Meeting Notes 2-12-2001

Present: Joanna, Marilyn, Yongjoo, Tom

In Today's meeting, we first discussed the step by step procedures of the "Comparing Distributions" lesson plan. What we outlined as the sequence of the lesson plan is following:

Preliminary discussion
Virtual Environment Exploration
Follow up Discussion
Calculating percentages and drawing bar graphs
Discussion

In our meeting today, we mostly discussed about the first preliminary discussion, which will be provided to help students understand the basic concepts of comparing distributions and develop their own metric of comparing distributions. In general, students will have a chance to compare distributions with the same populations first and develop their own comparison metric. Then, students will compare the distributions with different populations by normalizing the given data. Finally, by the end of the first discussion, students will be told a little story that would motivate them to go to the Virtual Environment and do the actual exploration and data collection.

In our meeting on 2-9-2001, we decided to give students distributions with the same population first before they get into normalization so that they can tackle one concept at a time. This decision was based on the Joanna and Marilyn's recommendation. If students have to normalize their data even before comparing distributions and developing their own metrics, they would have hard time to digest all the concepts at once. Thus, we decided to have students compare distributions with the same populations first and then go to distributions that require normalization later.
In our meeting on 2-9-2001, we thought that we would have students do the actual survey of "broccoli likeness" in their own classrooms to motivate their discussion. In today's meeting, we decided to give them raw numbers of two different distributions with the same populations because it is unlikely to have the same population number if students do the survey in their own classes. Before developing students' own metrics, students will be asked to compare the distributions qualitatively. They would say something like Group A's students like broccoli more than Group B's or Group A has more students who don't care about the broccoli. Students will be more engaged in this process of comparing distributions qualitatively. Then, we'll tell them something similar to following:

Scientists would like to compare things quantitatively. In other words, they 
would want to put numbers on everything. As being scientists by ourselves, we 
may want to do the same thing. How would you compare the two distributions 
about the broccoli numerically?

By saying the above to students, we hope that they are willing to create their own numeric metrics of comparing distributions. We agreed that developing their own metrics would be more appropriate as small group activities. Several transparencies with a table with raw numbers and bar graphs for each distribution will be provided to each group. Then, students will be asked to discuss how they are going to compare the given distributions quantitatively. We are hoping that they come up with an idea of measuring the actual difference of each bar of the bar graphs and sum them up as the final difference after overlaying the transparencies of two different bar graphs. For example, let's say Table 1 and 2 and Picture 1 and 2 are given to students. Then, we'll guide students to develop metrics of measuring the actual differences in the graph. Thus, students will get 10 unit differences between the group 1 and 2 by measuring the differences between single bars and sum up the results, i.e. |20 - 15| + |10 - 15| + |5 - 5| = 10. Note that students are dealing with absolute values here.

Table 1
Group 1
Like Broccoli Don't like Broccoli Don't Care
20 10 10

Table 2
Group 2
Like Broccoli Don't like Broccoli Don't Care
15 15 10

Table 1
	Group 1
Like Broccoli	Don't like Broccoli	Don't Care
20	10	10

Table 2
	Group 2
Like Broccoli	Don't like Broccoli	Don't Care
15	15	10

After they developed their own metrics, they are going to take one step further and will be asked to compare distributions with different populations. We'll probably provide another distribution, which has the doubled population number with doubled raw numbers for each category like Table 3, which is basically doubled Table 1. By comparing the raw numbers of students for each category to Group 1, we can hook the students into the issue of fairness. For example, we can say something similar to following:

Let's say we are trying to compare table 1 and 3. If we compare the 
number of students who like broccoli, group 1 has 20 and group 3 has 40. Since 
there are more students who like broccoli in group 3, we can say that group 3 
like broccoli more, right?

Students should reject this conclusion instantaneously. They should say something about the population when we make the above claims. If not, we have to guide them to realize the difference of total population in both groups. Then, we can show them the transparencies with percentages and help them understand that Group 1 and 3 actually have the same distributions if they normalize the distributions. Instead of referring this process as normalization, we are thinking of using the term "percentages" instead not to confuse students with new term. Since students have learned percentages before and knows percentages naturally, "percentages" seems to be more appropriate therm than data normalization. Joanna also suggested putting fractions on the graph would confuse students for the normalized graphs so we are going to put whole numbers instead. We also massaged the numbers to make them produce nice whole numbered percentages to simplify the students' understanding.

Table 3
Group 3
Like Broccoli Don't like Broccoli Don't Care
40 20 20

Table 3
	Group 3
Like Broccoli	Don't like Broccoli	Don't Care
40	20	20

At the end of the discussion after they develop their own comparison metrics and understand the concept of normalization, they are going to be told a little story similar to following:

We are working for the Department of Agriculture in United States. We have a 
big field, which is divided into 9 different sections and has different kinds 
of plants and flowers. Because the department will cut the funding for our 
research, we can only have one sector of the whole field next year. Thus, we 
are trying to find the best sector that represents the whole field and try to 
keep that sector. You guys will go out to the field and count all plants in 
your sector and will find out which sector represents the whole field best.

Something similar to the above story would be provided to motivate students to explore and count all plants in their sector. Then, they will do the actual exploration, which might be discussed in our next meeting.