This is the sixth in the series of questions, concerns, and issues related to NCTM's 1991 "Professional Standards for Teaching Mathematics" and the second in the set of "Standards for the Evaluation of the Teaching of Mathematics." Today we will focus on pages 88-103, Standards 4 and 5. Because all questions in the series are numbered consecutively for reference purposes, I'll begin with number 55. [If you have missed any of the earlier questions and want them, just contact me directly and I will forward them to you]
Ron Ward/Western Washington U/Bellingham, WA 98225 email@example.com
55. If you plan to use these standards as a framework to design or modify an observation instrument, you might want to ask yourself which of these standards (or parts of them) require a focus on the teacher, on individual students, on groups of students.
56. Standard 4: "Mathematical Concepts, Procedures, and Connections"--The primary emphasis in this standard is clearly content, and one could discuss any of the bulleted items with benefit. I am particularly interested in the evidence that the teacher "demonstrates a sound knowledge of mathematical concepts and procedures" and "represents mathematics as a network of interconnected concepts and procedures." To what extent do you think it is necessary for a teacher to understand such things as the nature of a mathematical concept [a logical construct], relational understanding [as opposed to instrumental understanding], the role of cognitive schema, reflective thought, and constructivism to meet this standard?
57. You might want to think about the CONNECTION between these evaluation standards and the first set of standards we discussed (pages 19-67). Also the CONNECTION to the original Curriculum Standards--especially #4 on "Mathematical Conncections," and the original Evaluation Standards--especially # 4-10 on student assessment, which discuss much of the same content, process and disposition matter. If you add all these ideas to the specific CONNECTIONS mentioned in this current standard, you should have an extensive "web" of related ideas.
58. I don't know if the authors intended the vignettes to serve as a "built-in addenda series" for the Professional Standards, but that is how they appear to me. As you go thru these evaluation standards, you should observe that some vignettes model teacher self-assessment while others suggest patterns of involvement for mentor teachers, peer assessors, colleagues as conferees, and so on. Perhaps some of you could share an evaluation model that is used in your school or district.
59. I am particularly interested in the nature of the teacher-evaluation role of any math supervisor, math coordinator, or math consultant in your district. Sometimes these postions involve no evaluation and sometimes, especially in "merit" school districts, their involvement is extensive. Also, to what extent are there individuals evaluating your teaching who have little or no background in mathematics?
60. Standard 5: "Mathematics as Problem Solving, Reasoning, and Communication"--The bulleted items in this standard make it clear that the teacher does not simply "facilitate" but rather has an important role to "model and demonstrate"--you will also find this emphasis in other standards in this evaluation series. How do you BALANCE this modeling role with the facilitating or orchestrating role?
61. Do you see "parallel roles" in Standards 4 and 5 for tasks and discourse?
62. I am interested particularly in what may be a subtle difference or just an unintended wording difference in one of the bulleted items. Instead of saying "models and emphasizes mathematical reasoning" they say "demonstrates and emphasizes THE ROLE OF mathematical reasoning." Do you see these as being different and the wording intentional?
63. I note that there is a recognition here that assessment should determine whether teachers and students are involved in appropriate activities OVER TIME. Does your district provide for longitudinal teacher evaluation by individuals as opposed to "one-shot" evaluation?
64. Would anyone care to comment on the meaning of this statement? "The very essence of studying mathematics is itself an exercise in exploring, conjecturing, examining, and testing--all aspects of problem solving." And how is that related to assessment?
65. It is very easy to get carried away by the content or process ideas themselves in these standards. So, note that in the "elaboration" text, from time to time, the authors provide specific statements about what the assessment should focus on in a particular area. It might be helpful to cull these statements into a smaller subgroup, especially if you are creating or modifying an evaluation instrument.
66. I notice that from time to time there are statements that clearly relate to issues that have been discussed on the NCTM listserve;for example, "The notion of communication emphasized in this standard cannot be fully realized in a lecture-oriented lesson or when students' responses are limited to short answers to lower-order questions." :)
67. I think there are good reasons for reading the vignettes carefully and critically. For example, in vignette 5.2, the peer mentor teacher "Doug Walker" makes a suggestion intended to help the teacher "Louise Knight." In my judgement, the adoption of his suggestion will result in the loss of something very valuable which was present in her original lesson plan. Do you see what it is?
That's all for this week folks. Please feel free to select one [or more] of these items that interests you and comment to the listserve about it. Or raise other questions, concerns, or issues of your own. Our purpose is to encourage everyone to read thru these standards--a little at a time so you can still work at your job!--and to provide a forum for their discussion with colleagues.