# Enhancing Your Students’ Conceptual Understanding of Statistics

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Enhancing Your Students’ Conceptual Understanding of Statistics
Michael Sullivan Joliet Junior College

GAISE (Guidelines for Assessment and Instruction in Statistics Education) Six Recommendations Emphasize statistical literacy I am going to expand this to statistical literacy, statistical reasoning, and statistical thinking Use real data Stress conceptual understanding rather than knowledge of procedures Foster active learning Use technology for developing conceptual understanding and analyzing data Use assessments to improve and evaluate student learning

A Recommendation on the First Day of Class
You are not in a math class!!!!!!

What is Conceptual Understanding?
The ability to interpret and explain results The ability to determine which techniques are appropriate De-emphasis of procedural methods Teach fewer concepts, and dig deeper We are trying to teach too much material in our statistics courses.

Conceptual Understanding
De-emphasize the use of procedures and emphasize the “big picture” idea. Two simple examples:

Conceptual Understanding
Test question: Draw two scatter diagrams for which r is close to 0.

Teach Others/Immediate Use
The Learning Pyramid Lecture 5% Reading 10% Audio Visual 20% Demonstration 30% Discussion Group 50% Practice by Doing 75% Teach Others/Immediate Use 80% Adapted from The Learning Triangle: National Training Laboratories, Bethel Maine ©mindServegroup 2005

May want to talk about how increasing sample size affects the precision of results as a prelude to the Law of Large Numbers. Also, compare sampling with nonsampling error.

The Golden Ratio Measure the height of your partner in centimeters. Call this y. Measure the height to your partners naval. Call this x. (We could also measure arm span) Draw a scatter diagram of the data treating height to naval (arm span) as the explanatory variable. Find the least-squares regression line treating height to naval as the explanatory variable. What is the slope? Interpret the slope. Does it make sense to interpret the intercept? Why?

Emphasize Statistical Literacy
Statistical literacy involves understanding and using the basic language and tools of statistics: knowing what statistical terms mean, understanding the use of statistical symbols, and recognizing and being able to interpret representations of data. Source: ARTIST website

Examples of Problems/Questions that Demonstrate Statistical Literacy
Suppose you have just constructed a 95% confidence interval. Name two options for increasing the precision of the interval. Draw a scatter diagram for which r = 1. According to popcorn.org, the mean consumption of popcorn annually by Americans is 54 quarts. The marketing division of popcorn.org unleashes an aggressive campaign designed to get Americans to consume even more popcorn. After two months, it was concluded that the marketing campaign was effective. Suppose, in fact, that the actual mean consumption of popcorn after the marketing campaign is 53.4 quarts. What type of error was committed? Why? What is an observational study? What is a designed experiment? Which allows the researcher to claim causation between an explanatory variable and a response variable? Explain what is meant by confounding. What is a lurking variable? What is resistance? Is the mean resistant? The median? Is the standard deviation resistant?

Emphasize Statistical Reasoning
Statistical reasoning is the way people reason with statistical ideas and make sense of statistical information. Statistical reasoning may involve connecting one concept to another (e.g., center and spread) or may combine ideas about data and chance. Reasoning means understanding and being able to explain statistical processes, and being able to fully interpret statistical results. Source: ARTIST website

Examples of Problems/Questions that Demonstrate Statistical Reasoning
In clinical trials of Nasonex, 3774 adult and adolescent allergy patients (patients 12 years old and older) were randomly divided into two groups. The patients in group 1 (experimental group) received 200 µg of Nasonex, while the patients in group 2 (control group) received a placebo. Of the 2103 patients in the experimental group, 547 reported headaches as a side effect. Of the 1671 patients in the control group, 368 reported headaches as a side effect. Is there significant evidence to conclude that the proportion of Nasonex users that experience headaches as a side effect is greater than the proportion in the control group? Are the results practically significant? The Food and Drug Administration sets Food Defect Action Levels (FDALs) for some of the various foreign substances that inevitably end up in the food we eat and liquids we drink. For example, the FDAL for insect filth in peanut butter is 3 insect fragments (larvae, eggs, body parts) per 10 grams. A random sample of 50 ten-gram portions of peanut butter is obtained and results in a sample mean of 3.6 insect fragments per ten-gram portion. Describe the sampling distribution of the sample mean.

Examples of Problems/Questions that Demonstrate Statistical Reasoning
Stocks may be categorized by industry. The following data represent the 5-year rates of return (in percent) for a sample of financial stocks and energy stocks ending December 3, Which sector is riskier? Does the sector with the higher risk, reward its investors? Why?

Examples of Problems/Questions that Demonstrate Statistical Reasoning
The following data represent the weights of plain M&Ms candies. Describe the distribution of weights. Which measure of central tendency is most appropriate to report? Which measure of dispersion is most appropriate to report? Justify your recommendations.

Examples of Problems/Questions that Demonstrate Statistical Reasoning
Suppose 100 different researchers wish to estimate the mean amount of time (in hours) 18 – 24 year old males spend watching television each week. Each researcher surveys a random sample of forty 18 – 24 year old males and constructs a 95% confidence interval for the mean time (in hours) 18 – 24 year old males watch television each week. How many of these intervals do we expect to capture the population mean?

