Written by Wendy B. Sanchez and Nicole F. Ice
(News Bulletin, July/August 2005)
Numerous groups, including NCTM and the National Research Council, recommend that classroom instruction should include mathematical experiences that require higherorder thinking. Research also points to the importance of including instructional activities that require higher-order thinking. For example, in a study of data from the National Assessment of Educational Progress, Wenglinsky (2000) found that students whose teachers emphasize higher-order thinking skills in mathematics outperform their peers by about 40 percent of a grade level.
There are many ways to engage students in higher-order thinking. Generally, for such thinking to occur, students need to be engaged in activities where they examine, reason, make conjectures, test hypotheses, solve problems, and communicate mathematically. A variety of instructional and assessment strategies can provide students with these types of experiences. The use of open-ended items is one relatively easy way for teachers to promote higher-order thinking, because there are many resources currently available that offer such items. In fact for the 2004–05 school year, we developed a project* to help teachers at Campbell High School in Smyrna, Georgia improve instruction and students’ mathematical understanding by using insights about students’ thinking that can be gained through the use of openended items.
Our goal was to create a professional development opportunity for mathematics teachers in their own classrooms, enabling them to observe how open-ended items could be used to help their own students. Project activities began with an initial workshop and included group meetings and sustained contact during the school year.
One surprising observation that participating teachers made was that it was not necessarily the students with the highest grades who performed well on open-ended items. Often students who were more difficult to engage during instruction were the very students who constructed the best responses to openended items. The teachers realized that some students were not engaged in traditional lessons and rote exercises because they were bored, not because they lacked understanding.
They also found that some students who were not adept at carrying out procedures were capable of understanding and communicating some of the more conceptual aspects of mathematics. One teacher said the project “opened my eyes to the idea that just because students make F’s [doesn’t mean] that they don’t know anything. Sometimes they are just bored with the procedural (mundane) nature of school.”
Teachers noted that using open-ended items increased the number of students who actively participated during the lesson. And they concluded that items that require students to think on a higher level are appropriate not only for honors classes, but for basic classes as well.
Another interesting observation that teachers made after participating in the project was that open-ended items challenged them to know the mathematics they teach more deeply. Evaluating student responses required the teachers to broaden their own thinking about the mathematics involved. One teacher said that the project “made me a better teacher because I have to think how to work the problem because I have to be ready to look at the students’ work.”
Project teachers reported that they have shifted towards a more conceptual approach to teaching as a result of participating in the project. One teacher said, “I check for depth of understanding as opposed to procedure. Can you explain why it works? Can you explain how? How does this connect?” Another teacher explained that her students’ “mathematical understanding has increased because they see the ‘why’ of the mathematical concept.” The teachers felt that they had made significant progress in using open-ended items during this school year, and many agreed to participate in the project for a second year.
We encourage readers to share the descriptions and results of any assessment initiatives they may be involved in to firstname.lastname@example.org for possible future publication in this column.
Wenglinsky, Harold. How Teaching Matters: Bringing the Classroom Back Into Discussions of Teacher Quality. Princeton, N.J.: Milken Family Foundation and Educational Testing Service, 2000. www.ets.org/research/pic/teamat.pdf.
* The project described in this column was supported through the
Improving Teacher Quality Grants Program.