**W**hy Use Open-Ended Questions?
Students aren't all alike. For example: Greg always comes up with a different way to approach a problem from anyone else's. He enjoys playing around with mathematics and finding particularly elegant or unusual approaches. Tasha wants to stay "in the box" where she feels comfortable. She prefers copying examples from the board, practicing at home, and reproducing procedures on a test. She does not like surprises. Once she has mastered a procedure, she does not want to cloud her mind with another approach. Alex, if not given explicit step-by-step instructions, will shut down. If you can get beyond Isabel's lack of confidence and her self-proclaimed lack of knowledge, she is often able to create a beautiful way to work a problem that you never considered. Students learn in a variety of ways, and the way they show their knowledge varies as well. How can our instruction and assessment address the needs of students as different as Greg, Tasha, Alex, and Isabel?

One way is to use open-ended questions. The nature of open-ended items allows students to approach problem solving however they choose. Students like Greg can really shine when faced with an open-ended question. Solving a problem for which a solution is not immediately apparent can give students like Isabel confidence in their mathematical knowledge. Even though students like Tasha resist open-ended items at first, they can become more comfortable with them through practice and through guidance about what constitutes quality responses. Students like Alex may not enjoy open-ended items, but if we only give "Alex-friendly" items, then we risk missing opportunities to engage the Gregs and Isabels in exciting mathematics.

Open-ended questions also help us address another need. Often we pay a great deal more attention to *how* to do mathematical procedures than to *when* to do them. We teach mathematics in discrete packages (section 2.1 today, section 2.2 tomorrow). Students learn a particular procedure for the unit test and quickly forget it. The context surrounding the procedures gets lost in carrying out the procedures. For example, consider the question *Can you use the Pythagorean Theorem to determine the length of the unknown side in the following triangle? Why or why not?*

One student responded: *Yes, you can use the Pythagorean Theorem to determine the unknown length, you can do this because you are told 2 of the 3 sides and all you have to do is plug it into the formula and figure it out.*

This student failed to point out that the triangle is not a right triangle and the Pythagorean Theorem is therefore not applicable. We often ask students to use the Pythagorean Theorem only in relation to right triangles. They never have to decide when the theorem is appropriate. The students know *how* to use the Pythagorean Theorem, but they do not know *when* to use it. Open-ended questions do not prescribe a solution in the asking of the question. When students are asked to make decisions, a higher level of thinking is required than when they are asked to mimic procedures. As Moon and Schulman (Heinemann, 1995) explain,

- Open-ended problems often require students to explain their thinking and thus allow teachers to gain insights into their learning styles, the "holes" in their understanding, the language they use to describe mathematical ideas, and their interpretations of mathematical situations. When no specific techniques are identified in the problem statement, . . . teachers learn which techniques the students choose as useful and get a better picture of their students' mathematical power. (p. 30)

Responses to open-ended questions give us insight into how our students think about and what they know about mathematics.

Students develop their own methods for getting right answers. Sometimes their methods are mathematically sound, other times they are not. Students can deceive us into thinking they understand something when in fact they do not. We should take care that our questions do not encourage this deception. For example, when students see a picture like the one following, they hardly need to read the question. They know that when all the sides of a figure have a number beside them, they should add the numbers.

Find the perimeter of the figure.

Does this question really enable us to assess whether a student knows anything about perimeter? Suppose we instead ask a student to draw a six-sided figure with a perimeter of 18. Responses to this question give us more information about a student's understanding of perimeter. Open-ended questions that ask students to generate examples that fit certain criteria enable us to get a better vision of students' understanding of mathematical topics.

Responses to open-ended questions also inform teachers about students' thinking and how they approach problems. This information can in turn influence instruction. For example, one teacher used a method for teaching percent problems that involved setting up the ratio . For example, the problem *What percent of 60 is 30?* could be solved using the following proportion: . She then asked an open-ended question about percentages and one of her students showed an alternative method for answering the question that involved relating the problem to 1%. The teacher realized that the student's method made more sense conceptually than her ratio method. The ratio method might help students get correct answers on percent problems, but did it give her students a conceptual understanding of percent? She decided to use the student's method when she next taught percent. Student-generated methods like this help us understand students' mathematical thinking rather than show us how well they can recite what we told them. We can then design instruction that begins with what the students already know and can do.