Mathematics 1111

Data, Probability, Statistics and Risk

Welcome to the web page for Mathematics 1111, an introductory college course on data, probability, statistics and decision-making offered through the Department of Mathematics at Louisiana State University, Baton Rouge primarily for students planning to teach mathematics in grades K-8. This page offers an overview of the course.

M1111 is presently going through the approval process. We anticipate that it will appear in the 2005 LSU Course Catalogue. The course has been under development for several years, and much of the curriculum developed for it has been---and is presently (spring 2004) being---taught in selected sections of Math 1100. For more information, see the short history of Math 1111 at the end of this page.

Read about M1111 below. | Click here to link to resources for teachers and students.

Acnowledgements. This course was develeped with support from the National Science Foundation under the Course, Curricuulum and Laboratry Improvement Program (NSF-CCLI), Grant #9980995, James Madden and Frank Neubrander PIs. Support for predecessor projects came from the Louisiana Collaborative for Excellence in the Preparation of Teachers (LaCEPT) and from the Louisiana State University Center for Faculty Development. Continuing support is being provided by the Gordon A. Cain Center at LSU and by the LSU MathVision Lab. The image above was drawn for James Madden by a Elvis, a first grade student at Bernard Terrace Elementary School, Baton Rouge, March 2003. Thanks, Elvis!

About Math 1111

Table of Contents

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• Proposed LSU Catalog Description
• Course Goals
• Content Outline
• Textbooks
• Assessment and Grading
• Intended Audience
• Relations to Other LSU Courses
• History of Math 1111

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Proposed LSU Catalog Description

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To be listed with the Mathematics courses, if approved:

1111 Data, Probability, Statistics and Risk (3) F, S Prereq: Math 1021. Obtaining and representing data, variables, tables and graphs, distributions and the parameters that describe them; randomness, probability theory, probability models, the binomial distribution; samples and populations, experimental design, correlation and regression, using probability models to assess statistical significance; decisions in the face of risks.

Course Goals

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The intent of the entire course is to strengthen students’ fundamental intuitions about the ideas implicit in questions such as: “How does one communicate efficiently about complex and highly variable phenomena?” “What does randomness mean and how does one reason about random phenomena?” “What can one infer about a large group from what one knows about a sample?” and “How can this kind of understanding guide practical decisions?”

The course is “concept-driven” not “procedure-driven.” Students will:

• communicate effectively about data,
• reason using the fundamental ideas of probability theory,
• draw correct conclusions from statistical information and explain the reasoning this involves,
• use probability and statistics to make judgments concerning personal and societal risk and safety.

Math 1111 was designed to meet the needs of pre-service K-8 teachers, but it was never intended to be exclusively for this audience. Mastery of a concept includes the ability to explain the concept to other people, especially those with different backgrounds or different levels of education. Therefore, the kind of education that is optimal for teachers is also optimal for anyone else who seeks a profound understanding.

Many students equate learning math with learning how to carry out procedures. A concept-driven course will challenge their expectations. For the instructor, a concept-driven course requires the development of new and more advanced methods of assessment. It is not enough to look at whether students carry out steps correctly. How do students perceive and interpret? What tools do they have for representing mathematical ideas? When it comes to solving problems, what kinds of conceptual resources have they acquired, and how do they apply them? Instruments capable of answering these questions are increasingly available from many sources, some of which are listed in section 5, below.

Content Outline

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1. Data. What is it good for? Where does it come from? Displaying and summarizing data. Tables, graphs. Data in the news. Experiments and observational studies. Ambiguity, uncertainty, validity and reliability.
2. Variables. Various meanings of the word “variable” (as letter in an equation versus feature of the world); quantitative (discrete and continuous), categorical and boolean variables. Comparison with concept of a function; reference class and value class. Numerous examples.
3. Distributions. Distributions of categorical variables: pie charts and bar charts. Distributions of continuous variables: bins, probability density functions. Measures of center and variability. Examples. Looking at distributions and drawing conclusions.

