Asian Studies

Statistics and Probability

Final report on the Mellon supported Enhanced Learning in Probability and Statistics project

For reference, please refer to the web page http://www2.kenyon.edu/People/hartlaub/MellonProject/mellon.html

We first recall our Project Goals from our proposal:

Develop three to six computer-based activity modules for the probability and mathematical statistics sequence.
Test the modules on students in the upcoming academic year.
Develop an Internet-accessible demonstration template that will allow the modules to be executed and viewed from the Internet.

Addressing our first goal, we developed a total of fifteen probability distribution demonstration/activity modules and two large projects. We will briefly describe these two types of modules in turn.

Probability Distribution Demonstration / Activity

A total of fifteen probability distributions have been featured on the web page with accompanying demonstrations and activities. Six of the probability distributions are of the discrete type (e.g. Poisson, binomial, etc.) and nine of the probability distributions are continuous (e.g. normal, exponential, etc.) The demonstrations/activities all follow a standardized format for consistency and aesthetic appeal. Each demonstration/activity explores four important aspects of a probability distribution:

Probability Distribution Function and Shape
Sample Probability Calculations
Properties of the Distribution (e.g. mean, variance, normal approximation, etc.)
Simulation (comparing empirical distribution and moments with theoretical)

These modules can be viewed as demonstrations with students using default parameter values, or these modules can serve as activities when students download the Maple worksheets and modify certain parameter values while running Maple locally.

Projects

Two large projects have been developed. Approximations to the Binomial Distribution is intended for a first semester probability course, and guides students through the normal and Poisson approximations of the binomial distribution using analytical, graphical, and numerical techniques. Parameter Estimation (The Tank Problem) is intended for a second semester mathematical statistics course and applies the methods of moments and maximum likelihood to an actual application and explores the important estimator properties of bias, variance, and efficiency.

Our second goal was met via formal uniform student evaluations at both Denison and Kenyon, administered in both the fall and spring semesters, and by the ongoing informal discussions with students in the classroom and lab. The Appendix (A) includes eight figures, one for each question on the evaluation form, providing a visual summary of the student feedback. Enrollments were small during the spring semester (3 students at each institution) so caution is necessary when making comparisons. The overall evaluation of Maple (Question 8) shows that the student ratings improved in the spring semester. The question with the most negative responses (Question 5) deals with ease of use. Although the students did not find the software particularly easy to use, they felt Maple was a useful tool (Question 6) that they could use outside of the classroom (Question 2). Although the Maple in-class demonstrations were easy to follow (Question 3), it does not appear that students want to see more in-class demonstrations (Question 4). On the other hand, the lab projects (Question 7) received high ratings, and we are hoping that the refined Maple worksheets and activities will lead to improvements in student understanding (Question 1) during the next few years. One student suggestion that will be implemented next year is a full day of instruction on the use of the statistics library in Maple.

We also invited student comments on the following items:

How might the use of Maple be improved for this course?
Please add any additional comments you may have concerning the use of Maple for in-class demonstrations, homework exercises, or lab projects.

From the students’ comments, three themes overwhelmingly dominated:

Use of Maple should be increased in class and on assignments
More in-class instruction is needed for students to fully understand the commands and value of the Maple Statistics Package
Students found the in-class Maple demonstrations valuable in learning and retaining course material

We are very pleased that our final goal of an Internet-based library of demonstration/activity modules and projects has been realized. Again I encourage you to point your web browser to our project web page

The Maple worksheets that comprise the demonstrations/activities and projects were exported as html files and placed on our project web page. Anyone with access to the web can view all the demonstrations in our library, and anyone with a local version of Maple can download the worksheets and change parameter values and re-execute the worksheet.

Although the intended audience for our web library is the students of the calculus-based probability and mathematical statistics courses, we feel our modules will prove useful in our introductory statistics, nonparametric statistics, and data analysis courses. Of particular value in these other courses are the Maple graphs and animations that can be viewed and understood without a calculus background.

Future Direction

Fortunately, this project has the potential for conference presentations, workshops, and papers. We plan to disseminate our work via the following media.

Submit a paper to the Journal of Statistical Education.
Offer a short course at either the annual meeting of the Mathematical Association of America or at the annual Joint Statistical Meetings. Travel support will be sought from the Denison-Kenyon Collaboration Mellon grant in a separate forthcoming proposal.
Investigators Hartlaub and Jones have been invited to write a paper for the web-based STAR Library (Statistics Teaching and Resource Library), which is geared toward the introductory statistics courses, and they have decided to use the parameter estimation project as the core activity in the paper. The paper will be comprised of an in-class activity for an introductory statistics course accompanied by a theoretical supplement for teachers.

We developed a brief evaluation form for student assessment of the use of Maple in the fall probability class. Some summary questions and the responses of the 7 students I had in probability class are given below.

Overall evaluation of the use of Maple in the course:

  Excellent  X
  Good       XX
  Fair       XXXX
  Poor
  Dismal

We wish there were less "Fair" responses, but other indicators lead us to believe that students want more Maple related activities; so we are optimistic that our project is in the right direction.

I would prefer to see more Maple in-class demonstrations:

  Strongly Agree     XX
  Agree              XXX
  No Opinion         XX
  Disagree
  Strongly Disagree

I think Maple is a useful tool for solving homework exercises:

  Strongly Agree     XX
  Agree              XXXX
  No Opinion
  Disagree           X
  Strongly Disagree

The lab projects were useful as a learning tool:

  Strongly Agree     X
  Agree              XXXXX
  No Opinion         X
  Disagree
  Strongly Disagree

We conclude from these results that students generally like Maple and appreciate its power as a tool for solving problems and gaining understanding --- and they wish to see Maple used more. In early spring semester we will be meeting at Denison to pool results from the evaluations at Denison and Kenyon.

Our tasks for spring semester include the wrap-up of the distribution demonstrations, the writing of another lab project (or two?), and the polishing of the interface between the web page and user.

Brian D. Jones, Kenyon College

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