Emphasize Statistical Thinking
Statistical thinking involves an understanding of why and how statistical investigations are conducted. This includes recognizing and understanding the entire investigative process (from question posing to data collection to choosing analyses to testing assumptions, etc.), understanding how models are used to simulate random phenomena, understanding how data are produced to estimate probabilities, recognizing how, when, and why existing inferential tools can be used, and being able to understand and utilize the context of a problem to plan and evaluate investigations and to draw conclusions. Source: ARTIST website

Examples of Problems/Questions that Demonstrate Statistical Thinking
What makes this a designed experiment? What type of experimental design is this? What is the response variable? Is it qualitative or quantitative? What factors are controlled in the experiment? In many experiments, the researcher will recruit volunteers and randomly assign the individuals to a treatment group. In what regard was this done for this experiment? Did the students perform better on the final exam in the fall semester? Can you think of any factors that may confound the results? Shows many facets of the statistical process

Examples of Problems/Questions that Demonstrate Statistical Thinking
Requires students to report the statistical process and contrast the study to what would occur in a designed experiment.

These are thinking problems because we don’t tell the student what type of test to use. The student must identify the correct procedure, verify the requirements of the procedure, do the test, and report the results.

What does GAISE say about being statistical literate?
See the GAISE report pages 5 – 7 Consider their carpentry analogy: In week 1 of the carpentry (statistics) course we learned to use various kinds of planes (summary statistics). In week 2 we learned to use different kinds of saws (graphs). Then we learned about using hammers (confidence intervals). Later we learned about the characteristics of different types of wood (tests). By the end of the course we had covered many aspects of carpentry (statistics). But I wanted to learn how to build a table (collect and analyze data to answer a question) and I never learned how to do that. Most of us probably use projects with the goal that students synthesize the material in the course and learn the statistical process. Personally, I have been disappointed in the students’ ability to create a project that demonstrates their understanding of the statistical process and their ability to recognize when certain procedures (conf int or hypothesis test) are desired, or applicable. Our students need more practice during the semester deciding what methods to use, or seeing how the various techniques are related to each other. “Many introductory courses contain too much material and students end up with a collection of ideas that are understood only at a surface level, are not well integrated and are quickly forgotten.” - Page 10, the GAISE report

Using Technology to Enhance Students’ Conceptual Understanding - Applets
Correlation by eye applet Draw a scatter diagram where the correlation of the data is about 0.8. Draw two scatter diagrams where the correlation of the data is close to 0. Draw a scatter diagram with about 6 points where the correlation is about Now add another point and move it around the Cartesian plane. How does this single point impact the value of the correlation coefficient? Regression by eye applet Draw a scatter diagram where the explanatory and response variable are negatively associated. Compare SSE for the regression line to an “eye-balled” line. Draw a scatter diagram where the explanatory and response variable are positively associated. Add another point that may be influential. How does this point impact the slope and/or intercept? Confidence interval applet Using either the proportion or mean confidence interval applet, illustrate the meaning of “level of confidence” Using either the proportion or mean confidence interval applet, illustrate the impact of not meeting the model requirements on the proportion of intervals that capture the parameter.

Using Technology to Enhance Students’ Conceptual Understanding - Simulations
Illustrate the sampling distribution of the sample mean using MINITAB.

Advantages of Personal Response Systems
Increased attention Increased attendance Increased retention Draper and Brown Students are twice as likely to attempt to construct an answer to a question using a PRS compared to a question that required them to raise their hand.

PRS Transmitter

Types of Questions Multiple Choice True/False Numeric Series
Short Answer Survey Be sure to show the audience how each question is constructed using the PRS PowerPoint plug-in.

Multiple Choice From what kinds of variables would side-by-side boxplots be generated? Qualitative only Quantitative only One qualitative and one quantitative Two quantitative Not sure

Free Response The reading speed of sixth-grade students is approximately normal, with a mean speed of 125 words per minute and a standard deviation of 24 words per minute. What is the probability that a randomly selected sixth-grade student reads less than 100 words per minute?

Series The following represent the steps in the statistical process. Put them in the correct order. Draw conclusions from the information Identify the research objective Organize and summarize the information Collect the information needed to answer the research questions

Camtasia Videos Ask students to watch the lecture at home…then class can be dedicated to developing the students conceptual understanding

Sources ARTIST website (https://app.gen.umn.edu/artist/index.html)
Chance, Beth L. (2002) Components of Statistical Thinking and Implications for Instruction and Assessment. Journal of Statistics Education 10, No. 3 delMas, Robert C. (2002) Statistical Literacy, Reasoning, and Learning: A Commentary. Journal of Statistics Education 10, No. 3 Draper, S. and Brown, M. (2004) Increasing interactivity in lectures using an electronic voting system. Journal of Computer Assisted Learning 20, 81 – 84 Draper, S. , Cargil, J. and Cutts, Q. (2002) Electronically enhanced classroom interaction. Australian Journal of Educational Technology 18, 13 – 23 Ebert-May, D., Brewer, C. and Allred, S. (1997) Innovation in large lectures—teaching for active learning Bioscience GAISE College Report (www.amstat.org/education/gaise/) Garfield, Joan (2002) The Challenge of Developing Statistical Reasoning. Journal of Statistics Education 10, No. 3 Hake, R. (1997) Interactive-engagement vs traditional methods: A six-thousand student survey of mechanics test data for introductory physics courses. American Journal of Physics Kennedy, G.E. and Cutts, Q.I. (2005) The association between students’ use of an electronic voting system and their learning outcomes. Journal of Computer Assisted Learning – 268 Rumsey, Deborah (2002) Statistical Literacy as a Goal for Introductory Statistics Courses. Journal of Statistics Education 10, No. 3 West, J. (Dec. 9, 2005) Learning outcomes related to the use of personal response systems in large science courses. Academic Commons. Wit, E. (2003) Who wants to be… The use of a personal response system in statistics teaching. MSOR Connections