R. Kirk Steinhorst and Carolyn M. Keeler

University of Idaho

Journal of Statistics Education v.3, n.3 (1995)

Copyright (c) 1995 by R. Kirk Steinhorst and Carolyn M. Keeler, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the author and advance notification of the editor.

**Key Words**: Conceptual learning; Authentic
assessment.

For traditionally trained statistics teachers, developing active learning material is difficult. We present representative active learning materials that we have used over the last several years. We also give examples of exam questions that we have used to test conceptual understanding gained through the class exercises.

1 In a preceding paper (Keeler and Steinhorst 1995), we argued that introductory statistics courses should involve active learning of conceptual statistical material using relevant examples. The statistical training of many statistics instructors today was traditional---lectures covering the mechanics of statistical methods and the theory of probability and mathematical statistics. In both methods and theory courses, students' involvement was limited to ``work the following problem'' assignments. For those of us trained this way, teaching introductory courses from a more conceptual, active learning viewpoint is a challenge. Certainly constructing questions that go beyond ``find the standard error of the mean for the following data'' is difficult.

2 With practice we can find exercises that get at what the student understands about statistics rather than what they know how to calculate. A good conceptual question will have just the right amount of ambiguity. The students must think through various possible responses.

3 In this paper, we report our experiences over the last four years in developing conceptual lecture material, exercises, and examination material for introductory statistics taught to students in the first two years of their university training. Some of the material we have converted from older, traditional material. Some of it is new. We use this material in a cooperative learning framework, but students need not work in groups. The specific format of the class---lectures, labs, group activities---is less important than the orientation. The aim is to develop a conceptual, rather than mechanical, understanding of statistics.

4 Higher education has long been dominated by the lecture method. This type of instruction tends to reinforce passive learning, particularly in inexperienced, unprepared students. A more constructivist view of learning challenges teachers to create environments in which students are encouraged to think and explore and construct their own understanding (Brooks and Brooks 1993). Garfield (1993) describes the constructivist approach as students bringing their existing knowledge, experiences, and beliefs to the classroom and constructing new knowledge in a way that makes sense to them. The instructor is challenged to involve students in the learning process by making them active rather than passive in the classroom. The end result is movement away from the traditional lecture approach. The instructor becomes a leader and facilitator rather than merely a lecturer.

5 Seminal work in active learning grew out of the experiential learning viewpoint of John Dewey and the student-centered instructional philosophy of cognitive psychologists Jean Piaget and L. S. Vygotsky. Dewey, Piaget, and Vygotsky do not see learning as receiving lecture material from an instructor, but as experiencing the material. In this setting, the teacher's ability to create a context where learners can discover and reconstruct knowledge becomes very important (MacGregor 1990).

6 Faculty members who are motivated to create classroom experiences where students are active learners soon realize the careful thought required to apply the concepts of active learning to their own discipline. In designing active learning experiences, both the students' current knowledge and emerging knowledge must be considered. This means that new information and concepts need to be at the appropriate level of difficulty. Activities or exercises must build on current knowledge in which students have confidence and simultaneously must challenge them to attempt the next step, develop a new understanding, or develop a new application of a concept.

7 In developing conceptual, active learning material for our courses we must not forget exams. Alan Mehler (1992), discussing improvement of courses in biochemistry, points out that often ``the last aspect to be considered, if it is considered at all, is the method of evaluating student performance.'' His premise is that the most influence that the instructor has on the student is through exams and that one must redesign exams to accomplish course goals.

8 Assessment must be authentic (see Wiggins 1993). Authentic learning and assessment has meaning only in relation to the learner. What is authentic is defined as a task that is representative of something the learner would experience outside of the school setting. Therefore, in statistics, the student might be asked to interpret an excerpt from a local newspaper article that describes a survey of residents' preferences for funding a new swimming pool.

9 One of the difficulties with authentic problems in statistics is that they are often messy or involve extensive computation. Careful selection of real world problems can help minimize the problem of messiness. If large datasets are provided in computer form, then standard statistics computer packages can be used to reduce the need for extensive hand calculation. In some cases, working with a few real numbers can be just as illustrative as working with a big dataset. We have found that working a small example by hand calculator and by using the computer package helps build confidence in the computer approach.

10 The material in this section can be used in a variety of ways. We have small groups of students answer the questions during class and turn in their answers to be graded (see Keeler and Steinhorst 1995). Alternatively, instructors can use the questions to stimulate general class discussion led by the instructor or can assign the questions as homework.

