Meets Wed. 3–3:50 PM in MHA 2623 Fall quarter, 2016

This Freshman Seminar is meant to free the minds of our students so that they approach problems with flexibility, intuition, and quantitative estimates. You think that's π you're smelling?

Students in physics classes often have problems transitioning from the lower division courses into upper division courses. Many reasons contribute, but one observation is that students have developed a habit of solving problems by pattern-matching. This can come in the form of equation-hunting or relying on exposure to essentially identical problems from the past.

Human beings are powerfully adept at pattern recognition, and this
versatile tool can get us rather far—all the way to
upper-division physics. At this point, *concepts* begin to rule, and
students must have a well-stocked toolbox of mathematical techniques
at the ready—and know which tools to grab for the problem at
hand. Sure, identifying relevant concepts and applicable tools also
constitutes a type of pattern matching, but at a higher, more flexible
level.

- Equation-hunting and formula-wedging are increasingly prevalent in physics majors: historically more characteristic of engineering students
- Over-reliance on pattern matching
- Stiff and formal relationship with numbers (relax, and let π = 3 = 10/3 = sqrt(10))
- Difficulty adopting concept-driven, first-principles approach
- Under-developed intuition; infrequent application of symmetry arguments
- Infrequent use of sanity checks, unit checks, estimation for scale
- Lacking intuition on basic units: how much is a N, J, W, C, F, etc.

- The educational system places emphasis on testing
- Many secondary education teachers don't really "get it"
- Changing population of majors: pragmatic, STEM-driven job-seekers who don't want to be left out of the emerging world
- Increasingly inaccessible world: urban environment with x-box and wifi vs. farm/woods, repair/dissection of simple equipment
- What I call the "Dora the Explorer Syndrome"—where
emphasis is on the
*idea*, not on challenges and practical feasibility of implementation. Kids come away thinking any hair-brained idea constitutes a victory.

So let's try re-programming the way students approach problems
*before* reaching the upper-division whiplash moment. That's
what this Freshman Seminar aims to do. No guarantees that this will
be successful, but let's try building comfort and versatility in
thinking about problems so that students have better intuition and
alternate ways to sanity-check their answers. Among the approaches,
we will:

- Learn that there are often various ways to skin the cat
- Learn to ask: "How do I know if this is
right?"—and to answer
*on your own*wihtout the aid of an outside source (like answers in the back of a book, or online solutions to a problem) - Build estimation skills to assist sanity checks
- Apply intuition to anticipate the outcome, and apply feedback to refine intuition when it is found to be off
- Check for symmetries or end-runs to potentially save time
- Check symbolic results in limiting cases to verify proper behavior
- Identify reliable, basic concepts (e.g., conservation laws) and build your own formulas from there
- Encourage exploration and daily life problem creation/solving

Famous physicists like Fermi and Feynman frequently forged formidable feats 'festimation. Less alliteratively, good physicists tend to have well-developed estimation abilities. Seldom are these skills learned in classes, but "learned on the mean streets," as my colleague George Fuller puts it. Maybe it's a selection effect: those who are good at estimation are more likely to succeed in physics/science.

For me personally, I learned at *least* as much in my
explorations outside of class as I did in the formal classroom. I
almost always had some problem I was chewing on, and my eagerness to
understand and make progress drove new discovery. What's more,
because I had problems in mind (some of which I got stuck trying to
solve), when new concepts and tools became available in the classroom,
I would think, "Aha! That's the piece I've been waiting
for!" and I could return to the problem and do it less clumsily
than before. By having context in mind already, I had carved out
space waiting to be filled by classes. By exploring the problem
before and after new techniques came along, I *owned* the
material. It was personal. It meant something to me.

Examples of problems that occupied me during high school and college appear in the list below (not chronological). I'm sure I am forgetting numerous lesser problems, but these are the ones that stand out in memory. And yes, my tendency to actually carry out these schemes brands me as an experimentalist. But just because kid-me did some stupid things does not mean that adult-me condones these activities!

**Why is the time between successive new moons not always 29.530589 days?**—led to a successful science fair project- In the process of working out the above problem, I needed
essentially instantaneous changes in some quantities, so
**I invented the derivative**to accomplish this. Granted, it was a crude quantitative construct formed by differencing two values as the independent variable (e.g., time) suffered a quantitatively very small change. Later that year when derivatives were introduced in my math class, you can imagine my excitement and personal connection. That whole "limit as delta-x approaches zero" business? Totally understood why. I owned it. **What math describes the distance between lines of longitude on the lunar map hanging over my desk?**—when I first encountered the sine function as a junior in high school, I had another Eureka moment: I had been waiting on the sine, and had an immediate friend.**What is the circumference of an ellipse?**— the formula for area is easy. The circumference is an introduction to*elliptic integrals*, unbidden.**Is the trajectory of a thrown rock a parabola or the end of a very long ellipse?**—and how to reconcile the math that says parabola with the intuition that says ellipse?**How does gravity vary as a function of latitude?**Can I reproduce the curve shown in my undergrad textbook by a combination of Earth's oblate shape and centrifugal force?**How should I throw a water balloon from the back seat of a roller coaster to hit the front car as we go around a loop?**Elliptic integrals again?!**Can I compute Sun and Moon positions accurately enough to produce an eclipse track to kilometer accuracy?**—associated with my journey to the 1991 solar eclipse.**Can I measure the radius of the Earth myself armed with a theodolite?**— yes, to about 10%. Variations in atmospheric refraction limited accuracy.**Can I use the same theodolite plus a timing signal get my position on the Earth using star sightings?**— yes: could tell whether I was in my yard or my neighbor's, after a few weeks of calculations!**What is the "muzzle velocity" of a three-man sling-shot?**—based on measuring "hang time" and figuring in drag**Building a telescope**also exposed me to many practical problems requiring thought, calculation, solutions, design, implementation.

The lesson should not be to copy what I did. Emulating a rock star's visible behavior is not effective (and will probably land you in jail!). Countless hours practicing and honing the basic skills is where it's at (which is utterly boring to most people). Concert pianists spent years perfecting excruciatingly uninteresting scales before spinning off masterpiece performances. Emulation is not in itself a likely path to success. Rather, find your own interests and ask your own questions. Let your curiosity guide you. Plunge in to solve problems or go as far as you can while you wait on pieces to fall in place. Don't be afraid to explore. It's stumbling into the dark that lets you build a picture of what you can't see by staring into that darkness.