I doubt very few if any of our students and probably few of you reading this would want to be thought of as “fools.” Though, of course, many--if not all of us--engage in foolish behavior from time to time (perhaps even at this very moment). I donon’t in any way mean the term as a repudiation or judgment, but something much more grand. To act the fool is not (or at least not only) to be a clown, but to enthusiastically follow your heart (“a fool for love”), your greatest passion (“a dancing fool; a Huskies basketball fool”), your deepest curiosity (“a questioning fool”). To act the fool in this sense is the opposite of “conventional” thinking that suggests mathematics is dull, dry, difficult and deadening. If math is to be an adventure that lures students to
with big ideas, it will depend greatly on how we, as teachers, play it.
Teachers of math have developed numerous ways to attack problem solving: illustrating, graphic organizers, strategies, and techniques, etc. We all have our favorites. Vertical teams might spend lots of productive time sharing and demonstrating methodologies and developing those, which could become familiar implements in a problem solvers’ “toolkit”. The advantage of “vertical” consideration of any strategy would be building in students both a sense of what the strategy or technique can be used for and in helping students gain confidence and sophistication for use of the strategy. (My four year old grandson knows what a hammer is and loves to use one, but when he is in high school if he is still holding and swinging a hammer the way he does now, I will really have failed him!)
Some of my own favorite “analytical tools” will be used in discussing the word problem sets that follow. I try to suggest as many different strategies as I can to students, often challenging them to explain or do problems several ways to help them see what works best for them. Among the familiar “logical” hobby horses I have been known to ride are: Venn Diagrams, Charting, Eliminating Possibilities, Trial and Error, Working Backwards, Vee Charting, Unit Conversion and Unitary Rates. The most useful tool for my algebra students is a “5-Part Problem Solving Page”10 which allows students to use a variety of “learning styles” simultaneously in working through a problem with:
Table of values
A ”multiple solutions” blank template for use with problems is included as Figure 2 in the appendix of this unit and use of the template is included in discussion of problem sets below.
All the various problem attack tools are useful, but mindful of what I said above about teaching procedures for just “getting answers” instead of for relational thinking and generalized understanding of the big ideas in math, it is possible to ‘lose sight of the forest (the big idea) while looking closely at the trees (a clever technique)’. A tool is only as good as the hand that holds it and the mind that knows when, where and how to apply it. (One of my favorite tools is a pair of pliers, but it makes a lousy hammer.) With this note of caution, I recommend highly another book series, which we examined in our seminar.