Math Applications for Living XX: When a Foot is NOT 12 Inches

Our Math Applications Course is finishing the semester with reviewing and the final exam.   Students have made considerable progress in problem solving and reasoning, although they often can not see their own progress.  One concept from the course continues to create problems, though … so I’ve been thinking about what causes this difficulty.

Here is an example of a problem dealing with the concept:

A poster is 18 inches by 24 inches (rectangular).  Find the area, and covert it to square feet.

Finding the area’s numerical value was easy; knowing that the unit is ‘square inches’ was simple enough.  Converting to square feet?  Not nearly so easy to see.  Most students kept saying “a foot is 12 inches”, so they divided the area by 12.  One student suggested that we convert the original numbers to feet, and find the area in square feet.  This suggestion was seen as being reasonable, so  we did that … and then came back to the original problem.

As we struggled with this problem, we went back to the ‘a foot is 12 inches’ statement.  After a bit, we drew a square on the board — one side labeled ‘1 foot’ and another ’12 inches’.  Yes, we said, those are the same.  We labeled two sides ‘1 foot’ and two sides ’12 inches’.  The area?  1 square foot, or 144 square inches (a few students then understood what to do with our problem).  Some did not see the implication for converting, so I started drawing 1-inch strips in the square.  That might have helped a little; perhaps not.

A foot is not a foot when we are talking about area (or volume). In some ways, this is another example of prior learning being built in an overly simple space … we say ‘1 foot is 12 inches’, instead of saying ‘1 foot long is the same as 12 inches long’.  Conditional statements are critical for accurate learning, and enable problem solving skills to develop; unconditional statements impede future learning as the price for short term results.

Where am I presenting learning without conditional statements, when there should be some?  I fear that my classes routinely omit qualifiers for statements, sometimes due to the focus on the present problem … sometimes out of relative ignorance of where else the concept is used.

Sometimes, we create our own problems by deliberately omitting “if” and “when” statements.  Yes, these statements can impede current results; yes, we can become obsessed with technical accuracy to the point that only mathematicians can understand what we are saying.  However, I suspect that the price for simplicity in the ‘now’ is a set of problems in the future.

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