Project-based learning

I was first introduced to the idea of organizing a course around a project or a driving question in my class on student motivation and course design.  It may sound like a weird combination, but it was about dealing with grade anxiety and student engagement through course design.  Anyway, the idea put forward was that one way to organize a course was to have a central project or question that the course would return to over and over and that would drive exploration of topics over the semester.  It provides context and motivation for the material to the students.

When I heard about project-based learning (PBL), I thought they were talking about the same thing.  However, PBL apparently does not require the entire course to be structured around one project or one question, but use projects to drive whatever is the current topic.  Which makes sense because I’ve been wracking my brains for single projects or questions that would encapsulate all/most of calculus or linear algebra.  Oh, there are plenty of applications, but it’s very hard to have one that touches upon all of the key concepts.  These courses are supposed to be toolboxes so that, when the need arises later on, the student can pick the right tool out of the box and fix the problem.  So to apply PBL to such courses, projects will probably have smaller, more localized scope.

While researching possible projects for courses, I came across this site with a number of linear algebra projects.  They are on a variety of applications of linear algebra, and the projects show a pretty good understanding of various linear algebra concepts.  However, upon further analysis, I believe most teachers would call these research papers rather than project.  See the syllabus.  Note: there is also a conflation of term when it comes to the word, “research.”  In graduate school, research refers to original thought, experiments, or analysis which expand the boundaries of human knowledge.  In high school, research usually means read a bunch of sources and sum it up in a paper. which in graduate school would be more of a literature review.  These papers are in the latter category, but at least it does require the students to understand a particular application in sufficient degree to explain them to their peers.

This bring up the question exactly what is a project.  To my mind, a project starts with a question or a problem.  In answering this question, the students are the ones who determine what they need to know and how to use them to address the question, though the teacher acts as a guide.  The students are the ones who decide what to do and how to do it; this not a sequence of activities laid out by the teacher that the students follow.  This provides room for students to explore and be creative.

I’m still trying to think of projects suitable for a college math course.  For now, I’m okay with a literature review / paper presentation to break the mold of lecture/homework/midterm.  Also, I’ve started compiling a list of historical/interesting math problems as material to discuss in class, as opposed to generic problems.

Famous/interesting math problems


The Problem of Points: Pascal and Fermat both put up 600 francs into a bet.  They will flip a coin, and if heads comes up Fermat gets a point.  If tails, Pascal gets a point.  The first person to get 10 points wins and gets all 1200 francs.  However, when Fermat is winning 8 to 7, he receives an urgent message to return to his hometown immediately and see his sick friend.  The game is stopped, but how should be the 1200 francs be split?

Matching problem: If you randomly assign N names to N people (who each have a unique name), what is the probability that none of them are correctly named?

Matching birthdays: Fun experiment.  How many students in the class do you need to see if at least two share the same birthday?

Buffon needle problem: A Monte Carlo method way of calculating pi.

Pass the beer: A counterintuitive stochastic process.

Two envelope paradox: A counterintuitive example brings up the idea of prior probabilities.

Monty Hall problem: Another thought exercise.

Deal or no deal: Risk aversion, and why expectations may not be the most accurate estimate of utility.  Also see St. Petersburg Paradox.

Medical tests and Bayes rule: The usual introduction to conditional probability.

Betting on different beliefs: If two people have different beliefs about an outcome, you can make bets that guarantee you come out ahead.  This also brings up the idea of prior probabilities (i.e. the gain comes from exploiting people who think their beliefs are certain when they could be wrong.).


Cylinders in spheres: (From Kepler) Inscribe a cylinder of maximal volume inside a given sphere.  (See Famous Puzzles of Great Mathematicians)

L’Hospital’s Pulley Problem: Let A and B be points on the ceiling that are 1 meter apart.  From B, let there be a string of length r < 1 with a pulley at the end.  From A, let there be a string of length l, such that l+r > 1, so that the string can go through the pulley to a mass.  Assuming the strings and pulley are massless and frictionless, at what distance from the ceiling does the weight end up?  (Hint: the mass ends up at its lowest point.)

