NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
Maths with attitude
Paul Glendinning
University of Manchester
NCETM, Manchester, December 2008
With corrections thanks to feedback from members of the audience.
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
Introduction
Aim:
Think about what it is to be a mathematician
Think about how this is taught
Think about what we teach without thinking
Maths with morals
Maths made personal
MATHS WITH ATTITUDE
Not a lot of mathematics, but a lot of culture of mathematics.
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
Oppositions and Tensions
syllabus/content
core maths
VIth form
being taught
intellectual maths
maths as maths
theorems
pure maths
Paul Glendinning
Maths with attitude
delivery/VLE
applicable maths
undergraduate
personal learning
functional maths
service maths
methods
applied maths
(and then there's statistics)
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
For today:
Not what is maths, but what
is a mathematician?
Two anecdotes:
Part III, Cambridge 1987
Conversation with the Dean, Manchester 2007
The hidden content of teaching (cf. previous slide).
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
Bad Maths in society
Case of Angela Cannings
Convicted of killing her own child because it was statistically implausible
that the same woman has three children dying of cot death. Appalling
that Sir Roy Meadows could make this statement, equally appalling that a
courtroom of intelligent people could not see the aw.
[I should have said more about conditional probability here...]
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
I am a mathematician
iamamathematicianiamamathematicianiamamathematicianiamamathem
aticianiamamathematicianiamamathematicianiamamathematicianiam
amathematicianiamamathematicianiamamathematicianiamamathemati
cianiamamathematicianiamamathematicianiamamathematicianiamama
thematicianiamamathematicianiamamathematicianiamamathematicia
niamamathematicianiamamathematicianiamamathematicianiamamathe
maticianiamamathematicianiamamathematicianiamamathematiciania
mamathematicianiamamathematicianiamamathematicianiamamathemat
icianiamamathematicianiamamathematicianiamamathematicianiamam
athematicianiamamathematicianiamamathematicianiamamathematici
aniamamathematicianiamamathematicianiamamathematicianiamamath
ematicianiamamathematicianiamamathematicianiamamathematician
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
I am a mathematician (contd)
Mathematics is always near me in the dusk, like something I
mean to say but can never quite remember. Its images glare
blindly through my sleep. I have pursued theorems over many
summers, but they are hard to nd and harder to see, being so
few and wary.
adapted from the extraordinary J.A. Baker, The Peregrine (1967):
Sparrowhawks were always near me in the dusk, like something I meant to
say but could never quite remember. Their narrow heads glared blindly
through my sleep. I pursued them for many summers, but they were hard
to nd and harder to see, being so few and wary. They lived a fugitive,
guerrilla life. In all the overgrown neglected places the frail bones of
generations of sparrowhawks are sifting down now into the deep humus of
the woods. They are a banished race of beautiful barbarians, and when
they died they could not be replaced.
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
Attitude
From The Oxford English Dictionary:
originally essentially the same as aptitude: appropriate for a purpose
early C18 technical term for the posture/position of an object(cf.
satellites)
later C18 poses of people and animals as well as objects.
C19: shift from using attitude to describe physical states, to using it
to describe mental states { particularly beliefs
leads to modern denition: a mental predisposition, or a state of mind
leading to expectations of actions or reactions, a trait of character.
New slang: attitude is used to reect a particular attitude (as it were) of
self-respect and condence, so attitude has become something a person
can be described as possessing, often with rebellious and
anti-establishment overtones.
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
Maths and attitude
Claim:
Knowledge and understanding of mathematics leads to certain mental
predispositions, i.e attitudes, towards the world in general.
These traits are associated with critical thinking, precision and
intellectual honesty.
Not the same as an attitude towards mathematics (do you like it? is
it fun?).
But mathematics has attitude as well. It is exciting, challenging, irritating
and amazing. It has no respect for any authority other than that of its
own making. Although my focus is on the attitudes inherent to a proper
understanding of mathematics, the attitude of mathematics is a
continuous undercurrent. This attitude is what makes doing mathematics
such an addictive activity.
