Gilligan, MOPE, and TiVo

Math 20
Mathematical Online Placement Exam
The ALM in Mathematics for Teaching Program

Gilligan, MOPE, and TiVo
Teaching activities at Harvard

Matthew Leingang

Harvard University
Department of Mathematics

University of California, Irvine
April 4, 2007

Matthew Leingang Gilligan, MOPE, and TiVo

Math 20

Outline

Math 20
1
History
Current syllabus
Examples

2
History
Implementation
Lessons learned

3
History
Example: Bayesian Decision Making


Math 20 History
Mathematical Online Placement Exam Current syllabus
The ALM in Mathematics for Teaching Program Examples

Outline

Math 20
1
History
Current syllabus
Examples

2
History
Implementation
Lessons learned

3
History


Math 20 History

Math 20: Introduction to linear algebra and
multivariable calculus

Taught since 2004
Original idea: stick
to the title
Almost no
applications
originally


Math 20 History

Syllabus for Math 20, Spring 2007
Foundational material

Systems of linear equations
Algebra

Gauss elim Inversion
Dot product
Vector Matrix Determinants

Eigenstuff

Function
Quad approx Partial derivative

Lin approx
Differentials


Math 20 History

Syllabus for Math 20, Spring 2007
Applications

Stationary points
Linear programming

Lag mult
Optimization
Game theory
Problems
Least squares
Assignment problem

Markov chains

Leontief


Math 20 History

Some fun problems you can solve

(Economics) which is better: sales tax or income tax?
(Linear programming) can you eat a healthy meal at
McDonald’s?
(Assignment problem) Match teaching fellows to time slots
to 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?


Math 20 History

A closed Leontief input-output system

Problem from Fall 2006 Final
Consider an island with a four-person economy:
Gilligan (agriculture) produces coconuts, palm fronds, and
bamboo poles by collecting them.
The Professor (manufacturing) produces shelter and
equipment by consuming raw materials and with the help
of the Skipper.
Mary Ann (service) takes coconuts and bakes delicious
coconut cream pies, upon which the entire island subsists.
The Skipper (labor) helps the professor with his projects.


Math 20 History

Problem continued
The distribution of products works like this:
Three-fourths of Gilligan’s raw materials go to the
Professor for his creations and the rest go to Maryann for
her pies.
Gilligan and the Skipper each use a sixth of the Professor’s
inventions. Mary Ann and the Professor himself use a third
apiece.
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 their
products.


Math 20 History

Solution

Find a solution to Ap = p, where

 Gilligan Professor Mary Ann Skipper

Gilligan 0 1/6 1/4 0
A = Professor  3/4 1/3 1/4 0
 
Mary Ann  1/4 1/3 1/4 0
Skipper 0 1/6 1/4 1


Math 20 History

Solution

Find a solution to Ap = p, where

 Gilligan Professor Mary Ann Skipper

Gilligan 0 1/6 1/4 0
A = Professor  3/4 1/3 1/4 0
 
Mary Ann  1/4 1/3 1/4 0
Skipper 0 1/6 1/4 1
T
p = 1 3.3 1.8 1 works.


Math 20 History

Results so far

Very happy students
Very high scores
Possible book in the
works someday


Math 20 History
Mathematical Online Placement Exam Implementation
The ALM in Mathematics for Teaching Program Lessons learned

Outline

Math 20
1
History
Current syllabus
Examples

2
History
Implementation
Lessons learned

3
History


Math 20 History

Status Quo

Pencil-and-paper exam
given on ﬁrst day of
Freshman week
Grade Report is three
numbers and a course
code: Math Xa, 1a, 1b,
or 21a


Math 20 History

Example placement information

HMPT1: 19 HMPT2: 10
HMPT3: 6


Math 20 History


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


Math 20 History


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


Math 20 History


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


Math 20 History


HMPT1: 19 HMPT2: 10
HMPT3: 6
Recommendation:
Math Xa
AP Calculus BC: 5
Recommendation:
Math 21a
Could be same person!


Math 20 History

Disadvantages of Status Quo

Students descend upon advisors to interpret these
numbers and give further guidance
Somewhat unnecessarily intimidating and impersonal
HMPT was designed in an era when high school student
exposure to calculus was limited


Math 20 History

Mathematical Online Placement Exam (MOPE)

Funded by Innovation Grant from the Provost’s Fund for
Instructional Technology
Goals
Give entering students more personal, more detailed
information for choosing a math course
Form part of a student-friendly web presence


Math 20 History

Features

Question database organized by mathematical topic and
type of question
A multitude of tests for qualiﬁcation or mastery
Can be taken any time
Topic-speciﬁc feedback, with granularity
Retakes after refreshing are allowed


