M358K (APPLIED STATISTICS): RELEVANCE TO FUTURE SECONDARY TEACHERS

What is statistics?

Statistics is not a branch of mathematics, but a mathematical science. It is sometimes described as the science of data. This begs the question: What are data? Data can be defined as numbers in context. So, for example, the numbers 35, 60, 50, 55, 48, 74, 67 are not themselves data, but if you know that they are the ages of a group of people, or the heights of a group of people, or the scores in games played by a certain basketball team, or the hours that a collection of light bulbs works before burning out, then they become data. The context brings the possibility of interpretation of the numbers.

Here are some more detailed answers to this question:

How does this course address the Texas State Board for Educator Certification (SBEC) Standards for Secondary Math Teachers?

Standards II (Patterns and Algebra), IV (Probability and Statistics), and V (Mathematical Processes) of the SBEC Secondary Math Standards include the following standards regarding the content of M 358K:

"The beginning teacher of mathematics is able to:

- (2.17s) work with patterns with random variations (p. 7) …
- (4.1s) investigate and answer questions by collecting, organizing, and displaying data from real-world situations;
- (4.2s) support arguments, make predictions, and draw conclusions using summary statistics and graphs to analyze and interpret on-variable data;
- (4.3s) communicate the results of a statistical investigation using appropriate language;
- (4.4s) investigate real-world problems by designing, administering, analyzing and interpreting surveys; …
- (4.6s) explore concepts of probability through data collection, experiments, and simulations; …
- (4.8s) use the graph of the normal distribution as a basis for making inferences about a population….
- (4.10s) investigate real-world problems by designing, conducting, analyzing, and interpreting statistical experiments;
- (4.11s) develop and justify concepts and measures of central tendency (e.g., mean, median, mode) and dispersion (e.g., range , interquartile range, variance, standard deviation) and use those measure to describe a set of data;
- (4.12s) calculate and interpret percentiles and quartiles;
- (4.13s) explore , describe, and analyze bivariate data using techniques such as scatter plots, regression lines, correlation coefficients, and residual analysis;
- (4.14s) explain and use precise probability language to make observations and draw conclusions from single variable data and to describe the level of confidence in the conclusions; …
- (4.16s) make inferences about a population using the binomial and geometric distributions. …
- (4.19s) organize, display, and interpret data in a variety of formats(e.g., tables, frequency tallies, box plots, stem-and-leaf plots, histograms) and discuss the advantages or disadvantages of a given format;
- (4.20s) apply linear transformations (translating, stretching, shrinking) to convert data and describe the effect of linear transformations on measure of central tend4ency and dispersion; …
- (4.22s) apply concepts and properties of discrete and continuous random variables to model and solve a variety of problems involving probability and probability distributions;
- (4.23s) describe and analyze bivariate data using various techniques (e.g., scatterplots, regression lines, outliers, residual analysis and correlation coefficients);
- (4.24s) transform nonlinear data into a linear form in order to apply linear regression techniques to develop exponential, logarithmic, and power regression models;
- (4.25s) describe and apply the characteristics of a well-designed and well-conducted survey or experiment;
- (4.26s) analyze and interpret statistical information from the media, such as the results of polls and surveys, and recognize valid and misleading uses of statistics;
- (4.27s) use the law of large numbers and the central limit theorem to describe the role of probability theory in the process of statistical sampling and inference; and
- (4.28s) use confidence interval arguments to formulate and test hypotheses. (pp. 13 - 16)
- (5.3s) use formal and informal reasoning to explore, investigate, and justify mathematical ideas;
- (5.4s) recognize examples of fallacious reasoning; (p.17)
- (5.7s) recognize that a mathematical problem can be solved in a variety of ways, evaluate the appropriateness of various strategies, and select an appropriate strategy for a given problem;
- (5.8s) evaluate the reasonableness of a solution to a given problem;
- (5.9s) use physical and numerical models to represent a given problem or mathematical procedure;
- (5.10s) recognize that assumptions are made when solving problems and identify and evaluate those assumptions;
- (5.11s) investigate and explore problems that have multiple solutions;
- (5.12s) apply content knowledge to develop a mathematical model of a real-world situation and analyze and evaluate how well the model represents the situation; (p. 18)
- (5.15s) explore problems using verbal, graphical, numerical, physical, and algebraic representations.
- (5.16s) recognize and use multiple representations of a mathematical concept;
- (5.17s) apply mathematical methods to analyze practical situations;
- (5.18) use mathematics to model and solve problems in other disciplines, such as art, music, science, social science, and business. … (p. 19)
- (5.21s) translate mathematical statements among developmentally appropriate language, standard English, mathematical language, and symbolic mathematics; …
- (5.23s) use visual media such as graphs, tables, diagrams, … to communicate mathematical information;
- (5.24s) use the language of mathematics as a precise means of expressing mathematical ideas." (p. 20 )

How does this course relate to the Texas Essential Elements for secondary mathematics?

- Statistical topics are listed in each grade level K - 8. (Click here to see a list of probability and statistics topics included in grades K - 8. Click here to link to examples of probability and statistics activities in grades K - 8.)
- Two of the nine basic Knowledge and Skills categories in the high school course Mathematical Models with Applications are statistics topics. (Click here for a description of the Knowledge and Skills categories. Click here for examples of activities involving these Knowledge and Skills categories. Look for activities labeled (2) or (3))
- Regression (a statistical technique) is included in the TEKS for Precalculus.
- AP Statistics is listed as a high school course. (Click here to go to the College Board AP Statistics home page.) Enrollment in AP Statistics has been growing recently, so there is a need for teachers prepared to teach it. (Click here to see the home page of an AP statistics course at St. Stephen's school in Austin.)

