Amy G. Froelich
W. Robert Stephenson
Iowa State University
William M. Duckworth
Journal of Statistics Education Volume 16, Number 2 (2008), ww2.amstat.org/publications/jse/v16n2/froelich.html
Copyright © 2008 by Amy G. Froelich, W. Robert Stephenson and William M. Duckworth 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 authors and advance notification of the editor.
Key Words:Activities, Introductory statistics, Statistical concepts
``Declare the past, diagnose the present, foretell the future; practice these acts. As to diseases, make a habit of two things - to help, or at least to do no harm.''
Hippocrates from Epidemics, Bk. I, Sect. XI.
The NSF grant that funded the development of the new course materials was for a ``proof of concept'' project. The focus of the project was to develop the materials and assess them to show that the concept was viable, that is that the materials could work in the laboratory setting and that the materials showed promise for improving students' learning. Through field testing and assessment we found that the materials worked well in the laboratory setting and that there was some improvement in student performance on the general topic of regression.
The materials were developed for a general introductory statistics course, Stat 101, at Iowa State University. Stat 101 is a four credit course designed for general majors on campus and is comprised of three hours of lecture and two hours of laboratory each week during the semester. Current laboratory activities are very prescribed. For example, the laboratory on the Normal model involves working through standard problems solving for the probability that one gets a value from a Normal model in a particular range or finding a cut-off value associated with a given probability. Similarly, the current lab involving regression looks at the basics of plotting lines and evaluating the strength of a relationship using the correlation coefficient. Students can, and often do, complete the current activities with not much thought given to the underlying statistical concepts.
The new materials we developed are lab activities that actively involve students in the design and implementation of data collection and the analysis and interpretation of the resulting data. Our overall goal was to have students begin to think like statisticians, to construct ways of thinking about data collection and analysis, to solve problems using data in context. Rather than follow steps set down for them, students would take ownership by making decisions about what data to collect and how to organize that data and interpret results within the context of the problem. We had identified several areas where students had struggled in the past and tried to approach those problem areas in new ways in the activities.
One of those problem areas involves the concept of a distribution. Students often struggle with what a histogram is really depicting and how numerical summaries of data relate to the shape and spread of a distribution. A second problem area is the Normal model. Again, many students struggle with the abstract concept of a model for the distribution of a population. Activity #2 was designed to allow students to explore distributions and the Normal model in the context of deciding how to appropriately label a Fun Size bag of M&Ms.
We view the ideas of correlation and regression to be of fundamental importance in an introductory statistics course. Students should understand how to interpret the least squares regression line within the context of the data. This interpretation should go beyond the simple algebraic definition of slope and intercept to include the statistical idea of variation and that the regression line fits within a meaningful context. Over the years we have seen many students have difficulty with correctly interpreting the slope and intercept, understanding the applications of correlation and regression and the limitations of these procedures. Activities #3, 4 and 5 explore various aspects of correlation and regression.
In the past students have been able to learn the terms related to experimental design but have had difficulty putting the ideas of experimental design into practice when asked to design an experiment to collect data. Activity #5 was developed to allow students to design a simple experiment and analyze the resulting data. Finally, we have noticed students struggle with the course project in Stat 101. This project combines development of an experimental plan with data collection and a full descriptive analysis of the resulting data. The activities were designed to give students more experience in conducting studies of this nature.
Because the activities were being developed and tested for the first time, it was necessary to come up with a plan that would facilitate the further development of the materials, field test the materials, allow for assessment of students, those who used the new materials and those who didn't, while meeting the constraints of teaching over 500 students in the introductory statistics course each semester. Here Hippocrates admonition to ``do no harm'' is an important consideration. We did not want to expose large numbers of students to untested materials in their first and, for most, only experience with statistics.
In Section 2 of this paper, we provide a short description of the new course materials, together with learning outcomes, that we developed for an introductory course in statistics. Section 3 discusses the plan for development and preliminary assessment of the materials. Section 4 gives a description of the items used to assess student learning and how they relate to learning objectives for the activities. In Section 5 we give a description of the participants involved in the study. In Section 6 we present the results of the assessments and a statistical analysis of those results. We discuss our findings in Section 7. Finally we indicate directions for further work in Section 8.
In the first activity, students determine the variables that could be used to describe the bags of M&Ms and then collect data on those variables. The learning outcome for this activity is to have students identify categorical and numerical variables that would be helpful in learning more about the bags of M&Ms and to carefully collect data.
