Polya: Part 1- In the Classroom

First of all, I must say I was not looking forward to reading this book. Just glancing through it, I was dreading how boring it was going to be. That being said, I really enjoyed this first part. I really feel that Polya has some great ideas and delivers it in a way that makes sense and isn't painful at all to read. I was actually a little bummed when I finished it!!

Through out this section, many things stood out to me: things that happen constantly in a math class that could be changed to help our students better solve "problems" and things that I never really thought of before. This blog is going to be a list of those things:

• Students should have a reasonable share of the work (pg. 1). I think often times as teachers we give students work, if it is too hard, we tend to completely dumb it down giving students not necessarily the answers, but pretty darn close. If the work is too easy to begin with, the students might complete it without help, but what's the point? they we not challenged and chances are had very little to think about.
• Purpose #4: Common Sense (pg. 3) Some times I feel like my students have none when it comes to math!! The suggestion of "Look at the unknown! And try to think of a familiar problem..." is so foreign to my students. They cannot take one problem and see connections between it and others and I think part of that is our fault as teachers (will revisit later in this post)
• Solving problems is a practical skill (pg. 4) To develop a skill takes practice and imitation. Our students learn to solve problems by doing many of them. I think a lot of times in math class we beat "kill and drill" type problems to death and every once and a while give them a story problem or application to work on just expecting them to know how to approach them. We need to condition our students and constantly give them practice in order to succeed.
• #6. Four Phases of Problem solving (pg. 5-6) Understand the problem, find connections, carry out our plan, review and discuss the solution. So important!! I am making this a list and hanging it up in my classroom. When we put a problem in front of our students and say solve this, most struggle to begin. Having a list of what to do can be so helpful to them.
• #7. Understanding the Problem (pg. 6-7) "It is foolish to answer a question you do not understand" yet another thing that will go up in my classroom this week! I know that I always try to get my students to answer questions like "What is the end goal?", "What does 'solve' mean?" or "How do we know we are done?", but I feel that they do not ever ask these questions of themselves. I feel that they get into robot-mode in math class: see a problem, attempt it with not goal or agenda in mind (basically regurgitate memorized information trying to copy what I just did).
• Good Ideas (pg. 9) "Mere remembering is not enough for a good idea, but we cannot have any good idea without recollecting some pertinent facts..." My students honestly think that math problems can only be solved by remembering steps about how to do each and every single one. They have a very hard time connecting bits and pieces to individual problems and I know I am guilty of perpetuating this. If students have a question about a problem, I tend to help them solve that problem and not really give them any tools to solve something similar.
• Forgetting "The Plan" (pg. 13) Like the last paragraph stated my students try to remember the plan for each specific problem and often times it is forgotten. Polya states that this tends to happen "if the student received his plan from outside, and accepted it on the authority of the teacher". I think this happens even to my students who perform well in classes. They are simply better at remembering the plan! We need to develop ways for students to find this plan on their own, and to make it theirs so that it can be applied to different problems and later on down the road.
• Accidental Questions (pg. 14) "...there is some danger that the answer to an incidental questions may become the main difficulty for the majority of the students". As teachers we need to be careful of the questions we ask. I have run into a situation where in trying to clarify or get my students thinking I have ended up thoroughly confusing my students.
• Good vs. Bad Questions (pg. 21) "The suggestions (we give our students) must be simple and natural because otherwise they cannot be unobtrusive". I think that 'obtrusive' questions are the norm in most math classes. We tend to led our students into answering a specific questions about a specific problem to get that specific answer. The students finish that problem, move on to the next and have the same difficulty with that one. Our questions should led our students into thinking through a problem on their own, guiding them into the right path.