Developing student understanding: Contextualizing calculus concepts

This study looked at the practice of one high school teacher who provided students with concrete examples from their physics class to give them a contextually rich environment in which to explore the abstractions of calculus. Students discovered connections between the physics concepts of position, velocity, and acceleration and the calculus concepts of function, derivative, and antiderivative. The qualitative study sought to describe several critical aspects of understanding: students' ability to explain concepts and procedures, to apply concepts in a physics context, and to explore their own learning. It included 32 seniors at a large, urban, comprehensive, religious school in a midwestern state. Samples of student work and reflections were collected by the teacher, as well as by students in individual portfolios. The teacher kept a reflective journal. This study suggests that making connections between calculus and physics can yield deep understandings of semantic as well as procedural knowledge.

The National Council of Teachers of Mathematics (NCTM, 1991) has called on teachers to facilitate students' deep understanding of mathematical concepts rather than apply old models that primarily emphasize mathematical manipulation and formal proof development. Wiggins (1993) supported this notion and stated,

Understanding is something different than technical prowess; understanding emerges when we are required to reflect upon achievement, to verify or criticize-thus to re-think and re-learn-what we know. Understanding involves questioning... the assumptions upon which prior learning is based. (p. 8)

Boix-Mansilla and Gardner (1997) concurred and noted, "Understanding is not merely a ... set of loosely organized actions. Rather, understanding is the ability to think with knowledge according to the standards of good practice within a specific domain such as math, history, ceramics or dance" (p. 382). New definitions of understanding in the mathematical sense accentuate students' ability to understand semantic knowledge (i.e., knowing facts and concepts and how they connect,) and relate it to procedural knowledge (i.e., knowing how and when to use skills and strategies) (Romberg & Carpenter, 1986).

Wiggins and McTighe (1998) identified several interrelated aspects of understanding, including explanation, interpretation, contextual application, perspective, empathy, and self-knowledge. The facets are not hierarchical or mutually exclusive; all of them are not applicable in every learning situation. Of these six components, explanation, contextual application, and self-knowledge can be consistently demonstrated in the realms of mathematics and science. The Calculus Consortium based at Harvard University (Kennedy, 1997), as well as NCTM (1991), supported the development of these areas. Students who can explain their ideas give evidence of understanding by making connections and inferences. Those who can apply knowledge demonstrate their ability to use what they have learned in complex situations. Finally, those who exhibit self-knowledge recognize the limits of their understanding.

In order for students to demonstrate depth of understanding, their learning experiences must provide them with the proper tools and contexts to do so. Instructional strategies closely tied tothe constructivist philosophy (Brooks & Brooks, 1993) provide the arena in which students develop skills related to explanation, application, and self-knowledge. When topics are generative, when goals about understanding are explicit, and when students encounter multiple performance opportunities, they are able to demonstrate a deep understanding of content (Perkins & Blythe, 1994).

Wiggins and McTighe (1998) identified the ability to see ways in which concepts and skills connect as essential to students' ability to explain their understanding. This ability to see connections is inherent in the other two facets of understanding, application and self-knowledge, as well. In authentic application, students are challenged to connect key ideas from a discipline to elements within a specific context. As evidence of self-knowledge, students articulate the connection between their performance and the particular demands of a task.

Both the NCTM Curriculum and Evaluation Standards (1989) and the Professional Standards for Teaching Mathematics (1991) emphasized the importance of connection in mathematics curriculum and instruction. Standard I of the Professional Standards for Teaching Mathematics, for example, recommends that teachers design tasks that "stimulate students to make connections and develop a coherent framework for mathematical ideas" (p. 25). Standard 4 states that a teacher should involve students in mathematical activities and discourse that facilitate an "understanding of mathematical concepts, procedures, and connections" (p. 89). Teachers should also "represent mathematics as a network of interconnected concepts and procedures; and emphasize connections between mathematics and other disciplines and connections to daily living" (NCTM 1991, p. 89).