Private: Learning Math: Data Analysis, Statistics, and Probability
Classroom Case Studies, Grades 3-5 Part B: Developing Statistical Reasoning (45 minutes)
The National Council of Teachers of Mathematics (NCTM, 2000) identifies data analysis and probability as a strand in its Principles and Standards for School Mathematics. In grades pre-K through 12, instructional programs should enable all students to do the following:
• Formulate questions that can be addressed with data, and collect, organize, and display relevant data to answer them
• Select and use appropriate statistical methods to analyze data
• Develop and evaluate inferences and predictions that are based on data
• Understand and apply basic concepts of probability
In grades 3-5 classrooms, students are expected to use appropriate statistical methods to do the following:
• Describe the shape and important features of a data set and compare related data sets, with an emphasis on how the data are distributed
• Use measures of center, focusing on the median, and understand what each does and does not indicate about the data set
• Compare different representations of the same data and evaluate how well each representation shows important aspects of the data
In grades 3-5, children readily notice individual data points and are able to describe parts of the data — where their own data falls on the graph, which value occurs most frequently, and which values are the largest and smallest. A significant development in children’s understanding occurs as they begin to think about the set of data as a whole. Our goal for children is for them to see a data set as a distribution of values with important features, such as center, spread, and shape.In grades 3-5 classrooms, students are expected to use appropriate statistical methods to do the following:
To focus students’ attention on the shape and distribution of the data, it is helpful to build from children’s informal language to describe where most of the data are, where there are no data, and where there are isolated pieces of data. The words clusters, clumps, bumps, and hills highlight concentrations of data. The words gaps and holesemphasize places in the distribution that have no data. The phrases spread out and bunched together underscore the overall distribution. Teachers must also continually emphasize and help students see that what they notice about the shape and distribution of the data implies something about the real-world phenomena being studied.
In grades 3-5, students learn to use measures of center to summarize a data set. Building on children’s informal understanding of what is the most, what is the middle, and what is typical, teachers can help students develop understanding about the mode, median, and the mean. But students need to learn more than simply how to identify the mode or median in a data set and how to find the mean: They need to develop an understanding of what these measures of center tell us about the data, and what each does and does not indicate about the data set. The emphasis in these grade levels should be on the median, with informal exploration of the mean. Children can see where the median is located among the data, but the mean is much more abstract, as it has no clear identity within the data themselves.
When viewing the video segment, keep the following questions in mind:
• Thinking back to the big ideas of this course, what are some statistical ideas that these students are developing?
• What questions could be posed to determine the extent of students’ understandings of what the mode, mean, median, and range do and do not indicate about the data set?
In this video segment, Suzanne L’Esperance selects a group of students to present their findings. Each group of students created a line plot of the class data on family size and determined the mode, median, mean, and range for the data set. Watch as the group of students takes turns presenting the summary information to the class.
Answer the questions you reflected on as you watched the video:
a. What statistical ideas are these students developing?
b. What questions could you pose to determine the extent of students’ understanding of what the mode, mean, median, and range do and do not indicate about the data set?
This line plot (or dot plot) below displays the family-size data collected by the students in Ms. L’Esperance’s fifth-grade classroom. Imagine yourself in a conversation with the children about this data. A key question you might ask the students is, “What do you notice about the data?” Using the informal language of clusters, clumps, bumps, hills, gaps, holes, spread out, or bunched together, write five statements that you hope students would make describing the set of data as a whole. See
Too often, children describe the data as numbers devoid of context. Another question you should frequently ask students regarding their observations is, “What does that tell us about the family size?”
For each of the five statements you wrote in Problem B2, indicate what that observation might imply about the real-world context of family size.
In another classroom investigation which reveals students’ understanding of the notion of “average,” students were given the following scenario (from Russell and Mokros, 1996):
We took a survey of the prices of nine different brands of potato chips. For the same-sized bag, the typical or usual or average price for all brands was $1.38. What could the prices of the nine different brands be?
Note that the language used — words like typical, usual, or average — keeps the discussion open to various ways that students might think about the notion of average.
