IBL approaches to Geometry and Probability for High School ... fileRationale Implementation...

RationaleImplementation

EvaluationSelf-evaluation and conclusions

IBL approaches to Geometry and Probabilityfor High School Teachers

Bret Benesh Matthew Leingang

Harvard UniversityDepartment of Mathematics

Legacy of R.L. Moore ConferenceAustin, TexasApril 12, 2007

Bret Benesh, Matthew Leingang IBL approaches to geometry and probability



Outline

1 RationaleInstructors’ backgroundGoals

2 ImplementationGeometry

ThemeClass Details

Probability

3 EvaluationQuestionsResults

4 Self-evaluation and conclusions




Instructors’ backgroundGoals

Outline



ThemeClass Details

Probability







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)





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: ???





Outline



ThemeClass Details

Probability







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

Maximize student learningImprove communication skillsMotivate studentsProvide a classroom model





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




GeometryProbability

Outline



ThemeClass Details

Probability






GeometryProbability

Platform for inquiry

Taxicab geometry

Compare and contrast with Euclidean




GeometryProbability

Platform for inquiry

Taxicab geometryCompare and contrast with Euclidean




GeometryProbability

Class Format

Meet once per week

Class length is two hoursMostly in-service high school teachers




GeometryProbability

Class Format

Meet once per weekClass length is two hours

Mostly in-service high school teachers




GeometryProbability

Class Format

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




GeometryProbability

Role of Instructor

Moderate discussion

RefereeAsk questionsNot an authority




GeometryProbability

Role of Instructor

Moderate discussionReferee

Ask questionsNot an authority




GeometryProbability

Role of Instructor

Moderate discussionRefereeAsk questions

Not an authority




GeometryProbability

Role of Instructor

Moderate discussionRefereeAsk questionsNot an authority




GeometryProbability

A typical day

Review

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




GeometryProbability

A typical day

ReviewWork on one problem

10% lecture45% small group work45% large group discussion




GeometryProbability

A typical day

ReviewWork on one problem10% lecture

45% small group work45% large group discussion




GeometryProbability

A typical day

ReviewWork on one problem10% lecture45% small group work

45% large group discussion




GeometryProbability

A typical day

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




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?




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?




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?




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?




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?




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?




GeometryProbability

Another example

A New Altitude

A = 12(2.3)(8.5) = 9.775




GeometryProbability

Grading

Mostly papers

Two examsClass participation




GeometryProbability

Grading

Mostly papersTwo exams

Class participation




GeometryProbability

Grading

Mostly papersTwo examsClass participation




GeometryProbability

Outline



ThemeClass Details

Probability






GeometryProbability

Probability course is TMM

I write problems

Students submit written up problemsStudents present solutionsI update notes with solutions




GeometryProbability


I write problemsStudents submit written up problems

Students present solutionsI update notes with solutions




GeometryProbability


I write problemsStudents submit written up problemsStudents present solutions

I update notes with solutions




GeometryProbability


I write problemsStudents submit written up problemsStudents present solutionsI update notes with solutions




GeometryProbability

Notes TOC

The FundamentalCounting PrinciplePermutationsCombinationsSet theoryAxioms of probabilityExpected valueConditional probability. . .




GeometryProbability

Notes TOC

The FundamentalCounting Principle

PermutationsCombinationsSet theoryAxioms of probabilityExpected valueConditional probability. . .

with

Provolo

ne

America

nW

hiz

without

Provolo

ne

America

nW

hiz




GeometryProbability

Notes TOC

The FundamentalCounting PrinciplePermutations

CombinationsSet theoryAxioms of probabilityExpected valueConditional probability. . .

AB

CD

AC

BD

AC

DB

AD

CB

AD

BC

AB

DC




GeometryProbability

Notes TOC

The FundamentalCounting PrinciplePermutationsCombinations

Set theoryAxioms of probabilityExpected valueConditional probability. . .

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




GeometryProbability

Notes TOC

The FundamentalCounting PrinciplePermutationsCombinationsSet theory

Axioms of probabilityExpected valueConditional probability. . .

A

B

C

A∪ (B∩C) = (A∪B)∩(A∪C)




GeometryProbability

Notes TOC

The FundamentalCounting PrinciplePermutationsCombinationsSet theoryAxioms of probability

Expected valueConditional probability. . .

