Learning Math: Data Analysis, Statistics, and Probability
Classroom Case Studies, Grades 3-5 Part D: Examining Children’s Reasoning (30 minutes)
As this course comes to a close and you reflect on ways to bring your new understandings 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 below describe scenarios from a classroom case study involving children’s developing statistical ideas. Some student comments are given for each scenario. For each student in Problems D1-D3, 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 students would ground their comments in the context? What further tasks and situations might you present for
each child to investigate? See
Ms. Johnson’s fourth-grade class was examining height. They measured their heights in inches and then displayed their data on the line plot below.
After plotting their data, here’s what the students had to say:
|a.||Damon: “The tallest person is 64 inches, and the shortest person is 52.”|
|b.||Juanita: “I think 64 is an outlier, because there’s a gap at 63.”|
|c.||Asher: “We don’t have a mode, because 55 and 56 are the same.”|
|d.||Larie: “Most of us are from 54 to 58 inches tall.”|
|e.||Michael: “The median is 58 because it’s the middle of the range.”|
|f.||Ali: “The range from 50 to 65 is 15.”|
|f.||Antrell: “I think the median is 57, because it’s the middle of our heights.”|
Ms. Johnson then told the class that they were going to measure the heights of the first-grade class. She asked the students, “What do you think will be true about the first graders?”
|a.||Ava: “I think they are going to all be shorter than us because they’re only in first grade.”|
|b.||Nichole: “Maybe their data will be more bunched together than ours because it seems like lots of first graders are about the same height.”|
|c.||Houa: “They’re going to be smaller than us so I would say [a typical height is] probably in the 40s.”|
|d.||Charles: “I think [a typical height is] maybe three feet, so that would be 36 inches.”|
The line plot shows the first graders’ height measurements:
As the fourth-grade students compared their height data to that of the first graders, they made the following comments:
|a.||Asher: “They are lots shorter than us.”|
|b.||Charles: “They are taller than I thought because they are more like three and a half feet tall.”|
|c.||Ali: “Most of [the first graders] are 42 inches tall, but most of us are 55 or 56 inches, so we’re 13 or 14 inches taller.”|
|d.||Nichole: “I think we’re 13 inches taller because our median is 56 and their median is 43.”|
|e.||Tarra: “Wow, I didn’t think any first graders would be as tall as us, but that kid is 52 inches tall.”|
|f.||Juanita: “Their heights are more spread out.”|
|g.|| Larie [responding to Juanita’s statement]: “I don’t really think their heights are more spread out because most of them are from 41 to 44 inches, and that’s only three
inches, but most of ours are more from 54 to 60, and that’s six inches.”
For more information about statistics problems for children like the problems in this session:
Russell, Susan Jo; Corwin, Rebecca B.; Rubin, Andee, Rubin; and Akers, Joan (1998). The Shape of the Data. Dale Seymour Publications.
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 to Problems D1-D3.
|a.||Damon is focusing on the extreme values and the range. The teacher could reinforce the meaning of range.|
|b.||Juanita incorrectly reasons that 64 is an outlier because there is no data point at 63. However, with 64 separated by only one inch from the other values, it is not unusual enough to be considered an outlier. The teacher might ask the rest of the class to discuss further the meaning of an outlier.|
|c.||Asher incorrectly reasons that there can be a mode only when one value has more data points than any other value. In this example, 55 and 56 are both considered modes. The teacher might use this as an opportunity to discuss the meaning of mode.|
|d.||Larie is looking at an interval of the range where most of the data is concentrated. The teacher might ask Larie to explain how she reached this conclusion and then ask the class to consider why it can be helpful to look at these smaller intervals of concentrated data.|
|e.||Michael incorrectly reasons that 58 is the median by finding the middle of the range from 52 to 64 instead of the middle of the data points. The teacher might begin by reviewing the meaning of median and then ask all the children to line up from shortest to tallest and, using themselves, find the median. Now the class would need to discuss the discrepancy between the median Michael proposed and the one the class found.|
|f.||Ali incorrectly thinks the range comprises all the numbers shown on the line plot, even when the numbers do not contain any values. This error arises if students are accustomed to seeing line plots that almost always begin and end with values that contain data. The teacher could use the following questioning to get Ali to focus on the data points: “What is the smallest height of someone in our class? Point to it on the line plot with your left hand. What is the tallest height of someone in our class? Point to it with your right hand. This distance from the lowest to the highest number is what we call the range.”|
|g.||Antrell correctly reasons about the median and finds it accurately. The teacher might want Antrell to show the class how he found the median and then ask the class to consider why it is important to find the median of a data set.|
|a.||Ava is thinking about the whole set of data about first graders in comparison to the fourth graders’ set of data. The teacher might ask other students whether they agree and to explain why or why not.|
|b.||Nichole is thinking about the spread of the data the class will collect. The teacher could ask the students to consider the spread of their own height data and to quantify their predictions for the range of height data of the first graders.|
|c.||Houa is thinking about the range of the data the class will collect. The teacher might ask the students to get out some measuring tapes and mark on the wall the heights from 40 to 50; once the children have looked at these heights, the teacher can ask them to react to Houa’s statement.|
|d.||Charles is thinking about a specific value, most likely the mode. The teacher might ask Charles if he thinks all first graders, a few of them, or most of them will be three feet tall.|
|a.|| Asher does not define “lots,” nor does she mention that one of the first graders is as tall as a fourth-grader. The
teacher could ask Asher to explain what she means by “lots” and ask whether all the first graders are shorter than
all the fourth graders.
|b.|| Charles is reasoning by focusing on the mode of 42 inches. The teacher could reinforce the meaning and use of the
|c.|| Ali is quantifying “how much taller” the fourth graders are than the first graders by comparing the modes of the
data sets. The teacher might ask Ali why she compared th.e modes and then ask the rest of the class to consider the
advantages and disadvantages of this approach.
|d.||Nichole is quantifying “how much taller” the fourth graders are than the first graders by comparing the medians of the data sets. The teacher might ask Nichole why she compared the medians and then ask the rest of the class to consider the advantages and disadvantages of this approach as compared to using the modes.|
|e.||Tarra is comparing the data sets and thinking about outliers as she reasons that it is unusual for a first grader to be 52 inches tall. This would be an opportunity to discuss further the meaning of outlier.|
|f.||Juanita is comparing the range of the two data sets. The teacher could ask Juanita to explain how she reached this conclusion and what this tells us about the heights of first graders.|
|g.||Larie is thinking about intervals of the data that contain most of the data. The teacher might use this as an opportunity to focus further attention on the importance of examining intervals when considering how the data are spread out or bunched together.|
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.