There is a growing knowledge base in mathematics education highlighting the importance of teachers’ cognizance about what and how they are noticing, namely considerations about their attending, interpreting, and decisions to respond on the basis of students’ thinking. However, less focus has centered on those in teacher leader positions and how those individuals may arrive at claims about students’ thinking and use evidence to support their assertions, meaning how they notice. As a result, a process to engage teacher leaders in considering students’ thinking was implemented for professional growth for those who support mathematics teachers. Findings highlight the degree of specificity of the teacher leaders’ noticing, the origin of the evidence they cite, and their connection between claims about students’ mathematical thinking, evidence, and broader implications for teaching and learning. Most teacher leaders included limited or robust evidence, with a majority citing transcript quotes as evidence for their claims about student thinking. Most teacher leaders were noticing at a mixed (Level 2) or focused (Level 3) level, based on a four-point scale. Implications for those working with mathematics teachers or teacher leaders to support the development of noticing and how to cite evidence are provided.
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Recognition to Enrique Galindo, Anderson Norton, Valerie Akerson, and Meredith Park Rogers for their support with the Iterative Model Building Project, from which this work was developed.
Clinical Interview: Organizing Question Creation/Conducting the Interview
Clinical interviews are constructed based on grade appropriate content, according to the Common Core State Standards for Mathematics. The following steps outline the process for designing your interview questions.
Select a student in grades K-12 (this could be a student you work with, a student in your neighborhood, or a student a local school).
Make arrangements to interview this student two times—once for Clinical Interview 1 and once for Clinical Interview 2.
Select appropriate mathematics content for the interview and complete the template provided. The content should align with the Common Core State Standards for Mathematics. For example, if you are working with a third grader, you may decide to focus the interview on unit fractions; if you are working with an eleventh grader, you may decide to focus the interview on applying trigonometry to general triangles or another grade-appropriate content area. This decision is left to your professional discretion.
Plan an overall goal for your interview (What do you want to know?)
Write a series of at least five problematic questions with follow-up questions that you could ask during an interview to gain information about the student’s mathematical thinking. Make sure these questions align with your overall goal for the interview. These questions may contain a series of related questions and may involve some type of mathematical tool or various forms of representation. Questions should likely either have more than one correct response, or more than one solution method. Design the questions so that they will provide information to help you learn about how your student understands the mathematical content selected for the interview.
Make arrangements and interview your selected student following the questioning protocol you wrote. Have means to audio record the interview with the student for later play-back. Collect any student work from the interview.
Listen to the recording and reflect on your interview experience. Save your recording in a secure location.
Clinical Interview Template:
In addition to submitting the questioning protocol, please respond to the following questions in a word document:
Give at least two specific examples of things your student said or did in the interview that help you understand more about how that student thinks. For each of your examples, describe the context of the task given to the student.
Based on one or both of the examples you gave above in question 1, what inferences can you make about how your student is thinking and reasoning? Explain how your inferences connect to your response to question 1.
Based on what you learned about how your student is thinking and reasoning from your Formative Assessment Interview, what would be an appropriate topic for a whole-class lesson if you were the teacher of this students’ class? Be specific in your suggestions and explain why you think they are appropriate in light of your responses to questions 1 and 2.
The Model Building assignment builds from the Clinical Interview. You will take the information gleaned from the Clinical Interview to construct a mathematical model of student thinking. The following steps outline the process for completing one Model Building assignment. You will repeat this process for the second Model Building assignment.
Listen to the audio recording from your Clinical Interview.
Select a clip from the audio recording, no longer than a few minutes, that highlights what the student says or does as evidence of how the student thinks about a particular idea.
Transcribe this segment of the audio recording verbatim. You will construct your model of student thinking based on this transcript.
Complete the Model Building Template, directions below:
Model Building Template directions:
There are eight important components that need to be included in the Model Building Template: logistical information, title, content about the Clinical Interview, start and end times of the video clip(s) selected, transcript, model, evidence, and table. These important components are described in more detail below.
Name and Date of Interview, grade level of student.
Create a descriptive title for the template.
Content about the Clinical Interview:
In this short paragraph, describe the goal of the Clinical Interview, the types of questions asked in the Formative Assessment Interview, and include a description and/or rationale for the selected clip.
This section should include the start and end times of the audio clip selected. When transcribing the clip, it is important to be sure to type what is said word for word, even if someone has misspoken or other people talk besides the interviewer and the student selected.
You should complete this table based on what the student knows, what the student does not know, and what additional evidence is needed to fully understand the student’s thinking.
A model is a statement or statements that capture the central aspects of how a student is reasoning about a particular math concept and any background information that is needed to contextualize the model so that it makes sense to the audience. The statement(s) should be general enough to apply to different problems and different settings, such as the classroom, but should be specific enough to provide insight into the individual student’s particular approach to problem solving. In particular, the model should be stated in such a way that it holds predictive value. In other words, others should be able to use the model to predict how that student would respond to a similar question or task. The model should also be consistent in that it can be used to explain all of the actions of the student, including his or her verbalizations.
The evidence is the portion of the transcript that documents how the student’s actions and verbalizations support the model. If there is information that seems to contradict the model it should be included, as it will pose some topics for consideration.
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Amador, J.M. Teacher Leaders’ Mathematical Noticing: Eliciting and Analyzing. Int J of Sci and Math Educ 18, 295–313 (2020) doi:10.1007/s10763-019-09956-5
- Clinical interview
- Teacher leader