nsf due-0126716 collaborative research: adaptation & implementation of activity & web-based...

NSF DUE-0126716

Collaborative Research:Collaborative Research: Adaptation & Implementation of Adaptation & Implementation of Activity & Web-Based Materials Activity & Web-Based Materials Into Post-Calculus Introductory Into Post-Calculus Introductory

Probability and Statistics CoursesProbability and Statistics Courses

Tracy Goodson-Espy

M. Leigh Lunsford

Ginger Holmes Rowell

NSF DUE-0126716

A Collaborative ApproachA Collaborative Approach

Athens State Univ.M. Leigh Lunsford

Middle Tenn. St. Univ.Ginger Holmes Rowell

Univ. of Alabama, HuntsvilleTracy Goodson-Espy

A&I Materials into Post Calculus Prob/Stat Courses

Provide Objective Independent Assessment of A&I

NSF DUE-0126716

Project* ObjectivesProject* 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&ICourses for A&I

ASU:•Applied Statistics & Probability I (3 hrs)

Clientele: CS, MA, Math. Ed. MajorsPrereq: Calculus II

MTSU:•Probability & Statistics (3 hrs)•Data Analysis (1 hr)

Clientele: CS, Math. Ed. MajorsPrereq: Calculus I

NSF DUE-0126716

The Materials for A&IThe Materials for A&I

• ““A Data-Oriented, Active Learning, Post-A Data-Oriented, Active Learning, Post-Calculus Introduction to Statistical Calculus Introduction to Statistical Concepts, Methods, and Theory (SCMT)”Concepts, Methods, and Theory (SCMT)”

• A. Rossman, B. Chance, K. Ballman• NSF DUE-9950476

• ““Virtual Laboratories in Probability and Virtual Laboratories in Probability and Statistics (VLPS)”Statistics (VLPS)”

• K. Siegrist• NSF DUE-9652870

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Statistical Concepts, Methods, and Theory Statistical Concepts, Methods, and Theory (SCMT): A Small Sample of Materials(SCMT): A Small Sample of Materials

Activity Context Concepts Description

Random-ization Test

Friendly Observers

Randomization, simulation, p-value

Uses cards (23 per student) and Minitab to simulate a randomization test to estimate p-value 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

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Virtual Laboratories in Probability & Virtual Laboratories in Probability & Statistics: An ExampleStatistics: An Example

Games of ChanceContents

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

• Monty Hall Experiment

Poker Experiment Applet

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Topics in ASU’s CourseTopics 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 VariablesUniform (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

NOTE: Topics

not covered linearly!

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An ExampleAn 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 TotalBeat threshold 3 8 11Do not beat threshold 9 4 13Total 12 12 24

*Journal of Personality and Social Psychology (Butler and Baumeister, 1998)

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An ExampleAn ExampleWeek 1: Activity with the Friendly Observers Experiment

•Randomness, random variable, empirical probabilitydistribution, p-value. (Hypothesis testing)

•Uses a tactile simulation (in-class) and Minitab (assignment) todetermine 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 andtheir 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 theactual p-value of the experiment.

•Uses the Ball and Urn Applet from the Virtual Labs.•Students finish report.

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Theoretical OrientationTheoretical 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 CycleTeaching Experiment Cycle

Teaching Hypotheses; Curricular & Instructional

Choices

Class Implementation

& Feedback

Instructors’ Reflections and Curricular Modifications

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The A&I CycleThe A&I Cycle

SCMTSCMT

VLPSVLPS

Materials

Use Materialsin Courses

Evaluate A&IRefine AdaptationProvide Feedback to

Material Developers

Use IndependentAssessment Results

to Improve A&I

Develop BestPractices for Use

of Materials Co-jointly

Receive Trainingon New Materials

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Assessment of A&I of MaterialsAssessment of A&I of Materials

• Will Use an Action Research ModelAction 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

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Project Goals and OutcomesProject 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 reform-oriented probability and statistics courses to improve overall student understanding of the subject.

