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As advertised, here is a post about Conversations from Algebra 1 and Research and Teaching.

There are several reasons this post has come a little later than anticipated. One of them is that I had a job change, some would even call it a career change, shortly after my last post. I am now working as an Instructional Designer for a company in Salt Lake City, Utah. Maybe someday I will post about why this change occurred, but not today.

CONVERSATIONS FROM ALGEBRA 1
During the two quarters I spent teaching this school year I worked with my Algebra 1 students on some problem sets obtained from the Philips Exeter Academy, a private school in New Jersey. The problem sets are designed to engage students in problem solving, not simply computing answers using methods shown by the teacher at the start of the class period. I was not sure how the students (or parents) would respond to this strategic shift. I did feel that it would be possible to get the students engaged and interested, and expected that I would have to encourage and prod to get some of them to work. All in all things turned out well.

Let me explain a little about the daily classroom procedures we followed. I would pass out a set of problems at the beginning of class and students would work on them. Problem sets varied in length from 7 to 12 problems. Almost all of the problems were what are generally referred to as 'story problems' among students and teachers. Students would work in groups or by themselves, whichever they preferred (some were initially opposed to talking to other humans, so they chose to work alone, though once they saw that everyone was struggling with the problems they quickly joined their peers). As the students worked on the problems I would stop by each group and ask more questions, get them to share their thoughts and ideas, and help foster discussion amongst themselves. I would then leave them alone to figure out a solution to the problem. Every fifteen or twenty minutes I would have students go to the board and share what they were thinking about a particular problem that was stumping the class. If the class had successfully solved a problem, I would ask several students to go to the board and show their solution. I would usually have two or three different students present solutions, and each would be unique. We would then talk as a class about why the presenters did the things they did. If a student made a comment that was inappropriate I would tell them that it was an inappropriate comment for the setting we were in. They quickly learned how to ask questions and challenge each other in a respectful manner. Students rarely got through more than five problems during our class time (89 minute class periods every other day).

One day we got into a discussion about positive numbers, negative numbers, and the number zero. The discussion may have been about opposites, it may have been about positive and negative numbers, I don't remember exactly what, but the students began asking each other questions and explaining why they felt zero was positive or negative or not a number at all. It was amazing to hear them carrying on a respectful, mathematics based conversation.

Several of the students in the class were students that did not get along well with other teachers. I taught in a high school, and many of the students were not academic over-achievers. They had learned to respect each other and to share their thoughts. I guess I was so impressed because it gave me a glimpse of what I think a mathematics classroom should be: a place where students are engaging with each other and with mathematics.

RESEARCH AND TEACHING
I think the teaching community has a long way to go before we can be called professionals. This is a strong statement, but I think it is an accurate one. I have heard recently the comments of professional athletes regarding their preparation for games. They talk about watching film, about studying other teams' players and plays. They use the latest research on training and health to keep their bodies in top physical condition. They are dedicated to improving themselves, and the better they get the more entertained the crowds that watch them, and ultimately pay them, are.

What about in teaching? Do we ever watch film of ourselves? Do we ever watch film of others teaching the lessons we teach? Do we know if the lessons we teach are effectively teaching the topics we think they are teaching? Do we know how the students see us, and more importantly, how they see the math?

There are some good things happening in math education. There are teachers and leaders who are working to create a quality environment so that their students can learn mathematics. But there is room for improvement.

I once sat in a meeting with those considered to be the leaders among math teachers in the school district where I taught. We were discussing ways to improve math instruction. An idea was presented that was in line with recent research on math education, and one of the veteran teachers in the group, remember this is a group of the leaders in the math teaching community of one of the largest school districts in the state of Utah, said something along the lines of "New teachers have too many other things to worry about." What other things was she talking about? Things like taking roll. Things like learning how to keep a classroom full of students busy so they didn't cause trouble. These are important things to learn as a teacher, but why not learn how to do them by becoming better at helping the students interact with the mathematics? Why not make mathematics education our primary concern, and figure out ways to use the mathematics to accomplish the things that we have to worry about as teachers?

It seems to me that there is not enough focus on, or knowledge of, recent research in mathematics education among secondary math teachers. Math teachers should be recording themselves teach, and then watching those recordings to see how students reacted to what was done. We should take what we learn from these videos and apply it in our teaching. We should have time to do these things, but currently we don't. Teachers, and those who ultimately make the decisions on spending for education, should be working on ways to make time.

CANS OF WORMS
I have probably opened several cans of worms above, more than I could eat in one sitting. I think many are aware of the current problems in education. I also believe that we should all do our part to come up with solutions and to solve these problems. I will write more on one proposed solution in days to come.

Zudents and Zune in the Classroom

So I am a Zune user...sort of. I picked one up the other day, and have been pretty impressed. The improvements in version 2.0 are very nice. The new feature I was most pleased with was the firmware upgrade. Before the update all my podcasts were listed as music files, which meant that when I played them they started from the beginning each time. The Teachers' Podcast has episodes that are long enough that it often takes me two or three sittings to listen to them. I listen here and there as I have time. It was pretty annoying to start from the beginning each time. Zune 2.0 puts podcasts in their own category and allows you to resume playback from where you were last listening.

And now on to the big news. I sent my first file to a Zudent (this is what I will call students with a Zune). I noticed he had a Zune, and so I turned my Zune on and sent him an episode of The Teachers' Podcast. The episode was quite long (about 50 minutes), but it took less than a minute (I would guess it was about 20 seconds) to send him the file. It was quick and easy.

So what would I do with this technology? Am I actually saying I will be encouraging the use of an mp3 player in my classroom? During class? Yes.

I am thinking that I could prepare a few different files and send each of my Zudents a different file. They could get in groups and listen/watch whatever it is I want them to listen/watch, and then discuss it as a group. We can then come together as a class and hold a class discussion on what was heard/learned.

I will hopefully have more Zudents come January (hopefully Zanta is coming to a lot of houses near me).

The race is on

It looks as if a tennis match of sorts is starting between Google and Microsoft. Microsoft's new OfficeLive product (http://officelive.microsoft.com/), set to be released soon, was seen as a response to Google's online document, spreadsheet, and presentation tools. From the looks of things the OfficeLive would be much nicer, as it would allow you to work with your files in their true format as office documents. According to this article (Google Plans Service to Store Users' Data) in the Wall Street Journal, Google now plans to offer a storage space for users' files.

One thing that will hopefully result from these new tools is that teacher collaboration will increase in quantity and quality. It will be up to the teachers to make it happen, but it seems a logical and easy way to collaborate on lesson plans.

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.