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Inquiry-Based Learning Opportunities for Secondary Teachers and Students
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Inquiry-Based Learning Opportunities for Secondary Teachers and Students

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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 …

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.

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  • My personal journey to IBL




  • Third time through a probability course for teachers

  • First time: team taught, disconnected

  • Second time: interesting for me, over their head


  • Third time: TMM







  • Bret’s journey:






  • Graduate work was in finite group theor y

  • Minored in math education

  • KTI Program


  • Core Plus and Connected Mathematics Project (CMP)


  • Goals:


  • Maximize student learning

  • Improve communication skills

  • Motivate students

  • Provide a classroom model


  • Platform for inquiry



  • Taxicab geometry

  • Compare and contrast with Euclidean


  • Class Format



  • Meet once per week

  • Class length is two hours

  • Mostly in-service high school teachers


  • Role of Instructor



  • Moderate discussion

  • Referee

  • Ask questions

  • Not an authority


  • Transcript

    • 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