I consider reading an essential part of my professional development. I enjoy a morning glance through a chapter or two, and like to wind down a winter’s day with a book and a cup of coffee. Sometimes reading is the only way to relax my mind at the end of a day. (Naturally, some professional literature is better at putting me to sleep than others). To this point in my young career, no book has changed my perspective on the teaching and doing of mathematics more than The Art of Problem Posing: Third Edition by Stephen I. Brown and Marion I. Walter. The duo writes quite a bit for “Mathematics Teacher” (the high school journal for the National Council of Teachers of Mathematics) as well. The processes introduced in the book have been crucial to the penning of many posts on this blog. The book creates a framework from which creative mathematics flows.
The book is filled with examples that illustrate its points. As a teacher, it is encouraging to see a practical link to the problems which I encounter and present to my students on a daily basis. As the book continues, I found myself anxiously guessing which direction the author was going to take. Many of the problems are directly transferrable into a high school atmosphere, but it is the broad process of creating new problems that changed how I approach every problem solving situation.
Brown and Walter focus on a process called “Problem Posing“. This process is designed to interweave the traditional problem solving process. The idea is that by asking questions and raising doubts about an original question, a student can learn about the intricacies of the original problem. In other words, playing with the question not only allows for various interesting pathways to be uncovered, it also reflects back upon the original attributes of the problem. The entire book, save a portion at the end dedicated to the teaching of mathematics education courses at the university level, is built around this theme. The process they use to introduce doubt into a problem is the “What If Not” method–or WIN.
A WIN begins with a problem. They use typical textbook problems, creative problem solving situations, and even objects like geoboards in the book. Any of these can be the subject for problem posing. The process begins by listing all the attributes about the problem or object. One particularly interesting example given in the text is that of the Fibonacci sequence:
1,1,2,3,5,8,13,21,34,55,89, …
With some practice, this process becomes less regimented. The questions begin to flow out as you change the problem’s attributes slightly. Brown and Walter’s list of attributes for this sequence is as follows:
1) We start with two given numbers.
2) The two starting numbers are both 1.
3) The first two numbers are the same.
4) We do something to any successive numbers to get the next number.
5) That something we do is an operation.
6) The operation is addition.
Although some of these attributes seem redundant (2 and 3 for example), it is important to list as many as you can. From here, each attribute is doubted, replaced, or eliminated. This creates a whole plethora of unique and interesting problems. It is a great way to get instant extensions for students who need enrichment for the day’s work. The new questions generated from this example could be:
1) What if we started the sequence with 3 numbers?
2) What if we changed the starting numbers?
3) What if the two starting numbers were different?
4) What if the operation that was performed was multiplication? exponentiation?
5) What if more than one operation was performed?
etc.
From here, the authors explore various pathways. Examples like these are repeated often throughout the book, and I find myself using the WIN framework in my own work with students in mathematics. I have made a habit of posing new problems using the WIN structure after I have solved a problem. This not only replenishes my store of new problems, it brings attention back to the original solution’s attributes.
This book comes highly recommended to any teacher or learner who is interested in more than solving problems. Problem posing is an integral part in the process of doing mathematics. We cannot fully understand the nuances of our work until we change the parameters and redouble our efforts. Problem posing is an excellent way to allow students to encounter the never-ending web of mathematical connections.
The Art of Problem Posing: Third Edition
Stephen I. Brown, Marion I. Walter
Routledge.
New York, NY,
2004.
NatBanting