In the second activity, students look at the distribution of the weight of the bags of M&Ms using the statistical software package JMP. The learning outcome for this activity is to reinforce the ideas of shape, center, spread and unusual values when describing the distribution of a numerical variable. It also serves to introduce the idea of using a Normal model to represent the distribution of the weights of bags of M&Ms. Students then use the distribution of the weight of the Fun Size bags to determine a reasonable label weight for these bags. Data on larger (labeled) bags of M&Ms are used to further motivate the use of a Normal model for package weights and students discover that the mean weight is not used as the labeled weight of a package. This leads to a discussion of how to use the Normal model to establish a label weight for the Fun Size bags. The learning outcome for this part of Activity # 2 is to see how to use a Normal model to approximate the distribution of a numerical variable and to use that Normal model in a practical application.
In the third activity, students use simple linear regression to predict the gross weight of a bag based on the number of M&Ms in the bag. The fourth activity has the students look at the regression of gross weight on net weight and of net weight on number to determine if estimates of the slope and intercept from these regression equations would give reasonable estimates of the weight of a single M&M and the weight of a single empty bag. In both activities, students apply the concepts of slope and intercept and discover when they are interpretable within the context of a problem. These activities also highlight the danger of extrapolation.
Finally, in the fifth activity, students design an experiment to determine the weight of a single peanut M&M and the weight of an empty bag using regression estimates of the slope and intercept. Through the previous activities the students have discovered difficulties with the observational study of existing packages of M&Ms especially with having the estimated slope and intercept give reasonable estimates of the average weight of an M&M and the average weight of an empty bag. In Activity # 5 they now use the ideas of experimental design to create a study that avoids those difficulties. A more complete description of the activities with learning outcomes and suggestions for extensions can be found in Froelich and Stephenson (2008) and at http://stated.stat.iastate.edu/NSF/ESSDactivities/index.html.
This special section was created, in part, to attract students with good math skills to the discipline of statistics. Students are invited to enroll in this section and invitations go to students with ACT math scores of 27 or higher. As such, students in this section have higher ACT math scores, on average, than students in the regular sections of Stat 101. This poses a problem of finding a suitable comparison group within the regular sections of Stat 101. We decided to have two ``Control'' groups for our study. The first consists of students in regular sections of Stat 101 with ACT math scores of 27 and above. This group would at least have a similar ACT math profile compared to the students in the special section. The second ``Control'' group consists of students with ACT math scores below 27. We wanted to collect data on these individuals to establish a baseline for the majority of students who currently enroll in Stat 101. Because we needed access to students records we obtained Institutional Review Board approval and only students who signed an informed consent form were included in our research study.
In the first year of the study draft activities were used in the special section of Stat 101. One of the authors was the instructor for this section. This instructor was present at the laboratory sessions where the new activities were first tried. If problems with the implementation of an activity arose, which happened with some frequency, the instructor was there to deal with the problem and suggest changes in the conduct of the activity for the next time it would be used. The control groups for the first year of the study were selected from the regular sections of the course during the same semester.
The goal for this first year was to establish the feasibility of the activities and look at preliminary assessment of the efficacy of the activities. This is analogous to Phase I - Feasibility and Phase II - Initial Efficacy in medical research, i.e. clinical trials. For a very nice discussion of the relevance of clinical trial methodology in statistics education research see ``Using Statistics Effectively in Mathematics Education Research'' ww2.amstat.org/research_grants/pdfs/SMERReport.pdf. Appendix A of that report shows how the medical model and the research model proposed in a RAND (2003) report relate to a proposed framework for education research. Specifically, Phase I/II in the medical model are consistent with the framework's phase - Frame and begin to Examine where small systematic studies are conducted. The next phase in the proposed framework is Examine and Generalize - where larger studies are conducted under varying conditions with proper controls.
One of the drawbacks to the first year's assessment plan is the potential confounding variable of instructor. The special section is taught by an experienced faculty member while the regular sections of Stat 101 are taught by experienced graduate teaching assistants. In order to control for this potential confounding variable, the control groups for the second year of the study included only students enrolled in an experienced faculty member's regular section of Stat 101 in the fall semester. That same faculty member also taught the special section the following spring where the new activities, with revisions from the previous year, were used. The special section again was chosen for the experimental group as additional activities (outside the scope of this paper) were introduced and field tested in the second year. The regular section in fall was not used to do a randomized comparative trial with the revised activities because of the inexperience of the graduate teaching assistants assigned to conduct the laboratory sections. Because many of our graduate students come from undergraduate mathematics programs and have had little or no exposure to the concepts presented in Stat 101, the fall semester is often as much a learning experience for the new graduate teaching assistants as it is for the students enrolled in the course. The goal of the second year was to look more carefully at the initial efficacy of the activities while controlling for the potential instructor effect. The second year's study would still fall into the Phase I/II medical model framework or the Frame and begin to Examine phase.
