TMI |
Thinking About Mathematics Instruction |
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Center for the Development
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Thinking About Mathematics Instruction
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| 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 BELIEFS coding scheme consists of six categories that span the continuum from a traditional, direct instruction position anchoring one end to a perspective at the other end in which instructional decisions are made by taking into account both the mathematical ideas of individual students and the assessment of the prevalent ideas within “the class” as a group. Though the categories are numbered from 1 to 6, we do not believe that they constitute an interval scale.
Categories of Belief with Indicators for Coding |
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| 1. Direct Instruction |
4. Attention to Students’ Mathematical Thinking |
| 2. Teachers Guiding Student Discovery | 5. Instructional Decisions Guided by Students’ Mathematical Thinking |
| 3. Teacher and Student Behaviors Associated with NCTM Standards |
6. Instructional Decisions Integrate Individual and Group Mathematical Thinking |
The indicators in each of the categories below characterize the attributes of a respondent’s writing. For example, responses categorized as “2” can be identified as such by the presence of indicators listed under category “2.” In order to be in a particular category, the response does not have to include all of that category’s indicators. A response may have individual sentences that are coded in several different categories. Because the pedagogy scoring scheme relates to systems of beliefs and these tend to change slowly and irregularly, the response as a whole should be coded according to the “preponderance of the data”; that is, the overall preponderance of codes of one category or another.
When assigning the overall code for a response we used a + or – to indicate that the response was very near the boundary of the category (i.e., close to the category above (+) or the category below (-).
| 1. | Direct Instruction | |
Responses in this category are based on a behaviorist view of learning. In this category respondents view teachers as presenting information that students absorb or providing opportunities for students to practice skills, with positive reinforcement for correct answers. Responses in this category indicate a preference for direct instruction and the primary focus is on the teacher. Responses are likely to indicate that the teacher should have followed the lesson plan, provided correct information, or avoided confusion. There is no attention to student agency – that is, there is no indication that the respondent thinks of students as taking an active part in their own learning. Responses indicate that the respondent is attending to the behavioral aspects of the scenario rather than the student thinking that is also present in the scenario. |
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Click here for Direct Instruction indicators and illustrative data |
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| 2. | Teachers Guiding Student Discovery | |
Responses in this category are still fundamentally based on a behaviorist view of learning; however, in this category student thinking is valued. Responses indicate that the respondent sees the teacher’s role as setting up situations that lead students to the specific ideas important to the lesson. This implies a greater sense of student agency than in Category 1 because students are being asked to “discover” ideas predetermined by the teacher. Unlike in later categories, once the work of discovery is underway, the teacher is less sensitive to the ideas that the students are generating and more sensitive to making sure the “discovery” doesn’t go awry. Unlike responses in Category “1,” there may be some attention to student thinking but the main focus of the response is the teacher. However, unlike responses in category “3,” where the teacher would be seen as “asking open-ended questions,” or “getting student strategies out,” responses in Category “2” will focus on whether the teacher is bringing students to the pre-determined right answer. If the response mentions student readiness, the concern will be for whether or not the students “get” the ideas that the teacher wants them to get, rather than a more nuanced interest in what students understand. If the response mentions confusion, it will be seen as something to be avoided. If the response includes mention of discovery/inquiry/constructivist learning/problem solving/critical thinking, and the like, these will be described as very general characteristics of the lesson. |
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Click here for Teachers Guiding Student Discovery indicators and illustrative data |
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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. |
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In the process/product school of thought about classroom observation, in which most administrators were trained, classroom observers used checklists to identify desirable teacher behaviors. Items on the checklists had been shown to have a strong correlation with high student outcomes on standardized tests and were directly observable teacher behaviors. That is, the checklists did not focus on student thinking or ideas. These checklists functioned as a set of rules for effective teacher behavior. Responses in Category 3 often appear to be guided by checklists and rules; respondents have simply substituted a new set of teacher behaviors to look for. |
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Click here for Teacher and Student Behaviors Associated with NCTM Standards indicators and illustrative data |
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| 4. | Attention to Students’ Mathematical Thinking | |
Here we see a real shift in what respondents appear to think is fundamental in a classroom. That is, classes are seen as fundamentally about ideas and thinking, rather than about teacher and student behaviors. Responses indicate that the respondent is attending to the student thinking in the scenario and sees the teacher’s behaviors as directed toward encouraging student thinking. Unlike responses categorized 3, in responses categorized 4, teacher actions are seen as connected to students as the generators of ideas and as sense-makers. Responses categorized 4 may be quite specific regarding the content of students’ ideas, or quite general. |
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Click here for Attention to Students' Mathematical Thinking indicators and illustrative data |
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| 5. | Instructional Decisions Guided by Students’ Mathematical Thinking | |
In this category, responses are now solidly based on a consideration of ideas as opposed to behaviors. Comments focus on how the teacher is working to get a sense of students’ thinking and how the teacher elicits students’ thinking to guide his or her teaching. If comments address the issue of “readiness,” they will consider whether students are at a point in their mathematical understanding to work with the ideas under discussion. If comments include reference to misunderstandings or confusions, they will be seen as important to probe. Unlike 6s, comments in this category about students’ knowledge will be focused on the group as a whole rather than on individuals. |
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Click here for Instructional Decisions Guided by Students' Mathematical Thinking indicators and illustrative data |
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| 6. | Instructional Decisions Integrate Individual and Group Mathematical Thinking | |
In this category responses are solidly based on a consideration of ideas. There are two pedagogically distinguishing characteristics of responses coded as 6. 1) Responses include mention of both individual and group thinking, i.e., respondents view the teacher as knowing enough about the thinking of the individuals in the class to be able to select individual students’ ideas for discussion because they know how others will also benefit. 2) Response indicates that respondent views the teacher as trying to find the “forward” place of where the students are. That is, the respondent views the teacher as knowing that students will only think about what they can think about, so she has to find the point at which student thinking and her agenda can come together. |
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Click here for Instructional Decisions Integrate Individual and Group Mathematical Thinking indicators and illustrative data |
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