Developing the TechMath Instructional Module
In 1945, George Polya published How to Solve It. A respected mathematician,
Polya wrote the book as a pre-World War II Hungarian to an audience that valued
mathematics. It was not until he immigrated to the United States, that the book
was published to an American audience that valued “how to” books. The book is a
classic. It has been translated into at least 17 different languages and has sold
over a million copies. In the book, Polya defines the stages of problem solving.
1. Identify the problem.
2. Develop a plan.
3. Implement the plan.
4. Look back.
The simplicity of the language veils the psychological depth of his analysis. For
example, “look back” includes a reflection of the problem-solving process, a metaanalysis of the solver’s cognitive processes. After publishing, How to Solve It,
Polya became a formidable leader in the post-World War II and Modern Math
movement in the United States. His focus for high school mathematics teachers
was helping them to teach mathematics through problem solving.
We can use Polya’s stages for problem solving and teaching mathematics, as a
framework for TechMath instruction. In a fundamental way, the monumental
movement in mathematics education research on mathematical learning and
teaching, curriculum development, and policy has roots or supports Polya’s work
with secondary mathematics teaching.
Polya’s Ten Commandments for Teachers
1. Be interested in your subject.
2. Know your subject.
3. Know ways of learning: the best way to learn anything is to discover it by yourself.
4. Try to read the faces of your students, try to see their expectations and difficulties, put yourself in
their place.
5. Give them not only information, but “know-how,” attitudes of mind, the habit of methodical work.
6. Let them learn guessing.
7. Let them learn proving.
8. Look out for scuh features of the problem at hand as may be useful in solving the problems to come—
try to disclose the general pattern that lies behing the present concrete solution.
9. Do not give away your whole secret at once—let the students guess before you tell it—let them find
out be themselves as much as is feasible.
10. Suggest it; do not force it down their throats.
Polya, G. (1962). Mathematical Discovery. New York: John Wiley, p.116.
Rose Sinicrope
1 of 4
TechMath Summer Institute 2009
The first step in developing TechMath instruction is to formulate a problem.
A TechMath problem has
a local business context;
develops or applies secondary mathematics; and
uses technology during the problem solving process as
part of the plan or
part of the “looking back.”
A TechMath problem is a problem that
students can understand;
interests students; and
is not an exercise.
A TechMath problem is instructionally rich; it is a “worthwhile mathematical task”
that:
engages students in “sound and significant mathematics”;
“develops mathematical understandings and skills”;
“stimulates students to make connections”;
“calls for problem formulation, problem solving, and mathematical reasoning”;
“promotes communication”;
“represents mathematics as an ongoing human activity”; and
“draws on students’ diverse…experiences and dispositions.”
Martin, T. S. (Ed.). (2007). Mathematics teaching today: Improving practice, improving
student learning, second edition. Reston, VA: National Council of Teachers of
Mathematics, pp. 32-33.
On the next page is an example of a “worthwhile mathematical task.” There are multiple methods
of thinking about the questions, different but equivalent algebraic representations for the nth
term, connections among slope, rate of change, and recursive definitions. You can extend the
question to other sequences in which the rate of change is not constant or to different geometric
patterns. It is also an opportunity to explore simplification of algebraic expressions. By allowing
students to approach the problem in different ways and by encouraging students to explain their
methods of thinking, students are able to develop their mathematical reasoning, communication,
connections, representations, and problem solving.
Rose Sinicrope
2 of 4
TechMath Summer Institute 2009
Stage 1
Stage 4
How many Square tiles
does it take to build
Stage 2
1. Stage 1?
2. Stage 2?
3. Stage 3?
4. Stage 4?
5. Stage 20?
6. Stage n?
Stage 3
Rose Sinicrope
Problem from NCTM 2007
3 of 4
TechMath Summer Institute 2009
Comparison of Instructional Development Frameworks
Tech-Math Module
Polya’s Problem Solving
Science 5E Lesson Structure
Strategies
1. Tell the business story
& pose the essential
question
2. Classroom Discussion
3. Think/pair/share
4. Classroom Discussion
Understand the Problem
Engagement
 Reasonable
approximations
 Reasonable plans
 Alert students to
subsequent lessons
Exploration
Develop a Plan
Instruction with Specific
 Hypothesis
Method /Concept
 Experiment
Development/Experiment
Explanation
Implement the Plan
 Analysis of Results
 Theory Building
Return to Essential Question
Elaboration
 Written communication
 Terminology introduced
of solution
 Similar problem with
Look Back
 Application to Other
given understanding
Problems
Evaluation
 Assessment
The above table presents a linear instructional development model, but instruction
and learning are dynamic. One method may suggest another method or we may
discover a different concept. We “look back” throughout problem solving and at
times return to understanding the question or developing another plan.
Explanations may result in further exploration.
The goal is not to finish, solve the problem, or answer the question; but to learn.
Instructional development is also a dynamic, iterative process. As we teach and
reflect on our instruction, we revise, enhance, and elaborate.
Rose Sinicrope
4 of 4
TechMath Summer Institute 2009