| Â Â Â Â Lesson 1B: Measuring Height |
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Overview
of the Lesson /
Preparing for
the
Lesson / Lesson
Activities / Assessment
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| Â Â Â Â Students'
Ways of Thinking |
    Students' Invented Displays |
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Here are a few of the displays that children invented to show what they noticed about flagpole measurements.
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Idiosyncratic.
One group of 6 th grade students claimed that there was no pattern to the set of measurements of the height of the school's flagpole (the teacher used the word pattern instead of trend when giving directions for creating the displays). As you see in the left side of the image, the measurements were listed randomly. The students said that the numbers did not “help each other.” They meant that the numbers could not be formed from any rule that they could detect (They gave examples of even vs. odd). They also objected that the idea of pattern did not fit into their previous experience with pattern blocks “square, triangle, circle.” The teacher suggested that the purpose of creating the display might be changed to “find a way to show how the data are alike and different.” Although this change in language might support invention of a different kind of display, for many students, the very idea of uncertainty implies an ontological category of “not mathematics.” Ironically, this only bothers students who are apt at generating and detecting patterns, and who understand the importance of generalization of the pattern. [See Graph]
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Noticing
order.
Some students tend to focus on the order of the data. Some values are greater than others.
A student ordered all measurements from least to greatest, and represented heights by bars. (Students often use lines instead of bars). “Typical” or modal values are indicated by plateaus, and variation by the differences among heights. [See
Graph]
This display orders the data and makes use of higher = greater value. It is analogous to the ordered value display but takes a different tact. Rather than dismissing the display as non-standard, it is important to discuss the design motivations of displays like these. For example, the student-designers are using space to order their data. They tried to evoke an image of stair-climbing as the values increased. [See Graph]
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Elaborating order.
Students ordered the cases and displayed their relative frequency as a square icon.
Students ordered measurements from least to greatest, and used squares to notate frequencies of each value. However, note that the interval between case values is not represented. When the teacher asked the students which values would not be likely to recur if they measured again, students pointed to the lowest value. The display made the multi-modal nature of these data visible. The statistics they computed represented residue from past classes—things that one did to batches of data. But they never referred to these statistics again. [See Graph]
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Constructing interval.
In this display, measurements were listed from least to greatest, but these sixth-grade students focused on the relative frequency of values missing from the interval. For example, 0 = 14 refers to the number of values in the interval between 30 feet and 66 feet for which there was no case. The 1 = 9 refers to the number of values in the interval for which there was only 1 case. The teacher used this display to highlight how displays could be structured to hide “holes” that this display drew attention to via the table. [See Graph]
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Grouping and ordering.
This display “bins” or groups the data by interval. It makes visible a different sense of the “shape” of the data. [See Graph]
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| Â Â Â Â Comparing Displays |
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We need two graphs!
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A team of students found that they needed to draw two displays. One was a bar graph with intervals of 10, and the other was a line graph of the values in the 40's of the same data. The bar graph hid the individual values but communicated the relative frequency within each interval. But because most of the values were in the 40's, students decided that these needed to be displayed in finer detail. But classmates found the juxtaposition of styles confusing.
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Relative frequency display vs. Interval display [See Graph] |
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The teacher asked students to compare the two displays. One student mentioned that he could see bins from the interval graph on the right, and another student said that it was easy to notice that there were many 40s. But the modes of particular values were easier to see with the display showing the frequencies of cases display. The teacher asked students to explain why the shapes were so different, yet the data were the same.
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Interval displays with and without holes [See Graph]
The graph on the left appends an extreme value (122) to the decade bin. The stem-and-leaf display on the right makes this hole in the data more visible.
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| Â Â Â Â Relating Display to Process |
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The lesson concludes with students working in pairs (or whole groups) to explain any patterns that they see in the interval display. What is it about the process of measurement that produces what they see? Why didn't everyone get the same value? [Students might advance the idea that the difference is due to differences in their heights. If so, pick a tall and a short student and guide discussion about how far away each person will be from the pole. Will shorter people be further or nearer to the pole in order to create a 45 degree angle?]
What might happen if we do it again?
The teacher asked students, “If we did the measurements again, would we get exactly the same numbers again?” This question was intended to let students think about any patterns in relation to measurement process. Some students suggested that sources of error, such as angle measure or distance measure, would lead to slightly different values. Others cited differences in their heights or shoe sizes, perhaps because they did not understand the triangle model, as leading to different outcomes. A significant portion of students suggested that of course they would get exactly the same values again, especially if they stood at the same spot on the playground. This debate foreshadowed the importance of continued emphasis on repeated process—what will happen if we measure again?
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