A

B

C

A∪ (B∩C) = (A∪B)∩(A∪C)




GeometryProbability

Notes TOC

The FundamentalCounting PrinciplePermutationsCombinationsSet theoryAxioms of probabilityExpected value

Conditional probability. . .

A

B

C

A∪ (B∩C) = (A∪B)∩(A∪C)




GeometryProbability

Notes TOC

The FundamentalCounting PrinciplePermutationsCombinationsSet theoryAxioms of probabilityExpected valueConditional probability

. . .




GeometryProbability

Notes TOC

The FundamentalCounting PrinciplePermutationsCombinationsSet theoryAxioms of probabilityExpected valueConditional probability. . .




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?




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?




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?




GeometryProbability

A typical day

I will have assigned a chapter’s worth of problems

I solicit volunteers to presentWe watch and question the presentersI stay seated




GeometryProbability

A typical day

I will have assigned a chapter’s worth of problemsI solicit volunteers to present

We watch and question the presentersI stay seated




GeometryProbability

A typical day

I will have assigned a chapter’s worth of problemsI solicit volunteers to presentWe watch and question the presenters

I stay seated




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




GeometryProbability

Grading

≥ 1 problem written per week, 0-4 scale

≥ 1 problem presented per week, 0-4 scaleTake-home final to come




GeometryProbability

Grading

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

Take-home final to come




GeometryProbability

Grading

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




QuestionsResults

Outline



ThemeClass Details

Probability






QuestionsResults

Questions

Influence thinking, teaching, or communicating?

Learn more than traditional format?Challenging? Rewarding?Take another class?Recommend class format?




QuestionsResults

Questions

Influence thinking, teaching, or communicating?Learn more than traditional format?

Challenging? Rewarding?Take another class?Recommend class format?




QuestionsResults

Questions

Influence thinking, teaching, or communicating?Learn more than traditional format?Challenging? Rewarding?

Take another class?Recommend class format?




QuestionsResults

Questions

Influence thinking, teaching, or communicating?Learn more than traditional format?Challenging? Rewarding?Take another class?

Recommend class format?




QuestionsResults

Questions

Influence thinking, teaching, or communicating?Learn more than traditional format?Challenging? Rewarding?Take another class?Recommend class format?




QuestionsResults

Outline



ThemeClass Details

Probability






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




QuestionsResults

Question 2

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


prob

geom

µ = 4.12

µ = 3.9




QuestionsResults

Question 3

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


prob

geom

µ = 4.07

µ = 4.15




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




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




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




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




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




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 thechallenging is explaining it well. Either way, it ischallenging.”“...it’s really the best way to learn math.”“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.”




QuestionsResults


“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 thechallenging is explaining it well. Either way, it ischallenging.”

“...it’s really the best way to learn math.”“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.”




QuestionsResults


“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 thechallenging is explaining it well. Either way, it ischallenging.”“...it’s really the best way to learn math.”

“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.”




QuestionsResults


“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 thechallenging is explaining it well. Either way, it ischallenging.”“...it’s really the best way to learn math.”“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.”




QuestionsResults


“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 thechallenging is explaining it well. Either way, it ischallenging.”“...it’s really the best way to learn math.”“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.”




QuestionsResults


“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 thechallenging is explaining it well. Either way, it ischallenging.”“...it’s really the best way to learn math.”“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.”




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.”




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.”




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.”




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.”




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.”




Our own reactions: Probability

Teaching TMM is harder than lecturing

When “drama” happens outside of class, class can beboringweakest students are involved





Teaching TMM is harder than lecturingWhen “drama” happens outside of class, class can beboring

weakest students are involved





Teaching TMM is harder than lecturingWhen “drama” happens outside of class, class can beboringweakest students are involved




Our own reactions: Geometry

IBL is more difficult to teach

IBL is more rewarding to teachStill difficult to keep weaker students involved





IBL is more difficult to teachIBL is more rewarding to teach

Still difficult to keep weaker students involved





IBL is more difficult to teachIBL is more rewarding to teachStill difficult to keep weaker students involved


IBL approaches to Geometry and Probability for High School ... fileRationale Implementation...

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