Gilligan, MOPE, and TiVo

Math 20Mathematical Online Placement Exam

The ALM in Mathematics for Teaching Program

Gilligan, MOPE, and TiVoTeaching activities at Harvard

Matthew Leingang

Harvard UniversityDepartment of Mathematics

University of California, IrvineApril 4, 2007

Matthew Leingang Gilligan, MOPE, and TiVo



Outline

1 Math 20HistoryCurrent syllabusExamples

2 Mathematical Online Placement ExamHistoryImplementationLessons learned

3 The ALM in Mathematics for Teaching ProgramHistoryExample: Bayesian Decision Making




HistoryCurrent syllabusExamples

Outline








Math 20: Introduction to linear algebra andmultivariable calculus

Taught since 2004Original idea: stickto the titleAlmost noapplicationsoriginally





Math 20: Introduction to linear algebra andmultivariable calculus

Taught since 2004Original idea: stickto the titleAlmost noapplicationsoriginally





Outline








Syllabus for Math 20, Spring 2007Foundational material

Vector Matrix

Function

Gauss elim

Determinants

Eigenstuff

Systems of linear equations

Inversion

Partial derivative

Lin approx

Quad approx

Differentials

Algebra

Dot product





Syllabus for Math 20, Spring 2007Applications

OptimizationProblems

Stationary points

Lag mult

Least squares

Markov chains

Leontief

Assignment problem

Game theory

Linear programming





Syllabus for Math 20, Spring 2007Applications

OptimizationProblems

Stationary points

Lag mult

Least squares

Markov chains

Leontief

Assignment problem

Game theory

Linear programming





Outline








Some fun problems you can solve

(Economics) which is better: sales tax or income tax?(Linear programming) can you eat a healthy meal atMcDonald’s?(Assignment problem) Match teaching fellows to time slotsto maximize TF satisfaction(Game theory) What percentage of the time should you say“Merry Christmas” versus “Happy Holidays” to strangers?(Markov chains) Will Detroit become an annular city?





A closed Leontief input-output system

Problem from Fall 2006 FinalConsider an island with a four-person economy:

Gilligan (agriculture) produces coconuts, palm fronds, andbamboo poles by collecting them.The Professor (manufacturing) produces shelter andequipment by consuming raw materials and with the helpof the Skipper.Mary Ann (service) takes coconuts and bakes deliciouscoconut cream pies, upon which the entire island subsists.The Skipper (labor) helps the professor with his projects.





Problem continuedThe distribution of products works like this:

Three-fourths of Gilligan’s raw materials go to theProfessor for his creations and the rest go to Maryann forher pies.Gilligan and the Skipper each use a sixth of the Professor’sinventions. Mary Ann and the Professor himself use a thirdapiece.Everyone shares Mary Ann’s pies equally.All of the Skipper’s labor goes to the Professor.

Find the equilibrium prices each should charge for theirproducts.





Solution

Find a solution to Ap = p, where

A =

Gilligan Professor Mary Ann SkipperGilligan 0 1/6 1/4 0

Professor 3/4 1/3 1/4 0Mary Ann 1/4 1/3 1/4 0Skipper 0 1/6 1/4 1

p =[1 3.3 1.8 1

]T works.





Solution

Find a solution to Ap = p, where

A =

Gilligan Professor Mary Ann SkipperGilligan 0 1/6 1/4 0

Professor 3/4 1/3 1/4 0Mary Ann 1/4 1/3 1/4 0Skipper 0 1/6 1/4 1

p =[1 3.3 1.8 1

]T works.





Results so far

Very happy studentsVery high scoresPossible book in theworks someday




HistoryImplementationLessons learned

Outline








Status Quo

Pencil-and-paper examgiven on first day ofFreshman weekGrade Report is threenumbers and a coursecode: Math Xa, 1a, 1b,or 21a





Example placement information

HMPT1: 19 HMPT2: 10HMPT3: 6

Recommendation:Math XaAP Calculus BC: 5Recommendation:Math 21aCould be same person!






