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Precalculus students applied the vertical-line test directly to the path of the caterpillar
Connecting Research to Teaching
How would your students respond to the following item?
A caterpillar is crawling around on a piece of graph paper, as shown below. If we wished to determine the creature's location on the paper with respect to time, would this location be a function of time? Why or why not?
Readers might be surprised to learn that 60 percent of the precalculus students who answered this item applied the vertical-line test directly to the path of the caterpillar to determine whether the caterpillar's location was a function of time, that is, that for every point in time, the caterpillar was at exactly one location. What does this response tell us about students' understanding of functions, the vertical-- line test, and interpreting graphs? Read further to find out.
The concept of function plays an important role throughout the mathematics curriculum. The typical mathematics definition of function from x to y is a correspondence that associates with each element of x a unique element of y.
The concept of function is central to students' ability to describe relationships of change between variables, explain parameter changes, and interpret and analyze graphs. Not surprisingly Principles and Standards for School Mathematics (NCTM 2000, p. 296) advocates instructional programs from prekindergarten through grade 12 that "enable all students to understand patterns, relations, and functions." Although the function concept is a central one in mathematics, many research studies of high school and college students have shown that it is also one of the most difficult for students to understand (Tall 1996; Sierpinska 1992; Markovits, Eylon, and Bruckheimer 1988; Dreyfus and Eisenberg 1982).
ALIGNMENT OF CONCEPT IMAGE WITH MATHEMATICAL DEFINITION
Vinner (1992) uses the term concept image to describe how students-as well as adults, for that matter-think about concepts. A person's concept image consists of all the mental pictures that he or she associates with a given concept. A student's concept image can differ greatly from a mathematically acceptable definition; and students' concept images are often very narrow, or they may include erroneous assumptions.
A student's concept image of function may, for example, be limited to the graph of a relation that passes the...