I can't do experiments. I am not even sure what the experiment would be if I could do it (and I am a very good experimenter). And mostly I am teaching concepts to elementary students which should help them only when they get to high school math. So I am stuck with subjective impressions. Yet...

An "Obvious" Sign of Learning

One day I was bemoaning the fact that I couldn't be sure my students had learned anything. In fact, I wasn't even sure what I was teaching them. It seemed hopeless.

Then I realized this. If one of the four boys I was teaching happened to miss a day, it was very obvious. I couldn't say what the other boys had learned, but they were obviously ahead of the boy who had missed.

The extreme example is this. For my first session with sixth graders one year, I started with a completely new topic, so that the two new students would not be at any disadvantage. However, as we did the exercise, it became obvious to me that the two new students were having difficulty with a concept that was not bothering the other students.

Aha. The previous year I did an exercise on measurement. At the time, I could not have said exactly what I was teaching, or what they were learning. But with the benefit of hindsight, they apparently had learned that an measurement system requires one completely arbitrary assumption about a unit of measure. I had not foreseen the relevance of this to the exercise I was doing. But without understanding this, the new students were having a difficulty that the other students were not having.

So, let's think about this. I didn't remind the students of last year's exercise. They didn't recall last year's exercise. They couldn't have known it was relevant to recall -- I didn't even realize it was relevant to recall. The exercise last year just gave them a better understanding of units of measure, and they spontaneously used that knowledge for this year's assignment. That's exactly how mental models are supposed to work.

College Experiences

In contrast, when I taught at the college level, I could not assume my students had learned anything. For example, introductory psychology was a pre-requisite for cognitive psychology. However, in teaching cognitive psychology, I had to explain every concept, even if I knew they had been exposed to it in introductory psychology.

Most tellingly, I taught a research methods course that had a statistics class as a pre-requisite. Very few students knew any statistics; I started from scratch, teaching them what they needed to know for my course, and the students were very grateful. I once had a student who did not take the statistics course and she was indistinguishable from the other students (except she seemed to enjoy doing statistical tests more).

And you need look no farther than current math textbooks to see how well students learn math. Repetition from year to year is the norm. The students are expected to be a little faster in relearning what they have previously learned, but they are not expected to actually remember it.
Ironically, we have built an educational system ideally designed for computers and poorly suited for human beings.