Learning Math: Data Analysis, Statistics, and Probability
Classroom Case Studies, Grades 6-8 Part A: Statistics as a Problem-Solving Process (20 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, nor of the real-world contexts from which the data were collected.
When viewing the video segment, keep the following questions in mind: See Note 2 below.
|•|| Think about each component of the statistical process as it relates to what’s going on in the classroom: What statistical question are the students trying to answer? How
were the data collected? How are the data organized, summarized, and represented? What interpretations are students considering?
|•||What connections among mathematics topics and across subject-area disciplines are apparent in this data investigation?|
|•||Thinking back to the big ideas of this course, what are some statistical ideas that these students are likely to encounter through their investigation of this situation?|
In this video segment, the teacher, Paul Sowden, applies the mathematics he learned in the Data Analysis, Statistics, and Probability course to his own teaching situation. He starts by asking his students to think about the relative amount of coins with each type of mint mark. The students then sort the coins into four groupings: Philadelphia, Denver, San Francisco, and no mint mark. They will now begin to analyze and interpret their 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 are the data organized, summarized, and represented?|
|d.||What interpretations are students considering?|
|e.||What connections among mathematics topics and across subject-area disciplines are apparent in this data investigation?|
|f.||What statistical ideas are these students likely to encounter as they investigate this situation?|
In this video segment, are the students working with quantitative data or qualitative data?
Questions may arise as students examine the nickels in this open-ended investigation. Formulate four statistical questions that students might ask about the nickels that would prompt further investigation.
Why is a circle graph an appropriate way to display this data? What characteristics of data are clearly shown through a circle graph?
The purpose in watching 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 his 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 the students are trying to answer is, “What is the relative frequency of each type of mint mark?”
b. The teacher brought a collection of nickels to the class so that students could examine the coins’ mint marks.
c. The students organized their data into a circle graph.
d. The students are developing conjectures about the relative frequency of each mint mark.
e. The students are using their knowledge of fractions as they explore this problem. The investigation of mint marks
involves connections to social studies.
f. Some statistical ideas are the nature of data, qualitative variables, variation, relative frequency, sampling, making a circle graph, and interpreting a circle graph.
The students are working with qualitative (categorical) data.
Answers will vary. One question might be, “Is this sample of nickels representative of the population of nickels?”
A circle graph is an appropriate way to display categorical data. Circle graphs show the fractional relationship of each category or part of data to the whole data set.
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.