It is unfortunate that in the United States mathematics has a reputation for being dry and uninteresting. I hear this more from adults than I do from children—in fact, I find that children are naturally curious about how math works and how it relates to the world around them. It is from adults that they get the idea that math is dry, boring, and unrelated to their lives. Despite what children may or may not hear about math, I focus on making instruction exciting and showing my students that math applicable to their lives.
Problem solving is one way I show my students that math relates to their lives! Problem solving is a fundamental means of developing students' mathematical knowledge and it also shows them that math concepts apply to real-world concepts.
In this article, we'll explore how a focus on finding “unknowns” in math will lead to active problem-solving strategies for Kindergarten to Grade 8 classrooms. Through the lens of George Polya and his four-step problem-solving heuristic, I will discuss how you can apply the concept of finding unknowns to your classroom.
WHO IS GOERGE POLYA
George Polya was a European-born scholar and mathematician who moved to the U.S in 1940, to work at Stanford University. When considering the his classroom experience of teaching mathematics, he noticed that students were not presented with a view of mathematics that excited and energized them. I know that I have felt this way many times in my teaching career and have often asked: How can I make this more engaging and yet still maintain rigor?
Polya suggested that math should be presented in the light of being able to solve problems. His 1944 book, How to Solve It contains his famous four-step problem solving heuristic. Polya suggests that by presenting mathematical thinking as a way to find “unknowns,” it becomes more engaging for students.
He even goes as far as to say that his general four-step problem-solving heuristic can be applied to any field of human endeavor—to any opportunity where a problem exists.
Polya specifically wrote about problem-solving at the high school mathematics level. For those of us teaching students in the elementary and middle school levels, finding ways to apply Polya’s problem-solving strategies as he intended forces us to rethink the way we teach.
Particularly in the lower grade levels, finding “unknowns” can be relegated to prealgebra and algebra courses in the later grades. Nonetheless, today’s standards call for algebra and algebraic thinking at early grade levels. The download for today’s post presents one way you can find unknowns at each grade level.
PRESENTING MATH AS A WAY TO FIND "UNKNOWNS" IN REAL-LIFE SITUATIONS
I would like to share a conversation I had recently with my friend Stu. I have been spending my summers volunteering for a charitable organization in Central America that provides medical services for the poor, runs ESL classes, and operates a Pre-K to Grade 6 school. We were talking about the kind of professional development that I might provide the teachers, and he was intrigued by the thought that we could connect mathematical topics to real life. We specifically talked about the fact that he remembers little or nothing about how to find the area of a figure and never learned in school why it might be important to know about area. Math was presented to him as a set of rules and procedures rather than as a way to find unknowns in real-life situations.
That’s what I am talking about here, and it’s what I believe Polya was talking about. How can we create classrooms where students are able to use their mathematical knowledge to solve problems, whether real-life or purely mathematical?
As Polya noted, there are two ways that mathematics can be presented, either as deductive system of rules and procedures or as an inductive method of making mathematics. Both ways of thinking about mathematics have endured through the centuries, but at least in American education, there has been an emphasis on a procedural approach to math. Polya noticed this in the 1940s, and I think that although we have made progress, there is still an over-emphasis on skill and procedure at the expense of problem-solving and application.
I recently reread Polya’s book. I can’t say that it is an “easy” read, but I would say that it was valuable for me to revisit his own words in order to be sure I understood what he was advocating. As a result, I made the following outline of his problem-solving process and the questions he suggests we use with students.
POLYA’S PROBLEM-SOLVING STEPS
STEP ONE: UNDERSTAND THE PROBLEM, AND DESIRING ITS SOLUTION
Restate the problem
Identify the principal parts of the problem
What is unknown?
What data are available?
What is the condition?
DEVISING A PLAN
Do you know a related problem?
Look at the unknown and try to think of a familiar problem having the same or similar unknown
Here is a problem related to yours and solved before. Can you use it?
Can you restate the problem?
Did you use all the data?
Did you use the whole condition?
CARRYING OUT THE PLAN
Can you see that each step is correct?
Can you prove that each step is correct?
Can you check the result?
Can you check the argument?
Can you derive the result differently?
Can you see the result in a glance?
Can you use the result, or the method, for some other problem?
POLYA'S SUGGESTIONS FOR HELPING STUDENTS SOLVE PROBLEMS
I also found four suggestions from Polya about what teachers can do to help students solve problems:
In order for students to understand the problem, the teacher must focus on fostering in students the desire to find a solution. Absent this motivation, it will always be a fight to get students to solve problems when they are not sure what to do.
A second key feature of this first phase of problem-solving is giving students strategies forgetting acquainted with problems.
Another suggestion is that teachers should help students learn strategies to be able to work toward a better understanding of any problem through experimentation.
Finally, when students are not sure how to solve a problem, they need strategies to “hunt for the helpful idea.”
Whether you are thinking of problem-solving in a traditional sense (solving computational problems and geometric proofs, as illustrated in Polya’s book) or you are thinking of the kind of problem-solving students can do through STEAM activities, I can’t help but hear echoes of Polya in Standard for Math Practice 1: Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary.
In the spirit of Polya’s original work, this I've created a download that shows how to focus on finding unknowns from Kindergarten to Grade 8.
We all know we should be fostering students’ problem-solving ability in our math classes. Polya’s focus on “finding unknowns” in math has wide applicability to problems whether they are purely mathematical or more general.
Our teaching practices might be very traditional or they might include activity-based learning and STEAM projects. Polya’s work can be applied to both situations.
Stay tuned to this blog as we continue to apply Polya’s problem-solving process to STEAM activities!