Learning Math: Data Analysis, Statistics, and Probability
Classroom Case Studies, Grades K-2 Part A: Statistics as a Problem-Solving Process (25 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, even very young 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: See
In this video segment, the teacher, Ellen Sabanosh, applies the mathematics she learned in the Data Analysis, Statistics, and Probability course to her own teaching situation by asking her students to analyze and interpret the data they collected earlier. (Each child was given two boxes of raisins; the children then counted and recorded the number of raisins in each box.) The children will now compile their data into a class line plot and discuss the distribution of the data.
Answer the questions you reflected on as you watched the video:
a. What statistical question are the students trying to answer?
b. How did the students collect their data?
c. How did they organize, summarize, and represent their data?
d. What interpretations are the students considering?
e. How does the teacher keep her students focused on the meaning of the data and the data’s connection to a real-world context?
f. What statistical ideas are these students beginning to develop?
As the students examined the data, Ms. Sabanosh asked several times, “What do you notice?” or “What else do you notice?” What are some reasons for asking open-ended questions at these points in the lesson?
Ms. Sabanosh gave each student two boxes of raisins for data collection. The students counted the number of raisins in each box separately and recorded both data values on the line plot. What were some advantages and disadvantages, mathematically and pedagogically, of her decision to give each student two boxes of raisins?
Ms. Sabanosh asked the students to analyze the data when only about half the data had been compiled onto the class line plot. How might early analysis of partial data, such as in this episode, support students’ evolving statistical ideas?
The purpose of the video segments is not to reflect on the methods or teaching style of the teacher portrayed. 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?
a. The question is, “How many raisins are in a box?”
b. The students collected the data by counting the number of raisins in each of the boxes of raisins they were given.
c. Students organized and represented their data by placing blue dots on a class line plot, and they summarized their data by finding the mode.
d. Students interpreted their data by reasoning that smaller numbers meant that they had bigger raisins.
e. The teacher asked the students to interpret their results by relating them back to the context.
f. Some statistical ideas the students touched on are the nature of data, quantitative variables, variation, range, mode as a summary measure of a data set, sampling, and making and interpreting a line plot.
Asking open-ended questions gives students more opportunities to engage in statistical problem solving and to construct their understanding of statistical ideas.
The main advantage is that giving students two boxes of raisins enlarged the sample, making the results slightly more representative of the population than if students had only been given one box. However, the overall sample size is still relatively small. One disadvantage in giving students two boxes of raisins is that the teacher and students had to carefully determine ways to organize their work environment so that each box was counted and recorded separately.
The early analysis of partial data encouraged students to begin thinking and making predictions about how the data might evolve.
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.