NSF DUE-0126716


• 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 331Applied Statistics & Probability (primarily an undergraduate probability course)

• Middle Tennessee State University - Math 2050/Statistics 5140Probability & Statistics (primarily an undergraduate course in inferential statistics)

• Middle Tennessee State University- Mathematics 6350Probability and Statistics for Teachers

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• 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 Mini-tab.

• 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.

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• 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


• Goal: The improvement and implementation of non-traditional 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.

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• Goal: A contribution to the mathematics community discussion/research concerning what topics/ materials/methods should be included in reform-oriented 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.

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Preliminary Survey ResultsPreliminary 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 ResultsSection 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 1Section One Results - Table 1

Topic MA 331 MA 2050

Sample Space 4.5 5

Set Notation 5 4

Venn Diagrams

How to CalculateProbability of anEvent

ConditionalProbability

Independence ofEvents

5

4.5

4.5

4.5

3.5

4

4

3

5 = High Knowledge

1 = Low Knowledge

NSF DUE-0126716



Multiplicative Lawof Probability

4 3.5

Additive Law ofProbability

4 3.5

Bayes’ Theorem

Discrete RandomVariable

BinomialProbabilityDistribution

4

4

4

3

4

4

5 = High Knowledge

1 = Low Knowledge

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HypergeometricProbability Dist.

4 3

NormalProbability Dist.

4 4.5

PDF of aContinuous RV

Central LimitTheorem

4

4

3.5

4.5

5 = High Knowledge

1 = Low Knowledge

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ConfidenceInterval for Means

3 4.5

CI for Proportions 3 4.5

StatisticalSignificance

(p-values)

4 4

As might be expected, the results seem to illustrate the different content area emphases in the two courses.

5 = High Knowledge

1 = Low Knowledge

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Section Two Results - Table 2Section Two Results - Table 2

ClassCharacteristic

MA 331 MA 2050

Class Dynamics 4 4.5

InstructionalStrategies

4 4.5

Amount ofTechnology Used

Usefulness ofTechnology

Class Pace

4.5

3.5*

4

4.5

4.5

5

*Due to cost and availability of Mini-tab

5 = Very Good

1 = Very Poor

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Section Three ResultsSection 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 3Section Three Results-Table 3

Which Class Activities Aided Learning?

MA 2050 (N=26)# of “Yes”

Descriptive Statistics (Fan CostIndex)

8

Equally Likely Probability (RandomBabies)

13

Basic Probability Rules

(Top 100 Films)

Conditional Probability

/Independence (Top 100 Films)

15

14

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Section Three Results-Table 3Section Three Results-Table 3Which Class Activities Aided Learning?

MA 2050 (N=26)# of “Yes”

Discrete RV (Random Babies) 15

Binomial Distribution (MC Exam) 10

Continuous Probability

(Penny Ages)

Normal Distribution

Sampling Dist. Of Sample Means

(Penny Ages)

11

9

6

NSF DUE-0126716

Section Three Results-Table 3Section Three Results-Table 3Which Class Activities Aided Learning?

MA 2050 (N=26)# of “Yes”

Interval Estimation (Ages ofMothers; Seat Belt Usage)

10

Review of Sampling Dist. ForProportions (M&M Sampling)

15

Confidence Intervals

(Beat the Threshold)

Statistical Tests for Proportions(Which Tire)

Hypothesis Testing (Which Tire &handout)

2

15

11

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Section Three Results-Table 3Section Three Results-Table 3

Which Class Activities Aided Learning?

MA 2050 (N=26)# of “Yes”

Comparing Two Means (SAT’s andBaseball Leagues)

9

Comparing Two Means (FishOil/Blood Pressure, Marriage Ages)

5

Comparing Two Proportions

(AZT and HIV)

15

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Section Three ResultsSection 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

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Further Data AcquisitionFurther 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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DisseminationDissemination

• 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 collaborative research: adaptation & implementation of activity & web-based...

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