A pretest exam was administered to all students during the first laboratory session of the semester. Questions on the pretest asked students to perform basic algebraic manipulations, calculate such numerical summaries as the mean and median, to describe a histogram of low temperatures for selected cities on a particular day, and to use the boiling point and freezing point of water to develop the equations relating degrees Fahrenheit to degrees Celsius. From the pretest questions, 11 items were assessed dichotomously. If the student answered the question related to an item correctly, the student was determined to have mastered that item. The 11 separate items were combined to form three main skill groups: ability to compute numerical summaries (Pretest Skill 1), ability to interpret a histogram and describe a distribution of numerical values (Pretest Skill 2) and ability in applied algebra (Pretest Skill 3).
Each student in the study completed common exam questions on the first and second exams during the semester. The first exam covered course material on describing and summarizing distributions and working with the Normal model. The second exam covered course material on describing the relationship between two quantitative variables using correlation and linear regression (no inference) and data collection either through sampling or experimentation. The specific problems used in the assessments for Years 1 and 2 of the study can be found in the Appendix. The same problems were not used in Years 1 and 2 because the Year 1 questions and answers had been released and appeared on web sites accessible to Year 2 students. For each assessment question, the authors developed a scoring rubric. The questions were blinded so that scorers would not be aware of group membership while grading. Two of the authors then scored all students' responses to each question. Any discrepancies in the scores obtained by the two authors for a particular student were resolved between the two authors. The agreed upon score for each student on each problem was then recorded.
The common group project required students to design an experiment to analyze the effect of a specific physical aspect of paper helicopters on an observable aspect of the helicopters' flight. A common choice for several groups was to vary the length of the helicopter wings to determine its effect on the flight time of the helicopters. Students were then required to analyze the resulting data using correlation and regression and to make a conclusion about the effect of the change in the helicopters on the change in the helicopters' flight. A scoring rubric, written by the authors, was used to evaluate the group projects. Students had general knowledge of the requirements of the project, but were not given the specific rubric. At the end of each year of the study, the course projects were blinded, randomly ordered and scored. In Year 1, an independent consultant scored all the projects using the project rubric. In Year 2, one of the authors (who was not involved in teaching the Stat 101 courses in Year 2) scored all the projects using the grading rubric. A complete description of the course project appears in the Appendix.
Table 1 summarizes the specific activities, learning objectives and corresponding assessment items.
|Activity||Learning Outcomes||Year 1||Year 2|
|#2||Distributions and Numerical Summaries||Exam 1: Q1 & Q2||Exam 1: Q1 & Q2|
|#2||The Normal Model||Exam 1: Q3 & Q4||Exam 1: Q3 & Q4|
|#3||Regression and correlation||Exam 2: Q1||Exam 2: Q1|
|#5||Experimental design||Exam 2: Q2||Exam 2: Q2|
|#3, #5||Connect experimental design |
with regression and correlation
For the Year 2 study, the SATS (Schau, et al., 1995) was used to assess student attitudes toward statistics. Each of the 36 items on the SATS is designed to measure one of six different components of student attitudes toward statistics:
Each item is measured on a Likert-like scale from 1 to 7 and is scored so that a higher response on the question indicates a more positive attitude toward statistics. A student's score on each of the six attitude components is the mean score on the items within that component. The SATS includes a pre-course version and a post-course version that differ only in terms of the tense of the question. The pre-course version of the SATS was given to all students during the first two weeks of the semester and the post-course was administered during the last week of the semester.
During Year 1 of the study, students from four of the regular sections of Stat 101 were invited to participate during spring. Each of these four sections was taught by a different graduate teaching assistant with at least one semester of experience teaching Stat 101. 377 students in the regular sections completed the course with a passing grade of D- or better. Of these students, 199 had agreed to participate in the study. Student characteristics: ACT Math score, ACT English score, ACT Composite score, High School Percentile Rank, Cumulative College GPA and Total Number of Hours Completed, were obtained from the university's registrar's office for all 199 students. 41 students of the 199 had ACT Math scores of 27 or higher. The 39 students with high math ACT scores who completed all assignments throughout the semester were designated the High Math Control Group (Control: H M). The remaining 158 students, whose ACT Math scores were 26 or lower, were candidates for the Regular Control Group. Due to limited resources for assessment, the number of students included in the Regular Control Group was reduced. The first reduction was made by removing the 39 students in the Regular Control Group who had completed the course project with at least one of the students from the High Math Control Group. A random sample of 50 students was selected from the 119 remaining students. Two of these 50 students chosen for the sample did not complete all assessments during the semester and were dropped from the study. Thus, the final Regular Control Group (Control: Reg) contained 48 students.
