Got (Math) Problem?

We like to believe that taking a mathematics class (or statistics) will improve a student’s ability to solve ‘problems’ with quantitative properties.  A basic flaw with this belief is that most of us (as math educators) do not like to present actual problems to our students — a problem is a situation where the solution is not just to be remembered.  There is a basic element of “have not seen this before” in a problem situation; at one extreme we have exercises (where memory can retrieve exactly what needs to be done) and the other extreme we have non-standard problems (where the presentation is different from experience and the solution involves synthesis).

We generally all hear variations on the phrase “I am not good at word problems”, though repetition has little to do with truth value.  Admittedly, many students are weak at solving problems; I’ve known quite a few colleagues who are not very skilled at this.  If you are interested, there is a wide body of literature of problem solving in mathematics over the past 40 years (or more).

My goal today is to share two ‘problems’ from my Math Literacy class.

As you’d expect, the concept of slope is central in Math Lit.  We begin working with linear and exponential patterns within the first two weeks of class.  Recognizing those patterns involves a first-order analysis of differences and quotients, and this is then presented as the concepts of slope and multiplier.  In the case of slope, students have experience in the homework (and in class) with both calculating and interpreting slope.

A bit later in class, we formalize the linear pattern with y=mx+b with still more work with identifying and interpreting slope.  Students get reasonably good at the routine problems.  However, I put this question on the ‘y=mx+b test’:

The cost to your company to print x paperback sci-fi novels is C=1200+3.50x where C is in dollars and x is the number of books.
A: What is the slope of this line?
B: Producing 298 books would cost $2243.  How much more will it cost to produce 299 books (compared to 298 books)?

Part A was ‘low difficulty’ (about 90% correct answers.  Part B had medium difficulty (~70% correct).  However, half of the students needed to calculate the new value and subtract … even though they had all ‘correctly’ interpreted a similar slope earlier.  Only about a third of the students could see the answer without calculation (with enough confidence to do the problem the ‘easy’ way).

Most ‘steps’ given for teaching problem solving are actually steps for solving exercises involving verbally statements.   The phrase ‘step by step’ is a admission that we are not doing ‘problems’; problem solving improves with more experience solving problems.

Related to slope … We also explore how to find the equation of a line from two points (or two data values).  Later, we learn how to write exponential functions given a starting value and percent change, or from 3 ordered pairs (conveniently with input values of 0, 1 and 2).

This week, I presented the class with a “problem” related to those two types of functions.

Using the data below, find the equation to find the value of a car ($$) based on the age in years.
Age         2             5              7                  10
Value    24400     19000     15400         10000

Coincidentally, students just completed a review problem on identifying linear and exponential patterns where the inputs were consecutive whole numbers.  We had also just calculated slope (again).  In this case, the situation became a problem because the inputs are not consecutive whole numbers.  Very few students could see (working in teams) how to solve the car value problem.  Most students understood with direct guidance (questioning) though one hopes that students will see what we see … that a linear pattern can be established by any consistent pattern of ‘equal slope’ values.

I am sharing these ‘problems’ and my observations in the hope that some people might be interested in exploring real problem solving in their math classes.  Developing problem solving capacity is not tidy, and often frustrating, but this type of work is rewarding to us and (I think) very helpful to students.

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