Hand and LemppMath 131Fall, 2006

Student Thinking Project[1]:

Investigating Students’ Understanding of Geometry

This project focuses on how students understand and use ideas from geometry to solve mathematics problems. You will learn to establish a setting in which you can examine a student’s thinking about mathematics, and you will develop skills at asking good questions, listening carefully, interpreting what you hear, and asking a good “next question.” The project has two parts. In Part 1, you will select a task (or series of tasks) that provides an opportunity for you to solicit students' ideas, and to see what they know, can do, and understand about the topic in geometry you have chosen. You will prepare to meet with these students by developing a plan for (1) creating a safe environment, and (2) probing students' ideas and skills. For Part 2, you will conduct your interviews, asking students to "think aloud" as they solve your problems. You will ask them for the reasons and meanings behind their mathematical moves. For Part 3, you will compare students' solution strategies, and report what you can and cannot say about their geometric knowledge and skills.

Part 1—Design the interview.

(a) Make arrangements at school. Prepare to meet with three students individually to explore their understanding of geometry. Consider carefully whom you select. You may want to invite three students who are recommended by a mathematics teacher for possessing different levels of understanding and skills in geometry. Or, you may want to select students from different schools or cultural backgrounds. Please confer with us about your choices.

It is a good idea to meet with each of the students before the interview to introduce yourself and the plan for the interview. Plan to spend approximately 20 to 40 minutes on each interview, depending on the age of the student.

(b) Study and select tasks. Select a number of geometry tasks around the same geometric topic from one of the Singapore textbooks or from another curriculum resource. Spend some time working through them with your group, and also on your own. Make sure you select some that you think will help you learn how your students understand and use ideas in geometry. Try to anticipate what your students might say, and prepare for how you could probe students' thinking. The more you have thought through possibilities, the more likely the work will be rewarding for you and the student.

(c) Make a plan for the session. Assemble the tasks/questions you have selected, and add annotations about the purposes you have for those questions, your anticipations of what your student might do or say, and what you will ask or do. (Design a chart or template to map out these aspects of the interview.)

Two items should be turned in as soon as you are ready.

  1. A description of the students you've selected, your reason for selecting these students, and when you will meet with them. Use a pseudonym for the students to protect confidentiality.
  2. Your plan for the session (as detailed in [c]).

Part 2—Conduct your sessions

At least two of you should meet with each student. Before you begin the session, you will need to explain to the student that you are asking them to do these problems for your geometry class, and to obtain their permission to keep written and/or audio records. You will also explain that you are not interested in assessing their work, but rather getting an idea of how they think through and solve problems in geometry. Ask them to "think aloud" as the work through your tasks. You may need to gently remind them to do this during the session. Be sure to probe their thinking to understand what they are doing from their perspective.

When you have finished a session, the two of you should individually prepare two draft assertions about the student's geometric reasoning and skills, along with the evidence to back your claims. You should then come back together to share your assertions. This is an opportunity to discuss your student's thinking and your conjectures with a colleague and to gain insight into your student from what the other notices or comments on. This can help to expand what you learn on your own about the student's thinking.

Part 3––Compare students' reasoning

(a) After all three sessions have been conducted, meet as a group to discuss the interviews, and what you learned about students' geometric thinking. Compare the analysis of each student's thinking about and work on the tasks. Use the following categories to guide your discussion.

(1) Description - Describe what happened during the meeting. Be sure to include details and quotes. What kind of environment did you create to foster interaction with your student? What sort of materials did you provide, and how did you arrange the space. How did the student seem to react?

(2) Analysis - Work through your data carefully and formulate at least two clear assertions about each student's understanding of the geometric topic, both ideas and procedures. One way to think about this analysis is to imagine that you are writing an especially detailed report for a parent to read. Explain what the child seems to know, understand, and know how to do. Illustrate your claims with examples.

For each assertion, state it clearly and explain what you mean and why you make it. Include both what you think and specific examples that back up what you think. I am interested in the richness of your description, the depth of your analysis, and your ability to connect your impressions with specific evidence from your investigations.

(3) Comparison - Examine the conjectures you've made about each student, and notice any patterns across them. Do they have different approaches to the same tasks? What do these approaches tell you about their strengths and weaknesses in certain areas? Are their areas where all three students share the same mathematical characteristics? What does this indicate about the nature of their mathematics instruction? What did the tasks you chose allow you to gauge about the students? What additional information might have allowed you to gain a fuller sense of each students' geometric reasoning and skills?

Student Thinking ActivityPage 1 of 2

[1] This project was adapted from an activity authored by Deborah Ball's activity, Analyzing Student Thinking.