How does this course relate to the National Council of Teachers of Mathematics' __Principles and Standards for School Mathematics__?

The __Principles and Standards__ include Data Analysis and Probability as one of their five content strands running through all grade levels. This strand includes statistics.

- Click here to see the Data Analysis and Probability overview.
- Click here to see the Data Analysis and Probability Standard for Grades Pre-K–2.
- Click here to see the Data Analysis and Probability Standard for Grades 3 - 5.
- Click here to see the Data Analysis and Probability Standard for Grades 6 - 8.
- Click here to see the Data Analysis and Probability Standard for Grades 9 - 12.

I didn't have much statistics in elementary, middle, or high school. Have things changed recently?

Yes, things have changed quite a lot recently. Statistics has increasingly become important in our modern world, for studying topics as diverse as the environment, medicine, the stock market, and technological innovations. In addition, modern software has made it easier to learn and do statistics. For a number of years, there has been a movement to include statistics in the curriculum. Statistics topics have been included in textbooks and curricula, especially in the middle grades, for quite a few years, but teachers have rarely covered the statistics topics, partly because they weren't familiar with the topics themselves. However, as new standards have been developed, there has been more progress in including statistics in what is actually taught. One big development was the start of Advanced Placement Statistics in 1998. The course has been growing steadily in popularity, and will probably continue to grow in enrollments, especially since many medical schools are now accepting statistics instead of calculus.

To help prepare UT's future math teachers to teach the statistics in the Texas school curriculum, the Department of Mathematics in 1998 started offering M358K: Applied Statistics, and this course became a required course for students seeking secondary certification in mathematics.

M358K is designed to include the topics (except those in the prerequisite course, M362K: Probability I) included in the AP Statistics syllabus, but M358K studies these topics in greater depth than they are studied in the AP Statistics course.

M358K has proved to be a good course for many other students besides those planning to be teachers, so it is not a "teachers only" course. This web page is designed to help supplement the course for future teachers and show how it is important for their background.

For more information on statistics in grades K - 12, see the above links to __the Principles and Standards for School Mathematics__ and the online article, "Statistics and Its Interface with the Secondary Mathematics Curriculum", by Gail Burrill, former president of the National Council of Teachers of Mathematics. See also the MISD/MMSTC Statistical DOE Project for a description of a course "Data Analysis and Research Methods" currently being taught to 9th graders at three magnet schools for mathematics, science and technology in Macomb County, Michigan.

What is the role of technology in this course?

In M 358K, you will be expected to use technology in creating graphs and statistical analyses. You will be given instructions in using Minitab statistical software to do this, but many things can also be done by using suitable calculators (such as the TI-83), Fathom (which you used in M 315C), or Excel. There are also lots of interesting web sites that can be helpful in learning statistics. Here are some:

- Histogram applet: How changing the size of the bins (i.e., the widths of the bars) affects the shape of a histogram of the duration (in minutes) for eruptions of the Old Faithful geyser in Yellowstone National Park.
- Correlation coefficient
- Guessing correlations
- Regression Applet
- Central Limit Theorem Illustrations
- List of Java Applets for Statistics
- Confidence Interval Simulation

How does this course relate to other courses I will be taking?

- M 362K (Probability I) is a prerequisite for this course. Material from M 362K will be used in M358K
- Some calculus topics will be used in M 358K
- You will probably find your technology experience in M 315C helpful in M 358K.
- One of the topics in the UTeach course Research Methods is the different uses of statistics in different areas of science.
- Statistics topics are good choices for your term project in the UTeach course Project Based Instruction.
- If you are working toward the BS-Teaching Option math degree, you will be required to take a depth course, which can be either M427K (Differential Equations) or M378K (Mathematical Statistics). If you think you would like to teach AP statistics, you should choose M 378K to gain additional background in statistics. M 358K will be good preparation for M 358K.
- If you think you would like to teach AP statistics and are pursuing the BS teaching option degree, you should choose your Supporting Course in an area using statistics. See the supporting course page for some suggestions. M 358K can usually count as the statistics prerequisite for these courses, but be sure to check with the instructor. ( If you are obtaining the BA, you can take one or more of these courses as electives.)

How can I get the most out of this course?

- Approach the course with an open mind. Statistics is in many ways different from mathematics, so expect some differences. For example, in statistics, answers are often inexact and uncertain. In fact, giving an estimate of how certain or uncertain answers are is part of statistics.
- Focus on conceptual understanding and developing statistical thinking skills. Statistics is a lot more than using formulas. For example, you need to decide which formulas to use where and how to interpret the results after you've obtained them -- and how to design and carry out a study so that the formulas you would like to use are appropriate to use.
- Read the SBEC, TEKS, and NCTM Standards mentioned above. Review them now and then to help keep your focus on developing the reasoning, problem solving, and mathematical communication skills you will need in teaching.
- Try to understand and be able to explain concepts from more than one perspective -- for example, using diagrams and using words. Practice recognizing incorrect, partially correct, and entirely correct ways of explaining a concept. These skills are important for teachers to develop.
- Be alert to technical use of vocabulary. Many everyday words have a more specific meaning in statistics and mathematics. Examples you will encounter in statistics include normal, significant, random.