One of the earliest challenges in a Probability I course is the introduction of random variables. At this point, the learning process already asks for a change in most students’ mindset. High-school knowledge to calculate means or intuition to calculate dice probabilities are no longer enough. The student confronts a new concept and needs time and a resourceful way of thinking to become familiar with it. Contrary to the dry application of formulas, the ability to draw connections becomes a must. The student needs to think creatively, to be able to recognize a Bernoulli random variable when the talk is about coin tosses or basketball free-throws or to model the binomial and geometric distribution from a seemingly vague text-exercise.
My enthusiasm when I taught for the first time Probability I was quickly superseded by a strong disappointment as I realized that I was losing most of my students after the first lectures into random variables. I felt that symbols such as P(X = x) were not clear to most of them and that the subsequent introduction of the density function f(x) or even worse the distribution function F(x), only added to their confusion.
A new formula may be learned and practiced rather quickly in the timeframe of a lecture. However, contrary to procedural knowledge, conceptual understanding is a complex process with interwined steps: one gets introduced to it, learns about its properties and works or experiments with it before hopefully getting a chance to master it. To compound the problem, these steps may occur at very different paces among the variety of students in large classrooms.
When confronted with this challenge, I focused on how this process of learning had worked on me in the past and discussed the issue with colleagues or more experienced faculty members from my department. I drew connections between the concept of random variables and the concept of functions that we had learned at high-school. I recalled that for many high-school years, I had to struggle with functions without having a clear understanding of what they meant. Perhaps this applied for many of my fellows. I also recalled that this limited understanding came together with a feeling of uneasiness, whenever I encountered the symbol f(x) in an exercise. A feeling that I certainly do not want to create in my students.
Since my first course as a tutor in Probability, I am striving to improve the learning experience for the students by deepening their understanding on random variables. To promote their motivation and engagement, a considerable time of my lectures is devoted to the applications of definitions, concepts and ideas in different settings or exercises that facilitate creative thinking. In the classroom, students are encouraged to present their thoughts on exercises that ask them to identify and model the underlying randomness through the choice of an appropriate random variable. The effects are twofold, as students learn from the mistakes of their peers on the one hand and how to properly define a problem in order to be able to solve it on the other hand, a practice particularly important in future graduate research.
As the background and educational needs of students from various disciplines differ, my main priority is to tailor each course to the specific students’ goals. To this end, I clarify from the outset the range and depth of knowledge that each class should aim at. Accordingly, I adapt the structure of assessments and reward opportunities in assisting the students’ progress. However, independently of the students’ mathematical background, it is always my main aspiration to communicate to them effectively that learning mathematics is not about the accumulation of inflexible knowledge or about the strict judgements and the bad/good grades.