During Year 2 of the study, students from a regular section of Statistics 101 for the fall semester, taught by the same instructor as the special section in the following spring semester, were used to form the control groups. 88 students in this section ultimately completed the semester course with a passing grade of D- or higher. Of these students, 72 had agreed to participate in the study at the beginning of the semester. 10 of these 72 students failed to complete one or more of the assessments throughout the semester and were dropped from the study. Student characteristics: ACT Math score, ACT English score, ACT Composite score, High School Percentile Rank, Cumulative College GPA and Total Number of Hours Completed, were obtained from the university's registrar's office for all students in the study. Six out of the remaining 62 students from the regular section of Statistics 101 did not have ACT Math scores on record. These students were dropped from the study, leaving 56 students in the control groups. These remaining students were divided into two groups based on their ACT Math scores. The 17 students with ACT Math Scores of 27 or higher were placed into the High Math Control Group and the other 39 students were placed in the Regular Control Group.
For both years we needed to check to see if the Experimental Group and the High Math Control Group were indeed similar in terms of ACT Math scores. Table 2 displays the numbers of students participating, means and standard deviations for ACT Math scores for the three groups in both years.
|Control: H M||A||39||28.97||2.134|
|Control: H M||A||17||29.35||1.539|
Groups with different letters have differences in mean values that are statistically significant. In both years, the Experimental Group has a slightly higher mean ACT Math score than the High Math Group but this difference is not statistically significant. In both years, the Regular Group does have a mean ACT Math score that is lower than either of the other two groups and this difference is statistically significant.
In addition, in Year 2, an ANOVA was used to determine if the mean scores of the three groups on the six components of student attitudes toward statistics (SATS) were significantly different. An ANCOVA was then used to determine if there was a significant difference in changes in attitudes (post - pre) between the three groups when including the pre-course attitude as a covariate. Finally, we included the six pre-course and six post-course attitude values as covariates in the ANCOVA models to determine if any aspects of attitudes were significant in predicting performance on the course assessments after controlling for student characteristics and group membership.
|Control: H M||A||39||39.90||5.60|
The null hypothesis of equal means among the three groups was rejected with a P-value <0.0001. On average, the Experimental and High Math Control Groups both scored significantly higher than the Regular Control Group. Although the overall mean for the Experimental Group was higher than that of the High Math Control Group, this difference was not statistically significant at the 5% level.
The final ANCOVA model for the Year 1 overall measure was highly significant with a P-value <0.0001. The factor of group membership (Experimental vs. Control) was not significant in the model. However, the covariates ACT Math score and Cumulative College GPA were highly significant in the model. This is consistent with the ANOVA model in Table 3. In the ANOVA, students with high ACT Math scores performed better on average than students with lower ACT Math scores. Summaries of the final ANCOVA model are given in Table 4.
|Cumulative College GPA||5.5152||<0.0001|
|Pretest Skill 1||1.7416||0.0205|
Overall performance on common exam questions during the Year 1 study was significantly related to ACT Math score but not significantly different for students exposed or not exposed to the new course materials. The new materials appear to ``at least do no harm.''
Even though the overall student performance was not significantly different for students exposed or not exposed to the new course materials, student performance on particular aspects covered during the semester could have differed for those students exposed to the new materials compared to those not exposed. To look at student performance on particular aspects of statistics, the scores on questions relating to learning outcomes for the specific activities were analyzed separately. The means and standard deviations of these scores for the Experimental, High Math Control and Regular Control Groups during Year 1 are in Tables 5, 6, 7 and 8. All ANOVAs were statistically significant with P-values less than or equal to 0.0002. Subsequent comparisons of group means was done using a Least Significant Difference approach with an individual comparison alpha level of 0.05. Groups with different letters have differences in means that are statistically significant.
|Control: H M||A||39||16.05||1.849|
We were very concerned about the results on the assessment questions dealing with distributions and numerical summaries. In this instance the Experimental Group was comparable to the Regular Control Group and significantly lower than the High Math Control Group. Closer examination of responses revealed that the Experimental Group missed the connection between the shape of the distribution and appropriate summary measures (e.g. symmetric shape - sample mean and sample standard deviation, asymmetric shape - five number summary or median and interquartile range). Although mentioned in lecture this idea was not reinforced by what students were doing in Activity #2. This lead us to revise the activity so as to reinforce this idea.
|Control: H M||A||39||4.15||1.461|
Average scores on questions dealing with the Normal model were similar for the Experimental and High Math Control Groups. These groups scored significantly higher, on average, than the Regular Control Group. Being able to solve Normal model questions relies on basic algebra skills and the ability to use the table of the standard normal distribution. These skills are more developed in students with higher ACT math scores.
|Control: H M||B||39||11.77||3.232|
The exam question on regression showed the largest differences between the groups in terms of average scores. The Experimental Group had the highest average score followed by the High Math Control Group with the Regular Control Group having the lowest average score. Examination of student responses revealed that the Experimental Group did better on interpretations of regression coefficients and R2. The High Math Control Group got lower average scores due to lower scores on these interpretations. The Regular Control Group tended to have difficulties with some of the calculations as well as the interpretations. All students in Stat 101 see the appropriate interpretations in lecture and in homework assignments. The fact that the Experimental Group did significantly better, on average, on this assessment was encouraging to us, indicating that Activities #3, 4 and 5 might be of some help.
|Control: H M||A||39||7.92||1.707|
The questions dealing with experimentation yield similar results to those dealing with the Normal model. There was no statistically significant difference between the Experimental and High Math Control Groups average scores. Both of these groups had scores that were significantly higher than the Regular Control Group.