Math 20 History

Portion of MOPE’s topic tree
composing
evaluating

inverse trig

evaluting trig fns (radians)
arc length; sector a
simplifying
evaluting trig fns (degrees) and circles
radian measure
sin2 + cos2 = 1

trig identities
sign and range of trig fns
angle-addition

Trigonometry
double-angle

law of sines
and triangles sinusoidal
law of cosines
graphs

trig fns from right triangles tan/cot


Math 20 History

Screenshot of sample question
https://mope.dce.harvard.edu:10000/authentication/index

MATH PLACEMENT TEST
TECHNICAL REQUIREMENTS FAQ
NAVIGATION HELP LOGOUT

TIME REMAINING: 43:56

QUESTION 9 22 questions left to answer SELECT YOUR ANSWER

Ø Á5˜ Ø Á 2˜
ø Øø Ø
If v = Ë ¯ and w = Ë ¯, what is the length of the vector v - w ?
Ë¯ Ë¯
Ë¯ Ë¯
È1˘ È-3˘

A. 2

B. 5

C. 3 TEST NAVIGATION

0 0
CLEAR YOUR ANSWER
NEXT QUESTION
D. 26 - 13 PREVIOUS QUESTION
NEXT BLANK
E. 7 FIRST QUESTION

GO TO QUESTION
10

SUBMIT YOUR ANSWERS
and end the test

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

Math 20 History

https://mope.dce.harvard.edu:10000/authentication/index.php?school=fas

MATH PLACEMENT TEST
TECHNICAL REQUIREMENTS FAQ
NAVIGATION HELP LOGOUT

Your Receipt

Last Name: Strozek

First Name: Lukasz

Email address: strozek@fas.harvard.edu

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 eager
to 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 placement
examination, 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


Math 20 History


You can take the test again in 1 hour. In the meanwhile
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.


Math 20 History

“Result”

Average of Math 1a First Midterm
HMPT1 HMPT1
all
passed
failed
MOPE failed 73.00 78.67 75.43
MOPE passed 89.50 N/A 89.50
all 78.50 78.67 78.56


Math 20 History

“Result”

Average of Math 1a First Midterm
HMPT1 HMPT1
all
passed
failed
MOPE failed 73.00 78.67 75.43
MOPE passed 89.50 N/A 89.50
all 78.50 78.67 78.56

Unfortunately, N = 2 here


Math 20 History

Math on the Web

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


Math 20 History

Chicken-and-egg problem

can’t be more widely adopted without greater credibility
can’t be more credible without better calibration
can’t be calibrated without more data
can’t get more data without more people taking it
can’t get more to take it without being more widely adopted


Math 20
History

Outline

Math 20
1
History
Current syllabus
Examples

2
History
Implementation
Lessons learned

3
History


Math 20
History

Background of the ALM program

Goal: better K-12
teachers in BPS and
area
Started in 2001 by
D. Goroff and P. Sally
Degree program since
2003
35 participants and
soon to graduate ﬁrst
Master’s class


Math 20
History

Objectives of the ALM program

Teach teachers the mathematics behind the rules, e.g.:
0.9999.... = 1
Division by zero is undeﬁned
Give resources to challenge their students
Demonstrate fun math learning activities


Math 20
History

Bayes’s Theorem

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

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


Math 20
History

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 know
P(Hj | Cα ∩ O), too.
Given a a “new” ω ∈ Ω O, if we can ﬁnd its observables
{Cαi }, what is the likelihood of ω being in any particular
state?


Math 20
History

Which brings us to TiVo

Ω is the set of all programs on
television
States Hj are your attitudes
toward programs
Observables {Cα } are
metadata about the programs
O is the set of shows you
have marked with thumbs
up/thumbs down.


Math 20
History

Preference Data from Math E-304 on March 6, 2006
Title Like Dislike Neutral Total
King of Queens 4 5 7 16
How I Met your Mother 5 0 11 16
2 and a half Men 3 3 10 16
Courting Alex 1 0 15 16
CSI: Miami 4 2 10 16
Wife Swap 3 3 10 16
Supernanny 3 4 9 16
Miracle Worker 0 0 16 16
Deal or no Deal 4 3 9 16
Apprentice 6 4 6 16
Medium 3 1 12 16
24 5 1 10 16
Total 41 26 125 192
Prob(each preference) 21.35% 13.54% 65.10% 100.00%

Math 20
History

Probability of class attitudes for each show (P(Hk | O))
Title P(like) P(dislike) P(neutral) Total
King 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%


Math 20
History

Frequency of attitude for each characteristic

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


Math 20
History

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

Characteristic Like Dislike Neutral Total
Drama 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%


Math 20
History

(Posterior) probability of class attitudes for shows
airing March 7, 2006
Title P(Like) P(Dislike) P(Neutral) Total
NCIS 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%


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