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Crossing the River with Dogs: Problem Solving for College Students

Ken Johnson, Ted Herr and Judy Kysh
Key College Publishing
Publication Date: 
Number of Pages: 
[Reviewed by
James Tanton
, on

 An impressive collation of mathematical questions and puzzles, all organized beautifully to illustrate a variety of problem solving techniques! In reading Crossing the River with Dogs I very much enjoyed working through, thinking about, and reviewing many of the chapter example problems along with the multitude of practice problems placed at the end of each chapter. These constitute the strength of the work. The authors cover all the classic problems — connect nine dots arranged in a grid with just four contiguous line segments, river transportation problems, the "sum 15" game, Monty Hall, and the like — and it is nice to see such gems collected in one volume. The text also contains some interesting non-standard problems: Is it always possible to draw a perfect square within a given triangle with all four vertices on the edges of the triangle? Is it possible to stack ten cards numbered one through ten so that if every second card is removed, working down the stack with "wrap around," the ten cards are plucked in order? (This is a clever variation of the Flavius Josephus problem.)

The authors arrange the text into 17 chapters, each reviewing a particular problem solving strategy — the value of diagrams, lists, and elimination techniques, for instance; the power of inductive reasoning, analyzing sub-problems, working backwards, using manipulatives, for example, and even the effectiveness of some simple dimensional analysis, techniques of algebra, and elementary graph theory. Each chapter contains detailed "real" examples of student discussions illustrating the thinking processes behind different problem solving approaches. The book is 490 pages long.

I do have to say that I am worried about the "pitch" of the text. At times the book discusses at great length very elementary mathematical concepts that are appropriate for a much younger and less sophisticated audience. (The examples in the "Make a Systematic List" and "Eliminate Possibilities" chapters, for instance, are too simplistic and the authors often don't move beyond this level of writing in later chapters.) The transcribed student conversations are often tedious to read and the mathematics discussed is not deep.

It is not immediate how one would use this text in a college-level course — even in a non-math majors' course. It may be that as a practiced math puzzle solver I am currently suffering from the "binary nature" of math — when you already know the problem and know how to solve it, then all appears "easy" and elementary; but if a problem or type of problem is new to you, then all appears mighty challenging, if not ridiculously hard. Maybe the authors have in fact done the right thing to keep the material very simplistic so that the focus remains on the strategies of the mind rather then the sophistication of the material. For a college course, perhaps the thing to do would be to only use the problems listed at the end of each chapter, hand them out to a class at the beginning of the week, say, and allow discussion of problem solving strategy to occur informally. Students could read the chapter discussions at their own volition. It would be nice for the instructor to also work with students on generalizing results and to explore with them methods of deductive reasoning and issues of proof. (For example, in the method of finite differences chapter, the authors came close to proving, but didn't quite establish, that if second/third differences are constant, then a relationship is indeed quadratic/cubic.) This, I suspect, would provide a satisfactory experience for all.

James Tanton left the college scene to take on the challenges of teaching mathematics in the secondary setting (and what a challenge it is!). He is currently the founding director of the St. Mark's Institute of Mathematics at St. Mark's School in Southborough, Massachusetts.

Draw a Diagram
Make a Systematic List
Eliminate Possibilities
Use Matrix Logic
Look for a Pattern
Identify Subproblems
Analyze the Units
Solve an Easier Related Problem
Create a Physical Representation
Work Backwards
Draw Venn Diagrams
Convert to Algebra
Evaluate Finite Differences
Organize Information
Change Focus
Visualize Spatial Relationship