When in the area of problem solving there are the “how to solve it” steps and, for educators, there are the “how to teach how to solve it” steps. The first set of steps was definitively established by George Pólya in his classic work How to Solve It, a book that should be required reading in all mathematics curricula, especially those with the goal of teaching teachers.
The second set of steps is a bit harder because learning is such an individual thing. There are cases in which what is clear to half of a class might as well, for the rest, have been stated in ancient Mayan. This is especially the case in mathematics classes, where some people have to overcome significant insecurities before they can even begin to understand and execute a problem solving strategy.
This book is a textbook designed to teach the educator how to pedagogically present the solving of problems. A small set of well known problems are presented and then thoroughly sliced, diced and dissected into minute segments. Some of the problems are:
How to measure out a volume using two jugs of different volumes.
Determining the last digit of a number to a very large power.
Sketching the graph of functions that involve the absolute value (ex. \(y = 4 |x|+ 5\))
Key heuristics, plans of attack, solutions and possible student responses or errors are given for each problem.
The materials in this book have been used in the Singaporean secondary schools. Given that Singapore traditionally ranks first or second in the world and has held that position for almost 20 years, this book should be taken very seriously. Sometimes there is no substitute for grinding away at a concept. That is what often must be done in mathematical problem solving. Repeating the basic tactics of problem solving for different problems is a necessity and this book contains all the necessary details.
Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.