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Characteristics of Open-Ended Questions

Open-ended questions require students to communicate their mathematical thinking, thereby providing teachers with valuable information that can inform their teaching. An open-ended item should:

  1. Involve significant mathematics. Assessment items, particularly open-ended ones, tell students what is valued and what is important. Consequently, it is critical that open-ended mathematics assessment items involve significant mathematics. Open-ended items often have several objectives, thereby giving students opportunities to demonstrate their understanding of connections across mathematical topics and how mathematics can model real world phenomena.

  2. Elicit a range of responses. Unfortunately, when the answer to a problem is a single number or mathematical object, students often conclude there is only one way to solve the problem. Items that require students to explain their thinking are more likely to encourage a wide range of responses because not all students think alike. Consider the question Can an equilateral triangle have a right angle? Why or why not? Typically, students focus on the angles and conclude that it is not possible, because all the angles of an equilateral triangle must have the same measure and a triangle cannot have 270 degrees. But one student with whom we worked concentrated on side lengths: It can't. 'Cause if it had a right angle it would have a hypotenuse. And that would be the longest side. But all the sides are equal. So it can't happen. What joy this student experienced when the teacher pointed out to the class this unique way of thinking about the problem!

  3. Require communication. One of the real strengths of using open-ended items is that, by design, students are given opportunities to communicate their thinking. Consider this question: Mary claims that you can find the area of any 30-60-90 triangle given the length of only one side. Is Mary correct or not? Justify your answer. Here's one response: Mary is right. If you know one side you can either divide by or 2 or multiply by 2 or . Then you can just multiply the height and the base, divide by 2, and you got it. Here's another response: Mary goofed. The angles are all different so the side lengths are all different. Knowing just one side is a start but you have to have two sides (base and height) to get the area. The first student sees the relevance of the relationship among the sides of a 30-60-90 triangle, whereas the second student, who may be aware of this relationship, does not see its relevance in the context of this problem. When students are required to communicate their reasoning processes, we have a better chance of understanding what they know and can apply it to a given problem.

  4. Be clearly stated. The fact that a question is open-ended should not blur its intent. The question should have a clear purpose even though there might be many possible responses. Further, students should know what is expected of them and what the teacher expects as a good and complete response. Many teachers find that sharing a variety of responses with their students and asking them to evaluate these responses helps students determine what constitutes a good response. Because students are often not used to explaining their thinking in writing in a mathematics class, it is important to help them develop their communication skills and their ability to analyze how well their writing communicates their reasoning.

  5. Lend itself to a scoring rubric. Every assessment item lends itself to at least a two-point scoring rubric: right or wrong. But the purpose of open-ended questions is to provide students with the opportunity to communicate their understanding in something other than a right vs. wrong scenario. The issue thus becomes one of whether it is possible to conceive of responses that have some value (better than a score of 0) but are not worthy of full credit. Giving students partial credit is a familiar notion, and using a rubric formalizes the process to help ensure fairness. One criterion for a good open-ended question is that it will elicit responses that are amenable to partial credit according to some established rubric. A question like Can an equilateral triangle have a right angle? does not, in itself, allow an assessment that involves partial credit. But the follow-up question Why or why not? allows for a variety of approaches.