Overview of the Lesson / Preparing for the Lesson / Students' Ways of Thinking / Assessment

1. Posing the Problem

We have found that our measurements do not all agree. There are different measurements, so how will we decide about our best guess of the real height of the flagpole? We can't be sure that one of us is a “super-measurer” who is always right while everyone else is wrong, so your job is to make a method (something to do) that someone else could follow that will result in the best guess for the real height of the flagpole.

When you have decided what to do, write it down so that someone else could follow it. (Students can use diagrams or other graphics.) You can use your display too if that helps.

Teacher Note: The point of this activity is to let students write descriptions of methods that are sufficiently clear and explicit so that other students can follow them. It is also helpful to press on generalization by asking students if their method would work with other, different sets of data. For example, in Lesson 1A, students measure arm-span, head-circumference, and area. If students develop their method with arm-span data, then it might be useful to have them try it out with head-circumference data. Teachers can also play the role of audience. Sometimes teachers role play “robots” who must be told exactly what to do or else they misunderstand and do silly things.

2. Comparing Methods for Finding the Best Guess

Teacher Note: It is important that students do not lose sight of the purpose of the method-to estimate the actual length of the arm-span or head circumference.

Students pass their solutions along to another student (or pair, or other small group), and students present the method of another student (or group), along with their guess about what the student was thinking about. (The number of presentations and content of presentations is determined by the teacher.) Teachers should be alert to thinking that seems to be guided by the “center clump” – perhaps asking students why they think the largest clump is at the center (most data sets collected in this manner will have a bell-shaped distribution).

Teacher Note: Students should be encouraged to consider traditional measures of central tendency (mean, median, mode) if none are proposed. However, the mean is a ratio measure (quantity per case), so if students are not familiar with ratio, then the mean is either an opportunity to introduce ratio or is to be avoided. (This is a judgment best made by teachers.)

Although the computation of the mean is fairly straightforward, students should be encouraged to explore the effects of extreme values on the mean. One way to accomplish this is to propose a measurer who is wildly inaccurate and to explore the consequences of this inaccuracy on the mean. A good value for the inaccurate measure is one that displaces the mean out of the center clump. Students should note that the same extreme value will have little effect on the median. This introduces students to the intuition of robust statistics. Students who reason that if more than one person measured and each obtained the same value, then that measurement is more likely correct, often advance the notion of mode as the best guess. Some measurement data will have multiple modes, which will make mode a poorer choice than a median. However, much depends on the nature of the measures.

 Teacher Note: The role of ordering values to obtain the median is often not well understood by children. One potential way of helping students see the value of order and of counting ordered values is to re-arrange the values so that the “middle” values of the list represent values found at the tails of the distribution. Then ask students if they believe that it is sensible to represent the actual length of the arm-span (or whatever attribute is being measured) by an extreme value. For example, if the values were 5 15 16 17 24, the list might be re-arranged to be 15 16 5 24 17. In this case, 5 splits the data into 2 groups of equal number, but 5 is a poor representation of the center.

For an odd number of cases, the median is one of the case-values, and occasionally students find the dual role as a case and as an indicator of best guess confusing. Students sometimes too suggest that only odd number of cases can have a median, because only then can the number of cases be split in half. It is important to emphasize that a median splits the cases into two 50% regions. Some students find it helpful to think about the median for an even number of cases as a “possible” value—a value consistent with the center clump of the data.

The following assessment helps reveal students' thinking: Example

Lesson 2 Supplement: Exploring the Median