Geoffrey Cowell
Geoffrey Cowell makes science exciting by showing how it is relevant to the everyday lives of the students in his Calculus and Statistics course — an integrated, computer-based course in calculus, statistics and physics for gifted Grade 12 students. "He has an uncanny ability to set up experiments in such a way that they seem realistic and fun, as though we are scientists out in the field," reports one student. "We know that the answer we arrived at is the true solution to a real problem, not a contrived question with the answer at the back of the book."
Under Geoffrey's guidance, enrollment in physics at W. J. Mouat Secondary School in Abbotsford, British Columbia has consistently been about 30 percent above the provincial average. In spite of this, his students' standardized performance on British Columbia-wide tests is also above the provincial average. The proportion of female students in his courses is significantly greater than the provincial average as well.
Approach to teaching
"I believe that the most exciting ideas occur at the boundaries between disciplines."
I strongly believe that at least one course at the high school level should be integrated - focusing on the threads that bind subjects together rather than emphasizing differences.
For example, an integrated science, calculus, statistics and computer course can offer students not only the chance to develop the problem-solving skills associated with calculus, but also the opportunity to be challenged by real-life situations and the use of analytical methods of statistics and spreadsheets. In this course, the formal examination has been largely replaced by real-life assignments, experiments and research topics to test the students' abilities.
In the same way that calculus helps students reach a new level in problem solving, statistics provides a fertile area for developing critical thinking. Happily, calculus leads right into statistics. Both disciplines lend themselves to the scientific method and to such topics as thermodynamics, forensic science, strobe lights, economics and even farming.
Technology has also changed these subjects. Years ago, courses in calculus and statistics introduced students to a few wonderful concepts and a mountain of computational skills. Today, one can focus more on showing students the power and applications of calculus and statistics, because the computer can deal with many of the computations normally required for real-life situations.
Transferable experience
I use many practical demonstrations and labs in my course to show how calculus, physics and statistics work together. Both of the following labs include calculus, and may involve spreadsheets and statistics.
Intensity of light with depth of penetration
In this lab, students use calculus to determine the intensity of light shining through various depths of coffee in a beaker.
Students set up the lab by placing a photocell directly beneath a beaker and positioning a light source above the beaker. The intensity of the light entering the photocell is turned into current that can be measured at intervals as the beaker is gradually filled with coffee.
The current (light intensity) decreases as the beaker is filled. The students measure the light intensity at various depths and plot their results on a graph to show the intensity of the light versus the depth of the coffee as well as the Natural Log of Intensity — Ln(I/Io) — versus depth.
Experimentally, the students derive this equation: I=Io ekx or, alternatively, Ln(I/Io) = kx. This equation is formed by integrating dI/dx = kI. This shows that the rate the intensity decreases with depth (dI/dx) is proportional to the level of the intensity (I) — that is, for every additional unit of depth (x) of coffee that is added to the beaker, the intensity drops by some fraction (k) of its original value. With this equation, the students plot the classic decay curve.
And then there were none…
A second lab I use in my integrated course can be easily set up with commonly available materials. It is based on the mystery novel And Then There Were None, by Agatha Christie, in which a man conspires to kill everybody on a small island and then commits suicide. The students are asked to determine the identity of the killer by using the body temperature of the victims to determine the time of death.
The victims are actually a collection of ornate liqueur bottles. These aren't chosen just for their looks; the different surface areas and volumes of the bottles produce varying cooling rates, just as different human bodies cool at different rates.
Each bottle is labelled with the name of a character from the novel — "Vera Claythorne," "Justice Wargrave." One or two of my Physics 11 students act as henchmen to commit the murders by pouring boiling water into each bottle at different times before the Calculus and Statistics students come to class.
When the students arrive, they are faced with 10 bottles each containing water of different temperatures. Some are cooling rapidly and others are cooling very slowly, depending on the shape and size of the bottle. To determine the order in which the victims died, the students apply a special decay curve known as Newton's Law of Cooling ((T - TRoom) = (To - TRoom)ekt) that they derived in calculus the day before.
Their first task is to find the cooling constant "k" for each victim, using the time between two temperature readings. The students then insert the "living" temperature into the equation to find the time of death. In this case I find it is better to start with water at an artificially high temperature of 92°C rather than at the real body temperature of 37°C.
I love the students' comments on this lab. "Couldn't have been dead long, this body is still HOT!" or, on finding the time of death, "If we had arrived in class six minutes earlier, we could have saved General MacArthur!"