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									                                               CMESG 2009, York
                                               CMESG 2009, York
THE TIES THAT BIND: COMPLEMENTARY AND
CONTRADICTORY AGENDAS
                             Peter Liljedahl
CMESG 2009, York
                   I apologize ...
                                                        CMESG 2009, York
• mathematics teacher education
   – elementary & secondary
   – pre-service & in-service
• teacher beliefs about mathematics, teaching
  mathematics, and learning mathematics ... and their
  interplay
• instructional design
   – problem solving
   – tasks
   – assessment



What are my interests?
                                               CMESG 2009, York
• creativity in mathematics
   – undergraduate students
   – pre-service teachers
                              MATHEMATICIANS
   – mathematicians
                               ON CREATIVITY




What are my interests?
                                                         CMESG 2009, York
•   BEd – major in mathematics specialization
•   BEd – minor in secondary mathematics education
•   BSc – major in mathematics education (in progress)
•   MSc/MEd – secondary mathematics education
•   MEd – elementary mathematics education
•   PhD – mathematics education

• PDP – Math 190, Educ 475, Educ 415
• Q’s – Educ 211, Educ 212, Educ 313, Educ 411


Mathematics education ... at SFU
                                                         CMESG 2009, York
•   BEd – major in mathematics specialization
•   BEd – minor in secondary mathematics education
•   BSc – major in mathematics education (in progress)
•   MSc/MEd – secondary mathematics education
•   MEd – elementary mathematics education
•   PhD – mathematics education

• PDP – Math 190, Educ 475, Educ 415
• Q’s – Educ 211, Educ 212, Educ 313, Educ 411


Mathematics education ... at SFU
                                               CMESG 2009, York
•   combinatorics    •   indigenous numeracy
•   probability      •   creativity
•   inequalities     •   dynamic geometry
•   proof            •   neuroscience
•   linear algebra   •   teacher education
•   infinity         •   problem solving
•   modeling         •   perseverance
•   functions        •   attitudes/beliefs

... graduate work at SFU
                                               CMESG 2009, York
•   combinatorics    •   indigenous numeracy
•   probability      •   creativity
•   inequalities     •   dynamic geometry
•   proof            •   neuroscience
•   linear algebra   •   teacher education
•   infinity         •   problem solving
•   modeling         •   perseverance
•   functions        •   attitudes/beliefs

... graduate work at SFU
                               CMESG 2009, York
‘We’ are different!
• this
• that
• the other way

• what constitutes
   – pre-requisite knowledge
   – curriculum


Conclusion #1
                      CMESG 2009, York
How many triangles?
                                                       CMESG 2009, York
Homework
• Find a 10 digit number (with the digits 0, 1, ...,
  9 used once each) such that the number
  formed by the first N digits is divisible by N.

Possible Project
• Consider X (X is a natural number). How
  many ways can X be written as the sum of
  consecutive natural numbers?

and then ...
                                                           CMESG 2009, York
• I assumed that they had no prior knowledge of
  triangular numbers, tetrahedral numbers, pyramidal
  numbers, or how to sum these!
• I did not allow the lesson to be built upon a reliance
  on prior knowledge!
• Knowledge gained in the lesson might be relevant for
  future activities/assignments!
• However, I did build the lesson on the assumption of
  capacities acquired in prior lessons!



Pre-requisite knowledge
                                        CMESG 2009, York
•   group work      •   curiosity
•   communication   •   wonder
•   reason          •   patience
•   inquiry         •   certainty
•   endurance       •   argument
•   perseverance    •   visualization
•   courage         •   thinking



Curriculum as capacities
                                                          CMESG 2009, York
• a priori
   –   capacities
   –   relationship with mathematics
   –   creative, intuitive, inductive, connected, messy
   –   feel like a doer (verb) vs. a knower (noun)

• emerging
   – certainty and the ... syndrome
   – articulation
   – chronological vs. logical

Curriculum goals - mathematics
                                    CMESG 2009, York
            3, 6, 9, 15, 24

            ?, ?, _, _, 100




Curriculum goals – math education
                                    CMESG 2009, York
Curriculum goals – math education
                                                 CMESG 2009, York
• a priori
   –   promoting student doing
   –   understanding student thinking
   –   instructional design
   –   teaching

• emerging
   – traditions of mathematics (ToM)
   – traditions of teaching mathematics (ToTM)



Curriculum goals – math education
                                            CMESG 2009, York
I am bound to mathematics!
• Mathematics content comprises between
   50% and 95% of what I teach.
• It is the context of all my activities.
• It is engages my students.
• It is my goal.




Conclusion #2
                          CMESG 2009, York
            shift gears




Teacher education ...
                                                               CMESG 2009, York
this is where the two fields collide
   –   tension between content and context
   –   knowledge for teaching mathematics (PCK) is
       fundamentally different than knowledge of mathematics
       (CK)
this is my interest
   –   knowledge vs. beliefs
   –   necessary vs. sufficient
   –   means to an end vs. an end unto itself (ToM, ToTM)
recent research
   –   Susan Oesterle


Mathematics for teaching
                                          CMESG 2009, York
•   do you do anything different?
•   do you plan differently?
•   do you teach differently?
•   do you assess differently?
•   do you have different expectations?
•   do you have different goals?




Teaching Mathematics for Teachers
                                     CMESG 2009, York
•   mitigate anxiety
•   increase comfort
•   increase confidence
•   improve attitudes
•   increase competency
•   create clarity
•   increase ways of understanding
•   increase ways of explaining

You do!
                                                 CMESG 2009, York
We are very similar!
• improve teachers
• use/foster capacities
• curiosities about student mis/understandings
• interest in student learning of ...
• benefits of knowledge about teaching
• student/teacher beliefs/efficacy



Conclusion #3
                                                CMESG 2009, York
You are bound to mathematics education!
• you teach so that students learn
• when they don’t ... you wonder about it
• when you wonder a lot you seek answers
• when answers can’t be found you create your
  own means of researching it
• when you get find something interesting you
  seek an outlet (CMESG)


Conclusion #4
                                          CMESG 2009, York
              refocusing




Mathematicians in mathematics education
                                                         CMESG 2009, York
• "I want to dispel two common myths. First, it
  is a common belief among mathematicians
  that attention to education is a kind of
  pasturage for mathematicians in scientific
  decline" (Bass, 2005).
• "Mathematics education is not mathematics.
  It is a domain of professional work" (Bass,
  2005).
  – it has its own history
  – it has its own canon of knowledge
  – it has its own theories, traditions, and practices


Mathematicians in mathematics education
                                                 CMESG 2009, York
•   mathematics for teachers
•   advanced/specialized mathematics
•   preparation of instructors (Natasha Speer)
•   co-supervision of graduate students
•   provide venue for research
•   voices in the media




Mathematicians in mathematics education
CMESG 2009, York
                   Thank you!
                                             CMESG 2009, York
Educ 212 - From bridges to balloon animals
                                                                     CMESG 2009, York
"Teaching Assistants’ Knowledge and Beliefs Related to
  Student Learning of Calculus"
• ... This study examined mathematics graduate student teaching
  assistants’ knowledge of student thinking for important concepts
  from calculus. Findings indicate that participating teachers did
  not possess rich knowledge of student thinking and that, in
  general, they were unable to generate solution paths other than
  the one they had used to solve the problem. In addition,
  teachers asserted a variety of (sometimes contradictory)
  relationships between students’ correct solutions and their
  understanding of the problems’ central concepts.

								
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