Reasoning with Units: Correct Wrong Answers

In the world of problem solving (as an academic endeavor), we talk about non-routine problems … ill-defined problems … and we talk about problem solving strategies beyond specific content issues.  When facing these types of situations, many students find great difficulty in transferring content knowledge; as mathematicians, we sometimes see this problem solving as the core outcome of learning mathematics.

Unfortunately, the teaching of mathematics often discourages broader reasoning.

I have been running in to a consistent error in thinking about units, which has led me to think about how this happens.  Here is the situation:

A desk has an area of 5 ft².  How many square inches is that?

Two bits of knowledge (neither correct) get in the way of solving this routine problem.  First, students equate 1 foot with 12 inches, whether we are talking about length or area or volume (they get 60 square inches).  Second, students treat the exponent (square) as affecting the 5 as well as the feet (300 square inches).  The first issue was addressed in an earlier post (see https://www.devmathrevival.net/?p=1471).  How about the exponent issue?

Misapplying the exponent could be caused by an over-generalized property of exponents.  However, I think the more likely error is a combination of two practices in mathematics education:

“find the area of a 4 inch square”

“just use the numbers in formulas, and write the correct unit with the answer — area is always squared”

The first practice, extremely common in early work with area, leads students thinking that something needs to be squared when they see a square indicated (like an exponent).  The second, more to my point today, leaves students with no reasoning about units.

For example, in last week’s Math Lit class, I asked the group what the formula for distance is (and got D=rt).  I asked how we usually measured distance; once we agreed on a context (a car) we agreed ‘miles’.  The next question — how do we measure speed?  This was much tougher, even though students deal with speed limit signs every day (usually without units 🙁 )  Once we got to ‘miles per hour’, we then wrote a typical calculation showing what happens to the units.

The next step:

A car has a speed of 40 miles per hour.  How far do they go in 20 minutes?

Many students see this as a trick question, saying that we should always give the time in hours (we would say ‘consistent units’).  However, including the units in the calculation makes it more obvious that we just need to change minutes to hours (they could do that).

Back to the square feet situation, few of us show the units in calculating area.  If we consistently did include units in calculations, students would have more experience in seeing where the ‘square’ came from (in ft²), and would be less likely to apply the square to the feet.

We have another instructional practice which discourages reasoning with units: the degree sign for temperatures.  By itself, the degree sign is not the unit — the unit for temperature must include the scale involved.  When we require units for temperatures, we should not accept just the degree symbol — 40° F is much different from 40° C, and nobody wants a household temperature of 40° K.  Even the simple conversion of F to C temperatures does not make sense if the scale is not included — the process becomes a black box of non-reasoning.

It is certainly true that “reasoning with units” will slow us down.  Our work is ‘cluttered’ by non-numerical information.  However, numerical information is the easier part for our students — it is the ‘clutter’ that needs to be seen and reasoned through if our students are to have any lasting benefit from our courses.

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