ConcepTests^TM: An Effective Tool for Peer Instruction

I have had the opportunity to teach college courses in both physics and mathematics. In physics, I came across a method of instruction called "peer instruction." This invaluable technique transforms the student's role from that of a passive observer to an active participant. The effectiveness of this method is well documented in physics through Eric Mazur's work at Harvard, and I have had similar success with it in mathematics.

In the 1999 academic school year, I taught an introductory single variable calculus course. Having used peer instruction to teach physics, I was reluctant to return to a standard lecture method. So, I adapted peer instruction for calculus. The results were extremely positive. In the beginning of the second semester I was able to test calculus students taught using peer instruction against those instructed in a more traditional manner. Testing demonstrated that the former were markedly better at solving problems and dramatically better prepared to answer conceptual calculus questions. These results are currently being published in Primus.

Basic Methodology
The peer instruction method is as follows: Given a weekly class meeting time of 160 minutes, the class is broken into two parts: concepts and problem solving. Approximately 120 minutes are spent on concepts while the remaining 40 minutes are spent on problems. The conceptual presentation focuses on a tool called a "ConcepTest." Each 15-minute segment of class is used to focus on a certain concept. The first 7-8 minutes of this segment are used to introduce the ideas via lecture. The students are then given a ConcepTest. The ConcepTest, presented on a transparency, is a multiple-choice question that is conceptual in nature. Students are given one minute of silence to read the question and decide on an answer, after which they vote for the answer they have chosen. Students are then given 2 minutes to try and convince their neighbor of the correctness of their approach. Students then vote again and the remainder of the 15 minutes is spent in discussing of the correct answer. The next 15-minute segment of class focuses on a new ConcepTest.

ConcepTest Development 
The most difficult aspect of this process is finding ConcepTests that correctly target the class's level of understanding after the introduction. For example, if a very small percentage of the class votes for the right answer, the discussion can be a fruitless endeavor. On the other hand, if all the students get the right answer, the discussion can be equally unproductive. If, however, approximately fifty to eighty percent of the students get the correct answer, a dramatic increase in correct answers is usually noted after discussion. To better understand what a ConcepTest is, consider the following. At the top of the transparency is a function, e.g.,, followed by the question, "Which of the following represents the derivative of the above function?" The remainder of the transparency shows the graphic representations of four different functions, labeled A through D. One of these graphs represents. The intention of this ConcepTest is to have the students arrive at the answer through a discussion of what it means to take a derivative. This ConcepTest was used as one in a series of similar queries. The latter ConcepTests in the series involved functions that would be very difficult to identify in their graphic forms. In adapting this method for use in calculus, it was necessary to develop sequences of ConcepTests for many areas of calculus. My intention was to use ConcepTests to probe every topic I covered. I have currently created over two hundred ConcepTests for two semesters of a single variable introductory course.

Other Peer Instruction Elements
Using peer instruction in physics as a model, I made several other adjustments to a standard Calculus course. Exams are broken into two parts: standard problem solving and written answers to conceptual questions. Equal worth is given to both sections of the exams. To prepare students for covering a topic in class, I make reading assignments from the text. Given a section of the book with several worked examples, students are quizzed on one or two of these examples. Since this material has not yet been covered in class, the examples on the quiz are identical to the examples worked in the book. A non-trivial portion of the grade is given to these quizzes. This aspect is invaluable to teaching a course like this because I do not have to spend time covering things in class that students can learn on their own. The class can focus on the concepts of calculus without getting too held back by the details. A common question is whether problem-solving skills suffer. As stated previously, students taught with this method have demonstrated more ability in problem solving than those students instructed in a standard lecture. While the testing has been limited in calculus, this same trend has been well documented in physics (Mazur, 1997). It is hypothesized that students' abilities to solve problems increase because they better understand why they are following certain procedures. This has also been demonstrated in physics to lead to better long-term retention of the subject. For me, one of the most valuable aspects of this teaching method is the constant feedback I get from the students. It enables me to better gear the course toward their needs, as their knowledge gaps are evident well before test time. Also, with a significant portion of the class time focused on concepts, I have noticed a shift in students' motivations. I see them start to focus more on why the answer is correct as opposed to just getting the correct answer. I believe this is another reason why students' problem solving abilities increase. Putting aside all of these benefits, it is a great deal of fun to discuss the ideas of calculus with a group of students who have become receptive. At the very least, they never fall asleep in class!

Some ConcepTest Examples
Consider the graph of the square root of x shown below. The square root of which of the following numbers can be most accurately approximated by the tangent line at x = 4? Is the approximation an underestimate, an overestimate, or neither? Why?

A. 4.2
B. 4.5
C. 3.9
D. 9.2
E. all of the above with equal accuracy

Choose all answers below that could be true from the following statement: If s(t) gives the position of an object at time t and the object is slowing down, then we could have

A. s'(t) negative and s''(t) negative
B. s'(t) negative and s''(t) positive
C. s'(t) equal zero
D. s'(t) positive and s''(t) positive
E. s'(t) positive and s''(t) negative

Which of the following graphs could represent the second derivative of the function below?

A

B

C

D

References
Eric Mazur, Peer Instruction: A Users Manual, Prentice Hall, 1997.

Editor's Note: For more about peer instruction and examples of ConcepTests in physics, visit http://galileo.harvard.edu. ConcepTest is a trademark of Eric Mazur.