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Open-Ended Items Better Reveal Students' Mathematical Thinking

Written by Wendy Sanchez and Nicole Ice
(News Bulletin, July/August 2004)

In this age of accountability, teachers need more—and more varied—data about their students' mathematical understanding than ever. One way to acquire these data is through the use of assessment items that are open-ended. Open-ended items "have more than one answer and/or can be solved in a variety of ways" (Moon and Schulman 1995, p. 25). In addition to producing an answer, students must also show their solution process and justify their answer. Open-ended items offer opportunities for students to demonstrate their mathematical thinking, reasoning processes, and problem-solving and communication skills. Because open-ended items invite a wider range of solutions and solution methods than more traditional assessment items, they are better at revealing students' understanding of mathematics.

Many of the mathematical questions that we ask students to answer allow them to reproduce memorized procedures without thinking about why the processes work and what the answer means. One of the authors recalls a test that she gave to second-year algebra students on solving systems of linear equations. Most teachers have given a similar test: first, students are asked to solve by graphing, then by substitution, then by elimination, and finally by any method that they choose. Once, after attending a workshop on assessment, the author added to her test the question "create a system of linear equations whose solution is (1,4)." Most students who were able to answer almost all the usual questions correctly were unsuccessful on the new question. These results made the author believe that her students had only a procedural understanding of systems of linear equations. They had memorized procedures that enabled them to produce correct answers, but they were not able to demonstrate an understanding of what the solution of a system of linear equations really means.

Although it is important to assess students' mastery of mathematical skills, it is also necessary to assess their conceptual understanding of mathematics. Often, just a little twist on the questions that we typically ask our students can yield the assessment opportunities we need. Consider the following original and revised items:



Each revised item asks students to provide an example that satisfies certain criteria rather than asking them to carry out specific procedures.

It is important to recognize what students need to know and understand to respond to the revised item that they would not necessarily have to know and understand for the original item. In the original item 1, students have to know the rules for integer addition, whereas in the revised item 1 they also have to know what integers are and the meaning of the word sum. The original item 2 requires students to use a procedure to reduce a fraction; revised item 2 offers students the additional opportunity to demonstrate their understanding of equivalent fractions. In original item 3; students substitute values of b and h into the area formula. However, in the revision, students must also show that they understand that the base and height are perpendicular. In the original item 4, students routinely graph the point (0, 4), move "down two, over one" and graph a second point to determine the line. The revised item requires an understanding of the meaning of slope and y-intercept and their relationship to the coordinate plane. The original item 5 requires students to set the denominator equal to zero to get x = 2 as a vertical asymptote. In the revised item, they also have to know what a rational function is and what it means for that function to have a vertical asymptote.

Try using open-ended items as warm-ups, as homework, and—after students are used to them—on quizzes and tests. The students' responses will provide evidence about their mathematical thinking that is different from the kind of information given by the more traditional tasks that we ask of our students. It can only be helpful to have more and different information about our students' mathematical understanding.

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Moon, Jean, and Linda Schulman. Finding the Connections: Linking Assessment, Instruction, and Curriculum in Elementary Mathematics. Portsmouth, N.H.: Heinemann, 1995.

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