Adapting Texts for Quality Use

I read a something (in Advanced Web-Based Training Strategies by Driscoll & Carliner) that caught my eye. The authors were discussing the problems associated with using off-the-shelf content as a broad solution to a training problem. They gave an example of a company that said they had to modify 75% of the off-the-shelf e-learning courses because the content would not work for their staff.

What about with teaching math in the classroom? I am currently using textbooks that are several years old. I assigned one of the story problems from the textbook to my students the other day. They were supposed to set up an equation that modeled the situation described. They struggled mightily. I changed the context to something that they were more familiar with, and they were immediately able to set up an equation to model the situation. The only reason they had been struggling with it in the first place was that the context was unfamiliar to them. And now the connection to the dilemma posed by Driscoll & Carliner.

Should I be adjusting about 75% of the textbook to make it work better with my students?

I think so.

Am I doing this?

No.

So what is going on? I know what I should be doing but I am not doing it as much as I should be. Jordan School District does great things with their math teacher trainings. We have the opportunity to collaborate and a very good structure in place to support this collaboration. I think there needs to be more, though. The vast majority of coordinated training efforts are done during the school year. The vast majority of the teachers have the summer off, at least from teaching. I propose allowing teachers the opportunity to spend their summers doing what the rest of the world does during their summer, namely working.

I realize some teachers may not take anyone up on the offer to work more, but I think that a lot of the young teachers will. We (the young) are in a position where we have to take second jobs to be able to afford necessities, such as a home, anyway. The low pay is what keeps a lot of qualified and quality teachers away from the profession. I often hear things like 'you don't get paid, but you get the summers off'. I may get the summer off from teaching, but it only because I have not yet found the opportunity to get a full-time summer job doing what it is that I am trained to do, namely teaching math or improving my ability to do so.

Are Common Assessments Valuable?

As we prepared for the start of the new year, my colleagues and I got into a discussion about common assessments and their value.

Our math department does quite a good job collaborating and using common assessments, and we were lauded as a good example by our administrators and others during the school-wide training meetings that happen the week before students come back. This was fine, but then we were informed that we did not meet AYP due to our math scores. In fact, we had a 47% pass rate for 2006 and a 34% pass rate for 2007. If I understand correctly, we were supposed to have improved by at least 10%, not dropped by 13%.

This was the news that prompted a few of us to start talking about what to do and how the results of the common assessments should be used. One fellow math teacher pointed out that as long as we are giving common assessments we should also be using the results of these assessments to guide our instruction. As we discussed the different approaches we are taking in our Algebra 1 classes she mentioned that having a common assessment may prove more valuable now that different approaches are being taken, as long as the results are being reviewed. If it turns out that one class does much better on the common assessments then we should consider the approach that teacher and class took, and the rest of us should adjust our teaching accordingly.

I am excited to see what happens. Last year we gave common assessments and, once they were given, we decided that the majority of students were doing poorly. Those of us teaching Algebra 1 were taking the same approach to teaching and getting the same results. We did nothing to change what we were doing, we just kept going because that was the way it worked. The common assessments didn't do much for us because we didn't do anything to respond to the results they showed. This year we will be looking at the results and seeing what was happening that made the difference between the successful classes and the less successful classes.

What is a 'Teacher'?

I was reading a little bit in 'E-Learning by Design' by William Horton, and came across a question that made me wonder about my role as a teacher. The question was one of a list meant to help the reader decide whether to skip the first chapter. The question reads, in part, "Do you lack either education and experience in instructional design? Perhaps you are a subject-matter expert or instructor..." So which am I, as a teacher? Am I a subject-matter expert? An instructor? An instructional designer? What does it mean to be any of these? Does an instructor have to be a subject-matter expert and/or an instructional designer?

My initial answers to these questions is that each of these are separate things. To be one is not necessarily to be the other. College professors, for example, may be subject-matter experts in their chosen field, but may not be able to instruct others in similar pursuits, and may not be able to design quality instruction. Is there value in becoming a professional with training in all of these areas? If one did obtain such training, would they be able to more effectively help students learn? Should all teachers receive specific training in each of these areas, or should teachers team up based on their strengths and have specific tasks assigned them? Should there be teachers working behind the scenes to create lessons for those who are actually in the classroom?

Chinese Philosophy and Math

I have been reading Fung Yu-Lan's book 'A Short History of Chinese Philosophy'. In the introductory chapter the author is talking about the spirit of Chinese philosophy. In the latter portions of the chapter he addresses why it is that Chinese philosophy is often written in brief and disconnected ways when compared with the Western philosophy and its more articulate style. In Chinese philosophy aphorisms, allusions, and illustrations are used to create this brief and disconnected style. Their use leads to suggestiveness, and Fung points out that this suggestiveness is less limiting than a more articulate approach. There are ideas that are suggested and understood that could not be articulated well without limiting the scope of thought.

It seems to me that the traditional methods of teaching math, namely memorization, drill, and practice, are more articulate (there is a certain way to solve this type of problem, and that method should be used to finish some set of exercises). Once a student has completed the exercises they should have a grasp on the mathematics.

