Private: Learning Math: Data Analysis, Statistics, and Probability
Classroom Case Studies, Grades 6-8 Part D: Examining Children’s Reasoning (30 minutes)
As this course comes to a close and you reflect on ways to bring your new understanding of data analysis, statistics, and probability into your teaching, you have both a challenge and an opportunity: to enrich the mathematical conversations you have with your students around data. As you are well aware, some students will readily grasp the statistical ideas being studied, and others will struggle.
The problems in Part D describe scenarios from a classroom case study involving children’s developing statistical ideas. Some student comments are given for each scenario. For each student, comment on the following:
• Understanding: What does the statement reveal about the student’s understanding or misunderstanding of statistical ideas? Which statistical ideas are embedded in the student’s observations?
• Next Instructional Moves: If you were the teacher, how would you respond to each student? What questions might you ask so that students would ground their comments in the context? What further tasks and situations might you present for each child to investigate? See Note 6 below.
Mr. Shapple teaches two sections of seventh-grade math. Students in both groups were trying to determine the typical height of a seventh-grade girl and a seventh-grade boy. They measured their heights in inches, combined their data, and then displayed the data on the line plots below:
After comparing the two data sets, here is what the students had to say:
a. Gregory: “The boys are taller than the girls.”
b. Marie: “Some of the boys are taller than the girls, but not all of them.”
c. Arketa: “I think we should make box plots so it would be easier to compare the number of boys and girls.”
d. Michael: “The median for the girls is 63 and for the boys it’s 65, so the boys are taller than the girls, but only by two inches.”
e. Paul [reacting to Michael’s statement]: “I figured out that the boys are two inches taller than the girls, too, but I figured out that the median is 62 for the girls and 64 for the boys.”
f. Kassie: “The mode for the girls is 62, but for the boys, there are three modes — 61, 62, and 65 — so they are taller and shorter, but some are the same.”
g. DeJuan: “But if you look at the means, the girls are only 62.76 and the boys are 64.5, so the boys are taller.”
h. Carl: “Most of the girls are bunched together from 62 to 65 inches, but the boys are really spread out, all the way from 61 to 68.”
The class then made box plots, building on Arketa’s suggestion. This new representation gave them another way to compare and discuss the variance between the two data sets as they analyzed the heights of seventh-grade boys and girls. Here are the box plots they made:
As they compared the box plots, students made the following comments:
a. Arketa: “There is a lot of overlap in heights between the boys and girls.”
b. Michael: “We can see that the median for the boys is higher than for the girls.”
c. Monique: “It looks like just 12.5% of the boys are taller than all of the girls, and maybe about 10% of the girls are shorter than the shortest boy.”
d. Gregory: “The boys are taller than the girls, because 50% of the boys are taller than 75% of the girls.”
e. Morgan: “You can see that the middle 50% of the girls are more bunched together than the middle 50% of the boys, so the girls are more similar in height.”
f. Janet: “Why isn’t the line in the box for the boys in the middle like it is for the girls? Isn’t that supposed to be for the median, and the median is supposed to be in the
To encourage students to discuss ideas of sampling and population, Mr. Shapple asked them to think about what they could say about other classes of seventh graders. Here are some of their responses:
a. Kassie: “I think we would get similar results if we collected data from all the seventh graders in the school district.”
b. Nichole: “I think our data would be spread out more if we got data from seventh graders from all over the country, because then there would be more short kids and more taller kids; we’re probably more in the middle.”
c. Charles: “I think the boys would still be taller, on average, than the girls.”
d. Carl: “I think our data would be similar to other seventh graders in our country, but I don’t think we can say much about seventh graders in other countries.”
If you’re working in a group, make a two-column chart with the labels “Understanding” and “Next Instructional Moves” for recording the group’s responses. You might also want to review the process of making a box plot.
Session 4, Part D: The Box Plot
a. Gregory does not quantify his statement and may only be looking at the upper extreme value. The teacher could ask Greg to determine “how much taller” the boys are than the girls.
b. Marie is comparing the variation in the data and notices overlap in the values. The teacher might ask her to quantify her response.
c. Arketa is considering other representations that might make certain patterns and relationships between the data sets more apparent. The teacher could ask the class to consider additional ways to represent the data that would make some comparisons more visible.
d. Michael correctly determines the median for each data set and quantifies “how much taller” the boys are than the girls by comparing the medians of the data sets. The teacher might ask the other students to react to Michael’s statement and then consider why it can be useful to compare the medians of two data sets.
e. Paul quantifies “how much taller” the boys are than the girls by comparing what he thinks are the medians of the data sets; what he found, though, was the middle of each range and not the middle of the data. This is an opportunity for the teacher to review the meaning of median as well as ways to find the median of a set of ordered data.
f. Kassie believes that she is comparing the modes of the data sets, but when three or more values have the same number of data points, such as the boys, the data is considered not to have a mode. The teacher can review the meaning of mode and ask the students to speculate as to why statisticians say that a data set doesn’t have a mode when three or more values have the same number of data points.
