which it mostly is), then the difference between our model's price and the market price captures the value of all those intangible attractions offered by a new car!

The heart of our model is to treat the decision (keep driving your current car, or buy a different car) as an optimal decision problem with risk. (Risk is introduced largely by the uncertainty of maintenance costs and the chance that the car will be scrapped due to accident or failure). With the reasonable simplification that you make this decision exactly once a year, this becomes a discrete-time problem that can be straightforwardly handled with matrix methods (we used Mathematica).

This project, like many economics questions, involved a fair amount of informal and meta-mathematical reasoning. For example, one component in our model aimed to capture the 'nuisance cost' of selling your old car and shopping for a new one. This process takes time and obviously provides a small but real incentive: holding on to that old car a bit longer means you can spend next weekend at the beach, or fixing your house, or working overtime -- not meeting buyers or fending off car salespeople. We estimated the amount of time involved, and valued that time as a proportional fraction of the average American's annual wages. (Note that it doesn't matter whether you would spend that time working, playing or even sleeping: since you've made an economic decision to work the number of hours you do, at this broad level we say that those activities are in equilibrium and are valued equivalently.) We also had to employ a variety of other informal and formal arguments to extract the data we needed (maintenance costs by age, scrappage rate, depreciation rates, etc.) from the data we had (overall household maintenance expenditures, market price, etc.).

In the end, we found that (in our simplistic mode) the optimal strategy is to initially purchase a five-year old used car, then sell it at fifteen years (if it makes it that long) to buy another five-year old used car. This strategy gives the lowest total cost (purchase price + total future costs) of any replacement schedule. Making an initial purchase of a car older or younger than that will incur additional costs of maintenance or depreciation. This, then, is a graph showing the premium in total future costs people will pay above the optimum initial purchase point:

Intangible/Irrational Value

Depending on your perspective, it measures some mixture of new-car allure, old-car cachet, buyer irrationality, or externalities not captured in our model.



Bouncy Ball Problem

(Classical Mechanics with Philip Morrison)
[Mathematica Notebook, PDF, MathML]

Here the problem was to analyze a seemingly straightforward problem: describe the 1-D motion of a perfectly elastic ball bouncing on a table moving up-and-down at a constant (sinusoidal) rate. Enrico Fermi proposed this as a model for the origin of cosmic radiation: here, the ball is a charged particle and the table is a moving magnetic field.

The assigned problem was simply to find a map from the ball's state at one collision to the state on its next collision. However, I was curious to find out what a computer simulation of the ball's motion could reveal about the global structure of the problem. (Specifically, I wanted to construct a picture of the 4-D phase space {hn, φn; hn+1, φn+1} for apex height h and table phase φ).

Since it's not immediately clear how the collision map can be solved in closed-form, one might like to use a computer approximate-root-finding algorithm. However, two problems present themselves: this is very, very slow, and is vulnerable to being trapped by false roots (near-misses): Near Miss

To understand why this is such a problem, you have to know a bit about the mathematics driving this technology. Computer root-finding algorithms generally find a solution that matches within a given tolerance. You can turn that tolerance smaller, but the search time grows faster than the accuracy improves -- and you cannot set the tolerance to zero.

Faced with this limitation, I turned back to mathematics. By non-dimensionalizing the system and making an observation (once the ball falls below h=A the maximum height of the table, it will collide within that period), I was able to reduce the parameter space and actually find a closed-form solution.

Then, in the regime where the table isn't pumping infinite energy into the ball, I used computer simulation to find the following highly interesting map of the phase space:

Phase Space

There are some initial conditions for the ball which give stable orbits: if the ball hits a little early, it will gain energy and travel farther (its next arrival will have a later phase); if it hits late it will lose a bit of velocity and thus phase. The phase and velocity remain stably constrained:

Periodic OrbitThere are other conditions for which the ball hops periodically among a few stable phase/velocity regions:

Complex Periodicand finally, there are starting situations for which the ball's motion becomes classically chaotic: after enough time, any two nearby starting positions will end up arbitrarily far apart:
Chaotic Motion

The phase space map above displays 500 iterations each of 50 initial conditions; the chaotic orbit I've highlighted shows 10,000 iterations of a single orbit. Each point involves computing the return map

δ_ (n + 1)
(δ_n + 2/γβ_ (n + 1)) mod 1
β_ (n + 1)
β_n - 2Sin[2 π δ_n]

It's safe to say that I would never have been able to generate the above graphs without the technological aid of a computer graphing program. However, it is only by understanding the limitations of the solutions this technology provides -- and the mathematical origins of those limitations -- that I was able to sensibly navigate the intricacies of understanding the mechanics (or in other words, to follow the bouncing ball).



Lock-In Amplifier Rescues Small Signals from a Sea of Noise

I designed and built software that uses a low-cost general-purpose data acquisition board to implement a lock-in amplifier. (The software is freely available under an open-source license from my website). A lock-in amplifier is a specialized piece of laboratory equipment capable of extracting a small, periodic signal buried within a sea of broadband noise. Since my design used general-purpose, low-cost hardware already on hand, we were able to teach and apply this useful device in the Modern Physics lab class at no additional cost.

Since the device was intended as a teaching tool as well as a scientific instrument, I took special care to make its operation as transparent and understandable as possible.

Specifically, I tried to make as clear as possible the connection between the time-domain and frequency-domain (Fourier transformed) representation of the signal as it passed through each stage of the process. Using LabView, students could easily open and manipulate a fully operational simulation of the lock-in. They could introduce and control the amount of broadband noise, signal power, and dc or harmonic noise in the inputs, and see (in time- or frequency-domain) the effects of multiplying, shifting and filtering the signal. Many difficult-to-grasp concepts -- like the ringing introduced near cutoff by a high-order filter, or the frequency shifts caused by mutiplying two periodic waveforms -- become clear when students are able to see both representations evolve, in real time and in response to their manipulations.


Why Study Math?

I know that one of the experiences many teachers dread is to hear their students ask "Why do I need to learn this?"

I want my students to ask this question and to understand the answer: I want them to be in charge of their educational path. So I wrote up my thoughts on the many compelling reasons to learn mathematics. I included arguments from educational, economic, cultural and historical perspectives.


Art in Mathematics, Mathematics in Art

(Functions and Modeling, Mark Daniels)

I chose four artworks from the Blanton Museum of Art and described connections between the artworks and relevant mathematical topics.