Some of the most impressive technological advancements are being developed for the field of mathematics. For example, the advent of programmable graphing calculators has added a new dimension to the learning and teaching of mathematical functions and concepts. Among their many features, these calculators quickly convert numeric sequences into graphical representations. This technology is touted for its ability to let students spend less time worrying about the actual calculations and more time interpreting and conceptualizing the ideas represented by the numbers, by a graph or by a set of functions. However, Stanic & Kilpatrick (1988) suggest that "problems have occupied a central place in the school mathematics curriculum, but problem solving has not" (p.254). Thus, although the technology affords the opportunity to spend time conceptualizing and interpreting, students still have difficulty understanding what the quantified solutions represent in terms of the real-world. Take for example the following modified question used in a high school freshman algebra class.

Football problem: When a football is punted, it goes up into the air, reaches a maximum altitude, then comes back down. Assume, therefore, that a quadratic function is a reasonable mathematical model for this real-world situation.

Let d= number of feet the ball is above ground.
a. When the ball was kicked it was 4 feet above the ground. One second later, it was 28 feet about the ground.
Two seconds after it was kicked, it was 20 feet up. Write the particular equation for expressing d in terms of t.

b. Find the d and t coordinates of the vertex, and tell what each represents in the real-world

c. Draw a graph of the function.

d. By looking at your graph and thinking about what it represents, figure out the domain for this function. Why would your model not give reasonable answers for d when the value of t is

i. below the lower bound of the domain?

ii. above the upper bound of the domain?

e. What influences in the real-world might make your model slightly inaccurate within the domain?

 

The students easily solved for the answers to each of the questions. The use of TI-82 calculators facilitated their efforts by decreasing the amount of cumbersome pen and pencil calculations once necessary to solve for 'x' or 'y'. The complexity in solving the problems has thus decreased while the complexity in interpreting the quantified explanations of such real-world events has increased. For instance, in solving for the correct answer when t=0, the right answer is -.09, given the equation, Y=16x²+40x+4. But, the answer while it was right, might not be reasonable when considering the real-world application. Yet, the students did not consider the right vs. a reasonable answer until the teacher prompted them. In essence, the students were not being problem solvers, but instead they simply solved the problem. Although a subtle distinction, the problem solver would question the validity of the right answer, -.09. In the real-world, -.09 signifies that the ball actually punctured the ground before being kicked by the punter. Only in extreme cases, will a football pierce the ground and only if the hike was fumbled and the punter recovered the fumble could -.09 be both the right and reasonable answer.

This problem was given to the students so they could analyze it so as to make the cognitive link between the numbers, the right answers and what is true in the real-world. However, it appears that there is still a gap in their knowledge base because they could not readily apply the complex math concepts to their understanding of the real-world. Research by Kaput & Roschelle (1996) has suggested that in the average math classes, teachers tend to present a reductive view of mathematical concepts. The complex math concepts are oversimplified in hopes that the students would better understand them. But as a result, the students tend to perceive math from a single perspective, the textbook perspective. The answers they compute represent the school-math perspective, not a real-world perspective. Consequently, when asked to judge the validity of the right answer in real-world terms, students lack the ability to transfer the answers into a qualitative, alternative real-world setting. As is seen with the football example, the students initially perceive the event in terms of numeric, school-math representations and not as a situated, real-world event.