I’ve had some response from a few of my mathematical colleagues to my posts on units (Units and Stoichiometry
and Cutting Mathematicians Some Slack
). One said “2 apples + 2 apples = 4 apples. What does that tell us about 2 oranges + 2 oranges? Nothing. So instead we teach 2 + 2 = 4.” A second objected to my characterization, and said that he did use units when he taught calculus but only insisted that his students explicate them at the beginning and end of the calculation and during a calculation to “keep the mathematics clean.” A third commented that he expected that students already should have been taught about units before they came to a calculus class and that “It's hard enough fitting in the calculus, let alone all the other stuff students SHOULD have been taught but weren't.”
These comments from three mathematicians (all of whom I know to be first-rate teachers of math) support my view that we need to continue to work on building an interdisciplinary dialog with mathematicians concerning service courses for science – and those of us who teach science need to better understand our students’ thinking on these issues. Over the last 25 years of teaching physics majors, engineers, and biology students, I have seen many of my students display serious difficulties about how they think -- or don't think -- about units.
My previous two posts suggested some theoretical reasons why students might get confused about units. We also need an articulation of our learning goals – what we want our students to learn about units – and some substantial research on how students interact with units and what their difficulties are. Physics Education Researchers – including me – unfortunately seem to have let this one slide. In any case, here's my summary of what I want my students to learn about units.
1. Many of the quantities we deal with in science are measurements, not numbers. They are intended to correspond to something physical and the particular number assigned to them depends on the choice of unit made. This means that equations like "1 inch = 2.54 cm" are correct and legitimate in science.
2. Because of point 1, not all mathematical operations or results that might be perfectly legitimate for numbers are allowed for physical quantities. So, for example, getting a negative result for a mass is a red flag, warning you that either something has been set up incorrectly or there is an error in the calculation.* While for a position coordinate, it is perfectly legitimate (usually -- though I give examples in which it is not** because of physical situations in order to get students thinking about these issues).
3. The "unit" should be treated as a part of the value of a symbol. It behaves like a mathematical symbol but one whose value is fixed and never changes. (Kind of analogous to the square root of minus one in complex numbers.) It should be manipulated in the same way as algebraic symbols. (Modulo grammatical equivalences -- "meter" in the numerator cancels "meters" in the denominator.)
4. It is preferable in learning to reason scientifically with mathematics not to put numbers in too soon, but to use symbols for both variables and constants and to do most of the manipulations with those. (One's choice of symbol should give a clue to the kind of quantity being described -- the dimension rather than the unit.) This permits students to learn a variety of useful "checking" tools to make sure one has not made a manipulation error -- dimension checks and taking limits as the constants get large or small. (It's hard to take the limit "as 4 goes to infinity.")
5. When actually putting numbers in, it is valuable to keep units, since often problems will often be phrased in mixed units. Although it is possible to change everything to common units first, failing to do this is a very frequent source of error for students and keeping the unit symbols in the calculations activates their awareness of the need to convert.
6. Note that in equations when we write something equals 0, that zero often has a unit that we tend not to express.
The mathematicians legitimately want the mathematics to proceed ”uncluttered by the units." I like to do that as well. But the scientifically correct and consistent way to get problems to be "uncluttered with units" is either to keep it as symbols rather than numbers, or to create natural scales and divide through so as to make equations dimensionless. To see what I mean by this last, check out my notes on how to do it for a Math Physics class: <http://www.physics.umd.edu/perg/MathPhys/content/2/pstruc/dimsDE.htm
In addition to suppressing units when we do pure math, we might also note that when writing computer code we often suppress units. (This led to a multi-million dollar failure at NASA.)
But any object oriented programming language can correctly retain (and even check) units. I doubt that classes in computation, either in computer science, engineering, or physics departments, teach objects with units – despite the fact that it would be incredibly useful for scientists.
I don't object to mathematicians leaving out units entirely. I would encourage them to introduce and talk about units, or even require their students to identify the proper unit at the end of a calculation. What I do object to is when they ignore units in a way that leads to equations that are incorrect if you fail to hold in your mind things like "the 1 in the numerator is a length and the 1 in the denominator is an area". Students' working memories are cluttered enough with trying to handle knowledge that they haven't yet organized. Having to keep track of the units of their numbers puts an additional burden on them. It leads them to ignore keeping track of dimensions in the middle of a calculation – and that becomes a habit that’s hard to break. And then in my class they write equations that both confuse them and hurt them on their science exams.
This is not a trivial point and it is not just one of "we do things in different ways." Service course should serve. Can't we work this out to find a set of approaches that is comfortable for both mathematicians and the scientists who need and value their classes?
* Or that you need to rethink some fundamental assumptions! Negative mass and square masses occur naturally in quantum field theory and have led to interesting new ways to look at the physical world.
** Position data taken by a sonic ranger is always positive since it can’t see “behind itself”.