HMPT1: 19 HMPT2: 10HMPT3: 6Recommendation:Math Xa

AP Calculus BC: 5Recommendation:Math 21aCould be same person!






HMPT1: 19 HMPT2: 10HMPT3: 6Recommendation:Math XaAP Calculus BC: 5

Recommendation:Math 21aCould be same person!






HMPT1: 19 HMPT2: 10HMPT3: 6Recommendation:Math XaAP Calculus BC: 5Recommendation:Math 21a

Could be same person!






HMPT1: 19 HMPT2: 10HMPT3: 6Recommendation:Math XaAP Calculus BC: 5Recommendation:Math 21aCould be same person!

















Disadvantages of Status Quo

Students descend upon advisors to interpret thesenumbers and give further guidanceSomewhat unnecessarily intimidating and impersonalHMPT was designed in an era when high school studentexposure to calculus was limited





Mathematical Online Placement Exam (MOPE)

Funded by Innovation Grant from the Provost’s Fund forInstructional TechnologyGoals

Give entering students more personal, more detailedinformation for choosing a math courseForm part of a student-friendly web presence





Outline








Features

Question database organized by mathematical topic andtype of questionA multitude of tests for qualification or masteryCan be taken any timeTopic-specific feedback, with granularityRetakes after refreshing are allowed





Portion of MOPE’s topic tree

Trigonometry

and circles

inverse trig

composingevaluating

arc length; sector area

radian measure

sign and range of trig fns

evaluting trig fns (radians)

evaluting trig fns (degrees)

and triangles

law of cosines

law of sines

trig fns from right triangles

trig identities

graphs

simplifying

sin2 + cos2 = 1

angle-addition

double-angle

sinusoidal

tan/cot





Screenshot of sample questionhttps://mope.dce.harvard.edu:10000/authentication/index.php?school=fas

1 of 1 8/22/05 9:21 PM

MATH PLACEMENT TEST TECHNICAL REQUIREMENTS

NAVIGATION HELP

FAQ

LOGOUT

TIME REMAINING: 43:56

QUESTION 9 22 questions left to answer SELECT YOUR ANSWER

TEST NAVIGATION

CLEAR YOUR ANSWER

NEXT QUESTION

PREVIOUS QUESTION

NEXT BLANK

FIRST QUESTION

GO TO QUESTION

1010

SUBMIT YOUR ANSWERS

and end the test

Answers: 0=>1 1=>4 2=>0 3=>2 4=>3 Correct answer: 1Question index: 779Question topic: 308

If vØ

=ÁËÈ

ËË5

1

˜¯˘

¯¯ and w

øØ=ÁËÈ

ËË

2

-3

˜¯˘

¯¯, what is the length of the vector v

Ø- wøØ

?

2A.

5B.

3C.

260

- 130

D.

7E.

We would appreciate if you reported any technical difficulties or mathematical inaccuracies to [email protected].





Screenshot of sample questionhttps://mope.dce.harvard.edu:10000/authentication/index.php?school=fas

1 of 1 8/22/05 9:22 PM

MATH PLACEMENT TEST TECHNICAL REQUIREMENTS

NAVIGATION HELP

FAQ

LOGOUT

Your Receipt

Last Name: Strozek

First Name: Lukasz

Email address: [email protected]

Test taken: Math-21a mastery

Test score: Your score is 7 out of 30

Placement: Placement not issued (test incomplete)

You can take the test again in 1 hours. In the meanhile you may want to review: Analytic geometry, Vectors and planes, Parametrization and vector fields, Optimization and extrema, Directional

derivatives, Double integrals, Differentiating functions of several variables, Gradients in the plane, Gradient and path-independent fields, Line integrals, and Applications of multiple integrals.

PRINT

CONTINUE

Results of this pilot version of the Online Placement Examination provide only one of several pieces of information to help you with course selection. The Mathematics Department is always eagerto meet you, to talk over your individual experience and goals, and to help formulate a plan that works for you. Please bring your scores on this and other tests (the pencil-and-paper placementexamination, SAT, AP, etc.) to any of the times and places specifically listed when advisors will be waiting to speak with you.

