by Danielle Champney, Education, SESAME
Teaching Effectiveness Award Essay, 2010
To a casual observer, Calculus II classes may appear to be a process of sorting through many methods to solve differential equations, infinite series, and integrals. When students view this course as a collection of methods to be learned, I can’t blame them for raising their hands and asking, “What method should I use for this problem?” From my perspective as a GSI, this question suggests a bigger problem, because I view Calculus II as more than just a solution-finding mission or strategy game. Students will learn little or resort to untested pattern-matching if I simply tell them what method to use each time they encounter a new problem! Learning how concepts in class are reflected in procedures used to solve problems is, to me, a core principle of the course. This suggests then that we can rebuild the procedures for ourselves if we have a robust understanding of the concepts of a particular unit of instruction and rely less on memorization. The body of research that demonstrates the importance of building understanding of procedures and concepts simultaneously (e.g., Rittle-Johnson and Alibali, 1999) provides evidence that when there is too much emphasis on memorizing procedures, students may decide that “understanding is not necessary when solving mathematics problems; one simply follows the procedure, whether it makes sense or not” ( Schoenfeld, 1988).
To work toward my instructional goal of using knowledge of concepts as a grounding for learning procedures, as well as to address students’ problem of knowing “what method to use” the very first time it comes up in the semester, I often start the Math 16B chapter on integration techniques with a very specific, structured intervention that spans two days. The intervention begins with a preliminary activity that requires groups of six to eight students to make predictions about generic types of integration problems. After making predictions, the students spend the rest of the class playing a game: Each student is given a mathematical function written on a card, and asked to pair up with as many of his or her group-mates as possible, two at a time, by multiplying the functions on their two cards and integrating the product. New cards are introduced if a group has exhausted all pairings. The range of resulting problems spans the chapter well, and it includes integrals that have only one viable solution path, integrals that can be solved with multiple methods, and integrals for which students’ set of methods is insufficient. This intervention is a game in the sense that it is the groups’ goal to get to as many problems, successfully, as they can. The first day’s homework is a structured, written reflection on the methods used to solve the problems and why they worked. Groups begin Day 2 revising their work based on instructor feedback. Following the revisions, students spend time comparing their reflections and revisiting the predictions they made at the beginning of the previous day, to discuss accurately predicted results and where or why some of their predictions fell short. This is the crucial aspect of the intervention that addresses the student question at the heart of this problem, “What method to use?” After requiring them to make predictions and complete practice problems that intentionally illuminate the concepts underlying the procedures, the act of revising and comparing predictions allows students to critically analyze their work, relate patterns back to core concepts, and make decisions about the appropriateness of the various methods.
Students’ written explanations allow the GSI to ask questions during the exercise; a strategy-based quiz a week later assesses students’ retention of the big ideas in the context of integration; and assessments throughout the semester tap into students’ reasoning about strategy choice in other Calculus II contexts. Finally, an end-of-semester evaluation invites the students to share their opinions of specific class activities. Sample responses from 2009 include the following: “I had a lot of trouble understanding the concept but . . . practicing with the game allowed me to understand the concept. I can easily say that without it, I probably would still be struggling.” “The most useful was the integral project because it helped us understand the nature of the problems.” “I got to see patterns and trends of integrals as well as good practice in tackling integration and choosing how to solve them.”
Thus this intervention both accomplishes one of my goals as a GSI and addresses a student issue that pervades Calculus II, first brought to light in the integration chapter. Encouraging students to explore procedures from a conceptual point of view demonstrates the connections in the mathematics, and it helps students make smarter choices when problem solving, rely less on memorization, and build skills that will help address that pesky question “What method do I use?” later in this and other courses.
Rittle-Johnson and Alibali (1999). “Conceptual and Procedural Knowledge of Mathematics: Does One Lead to the Other?” Journal of Educational Psychology 91, 175–89.
Schoenfeld (1988). “When Good Teaching Leads to Bad Results.” Educational Psychologist 23(2), 149.