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Understanding Theory with Simulations
Usually I teach upper-division classes with class size varying from 50 to 80 students. These students major in mathematics, economics, computer science, engineering, bioinformatics. I try to make the students like the subject. Also, I would like them to remember what I teach them without memorizing any formulas. My main goal is to focus on teaching the concepts, rather than on calculations or formulas.
In order to help students learn the concepts, I use simulations. First, of course, we discuss the theory--theorems, and proofs--but then we can verify the theory with simulations. For example, if we want to compute the probability of winning the lottery, we can do this mathematically or we can program a computer to play the game many, many times. It is like rolling a die, or tossing a coin, but we let the computer do this-- a thousand times. At the end, we record how many times we won the game. The number of times the game is won divided by the total number of times the game is played gives an estimate of the probability of winning. Certainly, if you figure the proof mathematically, nobody can argue about it, but seeing the theory in practice is another thing.
Statistics is interesting because of variation. Being able to understand and explain variation leads us to useful, interesting conclusions. Simulations again, allow us to see this variation in action. Sometimes we use graphs or other visual representations to look at how the results of the experiment are affected by the sample size. We might run the experiment ten times, or twenty times, or thirty times--the more we run the experiment, the more accurate the result is. The theory says, "as n goes to infinity." If we use a small n, we may not get exactly what the theory says. But during the simulation, we might find that we approach the predicted result as we increase the sample size gradually. So we can amend the theory to include the stipulation, "as n gets larger and larger."
One of the most important theorems in statistics is the Central Limit Theorem. It helps us to make inferences on population parameters--for example, about the mean of a population. Using the Central Limit Theorem we can calculate with 95% confidence that that the population mean will fall between two numbers. If students see how the Central Limit Theorem works in simulations, I believe that they do not forget it. It is a very powerful demonstration. We can do it on the chalkboard with the proof and the numerical example too, but seeing how it works on the computer, I think, stays with you.
We run the simulations using the statistical software Stata, which is available in the Statistics Computing Labs. Students automatically get accounts when they register for our courses that they have to activate at the lab. Sometimes we hold classes at the computer lab and ask students to perform the experiments. Also, many times I bring, on paper, the results of a simulation study and discuss with the students the results of the simulations. Outside the classroom, I communicate with students using various web tools. I use the class email function available through MyUCLA, and post homeworks on my own web site. In the Department of Statistics, we have course web sites which feature a Forum tool, developed by staff in the Statistics Computing Center, where students can add comments and questions. The instructor can then answer those questions and add comments that can be accessed by the students.
The response to these simulations has been very positive, so far. Usually I give out a questionnaire --class feedback--during the first few weeks, asking the students what they generally think of the course, and about the simulations in particular. I strongly believe that simulations help the students better understand the material. And I know other colleagues use them too and using simulations to teach statistics is a growing trend.
Sometimes the solution to a problem may be mathematically intractable. Its solution can be found using simulations. It is a very useful tool for teaching probability and statistics.
Oral Interview, April 2004