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Collaborative Research:
Adaptation & Implementation of
Activity & Web-Based Materials
Into Post-Calculus Introductory
Probability and Statistics Courses
Tracy Goodson-Espy
M. Leigh Lunsford
Ginger Holmes Rowell
NSF DUE-0126716
A Collaborative Approach
A&I Materials into Post Calculus Prob/Stat Courses
Athens State Univ.
Middle Tenn. St. Univ.
M. Leigh Lunsford
Ginger Holmes Rowell
Univ. of Alabama, Huntsville
Tracy Goodson-Espy
Provide Objective Independent Assessment of A&I
NSF DUE-0126716
Project* Objectives
• To improve post-calculus students
learning of probability & statistics.
• To provide students with better
preparation for their future careers in
mathematics & statistics, mathematics
education, and computer science.
*This project is partially supported by the National Science Foundation. The
project started in June, 2002 and continues through August, 2004.
NSF DUE-0126716
Courses for A&I
ASU:
•Applied Statistics & Probability I (3 hrs)
Clientele: CS, MA, Math. Ed. Majors
Prereq: Calculus II
MTSU:
•Probability & Statistics (3 hrs)
•Data Analysis (1 hr)
Clientele: CS, Math. Ed. Majors
Prereq: Calculus I
NSF DUE-0126716
The Materials for A&I
• “A Data-Oriented, Active Learning, PostCalculus Introduction to Statistical
Concepts, Methods, and Theory (SCMT)”
• A. Rossman, B. Chance, K. Ballman
• NSF DUE-9950476
• “Virtual Laboratories in Probability and
Statistics (VLPS)”
• K. Siegrist
• NSF DUE-9652870
NSF DUE-0126716
Statistical Concepts, Methods, and Theory
(SCMT): A Small Sample of Materials
Activity
Context
Concepts
Description
Friendly
Observers
Randomization,
simulation, p-value
Uses cards (23 per student) and Minitab to
simulate a randomization test to estimate pvalue from a 2x2 table for a psychology study.
Equal
Likeliness
Random
Babies
Sample space, long-run
relative frequency,
random variable,
expected value,
simulation
Uses index cards (4 per student) and Minitab to
simulate the matching problem and develops
probability calculations with equally likely
outcomes.
Fishers
Exact Test
Friendly
Observers
Counting rules,
hypergeometric
probabilities
Exact probabilities for simulation in
Randomization Test
General vs.
Specific
The Birthday
Problem
Applications of counting
techniques, complement
rule
Does calculations for the birthday problem
(using a spreadsheet) contrasting any birthday
vs. a specific birthday
Prob. Rules
100 top films,
2000 Michigan
primary
Variety of basic
probability rules
Discovery approach through two-way tables and
some Venn diagrams. HW is very interesting
Randomization Test
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Virtual Laboratories in Probability &
Statistics: An Example
Games of Chance
Contents
1. Poker
2. Poker Dice and Chuck-a-Luck
3. Craps
4. Introduction
5. Roulette
6. The Monty Hall Problem
7. Lotteries
8. Notes
Applets
•
Poker Experiment
•
Poker Dice Experiment
•
Chuck-a-Luck Experiment
•
Craps Experiment
•
Roulette Experiment
•
Monty Hall Game
•
Poker Experiment Applet
Monty Hall Experiment
NSF DUE-0126716
Topics in ASU’s Course
•Topics Included:
•Random Experiments, Sample Spaces, Random Samples
•Basic Descriptive Statistics (Mean, Var., Std. Dev., Sample Mean and Var.)
•Probability Theory:
Laws of Probability, Conditional Probability, Independence, Law of Total
Probability, Bayes’ Theorem
•Discrete and Continuous Random Variables
Uniform (discrete & continuous), Binomial, Hypergeometric, Geometric,
Normal, pdf vs. cdf.
•Expected Value
•Central Limit Theorem
•Confidence Intervals for Means & Proportions
•Basic Concepts in Hypothesis Testing
•Topics Not Included:
•Multivariate Distributions
•Moment Generating Functions
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An Example
Friendly Observers Experiment*:
•Researchers investigated a conjecture that having an observer with a vested interest
would decrease subjects’ performance on a skill-based task.
•Subjects given time to practice task.
•Subjects randomly assigned to one of two groups:
•Group (A) was told that the participant and observer would each win $3 if the
participant beat a certain threshold time.
•Group (B) was told only that the participant would win $3 if the threshold were
beaten.
•Threshold chosen to be a time that participant beat in 30% of their practice
turns.
A: observer shares prize B: no sharing of prize
Total
Beat threshold
3
8
11
Do not beat threshold
9
4
13
Total
12
12
24
*Journal of Personality and Social Psychology (Butler and Baumeister, 1998)
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An Example
Week 1: Activity with the Friendly Observers Experiment
•Randomness, random variable, empirical probability
distribution, p-value. (Hypothesis testing)
•Uses a tactile simulation (in-class) and Minitab (assignment) to
determine the empirical probability distribution for the random
variable X (the number of winners assigned to group A by chance).
