Transcript Doing mathematics in teacher preparation: giving space and

Mathematics, technology and
curriculum: a complex interaction
Frédéric Gourdeau
Université Laval, Québec, Canada
Outline
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Our students – a brief description
The doing of mathematics – a vignette (H. Bass)
Providing experiences for students
Technology : challenges and opportunities
Resistance to change
F. Gourdeau, SMC 2014
Our students
• Cegep : grades 12 and 13; calculus, linear algebra
• Engineering students
• Pre-service teacher education
• Mathematics major
– Cegep teaching
– Graduate work in maths or other subject area
F. Gourdeau, SMC 2014
The doing of mathematics
Hyman Bass (2011), A Vignette of Doing Mathematics:
A Meta-cognitive Tour of the Production of Some
Elementary Mathematics.
F. Gourdeau, SMC 2014
A brief description : when we do mathematics, we…
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Question
Explore
Represent
Look for structure
conjectures or new
questions.
• Consult, if we’re stuck: others, the literature, the
web.
F. Gourdeau, SMC 2014
• Connect with other mathematics, through our
research, reflection, analogies.
• Seek proof, to prove or disprove our conjectures.
Often this proceeds by breaking the task into smaller
pieces, for example by formulating, or proving,
related, hopefully more accessible, conjectures, and
showing that the main conjecture could be deduced
from those.
F. Gourdeau, SMC 2014
• Can be opportunistic, letting the mathematics guide
us if we see inviting trails.
• Prove, writing a finished exposition of the proof (if
one is found), using illuminating representations of
the main ideas, meeting standards of mathematical
rigor, and crafted to be accessible to the
mathematical expertise of an intended audience.
F. Gourdeau, SMC 2014
• Analyze proofs, which even if they are conceived of
as a means to an end, are a product worthy of note
and study, since the theorem typically distills only a
small part of what the proof contains.
• Use our sense of aesthetics and taste, associated
with words like elegance, precision, lucidity,
coherence, unity.
F. Gourdeau, SMC 2014
Reforming our curriculum
• For pre-service teachers, since the early 90’s
– Emphasis on the doing of mathematics
– Problem solving
– Culture
– Deconstructing packed formulae
– Finding accessible answers to “why?”
– And seeing when a “why” should be a “what”
F. Gourdeau, SMC 2014
What helped to make it possible
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CMESG
Few courses, did not impact of most Faculty
Flexibility in our institutional rules
People
– Hodgson
– Cassidy
F. Gourdeau, SMC 2014
New technologies
– Secondary school use it : it pushed us
– Fits well in the “doing”
– Presentation, team work: this has value
However
– Pre-service teachers: afraid of programming, and
have used surprisingly little technology
– Problems of assessment
F. Gourdeau, SMC 2014
Reforming our curriculum
• For maths majors, in the early 90’s
– Final year project
– History of mathematics
– Presentation
• Since
– More on communication, some work as a program
– Team work : very little
F. Gourdeau, SMC 2014
Personal impressions
• Resistance to change in the curriculum
– Mathematics is highly structured
– Mathematical language is very compact, built on
abstractions
– Being lost in mathematics is so easy
F. Gourdeau, SMC 2014
Personal impressions
• Resistance to a more important use of technology
– Assessment of the work
– Team work : who gets the grade
– Hard work for “us” if we change
• Time, recognition
– We are not used to be told how we should do
things in class : give us the content, let us do the
rest
– We are not used to adopt a program approach
F. Gourdeau, SMC 2014
Some successes
• R for engineers
– Technology to teach technology
• Maple/Matlab
– Important work in two service courses
F. Gourdeau, SMC 2014
Math majors
• Prepare for graduate work and/or for teaching at
Cegep
• More traditional approach
F. Gourdeau, SMC 2014
Technology
• Maple, Geogebra, Fractint, Matlab, R
• Used in many courses, not as an objective but as a
mean
F. Gourdeau, SMC 2014
Thanks
F. Gourdeau, SMC 2014