revisiting basic assumptions in mathematics teacher education · its relations to reality,...
TRANSCRIPT
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Rina ZazkisSimon Fraser University
Vancouver, Canada
Revisiting basic assumptions
in mathematicsteacher education
Based in part on:Zazkis, R. (2008). Examples as tools in mathematics teacher education.In D. Tirosh (Ed.) Tools in Mathematics Teacher Education. (in Handbook for Mathematics Teacher Education, Vol. 2, pp. 135-156). Sense Publishing.
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Rina ZazkisSimon Fraser University
Vancouver, Canada
Revisiting basic mathematical assumptions
inteacher education
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Rina ZazkisSimon Fraser University
Vancouver, Canada
Revisiting basic mathematical
assumptionsin teacher education
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• to improve/enhance teachers’personal understanding of mathematics
• to examine/introduce the variety of students’ possible understandings or misunderstandings of mathematics
Goals in teacher education
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Imagine there is a world map in the above rectangle
Mark Vancouver, BC Canada on the map
X
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X
Vancouver?
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What is your reaction to the following
statements?
• The sum of the interior angles in a triangle ABC in 280 degrees
• The graph of a function y=x is a parabola
• A number is divisible by 5 if and only if the sum of its digits is divisible by 5
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What is your reaction to the following
statements?• The sum of the interior angles
in a triangle ABC in 180degrees
• The graph of a function y=X2 is a parabola
• A number is divisible by 3 if and only if the sum of its
digits is divisible by 3
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And how about these?
• The sum of the interior angles in a triangle is always 180 degrees
• A graph of a function y=x is a straight line
• A number is divisible by 3 if and only if the sum of its digits is divisible by 3
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• The sum of the interior angles in a triangle ABC in 280 degrees
• The graph of a function y=x is a parabola
• A number is divisible by 5 if and only if the sum of its digits is divisible by 5
Look again
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• The sum of the interior angles in a triangle ABC in 280 degrees (possible, on a sphere)
• The graph of a function y=x is a parabola (indeed, in focus-directrix coordinate system)
• A number is divisible by 5 if and only if the sum of its digits is divisible by 5 (indeed, in base 6)
Look again
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Directrix coordinates
y=x
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Directrix coordinates
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Examples that “shake” our assumptions
• conventions
• shared understandings
• unintended constraints
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Examples that “shake” our assumptions
•conventions• shared understandings
• unintended constraints
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Conventions
• Base ten representation
• Cartesian coordinates
• Euclidian geometry (on a plane)
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Conventions
•Base ten representation
• Cartesian coordinates
• Euclidian geometry
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“if compared with mathematics resulting from pondering more profoundly the subject matter and its relations to reality, unorthodox positional systems are a mere joke”
Freudenthal (1983, p. 132)
“it is a good didactics to motivate pupils by jokes, and an unorthodox positional system may even be a good joke”.
Dines
“New Math”
Multi-base arithmetic:
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Working in bases other than 10
• Counting
• Conversion
• Operations
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count in base 4
1 , 2 , 3
410, 11, 12,
13, 14,
20, 21, 22, 23, 30, 31, 32, 33,
34,
40,
100, 101, 102, ...
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Add in base 5
33+14102
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Multiply in base 5
33x 14?
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convert 12.34 five
1x5 + 2x1 + 3x(1/5) + 4x(1/25) = 7+19/25 = 7.76
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1x5 + 2x1 + 3x(1/5) + 4x(1/25) = 7+19/25 = 7.76
12five = 7ten ; 34 five = 19 ten
12.34 five = 7.19 ten
7+ 34/25
1x5 + 2x1 + 3x(1/5) + 4x(1/5050)
convert 12.34 five
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play New Math
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FOR TEACHERS...From challenging extensions or extra-curricular activities
to
extending the boundaries of teachers’example spaces
ining renewed understanding of (automated)processes
– such as multi-digit addition or assigning place values –
embedding the conventional in an extended schema
gaining insight on student’s difficulties
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Examples that “shake” our assumptions
• conventions
•shared understandings
• unintended constraints
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Shared common understandings
Grandma baked 12 cookies for 3 of her grandchildren.
How many cookies will each child get?
280 students of ABC elementary school will go in a field trip by buses.
There are 40 seats on a bus.How many buses are needed?
Mary had 4 blouses, 3 skirts and 2 jackets.
How many outfits can she make?
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Shared common understandings
I ate healthy foods for 2 weeks and lost 7 pounds. How many pounds will I lose if I eat healthy foods for 20 weeks?
Jake bought a twelve-pack of beer and paid $10.44. He then decided he
needed two more cans of beer. How much will it cost him?
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FOR TEACHERS...From simple exercises, drill and practice
to
raising awareness of what is implicitly taken for granted and
can be an obstacle for a learner
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Examples that “shake” our assumptions
• conventions
• shared understandings
•unintended constraints
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Can you cut a cardboard square into 10 squares,
using all the material?
Can you plant 4 trees such that there is the same distance
between any 2 of them
play movie
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Assumptions as constraints
famous 9 dots
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Assumptions as constraints
famous 9 dots
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Can you plant 4 trees such that there is the same distance between each 2
of them
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Can you cut a cardboard square into 10 squares,
using all the material?
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FOR TEACHERS...
From riddles and brainteasers
to
identifying constraints in human thinking in support of problem solving
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Schema“to understand something means to assimilate it into an appropriate schema” (Skemp, 1973)How can one understand better what has
been already understood, that is, assimilated?
“to understand something better means to assimilate it in a richer or more abstract schema” (Zazkis, 20XX)
How can this happen?
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Black swans
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THANKS