It is about time for most active teachers to return to their classroom. While I am retired, the beginning of the school year still holds a special place in my routine. My schedule will change soon as tutoring requests pick up, workshops and math teacher meetings increase in regularity, and I start to get more questions about this and that.

Still, the first day is the most important day of the year, and I miss it. I always worked hard to have an interesting lesson that would surprise my students, excite them about the possibilities for the coming year, and give me a chance to meet them. I did not go over rules nor did I have non math getting to know you activities. We did math. I wanted to set the stage for them. It worked.

Still, the first day is the most important day of the year, and I miss it. I always worked hard to have an interesting lesson that would surprise my students, excite them about the possibilities for the coming year, and give me a chance to meet them. I did not go over rules nor did I have non math getting to know you activities. We did math. I wanted to set the stage for them. It worked.

I made sure the first lesson and every one thereafter was based on what I consider to be my most important in sight about how mathematics is learned and what I think teachers should be doing in their classroom.

We must cover content. We must teach students how to logically approach problems that they have not seen before. If we can also get students interested in math, then we are successful.

It took about ten years for me to figure this out, but the these tasks end up being connected and the outcome often leads to excitement and interest. The idea is to have students solve problems that lead to an understanding of the concepts and content we want them to master. Here is how I was able to manage this.

I gave students a problem to work on. It was one problem and everyone was expected to work on it. Often it was related to the work they did yesterday and flowed into the concepts I wanted them to learn today. It was one problem, one they could all get started on but perhaps could not solve, hence the phrase, work on instead of solve.

As they worked on it, I walked around and listened and watched. I did not help anyone but I paid careful attention to their progress. I tried to give them a problem they has not seen before. They understood that their task was to figure out what to do.

I encouraged them to work alone at first and then discuss their thoughts with neighbors. Sometimes they did arrive at a solution, sometimes they did discover an important idea, and those were exciting moments for all of us. I knew exactly which students had figured it out because I was watching and listening while they did.

Sometimes they did not see the breakthrough idea and could not solve the problem. When I observed that they were about to go off task, I might give a hint, or I might have a student who was on they way suggest a strategy, or I might even demonstrate a procedure for solving the problem. However it worked, I had their attention because they had worked on the problem and wanted to know what they might have done to solve it. contrast this with the traditional method of showing students how to do a problem, when they haven't worked on it. They probably don't even recognize the key idea because they never had a vested interest in solving the problem in the first place. When I start to explain, they want me to explain because they have exhausted their resources.

I believe one of the worst habits teachers have acquired is the belief that they have to explain something before a students can try to solve it. Another bad habit is to give students a worksheet with several problems on it. This encourages students to write down answers rather than ponder the problem at hand. They lose focus and really don't want to discuss the problem as much as they want to finish the worksheet.

It takes a lot of work to get students used to this style. they would complain that I wasn't explaining things. They were frustrated. They were confused about what they were expected to do. But I am stubborn and persistent and eventually there would come a day when I would tell them it was time to discuss the problem, and some of them would say, "No, wait. We almost have it." Then I knew that I had converted them from passive note takers into active problem solvers. I would just stand there and smile and tell them I guessed they could have a few more minutes.

One good problem, let them work on it, discuss it, consider alternate solutions and common errors, then move on to the next problem.

If this is an advanced class, there might only be two or three good problems in a given class period. In a more ordinary class, there might be fifteen in a class period, each a challenge for the students, each within reach.

Not only did it work, but teaching this way is great fun, because there is another unexpected part. Sometimes students think of an approach that I had not considered and I learn to become a better problem solver from them.

Have a wonderful journey with your students.