•Students start writing a report detailing their simulation results and
their empirical estimate of the p-value.
Week 3: Follow-up activity includes Friendly Observers Exper.
•Introduces hypergeometic probabilities and the hypergeometric
distribution.
•Students apply theory to Friendly Observers Experiment above
to compute the probability distribution of X and the
actual p-value of the experiment.
•Uses the Ball and Urn Applet from the Virtual Labs.
•Students finish report.
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Theoretical Orientation
• The project is based on an emergent constructivist
perspective,* meaning that mathematics learning can
be characterized as both a process of active individual
construction and a process of enculturation.
• This orientation emphasizes the importance of
analyzing students’ individual mathematical activities as
well as placing them in the context of the mathematical
community in which they were developed. This is done
through the use of a teaching experiment.
• Students’ individual constructive activities and their
roles in social processes in the classroom act in a
complementary way to enable student learning.
*1995, P. Cobb
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Teaching Experiment Cycle
Teaching Hypotheses;
Curricular & Instructional
Choices
Class
Implementation
& Feedback
Instructors’ Reflections and
Curricular Modifications
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The A&I Cycle
Receive Training
on New Materials
Develop Best
Practices for Use
of Materials Co-jointly
Materials
VLPS
Use Materials
in Courses
SCMT
Provide Feedback to
Material Developers
Evaluate A&I
Refine Adaptation
Use Independent
Assessment Results
to Improve A&I
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Assessment of A&I of Materials
• Will Use an Action Research Model*
– What is the problem? I.e., what is not
working in the classroom?
– What technique can be used to address the
learning problem?
– What type of evidence can be gathered to
show whether the implementation is
effective?
– What should be done next, based on what
was learned?
*1999 - R. delMas, J. Garfield, B. Chance
NSF DUE-0126716
Project Goals and Outcomes
•
The development of post-calculus probability and statistics courses that
produce well-educated students.
•
The integration of technology and group-based activity work into the
courses for the purpose of enhancing student learning.
•
The enhancement of student communication skills through oral and written
reports and presentations.
•
The improvement and implementation of non-traditional assessment
techniques for evaluating students.
•
A contribution to the mathematics community discussion/research
concerning what topics/materials/methods should be included in reformoriented probability and statistics courses to improve overall student
understanding of the subject.
NSF DUE-0126716
Project Goals and Outcomes
• Goal: The development of post-calculus probability
and statistics courses that produce well-educated
students.
• Outcome: The project uses SCMT and VLPS for
Three Different Types of Courses:
• Athens State University - Mathematics 331
Applied Statistics & Probability (primarily an undergraduate
probability course)
• Middle Tennessee State University - Math 2050/Statistics 5140
Probability & Statistics (primarily an undergraduate course in
inferential statistics)
• Middle Tennessee State University- Mathematics 6350
Probability and Statistics for Teachers
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Project Goals and Outcomes
• Goal: The integration of technology and group-based
activity work into the courses for the purpose of
enhancing student learning.
• Outcome: The project integrates selected SCMT and
VLPS materials into the courses as well as using Minitab.
• Using these materials, students work in groups on discovery-based
learning activities.
• The technology is integrated for three purposes 1) a tool to help
with computations, 2) a tool for visualization and 3) a tool for
simulation.
NSF DUE-0126716
Project Goals and Outcomes
• Goal: The enhancement of student communication
skills through oral and written reports and
presentations.
• Outcome: Each of the courses included in the project
requires multiple oral and written student reports and
presentations that explore student understandings of
the concepts covered in class and through the
individual and group activities.
NSF DUE-0126716
Project Goals and Outcomes
• Goal: The improvement and implementation of nontraditional assessment techniques for evaluating
students.
• Outcome: Each of the courses included in the project
uses assessment methods in addition to traditional
paper and pencil tests. Non-traditional assessments
include written and oral reports, homework including the
analysis of real data, written artifacts from lab activities
and group work.
NSF DUE-0126716
Project Goals and Outcomes
• Goal: A contribution to the mathematics community
discussion/research
concerning
what
topics/
materials/methods should be included in reformoriented probability and statistics courses to improve
overall student understanding of the subject.
• Outcome: While the project is on-going, preliminary
results indicate that extensive inclusion of real-world
examples, interactive lectures, student-active lab
assignments, carefully crafted web-activities, and
graded homework (that is connected to the previous
items) result in improved student retention and
mathematical understanding.
NSF DUE-0126716
Preliminary Survey Results
•
Students in the project classes were given mid-term
and final Class Activities Surveys. These surveys
consisted of three parts:
•
•
•
•
Section One asked a series of questions concerning student’s
beliefs concerning his/her understanding of specific mathematics
concepts covered in the course such as sample space, conditional
probability, independence of events, probability laws, etc. The
student was asked to rate their understanding on a 1-5 scale(L-H).
