In conceptual physics, students are offered three possible ways to justify an answer:

1. **Fact: **Write a fact verbatim from our fact sheet, then write connecting prose to figure out the answer;

2. **Calculation:** Write a relevant equation, make a table of known variables (including units) and do math to figure out the answer;

3. **Equation: **Write a relevant equation, indicate which variable is unchanging and how you know, then draw arrows indicating how the remaining variables increase or decrease.

Education professors might label these approaches as *verbal, quantitative, and semi-quantitative* reasoning.

I've always started the year with fact-based reasoning with reflection, refraction, and lens/mirror ray diagrams. (A properly drawn ray diagram counts as a "fact".) Then as we study waves, we do calculations with v=(lambda)f, v=(lambda)/T, and f=1/T.

A major emphasis of a first year physics class, though, is to get at semi-quantitative reasoning. "A wave speeds up as it moves from shallow to deep water. Does its wavelength increase, decrease, or stay the same?" First, use the fact that says "when a wave travels from one material to another, its frequency stays the same." Next, in the equation v=(lambda)f, f doesn't change. Since v and lambda are directly related - mathematically, when one increases, so does the other - an increased speed means an increased wavelength.

Thing is, no matter how hard I tried, 9th grade first-time physics students didn't understand this direct relationship. They randomly guessed at how v and lambda changed. Worse, they didn't at all see what the equation meant. I have them draw up-arrows to show that v and lambda increase, and a line over the f to indicate that frequency didn't change. But these arrows were clearly meaningless to a large portion of the class. Even when I drew every possible permutation of the equations and possible relationships on the board, it didn't help.

What was missing was the connection between the equation-in-variables and the underlying arithmetic. So I made that connection explicit.

Instead of teaching the equation-with-arrows approach to semi-quantitative reasoning, I taught the "false calculation". What's that? I'll show you with the above example.

"A wave speeds up as it moves from shallow to deep water. Does its wavelength increase, decrease, or stay the same?"

Using the relationship v=(lambda)f, make two different charts, one for the wave in shallow water, one in deep water. The question doesn't require a precise numerical answer for a wavelength - just how the wavelength changes. So MAKE UP NUMBERS that are easy to work with!

We know that the frequency is the same for each, because when a wave changes material, its frequency doesn't change - so call it 1 Hz for each. We know the speed is faster in deep water than shallow water - so let's call the speed 1 m/s in shallow water, and 2 m/s in deep water. (Any speeds/frequencies will work! The point is to make calculation simple.)

SHALLOW DEEP Using v = (lambda)f

v = 1 m/s v = 2 m/s

lambda = ? lambda = ?

f = 1 Hz f = 1 Hz

******************************

(1) = (lambda)(1) (2) = (lambda)(1)

**lambda = 1 m** ** lambda = 2 m**

Thus, the wavelength is greater in deep water.

*Evaluating the false calculation* I expect to see each of the following elements when a student uses a false calculation to answer a question involving semi-quantitative reasoning. You can "grade" a response that includes a false calculation by awarding one point for each of these items:

- The relevant equation is written clearly and used.
- The variable that does NOT change is indicated, with evidence as to why that variable is unchanged. This evidence usually includes either a fact from the sheet, or direct language in the problem statement. "The frequency doesn't change because it's constant" is not sufficient.
- Two charts, like the ones above, filled out with correct units on all values and a question mark indicating the unknown variable.
- The values in the chart plugged into the relevant equation, and a conclusion drawn.

That fourth point often is awarded *even if the calculation is executed incorrectly*. That is, if the student screws up the 5th grade math and gets lambda = 0.5 m and thus says the wavelength is greater in shallow water, that student will get full or nearly-full credit. The whole purpose of this methodology is to give beginning students a scaffolding to make predictions in a rigorous way rather than using guesswork.

Of course, my students are often using false calculations to make predictions in the laboratory. When they do the experiment and find out that the wavelength is in fact greater in deep water, the context is exactly right to show them that they did the math incorrectly - and they've advanced their understanding, which is the whole point of the exercise!