Private: Learning Math: Data Analysis, Statistics, and Probability
Classroom Case Studies, Grades K-2 Part C: Inferences and Predictions (30 minutes)
The NCTM (2000) data analysis and probability standards state that students should “develop and evaluate inferences and predictions that are based on data.” Appropriate beginnings for the concepts of inference and prediction in pre-K to grade 2 classrooms occur in conversations with children as they consider what the data are telling us, what might account for these results, and whether we’d get the same results in other similar situations. Inference and prediction are more advanced aspects of working with data, as they require some notion of the ideas of sampling and population.
Children’s first experience is often with census data — that is, the population of their class. When they begin to wonder what might be true for other classes in their own school and other schools, they begin to consider that many data sets are samples of larger populations. These considerations are the precursors to understanding the notion of inferences from samples. Children should be encouraged to develop conjectures based on data, consider alternative explanations, and design further investigations to examine their speculations. See
When viewing the video segment, keep the following questions in mind:
• How does Ms. Sabanosh encourage students to make inferences and predictions?
• An expectation for students at this level is that they will “discuss events related to [their] experiences as likely or unlikely” (NCTM, 2000). How is this expectation illustrated in this video segment?
In this video segment, Ellen Sabanosh asks her students to predict the number of raisins in an unopened box, based on the data they have collected. Watch them discuss both reasonable and unreasonable predictions.
Answer the questions you reflected on as you watched the video:
a. How does Ms. Sabanosh encourage students to make inferences and predictions?
b. How is NCTM’s expectation for students (i.e., that they will discuss events related to their experiences as likely or unlikely) illustrated in this video segment?
If you were using this investigation with your own students, what questions could you pose to further elicit and extend student thinking about notions of samples and populations?
Embedded in the students’ responses about “what would not be a good guess” are early notions of making conjectures and forming hypotheses. How could you extend this conversation to bring out more speculations and then formalize these notions into stated conjectures that could be investigated further? What questions would you ask? How would you expect children to respond? What are some conjectures that might result?
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. © 2000 by the National Council of Teachers of Mathematics. All rights reserved.
You might want to review the statistical ideas of samples and populations and the meaning of conjecture. Session 1, Part D: Bias in Sampling.
Here are two possible answers:
a. Ms. Sabanosh encouraged students to make inferences and predictions by asking them to think about how many raisins would be in one of the unopened boxes.
b. When the students discussed what would not be good guesses about the number of raisins in a box, they clearly suggested numbers that were not likely.
Answers will vary. One example of a question is, “If I went to the store and bought a different brand of raisins, but the same size box, how many raisins do you think would be in that box?”
Two conjectures that might result are, “A small [half-ounce] box of raisins cannot contain 100 raisins” and “A small box of raisins will contain between 14 and 35 raisins.” These could be formulated as the following question to be investigated: “What is the smallest and largest number of raisins that a box can hold?” The students could investigate this question by obtaining additional half-ounce boxes of raisins of the same brand, then counting the number of raisins in each box, and finally adding this information to their line plot. An extension would then be to count and analyze the number of raisins in half-ounce boxes of other brand names.
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.