### Tom Scavo and Byron Petraroja

Contents || Tom's Math Lessons

## Analysis

On the class data sheet alongside the area measurements, the teacher recorded the number of students in each classroom (see Table 1). These data were obtained from the school office, but could have been obtained directly as part of the primary measurement activity. The children soon realized that the number of students in each room was an important factor to be considered. There were twenty-four fifth graders in our room, for instance, and we felt cramped at times. What if three new classmates joined our group tomorrow? Would we be more or less crowded? Everyone understood that adding students to any given classroom meant less space per person. In fact, that's what we wanted to compute next: how much space did each student have?

The following activity was not only fun, but helped the children understand what we meant by "area per student." Everybody was asked to stand and form a two-dimensional human grid throughout the classroom. We held up a geoboard to illustrate the configuration we wanted to achieve. Students spread their arms and adjusted their position so that they were equally far from each of their neighbors; then they turned ninety degrees and adjusted their position again. We kept turning and adjusting until the students were approximately equidistant from one another. Then we estimated the area per student (by sight) and recorded our estimates.

Next we actually computed the area per student in our own classroom. We had already measured the area of our room, and of course everybody knew how many children there were. Now what should we do with these two numbers? Some students realized immediately that they should divide, while others did not. It's important at this stage to give students (or teams) enough time to realize this fact on their own. We didn't feel it was something that could be taught, but rather something that the students should discover for themselves.

Using calculators, each student computed the area per student in our classroom and recorded his or her answer on data sheets. Once again we emphasized the importance of also recording the units, that is, square meters per student , which we wrote as "". (Note: we considered the alternative notation "", but in the end decided to adopt "", which resembles the familiar notation for miles per hour or "mph".) Finally, we compared our computed answers to the earlier estimates.

We were now ready to compute the area per student in the remaining fifth and sixth grade classrooms. The students paired off, and each team was asked to compute the area per student using the measurement data obtained earlier. As each team completed its calculation, a team representative recorded the results on the class data sheet posted in front of the room (see the last column in Table 3). While the data were coming in, some students began to see differences with respect to area per student in various fifth and sixth grade classrooms. The bar graph in Figure 5 helped them to visualize these differences.

Similar to what we had done earlier (see Table 2), the children were asked to make additional area calculations. Specifically, we asked them:

1. What is the average area per 5th grade student?

2. What is the average area per 6th grade student?

3. What is the average area per 5th and 6th grade student?
These calculations gave us considerable difficulty. For instance, instead of computing
most of the students calculated
which is incorrect. Note that the two answers are nearly equal, which is unfortunate, since it made it difficult for us to convince the students that one was correct and other was wrong.

We also asked the following challenge question:

1. The principal is thinking of converting the teacher's lounge into a classroom. How many students should she assign to this new classroom?

Some of the students answered the latter question with another question, namely, what is the area of the teacher's lounge? This is exactly what we wanted them to say, since it indicated they could apply what they had learned to new situations.

The above average calculations confirmed what we observed in Figure 5: sixth graders have more space, on the average, than fifth graders. We discussed this as a class, and again someone singled out Room 112 as being very different. We wondered what would happen if Room 112 were out of the picture altogether. Would the fifth and sixth grade classrooms be more equitable in terms of area per student? This question led us to look at the data in different ways and to consider other numerical measures that are less sensitive to outliers. One such statistic is the median.

The next class period we proceeded to rank the area-per-student values from smallest to largest. From this ranking, the students easily identified the range of the data, but our immediate goal was to calculate the median. We decided to work initially with contrived data sets of various sizes, first an ordered data set of five numbers and then a set of six numbers. When asked to give the median of the latter data set, some of the students immediately (and correctly) suggested a value midway between the third and fourth numbers, but not everybody understood what we were doing. In fact, we had a difficult time explaining why one might want to "compute" a median in the first place.

We had hoped that looking at other measures of central tendency (median and mode) would help to strengthen the students' understanding of the data. Unfortunately, this was not the case. The students simply could not understand why we were sorting the numbers and picking out the middle one. For most of them, the average seemed good enough.

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