11 The four subsections correspond to the first four units in our course. Individual instructors might organize the material in a different order or adapt it to their particular course style. The first unit covers the design of statistical studies. The second unit covers probability from an introductory statistics viewpoint and replaces the traditional coverage of probability usually seen in first statistics courses. The third unit covers a conceptual view of descriptive statistics, and the fourth unit covers a conceptual view of sampling distributions.

12 Most of the introductory texts that the authors have used over the last 25 years have assumed that the data are already in hand. These texts overlooked the fact that the design of the study and the selection of an experiment, survey, or observational study as the vehicle for collecting data is central to the understanding of basic statistics. Newer texts include material on study design. Freedman et al. (1991) cover these concepts in chapters 1, 2, and 19.

13 To illustrate the concept of an experiment as defined by statisticians, we give the students the following written scenario:

Linus Pauling, the Nobel laureate chemist, has advocated taking large doses of vitamin C as a cure for the common cold and other ailments. It is obvious that modern medicine does not subscribe to vitamin C therapy. When you go to your doctor with a cold, she or he does not prescribe large doses of vitamin C. Experiments with large doses of vitamin C did not result in shortening the duration of a cold nor lessen the symptoms. Let us design a study to test the vitamin C therapy.

14 The lecture proceeds by involving the students in a discussion of how we might choose dorm students for our study and how we might make unbiased (double blind?) measurements of their cold experiences through the winter. The class usually (with some prompting) concludes that we need to select a fairly homogeneous group of students (perhaps from a singe dorm) and that we need to randomly assign students to treatments. We talk about the role of randomization in arguing causation. We discuss experimental units.

15 This mini-lecture/discussion takes about 30 minutes. We then ask students to work in groups to answer the following questions:

1. What is the population that we wish to study?

2. What is the experimental unit? Can it sometimes be a student and sometimes a dorm room?

3. Describe in a short paragraph how you would randomize the vitamin C and colds study using students from a single dormitory.

Since this is a very early group activity, most groups cannot get started until the instructor or teaching assistant helps them identify the population as college students living in dormitories. They then can proceed on their own. The randomization plans the students suggest are sometimes inventive, including schemes such as numbering students as they walk into the cafeteria, but common plans involve flipping coins or using numbered tokens in a hat.

16 After covering similar material on surveys and observational studies in other class periods, we ask students to work in groups to answer additional questions.

4. Design a survey to estimate the amount of time preschoolers spend watching television in a week. Carefully define the population, frame, sampling unit, and random selection plan.

Some students conclude that they must sample households, not individuals. Others consider sampling children from organized preschools in the region. We subsequently discuss how that plan misses children who do not attend formal preschool.

5. When scientists estimate a wildlife population such as elk, they say that they are conducting an ``animal census.'' Discuss the use of the word census from the perspective of this course.

6. Scientists wish to study the effects of listening to music while studying. Poulton (1977) found that music masks inner speech in working memory. Decide whether the following study is an experiment, a survey, or an observational study. Justify your answer.

- Researchers describe their upcoming study in the college newspaper. A few days later, they go room to room in dormitories and ask students about their music listening habits. They also obtain grade point averages of students who sign a release form.

Most students conclude that this is an observational study. Many students point out that not all students read the college newspaper and that those students who sign a release for their grades may be different from those who do not. Other students think this is a good example of an experiment or survey. They are chagrined to find out they are wrong.

17 The following test questions examine students' conceptual understanding of the material on experiments, surveys, and observational studies.

1. Classify each of the following studies as an experiment, survey, or observational study. Explain your reasoning in a short sentence.

(a) A study evaluating consumers' satisfaction with their current long distance telephone carriers.

(b) A study of differences between bull and cow elk foraging strategies.

(c) A study of the effectiveness of several antibiotics in controlling lung congestion in newborns.

(d) A study of radio stations to see what proportion use personal computer-based record keeping systems.

(e) An experiment station study of potato yields under center pivot irrigation and furrow irrigation.

2. The Salk polio vaccine study was _randomized controlled_. What does controlled mean in this context?

3. In an observational study of length of stay for childbirth in large versus small hospitals,

(a) What is the population?

(b) What is the observational unit?

(c) What can be randomized in this study?

A common incorrect answer identifies mothers as the population and a mother as the observational unit. Some students correctly identify large and small hospitals as the population and a hospital as the observational unit with clusters of mothers as subunits. Some groups also point out that large and small hospitals can be randomly selected from among all such hospitals, but clearly one cannot randomize ``large'' and ``small.''

