Learning Math: Data Analysis, Statistics, and Probability
Classroom Case Studies, Grades K-2 Part B: Developing Statistical Reasoning (40 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 pre-K through grade 2 classrooms, students are expected to do the following:
• Pose questions and gather data about themselves and their surroundings
• Sort and classify objects according to their attributes, and organize data about the objects
• Represent data using concrete objects, pictures, and graphs
• Describe parts of the data and the set of data as a whole to determine what the data show
Children at the K-2 grade level readily notice individual data points and are able to describe parts of the data — where their own data fall on the graph, which value occurs most frequently, and which values are the largest and the smallest. A significant development in children’s understanding of data occurs as they begin to consider the set of data as a whole. 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 holes emphasize places in the distribution that have no data. The phrases spread out and bunched together underscore the overall distribution. Teachers must 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 pre-K through grade 2 classrooms, students are expected to do the following:
The line plot (also commonly referred to as a dot plot in elementary classrooms) below displays the raisin data collected by the students in Ms. Sabanosh’s first-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 informal language (clusters, clumps, bumps, hills, gaps, holes, spread out, or bunched together), write five statements that you hope children would make describing the set of data as a whole.
Too often, children describe data as numbers devoid of context. Another key question you should frequently ask students regarding their observations is, “What does that tell us about the number of raisins in a box?”
For each of the five statements you wrote in Problem B1, indicate what that observation might imply about the real-world context of the number of raisins in a box.
What are some questions about the data that you would pose to the students in Ms. Sabanosh’s first-grade classroom to encourage them to further consider the statistical ideas of outliers, variation, center of a data set, and sampling?
Comparing data sets prompts students to look at and describe a data set as a whole in order to see how the characteristics of one group compare to the characteristics of another. Now we’ll examine data from another class not featured in the video. The line plot below displays the raisin data collected by Mr. Mitchell’s second-grade classroom: See
Imagine that Mr. Mitchell’s class and Ms. Sabanosh’s class compared each other’s raisin data to their own data.
a. What are four or five statements you would anticipate children might make as they compare the two data sets?
b. Which of these statements compare individual pieces or parts of the data, and which statements compare the data sets as a whole?
Data investigations begin by asking a question that can be answered by gathering data. It’s important to formulate a question with an understanding of the type of data you’ll need to collect. In this course, you studied two types of data, quantitative and qualitative. Quantitative data, such as the number of hours of television watched each week, are often referred to as numerical data; qualitative data, such as your favorite flavor of ice cream, are referred to as categorical data. The following statistical questions were gathered from pre-K through grade 2 classrooms:
|a.||How many raisins are in a box?|
|b.||How far can you jump?|
|c.||Who is your favorite author?|
|d.||How did you get to school today?|
|e.||Are you 6 years old?|
|f.||How many people are in your family?|
|g.||Will you go on the field trip to the zoo?|
|h.||How many pockets do you have today?|
|i.||How tall are you?|
|j.||Do you like chocolate milk or white milk better?|
|k.||What is your favorite restaurant?|
|l.||Which apple do you like best: red, green, or yellow?|
For each question above, identify the type of data that will be collected and an appropriate way to display the data (e.g., line plot, bar graph).
Formulate 10 questions that would be appropriate for your grade level and of interest to your students, five of which involve collecting qualitative (categorical) data, and five of which involve collecting quantitative (numerical) data.
Principles and Standards for School Mathematics (Reston, VA: National Council of Teachers of Mathematics, 2000). Standards on Data Analysis and Probability: Grades K-2, 108-115.
Reproduced with permission from the publisher. Copyright © 2000 by the National Council of Teachers of Mathematics. All rights reserved.
Using line plots (dot plots) in elementary school classrooms is a fairly new practice. Consider how you might use this graphical representation of data with your students. For example, when working with pre-kindergarten and kindergarten students, you might have the students begin with pasting the actual boxes on the charts instead of using dots. How does this compare with your current method of presenting data?
You may want to refer back to Interpreting a Line Plot in Session 2.
Comparing data sets is also a fairly new practice in elementary school classrooms. Think about why this step might be important for your students. You might also want to review the difference between quantitative and qualitative variables. Session 1, Part B: Data Measurement and Variation
Here are some possible statements that children might make:
|•||There is a clump of data from 29 to 35.|
|•||I can see a bump at 30.|
|•||There is big gap from 15 to 20.|
|•||There are four holes in the data.|
|•||The data are spread out from 14 to 35.|
|•||The clump from 29 to 35 tells us that lots of boxes had this number of raisins.|
|•||The bump at 30 is the number of raisins that occurred most often when we were counting how many raisins were in a box.|
|•||The gap from 15 to 20 tells us that no boxes had 15, 16, 17, 18, 19, or 20 raisins in them.|
|•||The holes in the data tell us that at each point where there’s a hole, there are no boxes with that number of raisins.|
|•||The “spread” of the data tells us that the smallest number of raisins (the “minimum”) in a box is 14, and the largest number (the “maximum”) is 35.|
To get at the idea of “outliers,” you might ask students whether there are any unusual data points on the graph. To focus on the variation in the data, you might ask students to talk about intervals where the data are clustered or concentrated, or where they are spread out. To introduce the concept of the median, you could ask students to try to find the center of the data set.
|a.||A sample answer is that the first graders are likely to notice that the mode for Mr. Mitchell’s class is 34, whereas their mode was only at 30. They are also likely to point out that they had a “really low number” of raisins at 14, which was “a lot lower” than the lowest number from the second-grade class, 26. Students might also notice that their raisin data is more spread out with a larger range than the raisin data from Mr. Mitchell’s class.|
|b.||The first observations compare individual pieces of the data. The second observation compares the whole data sets to each other.|
As data may be displayed in many ways, answers will vary; the representations listed here are only one option:
One example of a question that involves collecting quantitative (numeric) data and that often interests students in first or second grade is, “How many teeth have you lost?” An example of a question that involves collecting qualitative (categorical) data is, “What is your favorite pizza topping?”
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.