Finding Common Ground, Indianapolis, March 2006 Statistics, Data Analysis, and Probability Group Report

Working Group:
Christine Franklin, University of Georgia
Bert Fristedt, University of Minnesota
Bonnie Hagelberger,  Monroe Elementary School
Katherine Halvorsen, Smith College
Richard Scheaffer, University of Florida (emeritus)
Zsuzanna Szaniszlo, Valparaiso University
Dan Teague, NC School of Science and Mathematics

Every student at some time in their life will make decisions in which statistical reasoning skills are necessary.  This is true for all students regardless of their plans for post-secondary education.

STATISTICS

Mathematical aspects of statistics, including presentation of data and the development of statistical reasoning, should be part of the Pre K-12 mathematics curriculum.  The context of a statistical problem treated in a mathematics class needs to involve appropriate mathematics for the grade level in which it is used. Statistics is part of mathematics but the context (story) should not crowd out the mathematics. Statistical reasoning as taught in mathematics classes provides an essential foundation for the increasing use of statistics in other disciplines (sciences, social studies, English, etc.).

Statistical reasoning process
Statistics (Data Analysis in context)
1. Formulate questions (hypotheses) that can be answered with data
a. Categorical
b. Numerical (Measurement)
3. Represent data with tables, charts, and graphs
4. Analyze data (turn data into useful information)
a. Summarization of data: quantification of variability
i. Categorical: counts, proportions, bar graphs
ii. Measurements: plots on the real number line, describing distributions (shape, center, and spread)
iii. Bivariate association: Two-way tables and scatterplots; numerical summaries of association
b. Making conjectures, drawing conclusions, making generalizations, inference
5. Interpret results: Relate the inference to the original question
6. Probability’s role in data analysis
a. Random selection and margin of error
b. Random assignment and experimental error
c. Sampling distributions

Items 1 through 5 describe a process that should be repeated, with increasing sophistication through the grade levels.  For example, at the elementary school level students might collect data from the second grade class and, based on the data, make conjectures about the third grade class.  At the middle school level students should use random samples as the basis for evaluating conjectures.  At the high school level students may begin more formal inference using sampling distributions and margin of error.  At the high school level, statistics should be integrated into the curriculum, not limited to mathematics.

The study of statistics supports the learning of the arithmetic of whole numbers, fractions, and decimals in the elementary grades, and it supports the learning of proportional reasoning and algebra in the middle grades and beyond.

A further discussion related to these topics may be found in the Pre K-12 Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report at www.amstat.org/education/gaise/.  (See pages 12 – 15.)

PROBABILITY

The introduction of probability should come after students are exposed to fractions, although some preparatory notions of probability (more likely, less likely) should be introduced beforehand. Probability at the earlier stages meshes well with fractions; at later stages it meshes well with certain aspects of algebra. Throughout, probability supports important aspects of statistics.

Probability
0. Pre-probability
a. A sense of probability: more likely, less likely
b. Informal experiences with counting techniques: systematic listing of possibilities
1. Experimental (Empirical): Relative frequency
2. Theoretical (Mathematical): Deriving a probability from a model; e.g, equally likely outcomes.
a. Sample space
b. The arithmetic of probability (Addition and Multiplication Rules, Independence)
3. Representation: Tree diagrams, Venn diagrams, area models, interpreting odds as probabilities
4. Counting techniques based on the multiplication and the addition principles
5. Probability distributions
a. Binomial
b. Normal
c. Empirical (experimental) distributions
6. The role of probability in data analysis
a. Random selection and margin of error
b. Random assignment and experimental error
c. Sampling distributions

Allocation of Probability Topics to the K-12 Curriculum
Items 1, 2, 3, and aspects of 4 and 6 are appropriate for introduction at the middle school level.  Item 5 and other aspects of items 4 and 6 belong in high school.

Issues to be continued
• How much context should be included in the mathematics curriculum and how much belongs to other courses?
• Teachers are receptive to learning statistics and probability, but because these may not have been a part of their teacher preparation, they need in-service training and a lot of support in both content and pedagogy.

Tangentially related issues that deserve attention from the mathematics community
• Graph Theory – Discrete Math needs further discussion