Unformatted text preview: Pedagogical Mathematics John Mason
PIMS Celebration Lecture
SFU 2007
1 Definitions
Mathematical Pedagogy
– Strategies for teaching maths; useful constructs Mathematical Didactics
– Tactics for teaching specific topics or concepts Pedagogical Mathematics
– Mathematical explorations useful for, and arising
Mathematical
from, pedagogical considerations
from, 2 Perforations How many holes
for a sheet of
r rows and c columns
of stamps?
3 If someone claimed
there were 228 perforations
in a sheet,
how could you check? Possible Strategies
Watch What You Do
Watch – Specialise but attend to what your body does
Specialise
as way of seeing & as source of
generalisation
generalisation
Say What You See – Reveal/locate distinctions, relationships,
Reveal/locate
properties, structure
properties, 4 Holding Wholes (gazing)
Discerning Details
Recognising Relationships
Perceiving Properties
Reasoning on the Basis of Properties Perforations Generalised
Dimensions of possible
Dimensions variation:
variation:
For Each Stamp:
– number of horizontal perforations top & bottom
– number of vertical perforations, left & right
– number of perforations in the corners 5 Structural Generalisation: write down the number of
perforations for R rows and C columns and
characterise the numbers which can arise Structured Variation Grids
6 Pedagogical Offshoot
Pedagogical Vecten Holding Wholes (gazing)
Discerning Details
Recognising Relationships
Perceiving Properties
7Reasoning on the Basis of Properties Chords
Locus of midpoints of chords of a quartic?
Locus of midpoints of fixedwidth chords?
Chord Slopes family of chords with one end fixed: slope locus;
what happens as the length goes to zero? fixed width chords: slope locus;
what happens as the width goes to zero?
? chords of fixed length: slope locus;
what happens as the width goes to zero?
? envelope of slopes of chords through pt
8 Cubic Construction
Construct a cubic for which the
Construct roottangents are alternately perpendicular
roottangents
It seems a reasonable task, except that it is
It
impossible!
impossible!
Why is it impossible?
What sorts of constraints are acting?
What about quartics? 9 Discovery
Suppose a cubic has three distinct real
Suppose roots.
roots.
Then the sum of the cotangents of the
Then
rootangles is 0.
rootangles
More generally, for a polynomial of
More
degree d, the sum of the products of the
rootslopes taken d – 1 at a time is zero. 10 Extension
Suppose a line cuts a polynomial of
Suppose degree d > 1 in d distinct points.
degree
What is the sum of the cotangents of the
What
angles the line makes with the
polynomial at the intersection points?
polynomial 11 CuttingAngles (1)
• Let the line L(x) have slope m 1
cot(α ) =
ταν(θ − ϕ )
1 + tan(q ) tan(j )
=
tan(q )  tan(j ) =
12 1 + p ' ( rj ) m
p ' ( rj )  m α =θ ϕ
α QuickTimeª and a
TIFF (LZW) decompressor
are needed to see this picture. ϕ θ CuttingAngles (2)
Put f(x) = p(x)L(x)
d
We know that • ∑ • 1 δ
1
1
=0=∑
f '(rj )
(ϕ
1 π∋ ρ ) − µ • But the sum of the cots of the angles
between line and function is δ 1 + µ π ∋ ρϕ)
( ∑
1 13 µ +1
= µ ∑1 + ∑
= µδ
π ∋ ρϕ) − µ
(
(ϕ
1
1 π∋ ρ ) − µ
δ δ 2 Rootslope polynomial
δ
Given a polynomial p( x ) = α∏ ( ξ − ρ )
κ
κ=1 Define the rootslope polynomial of p to be
δ pρ ( ξ ) = α δ ∏ ( ξ − π ∋( ρκ ))
κ
=1 The constant term is (1)d times the discriminant
which is the square of the product of the
interrootal distances. 14 The coefficient of x is 0, which can be thought of
as the sum of the reciprocals of the rootslopes, times
the product of all the rootslopes. Chordal Triangles
Locus of centroids of chordal triangles?
Locus of Circumcentre of chordal triangles with
Locus
fixed chord widths?
fixed
Locus of area of triangles with fixed chord widths?
Limit of circumcentres?
Limit of excentres of chordal triangle?
Limit of centre of Bevan Circle?
Limit of area/product of chord widths?
15 Mean Menger Curvature
Mean
Given three points on a curve, the Menger
Given curvature is the reciprocal of the radius of
curvature
the circle through the three points
the
Given three points on a function, they
Given
determine an interval on the xaxis.
determine
Is there a point in the interior of that
Is
interval, at which the curvature of the
function is the Menger curvature of the
three points?
three
Try something easier first.
