An algorithm is a precise systematic method for solving a class of problems. It takes input, follows a set of rules and gives output that provides a conclusive answer. The accuracy of the answer depends on how expertly one applies the specific techniques or algorithms. Students practice algorithms using exercises from a textbook.
Problem solving is the process of applying previously learned rules to a situation which for the student is new and different. The common view in high school is that problem solving is a word problem. But problem solving is not just a word problem. Math does involve the translation of words into symbols and symbols into words, but if you have just taught students how to work word problem, that word problem is just an exercise. Problem solving must involve for the student something new and different never before encountered by the student. It requires the selection of a technique among various approaches.
Math has been developed to solve problems. We must teach students, not just how to use the tools they have been trained to do using exercises, but how to think and how to approach a never before encountered challenge. By practicing the student will learn different ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations. We must provide students with the opportunity to practice solving problems never before encountered. Math is the perfect subject with which we can teach strategies for solving problems such as drawing a diagram, working in reverse, generalizing a discovered pattern, adopting a alternate perspective, solving a simpler problem first and then generalizing, testing extreme cases to gain insight, etc. Students who learn to how to solve problems will be trained in methods of reasoning and will be ready to tackle problem encountered in real life.
I used to teach math classes with three levels of thinking:
1. learn the terms and definitions,
2. apply techniques taught to solve equations, and
3. learn to solve problems they had never before encountered.
I would hand out problems that could be solved using the techniques illustrated in class but that required various approaches that had not been discussed. Students would get excited trying to solve the problems, then frustrated and discouraged because they were not used to having to think. One Algebra 2 student grappled with a particularly difficult problem for the whole period and got nowhere. She used the white board drawing pictures and equations. She struggled and struggled with different approaches. At the end of the class she asked me how to solve the problem. I responded that I didn't know. She got angry and said she had wasted her time by working all period and failing to solve the problem. I pointed out the contrary, that she had learned how to think, how to approach the problem from different perspectives and how to struggle and not give up. Learning how to think is more important, and more fun, than getting the right answer.