Analysis of Covariance models were also run on each of the individual assessments. Summaries of the ANCOVA models are given in Table 9. All final ANCOVA models were highly significant with a P-value <0.0001. Group membership (Experimental vs. Control) and the covariates ACT Math score and Cumulative College GPA were all significant at the 0.1% level in the final ANCOVA model for Distributions and Numerical Summaries. The coefficient for the Group membership variable indicates that the Experimental Group had lower scores on this assessment than the Control Groups once you adjusted for ACT Math and Cumulative College GPA. This is consistent with what we saw in the Analysis of Variance. For the scores on the Regression question, the final ANCOVA group membership was also statistically significant at the 5% level. However, the sign of the coefficient indicates that the Experimental Group scored better, on average, than the Control Groups once the other variables are taken into account. Both of these results are consistent with what we saw in the Analysis of Variance. For scores on questions involving the Normal model and Experiments, group membership was not statistically significant. Again, these results are consistent with the results of the Analysis of Variance.
Cummulative College GPA
|45.8% ||ACT Comp
Pretest Skill 2
|Regression ||46.9% ||Group(C-E) |
Cumulative College GPA
Pretest Skill 1
|Experiments ||18.9% ||Cumulative College GPA
Pretest Skill 3
In all but one of the ANCOVA models, Cumulative College GPA was a highly significant covariate. Three of the four ANCOVA models also included a significant Pretest Skill (Pretest Skill 2 for the Normal model, Pretest Skill 1 for Regression and Pretest Skill 3 for Experiments). The significance of Pretest Skill 2 (ability to describe distributions using histograms) on student performance on the Normal model was not surprising. However, it was surprising this Pretest Skill did not show up in the ANCOVA model for Distributions and Numerical Summaries. For the Regression questions, the results of the ANCOVA indicate that even after controlling for ACT Math Score and Cumulative College GPA, the ability to compute numerical summaries (Pretest Skill 1) was still significantly related to students' scores.
For the course group project, students were randomly assigned to project groups in both the regular sections and special section of the course. There were a total of 7 projects from the Experimental Group, 28 projects from the High Math Group and 18 projects from the Regular Control Group. The 28 projects in the High Math Control Group were completed by groups containing only one or two students from the High Math Control Group. The rest of the project group members in the High Math Control Group were students with ACT Math scores below 27. The 18 projects from the Regular Control Group contained only students with ACT Math scores below 27. Table 10 gives the means and standard deviations for the project scores for the three groups.
|Control: H M||B||28||40.38||5.82|
As with the individual assessments, the null hypothesis of equal means between the three groups was rejected with P-value of 0.0037. The Experimental Group had the highest score on average, followed by the High Math Control Group and then the Regular Control Group. All pairs of comparisons among the three groups were statistically significant. We were encouraged by the performance of the Experimental Group on this assessment as, unlike the exam questions, the project requires students to put together ideas from various topics into a coherent statistical investigation.
|Control: H M||B||17||34.71||5.88|
The null hypothesis of equal means between the three groups was rejected with a P-value <0.0001. The Experimental Group scored higher on average than the High Math Control Group which scored higher on average than the Regular Control Group. These differences were statistically significant at the 5% level.
The final ANCOVA model for the Year 2 overall measure was highly significant with a P-value <0.0001. Again, the results of the ANCOVA are consistent with the results from the ANOVA. Group membership was significant in the ANCOVA even after you controlled for ACT Math score and Cumulative College GPA. The sign of the coefficient indicates that, on average, the students in the Experimental Group performed better than students in the Control Groups even after controlling significant covariates. Summary values of the final ANCOVA model for the overall measure of students' performance for the Year 2 study are given in Table 12.
Cumulative College GPA
Unlike in Year 1, the results from Year 2 indicate that students exposed to the new course materials did significantly better than those who did not use the new materials when using the overall measure of performance on common exam questions. In addition, after controlling for the significant covariates, the factor of group membership was significant at the 5% level. As in Year 1, ACT Math score and Cumulative College GPA turned out to be statistically significant covariates.