Consider how students might respond to this task and then develop three hypothetical student responses that are each based on a different measure of center — mode, median, and mean. See
The potato-chip task was presented to fourth-grade students in individual interviews to research students’ understanding of average. Here are some of the students’ responses:
a. Some students would put one price at $1.38, then one at $1.37 and one at $1.39, then one at $1.36 and one at $1.40, and so forth.
b. One student commented, “Okay, first, not all chips are the same, as you told me, but the lowest chips I ever saw was $1.30 myself, so, since the typical price is $1.38, I just put most of them at $1.38, just to make it typical, and highered the prices on a couple of them, just to make it realistic.”
c. One student divided $1.38 by nine, resulting in a price close to 15¢. When asked if pricing the bags at $0.15 would result in a typical price of $1.38, she responded, “Yeah, that’s close enough.”
d. When some students were asked to make prices for the potato-chip problem without using the value $1.38, most said that it could not be done.
e. One student chose prices by pairing numbers that totaled $2.38, such as $1.08 and $1.30. She thought that this method resulted in an average of $1.38.
For each response above, was the student reasoning about the “average” as a mode, median, or mean?
Read the article “What Do Children Understand About Average?” by Susan Jo Russell and Jan Mokros from Teaching Children Mathematics.
a. What further insights did you gain about children’s understanding of average?
b. What are some implications for your assessment of students’ conceptions of average?
Principles and Standards for School Mathematics (Reston, VA: National Council of Teachers of Mathematics, 2000). Standards on Data Analysis and Probability: Grades 3-5, 176-181
Reproduced with permission from the publisher. Copyright © 2000 by the National Council of Teachers of Mathematics. All rights reserved.
The potato-chip activity is adapted from Teaching Children Mathematics. Copyright © 1996 by the National Council of Teachers of Mathematics.
Used with permission of the National Council of Teachers of Mathematics.
Russell, Susan Jo and Mokros, Jan (February, 1996). What Do Children Understand About Average? Edited by Donald L. Chambers. Teaching Children Mathematics, 360-364.
Reproduced with permission from Teaching Children Mathematics. Copyright © 1996 by the National Council of Teachers of Mathematics. All rights reserved.
Using line plots (dot plots) in elementary classrooms is a fairly new practice. Consider how you might use this graphical representation of data with your students. How does this compare with your current method of presenting data?
a. Statistical ideas included using the mode, median, and mean as measures of center, and the range as an indicator of variation.
b. Answers will vary. Examples of questions would be, “You stated that the range is 3. What does this tell us about the data set?” or “You said that the median is 4. If this is the only information I had asked you to figure out, what wouldn’t I know about the data?”
Here are some possible statements that children might make:
• There is a bump at 4.
• The data are really bunched together.
• There is a cluster at 3 and 4.
• There’s a big gap from 6 to 11.
• The data are not very spread out
• The bump at 4 is the size of families that occurred most often for our class.
• Because the data are all bunched together, we know that families in our class are very similar in size.
• The cluster at 3 and 4 indicates that most families in our class have three or four people.
• The gap from 6 to 11 tells us that no families in our class have six, seven, eight, nine, 10, or 11 children.
• The lack of “spread” in our data tells us that our class’s families are similar in size.
• A response based on the mode might be to make the prices of all nine bags exactly $1.38. Another response based on the mode is to price four bags at $1.38 and the others at $1.30, $1.32, $1.36, $1.37, and $1.50. The reasoning is to place more bags at $1.38 than at any other price.
• A response that is based on the median is to make three bags cost $1.38 and the others cost $1.30, $1.30, $1.35, $1.40, $1.47, and $1.49. The reasoning is to put some bags at $1.38 and then to place an equal number of bags at prices lower and higher than $1.38. Here, three bags cost more than $1.38 and three bags cost less than $1.38.
• A response that is based on the mean is to make the bags cost $1.38, $1.37, $1.39, $1.36, $1.40, $1.35, $1.41, $1.34, and $1.42. Since there’s an odd number of bags, the reasoning is to place one bag at $1.38 and then add and subtract the same amount to create new prices. Here, 1 cent was subtracted from $1.38 to get $1.37, then 1 cent was added to $1.38 to get $1.39, and so on.
d. Mode or median
Answers will vary. You may want to use the suggestions for action research to assess your own students’ understanding of average. How would they respond to the potato-chip task?
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.