Change of Basis

1
Change of Basis

• A Case of
• Inquiry-Based Learning
• in Mathematics

2
A Taste of Honey

• Which would you rather do?...
• Tedious and seemingly pointless row-reductions,
  matrix inversions, and eigenvalue computations,
  or
• The Markov Dance?

3
Learner-Centered Methodologies

• are useful in order to
• more truly approximate real-world situations,
• enliven classroom activities,
• encourage alternative interpretations of course
  material,
• and provide hands-on experience in relevant
  problem-solving skills.

4
Breaking with Tradition

• Inquiry-Based Learning (IBL)
• Problem-Based Learning (PBL)
• Guided Discovery
• Application-Based Learning

5
Design Considerations

• Much credit is due to L. Dee Fink, Creating
  significant learning experiences.
• Fink outlines a 12-step process for designing,
  implementing, and assessing the effectiveness of
  nontraditional college courses.

6
Situational Factors

• Class size, level, frequency of meeting?
• Expectations, from within and from without?
• Student characteristics age, experience,
  motivations?
• Teacher characteristics experience, comfort
  level, competence?

7
Special Pedagogical Challenge

• For my class, I chose
• overcoming the expectations that both I and the
  students have regarding how a math course is
  supposed to be taught

8
Learning Goals

• The next step was to identify learning goals for
  the course, organizing them roughly into six
  categories.
• For MATH 365, I identified 15 such goals,
  ranging from mastery of foundational knowledge to
  becoming aware of ones learning style.

9
Paradigm Shift

• It was soon time to choose a teaching
  strategy, or paradigm, for the manner in which
  the course would be led.
• For MATH 365, I opted to integrate team-based
  learning and application-based learning.

10
Applications Take Coordination

• The course was to be centered around
  applications where does linear algebra arise,
  and how do we apply it in practice?
• Finding a large variety of appropriate
  applications takes time!

11
Rewriting History

• I had trouble finding a text appropriate to the
  chosen pedagogical perspective.
• Ultimately, I ended up writing dozens of
  stand-alone exercises, projects, activities, and
  worksheets anyway.

12
Time Aint on My Side

• As one can imagine, this takes a tremendous
  amount of time.
• Many of the difficulties in teaching in this
  manner stem from time considerations. For
  instance, theres the issue of

13
Pitfalls!

• 1. Coverage.
• Are you wedded to the notion that youve got to
  cover everything on your departmental syllabus?

14
Pitfalls!, cont.

• 2. Patience.
• Can you stand by silently and let students
  figure something out for themselves?

15
Pitfalls!, III

• 3. Humility.
• Are you willing to admit that you are capable of
  making mistakes?

16
Son of Pitfalls!

• 4. Compromise.
• Are you willing to adjust your course design to
  take into account unforeseen circumstances?

17
Triumphs!

• 1. Tools of the trade.
• You would be amazed the kinds of teaching
  techniques youll discover when you think outside
  the box! (Blogs, assessment tools, et cetera)

18
More Triumphs!

• 2. Communication.
• The personal attention given each student leads
  to open lines of communication and to dialogue
  concerning the course and the material.

19
Yet More Triumphs!

• 3. Trust.
• Closer communication and more meaningful
  interaction leads to an increased level of trust
  and mutual respect.

20
Triumphs Forever!

• 4. Awareness of Learning.
• Students develop a better sense of how it is
  that they learn.

21
Triumphs vs. The Creature From Planet X!

• 5. Comprehension and Retention.
• Students learn to assimilate and apply
  foundational knowledge, and retain that
  knowledge better.

22
Sine qua non

• Before adopting this course design, one might
  want to make sure that
• ones energy level is high,
• ones willingness to experiment is great,
• ones time is flexible, and
• ones class size is SMALL!

23
Looking Back

• This course has turned out very different from
  the way I had envisioned it. Its been
• less romantic,
• more work, and
• more subtle in working its magic.