To look at differences in student performance over the course of the semester on items tied to the activities, separate analyses of scores on common exam questions dealing with Distributions and Numerical Summaries, the Normal model, Regression and Experiments was performed. Table 13 contains the means and standard deviations of the scores for the Experimental, High Math Control and Regular Control Groups for these areas, respectively.
|Distributions and Numerical Summaries |
(Exam 1, Questions 1 & 2, out of 16 points)
|Control: H M||A||17||12.53||2.54|
|Normal Model |
(Exam 1, Questions 3 & 4, out of 10 points)
|Control: H M||A||17||8.76||2.36|
(Exam 2, Question 1, out of 12 points)
|Control: H M||B||17||7.53||2.05|
(Exam 2, Question 2, out of 7 points)
|Control: H M||A||B||17||5.88||0.91|
For each assessment, the null hypothesis of equal means for the three groups was rejected with P-values of <0.0001, 0.0006, <0.0001, and 0.0064 for the Distributions and Numerical Summaries, Normal Model, Regression and Experiments assessments, respectively. With all of these analyses, the Regular Control Group had significantly lower mean scores than the Experimental Group. The Experimental Group had higher average scores than the High Math Control Group on each of the four assessments. However, for all except the assessment question on Regression, there was no statistically significant difference between the Experimental Group and the High Math Control Group. The accumulation of differences was enough to create the statistically significant difference between these two groups when looking at the overall measure of performance on common exam questions.
The ANCOVA results mirror the findings above. All final ANCOVA models were statistically significant with P-values <0.0001, <0.0001, 0.0002, and <0.0001 for the Distributions and Numerical Summaries, Normal Model, Regression and Experiments assessments, respectively. The only time Group membership was statistically significant in the ANCOVA model was for the question on Regression. The sign of the coefficient was consistent with the fact that the Experimental Group scored higher, on average, than the Control Groups on this assessment question. The three other assessment areas produced ANCOVA models that did not include a Group membership variable. Summary values for the final ANCOVA models for the three exams are given in Table 14.
|Distributions and |
|58.1% ||ACT Math
Pretest Skill 3
|34.4% ||Cummulative College GPA
Pretest Skill 3
|Regression ||52.6% ||Group (C-E)
Cumulative College GPA
|Experiments ||21.8% ||ACT Math
Cumulative College GPA
For the course group project, students were randomly assigned to project groups in both the regular and special section of the course. There were 6 projects completed from the Experimental Group. From the two control groups, there were a total of 21 different projects. From prior experience, high math ability students play a significant role in the completion of the group course project. Therefore, a project was assigned as a High Math Control Group project if the project was completed by at least one student from the High Math Control Group. This assignment also matches the way projects were treated in the Year 1 study, in that only Regular Control Group students contributed to the projects in their own group, while both High Math and Regular Control Group members contributed to projects in the High Math Control Group. This division resulted in 10 projects for the High Math Control Group and 11 projects for the Regular Control Group in the Year 2 study. Table 15 gives the mean and standard deviations for the course project for the three groups (Experimental, High Math Control, Regular Control).
|Control: H M||A||10||31.80||8.57|
Unlike in Year 1, there were no significant differences in means among the three groups (P-value = 0.2734). The scores on the project were generally lower in Year 2 compared to Year 1. Also, except for the Regular Control Group, there was quite a bit more variation in scores for the Year 2 project compared to that in Year 1. The magnitude of the differences between groups was substantial but finding statistically significant differences was hampered by the larger variation and smaller sample sizes.
The SATS was used to assess differences in student attitudes about statistics at the beginning of the semester. The means and standard deviations of the scores on the six components of the SATS from the three groups is given in Table 16 below.
|SATS Component||Group||Mean||Std. Dev||Mean||Std. Dev.|
Control: H M
|Cognitive Competence ||Experimental
Control: H M
Control: H M
|A || |
Control: H M
| ||3.95 |
| ||4.28 |
Control: H M
|A || |
Control: H M
There were significant differences in mean attitudes between the three groups on five of the six components on both the pre-course and post-course SATS. The Experimental and High Math Control Groups had significantly higher mean scores than the Regular Control Group on the Affect and Cognitive Competence components of the pre-course SATS. Many students, especially at the beginning of a semester, see the introductory statistics course as essentially a mathematics course. It is not surprising then that students with strong mathematics backgrounds would have a better feeling about learning statistics (Affect) and would be more confident in their abilities to learn the course materials (Cognitive Competence). By the end of the course, the same patterns emerged with slightly higher means scores for the Experimental and High Math Groups and slightly lower mean scores for the Regular Group. The Experimental Group had significantly higher mean scores on the Value and Interest component of the pre-course SATS than either control group. This result is not unexpected given the nature of the Experimental Group. These students were interested in learning statistics and valued the subject enough to enroll in a special section of the course. By the end of the semester, the difference between the High Math Group and Experimental Group was still present but no longer statistically significant. Interestingly, the Experimental Group and the Regular Control Group had statistically higher mean scores for the amount of Effort they would spend on learning statistics and on the course itself than the students in the High Math Control Group. This result may be attributed to the fact that the Regular Control Group expected to work harder on the course possibly due to perceived lack of mathematics ability while the Experimental Group expected to work harder possibly due to enrollment in a special section of the course. Again, by the end of the semester, the difference between the High Math Group and the Experimental Group had narrowed and was no longer statistically significant. The means for Difficulty are consistent with those for Effort, both pre- and post-course, but no statistically significant differences are seen among any of the groups.