Anyone considering courses like Math 23 or Math 25 should especially plan on consulting with Professor Taubes during his office hours.

Aug 22 2005 21:22:30 #58791-60547-10506-01628

We would appreciate if you reported any technical difficulties or mathematical inaccuracies to [email protected].





Screenshot of sample question

You can take the test again in 1 hour. In the meanwhileyou may want to review: Analytic geometry, Vectors andplanes, Parametrization and vector fields, Optimizationand extrema, Directional derivatives, Double integrals,Differentiating functions of several variables, Gradientsin the plane, Gradient and path-independent fields, Lineintegrals, and Applications of multiple integrals.





“Result”

Average of Math 1a First MidtermHMPT1failed

HMPT1passed

all

MOPE failed 73.00 78.67 75.43MOPE passed 89.50 N/A 89.50

all 78.50 78.67 78.56

Unfortunately, N = 2 here





“Result”

Average of Math 1a First MidtermHMPT1failed

HMPT1passed

all

MOPE failed 73.00 78.67 75.43MOPE passed 89.50 N/A 89.50

all 78.50 78.67 78.56

Unfortunately, N = 2 here





Outline








Math on the Web

Very challenging problem!Originally we converted TEX to MathMLLater went to images (no MathML support)





Chicken-and-egg problem

can’t be more widely adopted without greater credibilitycan’t be more credible without better calibrationcan’t be calibrated without more datacan’t get more data without more people taking itcan’t get more to take it without being more widely adopted




HistoryExample: Bayesian Decision Making

Outline








Background of the ALM program

Goal: better K-12teachers in BPS andareaStarted in 2001 byD. Goroff and P. SallyDegree program since200335 participants andsoon to graduate firstMaster’s class





Objectives of the ALM program

Teach teachers the mathematics behind the rules, e.g.:0.9999.... = 1Division by zero is undefined

Give resources to challenge their studentsDemonstrate fun math learning activities





Outline








Bayes’s Theorem

Theorem (Bayes)Let Ω be a probability spacewith probability measure P.If A and B are events, then

P(B | A) =P(A | B)P(B)

P(A)

Proof.

P(B | A)P(A) = P(A ∩ B) = P(A | B)P(B)





Bayes and partitions

If Ω = H1 ∪ H2 ∪ . . . ∪ Hn is a partition, and E is any event, then

P(Hi | E) =P(E | Hi)P(Hi)

P(E)

=P(E | Hi)P(Hi)

P(E | H1)P(H1) + · · ·+ P(E | Hn)P(Hn)

If P(E) and P(E | Hj) can be estimated, then so can P(Hi | E).





Bayes and partitions

If Ω = H1 ∪ H2 ∪ . . . ∪ Hn is a partition, and E is any event, then

P(Hi | E) =P(E | Hi)P(Hi)

P(E)

=P(E | Hi)P(Hi)

P(E | H1)P(H1) + · · ·+ P(E | Hn)P(Hn)

If P(E) and P(E | Hj) can be estimated, then so can P(Hi | E).





Observations and Observables

Suppose O ⊂ Ω is a “representative” sample:P(E | O) ≈ P(E) for all events E .Suppose we know what P(Hj | O) are.Suppose also we have sets Cα and we knowP(Hj | Cα ∩O), too.Given a a “new” ω ∈ Ω \O, if we can find its observablesCαi, what is the likelihood of ω being in any particularstate?





Don’t look at this all at once

P(Hi | Cα1 ∩ Cα2 ∩ . . . ∩ Cαm )

=P(Cα1 ∩ Cα2 ∩ . . . ∩ Cαm | Hi)P(Hi)∑n

k=1 P(Cα1 ∩ Cα2 ∩ . . . ∩ Cαm | Hk )P(Hk )

!≈

(∏mj=1 P(Cαj | Hi)

)P(Hi)∑n

k=1

(∏mj=1 P(Cαj | Hk )

)P(Hk )

≈

(∏mj=1 P(Cαj | Hi ∩O)

)P(Hi | O)∑n

k=1

(∏mj=1 P(Cαj | Hk ∩O)

)P(Hk | O)

But everything at this stage is known.Matthew Leingang Gilligan, MOPE, and TiVo




Which brings us to TiVo

Ω is the set of all programs ontelevisionStates

Hj

are your attitudestoward programsObservables Cα aremetadata about the programsO is the set of shows youhave marked with thumbsup/thumbs down.





