Two non-traditional content courses for in-service high school teachers at the Harvard Extension School

The ALM ProgramRationale for the courses

ImplementationEvaluation

Conclusions

Two non-traditional content courses forin-service high school teachers at the

Harvard Extension School

Bret Benesh Thomas Judson Matthew Leingang

Harvard UniversityDepartment of Mathematics

Critical Issues in Education: Teaching Teachers MathematicsMathematical Sciences Research Institute

Berkeley, CaliforniaMay 31, 2007

Bret Benesh, Thomas Judson, Matthew Leingang Non-traditional content courses for in-service teachers

http://www.msri.org/calendar/workshops/WorkshopInfo/430/show_workshop

http://www.extension.harvard.edu/math/



Conclusions

On Deck

The Harvard ExtensionSchool’s Master ofLiberal Arts (ALM) inMathematics forTeachingGeometry andProbability coursestaught this yearEvaluations andReflections






Conclusions

On Deck

The Harvard ExtensionSchool’s Master ofLiberal Arts (ALM) inMathematics forTeaching

Geometry andProbability coursestaught this yearEvaluations andReflections






Conclusions

On Deck

The Harvard ExtensionSchool’s Master ofLiberal Arts (ALM) inMathematics forTeachingGeometry andProbability coursestaught this year

Evaluations andReflections






Conclusions

On Deck

The Harvard ExtensionSchool’s Master ofLiberal Arts (ALM) inMathematics forTeachingGeometry andProbability coursestaught this yearEvaluations andReflections






Conclusions

Outline

1 The ALM Program2 Rationale for the courses

Instructors’ backgroundGoals

3 ImplementationGeometry

ThemeClass Details

Probability4 Evaluation

QuestionsResults

5 Conclusions






Conclusions

History and Purpose of ALM Program

Paul Sally’s Seminarsfor ElementarySpecialists andMathematics Educators(SESAME)Meet state standardsfor mathematics contentIn-service secondaryschool teachers andpeople consideringcareer change






Conclusions


Paul Sally’s Seminarsfor ElementarySpecialists andMathematics Educators(SESAME)

Meet state standardsfor mathematics contentIn-service secondaryschool teachers andpeople consideringcareer change






Conclusions


Paul Sally’s Seminarsfor ElementarySpecialists andMathematics Educators(SESAME)Meet state standardsfor mathematics content

In-service secondaryschool teachers andpeople consideringcareer change






Conclusions


Paul Sally’s Seminarsfor ElementarySpecialists andMathematics Educators(SESAME)Meet state standardsfor mathematics contentIn-service secondaryschool teachers andpeople consideringcareer change






Conclusions

Whom are we teaching?

In-service teachers come from all kinds of Boston area schools:

from Boston Latinto Boston Public






Conclusions



from Boston Latin

to Boston Public






Conclusions



from Boston Latinto Boston Public






Conclusions

Description of ALM Program Requirements

Students must take 10courses, up throughone year of calculusOne of the coursesmust be on pedagogyStudents mustcomplete a master’sthesis






Conclusions


Students must take 10courses, up throughone year of calculus

One of the coursesmust be on pedagogyStudents mustcomplete a master’sthesis






Conclusions


Students must take 10courses, up throughone year of calculusOne of the coursesmust be on pedagogy

Students mustcomplete a master’sthesis






Conclusions


Students must take 10courses, up throughone year of calculusOne of the coursesmust be on pedagogyStudents mustcomplete a master’sthesis






Conclusions

ALM Courses

“Standard" math courses (calculus, discrete math, etc.)

Courses designed for the secondary school teacher

Math E-300 Math for Teaching ArithmeticMath E-301 Math for Teaching Number TheoryMath E-302 Math for Teaching GeometryMath E-303 Math for Teaching AlgebraMath E-304 Inquiries into Probability and CombinatoricsMath E-306 Theory and Practice of Teaching Statistics






Conclusions

ALM Courses

“Standard" math courses (calculus, discrete math, etc.)Courses designed for the secondary school teacher







Conclusions

ALM Courses

“Standard" math courses (calculus, discrete math, etc.)Courses designed for the secondary school teacher







Conclusions


Outline




ThemeClass Details


QuestionsResults

5 Conclusions






Conclusions


Bret’s backgroundWhat living in Madison can do to you

Graduate work was in finite group theoryMinored in math educationKTI ProgramCore Plus and Connected Mathematics Project (CMP)






Conclusions


Matt’s backgroundHow on Earth did I get so jaded?

