Problem Solving … and Learning Mathematics

Our Math – Applications for Living course is sometimes used as a last option; students try passing the intermediate algebra class, and (after 2 or 3 tries) an adviser says that they have another option.  This is not true for all students in the course, though it is a common path to my door.  The result is a class with some very anxious students, and many who doubt their ability to solve ‘word problems’.

Math – Applications for living is all about problem solving; all topics are verbally stated.  We had an interesting experience last week when we did an example with a simple statement:

The distance from the Moon to the Earth is 3.8 x 10^5 km.  A light-year is 9.5 x 10^12 km; in one second, light travels 3 x 10^8 meters. How long does it take light to travel from the Moon to the Earth?

The problem presents to issues to resolve: the operation to perform, and making the units consistent (meters and km).  A few students knew to divide distance by speed to get time; if they did not already know this, it did not help much to solve the D=rt formula for t.  We explored the problem by working with rates (as we have been doing for most unit conversions); this helped a little more.

We got frustrated, however, with the km and meter conversion in the same problem.  After about 10 minutes of discussion, some progress was made.

In working through these struggles, more than one student said something like:

Can’t you just show us how to solve these in a way that we already understand?

Of course, it is exactly this gap between current understanding and present need that causes learning to happen.  As a problem solving issue, this is essentially a statement of what problem solving is … as opposed to exercises.  In the most encouraging manner, I told the class that this tension they are frustrated with — is the zone where we will learn something.  I stated, with emphasis, that if I did not create situations where there was a gap like this that they would leave the course with the same abilities as when they started.

I’ve been talking with faculty in some other programs at my college about the mathematical needs of their students.  The first thing they say is always ‘problem solving’, and they don’t mean solving a page of 20 ‘problems’ using the same steps.  The second thing they say depends on their program, and a surprisingly large number of them say ‘algebra’ is the next priority — in spite of the fact that algebra is often de-emphasized outside of the STEM-path.  In the Math – Applications for Living course, we use algebraic methods when useful, as it is when solving problems with percents.

In the larger context, all learning is problem solving.  A learner faces a situation where existing knowledge is not sufficient, and the gap is completed by some additional learning.  I believe that this statement is true regardless of the pedagogy a teacher uses, whether active or passive for the learner.   I do not agree with a constructivist viewpoint, especially the more radical forms; however, there is a basic element in the constructivist view that is true, I believe — knowledge is built as a result of gaps.  I believe that teachers can (and should) model the process of filling the gaps, and explaining the reasoning behind ideas that can help.  Learning math does not need to involve students stumbling through to discover centuries of mathematics; we can both guide and be a sage in the process.

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