My unsolved problem is the Circular Window Problem (click here for the description). I have yet to consider the hint given on the web page. However, I feel that I am close. The only difficulty is making the three small circles tangent to one another!
The main goal is to find the centers of the the inscribed circles. I have tried several sequences of constructions but have been unable to find a solution. My difficulty stems from my lack of experience with problem solving in geometry. I know many theorems, constructions, and axioms but these these do not help if I haven't learned how to fit these together through practice. I know that I can solve this problem by looking at the hint provided on the web page, but I would rather work with someone to figure out the solution. In this way, I'll remember the ideas and connections better because someone has helped show how they approach the problem. Thus, I have not learned much yet, but I haven't stopped my attempt to find a solution.
I have a problem with the above proposition. In reading between the lines, this statement is an either/or proposition. That is, we can either have problem solving as a central part of secondary school mathematics or we can relegate problem solving to the periphery of the curriculum. The way the statement is proposed, it seems that there is no middle ground. Thus, with teachers under the expectation (whether theirs or someone else's) to cover a given amount of material, problem solving will take a back seat in favor of the traditional, mathematical concepts.
I will argue a position against the above proposition. One of the first advantages of having problem solving as a part of the mathematics curriculum is something that I have noticed in my approach to EMT725. My background in calculus and algebra is much stronger than my background in geometry. Thus, my first instinct is to find some function to fit each problem. However, I have noticed that this is far from efficient for many of the problems posed for this course. There are many different and important tools for students to use in attempting to solve mathematical problems. Thus, through problem solving, it is possible for students to see the connection of the various branches of mathematics. Whereas in a typical mathematics textbook or classroom, the exercises and mathematics concepts are presented by repetition of a certain sequence of steps to get students learn a specific concept. This type of work hides the 'truer ' nature of mathematically discovery. Through a curriculum that provides access to problem solving ideas, students will gain skills that will enable him/herself to approach a variety of different problems. The students will be less intimidated by a problem, mathematical or otherwise. They will be able to 'attack' the problem using various strategies instead of the prescribed set of steps in order to be successful. Lastly, problem solving due to its diverse methods, its interesting problems, the possible extensions can lead to intrinsic motivation in students. The issues discussed above have been considered important to the mathematics education community (i.e., they are included in adopted standards).
The first disadvantage of giving problem solving a role in a mathematics classroom is that the teacher will not be able to cover all the curriculum and thus the teacher may be considered as having not prepared the students. The common argument goes that the students need to have certain mathematical knowledge when they take standardized tests such as the ACT or SAT. Due to the importance placed on these tests, students have the opportunity to enroll in one of many prep courses available to help increase their scores. However, these courses teach not concepts but strategies in approaching the problems. Dr. Wilson even presented in class some classic problems that were once on a test of this type. These strategies would be strengthened by having problem solving within the curriculum. It would be interesting to perform a study of how knowledge of problem solving affects one's score on these standardized tests. Secondly, teachers may not be comfortable with methods to 'teach' problem solving to their students or they may feel that they're not prepared enough to sort through their math skills to find an answer to a problem. Many teachers who may not be comfortable with their mathematics skills will probably teach from the textbook. To teach problem solving, one needs a great deal of time to prepare for the lesson and try to imagine the various lines of reasoning and questioning that students will pursue. It seems this would not be an issue if the teachers were better prepared for teaching problem solving. Lastly, it is more difficult to show outcomes within the classroom. This is a concern for politicians, administrators, school boards, parents and thus teachers. But they can be shown the value of problem solving over the traditional method. We must involve them in the classroom to help them understand the benefits of problem solving. They must be shown that even though a student may not know a specific concept, s/he may still be able to solve this and many other types of problems using various methods.
Thus, I feel the benefits of including problem solving in the curriculum outweigh the disadvantages. The drawbacks that I have discussed are the ones that I have heard throughout my career. These are also easily remedied with a small amount of effort. I feel more confident as a math educator with skills in problem solving.