It’s common knowledge among ESL teachers that any activity should be prefaced with a pre-activity. Not only textbooks, but handouts, powerpoint presentations, and even off-the-cuff improvisations by the teacher are prefaced by some schemata-activating questions, discussion points or pictures, the theory being that students are better able to engage with the main activity after their brains have all the context-appropriate neurons firing.

I have never seen any evidence that the principle of preparing students for any activity should only apply to *main* activities, however, so it stands to reason if we are right about the importance of schema activation that these pre-activities could use pre-activities of their own. If you follow my logic, responsible pedagogy *should *involve a pre^{n+1}-activity before any pre^{n}-activity.

This presents a philosophical and practical pedagogical problem, as responsible language teaching now seems to entail an infinite series of increasingly small pre^{n}-activities, which in an echo of Zeno’s arrow, mean that we can never actually physically reach the start of our main activity.

With an eye toward helping my fellow language teachers out of this conundrum, I would like to propose a pragmatic (no pun intended) solution to the pre^{n}-activity dilemma, which is this:

Teachers should not have pre-activities whose length would be shorter than the time it takes for light to travel from the teacher to the nearest student. The last pre-activity whose length is longer than this time will be called the

shortest physically productive pre-activity.

I will illustrate this principle by assuming a few values:

- The speed of light is 3.08 * 10
^{8}m/s - Our main activity is planned for 20 minutes (1200 seconds).
- Our pre
^{1}-activity is 5 minutes, or 1/4 the main activity, and schemata-activating pre-activities for other pre-activities will also last 1/4 as long as the activity that they prepare for. - For the sake of simplicity, the nearest student is seated 3.08 m from the teacher.

Given our values for the distance between the teacher and the nearest student, it takes 1/10^{8} seconds for light to travel from the teacher to the nearest student. Any pre^{n}-activity that takes less time than that will be over before the last one can be physically sensed by the students.

It’s worth pointing out that light is the fastest known physical phenomenon in the universe; no cognitive activity (or any activity with a physical substrate) can outpace it, no matter how “quick” the student. The speed of light is, therefore, a crucial property to consider when planning pre^{n}-activities whose length is measured in millionths of seconds.

The question then is what value of *n* in a pre^{n}-activity yields an activity whose length is less than 1/10^{8} seconds. I solved for *n *by plugging in the values above:

1200/4^{n} = 1/10^{8}

1200=4^{n}/10^{8}

120000000000=4^{n}

120000000000^{1/n}=4

Doing these calculations the old-fashioned way, I come up with a value of 18 for *n *as the last pre^{n}-activity whose length is longer than the time it takes for light to travel from the teacher to the student. Therefore, with the assumptions above, a main activity should be preceded by exactly 18 pre-activities, the 18th pre-activity being the *shortest physically productive pre-activity.*

It is to be hoped that teachers integrate this knowledge into their lesson planning thoughtfully and responsibly.