Geometer by training, teacher by tradeThird time through a probability course for teachersFirst time: team taught, disconnectedSecond time: interesting for me, over their headThird time: ???






Conclusions


Goals for Math E-302“Math for Teaching Geometry”

Maximize student learningImprove communication skillsMotivate studentsProvide a classroom model






Conclusions


Goals for Math E-304“Inquiries into Probability and Combinatorics”

Build a discipline from the ground upTeach students what they’re ready to learnDevelop ability to read, write, and criticize mathematicalarguments






Conclusions

GeometryProbability

Outline




ThemeClass Details


QuestionsResults

5 Conclusions






Conclusions

GeometryProbability

Platform for inquiry

Taxicab geometry

Compare and contrastwith Euclidean






Conclusions

GeometryProbability

Platform for inquiry

Taxicab geometryCompare and contrastwith Euclidean






Conclusions

GeometryProbability

Class Format

Meet once per week

Class length is two hoursMostly in-service high school teachers






Conclusions

GeometryProbability

Class Format

Meet once per weekClass length is two hours

Mostly in-service high school teachers






Conclusions

GeometryProbability

Class Format

Meet once per weekClass length is two hoursMostly in-service high school teachers






Conclusions

GeometryProbability

Role of Instructor

Moderate discussion

RefereeAsk questionsNot an authority






Conclusions

GeometryProbability

Role of Instructor

Moderate discussionReferee

Ask questionsNot an authority






Conclusions

GeometryProbability

Role of Instructor

Moderate discussionRefereeAsk questions

Not an authority






Conclusions

GeometryProbability

Role of Instructor

Moderate discussionRefereeAsk questionsNot an authority






Conclusions

GeometryProbability

A typical day

Review

Work on one problem10% lecture45% small group work45% large group discussion






Conclusions

GeometryProbability

A typical day

ReviewWork on one problem

10% lecture45% small group work45% large group discussion






Conclusions

GeometryProbability

A typical day

ReviewWork on one problem10% lecture

45% small group work45% large group discussion






Conclusions

GeometryProbability

A typical day

ReviewWork on one problem10% lecture45% small group work

45% large group discussion






Conclusions

GeometryProbability

A typical day

ReviewWork on one problem10% lecture45% small group work45% large group discussion






Conclusions

GeometryProbability

A typical problem

What is the definition of acircle in Euclidean geometry?What does a circle look like intaxicab geometry?What is the diameter of acircle in taxicab geometry?What is the circumference intaxicab geometry?What is π in taxicabgeometry?






Conclusions

GeometryProbability

A typical problem

What is the definition of acircle in Euclidean geometry?

What does a circle look like intaxicab geometry?What is the diameter of acircle in taxicab geometry?What is the circumference intaxicab geometry?What is π in taxicabgeometry?






Conclusions

GeometryProbability

A typical problem

What is the definition of acircle in Euclidean geometry?What does a circle look like intaxicab geometry?

What is the diameter of acircle in taxicab geometry?What is the circumference intaxicab geometry?What is π in taxicabgeometry?






Conclusions

GeometryProbability

A typical problem

What is the definition of acircle in Euclidean geometry?What does a circle look like intaxicab geometry?What is the diameter of acircle in taxicab geometry?

What is the circumference intaxicab geometry?What is π in taxicabgeometry?






Conclusions

GeometryProbability

A typical problem

What is the definition of acircle in Euclidean geometry?What does a circle look like intaxicab geometry?What is the diameter of acircle in taxicab geometry?What is the circumference intaxicab geometry?

What is π in taxicabgeometry?






Conclusions

GeometryProbability

A typical problem

What is the definition of acircle in Euclidean geometry?What does a circle look like intaxicab geometry?What is the diameter of acircle in taxicab geometry?What is the circumference intaxicab geometry?What is π in taxicabgeometry?






