Here is the third in the series of questions, comments, concerns, and issues related to NCTM's 1991 "Professional Standards for Teaching Mathematics." The focus this week will be on pages 34-54, Standards 2-4 on Discourse. [If you missed the posting on either 9/22 or 9/29, let me know personally, and I will send you a copy]. Because all questions in the entire series will be numbered consecutively, I will start today with # 19.
Ron Ward/Western Washington U/Bellingham, WA 98225 email@example.com
19. In the statement "Discourse is both the way ideas are exchanged and what the ideas entail," does the inclusion of the second phrase sound novel to you? Do you think of discourse as "what the ideas entail"? In what way? I found it a useful exercise, as I read thru the three related standards, to note every illustration of this phrase.
20. There are any number of really good questions in this material, whose answers would be beneficial to us all, I think. Here are just a few: a. What makes something true or reasonable in mathematics? b. How do we figure out whether or not something "makes sense"? c. Why is it so important for classroom discourse to be founded on mathematical evidence?
21. There is a statement here which ties this commentary to an earlier question [# 3] about mathematical communities: "When students make public conjectures and reason with others about mathematics, ideas and knowledge are developed collaboratively, revealing mathematics as constructed by human beings within an intellectual community." Perhaps some of you could share how you develop your classroom into such a community.
22. Standard 2: "The Teacher's Role in Discourse" lists a number of skills that teachers must possess. I was particularly struck by the number of critical decisions that teachers must be able to make "on the spot"--How do teachers learn to do this? What kind of knowledge must they have to make the right decisions?
23. It seems to me that many experienced teachers will find it difficult to break some old habits;e.g., following a student statement with "Why?" only if the statement is wrong, or endorsing or dismissing student suggestions. What other habits must be broken in order to embrace the vision of teacher's role presented here?
24. The authors correctly point out that some students, "particularly those who have been successful in more traditional mathematics classrooms, may be resistent to talking, writing, and reasoning together about mathematics." How do you overcome such resistance?
25. This standard identifies at least three aspects of the teacher's role [to provoke students' reasoning about mathematics;to encourage and expect students to do virtually all the talking, modeling, and explaining themselves;and to monitor and organize students' participation]. I believe it would be helpful for teachers to share their techniques for addressing these aspects.
26. Now, as a teacher who resists "covering ground," and who tends to move rather slowly thru material, milking it for all it's worth, I hate myself for even asking this next question: Doesn't it seem that taking the approaches suggested here will drastically reduce the amount of material that will be learned? Isn't there a point where the "less is more" idea really does translate into too little?
27. When you think about ways that students can contribute to the classroom discourse, other than by talking or writing, what comes to mind?
28. When there is an emphasis on students "constructing" their own understanding, when a teacher remains neutral about the correctness of what is being said, is there a danger that kids will construct incorrect concepts? Have to subsequently unlearn an incorrect procedure?
29. As usual, I encourage readers to react to the four vignettes included in this standard.
30. One of the things I noticed as I read the vignettes, is that there are several places where some form of assessment is occurring. I would certainly invite readers to comment on any of these in light of the 1995 Assessment Standards;however, I will refrain from posing related questions myself since it is challenging enough just to get subscribers to read one of these documents, much less two or three at once! :)
31. Standard 3: "Student's Role in Discourse" makes it quite clear that students have a responsibility in the classroom. But do they accept it? If not, how can you as a teacher encourage it?
32. It would be worthwhile, I think, to take any one of the bulleted responsibilities (skills) required of students and discuss it. For example, one statement mentions a "variety of tools" that students should use to reason, make connections, solve problems, and communicate. They don't initially identify any of these, so you might ask yourself what "tools" are appropriate for this purpose. Later, as you read the various vignettes, you will encounter a number of them, but they are not labeled as such. [You can find a pretty complete checklist, though, when you get to Standard 4 :) ]
33. There are three more vignettes in this standard to which I invite reader reaction. I was particularly struck by illustrations of the teacher's role as a "facilitator" of learning. What are the hallmarks of a good facilitator?
34. Standard 4: "Tools for Enhancing Discourse" presents a rather extensive list of tools for student use. It occurred to me that it would be helpful to have some way of continuously reminding students of all the tools to which they have access. Any suggestions? Visual possibilities?
35. Once again, please feel free to react to the vignettes in this standard. I will just say that as I have read thru all the vignettes up to this point, I have occassionally found mathematical errors. I am assuming that these should be fairly apparent to the readership and am not pointing them out. But you should read carefully!
Well, that's it for another Friday. Once again, I have received several responses directly to me that were not posted to the listserv. At the end of the first set of standards, I will summarize all of these. But I also note that individuals are now starting more focused discussions centered around specific items mentioned in these standards. For example, Jason Sustarsic's "Model Teacher" heading touches on a question raised back in Question 2: Do teachers teach as they were taught or as they were taught to teach? Ted Panitz has introduced a discussion related to "Getting Students to take Responsibility." This extends nicely the question in Question 4 about getting students to take charge of their own learning. And it dovetails beautifully with the current issue raised in Question 31. So I encourage others among you to select issues which interest you and participate similarly. It is my feeling that we can benefit greatly from the real-life experiences of the teachers on this listserv to augment, supplement, and complement the written material in the Professional Standards.