Using Research Projects to Help Develop High School Students' Statistical ThinkingAuthor(s): Randall E. Groth and Nancy N. PowellSource: The Mathematics Teacher, Vol. 97, No. 2 (FEBRUARY 2004), pp. 106-109Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20871523 .

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Randall E. Groth and Nancy N. Powell

Using Research Projects to Help Develop

High School Students' Statistical Thinking

s

Vf e need to engage

students in all phases

of the

investigative cycle of

statistics

tatistics plays a key role in shaping policy in a

democratic society, so statistical literacy is essential for all citizens to keep a democratic government strong (Wallman 1993). However, fostering the sta tistical thinking is a complex endeavor We ulti

mately need to engage students in all phases of the

investigative cycle of statistics, including data gath ering, data analysis, and inference.

This article describes two projects in which we

attempted to help Advanced Placement (AP) statis tics students become proficient at moving through the investigative cycle. Although the projects were

implemented in an AP class, they could be included as part of any class in which linear equations and best-fit lines are studied. As we describe the two

projects, we also discuss some of our reflections on

the extent to which the projects helped our stu dents become more proficient. In the first project, the students focused on the data analysis and infer ence phases. In the second project, the students had the opportunity to experience all phases of the

cycle.

PROJECT 1 : ANALYZING AND DRAWING CONCLUSIONS FROM EXISTING DATA FILES

Project 1, shown in figure 1, was assigned during the first semester of the school year. At the time, students were using their textbook (Yates, Moore, and McCabe 1999) to study concepts related to cor relation. Project 1 helped students develop deeper understandings of some of the concepts that they were studying, and it helped them become comfort able using the computer program Fathom (Finzer, Erickson, and Binker 2001). We saw Fathom as a

powerful tool for helping formulate and investigate statistical conjectures, since it allows users to

quickly import data, calculate summary statistics, test hypotheses, and build graphs and tables. Stu dents had some freedom to choose the problem that

they would study; we encouraged them to explore some of the data files contained in Fathom. Because Fathom comes packaged with existing data files, we

were able to move the focus of the project away from the planning and data-collection parts of the

investigative cycle and toward the analysis and conclusion parts of the cycle.

In assessing students' work on the first project, we gained several important insights about stu dents' thinking while they engaged in analyzing data. These insights were valuable for guiding future instruction. We found that some students tried to make the intercepts of the regression line have meaning even when they did not. Some stu dents tried to make sense of a negative regression line intercept within the context of plotting SAT scores versus grade-point average.

Perhaps partially because of the stated require ments of the project, students attempted to force

meaning on the intercept when it really had no use ful physical interpretation. Some students also did not seem to realize that it was dangerous to force

meaning on intercepts, or any points, that lie well outside the cluster of points used to produce the

regression line. After identifying these gaps in stu dents' understanding, we were able to address them in future instruction.

Although many students identified the slope of the regression line as a rate of change, quite a few had difficulty giving the specific units of the rate of change. In addition, some students confused the

slope of the regression line with the correlation coefficient and stated that the slope of the regres sion line indicated the strength of the linear rela

tionship between the two variables. Finally, we

noticed that some students struggled with the idea

Randall Groth, regroth@salisbury.edu, teaches mathe matics education at Salisbury University in Salisbury, MD 21804. His interests include teaching mathematics methods courses for preservice teachers and researching high school students' statistical thinking. Nancy Powell, powelln@district87.org, is a 1992 Presidential Awardee and the lead teacher for mathematics at Bloomington High School in Bloomington, IL 61701. She is interested in helping students make mafamatical connections and

enjoys giving workshops for teachers of grades K-12.

Photograph by Nancy Powell; all rights reserved

106 MATHEMATICS TEACHER

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that two variables can have a correlation close to 1 or -1, yet not cause each other. Identifying these

problems in thinking proved to be an important first step in dispelling them.

Since the assignment involved completing a writ ten report, we helped students learn to communi cate the results of statistical analysis. In assessing the students' work on the projects, we noticed that several students used the formal term outlier to describe "unusual" data points in a bivariate situa

tion, even though they had only studied formal pro cedures for identifying outliers in univariate situa tions. We then had a class discussion of the

importance of being careful in using formal terms when reporting statistical results. We also noticed that students tended to say in their reports that

they were using the regression line to "find values," when a more precise wording would have stated that the regression line can be used to "predict approximate values." These imperfections in com munication allowed us to emphasize that even if statistical analysis is correctly done and the conclu sions are appropriate, the study is of limited value if the data-analysis process and the conclusions are not communicated properly.

PROJECT 2: EXPERIENCING ALL PHASES OF THE INVESTIGATIVE CYCLE

As a final project for our AP statistics course, stu dents designed their own statistical projects and

posters. We instructed students to follow the guide lines given by the American Statistical Association (ASA) at www.amstat.org/education/posterl.html, and we allowed them to do joint projects with class mates who had common interests. The ASA guide lines helped students experience all phases of the

investigative cycle. We set aside approximately two weeks to allow everyone in class to identify a quan tifiable problem of interest, make a plan for investi

gating it, gather data, analyze the data, and draw conclusions.

