 is the probability that they will get the same score?
Students can figure out that the answer is a very complicated sum of fifty binomial coefficients that would take forever to calculate. Guided by suggestions from the teacher, students can analyze the general form of the expression: a sum of products. Where else have they seen sums of products? "That's what you get for coefficients when you multiply polynomials," a student remarks. After further analysis students discover that the sum they want is the center coefficient in a polynomial of degree 100. Then they can use the binomial theorem to figure out the probability.
Now the teacher can open a discussion about what enabled them to solve the problem. The answer, of course, is the beautiful connection between polynomial algebra and sums, which they could use because of the connection between discrete mathematics and arithmetic.
Once students learn to appreciate connections, virtually everything can be covered from the point of view of establishing further connections. Logarithms can be taught as a way to connect sums with products. Much of elementary geometry can be taught as a way to connect side lengths and angles. Transformations can be taught as a way to connect arbitrary geometric figures with simpler ones, so as to transfer information from simple figures into more complex ones. Much of mathematics can be taught through problems that require the connections for solutions. After a while, students would learn the intrinsic value of the connections, and would be able to appreciate establishing a connection as a valid goal in itself.
Aleksandr Khazanov is a graduate student in mathematics at Pennsylvania State University. Individuals interested in further details are invited to correspond with him by e-mail to email@example.com.