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(1887-1985), a Hungarian mathematician, wrote "How to solve it." for high school students in 1957. Here is his four step method. Read the problem over carefully and ask yourself: Do I know the meaning of all the words? What is being asked for? What is given in the problem? Is the given information sufficient (for the solution to be unique)? Is there some inconsistent or superfluous information which is given? By way of checking your understanding, try restating the problem in a different way. In essence, decide how you are going to work on the problem. This involves making some choices about what strategies to use. Some possible strategies are: -- making a picture which relates the information given to what is asked for can often lead to a solution. -- this is a strategy which is especially useful in problems where you need to count the members of a set. -- almost any problem can be made simpler in some way. By working out simpler versions, you can often see patterns which help solve the original problem. -- Many problems can be broken into a series of smaller problems. This strategy can turn a problem which on first glance seems intractable into something more doable. -- the method of algebra. Very useful in a lot of problems. : Spend a reasonable amount of time trying to solve the problem using your plan. If you are not successful, go back to step 2. If you run out of strategies, go back to step 1. If you still don't have any luck, talk the problem over with a classmate. After you have a proposed solution, check your solution out. Is it reasonable? Is it unique? Can you see an easier way to solve the problem? Can you generalize the problem?
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April 19, 2023 3-5-operations-and-algebraic-thinking , k-2-operations-and-algebraic-thinking , 6-8-expressions-and-equations
Polya’s problem-solving process: finding unknowns elementary & middle school, by: jeff todd.
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. Plus, download my Finding Unknowns in Elementary and Middle School Math Classes Tip Sheet .
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 a fundamental means of developing students' mathematical knowledge and it also shows them that math concepts apply to real-world concepts.
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.
Who Is George 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 suggested that math should be presented in the light of being able to solve problems...that by presenting mathematical thinking as a way to find “unknowns,” it becomes more engaging for students.
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 process 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 Mathematics 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 Process
1. understand the problem, and desiring the solution .
- Restate the problem
- Identify the principal parts of the problem
- Essential questions
- What is unknown?
- What data are available?
- What is the condition?
2. Devising a Problem-Solving Plan
- 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?
3. Carrying Out the Problem-Solving Plan
- Can you see that each step is correct?
- Can you prove that each step is correct?
4. Looking Back
- 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:
Suggestion One 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.
Suggestion Two A second key feature of this first phase of problem-solving is giving students strategies forgetting acquainted with problems.
Suggestion Three Another suggestion is that teachers should help students learn strategies to be able to work toward a better understanding of any problem through experimentation.
Suggestion Four 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 Conclusion
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.
Grab my download and start applying Polya’s Four-Step Problem-Solving Process in the lower grades!
| described the experience of problem solving in his book, , p. v: These and other general mathematics problem-solving strategies, or rules of thumb for successful problem solving, are called |
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Polya's 4-Step Process. George Polya was a mathematician in the 1940s. He devised a systematic process for solving problems that is now referred to by his name: the Polya 4-Step Problem-Solving ...
Is this problem similar to another problem you have solved? Step 2: Devise a Plan: Below are some strategies one might use to solve a problem. Can one (or more) of the following strategies be used? (A strategy is defined as an artful means to an end.) 1.
This four-step process consists of understanding the problem, devising a plan, trying the plan, and revisiting the solution. (Polya, 1947) In order to focus on the skills and knowledge of the problem-solving process, I began by using tasks where the mathematical processes were obvious. This allowed me to focus on the problem-solving process ...
Much of the advice presented in this section is based on a problem-solving process developed by the eminent Hungarian mathematician George Polya (see the historical high-light at the end of this section). We will now outline Polya's method. George Polya's Problem-Solving Method Step 1: Understand the problem.
Polya's Four Phases of Problem Solving The following comes from the famous book by George Polya called How to Solve It. 1. Understanding the Problem. ... Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct? 4. Looking Back. Examine the solution obtained.
Math in the Modern World Playlist: https://www.youtube.com/watch?v=vb-hDOm3GzU&list=PLbZl6MGLeYnsoaxa2L-xouDPHcoe9z23x&index=3A more logical answer to 13:36 ...
Carry out the plan: Spend a reasonable amount of time trying to solve the problem using your plan. If you are not successful, go back to step 2. If you run out of strategies, go back to step 1. If you still don't have any luck, talk the problem over with a classmate. Look back: After you have a proposed solution, check your solution out.
Explanation of Polya's Four Step problem solving technique. The four steps are explained in simple terms with an example of applying Polya's method.
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.
3. Carry out the plan— If the plan does not seem to be working, then start over and try another way. Often the first approach does not work. Do not worry, just because an approach does not work, it does not mean you did it wrong. You actually accomplished something, knowing a way does not work is part of the process of elimination.
To begin this task, we now discuss a framework for thinking about problem solving: Polya's four-step approach to problem solving. Polya's four-step approach to problem solving 1. Preparation: Understand the problem Learn the necessary underlying mathematical concepts Consider the terminology and notation used in the problem: 1.
Learn about Polya's problem-solving process and the steps involved. See what is the final step in the problem-solving process and how it provides a...
POLYA'S FOUR STEP PROBLEM SOLVING PROCESS Understand Devise a Plan Carry out Plan Look Back PROBLEM SOLVING STRATEGIES (exmples) Making a Drawing Guessing and Checking Making a Table Using a Model Working Backwards §1.2 KEY IDEAS Patterns in Nature / Number Patterns
Polya's Four Step Problem Solving Process . ... Usually, a problem is stated in words, either orally or written.Then, to solve the problem, one translates the words into an equivalent problem using mathematical symbols, solves this equivalent problem, and then interprets the answer.This process is summarized in Figure 1.1. ...
Intermediate Algebra Tutorial 8. Use Polya's four step process to solve word problems involving numbers, percents, rectangles, supplementary angles, complementary angles, consecutive integers, and breaking even. Whether you like it or not, whether you are going to be a mother, father, teacher, computer programmer, scientist, researcher ...
This video walks you through using Polya's Problem Solving Process to solve a word problem.
in 1) and describe its positive aspects.Formulate problems that might occur due to the u. lem lies in asking the right questions"Come up with a question related to the problem(s) in 3) and answer it by using P. ya's four step problem solving method.Create a presentation w. solving method:1Understand the qu.
16. Do a simulation. 7. Draw a diagram. 17. Use a model. 8. Use direct reasoning. 2.3.1: George Polya's Four Step Problem Solving Process is shared under a license and was authored, remixed, and/or curated by LibreTexts.
The Four-step Problem-solving Process George Polya described the experience of problem solving in his book, How to Solve It , p. v: A great discovery solves a great problem but there is a grain of discovery in the solution of any problem.
Is there enough information? Is there extraneous information? Is this problem similar to another problem you have solved? Step 2: Devise a Plan: Below are some strategies one might use to solve a problem. Can one (or more) of the following strategies be used? (A strategy is defined as an artful means to an end.) 1.
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This is not one of Polya's steps. 1. The Second Step in Polya's Four-Step Problem Solving Process is what? Carry out the plan. Document your solution. 2. Identifying all relevant and irrelevant ...