As students began to select their problems, sev eral interesting ideas for projects emerged. One student wished to learn whether a relationship existed between a student's grade level and the number of steps that he or she took in a day. A pair of students decided to investigate the relationship between the types of music that students listened to and their grade-point averages. Another pair of students wondered how the distribution of types of cars driven in town compared with the distribution of types of cars driven in the entire United States. Yet another pair became interested in the relation

ship between brain size and intelligence. A larger group of students wanted to determine whether boys could generally throw a softball faster than girls. An other group decided to compare the blood pressures

Project Choose one pair of variables from the "Data in Depth" folder in Fathom that interests you and that seems to be correlated. Write a short paper organized in the following manner:

Introduction (10 pts.) Why would someone be interested in investigating the relationship between the two variables that you have chosen?

Background (20 pts.) Does one of the variables that you have chosen actually influence the other? If you think so, make an argument that one does influence the other. If you do not think so, explain why the variables that you have chosen are correlated but one does not "cause" the other.

What are some interesting questions you have that will be answered when you analyze your data set? Include questions that can be answered by finding the equation of the regression line, the slope and intercepts of the regression line, and the strength of the linear relationship between the variables. You will answer these questions in the "Results" section.

Method (20 pts.) What kinds of technology did you use to investigate the relationship between the two variables (for example, Fathom, -83, other)? What did the technology do for you (or what did you make it do for you)? What are some other tools or methods (if any) that you decided to use to investi gate the relationship?

Data Analysis (20 pts.) Copy and paste all Fathom output into this section. Once you have your Fathom output pasted into a Word Document, you can move the output around in the docu ment even on a computer that does not have Fathom. You might want to copy and paste your output before writing the other sections of this report. Are there any "unusual" values in the data set? What makes them unusual? Are you choosing to throw out any of the unusual values before you plot a regression line? If so, justify the decision to throw out unusual values.

What is the equation of the regression line? What is its slope and intercept? What is the correlation coefficient r? What are some of the graphs that you analyzed to investigate the relationship?

Results (20 pts.) How strong is the relationship between the two variables? How do you know how strong the relationship is? Are they positively or negatively associated? Since you know the strength of the relationship, what are some of the questions that you can answer? What are the answers to these questions, and how did you find them? What valuable information is provided by the slope and intercepts of the regres sion line? What are some questions that you can now answer since you know the slope and intercepts of the regression line? What are the answers to these ques tions, and how did you find them? What valuable information is provided by the regression line? What are some questions that you can answer now that you know the regression line and its equation? What are the answers to these questions, and how did you find them? Discuss the answers to any of the other questions that you wrote in the "Back ground" section.

Conclusion (10 pts.) What did you learn (for example, about statistics, technology, baseball, cars, build ings) that you did not previously know? Is there anything you would do differently if you were to write the paper again? What are some other interesting studies that could be done that are related to what you did in this paper?

Fig. 1

Project involving the analysis of existing data

Vol. 97, No. 2 ? February 2004 107

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Helping students

develop statistical thinking is

extremely

challenging

of athletes to those of nonathletes. Still another

group decided to investigate the relationship between the stock price of a company that produced fatty foods and the amount of obesity in the United States. Even though some groups took one or two

class periods to settle on a question that interested

them, allowing the students to identify their own

problems to investigate helped them develop feelings of ownership of the problems at the outset.

Because of the diversity of the questions posed by members of our class, students chose a number of different types of plans for completing the proj ects. The students who wanted to examine the rela

tionship between the types of music and grade point averages for students at the high school decided that a survey would be the most efficient way of

gathering information for their project. The students who were investigating the number of steps taken in a day, the types of cars driven in town, softball

throwing, and blood pressure all chose observation al studies as their mode of inquiry. Students inves

tigating the questions about blood pressure, brain

size, and stock prices of a company chose to use

preexisting data to help answer the questions. The

diversity of modes of inquiry used by our students in answering the questions that they had posed re

flected some of the methods of data production that

they had studied during the AP statistics course.

