## Turnkey material for learning how to solve problems

It is useful to bookmark this page. It serves as the benchmark for developing a coherent approach to problem solving across the curriculum.
The following gives links to material which form organized lessons in problem solving. They lend themselves to thematic instruction. These lessons will open in a separate window. Examine as many successive pages as you wish. Return to this page by closing the last window opened.

An evolving problem solving text book
The book material develops the philosophy on which reliable problem solving is based. Examples are used to illustrate the ideas. The use of the material is appropriate at all grade level. At the elementary grades it introduces sound, comfortable ideas associated with problem solving. It shows how problem solving is a natural process closely associated with everyday life. At more advanced levels it provides a way for students to overcome preconceptions about problem solving that have become imbedded in earlier years.

A Short Course in Problem Solving
The lesson provides a sequence of screens showing the principles of reliable problem solving by means of examples. The sequence starts with the simplest ideas and evolves to handle complex problems reliably. The resource can be introduced in early grades and returned to in successive grades. There is an index to select appropriate entering points on successive use of the material.

K4-K12 Problem Set
This is an extensive set of algebra problems starting with quite elementary problems. The problems are ordered in order of increasing grade level. This problem set forms the core of problem solving in algebra at all grade levels. Some of the problems are interactive.

Problem Solving Lesson Plans
This is a set of lesson plans that serve to introduce and embed the problem solving process. It fills a serious need for lessons plans that form a sound basis for problem solving. This will serve you for a life time.

Problem Solving Manipulatives
Teaching problem solving is not complete without hands-on, minds-on material. This lesson provides problem solving manipulatives which gives even the most reticent student an exciting and meaningful tool for learn to solve problems reliably. These manipulative work together with the Lesson Plans described above to provide a complete approach to learning problem solving.

Making Sense of Story Problem
This inexpensive book has two parts. The Work Book and the Answer Book. The Answer Book serves as guide for the teacher. Students develop solutions to problems in the Work Book. The problems range from very simple problems to upper middle school problems. The emphasis is on embedding the basic problem solving principles. Once this is done, problems in more advanced courses can be solved reliably. The link above causes one of the more advanced problems from the answer book to be displayed.

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Guest Book
Comments in the the guest book show why the problem solving principles shown at 21st Century Problem Solving have become the world standard for problem solving.

Interactive lessons.
The lesson links below provide particular interactive material for learning the problem solving process. These provide ideal turnkey online lessons on problem solving.
An excellent lesson providing exercise in reliable problem solving
How to start a problem solution

SureMath
SureMath is a symbolic algebra program designed for problem solving using modern problem solving processes. In environments in which there is a commitment to including reliable problem solving in the curriculum SureMath greatly expands the students opportunities to learn not only problem solving but subject matter in all subjects, at all levels, that use mathematics for problem solving. Once you have made the move to using SureMath for problem solving you will find the savings in time and the satisfaction in solving problems reliably will be the most satisfying educational experience of your lifetime.

SureMath is a Macintosh application. Those who are limited to using Windows can easily extend their resources. Click here to see how. Using Suremath for problem solving is a step you will never regret.

The links above provide a systematic approach to implementing problem solving across the curriculum. Other material on the site provides additional problem solving experience. You will find it rewarding to explore the site. This is done by clicking on the image bar at the top of this page. Material in physics and chemistry is available. Of particular importance are the How To pages which provide deeper insight into problem solving. Problem solving in Physics and Chemistry are also available.

Though this site deals specifically with problem solving in situations that use mathematics, the problem solving principles used apply equally well to problem solving in areas that use tools other than mathematics for problem solving.

Students: Using the simple problem solving principles described in these web pages will make you a happy camper in the forest of problem solving.

## Why learn to solve problems?

Simply because it is fun and satisfying.

The National Council of Teachers of Mathematics puts it this way:

"Problem solving - which includes the ways in which problems are represented, the meanings of the language of mathematics and the ways in which one conjectures and reasons - must be central to schooling so that students can explore, create, accommodate to changed conditions, and actively create new knowledge over the course of their lives."

## A quick summary of logical problem solving

1. Start the problem solution by identifying what is asked for.
• What is asked for can quickly be determined by scanning the problem statement, looking for such words as find, show that, how many, what, where and the like. The words following these words define what is asked for. The words stating what is asked for are often ended with a question mark (?).

• Make no attempt to understand the problem. Trying to understand a problem is wasteful of time and effort and can be scary. It is not possible in any but the simplest of problems. It is the process of solving a problem that brings about the understanding of the problem. An understood problem is, of course, a solved problem. It is, then, not possible to understand a problem before solving it. When you are reading a story, can you understand a sentence before you read it? Of course not. The same is true of problems. They have to be worked step-by-step just as one reads a sentence word-by-word. The understanding of either a problem or a sentence evolves as either is processed.

2. Respond to what is asked for.
• First, read the problem, stopping at the first statement that provides an answer to what is asked for.

• If a response to what is asked for is found in the problem statement, convert the words found into an equation. This forms the starting equation.

• If a response is not found in the problem statement, the response must be found in knowledge space. Knowledge space consists of concepts of particular courses, textbooks, reference books, encyclopedias, dictionaries, the world wide web, experience, common sense and so forth. Using knowledge space will provide the starting place for solving the problem. In problems which can be solved by mathematical methods, an equation can be written representing the concept which was determined to be the starting point. This is the starting equation for the problem.

3. Process equations left-to-right.
• Each quantity encountered in processing an equation will make a request for information.
• Respond to each request. Look first to the problem statement for a response. If it is not present in the problem statement, the response must be found in knowledge space.

• Responding to these requests leads to a result. The solution of a problem consists of recursive use of the request-response-result process. In introductory algebra and pre-algebra the response is quite frequently in the problem statement. In more advanced algebra and subject matter courses such as geometry, calculus, physics and chemistry, the responses are more frequently found in knowledge space.

• The response-request-result sequence leads to a hierarchical structure for a problem solution. This provides a reliable, efficient process for solving problems just as organizing knowledge hierarchically provides a reliable, efficient structure for accessing knowledge.

• When the mathematical functions get too complex to handle easily, order your own student copy of SureMath It will last you a lifetime.

## What is a problem?

A problem is a request for an outcome subject to a number of conditions that must be simultaneously satisfied.

In the typical end-of-chapter problems the outcome requested and the conditions are well defined. In so-called real life problems the conditions are usually a mixture of well defined and poorly defined conditions. Some conditions may not even be known and the outcome requested may be inadequately defined. Nevertheless, the problem solving process has the hierarchical structure described above. Solutions of real life problems use the same rules as end-of-chapter problems.

For more details,

1. send e-mail to SureMath Publishing, Inc.
• for a free Macintosh disk demonstrating SureMath