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Assessing Geometric and Measurement Understanding Using Manipulatives

Using a Journal Article as a Professional Development Experience
Learning

Title:          Assessing Geometric and Measurement Understanding Using Manipulatives
Author:      Marilyn E. Strutchens, Kimberly A Harris, and W. Gary Martin
Journal:     Mathematics Teaching in the Middle School
Issue:        March 2001, Volume 6, Issue 7, pp. 402-405

Rationale for Use

The authors express concern that students often have very little understanding of geometry and measurement concepts.   Through a consideration of student’s response to a selected question from NAEP, the authors ask us to reflect on the depth of students’ conceptual understanding of geometric properties and measurement.

This article reinforces the statement made in the The Learning Principle (NCTM 2000) that “In recent decades, psychological and educational research on the learning of complex subjects such as mathematics has solidly established the important role of conceptual understanding in the knowledge and activity of persons who are proficient”, page 20.

Procedures

Participants work in pairs to find solutions to problems 1, 2, and 3 on page 402 of the article. They have cut-outs of shapes as used in the problems. Rulers not allowed. NOTE:  Facilitator should prepare the three shapes shown on page 403, using lightweight cardboard.  Inform participants that these three problems appeared on the mathematics portion of the 1992 and 1996 National Assessment of Educational programs (NAEP), given to 4th, 8th, and 12th grade students.

• Discuss solutions and strategies used to get solutions, reflecting in particular on:
• Whether procedural or conceptual knowledge was used to find solutions;
• The role of manipulatives in solving the problems;
• The difference in complexity of the three problems; and
• If finding solutions would be different if rulers were used and why.
• Ask participants to predict and discuss the challenges these three problems may present to their students. What errors or misconceptions might emerge as students do the problems?

1. Problem 2 in the set was only given to grades 8  and 12 students. The authors state that 6% of the eighth graders and 12% of the 12th graders gave completely correct answers (correct response must identify which shape has the longest perimeter and must include an explanation of the answer).

Have participants:
• Discuss why they think students had such difficulty with this problem?
• Discuss the level of the student’s understanding of geometric properties and measurement as indicated in three student’s responses and explanations. Shown in Fig. 3 on page 404.
• Repeat the previous bulleted task for student’s response to Problem 3, shown at the bottom of page 404.

Next Steps/Extensions

• To assign the three problems to their students and have them share solutions with the class.  Record responses to share in a follow-up session. * Reflect on the following issues in regard to the students sharing of solutions in their classrooms:
• The role of the shape cut-outs in finding a solution;
• The student’s responses and explanations as compared to the responses shared in the article; and
• As in NAEP, did your students find Problem 1 easier than Problems 2 and 3. Why do you think that is so?
2. After sharing experiences and reflections of student’s responses to the problems,  have participants discuss the following:
• What questions in regard to your student’s conceptual understanding of mathematical ideas are you still wondering about?
• What questions surprised you in regard to your student’s conceptual understanding of mathematical ideas?
• Did your students demonstrate a conceptual understanding, a procedural understanding or both?
• What would you like to discuss further in regard to conceptual understanding with your colleagues?
• How did your students react to using the manipulatives provided? Did they contribute in any way to students getting a successful solution to the problems?
3. Ask participants to share with each other, the types of learning experiences they might design to help students attain a deeper knowledge and understanding of perimeter and area, based on the following:
• The Learning Principle (NCTM, 2000, p20) states, “the kinds of experiences teachers provide clearly play a major role in determining the extent and quality of student’s learning”.
• Suggestions in the above article that the results of NAEP indicate that students need the opportunity to develop deeper conceptual knowledge of perimeter and area.
• Your student’s responses to problems 2 and 3.

Connections to Other NCTM Publications