The Sorcerer’s Apprentice, or Never Use a Formula You Don’t Understand

In my grade 10 Science class I recently gave my students an introductory microscope lab, and in my haste I used a “canned” lab from a textbook. Although there are some good activities in this lab, students are presented with a number of equations for determining FOV and magnification, including:

These equations, at face value, are straightforward – in other words they can plug in the numbers and get an answer. But there I something subtly insidious about them – they are just confusing enough that students are unable to apply these formula correctly later. Why? Because they are overly scripted, making the calculations look more complicated than they are, implying that without the formulas, they would not be able to achieve the “correct” answer. They build a reliance on formula, rather than concepts – and formulas without knowing what they mean can lead to trouble – much like poor Mickey’s spell in The Sorcerer’s Apprentice.*

So after an abysmal assessment (which was in part a setup – I could see they were becoming formula dependent), I gave them the following question:

Both images represent the view through the same microscope, with exactly the same settings. How big is the object in the second image?

Their first question? “Which formula do we use?”

My response was a shrug.

I watched as they struggled – one or two figured it out pretty quickly, but others tried dividing the object width (~12mm) by 7 (and some by 8!), some multiplied by 7, some divided by 40 (the circle diameter), but it was clear they were searching for a magic formula. Some, after scowling for a good long time, finally asked “which units do we use? Millimeters or UM’s?” (Aaaagh! That’s not a U! That’s a µ!)

It was challenging to subtly hint at how to simply measure the object without “giving” them the answer, because I didn’t want them to revert to the mindset of me, the teacher, as the sole gatekeeper of knowledge. Eventually they worked it out. Some estimated, some marked off the length of the object on a pencil or sheet of paper and held it to the millimeter scale, and the cleverer ones borrowed a friend’s sheet and held them together in front of a light. (And those that just used someone else’s numbers, well, I had multiple versions of the sheet, so they invariably had to redo it anyway!)

The next question was a bit more involved. I said the view in the image above was through a microscope with 10x ocular and 2x objective. I then asked what the FOV would be using a 20x objective. Despite my earlier warning stay clear of equations for this exercise, I saw many pulling out the equations from the previous lab. And that’s where they really got into trouble…

Numbers were thrown willy-nilly into the equations in the hope that somehow they were correct. Several students, despite correctly identifying the magnifications as 20x and 200x, wrote out

40 / 200 = 7mm / x

When I asked where the 40 came from, they said “low power on a microscope is 40x”.

“All microscopes?” I asked. That threw them.

Eventually I helped them work out that the higher magnification was ten times the lower magnification, so the view would be zoomed in ten times as close. The FOV should then be 10x as small (which is in itself a tricky concept, students are tempted to say 10 times the magnification means bigger, so the FOV is 10 times bigger). For most it eventually clicked that 10x the magnification means the field diameter is 10x smaller. Simple and no formulas to memorize.

It was remarkable, in a way, that a simple set of four of these questions took them a full 80 minute period – but that was mainly because I wouldn’t let them get away with wrong answers. One could call it a waste of a period, but I would not. It was absolutely necessary.

This is exactly the kind of thing Eric Mazur talks about. I will definitely be doing more of these exercises in the future!

*I mean the Fantasia version. Though that scene is included in the recent Nicholas Cage film.