Unit Conversions as Gauge Transformations?

I recently read through a very nice blog post, references of which led me to reread Redish’s 2006 paper about how math is used in physics.  The following quote jumped out at me:

In introducing the concept of units to my introductory classes, I tell them that a quantity has a unit when we have an operational definition for assigning a number to it and that the number that results depends on the choice of an arbitrary standard. Since the standard is arbitrary, we may only equate quantities (or add them) if they change in the same way when we change our standard. Otherwise, a numerical equality that we obtain might be true for one choice of a standard but not for another. An equation that has physical validity ought to retain its correctness independent of our arbitrary choices.

Many physicists will recognize in the above the “scent of Einstein” – the idea that “a difference that makes no difference should make no difference.” Einstein, Poincaré, and others around the turn of the last century introduced the idea of analyzing how physical measurements change when you change your perspective on them – rotate your coordinates (in the case of identifying vectors and tensors), hop onto a uniformly moving frame (in the case of special relativity), or make a general non-linear coordinate transformation (in the case of general relativity).

Of course, the transformations that we use with units do not depend on spatial coordinates, but considering the symmetry transformation explicitly punched me in the gut because I realized that we don’t teach this explicitly to students in science classes.  Students trying to to convert units typically try either of two strategies where I work.  The first (used with metric conversions) is to consider how many factors of 10 there are between the units (like km and cm) and then move the decimal place that many times.  If students remember this procedure, they as often move the decimal point the wrong direction.  The second is to use the unit-factor method, wherein they often try “fun” factors like \frac{100\,\textrm{m}}{1\,\textrm{cm}}.

At least for proportional units—let’s leave affinely related units for another time—we can regard a quantity as a pair (n,u), where n lives in some mathematical space, and u is our unit, a physical “arbitrary standard”, as Redish calls it, which lives in some torsor for the multiplicative real numbers \mathbb{R}^\times.  On this space, we can define an equivalence relation for physical equivalence, (n,u)\equiv(\Lambda^{-1}\cdot n,\Lambda\cdot u), where \Lambda\in\mathbb{R}^\times is a scale-factor.  In my mind this leads to the following exercise:

The width of a desktop is measured in units of cm and found to be 134.4 multiples of 1 cm, or in other words, 134.4 cm.  If, instead, one had measured it in a different unit, how many multiples of the new unit would the length be?  Try for as many different units as you can.

This gets at two ideas.  Students don’t always think in terms of 3 cm being 3 multiples of a centimeter length.  They also don’t often get to choose what measurements they use, or the teacher has them convert in predictable ways.  Here they can see that (134.4, cm)≡(134.4/10,cm*10)≡(134.4*10,cm/10).  Would this help them build a chain of understanding of units?  I realize that when I convert, I explicitly play the epistemic game in my head, “m is a bigger unit than cm, so it takes less of that length”.  Do all students have that understanding?

The other epistemic game I play in my head is to think, “134.4 cm is about 100 cm.  I know 100 cm is the same as a meter, so 134.4 cm is a little bigger than a meter.  It’s definitely less than 2 meters, since 2 meters is the same length as 200 cm.  I’m pretty sure that 134.4 cm works out to 1.344 m.”

How do you teach students to understand units in elementary, middle, and high school?  I’d love to know what conceptual tools or epistemic games you give students.