A problem that is to be used to teach mathematics cannot be this articulate. There needs to be more suggestiveness, thus allowing the students to use whatever mathematically accurate approaches they see fit. Having a classroom discussion on the ideas that come from the various solutions will help the students define certain mathematical concepts in ways that will be beneficial to themselves.

Fung quotes a passage from the Chuang-tzu, a significant book of philosophy, which may help distinguish between ideas suggested and methods articulated, and how ideas suggested can be the more powerful of the two.

"A basket-trap is for catching fish, but when one has got the fish one need think no more about the basket. A foot-trap is for catching hares; but when one has got the hare, one need think no more about the trap. Words are for holding ideas, but when one has got the idea, one need no longer think about the words." (pg 12)

Later on Fung says that 'words are something that should be forgotten when they have achieved their purpose. Why should we trouble ourselves with them any more than is necessary?' (pg 13) It seems to me that the traditional methods of teaching math are having our students deal with the words. The formulas they memorize and the specifically formatted equations they learn to solve are the words used to describe the ideas of mathematics. Our students, however, don't get to experience the ideas of the mathematics because we are spending so much time focusing on the words.

The last paragraph of the first chapter of Fung's book says the following.

"Kumarajiva, of the fifth century A.D., one of the greatest translators of the Buddhist texts into Chinese, said that the work of translation is just like chewing food that is to be fed to others. If one cannot chew the food oneself, one has to be given food that has already been chewed. After such an operation, however, the food is bound to be poorer in taste and flavor than the original."

Do we assume our students are not capable of chewing the food of mathematical ideas? Are we continually chewing this food for them? I think we are. It is no wonder that our students are not enjoying math classes as much as we would like them to, or even as much as they would if we would let them.

More Thoughts on Problem Solving

We have finished up our second group project focused on using problem solving to teach mathematics. As we wound it up we were presented with a couple of thoughts that, I think, show us where we are today in our understanding of such a task.

The first thought was that, in 1980, NCTM released a document titled 'An Agenda for Action: Recommendations for School Mathematics of the 1980s'. Akihiko Takahashi, our instructor, discussed with us a couple of the phrases from that document on how problem solving should be used in a mathematics classroom. He then noted that a major reform in Japanese mathematics teaching took place as a result of this document. They, the Japanese, took what was said and began implementing it. Today they are seen as leaders in the teaching of mathematics in their k-12 schools. What about those of us in the U.S? Where are we? It seems that we missed out on the general ideas of this document. It was our organization that crafted this document, but somehow, it seems, it did not lead to widespread change in the teaching of mathematics. I may be seeing it wrong, but from what I have read, studied, and experienced, I think I am right in my assessment.

The second thought I had relates to teachers working together to create problems and lessons that will help students learn mathematics through problem solving. As we struggled with each other, or more accurately against each other, while trying to put two such lessons together in two weeks, there were a lot of groups that struggled to work well together. We were told that we needed to spend more time learning each other, rather than trying to teach each other. Apparently we were busy trying to tell each other the way it should be and why other ideas would not work.

The last thought I had came as we discussed why we had been given such a task. Aki reminded us that we are often bombarded with great activities and problems at conferences, but never take the time to figure out what mathematics the students could learn from such an activity. We simply go and use the activity as is, and as a result we often have a less than desired outcome. In short, we need practice turning good problems into tasks where students learn mathematics.

Finally, although at times I was frustrated with these tasks and with the dynamics of the groups I was working with, I did learn a few things that I would like to implement in my teaching. I do think that mathematics as is commonly taught is more boring than it should be. I hope to figure out some of this stuff in time to try a few things in the classroom this year, and then record what happened and use that to improve in the coming years.

Video Recorder In The Classroom?

I am for it. A few schools in the Seattle are participating in what they call a Video Club. Several teachers get together once a month for a couple of hours to discuss a 5-7 minute video clip from one of their classrooms.

The main focus of the meetings is the mathematics that the students are doing. The first topic of discussion is the math of the problem. Once the problem has been discussed they move on to possible misunderstandings the students could have. Once this has been done they watch the video. Then comes discussion of the what evidence the video provided about what the students understood about the math. Finally there is some discussion about questions that are still open.

All of the teachers said that they have gained insight into their own teaching as they have participated in the video club. It makes sense. Coaches at all levels watch video of their athletes in order to understand how best to help them. There are things you see in a video that are much harder to see in real time. If this practice has helped so many coaches to help athletes over the years, it should also help us help our students each year.

Mathematical Eye?

Today we were told about the early morning fish markets in Japan. Those bidding in the tuna auction have to decide what is on the inside of the tuna before it is cut open. All they have to go by is as cross section where the tail is cut off. Those with a good tuna eye are able to pick the good fish and so end up making more money. Those without a good eye for what is on the inside don't get the money.

What's this got to do with math education? It goes back to the idea of using problem solving to teach mathematics. In order to use a problem in this way you must be able to see what math can be learned, not just what math can be used to solve it. Our ability to distinguish between these two ideas will determine our success when using problem solving in such a powerful way. This ability is a talent. For the sake of my students I hope this talent can be learned.