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Solving Problems - in multiple ways

Proposition.  Students should learn to solve problems in multiple ways.  Can this be so or are we asking yet again our students learn even more than what is needed?

The short answer is "yes." Let me explain.  First, the math teaching community has embraced, I think correctly, the idea of multiple representations.  This means looking at data and functions in multiple ways graphs, tables, formulas, and the like.  In fact, I've written on this. See http://disted6.math.tamu.edu/newsletter/newsletters_new.htm#current_issue for the three articles. Second and more generally, the more facets of the same thing a person is familiar with, the better is his knowledge of it.

When it comes to problem solving, the same rule applies.  If a person can solve a problem in two or more different ways, this is an indicator of their understanding of the problem and techniques to solve it.  If they can solve it in only one way, this is an indicator that they have a single method in their mind.  Certainly, this method may work some of the time, but it delimits their problem solving ability all around.  Indeed, in your area and mind, the more successful practitioners are those that can approach a problem from multiple directions and have the tools to proceed.  In the business world, for example, multiple solution methods are worshiped as "innovation."

Many, many Nobel prizes have been won for exactly this skill.  Remarkably, many so called new ideas are nothing more that looking at a problem in a different way - using the same tools that have been available to all.

In the schools the downside comes in two forms: (1) The teacher must be able to do the same, and often in algebra courses and up they cannot.  (2) This will encourage students to think beyond just what is taught and to invent new techniques and explore new ideas, and this will necessitate the teacher to recognize, understand, and grade them.  Again, many teachers cannot.  At least in math, many teachers understand little more than what they teach.  Not all teachers to be sure.  This highlights one problem in education today.  The core of teachers is so mixed in content skills that teaching beyond rote seriously taxes many while celebrated by many more.

We here in the University appreciate students' abilities to solve problems in multiple ways.  It makes what we do special, and often very much different from the rigor and dynamics found in the schools - and their administrations.
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