One of the major obstacles in teaching at an advanced level is that we often have students with preconceived ideas. (The term "students" should be intepreted in the broader sense. It may include ourselves, for example.) Here is an example I have come across to many times, in the teaching of probability.
I ask the students to prove that if we take the points of a homogeneous Poisson process on the real line and translate each one of them by an independent random variable then we get again a Poisson process with the same rate.
For example, we may perform the translations by independent standard normal random variables.
The student often has a preconceived idea that a Poisson process is a random function of time $(N_t, t \ge 0)$ such that
- It starts from zero: $N_0=0$
- It is right-continuous, non-decreasing with values in $\{0,1,2,\ldots\}$
- It has stationary and independent increments.
One problem that the student faces (if he or she does not remember the definition I gave in class) is that the question above says to consider a Poisson process on the real line. So, after some thought, the student realizes that the following definition works:
Let $N'$ be an independent copy of $N$. Extend $N$ on $t \in (-\infty, 0)$ by letting $N_t := N'_{t-}$. It can then be checked that the process $(N_t, -\infty < t < \infty)$ satisfies
- $N_0=0$
- It is right-continuous, non-decreasing with integer values
- It has stationary and independent increments.
Rightly then, one can say that $(N_t, -\infty < t < \infty)$ is a Poisson process on the real line. Next, the student attempts to solve the problem by, say, doing this. They define the times of discontinuities of $N$ by $\cdots < S_{-1} < S_0 < S_1 < S_2 < \cdots$, agreeing, for instance, that $S_0 \le 0 < S_1$ (after all, the origin of time must be placed somewhere--and this agreement is up to us), then define $T_n := S_n +X_n$, for each integer $n$, where the $X_n$ are independent identically distributed random variables, and then try to show that the process with discontinuities at the times $T_n$ has exactly the same law as $N$.
This is an almost impossible task. The reason is that the student immediately realizes that the new points are not even ordered in the same way as their indices (something that was true for the old points). In fact, the ordering of the new points is random! Definitely, there is a first new point to the right of the origin of time, but this is not necessarily the point $T_1$. In fact, if the $X_n$ are standard normal, the first point to the right of $0$ could be the point $T_{517}$ with some probability, or the point $T_{-29}$, etc.
The student then may ask me for a hint. I try to bring in to them the idea that the above definition may be OK for some purposes, but that it has fundamental drawbacks. It is much better to first define a Poisson process as a random discrete subset of the real line such that the number of points contained in fixed (nonrandom) disjoint subsets of the real line are independent random variables. From this, it follows that the number of points in a set is a Poisson random variable with mean depending on a deterministic function of the set (this function being a nonnegative measure which, in the homogeneous case, is a multiple of the Lebesgue measure). From this definition it is not hard to show that the previous one is a theorem. Moreover, this is a definition which extends to higher dimensions and even to infinite dimensions.
But getting rid of preconceived ideas is very hard. Especially when teachers insist on a traditional way of approaching things.
One should not, actually, underestimate the fact the inertia of many teachers (as I said, the term "student" includes ourselves) to unlearn something and learn it from a different point of view. This is a major obstacle in the teaching of Mathematics.
