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© Education Development Center, 2006-2008

Thinking About Mathematics Instruction
Leadership Content Knowledge
Elementary and Middle School Principals’ Survey

Table of Contents

1. Introduction
2. Overview and comparison of the Pre- and Post-surveys
3. Pre-Survey
4. Post-Survey
5. Assembling Your Own Mathematics Content Knowledge Section
6. Coding Schemes and Sample Responses
7. Validity and Reliability Considerations

Indicators and Illustrative Data for Category 3: Teacher
and Student Behaviors Associated with NCTM Standards

Click here to return to the complete Coding Scheme for A Classroom Reflection: BELIEFS. To view illustrators for other categories, click on the appropriate link below.
Category 1	Category 2	Category 3	Category 4	Category 5	Category 6

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3.	Teacher and Student Behaviors Associated with NCTM Standards
	Respondents write about the characteristics of a standards-based pedagogy, mentioning things such as group work, discussion, sharing strategies for solving problems, using manipulatives, and the like.  Compared to a response categorized as 2, in a Category 3 response there often is a greater sense of openness about the range of student thinking that is appropriate.  For a Category 3 response, student agency comes to the fore, in that category 3 responses focus on particular teacher behaviors and instructional methods or activities that give students the opportunity to express their ideas. However, respondents do not connect these activities to the thinking the students would do when engaging in them. Responses in Category 3 often have the feel of a check-list of desirable teacher and student behaviors. Responses in this category also tend to over-generalize, stating, for example, that teachers should not tell the answer.  They value students having the opportunity to express their thinking, but they do not get into the details of students’ ideas that are described in the scenario. While responses will focus mostly on teacher and student behaviors, it is a more progressive set of behaviors than in Categories 1 and 2. Their comments about students are usually byproducts of comments about the teacher.
	Indicators and Illustrative data
	Response indicates that in interpreting the scenario, the respondent’s writing reflects that the following is important to good classroom practice: Students inquire and think, explore, share strategies, come up with answers, prove their thinking, and the like.  These student behaviors are simply named, without specific connection to the content of the student thinking. Example: “Students were inquiring and thinking …”  Teacher asks open-ended questions, encourages students to come up with answers, asks students to explain their thinking, uses manipulatives, and the like.  These teacher behaviors are simply named, without connection to what either students or teachers might learn from engaging in the activity. Example: “Asking probing questions is the technique that was being used.”  Example: “Asking questions to clarify or prove a statement.” The students’ thinking process is what matters, not whether or not they got the correct answer. A corollary is that teachers should encourage many different ways of thinking. Example: “She was getting them to challenge preconceived notions and look at problems as processes with varied outcomes.  Solutions will vary due to the problem, and that’s ok.  Math is not an exact science.” Example: “A solution was not the important part of the lesson.  Thinking and brainstorming was more important.” Teacher should follow the rules of “good constructivist” practice.  Interpretation of these rules may be overgeneralizations. Example: “In a constructivism approach to teaching math a teacher must ask inquiry-based questions and let students do most of the talking.” Example: “Good teaching is letting the students do the work.”
	Sample Responses for the Category 3 Scoring Scheme (Note that all sample responses represent a principal’s entire response and are taken verbatim from study participants’ responses.)
	a. The teacher was using questions to have the students think about the problem. A solution was not the important part of the lesson. Thinking and brainstorming was more important. This way the students were analyzing and making predictions. They developed concrete problems to visualize what the problem would look like. b. She was using questioning techniques to develop higher order thinking. Yes, it was good teaching because she had the students think and figure out possible solutions. There was no right or wrong answer. They were thinking of concrete possibilities. c. The teacher was giving alternative probabilities. The students were given the same format but with different units. Students had to look at the problem and consider the possibilities. With the pizza’s there was division and multiplication to consider in finding a solution.