The next concept involved the average area of fifth and sixth grade classrooms. In our experience, students have little difficulty computing an average, but they invariably have problems interpreting what an average represents. For our purposes, the area bar graph in Figure 3 provided a convenient graphical aid with which to discuss averages. As a class, we did the following thought experiment: with a scissors, we cut off the tops of the taller bars and stacked them on top of the shorter ones so that all the bars were the same height. This uniform height was the average area of the fifth and sixth grade classrooms, the same value obtained numerically by adding and dividing (which we were about to verify). We found that the students had little difficulty in performing this thought experiment, and in fact, the class as a whole experienced an "aha!" sensation when shown this graphical interpretation of average.
Since the above visual demonstration was so effective, we chose not to make this concept more concrete. Using construction paper and scissors, however, this could easily be done, and in fact, we plan on doing it next year so that tactile learners can benefit as much as visual learners.
With a better understanding of average, the children were asked to compute the average area and the average number of students in fifth and sixth grade classrooms. This they did rather easily using calculators (see the first two rows of Table 2). To keep the level of understanding high, and for the purposes of evaluation and assessment, each student was also asked to show his or her work, which could be organized vertically or horizontally. For example, we wanted them to write something like
with the approximate areas being obtained by calculator. Some of the students resisted writing the problem on paper, however, arguing that the calculator made it unnecessary. We believe just the opposite is true, and therefore insisted on seeing their work.
Table 2: Average Area and Average Number of Students
For homework, we asked the students to compute averages for the fifth and sixth grades combined. This they did (see the last row of Table 2), but instead of adding all the values and dividing by 14, some of them (incorrectly) computed the average of the two averages. The answers obtained by the two methods did not agree since there were different numbers of fifth and sixth grade classrooms. Computing the average of the two averages, they found the average area of all the classrooms to be
whereas the correct calculation is
using the sums obtained earlier. The fact that the two values turned out to be nearly the same is unfortunate since some of the students thought it was a minor discrepancy due to round-off error.
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