Conclusions

GeometryProbability

Another example

A New Altitude

A = 12(2.3)(8.5) = 9.775






Conclusions

GeometryProbability

Grading

Mostly papers

Two examsClass participation






Conclusions

GeometryProbability

Grading

Mostly papersTwo exams

Class participation






Conclusions

GeometryProbability

Grading

Mostly papersTwo examsClass participation






Conclusions

GeometryProbability

Probability course by the Moore Method

No textbooks at all; Iwrite problems directedtowards the courseobjectivesStudents submit writtenup problemsStudents presentsolutionsI update notes withsolutions






Conclusions

GeometryProbability


No textbooks at all; Iwrite problems directedtowards the courseobjectives

Students submit writtenup problemsStudents presentsolutionsI update notes withsolutions






Conclusions

GeometryProbability


No textbooks at all; Iwrite problems directedtowards the courseobjectivesStudents submit writtenup problems

Students presentsolutionsI update notes withsolutions






Conclusions

GeometryProbability


No textbooks at all; Iwrite problems directedtowards the courseobjectivesStudents submit writtenup problemsStudents presentsolutions

I update notes withsolutions






Conclusions

GeometryProbability


No textbooks at all; Iwrite problems directedtowards the courseobjectivesStudents submit writtenup problemsStudents presentsolutionsI update notes withsolutions






Conclusions

GeometryProbability

Notes Table of Contents

The FundamentalCounting PrinciplePermutationsCombinationsSet theoryAxioms of probabilityExpected valueConditional probabilityFamous probabilitydistributions






Conclusions

GeometryProbability


The FundamentalCounting Principle

PermutationsCombinationsSet theoryAxioms of probabilityExpected valueConditional probabilityFamous probabilitydistributions

with

Provolo

ne

America

nW

hiz

without

Provolo

ne

America

nW

hiz






Conclusions

GeometryProbability


The FundamentalCounting PrinciplePermutations

CombinationsSet theoryAxioms of probabilityExpected valueConditional probabilityFamous probabilitydistributions

AB

CD

AD

CB

AC

DB

AB

DC

AC

BD

AD

BC






Conclusions

GeometryProbability


The FundamentalCounting PrinciplePermutationsCombinations

Set theoryAxioms of probabilityExpected valueConditional probabilityFamous probabilitydistributions

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1






Conclusions

GeometryProbability


The FundamentalCounting PrinciplePermutationsCombinationsSet theory

Axioms of probabilityExpected valueConditional probabilityFamous probabilitydistributions

A

B

C

A∪ (B∩C) = (A∪B)∩(A∪C)






Conclusions

GeometryProbability


The FundamentalCounting PrinciplePermutationsCombinationsSet theoryAxioms of probability

Expected valueConditional probabilityFamous probabilitydistributions

A

B

C

A∪ (B∩C) = (A∪B)∩(A∪C)






Conclusions

GeometryProbability


The FundamentalCounting PrinciplePermutationsCombinationsSet theoryAxioms of probabilityExpected value

Conditional probabilityFamous probabilitydistributions

A

B

C

A∪ (B∩C) = (A∪B)∩(A∪C)






Conclusions

GeometryProbability


The FundamentalCounting PrinciplePermutationsCombinationsSet theoryAxioms of probabilityExpected valueConditional probability

Famous probabilitydistributions






Conclusions

GeometryProbability


The FundamentalCounting PrinciplePermutationsCombinationsSet theoryAxioms of probabilityExpected valueConditional probabilityFamous probabilitydistributions






Conclusions

GeometryProbability

Fun problems

Give them a menu; askhow many combinationplates can be ordered

Verify the publishedprobabilities for winningvarious lottery gamesWhy can we multiplyprobabilities of“consecutive” events?






Conclusions

GeometryProbability

Fun problems

Give them a menu; askhow many combinationplates can be orderedVerify the publishedprobabilities for winningvarious lottery games

Why can we multiplyprobabilities of“consecutive” events?






Conclusions

GeometryProbability

Fun problems

Give them a menu; askhow many combinationplates can be orderedVerify the publishedprobabilities for winningvarious lottery gamesWhy can we multiplyprobabilities of“consecutive” events?