The students who gathered their own data came

to appreciate that gathering data for a statistical

study can be a complex endeavor. The pair using a

survey to gather information realized that the man ner in which they asked the survey questions could

easily influence the manner in which people respond ed. The student who investigated the relationship between the number of steps that a student takes in a day and his or her grade level used an electronic device called a Digiwalker, which we provided. However, he quickly discovered that the device

reported inaccurate data. The group investigating the softball-throwing question needed to learn how to operate a radar gun that they borrowed from a

policeman. The pair of students gathering informa tion about blood pressure discovered that finding a

truly random sample of students was impossible, since some teachers would not let students out of class to have their blood pressure taken. The com

plicated and often imperfect nature of gathering data was conveyed to students in a manner that was far more memorable than conventional text

book instruction. Once the data had been gathered, many of our

students used both Fathom and Excel to help ana

lyze the data. Using Fathom and Excel enabled stu

dents to quickly produce such graphical displays as

scatterplots, histograms, and dot plots. Some of the

graphs that our students created are shown in fig ures 2 and 3. More student work is displayed

online at www.district87.org/staf?7powelln/Statistics /EndofYearProject/Default.htm. The software

produced graphical displays served as tools to help students think about the validity of the conjectures that they made at the start of the project and some times led students to pose new questions that they had not asked at the outset. Fathom also allowed students to produce summary statistics, such as the

mean, median, mode, and correlation coefficients, as needed. These summary statistics played a role similar to that played by the graphs, since they led our students to think about their initial conjectures and formulate new conjectures that were based on

the data. Project Ts focus on using Fathom during the analysis phase of the investigative cycle cer

tainly paid dividends when our students engaged in this end-of-year project.

Collection 1 150+

750000 850000 950000 1050000 MR! count

FSIQ = 0.000119 MRI count + 5.2; 2 = 0.13

Fig. 2

Fathom output comparing IQ scores to brain size

(measured by MRIcount)

Av#fiQ# Stop> According to Grade

Stop*

Fig. 3 Excel output comparing number of steps taken

in a day to grade level

To wrap up the project, each group presented its

findings to the entire class. We believed that evalu

ating and discussing the validity of each group's conclusions were important. We designed a written feedback form, shown in figure 4, for students to

fill out to critique each presentation. We also

encouraged students to ask questions at the end of

each presentation. Although students were initially hesitant to ask questions about the work of their

108 MATHEMATICS TEACHER

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classmates, several good class discussions did even

tually take place as presentations proceeded. For

instance, the group that was trying to determine whether boys threw softballs faster than girls did used a two-sample t-test to analyze the data that

they had collected. The class then discussed whether that particular test was appropriate for the given situation. At the end of its presentation, another group stated that the increasing consump tion of a certain fatty food caused the increasing instances of obesity in the United States. The class

challenged this conclusion, since some students

argued that several other factors could contribute to increasing obesity. The group investigating the

relationship between types of music that students listened to and their grade-point average was also

questioned. Some students asked why the group listed the various types of music on the question naire instead of letting the students fill in the types of music they enjoyed. The questions and chal

lenges from classmates helped students in each of these groups reflect on and evaluate the various

phases of the journeys they had taken through the

investigative cycle.

CONCLUSION As noted at the beginning of this article, helping students develop statistical thinking is extremely challenging. In addition to helping students master the mathematics needed for the analysis and infer ence phases of the investigative cycle, teachers need to help students master some of the nonmath ematical elements of the cycle involved in identify ing a problem, creating a plan of attack, and gath ering necessary data. Our strategy identified and addressed some of the difficulties students had in the data analysis and inference phases of the cycle in one project and then immersed them in all phas es of the investigative cycle in a later project. We

hope that the projects and experiences that we have described will add to the dialogue concerning how best to develop and foster statistical thinking.

REFERENCES Finzer, William, Tim Erickson, and Jill Binker. Fath

om. Emeryville, Calif.: Key Curriculum Press, 2001. Wallman, Katherine . "Enhancing Statistical Litera

cy: Enriching Our Society." Journal of the American Statistical Association 88 (March 1993): 1-8.

Yates, Daniel S., David S. Moore, and George P. McCabe. The Practice of Statistics: TI-83 Graphing Calculator Enhanced. New York: W. H. Freeman & Co., 1999. Mr

Audience Evaluation Evaluator's first initial/last name:.

Group topic/name of presentation:_

Team members:_ Write first name, last initial for each member

Respond yes or no to each of the following questions. Write a sentence or two to justi fy each of your responses. Use the back of this page if necessary.

1. Did you understand the topic being studied? Was the question that the group stud ied interesting?

2. a) Was the presentation informative or creative?

b) Was the presentation done in a professional way?

3. Was the use of visuals and technology (visual aids such as graphs, posters, Web

pages, PowerPoint presentations) correct, and did they add to the understanding of the material presented?

4. a) Were the data-gathering methods appropriate and reliable? Were they clearly explained?

b) Did the statistical analysis performed by the group justify and support the con clusions made by the group?

g) Were there any errors or omissions in the statistical analysis?

5. What questions or comments do you still have for the group after the presentation?

Use the table below to rate the presentation in each of the following categories.

Really Good Points

Criterion_Awesome Good Okay Start Awarded Did you understand the question? 4 3 2 1 _

Was the presentation professional, informa

tive, and creative? 4 3 2 1

Was the use of techno

logy and visual aids effective? 4 3 2 1

Were the statistics cor rect and reliable? 4 3 2 1

Overall rating 4 3 2 1

Total points awarded

Fig. 4 Audience evaluation form for project 2

Vol. 97, No. 2 ? February 2004 109

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