4. I would like to know the various proportions of single rooms, double rooms, and multiple (more than two people) rooms in the dormitories, fraternities, and sororities on campus. I am going to conduct a statistical survey.

(a) Define the sampling unit. (Hint: It is not a person.)

(b) Define the population.

It is obvious that there is not always a single best answer to some of these questions. That is why the students must write a sentence or two explaining their position. Learning occurs as they think through the possible answers and compose a defense for the one they choose.

18 Many introductory statistics texts have a unit on probability that does not relate to the rest of the course. We try to limit our discussion of probability to those skills that the students need to successfully complete the statistics chapters. Probability mass functions (pmf's) and probability density functions (pdf's) are usually not covered in the first course. We think covering pmf's and pdf's in a nonmathematical fashion accomplishes several important goals.

- It helps to reinforce the idea of a population.
- It establishes PARAMETERS as rational descriptors of populations.
- It allows us to think about random sampling and random variables.

The use of uniform density functions and (asymmetric) triangular distributions allows students with only an algebra background to find areas under the curve and the population MEAN and MEDIAN. (Words in uppercase letters denote population parameters.)

19 The elementary rules of probability can be covered in the context of drawing a unit at random from a population with given pdf or pmf. The probability that the observed response is below a specified level can be found by finding the appropriate area under the curve. The density can also be used to illustrate disjoint events and the complement of an event.

20 The group exercises on this unit take the following form:

1. If half of voters support Clinton's health care plan (1) and half do not (0), draw the appropriate mass function and carefully label the axes.

2. Describe a response variable, y, of interest in (one of) your field(s). Draw a likely mass or density function for this variable. Mark the population mean with the symbol \mu and the population median with an M.

This is a difficult question for many students because they have to produce a relevant example from their own experience.

3. Suppose that average grades in this course are uniformly distributed between 60 and 100. [Students are given a graph of the probability density function.]

(a) What is the probability that a randomly chosen student will make an A (that is, have an average grade above 90)?

(b) The vertical axis is labeled ``relative density.'' What does this mean?

Whether or not students correctly find P(A) = 1/4, we give them partial credit for properly shading the area above 90 and starting the calculation. Sometimes we identify the height of the curve as 1/40; sometimes we do not. When we do not, the students have a more difficult time completing the task in the time allotted.

21 Appropriate exam questions over material of this type include the following.

1. If an observational study is conducted to determine the amount of time three and four-year-old children watch TV per week, then the distribution of y = time per week is a (mass, density) function. (Circle one.) Why?

2. If people are ambivalent about requiring medical personnel to disclose whether or not they are HIV positive, they might answer an appropriately worded question with equal proportions of 1 (strongly agree), 2 (agree), 3 (neutral), 4 (disagree), and 5 (strongly disagree). The mass function appears below. [Students are given a graph of the pmf.]

(a) What is the MEAN?

(b) What is the height of each of the ``sticks''?

The skills required here are understanding and visualization, not computation.

3. The heights of college females are normally distributed with MEAN = 5'4'' and VARIANCE = 4''. What is the corresponding SD and what does it tell you?

``What is the corresponding SD ...?'' is a mechanical question. ``What does it tell you?'' adds a conceptual component.

22 In our coverage of descriptive statistics, we emphasize ideas rather than computation and suggest that one use a computer package to do the calculations. We ask students to analyze data about themselves that they generated during the first week of class. We use a variation of the Activity-Based Statistics Project exercise, Getting to Know the Class (Scheaffer, Gnanadesikan, Watkins, and Witmer, in press). The students are asked to provide answers to questions like, ``How many sisters and brothers do you have?'' or ``What time did you go to bed last night?'' The idea is to generate data of differing types in which the students will have a vested interest. We use MYSTAT or SYSTAT to analyze the data.

23 Group exercises include the following questions:

1. For the data on students' pulse rates, how many bars should the histogram have? Explain.

The old emphasis used to be on mechanics---bins and drawing. The more interesting questions relate to the uses of a histogram and how the visual information is changed by varying the number of bars.

2. For the data on bedtimes, is the mean or median preferred as a measure of the ``middle'' of the data? Explain your reasoning.

3. Draw lines connecting items in column A to the most related item in column B.

A B mean histogram probability density function MEAN(Y - MEAN)^3/SD^3 MEDIAN MEAN skewness median sd \sqrt{variance}

To help groups work through this exercise, we have to help them overcome their initial reticence. If we can get them to connect mean and MEAN, then they usually can complete the remaining connections.