The brachistochrone problem: Given two points A and B, construct the curve such that, if a mass is only acted on by gravity, the mass starts at rest at A and reaches B in the shortest amount of time.

Suspended Rope: What is the curve of a suspended string?

Sum of reciprocal powers:

Wallis’ Formula:

Beads and holes: Take two beads with a different radius.  Drill a hole in each so that height of the remainder is the same.  Which shape has the most volume left?

Surface area of sphere slices: Make equally thick horizontal slices of a sphere.  The surface of each slice is the same.

A quick way to find square roots: Based off Newton’s method. Let y be the number you are trying to find the square root of.  Starting with x_0, let x_{n+1} be the average of x_n and y / x_n.  This actually converge pretty fast.

The black hole: The rotation of y(x) = 1/x, x > 0, about the x-axis has finite volume but infinite surface area.  How can you paint it?

Circles and arcs: Consider a unit circle and an arc of length s in the first quadrant.  Create the shape bounded by the arc and its projection onto the x-axis.  Ditto the y-axis.  The sum of the area of these two regions is always the same, no matter where the arc is located, and their sum is s.

Volume of a pyramid: Can probably adapt this to find the volume of a 3d pyramid with arbitrary base.

Fourier series: An alternative to Taylor/Maclaurin series.

Relationship between volume and surface area of spheres: Differential geometry.

Cantor set: For infinite series.  The Cantor set is an uncountable set which has Lebesgue measure zero.

Thinned-out harmonic series: A weird way to thin out harmonic series so it converges.

How to integrate a bell-curve:

Weirstrass’ pathological function: Continuous everywhere, differentiable nowhere.

Does order of addition matter?:

Complex numbers

Chords of a circle: Take N equidistant points on the unit circle. Pick one of those points, then draw chords from it to all other points on the circle.  For any N, the product of the lengths of those chords is N.

i-i: i to the i is a real number.  Also, ‘aye’ means yes, and ‘aye aye’ means I have heard the order and will carry it out.

Life imitates art: averted

(Life imitates art: averted)

Let me tell you a story, for it begins just like a story.

Three years ago, some ballroom people and I drove down to LA for Camp Hollywood, a swing dance camp.  Among other things, I met and danced with a pair of shy red-headed twins.  Now, I have a thing for shy girls and redheads.  (I also have a thing for women who wear glasses and have advanced degrees in a math/science field.)  However, when I asked one where she was from (people had come from all over, so you never know…), she said she was from SoCal (oh).  And since I was to drive north eight hours the next day, I decided to save myself the eventual disappointment and not pursue things any further.

Well, about a year ago, I had one of my periodic moods to be more active on OkCupid, when I found a pair of red-headed twins who liked dancing.  I messaged both, one wrote back.  She said she remembered dancing and talking to me.

See?  This sounds like the charming story someone tells about how their grandparents met.

Anyway, we message each other.  We IM each other.  We share personal things.  We talk about relationships.  We decide to meet.  And that’s probably where life diverged from art.  I went down to SoCal, and it wasn’t a bad trip, but I still wasn’t sure about yet another long-distance relationship.  See, I was scheduled to graduate within six months, and who knows where I’d end up afterwards.  She just transferred to a new school and definitely would be there for 2-4 years.  Things didn’t click enough for me to, say, restrict my first job to whatever I could find in her immediate area so we could give it a shot.

We talked some more.  Cracks were beginning to form.  Then she visited me, and during that trip I decided it wouldn’t work out in the long term.  Yes, despite the red hair; I’m not that superficial.  We’re now trying to stay IM buddies.

Since I’m trying to prevent this blog from becoming some weepy, angsty Livejournal, the thing is this: earlier this year, I decided it wasn’t worth it to pursue a relationship until after I graduated.  It’s very likely I’d move after graduation, so what would be the point?  Save myself the worry and concentrate on graduating.  But then this… story happened, and I knew I would regret it if I didn’t give it a chance.  So I did, and I don’t regret it.

The End