16 RolleLagrange MeanParabola
Given three points on a function but not
Given on a straight line, there is a unique
quadratic function through them.
Is there a point in the interval spanned,
Is
to which some point on the parabola can
be translated so as to match the function
in value, slope and second derivative at
that point?
that 17 Why ‘Mean Value’?
b Let f be integrable on [a, b].
The average (mean) value of f on [a, b] is ∫f
a β − α A sensible ‘Mean Value’ property (MVP):
b There exists c in [a, b] for which ∫ f =f (c) (b  a) a The usual ‘Mean Value’ Theorem(s) then
The become
become 18 – Derivatives (being continuous) satisfy the MVP
Derivatives
on every subinterval
on
– Integrals of continuous functions satisfy the
Integrals
MVP on every subinterval
MVP Mean Menger Curvature
Given a circle through three points (Menger circle), is there a point
Given
on the spanned interval with the same curvature?
on
Suppose f and g are both twice differentiable on [a, b],
that k(f) <k(g) on [a, b], and that f(a) =g(a).
If f Õa ) ² gÕa ) th e n
(
(
f or a ll x in (a , b ], QuickTimeª and a
TIFF (Uncompressed) decompressor
are needed to see this picture. QuickTimeª and a
TIFF (Uncompressed) decompressor
are needed to see this picture. QuickTimeª and a
TIFF (Uncompressed) decompressor
are needed to see this picture. f (x ) g (x ): QuickTimeª and a
TIFF (Uncompressed) decompressor
are needed to see this picture. QuickTimeª and a
TIFF (Uncompressed) decompressor
are needed to see this picture. QuickTimeª and a
TIFF (Uncompressed) decompressor
are needed to see this picture. If f Õa ) ³ gÕa ) th e n (
(
f or at most one x in (a , b ], f (x ) = g(x ). If g has constant curvature (is part of a circle) and
k(f) ≠k(g) on [a, b] then f and g can intersect
at most twice. 19 Therefore there m
ust
be a point s on [a, b]
at which
k(f(s)) =Menger Curvature
of the points a, b, c. Cauchy Mean Value Theorem
Augustin Cauchy 17891857)
Let [f(t), g(t)] trace a differentiable curve
trace in the plane
in
– in each interval [a, b] there exists a point s at
there
which
[g(b) – g(a)] f’(s) = [f(b) – f(a)] g’(s)
)] f’
)] g’ 20 Cauchy Mean Menger Curvature?
Is there a Cauchy version of curvature
for curves in the plane?
NO! Counter Example: QuickTimeª and a
TIFF (Uncompressed) decompressor
are needed to see this picture. 2 +t 2
2 3/ 2 (1+t ) QuickTimeª and a
TIFF (Uncompressed) decompressor
are needed to see this picture. 21 ProceduralInstrumental
ConceptualRelational
Human psyche is an interweaving of behaviour
Human
behaviour
(enaction)
(enaction)
emotion (affect)
awareness (cognition)
Behaviour is what is observable
Teaching:
transposition
22 – Expert awareness is transposed
Expert
into instruction in behaviour
into
– The more clearly the teacher indicates
The
the behaviour expected,
the easier it is for learners to display it
without generating it
from and for themselves
from didactique didactic tension Tasks & Teaching
Tasks are only a vehicle for engaging in
Tasks mathematical thinking
mathematical
Learners need to be guided, directed,
Learners
prompted, and stimulated to make sense of
their activity: to reflect
their
– To manifest a reflection geometrically as a
To
rotation, you need to move into a higher
dimension!
dimension!
– The same applies to mathematical thinking!
23 Pedagogical Mathematics
‘Problems’ to explore probe learner
Problems’ 24 awareness and comprehension, and afford
and
opportunity to use their own powers to
experience mathematical thinking
experience
These may not be at the cutting edge;
Often they may be hidden in the
Often
undergrowth of mathematics of the past
undergrowth
but they can be intriguing!
Effective teaching of mathematics results in
Effective
learners with a disposition to explore for
themselves.
themselves. Further Reading
Mathematics Teaching Practice: a guide for university and college lecturers, Horwood Publishing. Mason J. (2002). Mathematics as a Constructive Activity: learners generating examples. Erlbaum. Watson A. & Mason J. (2005). CounterExamples in Calculus. manuscript, Klymchuk, S. & Mason, J. 25 http://mcs.open.ac.uk/jhm3 ...
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 Winter '08
 JARVIS
 Math, Calculus, Derivative, Chords, Menger Curvature

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