Tom Butts, Ph.D.
A Polya Disciple
"A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime." George Polya How to Solve It , 1943
Although I never met him, I consider myself a disciple of George Polya, the "father" of modern mathematical problem solving. Since first reading How to Solve It many years ago, I have tried to embody and refine his principles of learning and teaching mathematics in my own writings and teaching. I consider myself a 'communicator' of mathematics and ideas about mathematics teaching.
“Thus, a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking. Also a student whose college curriculum includes some mathematics has a singular opportunity. This opportunity is lost, of course, if he regards mathematics as a subject in which he has to earn so much credit and which he should forget after the final examination as quickly as possible. The opportunity may be lost even if the student has some natural talent for mathematics because he, as everybody else, must discover his talents and tastes; he cannot know that he likes raspberry pie if he has never tasted raspberry pie. He may manage to find out, however, that a mathematics problem may be as much fun as a crossword puzzle, or that vigorous mental work may be an exercise as desirable as a fast game of tennis. Having tasted the pleasure in mathematics he will not forget it easily and then there is a good chance that mathematics will become something for him: a hobby, or a tool of his profession, or his profession, or a great ambition."
Problem Solving Philosophy of Teaching
Since "thinking is driven by questions, not answers", one of the cornerstones is in my adaptation of this problem-solving philosophy is asking “stimulating questions” through a six-step cycle:
- Ask Questions
- Goto 1
Questions may be written or oral. Listening could be reading solutions and comments. Responses can be oral or written.
The second key component is to allow mathematics to be problematic by posing problems that “proportionate to the student’s knowledge”, allow them to struggle to find solutions [contrary to the popular belief that the teacher’s role is to remove the struggle], and then examine the methods they have used. Research says that “most, if not all, important mathematics concepts and procedures can best be taught through problem solving. … Students are able to learn new skills and concepts while they are solving challenging problems. In fact, students who develop conceptual understanding through problem solving early perform best on procedural knowledge later.”
The key ingredient in this process is the ability to pose problems, often using the “2 out of 3” Principle. A problem can be thought of as consisting of three parts [the Anatomy of a Problem].  Data; information; conditions;  Question[s]; and  Answer[s] and Solution[s]
In a traditional problem, of course, the two given parts are the data/information, and the question. and the unknown third part is a single numerical answer. Many stimulating problems can be constructed by giving two other parts [or portion thereof] and asking the solver to find alternatives for the third part.
Polya’s Ten Commandments of Teaching Mathematics
Four of these commandments are essential parts of this teaching philosophy: 3. The best way to learn anything is to discover it by yourself. This applies to you as well as your students. 6. Let them [the students] learn guessing. 7. Let them learn proving [giving convincing arguments] 9. Do not give the whole secret at once - let the students guess before you tell - let them find out for themselves as much as possible
Research and Outreach
Since 1986, my writing has largely been devoted to textbooks and other resource materials for students and teachers grades 6 – 14. Included are
- a High School series on Integrated Mathematics,
- writing activities for “We All Use Math Every Day” – the outreach program for the NUMB3RS TV shows coordinated by Texas Instruments.
- many contributions to Figure This - a national Media Grant of the National Council of Teachers of Mathematics,
- serving three years on the NCTM [National Council of Teachers of Mathematics] Student Math Notes committee.
Speaking at various meetings is a way of interacting with teachers and/or students. Recent talks include :
- "Learning Through Questioning"
- “Teaching Algebra and Geometry Through Problem Solving”
- “Art of Questioning: Twenty Questions [or More]”
- “Teaching Mathematics Through Problem Solving: A Personal Perspective”
- "Conversely Speaking: the Power of Thinking Backward
- “Using Mathematics: From Cryptography to Catching Criminals
Since retiring in June, 2011, I have kept active by working on several projects including being a coauthor of a HS textbook on advanced quantitative reasoning and serving on several Texas education committees involving student assessment and teacher professional development.
Mathematical competitions have always been dear to my heart since I participated in them as a student. I have been involved with the American Mathematical Competitions since 1980, serving as chair of AMC 8 competition for seven years in the 1980's and now on the Advisory Panel. In addition, I was a writer on the MATHCOUNTS Question Committee for three years and worked on several local competitions for many years.
My favorite hobby is listening to and collecting OTR – old time radio. These shows first aired between the late 1930’s to the early 1960’s when dramatic radio ceased to exist and gave way to the present music and talk formats.
- Updated: March 26, 2012