Example 6. Figure 7 shows a problem asking students to complete a table showing conversions of angle measurements -- it illustrates a matching question of type (i). The figure shows two of the correct answers already selected from drop down lists.

Figure 7: Matching question – unknown to answers

This task requires calculation. Students need to know the meanings of and relationships among revolution, degree measure, and radian measures and be able to apply these relationships in particular situations. According to Bloom’s taxonomy, this illustrates an application-level task. We expect that the students have developed a good understanding of the relationships among revolutions, degrees, and radians and are able to apply that understanding in particular situations. Of course, if students just memorize the formulas for conversion between different measures, the task is no more than at the knowledge level.

Example 7. Figure 8 presents a problem concerning the parametric equations of a line segment. It illustrates a situation in which students would need to perform some algebraic calculations to find the answer. Also, they need to identify the graph that corresponds to a given set of parametric equations and then identify the xy-equation of that graph. Based on Bloom’s taxonomy, this is an application-type task.

Figure 8: Matching question – graphs and algebraic manipulation

Example 8. The problem shown in Figure 9 illustrates a type (i) matching question in which students need to use the relationship between two coordinate systems (one a rotation of the other) to find an equation of a parabola in the xy-coordinate system before proceeding with some of the other parts of the equation. The figure shows the correct answers already selected for some parts of the problem.

Figure 9: Matching question – answers to unknowns

This is a multi-step task at the analysis level, according to Bloom’s taxonomy. It requires students to understand a transformation of the coordinate system (in this case, rotation) and to find the formula of the conic section in the transformed system. Furthermore, the task requires students to use algebraic properties of parabola to find the coordinates of the focus and of x- and y-intercepts. Note that the answers to the questions are listed on the left column together with multiple distracters. Choices in the right-hand column illustrate the possibilities for students to choose from for any answer in the left hand column.

Example 9. Figure 10 illustrates how a matching question may be used for a problem concerning a mathematical proof of a trigonometric identity. This is an example of our second type of matching question.

Figure 10: Matching question – proof

The cognitive activity involved may not be identical to that of writing a proof, but it does involve several items that are important to reading and understanding proofs. Students must examine the overall form of a verification of a trigonometric identity, and they must consider how the different steps are related to one another. According to Bloom’s taxonomy paradigm, this is a high level task -- synthesis -- which requires a student to correctly assemble the steps of a proof, by transforming trigonometric expressions to determine the relationship between steps and to recall relevant definitions. If students were simply asked to write a proof of the trigonometric identity, they would also have to construct the parts by applying algebraic techniques. Often students' algebraic skills prevent them from reaching the synthesis portion of the task. By supplying the necessary expressions, this problem provides a learning experience for such students.

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