The Harvard ALM in Mathematics for Teaching in Extension degree is described, along with two inquiry-based learning courses taught in that program. Also covered is the Harvard Secondary School Program and an IBL course taught in it.

1. Inquiry-Based Learning
Opportunities for Secondary
Teachers and Students
Bret Benesh, College of Saint Benedict
Andrew Engelward, Harvard University
Thomas W. Judson, Stephen F. Austin University
Matthew Leingang, New York University
AMS-MAA Special Session on Inquiry-Based Learning
Washington, DC
January 7, 2009

2. On Deck
• The Harvard Extension
School’s Master of Liberal
Arts (ALM) in Mathematics
for Teaching
• Geometry and Probability
Courses
• Harvard’s Secondary
School Program (SSP)
• A Topology Course
• Questions and Future
Directions

3. Master of Liberal Arts
(ALM) in Mathematics for
Teaching in Extension

4. ALM Program
• Talented group of teachers with a strong desire for outreach to
public schools (Harvard Mathematics Department)
• Experience providing mathematical content for public school
teachers (Prof. Paul Sally’s SESAME Program)
• Need for mathematically rich courses for teacher training/
development and professional licensure requirement for public
school teachers (Boston Public Schools)
• Harvard Extension School

5. Program Goals
• Solidify teachers’ knowledge base of middle and high
school mathematics
• Provide courses in number sense, algebra, geometry,
probability
• Expose teachers to a variety of teaching approaches/
classroom environments
• Require teachers to re-encounter the learning process
from a student’s perspective
• Reenergize teachers’ passion for teaching

6. Who are the ALM
Students?
• Middle and high school
teachers from Boston
area schools
• Boston Latin to Boston
Public
• People looking for a
career change

7. Who are the ALM
Students?
• Middle and high school
teachers from Boston
area schools
• Boston Latin to Boston
Public
• People looking for a
career change

8. ALM Program Requirements
• Students must complete one year of calculus
• Take at least three mathematical theory courses
• Take at least one pedagogy and lesson study
course
• Take four electives and complete a master’s
thesis OR take six electives and complete a
capstone course

9. ALM Courses
• Math E-300 Math for Teaching Arithmetic
• Math E-301 Math for Teaching Number
Theory
• Math E-302 Math for Teaching Geometry
• Math E-303 Math for Teaching Algebra
• “Standard” math courses • Math E-304 Inquiries into Probability and
(calculus, linear algebra, Combinatorics
discrete math, etc.)
• Math E-306 Theory and Practice of
• Teaching Statistics
Courses designed for the
secondary school teacher

10. Use of Inquiry-Based Learning
in the ALM program
• Expose each ALM candidate to at least one full IBL course
experience
• Goals for IBL exposure:
• Require teachers to participate in creating mathematics
• Erode teachers’ conception of mathematics as a
monolithic sequence of rules/algorithms
• Provide alternative model for classroom teaching/
learning for teachers to use in their own classrooms

11. IBL courses in the
ALM program

12. Why teach IBL?

13. Geometry

14. Geometry

15. Typical Problem Set
• What is the definition of a circle in Euclidean
geometry?
• What does a circle look like in taxicab
geometry?
• What is the diameter of a circle in taxicab
geometry?
• What is the circumference in taxicab geometry?
• What is π in taxicab geometry?

16. Combinatorics

17. Probability

18. How has this course affected the
way you think about mathematics?
Probability (µ=4.21) Geometry (µ=4.3)
3: No change
3: No change
10%
14%
5:Very positively
5: Very positively
36%
40%
4: Somewhat positively
4: Somewhat positively 50%
50%

19. How has this course affected the way you
think about teaching mathematics?
Probability (µ=4.12) Geometry (µ=3.9)
3: No change 3: No change
14% 5:Very positively 20% 5: Very positively
29% 30%
4: Somewhat positively 4: Somewhat positively
57% 50%

20. Would you recommend a course taught in
this format?
Probability (µ=4) Probability (µ=4.15)
3: Undecided 5:Yes! 3: Undecided
14% 14% 20%
5: Yes!
35%
4: Sure
4: Sure
45%
71%

21. I have always found
proofs difficult and
intimidating. Now I
feel more comfortable
with them.
It’s really the
best way to learn
Comments
math.

22. I think a
little more
teacher-based
instruction would
allow for a more
rigorous pace, which
pushes students and can
lead to more of a need
for interaction and
discussion by
necessity.

23. Waiting
for the other
students to finish is a
bit of a waste of
time.

24. I wish there
was more
concrete
learning.

25. I see more value in working in
groups as an ongoing strategy [for
teaching]. It takes a while to build trust,
but once it’s established the outcome
in class thinking is fantastic!
I leave [class]
excited and
bewildered.

26. Reflections

27. π in taxicab geometry

28. π in taxicab geometry
•C=8
2 2
2 2

29. π in taxicab geometry
•C=8
2 2
2
•d=2
2 2

30. π in taxicab geometry
•C=8
2 2
2
•d=2
•π=C/d = 4
2 2

31. The Harvard SSP

32. Harvard’s Secondary
School Program (SSP)
• Every summer, more than 1,000 high school
students who have completed their 10th,
11th, or 12th year attend the Harvard
summer session
• Alongside undergraduate students, they
explore subjects not available at their high
schools and earn college credit in college-
level courses.

33. Math S-101
• Spaces, Mappings, and Mathematical
Reasoning: An Introduction to Proof
• A point-set topology course leading to a
proof of the Brouwer Fixed Point Theorem

34. Math S-101 Goals
• To gain an appreciation of mathematical reasoning
and proof
• To develop skills in structured mathematical
reasoning and proof
• To develop a basic understanding of sets and point
set topology
• To improve skills in learning and communicating
mathematics with respect to the spoken and written
word

35. Math S-101 Learning
Objectives
• Apply the ideas of mathematical proof and
reasoning in more advanced mathematics
courses.
• Understand and apply the basic ideas of point
set topology, including closure operators,
continuity, connectedness, and mappings.
• Understand and be able to apply the Brouwer
Fixed Point Theorem

36. Features of the Course
• Students work from a
set of notes (Danny
Goroff)
• Students present
solutions to problems at
the board
• Students turn in a
notebook every week
• Online discussion board

37. The 2008 Student
Population
• Two graduate students in the ALM program
• Three Harvard undergraduate students
• One LSE undergraduate
• Six high school students

38. Student Reactions
I can take
AP calculus
at my high
I felt well-
school next
prepared to take
year, but my
advanced mathematics
high school does courses [graduate
not offer any student at LSE]
courses like
Math S-101
[high school
senior]
I had a better idea of how to
write a proof than the honors
calculus students [Math 122
student at Harvard]

39. Questions and Future
Directions
• Does IBL really work?
• How do we train IBL
teachers?
• How do we attract
more high school
students and teachers to
IBL courses?

40. Thank You

41. Photo Credits
Foraggio
Cyndie@smilebig! Jesus V
Fotographic
Thomas Hawk William Fawcett Sean Dreilinger
Christopher
B. Romeo
Sunflower Central Stock.XCHNG
Photography
Oriano
Nicolau Plasticrevolver