Conclusions

GeometryProbability

A typical day

I will have assigned a chapter’s worth of problemsI solicit volunteers to presentWe watch and question the presentersI stay seated (referee)






Conclusions

GeometryProbability

Grading

≥ 1 problem written per week, 0-4 scale≥ 1 problem presented per week, 0-4 scaleTake-home final






Conclusions

QuestionsResults

Outline




ThemeClass Details


QuestionsResults

5 Conclusions






Conclusions

QuestionsResults

Questions

We surveyed the E-302 andE-304 students.

Influence thinking,teaching, orcommunicating?Learn more thantraditional format?Challenging?Rewarding?Take another class?Recommend classformat?






Conclusions

QuestionsResults

Questions


Influence thinking,teaching, orcommunicating?

Learn more thantraditional format?Challenging?Rewarding?Take another class?Recommend classformat?






Conclusions

QuestionsResults

Questions


Influence thinking,teaching, orcommunicating?Learn more thantraditional format?

Challenging?Rewarding?Take another class?Recommend classformat?






Conclusions

QuestionsResults

Questions


Influence thinking,teaching, orcommunicating?Learn more thantraditional format?Challenging?Rewarding?

Take another class?Recommend classformat?






Conclusions

QuestionsResults

Questions


Influence thinking,teaching, orcommunicating?Learn more thantraditional format?Challenging?Rewarding?Take another class?

Recommend classformat?






Conclusions

QuestionsResults

Questions


Influence thinking,teaching, orcommunicating?Learn more thantraditional format?Challenging?Rewarding?Take another class?Recommend classformat?






Conclusions

QuestionsResults

The results: Question 1

How has this course affected the way you think aboutmathematics?

5=Very positively4=Somewhat positively3=No change2=Somewhat negatively1=Very negatively

prob

geom

µ = 4.21

µ = 4.3






Conclusions

QuestionsResults

The results: Question 1

How has this course affected the way you think aboutmathematics?


prob

geom

µ = 4.21

µ = 4.3






Conclusions

QuestionsResults

Question 2

How has this course affected the way you think about teachingmathematics?


prob

geom

µ = 4.12

µ = 3.9






Conclusions

QuestionsResults

Question 3

How has this course affected the way you think aboutcommunicating in mathematics?


prob

geom

µ = 4.07

µ = 4.15






Conclusions

QuestionsResults

Question 4

Do you think that you learned more, less, or as much as youwould have in a more traditionally taught course?

5=Much, much more4=A little more than usual3=No change in learning2=A little less than usual1=A lot less than usual

prob

geom

µ = 3.78

µ = 3.38






Conclusions

QuestionsResults

Question 5

How challenging is this course?3=Very challenging. I had to think much harder than Inormally do.2=Sort of challenging.1=Not challenging at all. I could do this in my sleep.

prob

geom

µ = 2.21

µ = 2.3






Conclusions

QuestionsResults

Question 6

How rewarding is this course?4=Ridiculously rewarding. Math is more fun than watchingDancing with the Stars!3=Sort of rewarding2=I don’t get anything out of it1=I feel like this class saps my will to live.

prob

geom

µ = 3.14

µ = 3.28






Conclusions

QuestionsResults

Question 7

Would you like to take another course taught in this format?5=Yes! Where do I sign up?!?4=Yes, with some reservation3=Undecided2=No1=Hell no

prob

geom

µ = 3.85

µ = 4.17






Conclusions

QuestionsResults

Question 8

Would you recommend a course taught in this format?5=Yes! I want to share the love!4=Sure, it was pretty good.3=Undecided2=No.1=Yes, but only to my worst enemy.

prob

geom

µ = 4

µ = 4.15






Conclusions

QuestionsResults

Some quotes from the probability class

“I have always found proofs difficult and intimidating. Now Ifeel more comfortable with them.”“Either a problem is challenging/hard, or it is easy and thechallenge is explaining it well. Either way, it is challenging.”“...it’s really the best way to learn math.”






Conclusions

QuestionsResults

More quotes from the probability class

“I think a little more teacher-based instruction would allowfor a more rigorous pace, which pushes students and canlead to more of a need for interaction and discussion bynecessity.”

“Waiting for the other students to finish is a bit of a waste oftime.”“I don’t necessarily like the experience, but at least it waspedagogically interesting.”






