Mathematical Problem Solving

Solving problems is an important role in mathematics. Mathematics teaching and learning is aimed at developing children's ability to solve various mathematics problems.

There are various techniques for solving mathematical problems and their application vary from one mathematical problem to the other. The key to successfully solving mathematical problems is flexibility and creativity. One should be able to change methodology if it is not effective for a mathematical problem.

Brainstorming techniques are quite useful in mathematical problem solving. Important information relevant for finding a solution to a problem has to be separated from the rest. Knowledge of techniques in algebra and geometry must be gained, as well as acquaintance with statistical ideas, logic and logical concepts, number theory and arithmetic basic operations. A deep understanding of binary number systems, counting techniques, linear programming techniques and matrix algebra are also crucial for solving mathematical problems. Important techniques are also probability laws, calculus ideas such as differentiation, integrals and different equation solving techniques, complex numbers, probability distributions and solid geometrical concepts and ideas.

After gaining knowledge, suitable approaches to problem solving must be selected, along with techniques which can help solve the problem fast and save time. Different techniques have not the same effectiveness and one must find the most effective. The intensive practice of problem solving makes a person able to quickly choose the technique in case of an examination. The exercises speed up the cognitive retrieval and one is able to quickly choose the most suitable technique.

The ability to solve mathematical problems can be developed only though the continual practice of problem solving. Some textbooks in the United States offer linear models for solving mathematical problems. These models involve reading, deciding, solving and examining the result. However, such models are often inconsistent and cannot be practically used. Linear models do not encourage students to think. Such traditional models present problem solving as a series of steps and a process that has to be remembered. One has to be able to pose or formulate a problem in order to solve it. This area has been subject to studies in the United States in recent years.

Garofola and Lester suggest in their article, Metacognition, cognitive monitoring, and mathematical performance,that problem solving is a series of processes of which students are unaware. Students with a good knowledge base are more able to use heuristics in solving geometry problems. The successful problem solvers are using mathematical structure similarities to categorize math problems.

Algorithms are important in solving math problems and they must be developed. However, using an algorithm is still not problem solving. The process of creating an algorithm is problem solving. Problem solving can be taught by teaching children to create their algorithms.

A useful method in mathematics is also heuristics. These are types of information that help students find a solution to a math problem. Heuristics are similar to strategies, techniques, and rules-of-thumb. These techniques are not very valuable out of the problem context, but in doing mathematics they are very helpful. Heuristics are considered very important in various theories in mathematics. However, despite the use of explicit instructions and heuristics, problem solving is not that simple as such a simple analysis is limited. Theories must include classroom contexts, past knowledge and experience. Several studies have been focused on the research of heuristic processes. These studies found that task specific heuristic instruction was more effective than general heuristic instruction. The heuristic of subgoal generation also enables students to make problem solving plans. Thinking aloud, acting as a teacher and direct instruction enhanced the development of students' abilities to generate subgoals.

In the past, problem solving research involving technology has often dealt with programming as a major focus. This research has often provided inconclusive results. Indeed, the development of a computer program to perform a mathematical task can be a challenging mathematical problem and can enhance the programmer's understanding of the mathematics being used. Too often, however, the focus is on programming skills rather than on using programming to solve mathematics problems. There is a place for programming within mathematics study, but the focus ought to be on the mathematics problems and the use of the computer as a tool for mathematics problem solving. Further basic mathematical problem solving techniques are the use of graphs, patterns and logic concepts. Counting strategies such as tree diagram, binomial expansion and pascal triangle can also enhance problem solving. Trigonometry is used for finding relationships between angles and sides, along with properties waves and simple harmonic motion, as well as projectile motion and circular motion and elliptical motions.

Mathematical Problem Solving: Selected full-text books and articles

Mathematical Thinking and Problem Solving By Alan H. Schoenfeld Lawrence Erlbaum Associates, 1994
Toward a Unified Theory of Problem Solving: Views from the Content Domains By Mike U. Smith Lawrence Erlbaum Associates, 1991
Librarian's tip: Chap. 5 "A View of Mathematical Problem Solving in School"
Problem Solving and Comprehension By Arthur Whimbey; Jack Lochhead Lawrence Erlbaum Associates, 1999 (6th edition)
Librarian's tip: Chap. 11 "Solving Mathematical Word Problems"
Mathematics Education: Models and Processes By Lyn D. English; Graeme S. Halford Lawrence Erlbaum Associates, 1995
Librarian's tip: Chap. 8 "Problem Solving, Problem Posing, and Mathematical Thinking"
Fundamental Constructs in Mathematics Education By Sue Johnston-Wilder; John Mason RoutledgeFalmer, 2004
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