Bovine Math...

I have created a collection of mathematical puzzles that, I hope, address (partially!) a need that has been created by a wrong-headed approach to curriculum design. I am speaking about Ontario, but I suspect that these comments apply to many other jurisdictions.

If you study the Ontario Mathematics policy document, you'll see that the teaching of algebra has become a set of utilities. Quadratic equations are dealt with primarily as a way of finding points of intersection. Linear systems are taught to enable students to solve inane word problems. There is precious little else. I suppose that when you consider that the treatment of Calculus is little more than a joke, maybe our students don't really need much more than this in the way of algebraic preparation. But I don't believe this.

I believe that our students are capable of, and would welcome a much more interesting and challenging curriculum.

This is a collection of mathematical puzzles that all look the same. The "mechanics" can be quite different, but the central idea doesn't change. In the diagrams you have bridges, fields and cows. The cows are browsing in the fields and the bridges hold clues and other bits of information about the numbers of cows. It is hoped that familiarity with this problem structure will help students concentrate on the actual mathematics involved with each new set of instructions. In fact, the problem descriptions have been intentionally kept brief. In this way, students (and teachers) will not have to de-code strange situations and de-cipher new or obscure contexts. The "nuts and bolts" encountered in these problems will be challenging enough. Understanding the mathematical language and creating problem solving strategies become the central concerns. This is not to say that context isn't important, rather, it is hoped that these problems can act as a staging area somewhere between learning algebraic skills and tackling relevant or world-style problems.

It is intended that students will encounter these problems as enrichment or re-enforcement, over a number of years. The early problems involving simple addition can easily be introduced to a grade 5 or 6 class, while the problems involving differences require more maturity. The problems involving consecutive numbers and migrations might better be left until students have been introduced to some algebraic techniques … but not for long. Some of these problems would be ideal for students just coming to grips with solving systems of equations. Essentially, most of these problems can be introduced just as students mathematical learning is progressing from generalizing arithmetic to algebraic thinking. While it is important that students acquire arithmetic skills, it is equally, if not more, important that they foster a mathematical sense about these skills. This is to say that they must be able to solve problems that require more than a passing familiarity with simple arithmetic and algebra that has been taught in order to solve specific and limited problem types with a narrow range of application.

Cow-Friendly Numbers: The puzzles in this collection involve only whole numbers! All answers can be expressed as a number of cows, and in the spirit of compassion and kindness to our animal friends, no "fractional" cows will be considered. Also, to avoid “bovine ennui”, empty fields are to be avoided. There are some exceptions to this. The last few puzzles will necessitate the use of "fractional" cows during the process of the "migration". However, by the time the problem has been solved, all of the cows will have "been put back together again"…. unlike poor old Humpty Dumpty.

Communication:Hopefully, students will find these puzzles enjoyable. In the years that I have been using these exercises, I have found that they engender a positive atmosphere and generate a lot of fun. They provide ample opportunity for students to discuss, debate and debunk each others' ideas. I believe that this is an integral part of a successful mathematics program. Students should be shown how to write concise, correct and convincing solutions and then they should be required to do so. Many of the puzzles in this collection are complex enough that they require reasonably complex, comprehensive and comprehensible written solutions.

Transition from Arithmetic to Algebra: From Experiment to Technique: A students' first encounter with algebra is often quite mystifying. After some drill and repetitious exercises, she may start to perform her tasks correctly with some regularity and even develop a degree of confidence. Alter the task or require an explanation and all too frequently the wheels start to wobble and fall off. This is because students are rarely given the time and sort of practice that is necessary if the mysteries of algebraic expression are to be unraveled. By actually using these newly acquired skills in genuine problem solving situation, students will internalize these methods and this sort of thinking. This is an important moment in the mathematical development of a student and should be handled with great care. The puzzles in this collection have been designed to provide students with appropriate opportunities to wrestle with ideas found in the grey zone between the worlds of Arithmetic and Algebra.

Solutions: Many of these puzzles can be solved in a variety of ways, from naïve “guess and check” to the use of formal algebraic techniques.