Conclusions

QuestionsResults


“I think a little more teacher-based instruction would allowfor a more rigorous pace, which pushes students and canlead to more of a need for interaction and discussion bynecessity.”“Waiting for the other students to finish is a bit of a waste oftime.”

“I don’t necessarily like the experience, but at least it waspedagogically interesting.”






Conclusions

QuestionsResults


“I think a little more teacher-based instruction would allowfor a more rigorous pace, which pushes students and canlead to more of a need for interaction and discussion bynecessity.”“Waiting for the other students to finish is a bit of a waste oftime.”“I don’t necessarily like the experience, but at least it waspedagogically interesting.”






Conclusions

QuestionsResults

Some quotes from the geometry class

“I see more value in working in groups as an ongoingstrategy [for teaching]. It takes a while to build trust, butonce its established the outcome in class thinking isfantastic!”

“I have thought more about this ‘stuff’ than I have thoughton other courses.”“It is tiring to think this hard consistently, but good still.”“I wish there was more concrete learning.”“I leave excited and bewildered.”






Conclusions

QuestionsResults


“I see more value in working in groups as an ongoingstrategy [for teaching]. It takes a while to build trust, butonce its established the outcome in class thinking isfantastic!”“I have thought more about this ‘stuff’ than I have thoughton other courses.”

“It is tiring to think this hard consistently, but good still.”“I wish there was more concrete learning.”“I leave excited and bewildered.”






Conclusions

QuestionsResults


“I see more value in working in groups as an ongoingstrategy [for teaching]. It takes a while to build trust, butonce its established the outcome in class thinking isfantastic!”“I have thought more about this ‘stuff’ than I have thoughton other courses.”“It is tiring to think this hard consistently, but good still.”

“I wish there was more concrete learning.”“I leave excited and bewildered.”






Conclusions

QuestionsResults


“I see more value in working in groups as an ongoingstrategy [for teaching]. It takes a while to build trust, butonce its established the outcome in class thinking isfantastic!”“I have thought more about this ‘stuff’ than I have thoughton other courses.”“It is tiring to think this hard consistently, but good still.”“I wish there was more concrete learning.”

“I leave excited and bewildered.”






Conclusions

QuestionsResults


“I see more value in working in groups as an ongoingstrategy [for teaching]. It takes a while to build trust, butonce its established the outcome in class thinking isfantastic!”“I have thought more about this ‘stuff’ than I have thoughton other courses.”“It is tiring to think this hard consistently, but good still.”“I wish there was more concrete learning.”“I leave excited and bewildered.”






Conclusions

Outline




ThemeClass Details


QuestionsResults

5 Conclusions






Conclusions

Reflections

Costs/benefits of IBLmethods vs. lecturingDifferent kind of dramawith a TMM courseThe challenge ofinvolving weakerstudentsReactions to the finalexam






Conclusions

Reflections

Costs/benefits of IBLmethods vs. lecturing

Different kind of dramawith a TMM courseThe challenge ofinvolving weakerstudentsReactions to the finalexam






Conclusions

Reflections

Costs/benefits of IBLmethods vs. lecturingDifferent kind of dramawith a TMM course

The challenge ofinvolving weakerstudentsReactions to the finalexam






Conclusions

Reflections

Costs/benefits of IBLmethods vs. lecturingDifferent kind of dramawith a TMM courseThe challenge ofinvolving weakerstudents

Reactions to the finalexam






Conclusions

Reflections

Costs/benefits of IBLmethods vs. lecturingDifferent kind of dramawith a TMM courseThe challenge ofinvolving weakerstudentsReactions to the finalexam






Conclusions

Final Thoughts

Please let us know about research into effectiveness of IBL(or analogous) methods

ALM URL:http://www.extension.harvard.edu/math/

Great thanks to the Educational Advancement Foundationfor support







Conclusions

Final Thoughts

Please let us know about research into effectiveness of IBL(or analogous) methodsALM URL:http://www.extension.harvard.edu/math/








Conclusions

Final Thoughts

Please let us know about research into effectiveness of IBL(or analogous) methodsALM URL:http://www.extension.harvard.edu/math/






Economy & Finance

Two non-traditional content courses for in-service high school teachers at the Harvard Extension School