1. Randomness. Examples of random phenomena (coin flips, games of chance, etc.). Producing and recognizing random data; biases and misconceptions. Describing random phenomena. Probability and relative frequency. Exploring simulations.
2. Probability models. Ad hoc models (reference class models, tree models); Probability theory: sample space, outcomes, events, independence, conditional probability.
3. The binomial distribution. Pascal’s triangle. Various interpretations of the binomial coefficients. Applications to probability. The binomial and the normal distribution.

C. Statistics

1. Statistical significance. Distinguishing signal from noise, distinguishing between systemic and random effects.
2. Design of Experiments. Null hypothesis, p-value, “lady tasting tea”.
3. Population parameters from sample statistics. Population proportion, average; random samples; sample size, margin of error and level of confidence; central limit theorem.
4. Associations between variables. Two-by-two tables. Linear regression.

D. Risk

1. Expected value. Gambling and lotteries.
2. Decision under uncertainty. Allocating resources to meet personal and societal risks in safety, security, health, finance, etc.
3. Psychology of risk. The work of Kahneman and Tversky.

Textbooks

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Any one of the following textbooks is suitable for use in this course. The table below, intended as a rough guide only, shows how the contents of the course are supported by chapters and/or sections of each of the recommended texts. Topics for which supplemental material will be needed are marked with ***.

• J. O. Bennett, W. L. Briggs, and M. F. Triola, Statistical Reasoning for Everyday Life, second edition. Addison-Wesley. 2003. Paperback, 528 pages. ISBN: 0-201-77128-4.
• David S. Moore, Statistics: Concepts and Controversies, 5th edition. W H Freeman & Co. 2000. Paperback, 557 pages. ISBN: 0716740087.
• Allan J. Rossman and Beth L. Chance, Workshop Statistics, second edition. Springer-Verlag. 2000. Paperback, 648 pages. ISBN: 1930190034.
• Jessica M. Utts, Seeing Through Statistics, second edition. Brooks/Cole. 1999. Paperback, 465 pages ISBN: 0-534-35786-5.

Clearly, some topics are more extensively supported by some texts than by others. Especially in relation to content sections C. 2, 3 and 4, some books include much more material than can reasonably be included. None of the textbooks listed contains complete coverage of all the topics in the Content Outline. Supplementary materials will be available at the course web site.

Assessment and Grading

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Assessment tools appropriate for a course of this kind are being developed in a number of settings. Here are some places to begin looking:

• Field-tested Learning Assessment Guide (FLAG) web site.
• The Journal of Statistics Education.

In the latter, see especially:

• Joan B. Garfield. "Beyond Testing and Grading: Using Assessment to Improve Student Learning." Journal of Statistics Education Volume 2, Number 1 (1994).
• Beth L. Chance, "Components of Statistical Thinking and Implications for Instruction and Assessment." Journal of Statistics Education Volume 10, Number 3 (2002).

Sample grading policy statement.

Intended Audience

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Math 1111 will provide a thoughtful, conceptual treatment of data, probability, statistics and the theory of risk. It satisfies a documented need that is currently unmet at LSU in pre-service teacher education programs. It also meets needs of other programs, such as mass communication, where an understanding of statistical methods is needed, yet where proficiency with specialized computational methods is of lesser importance. Finally, Math 1111 may be useful in providing background for students who go on to other statistics courses where the intent is to develop more advanced proficiency with specific techniques.

Math 1111 will provide future K-8 teachers with the understanding needed to: 1) deal competently with the occurrences of data, probability and statistics in the curricula they will teach and 2) understand statistical methods used to assess students, schools and/or educational programs. No other course currently offered at LSU meets the needs of the pre-service teacher audience that occur in this connection.

