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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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Coding Scheme for A Classroom Reflection: MATH-IN-USE

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

The Math-in-Use scoring scheme consists of two indices: one that codes the amount of mathematics the respondent used when interpreting the scenario and one that codes the correctness of the mathematics described. The index that measures the amount of mathematics uses a 5-point scale, anchored at one end by responses that make no mention of mathematics in the interpretation of the scenario and at the other end by responses that discuss the mathematics in some detail and how the teacher and students are working with it.

The Correctness Code index takes stock of the accuracy of the respondent’s comments about the mathematics. It has three categories; “unscorable” for responses that do not provide enough information to determine whether the respondent understands the math, “correct” for responses that accurately describe the mathematics, and “incorrect” for responses that describe the mathematics inaccurately.

One final aspect of the Correctness Code index looks at what principals say about the kinds of mathematical learning opportunities children in the scenario are afforded. These questions (Questions 4 & 5)  follow the other three about teacher actions. We used Questions 4 & 5 to note whether principals’ responses to these learning opportunities contained more (+), less (-), or the same (=) amount of mathematics contained in principals’ interpretations of the teaching in the scenario in Questions 1-3. Click here to view additional information about, and indicators of, our correctness coding.

Categories of Belief with Indicators for Coding
0. Blank Entry	C. Expanding MIU
A. No Math	D. Attending to Mathematical Thinking
B. Modest MIU	E. BIg Picture

CATEGORIES (with indicators)

The numbered statements under each category below are indicators of the category. Responses do not necessarily need to include them all to be scored in that category. Note that Box A in A Classroom Scenario (corresponds with the first underlined question by the teacher) tends to elicit general comments from respondents rather than comments that contain mathematical ideas. Boxes B and especially C tend to elicit more mathematically laden comments. Because of this, we learned that it was important to read the entire response over before commencing to code.

Unlike the pedagogy scoring scheme, the MIU coding scheme does not tap into a system of beliefs; rather, it is designed to provide a sense of the kind of mathematical thinking a principal is able to demonstrate. Because of this, we coded responses for math-in-use using a leading edge strategy, rather than a “preponderance of the data” as when we coded the response for pedagogical beliefs. Thus, when coding for math-in-use, if a respondent made a statement at a higher level than most of his/her other statements, we thought it to be appropriate to base the score on that statement alone.

We coded by idea rather than word by word. If an idea such as division is mentioned more than one time, we counted it one time only.

If there is only one statement in a higher category, we coded the response in the higher category followed by a minus sign. For example if all the statements are As and Bs, except for one statement which is a C, the response was coded a C-.

0.	Blank Entry
A.	No Math
	Comments are not grounded in mathematics. They could be about language arts, social studies, science, etc. No mention of math at all Respondent says words like math, numbers, problems, number sense, problem-solving, manipulatives. These words are isolated, not in context and not part of a mathematically meaningful statement. Respondent provides a non-mathematical rationale for using the pizza as an example, such as real-world context is a good teaching move.
	Click here for samples of the No Math category
B.	Modest MIU
	Respondent makes general references to math.  Comments reflect back or repeat the mathematics mentioned in the scenario.  Comments tend to either be quite general about the mathematics, or they use the same words as those that appear in the scenario. Respondent notes that this is a lesson about division, division with remainder, division of a smaller number divided by a larger number, or says something about reversing numbers or the problem was read incorrectly but response is not specific; it just describes what happens in the scenario without interpretation. Respondent makes general, unsupported comments about the processes involved in doing mathematics, e.g., use of manipulatives, proof, claims, conjectures, estimation, and mathematical habits of mind.  To be coded B, these comments have to be made in context of math in the scenario.  For example, “use manipulatives to show that you can divide a smaller number by a larger number.”
	Click here for samples of the Modest MIU category
C.	Expanding MIU
	Respondents’ comments about the math do more than just reflect back what is in the scenario. Respondents describe in their own words what is going on mathematically, and in doing so, they demonstrate a degree of engagement with the mathematics. While they may make references of a general nature to the mathematical thinking of the students or teacher, they do not provide detail about or unpack the general statements they make. Respondents refer to math that is not already part of the scenario, e.g., · Fraction · Decimal · Percents · A piece of a whole · Parts of a group · Quotient, dividend, divisor · Numbers less than 1 · Fair shares, partitive, quotitive · Order of division · Whole numbers being divided into parts · Ratios   Respondent writes about how using the pizza as an example can further the students’ mathematical understanding of this problem. Respondent provides a basic mathematical reason why (such as pizza can be divided into parts). Respondent only notes that the kids’ examples of objects to divide do not work well mathematically; respondents do not say why they don’t work well. Respondent indicates that 1 divided by 4 is easier numerically to divide than 5 divided by 39, but does not recognize that 5 divided by 39 is a different challenge in division. Respondent might say that kids often have trouble with X (a specific math topic).  But, the respondent would not provide a perspective on why they have trouble with it or what to do about it. Respondent writes about math process (conjecturing, proving, claiming, etc). Example: "The teacher continued to ask probing questions.  She asked, 'Is it true?' – give me some proof – she wanted the students to mathematical [sic] defend their answer."
	Click here for samples of the Expanding MIU category
D.	Attending to Mathematical Thinking
	Respondent’s comments reflect an understanding of the mathematics involved coupled with attention to how the teacher and students interact with the mathematics at play in the classroom. Respondent’s comments provide some detail about or unpack how the teacher and students might be making sense of or understanding the math. Respondents make conjectures about the thinking of the students and or teacher and provide a rationale as well. In addition to describing the math, respondents may comment on the nuances of teaching it, the challenges of learning it, and possible misconceptions.  They are explicit about the mathematical usefulness of the representation. Respondent writes about the relative mathematical usefulness/merits of the representations that students and the teacher bring up (dollars, desks, candy, pizza) and how the students and/or teacher are thinking about the math. Respondent writes about how to work with students’ thinking around their misunderstandings. In addition to writing about how 1 divided by 4 is easier numerically to deal with than 5 divided by 39 and that it more easily illustrates that a smaller number can be divided by a larger number, the respondent recognizes that 5 divided by 39 is a different challenge in division; solving 1 divided by 4 may not help students to compute 5 divided by 39. Respondent might say that kids often have trouble with X (a specific math topic).  But, the respondent would not provide a perspective on why they have trouble with it and what to do about it. Respondent writes explicitly about the consequences to students' mathematical thinking of a teacher's move.  For example: “This strategy is important to guide students to thinking about the mathematical meaning of the position of numbers in operations.”
	Click here for samples of the Attending to Mathematical Thinking category
E.	Big Picture
	In addition to attending to the mathematical thinking in the scenario, respondents have a broad perspective on the important mathematical ideas of K-8 mathematics and what the teaching and learning challenges are. In category C or D, a respondent might say that kids often have trouble with X (a specific math topic).  In category E, respondent would provide a perspective on why they have trouble with it and what to do about it. Respondent has a broad perspective on what kids usually struggle with in understanding division with fractions in the quotient and strategies teachers can use with this topic. Respondent refers to the teacher’s long-term mathematical agenda demonstrating a very fluid understanding of how to build kids ideas in math.
	Because no study participants’ responses fell into this category, we cannot provide a sample response.