Section Two included a series of questions that asked students to
rate the functioning of the class in terms of class dynamics, group
dynamics, instructional strategies used, amount of technology
used, and the effectiveness of the technology for conveying ideas.
Section Three included open-ended questions that solicited
student opinions.
This survey was also given to pre-project classes in Spring 2002.
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Section One Results
• Table 1 shows the median student responses to
Section One of the survey concerning student
beliefs about their understanding of core material.
(N=14 in MA 331 at ASU; N=26 in MA 2050 at
MTSU)
• In order to test the validity of the students’
answers, students were also asked to evaluate
their understanding of topics that were not covered
in the course.
We believe 95% of survey
respondents provided valid answers concerning
their beliefs about their own understanding.
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Section One Results - Table 1
Topic
Sample Space
MA 331 MA 2050
4.5
5
Set Notation
5
4
Venn Diagrams
5
3.5
5 = High Knowledge
How to Calculate
Probability of an
Event
Conditional
Probability
Independence of
Events
4.5
4
4.5
4
4.5
3
1 = Low Knowledge
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Section One Results - Table 1
Topic
Multiplicative Law
of Probability
Additive Law of
Probability
Bayes’ Theorem
MA 331 MA 2050
4
3.5
4
3.5
4
3
5 = High Knowledge
1 = Low Knowledge
Discrete Random
Variable
Binomial
Probability
Distribution
4
4
4
4
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Section One Results - Table 1
Topic
MA 331 MA 2050
Hypergeometric
Probability Dist.
Normal
Probability Dist.
4
3
4
4.5
PDF of a
Continuous RV
4
3.5
Central Limit
Theorem
4
4.5
5 = High Knowledge
1 = Low Knowledge
NSF DUE-0126716
Section One Results - Table 1
Topic
MA 331 MA 2050
Confidence
Interval for Means
3
4.5
CI for Proportions
3
4.5
Statistical
Significance
(p-values)
4
4
5 = High Knowledge
1 = Low Knowledge
As might be expected, the results seem to illustrate the
different content area emphases in the two courses.
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Section Two Results - Table 2
Class
MA 331
Characteristic
MA 2050
Class Dynamics
4
4.5
Instructional
Strategies
4
4.5
5 = Very Good
1 = Very Poor
Amount of
Technology Used
Usefulness of
Technology
Class Pace
4.5
4.5
3.5*
4.5
4
5
*Due to cost and availability of Mini-tab
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Section Three Results
•Table 3 indicates the responses of the MA
2050 class concerning the activities that
they found to be most useful.
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Section Three Results-Table 3
Which Class Activities
Aided Learning?
MA 2050 (N=26)
# of “Yes”
Descriptive Statistics (Fan Cost
Index)
8
Equally Likely Probability (Random
Babies)
13
Basic Probability Rules
(Top 100 Films)
15
Conditional Probability
/Independence (Top 100 Films)
14
NSF DUE-0126716
Section Three Results-Table 3
Which Class Activities
Aided Learning?
MA 2050 (N=26)
Discrete RV (Random Babies)
15
Binomial Distribution (MC Exam)
10
Continuous Probability
(Penny Ages)
11
Normal Distribution
9
Sampling Dist. Of Sample Means
(Penny Ages)
6
# of “Yes”
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Section Three Results-Table 3
Which Class Activities
Aided Learning?
MA 2050 (N=26)
# of “Yes”
Interval Estimation (Ages of
Mothers; Seat Belt Usage)
10
Review of Sampling Dist. For
Proportions (M&M Sampling)
15
Confidence Intervals
(Beat the Threshold)
2
Statistical Tests for Proportions
(Which Tire)
15
Hypothesis Testing (Which Tire &
handout)
11
NSF DUE-0126716
Section Three Results-Table 3
Which Class Activities
Aided Learning?
MA 2050 (N=26)
# of “Yes”
Comparing Two Means (SAT’s and
Baseball Leagues)
9
Comparing Two Means (Fish
Oil/Blood Pressure, Marriage Ages)
5
Comparing Two Proportions
(AZT and HIV)
15
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Section Three Results
The MA 331 (N=14) class reported the
following activities related to the following
concepts to be the most helpful to their
learning:
• Basic Probability Rules
• Conditional Probability
• Hypergeometric probabilities
• Bayes’ Theorem
• Central Limit Theorem
NSF DUE-0126716
Further Data Acquisition
• During the spring term 2003, each project
class will be observed repeatedly by the
project evaluator. Individual videotaped
teaching interviews will be conducted with
selected students from each class and
case studies will be developed from these
interviews and the written artifacts of
student work including, tests, homework,
and reports.
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Dissemination
• Presentations at Professional Conferences
• In-Service Training for High School Statistics
Teachers (Spring 2004)
• Summer Workshop for College Faculty
(Summer 2004)
• Papers in Mathematics Education Journals
• Project Website:
http://www.athens.edu/NSF_Prob_Stat/
NSF DUE-0126716