Students responses on both the pre-course and post-course versions of the SATS were used to look at potential differences in the change in attitudes over the course of the semester between the three groups. Six ANCOVA models, one for each component of the SATS, were used to look at differences in changes in attitudes with the pre-course attitude score on the component included as a covariate in the model. Differences in the change in attitudes in the three groups were statistically significant at the 5% level only for the Affect and Cognitive Competence components. In both cases, the significant change occurred between the Regular Control Group and the other two groups. And in both cases, the sign of the change was negative, indicating a more negative change in attitudes on these two components across the semester for the Regular Control Group.
Finally, student responses on the SATS were used to determine if attitudes toward statistics were significantly related to overall performance in the course after controlling for group membership and student characteristics. Using the overall measure of student performance on the common exam questions, student mean scores from the six components of the pre course SATS and the 6 components of the post course SATS were added to the list of potential variables that could be included in an ANCOVA model. Only one covariate, post course Cognitive Competence was a statistically significant addition to the ANCOVA model (P-value of 0.0006 for full versus reduced model). The coefficients and corresponding P-values for the SATS ANCOVA model are given in Table 17.
Cumulative College GPA
Post Cognitive Competence
>From the point of view of a ``proof of concept,'' this study was a success. We were able to develop new lab activities, field test and make adjustments to them, so as to ``do no harm.'' The preliminary results on assessing student learning are especially encouraging for the topic of regression. Students exposed to the new activities performed better on the assessment of regression compared to students that were not exposed even after adjusting for significant covariates. This may be due, in part, to an additional laboratory activity (Activity #4) that tries to get students to think about what might be reasonable estimates for the y-intercept and slope coefficient given the context of the explanatory and response variables.
In analyzing the statistical results above, several patterns emerge. Students with strong math backgrounds performed better on average than the other students. In every case, the Regular Control Group performed no better and often times significantly worse on average than either the High Math Control Group or the Experimental Group. Particularly at the beginning of the semester, students with high math abilities appear to use their previous mathematics skills to help them learn concepts about distributions. The questions dealing with distributions and numerical summaries require students to have basic skills in statistics along with good numerical literacy. It is not surprising then that students with stronger mathematics backgrounds would score higher on these questions.
Except for the Year 1 assessment of distributions and numerical summaries, the Experimental Group scored, on average, the same as or in many cases significantly higher than the High Math Control Group. The performance of students in the special section was noticeably lower on the questions which required students to describe characteristics of a distribution (Questions 1 & 2 - Year 1 in the Appendix). Many times, students in the special section left out some of the characteristics of the distribution and thus scored lower on the problem as a whole. This result could be due to the activity or to differences in the emphasis of the instructors when teaching this material. In the Year 2 study, which controls for instructor differences and used a revised Activity #2, the mean scores of the High Math Control Group and the Experimental Group on assessment of distributions and numerical summaries are not statistically different.
In both Year 1 and Year 2 of the study, the Experimental Group performed much better, on average, on the assessment of regression than either of the control groups. Regression concepts, such as slope and intercept, appear many times in high school algebra courses. However, in teaching this material as it relates to regression in statistics, our experience has been that students, even ones with strong math backgrounds, do not easily transfer their previous mathematical knowledge to this new application. The new course materials put a great deal of emphasis on regression and correlation concepts. The students in the special section of the course were exposed to these concepts repeatedly over the course of several labs. While the amount of class time used to cover these topics between the two sections was approximately the same, the regular section only completed one lab on regression and correlation as opposed to two labs for the special section.
While the differences in performance on the regression assessment carried over into the course group project in the Year 1 study, no significant difference in performance between the three groups was found for the projects in the Year 2 study. The observed mean differences between the three groups were roughly equal between Years 1 and 2. However, the standard deviations of experimental and high math control groups were higher and the number of projects was lower in Year 2 than in Year 1. These differences lead to a lower power for the Year 2 study. Also, in studying the Year 2 projects, we found that students had difficulties in understanding the concept of replication - having several experimental units in each treatment group. Many project groups in both the special and the regular section simply conducted the experiment by applying the treatment to each experimental unit several times. The labs on experimentation used in the two sections of Statistics 101 have since been rewritten to help students better understand the idea of replication within an experiment. The effect of these revisions on project scores has not yet been tested. Finally, in discussing these results, we have discovered that the two instructors for the special section in this study approach the concept of experimental units differently when teaching the course. This discussion has lead to more consistent instruction of this topic.
