I love mathematics. It is austere, beautiful, rigorous and logical. It is one of the few truly honest disciplines-- premises are defined, and deductions follow clearly from first principles. Logic chases truth up the tree of grammar, and ultimately we develop useful conceptual tools that enable feats of apparent sorcery. Need to shout a message in public that only your intended recipient can understand? We can do that. Curious if socio-economic and religious indicators can predict political alignment? we can do that too.
I was also incredibly lucky to learn mathematics at Reed College, where a proof based approach is dominant among the faculty. This is good because as a result I was rarely asked to be a human computer. I certainly computed my fair share of singular value decompositions (I think two is a fair share) in linear algebra, and path integrals in complex analysis, but for the most part problem sets consisted of a series of proofs.
One problem with this approach is I rarely felt comfortable applying these concepts. A friend glibly asserted: "I can't build you a pridge, but I could prove it exists in arbitrarly many dimensions."
After a working at Periscopic, and scripting for about a year, it occurred to me that mathematics classes at Reed were structured like software libraries -- meaning we started with simple basic concepts and used them to build more complicated ideas over the course of the class, which also got me thinking about ways to give some classes more concrete contours without loosing that lovely, intimately theoretical approach.
Build a library implementing the core concepts taught in class.
For example, I could imagine a linear algebra class that proceeded along much
the same lines as it did at Reed. We would start with the axioms of
a vector space: associativity, commutativity, and inverses for vector addition;
vector 0; and rules for scalar multiplication. Then, we could implement vectors
as a sublcass of, for example, Python lists. Students could test their
implementation against the axioms to demonstrate that they have indeed made
vectors over their chosen field. The teacher could provide alternate
implementations that are also consistent with the vector space axioms, or use the
standard implementation over a different field to demonstrate the flexibility
of the idea. For example, the students could design a vector space over the
field of reals, and the teacher could demonstrate that polynomials of degree
n are a vector space over the reals as well, by doing something as simple as
__repr__ method  of the vector object.
Vector multiplication is a simple algorithm that the students will eventually extend to matrix multiplication. This will provide them with easy access to visualization software that will permit clear demonstration of the affects of rotation or translation matrices. A brief forray into technical applications could be made - for example SVG transfromations - accept a matrix which completely specifies the desired transformation.
The students should be made to suffer through matrix inversion when they first encounter it, and then encouraged to explore and implement any one of a variety of matrix inversion algorithms. They should be asked to prove properties the matrix inverse, and then use those properties to test their implementation.
Their computer processes could also be used to provide a compelling introduction to a number of linear algebra applications. I personally found Linear Algebra to be one of the less interesting courses I took in college, but it is certainly the most useful - linear algebra concepts pop up all over the place - so this subject would especially benefit from brief forays into application.
One example subject could be finding a Support Vector Machine (this would require multivariable calculus as a prerequisite) for an interesting prediction problem. The class could even take a Kaggle competition as its target, and learn the mathematics necessary to implement logistic regression, or a k-nearest-neighbors learning algorithm, and see how it matches up against other competitors solutions. The cash reward could provide an extra little bump to student's investment in the course.
It's true that repeatedly executing an algorithm by hand leads to mastery of
that algorithm, but it is also true that the interesting mathematics does not
lie in the mechanics of these concepts, but in their application to problems
that would be difficult or impossible without them, and in proving interesting
facts about these algorithms. It's unlikely that any undergraduate will ever
have to implement even a t-test by hand in a career in science or statistics.
As such, rather than learn the mechanics themselves, they should learn how to
tell a computer to do the mechanics for them. They should learn how to test the
computer's results, and how to write good, reusable code. All of these skills
will ultimately be more useful than knowing how to invert an
n x n matrix by
hand, for example. Moreover, I think they will help the student think
conceptually about mathematics as a process, and help them see new advances in
context of old algorithms.
__repr__ method determines what happens when