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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

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Strand-coding Scheme for Static Scores on
A Classroom Reflection: Strand-by-Strand Description

Coding Scheme for A Classroom Reflection: BELIEFS

Coding Scheme for A Classroom Reflection: MATH-IN-USE (MIU)

Strand-Coding Scheme for Static Ccorers on A Classroom Reflection

Strand by strand description with examples

ATTENTION TO STUDENTS

ATTENTION TO TEACHING

RATIONALE

For an overview of the strand-coding scheme, click here.
Click here to skip to the Identifying Shifts in Thinking section.

1.	ATTENTION TO STUDENTS
	This strand provides an indication of whether or not principals are focusing on what students are doing and thinking. Often what the teacher is doing consumes the totality of a principal’s attention during an observation. As principals are learning to look at classrooms in new ways, their focus tends to shift to include the students as well. Responses that fall into this category comment on what the students are doing in the scenario and convey the sense that the principal had actually been in the classroom because of how engaged the principal seems to be with the student activity. Examples of data that indicate that the principal was focusing on the students: One student was allowed to explain his reasoning… In certain problems (like this one) you need to understand what is being asked for …this worked out well when one student seemed to understand this would be a very small number for the answer. The teacher seemed to have jumped ahead in that there were a variety of answers given by students in this scenario. Examples of data that do not indicate that the principal was focusing on students when analyzing the scenario because, although the principal mentions students, he or she did not actually reference student activity depicted in the scenario: Yes, students must know that there is order in mathematics. She was trying to teach by having the students discover the true and correct understanding of the problem. Good teaching is having the students become problem solvers/thinkers. In the traditional band for this strand, the principal uses language such as, “students learn,”, “think critically,”  “use higher order thinking skills.” In the constructivist band, responses are about students making sense for themselves and handling abstractions. In the mixed position, respondents use reform language and write about how students explain, share ideas, explore, and the like.
2.	ATTENTION TO TEACHING
	2A.	What the teacher does
		This strand picks up the nature of the respondent’s comments about the teaching and provides an indication of whether the respondent’s ideas about what the teacher does reflects a shift from a more traditional position to one that supports socio-constructivist learning. Traditional band. Responses that reflect traditional views about the teaching are those that indicate a preference for direct teaching and providing answers. If respondents comment about the teacher checking for students’ understanding, in the traditional band these comments tend to reflect the idea that students either “get” a concept or they don’t. In other words, respondents whose comments fall into this band do not seem to be thinking of “getting it” as an ongoing process of building understanding of the different mathematical ideas within a given concept. This strand also takes into consideration responses that address the effect of the teacher’s actions on students. In the traditional band, the principal uses language that refers to the teacher getting students to “think critically,” “use higher order thinking skills,” or “problem solve.”  Examples of responses categorized as traditional: She was …wording her question in a way to direct them. The dialogue ultimately will lead students to the understanding that answers do not necessarily come in whole number packages. She was checking for understanding of the division process. Continues with critical thinking. Constructivist band. Responses that are aligned with socio-constructivist views of learning are comments that indicate a preference for students making sense of mathematical ideas and for using students’ ideas to guide instruction. In this band are comments that reflect an understanding of the complex nature of both mathematical concepts and of students’ thinking about them. In terms of responses in the constructivist band that address the effect of the teacher’s actions on students, the teacher’s actions are seen as providing the opportunity for students to make sense for themselves. Examples of responses categorized as constructivist: She gave a concrete example that students had experience with and could make meaning from. …but she realized that Joes’ misinterpretation is common to others and chose to challenge kids thinking. The teacher was allowing the students an opportunity to work through possible solutions themselves. Mixed band. Responses that indicate mixed views about the teaching are comments that indicate a preference for the teacher encouraging reform classroom activities such as exploration and sharing ideas without being guided by students’ thinking. In terms of responses in the mixed band that address the effect of the teacher’s actions on students, the teacher’s actions are seen as providing the opportunity for students to think out loud, conceptualize, explain, and other reform behaviors. Examples of responses categorized as mixed: Her question allowed for further conversation and dialogue about students knowledge. This opportunity to discuss problems or statements is essential to understanding. Wanting the students to think, discover, problem solve.
	2B.	View of Confusion
		This strand picks up the respondent’s ideas about the role of confusion in the classroom and provides an indication of whether the respondent’s view about confusion shifts from a belief that confusion is something to be avoided to one that reflects an understanding of confusion as a normative aspect of standards-based math learning.  Example of a response that reflects a view of confusion as something to be avoided: If the objective of the lesson is for students to be dividing whole #’s, then the teacher was confusing the students and should not have pursued this line of thought.  Example of a response that reflects an understanding of confusion as a normative aspect of standards-based math learning: On one hand it was good teaching because the teacher took a misunderstanding of a student and pursued the thinking behind the confusion as opposed to ignoring or just correcting him.
	3.	RATIONALE
		This strand picks up whether a respondent’s comments about the teacher’s motivation for the pedagogical moves she makes shifts from reasons that do not take the forwarding of students’ mathematical thinking into consideration (e.g., the teacher is motivated by a desire to engage all students in the activity or by a desire to promote her students’ self-esteem) to reasons that do take the forwarding of students’ mathematical thinking into consideration (e.g., the teacher is motivated by a desire to hear the thinking of a broad range of students in order to use particular students’ ideas for the benefit of the learning of everyone). Examples of data that reflect non math-based reasons for the pedagogical moves the teacher makes: Great way to encourage participation. Yes, this also encouraged thinking skills and all responses are OK. No threat of wrong answers. This is good teaching because she has provided practical application of the problem in a situation kids can relate to. Examples of data that reflect math-based reasons for the pedagogical moves the teacher makes: It was good teaching because order makes a difference in math [i.e., division is not commutative]. It is good teaching because again, it forces them to think more deeply about what they understand about numbers and the rules of divisibility. This was good teaching because she allowed time for students to work through misconceptions and waited for a logical explanation from a student.

Identifying shifts in thinking

After coding the data, we tallied the results on a Scoring Sheet and used the following guidelines to determine shifts in thinking:

For 1: Students
We attributed change in beliefs if, to a greater extent than in the pre-survey, a principal’s comments on the post-survey reflected increased attention to what students were doing and thinking and/or to a greater extent than in the pre-survey, a principal’s comments on the post-survey attended to the active participation of students in constructing their own understanding.

For 2A: What the teacher does
We attributed change in beliefs if, to a greater extent than in the pre-survey, a principal’s comments on the post-survey reflected that he or she values teaching strategies that support socio-constructivist learning.

For 2B: View of confusion
We attributed change in beliefs if,to a greater extent than in the pre-survey, a principal’s comments on the post-survey reflected an understanding of confusion as a normative aspect of standards based math learning.

For 3: Math rationale
We attributed change in beliefs if a principal’s comments about the teacher’s motivation for the moves she is making shifted from non math-based reasons on the pre-survey to math-based reasons on the post-survey.

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