The Nature of Open-Ended Assessment Items

The Nature of Open-Ended Questions

Many of the questions we traditionally ask students call for a single number, figure, or mathematical object. For example, asking what the prime numbers between 10 and 20 are requires students to identify specific numbers. Similarly, asking students which triangles in a set of triangles are congruent requires them to identify specific objects. These kinds of questions are closed ended because the expected answers are predetermined and specific.

In contrast, open-ended questions allow a variety of correct responses and elicit a different kind of student thinking. Consider the following questions:

Suppose you forgot what 8 x 6 is, but you remembered that 5 x 6 is 30. How could you use this fact to figure out what 8 x 6 is?

Divide and label the garden plot below so that 50% of the garden is planted in peas, 25% is planted in beans, 15% is planted in corn, and 10% is planted in carrots.

Give a reasonable equation for g(x) in the form g(x) = a(x - h)² + k. Explain why your equation is reasonable.

The following elementary students' responses to the first question clearly indicate that the two students conceptualized the problem differently yet correctly.

Response 1:
Response 2:

Both responses demonstrate the ability to decompose the original multiplication problem into subproblems. The open-ended nature of the question allows students to demonstrate their own ways of solving the problem.
Similarly, consider these two responses to the second question.
Response 1:

Response 2:

Although both responses are correct, each student made a different decision about how to subdivide the rectangle.
There are many correct responses to the third question, as long as -1 < a < 0, h > 0, k > 0, and h is approximately four times k (assuming the scaling on the x and y axes is the same). One correct response is shown below.

Both closed-ended and open-ended questions are appropriate for assessing students' mathematical thinking. A test consisting solely of open-ended questions would take an inordinate amount of time to grade and might not cover the curriculum adequately. Closed-ended questions are a reasonable way to sample students' understanding of a broad range of topics. But closed-ended questions do not allow students to reveal their thinking processes as well as open-ended questions.