4. If answers to the question ``Do you smoke?'' are coded as 1 = yes and 0 = no, show that the sample mean is the observed proportion of yeses.

Students initially balk at this question because it looks like a proof. However, the answer is quite straightforward, and, with a little support from the instructor or teaching assistant, most groups put down a coherent argument.

24 Descriptive statistics appear naturally in this class because the students know about collecting data and describing populations. The sample statistics---mean, median, variance, sd---become obvious estimators of their population counterparts---MEAN, MEDIAN, VARIANCE, SD. Students accept histograms, stem-and-leaf plots, and dot plots as natural estimators of density or mass functions. We define the sampling distribution of sample means and the PARAMETERS MEAN(mean), VARIANCE(mean), and SE(mean). But more importantly (and where we differ from most beginning texts), we also talk about the sampling distributions of the median, sd, and range and their corresponding PARAMETERS MEAN(statistic), VARIANCE(statistic), SE(statistic), and MEDIAN(statistic). Only in this way do students get the big idea.

25 We use the following group exercises:

1. The handout sheet represents 100 apartment buildings. The number of squares in a group denotes the number of children requiring after school care for that building.

(a) Look at the sheet and estimate (by eye) the average number of children needing care per building.

(b) Pick five ``representative'' buildings and calculate the average number of children needing care for the five representative buildings that you selected.

(c) Using the random number table provided, pick five buildings at random and calculate the average number of children needing care.

This is a variation on the ``random rectangle'' exercise from the Activity-Based Statistics Project (Scheaffer et al., in press). It is an effective exercise for helping students discover the worth of random sampling. A dot plot of the estimates from the eyeball, representative, and random sampling procedures along with the known population distribution and the known distribution of means from random samples of size five is compelling for students.

2. ``Randomly'' select classmates and form several samples of size three. For the variable y = distance you live from this classroom building, calculate the sample mean of each sample.

The instructor then collects all of the sample means and constructs a dot plot to give the students a sense of the sampling distribution of the mean. Repeat the exercise using the median or standard deviation as the statistic calculated.

3. The notation mean(MEAN) makes no sense. What is wrong with it?

We want the students to conclude that MEAN is a constant and hence one cannot produce numbers that can be averaged in the sense of mean( ). A common answer is, ``mean(MEAN) makes no sense because you cannot take the sample mean of the MEAN.'' We give them partial credit for paraphrasing the question and mark the answer with ``Why?'' in the margin.

4. Explain the difference between MEAN and mean.

5. Explain the difference between MEAN and MEAN(mean).

Common wrong answers include ``MEAN is the mean of `the population' while MEAN(mean) is the mean of a subpopulation of means and will have some value because of estimation'' and ``MEAN is the mean of `a population' while MEAN(mean) is the mean of a sample of means.'' It is common for them to understand what a population mean is, but to misconstrue what MEAN(mean) is. However, the answers often show that they are trying to understand the concepts.

26 Exam questions on this material include the following.

1. The ACT test has a MEAN of 19 and an SD of 1.1. If we sample 50 high school seniors who have taken the ACT, find

(a) MEAN(mean) = ____________

(b) VARIANCE(mean) = ____________

(c) SE(mean) = ____________

(d) MEAN(variance) = ____________

(e) VARIANCE(variance) = ____________

2. An analysis of cholesterol data for 106 adults appears below. In the ``sample'' column write the name of the statistic and give its value.

Systat analysis of blood serum cholesterol data

CHOLESTR N of CASES 106 MINIMUM 120 MAXIMUM 414 MEAN 217 STANDARD DEV 53 MEDIAN 216 POPULATION sample MEAN e.g., mean = 217 MEDIAN VARIANCE SD RANGE

This exam question reinforces the connection between PARAMETERS and statistics. It also reinforces our predilection to encourage students to use computer packages for real statistical work.

3. The U. S. Center for Health Statistics collects data on the daily intake of selected nutrients by income level. Suppose we collect data on protein intake for 15 people randomly selected from individuals whose incomes are below the poverty level and 10 people randomly selected from individuals whose incomes are above the poverty level. A computer summary of the data appears on the back of the previous page. Provide the following:

(a) mean_1

(b) sd_2

(c) mean_1 - mean_2

(d) MEAN(mean_1)

(e) VARIANCE(mean_1)

(f) SE(mean_1)

(g) variance(mean_1)

(h) se(mean_1)

27 The computer printout provided consists of sample size, mean, variance, standard deviation, and standard error by group. For mean_1, sd_2, and se(mean_1), the student merely has to identify the appropriate number on the printout. The variance(mean_1) is the square of se(mean_1). The difference between sample means requires identifying the two numbers and subtracting them. MEAN(mean_1), VARIANCE(mean_1), and SE(mean_1) are difficult. The correct answer is a formula, not a number. Of course, the students will have worked a similar example in class, so they should know how to answer the question. Nevertheless, many students feel compelled to put down a number.

