Learning Math: Data Analysis, Statistics, and Probability
Classroom Case Studies, Grades 3-5 Part A: Statistics as a Problem-Solving Process (30 minutes)
A data investigation should begin with a question about a real-world phenomenon that can be answered by collecting data. After the children have gathered and organized their data, they should analyze and interpret the data by relating the data back to the real-world context and the question that motivated the investigation in the first place. Too often, classrooms focus on the techniques of making data displays without engaging children in the process. However, it is important to include children in all aspects of the process for solving statistical problems. The process studied in this course consisted of four components:
Children often talk about numbers out of context and lose the connection between the numbers and the real-world situation. During all steps of the statistical process, it is critical that students not lose sight of the questions they are pursuing and of the real-world contexts from which the data were collected.
When viewing the video segment, keep the following questions in mind:
• How do the students in this classroom apply the first two components of the statistical process? What statistical question are the students trying to answer? How were the data collected?
• As the fifth graders move onto the next two components of the statistical process — analysis and interpretation — what issues do you think will come up?
• Thinking back to the big ideas of this course, what are some statistical ideas these students are likely to encounter through their investigation of this situation?
See Note 2 below.
In this video segment, teacher Suzanne L’Esperance applies the mathematics she learned in the Data Analysis, Statistics, and Probability course to her own teaching situation. She starts by establishing the context for her students to investigate family size, telling them about her friend who is in construction and how he needs help from the students in the class. The students consider the context and then begin to collect the data.
Answer the questions you reflected on as you watched the video:
• How do the students in this classroom apply the first two components of the statistical process?
• What statistical question are the students trying to answer?
• How did the students collect their data?
• As the students move on to analysis and interpretation of their data, what issues do you think will come up?
• What statistical ideas are students likely to encounter as they investigate this situation?
In this video, Ms. L’Esperance establishes a rich and elaborate real-world context to situate the students’ investigation of family size. How do you think the class would have responded if she had not constructed a context for the investigation and instead had simply said, “Today we are going to investigate family size; how many people are in your family?” What is the impact on the students’ level of engagement?
Too often, students lose the connection between the numbers and the real-world situation once they have gathered their data. How might the richer context provided by Ms. L’Esperance reinforce the connection between the data and the real-world phenomenon being studied, and prevent students from working with mere numbers out of context?
What are some ways in which this richer context will support students’ reasoning as they “interpret the results”?
Why do you think Ms. L’Esperance phrased the question about family size as “How many people live in the house that you slept in last night,” as opposed to simply “How many people are in your family?” With your own students, how would you define “family”? See
When engaging students in the process of statistical problem solving, students must consider what to measure and how to measure it to ensure accuracy in collecting their data. In this lesson, Ms. L’Esperance defined “family” for her students. But, it is also important to give students a chance to form — or to help form — their own definitions for the purpose of their investigations.
How would you facilitate a discussion with your students on what constitutes a “family”? Describe some of the sensitive issues that might arise and how you would handle them.
Before examining specific problems at this grade level with an eye toward statistical reasoning, you will watch a teacher (who has also taken the course) teaching in her classroom. The purpose in viewing the video is not to reflect on the teacher’s methods or teaching style. Instead, look closely at how the teacher brings out statistical ideas while engaging her students in statistical problem-solving. You might want to review the four-step process for solving statistical problems. What are the four steps? What characterizes each step?
You might want to review the four-step process for solving statistical problems. What are the four steps? What characterizes each step?
The 2000 United States Census defines a household as one or more persons living in a housing unit. One person who owns or rents the residence is designated as the householder. For the purposes of examining family and household composition, two types of households are defined: family and non-family. A family household has at least two members related by blood, marriage, or adoption, one of whom is the householder. A non-family household can either be a person living alone or a householder who shares the housing unit with non-relatives only — for example, boarders or roommates. The non-relatives of the householder may be related to one another.
a. The question is, “How many people are in a family?”
b. Each child was told to use connecting cubes to show the number of people that live in his or her house.
c. One might expect that the issue of “center” (median) would arise when determining the typical size of families, as well as the issue of variation in the data.
d. Some statistical ideas are the nature of data, quantitative variables, variation, range, measures of center, sampling, making a line plot, and interpreting a line plot.
It is likely that this rich context more fully engages children because there is a clear purpose for investigating family size. The context also increases the authenticity of the task. There is a real-world rationale for why a person in a specific profession would need to know the size of families or households in a particular area. While children might also be curious about the number of people in each of their families, they do not have a reason — other than their interest — to extend the investigation beyond their classroom.
The rich context grounds students more firmly in the situation. With the clear purpose of examining family size and the fact that the data are about them, they more readily analyze the data with the real-world situation in mind than they would if they were just thinking about numbers out of context.
Again, the rich context grounds students in the situation. They are invested in helping the teacher’s friend and have a clear purpose for interpreting the data so that they can make an appropriate recommendation. The students are also able to draw on their own knowledge about the neighborhood and about family size. Thus, as they interpret the results, they are more likely to raise issues, think beyond their own classroom sample, and become curious about the larger population.
The teacher’s phrase resembles the definition of a “household” as set by the 2000 United States census. Answers to the second question will vary, but might include “all the people you spend holidays with,” “the people you’re related to,” and “your mother, father, sisters, and brothers.”
Sensitive issues might involve brothers and sisters who no longer live in the same household, parents or siblings who have died, single- and dual-parent households, same-sex and different-sex guardians, or joint-custody situations. These are all aspects of people’s lives, and are good examples of the importance of definitions when collecting data.
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.