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 , 1945
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.

Problem Solving Philosophy of Teaching

Research says” “”thinking is driven by questions, not answers”, and “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. Allowing mathematics to be problematic becomes the key to implementing this philosophy.
A teacher should
• pose problems that are just within the student’s reach,
• allow them to struggle to find solutions, and
• examine the methods they have used.
The underlying belief that all students need to struggle with [challenging] problems if they are to truly learn mathematics is in direct conflict with what many mathematics teachers feel is their main goal: to step in and remove the struggle and the challenge.

One of the cornerstones in my adaptation of this problem-solving philosophy is asking “stimulating questions” through a seven-step cycle:

  1. Ask
  2. Anticipate
  3. Wait
  4. Listen
  5. Respond
  6. Assess
  7. Goto 1

Questions may be written or oral. Anticipate possible student responses and observe students as you wait. Listening could be reading solutions and comments. Responses can be oral or written.

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]. [1] Data; information; conditions; [2] Question[s]; and [3] Answer[s] and Solution[s]. In a traditional problem, of course, the two given parts are the data/information, and the question. while 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

From 1986 to my “retirement” in 2011, my writing was largely devoted to textbooks and other resource materials for teachers and students. 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 promotes interacting with teachers and/or students. Talks included :

“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”

I served on several Texas education committees involving student assessment and teacher professional development.

Current and Future Plans

Since 2009, my main interest has been in quantitative reasoning. I am one of two senior authors of Advanced Quantitative Reasoning: Mathematics for the World Around Us – a high school textbook for quantitative reasoning courses. The Common Core edition was published in 2014, the Texas edition in 2015. There is an active website associated with these books for which I contribute materials and perform other tasks. In addition I work with several school districts on curriculum projects. Writing a book on mathematical thinking and reasoning is a long-term goal.

Mathematical Competitions

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) and watching old movies. These radio shows originally aired from 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.