Learning Math: Data Analysis, Statistics, and Probability
Classroom Case Studies, Grades 6-8 Part C: Inferences and Predictions (35 minutes)
The NCTM (2000) data analysis and probability standards state that students should “develop and evaluate inferences and predictions that are based on data.” In grades 6-8 classrooms, students are expected to develop and evaluate inferences and predictions in order to do the following:
• Use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken
• Make conjectures about possible relationships between two characteristics of a sample on the basis of scatter plots of the data and approximate lines of fit
• Use conjectures to formulate new questions and plan new studies to answer them
When viewing the video segment, keep the following questions in mind:Inference and prediction are more advanced aspects of working with data, as they require an understanding of sampling. Students in grades 6-8 are developing an understanding of the idea of sampling. They often still expect their own intuition to be more reliable than the information they are obtaining from the data. Students begin to develop an understanding of these statistical ideas through conversations as they consider what the data are telling us, what might account for these results, and whether this would be true in other similar situations. Students’ early experiences are often with census data — e.g., the population of their class. When they begin to wonder what might be true for other classes and other schools, they begin to develop the skills of inference and prediction. In the later middle grades and in high school, students begin to learn ways of quantifying how certain one can be about statistical results.
• How does Mr. Sowden encourage students to make inferences and predictions?
• What are some of the students’ preliminary conclusions?
• Which of the students’ inferences are based on the data, and which are based on their own personal judgement?
In this video segment, Paul Sowden asks the students to look for patterns in the four line plots and to try to determine where the coins with no marking were minted. Students discuss the variance in the data and speculate on why the coins have no mint marks and about where those coins might have been minted.
Answer the questions you reflected on as you watched the video:
a. How does Mr. Sowden encourage students to make inferences and predictions?
b. What are some of the students’ preliminary conclusions?
c. Which of the students’ inferences are based on the data, and which are based on their own personal judgement?
In the video segment from Part A, students considered why this set of coins contained more coins from Philadelphia. One student hypothesized that this was because Philadelphia is the closest U.S. Mint. This student was beginning to think about the sample of coins the students were using. How might you facilitate a discussion with your students about bias in data and the extent to which a data set can be representative? What questions would you pose? What issues would you raise?
If you were teaching this lesson on investigating nickels and their mint marks, what questions might you ask to focus students’ attention on each of the following central elements of statistical analysis: See
• Defining the population
• Defining an appropriate sample
• Collecting data from that sample
• Describing the sample
• Making reasonable inferences relating the sample and the population
In the video, the students used circle graphs and line plots to examine the variation in their data. The teacher plans to continue analyzing the variation in the data, using different types of representations. What other types of representations might he use to examine the data?
How could you extend the discussion in this video segment to bring out more speculations about the nickels? How might you formalize these notions into stated conjectures that could be investigated further? What are some conjectures that might arise? How could you formulate them into new questions? How could these questions then be investigated?
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.
You might want to review the statistical ideas of samples and populations.
Session 1, Part D: Bias in Sampling
Here are some possible answers:
a. The teacher asks the students to consider reasons why some coins do not have mint marks. Then he pushes them to think about which mints can likely be eliminated based on the information they see across the four line plots.
b. Some students think the coins with no mint marks were just mistakes. Others think that those coins are likely to be from Philadelphia, because so many nickels contain the Philadelphia mint mark. When thinking about which coins could be eliminated, the students eliminate Denver first and then San Francisco.
c. At first, students are reasoning more from their own personal judgment, such as when one student says that the missing mint marks are a mistake because the machines were working too fast. Other students’ comments were more grounded in the data.
Two questions you might ask are, “Where do you think I got these coins?” and “How might that affect our results?”
• How well do you think our sample of nickels represents the population of nickels in this country?
• If we lived near San Francisco and collected nickels, how might our results be different? How about if we lived near Denver?
The teacher might have the students make box plots for each category of coin mint mark.
One example of a conjecture that students might make is “Philadelphia produces more nickels than the other mints.” This could be formulated as a new question to be investigated: “Does Philadelphia produce more nickels than the other mints?” The students could investigate this question in several ways. They might want to just enlarge their own sample of nickels, with each student collecting nickels over the next week for further analysis. This could also evolve into an Internet project in which your students contact students in other parts of the country, especially those who live closer to the other mints. Each group of students could collect and analyze a sample of nickels. They could then compare across samples and finally combine them into one larger sample.
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.