Problem+Solving+Resources+and+Links





Problem Solving
From: http://standards.nctm.org/document/chapter3/prob.htm

Instructional programs from prekindergarten through grade 12 should enable all students to--
• build new mathematical knowledge through problem solving; • solve problems that arise in mathematics and in other contexts; • apply and adapt a variety of appropriate strategies to solve problems; • monitor and reflect on the process of mathematical problem solving. Problem solving is an integral part of all mathematics learning. In everyday life and in the workplace, being able to solve problems can lead to great advantages. However, solving problems is not only a goal of learning mathematics but also a major means of doing so. Problem solving should not be an isolated part of the curriculum but should involve all Content Standards. Problem solving means engaging in a task for which the solution is not known in advance. Good problem solvers have a "mathematical disposition"--they analyze situations carefully in mathematical terms and naturally come to pose problems based on situations they see. For example, a young child might wonder, How long would it take to count to a million? Good problems give students the chance to solidify and extend their knowledge and to stimulate new learning. Most mathematical concepts can be introduced through problems based on familiar experiences coming from students' lives or from mathematical contexts. For example, middle-grades students might investigate which of several recipes for punch giving various amounts of water and juice is "fruitier." As students try different ideas, the teacher can help them to converge on using proportions, thus providing a meaningful introduction to a difficult concept. Students need to develop a range of strategies for solving problems, such as using diagrams, looking for patterns, or trying special values or cases. These strategies need instructional attention if students are to learn them. However, exposure to problem-solving strategies should be embedded across the curriculum. Students also need to learn to monitor and adjust the strategies they are using as they solve a problem. Teachers play an important role in developing students' problem-solving dispositions. They must choose problems that engage students. They need to create an environment that encourages students to explore, take risks, share failures and successes, and question one another. In such supportive environments, students develop the confidence they need to explore problems and the ability to make adjustments in their problem-solving strategies.



Problem Solving
From: http://en.wikipedia.org/wiki/Problem_solving

Problem solving forms part of thinking. Considered the most complex of all intellectual functions, problem solving has been defined as higher-order cognitive process that requires the modulation and control of more routine or fundamental skills (Goldstein & Levin, 1987). It occurs if an organism or an artificial intelligence system does not know how to proceed from a given state to a desired goal state. It is part of the larger problem process that includes problem finding and problem shaping.



Creative problem solving
From: http://en.wikipedia.org/wiki/Creative_problem_solvingia

Creative problem solving is the mental process of creating a solution to a problem. It is a special form of problem solving in which the solution is independently created rather than learned with assistance. Creative problem solving always involves creativity. However, creativity often does not involve creative problem solving, especially in fields such as music, poetry, and art. Creativity requires newness or novelty as a characteristic of what is created, but creativity does not necessarily imply that what is created has value or is appreciated by other people. To qualify as creative problem solving the solution must either have value, clearly solve the stated problem, or be appreciated by someone for whom the situation improves.[1] The situation prior to the solution does not need to be labeled as a problem. Alternate labels include a challenge, an opportunity, an improvable situation, or a situation in which there is room for improvement.[2] Solving school-assigned homework problems does not usually involve creative problem solving because such problems typically have well-known solutions.[3] If a created solution becomes widely used, the solution becomes an innovation and the word innovation also refers to the process of creating that innovation. A widespread and long-lived innovation typically becomes a new tradition. "All innovations [begin] as creative solutions, but not all creative solutions become innovations."[4] Some innovations also qualify as inventions.[5] Inventing is a special kind of creative problem solving in which the created solution qualifies as an invention because it is a useful new object, substance, process, software, or other kind of marketable entity.[6]

Some problem-solving techniques
From: http://en.wikipedia.org/wiki/Problem_solving

There are many approaches to problem solving, depending on the nature of the problem and the people involved in the problem. The more traditional, rational approach is typically used and involves, e.g., clarifying description of the problem, analyzing causes, identifying alternatives, assessing each alternative, choosing one, implementing it, and evaluating whether the problem was solved or not. Another, more state-of-the-art approach is appreciative inquiry. That approach asserts that "problems" are often the result of our own perspectives on a phenomenon, e.g., if we look at it as a "problem," then it will become one and we'll probably get very stuck on the "problem." Appreciative inquiry includes identification of our best times about the situation in the past, wishing and thinking about what worked best then, visioning what we want in the future, and building from our strengths to work toward our vision. [2] 1. divide and conquer: break down a large, complex problem into smaller, solvable problems. 2. Hill-climbing strategy, (or - rephrased - gradient descent/ascent, difference reduction) - attempting at every step to move closer to the goal situation. The problem with this approach is that many challenges require that you seem to move away from the goal state in order to clearly see the solution. 3. Means-end analysis, more effective than hill-climbing, requires the setting of subgoals based on the process of getting from the initial state to the goal state when solving a problem. 4. Trial-and-error (also called guess and check) 5. Brainstorming 6. Morphological analysis 7. Method of focal objects 8. Lateral thinking 9. George Pólya's techniques in How to Solve It 10. Research: study what others have written about the problem (and related problems). Maybe there's already a solution? 11. Assumption reversal (write down your assumptions about the problem, and then reverse them all) 12. Analogy: has a similar problem (possibly in a different field) been solved before? 13. Hypothesis testing: assuming a possible explanation to the problem and trying to prove the assumption. 14. Constraint examination: are you assuming a constraint which doesn't really exist? 15. Incubation: input the details of a problem into your mind, then stop focusing on it. The subconscious mind will continue to work on the problem, and the solution might just "pop up" while you are doing something else 16. Build (or write) one or more abstract models of the problem 17. Try to prove that the problem cannot be solved. Where the proof breaks down can be your starting point for resolving it 18. Get help from friends or online problem solving community (e.g. 3form, InnoCentive) 19. delegation: delegating the problem to others. 20. Root Cause Analysis 21. Working Backwards (Halpern,2002) 22. Forward-Looking Strategy (Halpern, 2002) 23. Simplification (Halpern, 2002) 24. Generalization (Halpern, 2002) 25. Specialization (Halpern, 2002) 26. Random Search (Halpern, 2002) 27. Split-Half Method (Halpern,2002) 28. The GROW model