Students' Invented Displays

Noticing similar values.

Less often, students notice that values tend to cluster. Their displays reflect this organization. Two examples are provided below. Notice that the first orders the groups or "bins" of data from greatest to least (however imperfectly). The second is closer to convention and allows viewers to see a hill in the data.

Different senses of shape afforded by case value graph vs. grouped-value display

A group of students compared the case-value graph and the grouped-value based display above. A student said, “I think theirs (the case-value graph) are started from the bottom and goes up but doesn't come back down because they didn't do it like ours. They just did bars instead of doing different columns like what we were doing, 90s, 80s, 70s, 60s, and so on.” The student explained how design choices made differences in shapes of the two graphs. The case-value graph went up like stairs because creators ordered measurements from least to greatest and used lines to represent magnitude of each measurement. However, the grouped-value graph made a “mountain shape” because its creators used bins of 10s.

To enhance students' understanding of each graph and emerging shapes of graphs, the teacher asked what-if questions. For example, “What would the graph (the case-value graph) look like if the whole class got 193?” A student answered, “It would just be the same line all across.”

The teacher found that it useful to talk about how similar values could be identified from both graphs, so the teacher asked students, “If I want to find this group in there (grouped-values graph), what do I look for here (the case-value graph) ? Can I find them easily?” A student said that it was very difficult because they had to go across to read each value from the y-axis. Another student said that she would look for plateaus to find similar values.

Chumps and Holes [See graph]

A group of students were interested in values that were missing, so they drew a number line and differentiated observed values and missing values by using colors and text sizes. Also, the creators put Xs to represent frequencies of each observed value. Students said that the graph was different from other graphs because it used the number line and Xs without noticing the implication of the graph. The teacher asked students, “What does the number line help us see something we haven't seen so far from other displays?” A student answered that the creators put numbers in between so she could see how far they went. The teacher went on to emphasize how the display helped everyone notice gaps and holes in the measurements.

Using the grouped-value displays, the teacher asked students about their best guess of the length of the arm-span. Students were drawn to the center clump evident in the grouped-value display, suggesting that the “real length” of the arm span was in the 150s, because most values were in that bin. The teacher asked, “Why are we getting graphs that look like this (normal curve)? Some values are under 150s and some values are over 150s.” Students' reasoning tended to focus on their perceptions of “mistakes” when measuring. For example, some students left gaps when they iterated (moved) the ruler, others overlapped the rulers, and some others miscounted or measured with different units. The teacher asked students to consider each form of error and to judge whether or not it would lead to an overestimate of the length (overlaps) or an underestimate of the length (gaps).

 

The teacher also asked: “What will happen if we measure again, the same way using the same tool? Students answered that they probably got different measurements. One student suggested that students who were in 170s or 180s would be more careful what they were doing second time and so they would get better measurements like 140s, 150s, or 160s. The same student said that students who underestimated the first time would also tend to measure more carefully, so their measurements would more likely be in the middle bins of the data.