The ANCOVA results are consistent with the ANOVA results. The results of significant mean differences in scores on the assessments between the experimental and control students from the ANOVA analyses are still present even when controlling for other student characteristics. The variable Cumulative College GPA is significant in all but one ANCOVA model. Controlling for other student characteristics and their group membership (either Control or Experimental) students with higher GPAs are scoring higher on average on these assessments. ACT Math is significant in four of five ANCOVA models from Year 1, and in three of five ANCOVA models from Year 2. Students with higher ACT Math scores tend to do better on these assessment items.
In terms of attitudes towards statistics, as measured by SATS, there was no significant change in the patterns of means for the three groups from pre- to post-course. According to the ANCOVA models, only Affect and Cognitive Competence changed, for the worse, for the Regular Control Group. Again, the new activites ``did no harm'' for the experimental group.
Field testing the activities in the special section solved a logistical problem but introduced the problem of finding a comparable control group. Having a control group with a similar ACT Math profile addressed this difficulty. Due to constraints on teaching assignments for Year 1 of the study, group membership (either experimental or control) was confounded with course instructor. The special section of the course was taught by an experienced professor while the regular sections of the course were taught by relatively inexperienced graduate teaching assistants. Differences in teaching styles, methods, and emphasis between the five instructors could affect student learning. While this limitation is not present in Year 2 of the study (the same instructor taught all students), the regular and special sections of Statistics 101 were structured differently in both years. The enrollments of the special sections were around 20 students while the enrollments in the regular sections were around 100. The lab section for the special sections was led by both the course instructor and a graduate student teaching assistant while the labs for the regular sections were each run by a single graduate student teaching assistant.
Now that the development and initial assessment has been performed and we are satisfied that the new materials will not do harm we plan to proceed to a randomized clinical trial (Phase II) so as to include a proper randomized control group. Because of the restrictions mentioned earlier, we intend to conduct this trial in a different introductory statistics course where the laboratory is taught by the course instructor, rather than a graduate teaching assistant. There are multiple sections of this course that will act as blocks in our design. Students from sections of this introductory statistics course who agree to participate will be randomly assigned to treatment groups. One group will use the activities we developed in lab and the other group will use the current laboratory activities. With the availability of the ARTIST materials (www.gen.umn.edu/artist) we hope to use these in future assessment so as to be able to compare performance with other statistics students at different institutions.
For a sample of size 20, which is affected more by a single outlier, the mean or the midrange? Explain your answer.
Experimental Group: Intelligence Quotient (IQ) scores are normally distributed with a mean of 100 points and a standard deviation of 15.
Grades will be determined on:
Brooks, J.G. and Brooks, M.G. (1993), In Search of Understanding: The Case for Constructivist Classrooms, Association for Supervision and Curriculum Development, Alexandria, Virginia.
Cobb, G.W. (1993), "Reconsidering Statistics Education: A National Science Foundation Conference," Journal of Statistics Education [On-line], 1(1). ww2.amstat.org/publications/jse/v1n1/cobb.html
Froelich, A.G. and Stephenson, W.R. (2008), "How Much Does an M&M Weigh?" submitted for publication.
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) ww2.amstat.org/publications/jse/v3n2/keeler.html
Kvam, P.H. (2000), "The effect of active learning methods on student retention in engineering statistics," The American Statistician, 54(2) , 136-140
Mvududu, N. (2003) "A Cross-Cultural Study of the Connection Between Students' Attitudes Toward Statistics and the Use of Constructivist Strategies in the Course," Journal of Statistics Education [Online], 11(3) ww2.amstat.org/publications/jse/v11n3/mvududu.html
Quinn, R.J., and Wiest, L.R. (1998), "A constructivist approach to teaching permutations and combinations," Teaching Statistics, 20, 75-77.
Schau, C., Stevens, J., Dauphinee, T. L., and Del Vecchio, A. (1995), "The development and validation of the Survey of Attitudes Toward Statistics," Educational and Psychological Measurement, 55(5), 868-875.
Steinhorst, R. K., and Keeler, C. M. (1995), "Developing Material for Introductory Statistics Courses from a Conceptual, Active Learning Viewpoint," Journal of Statistics Education [Online], 3(3) ww2.amstat.org/publications/jse/v3n3/steinhorst.html
Weinberg, S. L., and Abramowitz, S. K. (2000), "Making General Principles Come Alive in the Classroom Using an Active Case Studies Approach," Journal of Statistics Education [Online], 8(2) ww2.amstat.org/publications/jse/secure/v8n2/weinberg.cfm
Amy G. Froelich
Department of Statistics
Iowa State University
Ames, IA 50011-1210
W. Robert Stephenson
Department of Statistics
Iowa State University
Ames, IA 50011-1210
William M. Duckworth
Department of Decision Sciences
Omaha, NE 68178
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