g. DeJuan correctly calculates the means and quantifies “how much taller” the boys are than the girls by comparing the means of the data sets. The teacher could now have the students compare the medians and means of the two data sets. What does each tell us about the data? In this situation, is one comparison more appropriate than the other one? Why or why not?
h. Carl is comparing intervals of the two data sets that contain the most data. The teacher could take this opportunity to focus further attention on the importance of examining intervals in considering how the data are spread out or bunched together.
a. Arketa is comparing the variation by looking at the range of each data set. The teacher might ask her to quantify her response.
b. Michael compares the data sets by looking at the medians. The teacher could ask Michael to point to the median on each box plot and then review that 50%, or half, of the data box plot is on each side of the median.
c. Monique incorrectly reasons that one can further subdivide the lines (or boxes) and that a fractional part of a line reflects a fractional part of the data. The teacher should ask Monique how she arrived at those percentages and then show this same finding on the line pot to see if she recognizes the discrepancy.
d. Gregory is correctly reasoning about the box plots with quartiles. The teacher might ask the rest of the class to evaluate Gregory’s statement for its accuracy.
e. Morgan is correctly reasoning about the spread of the middle 50% of the data on the box plots. The teacher might ask the rest of the class to evaluate the accuracy of Morgan’s statement.
f. Janet does not understand how the box represents quartiles of the data. The teacher could go back to the line plots of the data and actually draw the box plot directly over the data so that Janet can see the distribution of the data within the quartiles of the box plot.
|a.||Kassie thinks that her class’s sample is representative of the district if the district were defined as the population. The teacher could ask the other students to give reasons for supporting or refuting Kassie’s conjecture.|
|b.|| Nichole uses personal judgment about her class’s data probably being “more in the middle,” but she is correct in
thinking that a larger sample would increase the range of the data, as a larger sample might reveal that her
classmates are more homogenous in comparison to other seventh-grade classes, if the population were defined as
the country. The teacher might ask Nichole why she thinks her class is “more in the middle,” and then ask the rest
of the class to react to her conjecture.
|c.|| Charles is thinking about measures of center. The teacher might ask Charles to explain further what he means by
“on average.” Is he referring to the mode, the median, or the mean?
|d.|| Carl thinks that their sample is representative of the country if the country were defined as the population, but not if the population included seventh-graders from other countries. The teacher might use this as an opportunity to
discuss further reasons for defining the population being investigated.
Session 1 Statistics As Problem Solving
Consider statistics as a problem-solving process and examine its four components: asking questions, collecting appropriate data, analyzing the data, and interpreting the results. This session investigates the nature of data and its potential sources of variation. Variables, bias, and random sampling are introduced.
Session 2 Data Organization and Representation
Explore different ways of representing, analyzing, and interpreting data, including line plots, frequency tables, cumulative and relative frequency tables, and bar graphs. Learn how to use intervals to describe variation in data. Learn how to determine and understand the median.
Session 3 Describing Distributions
Continue learning about organizing and grouping data in different graphs and tables. Learn how to analyze and interpret variation in data by using stem and leaf plots and histograms. Learn about relative and cumulative frequency.
Session 4 Min, Max and the Five-Number Summary
Investigate various approaches for summarizing variation in data, and learn how dividing data into groups can help provide other types of answers to statistical questions. Understand numerical and graphic representations of the minimum, the maximum, the median, and quartiles. Learn how to create a box plot.
Session 5 Variation About the Mean
Explore the concept of the mean and how variation in data can be described relative to the mean. Concepts include fair and unfair allocations, and how to measure variation about the mean.
Session 6 Designing Experiments
Examine how to collect and compare data from observational and experimental studies, and learn how to set up your own experimental studies.
Session 7 Bivariate Data and Analysis
Analyze bivariate data and understand the concepts of association and co-variation between two quantitative variables. Explore scatter plots, the least squares line, and modeling linear relationships.
Session 8 Probability
Investigate some basic concepts of probability and the relationship between statistics and probability. Learn about random events, games of chance, mathematical and experimental probability, tree diagrams, and the binomial probability model.
Session 9 Random Sampling and Estimation
Learn how to select a random sample and use it to estimate characteristics of an entire population. Learn how to describe variation in estimates, and the effect of sample size on an estimate's accuracy.
Session 10 Classroom Case Studies, Grades K-2
Explore how the concepts developed in this course can be applied through a case study of a K-2 teacher, Ellen Sabanosh, a former course participant who has adapted her new knowledge to her classroom.
Session 11 Classroom Case Studies, Grades 3-5
Explore how the concepts developed in this course can be applied through case studies of a grade 3-5 teacher, Suzanne L'Esperance and grade 6-8 teacher, Paul Snowden, both former course participants who have adapted their new knowledge to their classrooms.