Private: Learning Math: Data Analysis, Statistics, and Probability
Classroom Case Studies, Grades 6-8 Part B: Statistics as a Problem-Solving Process (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 the grades 6-8 classroom for data analysis and statistics, students are expected to do the following:
|•||Formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one population|
|•||Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatter plots|
|•||Find, use, and interpret measures of center and spread, including mean and interquartile range|
|•||Discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem and leaf plots, box plots, and scatter plots|
In grades 6-8, students examine relationships among populations or samples, and they examine two variables within one population, such as comparing arm spans and heights. They learn to use new representations, such as box plots and scatter plots, to help them examine these relationships. Students also use measures of center to summarize and compare data sets. Building on informal understandings of what is typical, what is usual, what is the most, and what is the middle, students develop understanding about the mode, median, and mean. However, 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 indicates about the data set. The mean receives increased emphasis in these grade levels, but students also continue to use the median, especially in creating Five-Number Summaries for making box plots.
Consider topics of interest to students in grades 6-8 that involve collecting data about a characteristic shared by two populations. Formulate five questions that involve collecting qualitative (categorical) data and five questions that involve collecting quantitative (numerical) data. For each question, identify the type of data that will be collected and an appropriate way to display the data (e.g., line plot, bar graph, histogram, circle graph, stem and leaf plot, box plot). See Note 3 below.
Consider topics of interest to students in grades 6-8 that involve collecting data about different characteristics within one population, and then formulate five questions that involve collecting quantitative (numerical) data. For each question, identify an appropriate way to display the data (e.g., line plot, bar graph, histogram, circle graph, stem and leaf plot, box plot, scatter plot) and describe how each display would be used to highlight potential relationships between the two characteristics.
The next classroom investigation reveals students’ understanding of the notion of average. Here’s the 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 students in individual interviews to research students’ understanding of average. Here are some of the students’ responses:
|a.||Some student 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?|
|c.||What would be an example of a “construction” task and an “unpacking” task?|
|d.||Why might you want to include some “construction” and “unpacking” tasks into your instructional program?|
Principles and Standards for School Mathematics (Reston, VA: National Council of Teachers of Mathematics, 2000). Standards on Data Analysis and Probability: Grades 6-8, 248-255.
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.
If you’re working in a group, make a three-column chart with the labels “Question,” “Type of Data,” and “Appropriate Data Display” for recording the group’s responses to Problems B1 and B2.
If you’re working in a group, discuss ideas for using different measures of center to solve the potato-chip task. You might also want to review the statistical ideas of median and mean.
Session 2, Part D: The Median
Session 5: Mean and the Median
Answers will vary. Two questions that involve qualitative (categorical) data that could be displayed with bar graphs and might interest students are, “What is your favorite type of music?” and “What is your favorite musical group?” Two questions involving quantitative (numerical) data that could be displayed on line plots are, “How many hours do you spend per week in chat rooms?” and “How much money do you spend on CDs each month?”
One such question might be, “What is the relationship between grade point average and the number of hours a student studies?” A scatter plot would be an appropriate way to display this data.
• 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.