Pursuant to state mandates, LSU’s teacher preparation programs have recently been redesigned, and currently there are separate tracks for preK-3, 1-6 and 7-12. Plans for a separate 4-8 track have been considered. Math 1111 is ideal for all tracks. (Intending secondary math and science teachers may have access to the contents of Math 1111 (and more) through upper-division statistics and probability courses. Arrangements whereby an intending secondary math or science teacher could earn advanced math credit by interning as an assistant instructor in Math 1111 are under consideration.)

Relations to Other LSU Courses

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Other courses with related content and no additional prerequisites include: Math 1029, Math 1100, Experimental Statistics 2201 and Sociology 2201. Psychology 2011 is related, but has additional requirements.

• Math 1029 and Math 1100 both cover a range of topics, and therefore they do not cover the specific topics of Math 1111 with the depth and detail needed by teachers and desirable for other potential audiences. Math 1029 and Math 1100 also have potential overlaps with other specialized courses for teachers, and therefore they are not suited for the teacher-preparation programs.
• Experimental Statistics 2201 Introduction to Statistical Analysis is a 4-hour course, with 3 hours of lectures and 2 hours of lab. It covers material that is related to the data analysis and statistics that occurs in Math 1111. It goes much deeper into technical aspects of statistical tests than Math 1111 will, and it places greater emphasis on developing proficiency with numerous techniques. On the other hand, Experimental Statistics 2201 treats neither probability theory nor risk theory, so at least half of the content of Math 1111 is absent from this course.
• Sociology 2201 Introduction to Statistical Analysis is a 4-hour course, with 3 hours of lectures and 2 hours of lab. It is specialized for sociology majors and it does not include probability or decision-making.
• Psychology 2011 General Statistics is specialized for psychology majors and requires departmental permission for students not majoring in psychology.

History of Math 1111

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Math 1111 was developed with funding from NSF in order to meet the needs of future K-8 teachers for solid, conceptual understandings of data analysis, probability and statistics. As originally conceived, the NSF project had aimed to reform the LSU course Math 1100: The Nature of Mathematics. However, the changes that were finally deemed necessary were so great that an entirely new course was designed.

When first introduced in the late 1980's, Math 1100 was offered primarily as a way for humanities majors to fulfill college requirements. It treated an assortment of topics including sets, Venn diagrams, propositional logic and truth tables, combinatorial probability, the concepts of average and standard deviation and a few other things at the choice of the instructor. In the late 1990s, Math 1100 became one of four required courses for all elementary education majors. With this change, the mathematics department began to think more carefully about the content. In 1999, a panel of instructors proposed a model in which the course would be organized around four main topics: I) The Language of Mathematics, II) Probability, III) Statistics; and IV) Finance. Exactly what these categories were to mean was left flexible. The Language of Mathematics, for example, could include the topics on set theory and logic that were traditionally part of the course. A more modern course would put the emphasis on data analysis, addressing questions such as: How is data collected or created? How is it displayed, interpreted and conceptualized? How is it used? The underlying ideas in logic and set theory that occurred in the old course would still be very much present, but the way they would be treated and discussed would change. For example, in the old design Venn diagrams were treated as a central topic, important for their own sake. In the new course, they would arise in context. Certain kinds of exercises that were popular in the old course would be omitted to make way for material with more immediate relevance to data analysis. Probability and some selected topics in statistics were always in Math 1100. They would become more central in the newer version of the course.

This vision for Math 1100 was the basis for NSF National Science Foundation Grant 9980995, “Using the LaCEPT Model to reform an Elementary Statistics Course”. This grant supported curriculum development for the course, envisioning a course that would change and adapt under the care and dedication of those responsible for it. The present design for Math 1111 is the result of the process of curriculum experimentation that has been supported by NSF. LSU instructors who have contributed curriculum ideas include: Robert Kelso, Phoebe Rouse, Julia Ledet, John Koehl, DesLey Plaisance and Cynthia Sullivan.

James J. Madden
Department of Mathematics
Louisiana State University
Baton Rouge, LA 70808
madden@math.lsu.edu

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