to foreshadow the study of integral calculus.
A completely different approach available to all students would involve computer simulation to generate random points contained within the circumscribed rectangle. This method is based on a probabilistic model that assumes that as the number of randomly generated points increases, the ratio of points will closely approximate the ratio of areas.
In theory, this is a perfect lesson. Students learn about quite deep connections between geometry, algebra, rudiments of calculus, statistics and discrete mathematics. But in practice it wouldn't work. There is no reason to expect that having found one solution to the problem students would be interested in finding others. And if a teacher proposes the second solution, there is no reason to expect that students would appreciate its beauty and significance. Only if students are already mathematically mature enough to appreciate deep mathematical connections and their fundamental role in mathematical arguments will it suceed in enhancing such appreciation.
Paradoxically, to develop mathematical maturity in students we need to make the problems we offer more challenging. We should, therefore, offer problems that absolutely cannot be solved without using deep mathematical connections. These problems would instill in students the idea that mathematical connections are not merely things that teachers like to bother them with, but are absolutely fundamental tools in mathematics. This would force students to pay attention, and once they do a teacher could emphasize the role of mathematical connections in solving problems.
Here is an example of a such a lesson for students who know something about probability and binomial coefficients: Laura and Steve are taking a 50-question test with all questions worth 2 points and no partial credit. Assume that Laura and Steve have the same capability of performing well on the tests, that all the problems are equally difficult for these students, and that their expected score on the test is 50. What 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.