28 We have illustrated our methods using material on design, populations and probability, descriptive statistics, and sampling distributions. We use similar material for inference in finite populations, including confidence intervals, basic hypothesis testing, and simple linear regression and correlation.

29 Through the use of this material, we have been successful in dealing with statistics in a conceptual fashion at the lower division level. We feel there has been too much emphasis in the past on rote learning and mechanics. It is possible to reformulate traditional materials on probability, descriptive statistics, and sampling distributions in a conceptual, yet straightforward, way that engages students and helps them to understand the ideas of statistics without getting lost in the details. With the wide availability of statistical software on a variety of platforms, there is no reason to dwell on the mechanics of statistical methods. Why teach students to be $10 calculators? Help them discover statistical thinking and the ability to solve real-world problems.

30 We have developed material that is not part of the traditional course. The unit on experiments, surveys, and observational studies above illustrates how important ideas that have been left out of traditional introductory courses can be added at a level that college students can understand.

31 Another topic that we have added is basic inference in finite populations. Most of our students will be faced with interpreting sample surveys in their technical and personal lives. They will see surveys every day. After our unit on simple, stratified, cluster, and systematic sampling and our formulation of basic inference for simple random sampling, students can understand the graphic over Dan Rather's shoulder that says that the margin of error is plus or minus 3%. The conceptual development of statistics has many rewards.

32 Instructors can use the conceptual approach with groups or individuals. We lean toward cooperative group learning. Our experiences with this approach in introductory statistics courses have been extremely positive. We think this approach directly addresses the concerns about the first course in statistics discussed by Hogg (1991), Watts (1991), Cobb (1993), and Snee (1993), among others. Students' attitudes are better and test scores have improved (Keeler and Steinhorst 1995).

Brooks, J. G. and Brooks, M. G. (1993), In Search of Understanding: The Case for Constructivist Classrooms, Alexandria, VA: Association for Supervision and Curriculum Development.

Cobb, G. (1993), ``Statistical Thinking and Teaching Statistics,'' UME Trends, November.

Freedman, D., Pisani, R., Purves, R. and Adhikari, A. (1991), Statistics (2nd ed.), New York: W. W. Norton and Company.

Garfield, J. (1993), "Teaching Statistics Using Small-Group Cooperative Learning," Journal of Statistics Education [Online], 1(1). (http://jse.amstat.org/v1n1/garfield.html)

Hogg, R. V. (1991), ``Statistical Education: Improvements Are Badly Needed,'' The American Statistician, 45(4), 342-343.

Keeler, C. M. and Steinhorst, R. K. (1995), ``Using Small Groups to Promote Active Learning in the Introductory Statistics Course: A Report from the Field,'' Journal of Statistics Education [Online], 3(2). (http://jse.amstat.org/v3n2/keeler.html)

MacGregor, J. (1990), ``Collaborative Learning: Shared Inquiry as a Process of Reform,'' New Directions for Teaching and Learning, 42, 19-30.

Mehler, A. H. (1992), ``Integration of Examinations and Education,'' Biochemical Education, 20, 10-14.

Poulton, E. C. (1977), ``Continuous Intense Noise Masks Auditory Feedback and Inner Speech,'' Psychological Bulletin, 84(5), 977-1001.

Scheaffer, R., Gnanadesikan, M., Watkins, A., and Witmer, J. (in press), Activity-Based Statistics, New York: Springer-Verlag.

Snee, R. (1993), ``What's Missing in Statistical Education?,'' The American Statistician, 47(2), 149-154.

Watts, D. G. (1991), ``Why Is Introductory Statistics Difficult to Learn? And What Can We Do to Make It Easier?,'' The American Statistician, 45(4), 290-291.

Wiggins, G. P. (1993), Assessing Student Performance, San Francisco, CA: Jossey-Bass.

R. Kirk Steinhorst

Division of Statistics

University of Idaho

Moscow, ID 83843

Carolyn M. Keeler

Educational Administration

UI Boise Center

800 Park Blvd.

Boise, ID 83712

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