Preference Data from Math E-304 on March 6, 2006Title Like Dislike Neutral TotalKing of Queens 4 5 7 16How I Met your Mother 5 0 11 162 and a half Men 3 3 10 16Courting Alex 1 0 15 16CSI: Miami 4 2 10 16Wife Swap 3 3 10 16Supernanny 3 4 9 16Miracle Worker 0 0 16 16Deal or no Deal 4 3 9 16Apprentice 6 4 6 16Medium 3 1 12 1624 5 1 10 16Total 41 26 125 192Prob(each preference) 21.35% 13.54% 65.10% 100.00%





Probability of class attitudes for each show (P(Hk | O))

Title P(like) P(dislike) P(neutral) TotalKing of Queens 25.00% 31.25% 43.75% 100.00%How I Met your Mother 31.25% 0.00% 68.75% 100.00%2 and a half Men 18.75% 18.75% 62.50% 100.00%Courting Alex 6.25% 0.00% 93.75% 100.00%CSI: Miami 25.00% 12.50% 62.50% 100.00%Wife Swap 18.75% 18.75% 62.50% 100.00%Supernanny 18.75% 25.00% 56.25% 100.00%Miracle Worker 0.00% 0.00% 100.00% 100.00%Deal or no Deal 25.00% 18.75% 56.25% 100.00%Apprentice 37.50% 25.00% 37.50% 100.00%Medium 18.75% 6.25% 75.00% 100.00%24 31.25% 6.25% 62.50% 100.00%Prob(each attitude) 21.35% 13.54% 65.10% 100.00%





Frequency of attitude for each characteristic

Characteristic Like Dislike Neutral TotalDrama 12 4 32 48Comedy 13 8 43 64Reality 12 11 41 64Game Show 4 3 9 16Male Lead 22 16 42 80Female Lead 22 16 42 80Ensemble 9 2 21 32TV-PG 26 18 84 128TV-14 15 8 41 64Totals 135 86 355 576





Conditional probability of each characteristic, givenattitude and observed (P(Cα | Hk ∩O))

Characteristic Like Dislike Neutral TotalDrama 8.89% 4.65% 9.01% 8.33%Comedy 9.63% 9.30% 12.11% 11.11%Reality 8.89% 12.79% 11.55% 11.11%Game Show 2.96% 3.49% 2.54% 2.78%Male Lead 16.30% 18.60% 11.83% 13.89%Female Lead 16.30% 18.60% 11.83% 13.89%Ensemble 6.67% 2.33% 5.92% 5.56%TV-PG 19.26% 20.93% 23.66% 22.22%TV-14 11.11% 9.30% 11.55% 11.11%Totals 100.00% 100.00% 100.00% 100.00%





(Posterior) probability of class attitudes for showsairing March 7, 2006

Title P(Like) P(Dislike) P(Neutral) TotalNCIS 23.98% 9.87% 66.14% 100.00%The Unit 22.24% 2.80% 74.96% 100.00%Amazing Race 14.58% 14.46% 70.96% 100.00%According to Jim 19.30% 14.67% 66.03% 100.00%Sons & Daughters 18.47% 4.29% 77.24% 100.00%Boston Legal 25.33% 2.45% 72.22% 100.00%Joey 19.30% 14.67% 66.03% 100.00%Scrubs 21.20% 3.79% 75.00% 100.00%Law & Order: SVU 25.33% 2.45% 72.22% 100.00%American Idol 20.69% 6.59% 72.72% 100.00%House 27.40% 8.69% 63.92% 100.00%
