Learning Math: Data Analysis, Statistics, and Probability
Classroom Case Studies, Grades 3-5 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.” In grades 3-5 classrooms, students are expected to develop and evaluate inferences and predictions, to propose and justify conclusions and predictions that are based on data, and to design studies to further investigate their conclusions or predictions.
Inference and prediction are more advanced aspects of working with data, as they require some notion of the ideas of sampling and population. Students in grades 3-5 are only beginning to develop an understanding of sampling. They often trust their own intuition more than the information they are obtaining from the data. Children 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. Children’s 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 in their own and other schools, they begin to develop the skills of inference and prediction. It is not until the later middle grades and high school that students begin to learn ways of quantifying how certain one can be about statistical results. See
When viewing the video segment, keep the following questions in mind:
• How does Ms. L’Esperance encourage students to make inferences and predictions?
• What are some of the students’ preliminary conclusions?
• How are the ideas of sampling and population embedded in this conversation?
In this video segment, Suzanne L’Esperance facilitates a whole-class discussion as the students consider potential conclusions to the original problem on how large to build the house. Students discuss the variance in their data, the limitations of their small sample, and the need for additional data.
Answer the questions you reflected on above as you watched the video:
|a.||How does Ms. L’Esperance encourage students to make inferences and predictions?|
|b.||What are some of the students’ preliminary conclusions?|
|c.||How are the ideas of sampling and population embedded in this conversation?|
Based on the family-size data gathered by the class and shown in the line plot below, how would you respond to Ms. L’Esperance’s initial question: What size should she tell her friend to build his homes? What reasons can you offer to support this response, and how are they related to the ideas you have studied in this course? Are your reasons based on the data collected, or did you also bring in some of your own judgements?
Children are expected to develop and evaluate inferences and predictions. Evaluate each of the responses below by commenting on the following:
|•||why the response makes sense (or doesn’t) based on the data; and|
|•||the limitations of each response. In other words, what statistical ideas are the children not taking into account?|
The children’s responses to the question of how big to build the homes were as follows:
|a.||“He should build homes for four people.”|
|b.|| “You can tell him to put in a couple of each, because some people live with two people, so he should put more fours
and threes, but put some of the other kinds also.”
|c.||“He should build them for three people and four people.”|
|d.||“I know some people that have six and eight people in their families, so he should build some larger houses too.”|
In thinking about the data that were collected, in what ways might the students’ sample be biased? How might you facilitate a discussion with the students about bias in data? What questions would you pose? What issues would you raise?
According to the 2000 census, the average size of households in the United States is 2.62 people. How might your students respond to this information in light of their own data? What statistical ideas would you want to surface in this discussion?
If you were teaching this lesson on investigating family size, what questions could you ask students to encourage them to focus on each of these central elements of statistical analysis?
|•||Defining the population|
|•||Defining an appropriate sample|
|•||Collecting data from that sample|
|•||Describing the sample|
|•||Making reasonable inferences relating the sample and the population|
A student commented that the class should “wait until we get more information” before making a recommendation to Ms. L’Esperance’s friend. How could you extend this conversation to bring out more predictions and then formalize these notions into stated conjectures that could then be investigated further? What questions would you ask? What are some conjectures that might result? How could these be investigated?
Principles and Standards for School Mathematics (Reston, VA: National Council of Teachers of Mathematics, 2000). Standards on Data Analysis and Probability: Grades 3-5, 176-181.
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 bias in measurement, samples, and populations, and the meaning of “conjecture,” possibly providing one or two examples.
Session 1, Part D: Bias in Sampling
Here are some possible answers:
|a.||Ms. L’Esperance encourages students to make inferences and predictions by focusing their attention on the problem context and asking them to make suggestions regarding what she should tell her friend about how big to build his homes.|
|b.||Many of the children concluded that Ms. L’Esperance should tell her friend to build homes for four people. However, other children took into account the variance in the data and concluded that, while he should build some homes for two people, he should build the most homes for three or four people.|
|c.||When the teacher asks the students to consider the number of data points collected (the sample size), she implicitly encouraged them to consider ideas of sampling and population.|
The data make a strong case that homes should be built for families of size two, three, four, and five. You may agree with the students that four is an appropriate conclusion, but you probably also realize that this sample is very small and that more data should be gathered.
|a.||This response makes sense in that it is based on the mode; the limitation is that it does not take into account the variation in the data.|
|b.||This response takes into account the variation in the data.|
|c.||This response is based on the two values with the greatest number of responses, so the student does consider variation in a narrow sense but does not take into account the limited sample.|
|d.||In this context, this response doesn’t make sense; the student has gone beyond the actual data involved and is considering issues of sampling and population.|
The sample is biased in that, as children, they all live in households that contain at least two people; thus, households in which one person lives are not considered. Some questions a teacher might pose include, “Why doesn’t our line plot show any families of size one?” and “Does anyone in your neighborhood live in a household with only one person?”
The students are likely to wonder why the average size of households is so much smaller than what their data indicated. You would want students to think about how their sample was collected and the bias or limitations inherent in their sample.
Here are some questions you might ask:
|•||What should we tell my friend about where this information came from and the part of our city in which he should build homes of this size?|
|•||If my friend decides to build houses in another city, should they be the same size as the houses we think he should build here?|
Two conjectures that might result are, “The typical family size in our area is four people” and “You will not find families in our area that have 10 people.” These could be formulated as new questions to be investigated: “What is the typical family size in our area?” and “What is the range of family size in our area?” The students could investigate this question in several ways. They might want to survey students in other classes and grade levels in their school on family size, they might want to have each student survey 10 neighbors, or they might want to locate census data for their community.
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.