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.Teachers, administrators and curriculum developers will find it useful to bookmark this page. It serves as the benchmark for developing a coherent approach to problem solving across the curriculum. There is an urgent need to embed mathematics in the K-12 curriculum. These web pages extend the ongoing reforms in education to include using mathematics in problem solving.
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 the student to 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.
How to solve problems
The pages available from this menu describe the basics of problem solving in detail. Why students can't solve problems reliably is explained and the resolution of the difficulty is addressed. The cure of the "I can't solve word problems" syndrome and the Hierarchical nature of problem solving extend your understanding of problem solving. The menu on the page About Problem Solving leads to additional instruction in the concepts involved in reliable problem solving.
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 the student 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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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.
The lesson links below provide particular interactive material for learning the problem solving process. These provide ideal turnkey online lessons on problem solving.
Adapted from Quest 2000, Addison-Wesley, 1995.
An excellent lesson providing exercise in reliable problem solving
How to start a problem solution
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. Click here to make that move into the future.
SureMath is a Macintosh application. Those who are limited to using Windows can easily extend their resources. Click here to see how. Adding SureMath to your teaching resources 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. Encourage your students to explore the site. This is done by clicking on the image bar at the top of this page. Problems and their solutions in algebra, physics and chemistry are 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.
Evolving technology has made it possible extend our problem solving creativity by using electronic helpers. The basic resources available for this purpose areThese work together for creative problem solving.
The problem solving methods shown in these pages rely on meaningful use of mathematics that actively engage the student, individually or in groups, directly in the problem solving process. This produces not only reliable problem solving skills but also a genuine appreciation of mathematics. Let's work together to help students discover problem solving as an enriching experience. The global nature of the problem solving process described in the pages of this web site provide a means of doing this.
The objectives of the National Council of Teachers of Mathematics outlined below.
Problem solving must be the focus of school mathematics.
The National Council of Teachers of Mathematics has stated this position in its Curriculum and Evaluation Standards for School Mathematics and in An Agenda for Action .
Achieving these goals of NCTM is accomplished by the problem-solving approach described in these web pages and the SureMath software program.
NCTM's recognition of the importance of computers in teaching mathematics is stated in these two short quotes, also from the NCTM Standards:
"Computers are powerful problem-solving tools. The power to compute rapidly, to graph a relationship instantly, and to systematically change one variable and observe what happens to other related variables can help students become independent doers of mathematics."
[Computers and other technologies] "free students from tedious computations and allow them to concentrate on problem solving and other important content. ...As paper-and-pencil computation becomes less important, the skills and understanding required to make proficient use of calculators and computers become more important."
The problem-solving principles presented in these pages have solved the problem of solving word problems. The SureMath software facilitates handling the mathematical functions. The combination empowers the student to discover the vital interrelationships of their academic subjects.
All examples in these web pages were solved using SureMath .
Problem solving skills developed at early grade levels will be valuable at later grade levels and in all subjects using mathematical methods for problem solving.
With problem-solving principles already understood, more in-depth exploration of later subject matter and more advanced mathematics is possible without spending additional time and resources on teaching problem solving methods limited to each subject.
Problem solving can be integrated across the curriculum in all subjects that use mathematics for solving problems . This is of particular advantage in content-oriented subjects. Students learn to use what they have learned in mathematics classes. The words change, the symbols change, but the problem solving process remains the same.
|A primary objective is the development of creative thinking skills. Creative accomplishment in any endeavor requires skill in using the appropriate tools. The tools for reliable problem solving are described in detail in these web pages. Once these tools are learned problem solving can be applied creatively in a vast array of situations. It is not simply a matter of learning problem solving for the sake of solving problems. Instead it empowers the student to think creatively. It makes it possible for student to move from using arithmetic to solve problems to using mathematics to solve problems. It extends the student's reach.|
There is a common denominator for solving all problems, resulting from the definition of a problem:
In typical end-of-chapter problems the outcome requested and the conditions are well defined. In so-called real life problems the conditions can be a mixture of well defined and poorly defined conditions. Some conditions may not even be known and the outcome requested may be inadequately defined.
For poorly defined problems and conditions, a situation that sometimes exists in real-life problems, the process is the same. In this case one becomes aware of need for more precise specification of the problem, the outcome and the conditions. This often calls strongly on use of general knowledge, inventiveness and originality to produce an outcome that satisfies the given conditions and implied conditions. It is a discovery process within a guiding framework of the logical problem solving process described in these web pages. Indeed one may discover that the problem and outcome differ from that previously supposed. Systematic problem exploration and solving is a powerful process.
Numerous examples showing the commonalty of problem-solving structures in different situations are shown in these web pages. Students have the opportunity to discover how mathematics and subject matter fit together.
For Windows users, NuCalc 2.0 (The Graphing Calculator for Windows) includes capabilities similar to SureMath.