The Teaching Gap & Five Practices

This past week as part of our home workshops for our content area seminar we continued reading The Teaching Gap as well as learning about the Five Practices for Orchestrating Mathematical Discussion. The Five Practices include first anticipating what strategies students will use while they are trying to solve the problem. Then while they are actually working you monitor what they are doing and then select different strategies for students to share. Once you have selected who you want to share then you need to sequence them in a meaningful order. As these are presented you should make sure students can see the connections between the strategies that are being used. The Teaching Gap talks about how the “teaching script” is different between the United States, Germany, and Japan. The latest chapter we read talked about the use of the chalkboard in class. They talk about how in a typical American class some sort of overhead projector is used to focus students attention. However, teachers in Japan would never use an overhead projector for that purpose, they use the board as a way to keep a record of what was talked about during the class period. The pattern of lessons was also discussed, talking about how the day was structured. All three start with some type of review of what was happening the day before. In Germany they then then give the problems and topic for the day, spend time developing the procedures for the problems, and then practice the problems. In Japan they would then present the problem for the day, and then allow the students time to work individually or in groups, then discuss the solution methods, and then summarize the day. In the US after the introduction there is demonstration on how to solve the problem, then practicing, and next assigning homework.

As I was reading I was thinking about how I see some of this in the way I teach and the way that I have learned throughout my education. I definitely have seen the American teaching script in my math classes. Reading through the routine it sounded a lot like a vast majority of the math courses I have taken. I would say typically it is the way that I teach although we are trying to get more discovery based learning activities in our classroom this semester. Before reading this I never really thought of the way math was taught was bad, I mean I’ve always done well. It’s really hard to even think about changing because this is all that I know. In my placement this semester we have worked on trying to having our students do some of the discovering on their own but after reading TTG I wonder is this really all that different from the standard American script. I’m really hoping that as I finish the rest of the book I will be able to see what it means to teach mathematics in these countries and how I can incorporate those ideas into my teaching in a meaningful way.

When they started talking about the use of the overhead projector as a way to focus students in an American classroom I was taken aback. I have a document camera and overhead projector that I use on a daily basis in my teaching and have had the thought what would I do without it. I have used it so that I can work through the same thing that students have in front of them. We work through the page and I write on the board what they need to write to write down and when I finished that portion I erase and move on to the next portion of the notes. We continue this until we are finished and even thinking back about my education this seemed to be typical. They talked about  how in the Japanese classroom the board is used as a record keeping device. I thought this was a really cool idea as I think it would be much easier to have meaningful discussion as it would be easier to refer back to things. I can’t say how many times I try refer back to something earlier in the notes and its hard because I don’t have it written and sometimes its hard for my students to find what I’m talking about. I would like to find a way to still use the overhead projector but also keep some sort of the record of what it going on. Once again I think it will be hard work to incorporate this because the American script is all I know.

Now on to orchestrating mathematical discussions…Before I read about these five practices I would say that this was something that I did or tried to do but it was really never as connected as the five practices suggest. When trying to get discussion going I do try to check with every group and tried to have an idea of who I wanted to ask to share. However, I never really thought about sequencing them in a way that made sense I always seem to pick and chose groups to share. I did try to make connections between the groups ideas but I think it could be easier to do this if I had sequenced them correctly. After reading about these and thinking about how my discussions have gone in the past I want to try and think more about these thing before we share as a whole group. As for my experiences with this I’m not really sure what my past teachers have done in terms of these and I’m honestly not sure about my CT this semester. I would say that my CT seems to be better able to do this, I’m not sure if its the five practices or more of the fact that she knows the students much better and know what they are going to have trouble with.

I think these sections helped me to start to think about how to make sure what I am doing is effective. Having the ideas that come from the five practices should allow me to make sure my discussion are meaningful and I don’t feel so scattered brained during them. I also would like to think about how to make it easier to go back to items we already talked about that isn’t just “on the last page we talked about this” so students can see quickly what I am talking about.



  1. You wrote:

    >>”In my placement this semester we have worked on trying to having our students do some of the discovering on their own but after reading TTG I wonder is this really all that different from the standard American script.”

    This really struck me as quite a profound observation. The question now is: where do you go from there?

  2. Nice blog post. I was curious when you were talking about the different classroom routines that are used in different countries. I was particularly interested in Japan, where you said they had a “problem of the day” that they worked on and they would use the board, but not an overhead projector. How much do you think types of representations that are created/promoted play a role in what medium is used to present/display mathematics? In a traditional algebra curricula, it is no surprise that symbolic representation is the defacto representational leader. Yet, in an inquiry-based lesson (maybe like Japan’s problem of the day, I’m really not sure here because I didn’t read the article but I’m assuming that a single daily problem would be inquiry-based to some degree), there may be numerous entry points into the problem which may produce different representations that could be created and discussed. A benefit of using a board is to show them all in tandem.

    So one thing that I think about here is how much the types of tasks and possible representation(s) produced play a role in the types of technologies used. What is the relationship between the two?

    1. Thanks! I think you bring up a good point about the representations and what type of technology is used. To be honest I don’t have an answer about that in the context of the different countries. I do wonder if it is any different in geometry lessons where I feel as the technology lends its self well to the subject matter rather than other more symbolic representations. I don’t know if The Teaching Gap will answer this question in later chapters but the impression I’m getting so far is that they do not have the projectors. Whether this is just classroom level or a school wide level I’m not sure.

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