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
Three Moral Stories (or stories with morals)
1. Simon Jenkins and the quadratic equation
2. Robert Hooke and Hooke's Law
3. My latest research (always the most exciting).
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
In two near-identical articles (The Times, April 2003; The Guardian, June
2008) Simon Jenkins claims that at age 16 he `breakfasted on quadratic
equations...' but follows this up with a stark sentence:
IT WAS A WASTE OF TIME.
His argument is that maths is never useful (except possibly to a few
scientists and engineers who can be taught it later) and that it is hard,
and so it should be dropped in favour of the `humanities and social
sciences which are clamouring for a place on the curriculum'.
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
The quadratic formula
The equation
p
y2 = p
is easy to solve: y = p provided p 0.
Now use completing the square to get the corresponding formula for the
general quadratic:
ax 2 + bx + c = 0
as
a 6= 0, b2 4ac.
Paul Glendinning
Maths with attitude
p
b b2
x=
2a
4ac
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
What we learn/teach
the formula (knowledge)
application of completing the square (method/technique)
importance of easy cases (method/technique)
extension to imaginary numbers (extension/play)
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
and attitude?
knowing where something comes from, not relying on memory or
accepting someone else's word for it (important even for graduate
students)
appreciating need for proof and for precision
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
Robert Hooke
Robert Hooke was one of the
great experimentalists of the
seventeenth century. He spent
forty years as Curator of
Experiments for the Royal
Society (rst in this post) { in
1660 he considered eect on
extension of a spring due to a
suspended weight.
Figure from the frontispiece of
Hooke's De potentia
restitutiva (1678).
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
Hooke's Law
His conclusion:
or
Force
/ extension
F = kx
where F is force, x is extension, and k is the spring constant.
What is the status of this?
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
Hooke's Law?
Taken literally (and how else should we take a law) this is clearly rubbish
usually the spring is in compression and need F > F0 to get any
extension
if F too large the spring loses its restorative properties { slipping or
plastic phase.
to a mathematician F F0 kx looks like the beginning of a Taylor
series!
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
Moral
Huge gulf in approach between mathematicians and scientists, even when
the common language of mathematics is being used.
For a mathematician a result (theorem) should be true. There should be
no exceptions, no special cases. (Finding such leads to a revision of the
theorem, a sense that it was wrong as previously stated.)
For scientists a `law', even when expressed mathematically, is just a
current best guess, a tool which can be used when appropriate. The fact
that it is obviously wrong is not a problem { it simply reects the fact that
it has been used in an `inappropriate' setting.
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
3. STRANGE NONCHAOTIC ATTRACTORS
xn+1 = xn + !; mod 1
! 2= Q
yn+1 = 2 cos(2 xn ) tanh yn
If 0 < < 1
(1)
0.3
0.2
0.1
0.0
−0.1
−0.2
−0.3
−0.4
−0.5
0.0
0.1
Paul Glendinning
Maths with attitude
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
3. STRANGE NONCHAOTIC ATTRACTORS
xn+1 = xn + !; mod 1
! 2= Q
yn+1 = 2 cos(2 xn ) tanh yn
If > 1
(1)
4
3
2
1
0
−1
−2
−3
−4
0.0
0.1
Paul Glendinning
Maths with attitude
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
WHAT WAS THAT?
A strange nonchaotic attractor
WHAT IS THAT?
STRANGE = complicated geometry
NONCHAOTIC = not chaotic
(i.e. transverse Liapunov exponents negative)
ATTRACTOR = something that attracts
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
In other words....
Their geometric structure is fractal like that of typical chaotic
attractors, but they are not sensitive to initial conditions,
because the Lyapunov exponents of typical trajectories are
negative.
Neumann & Pikovsky, 2002
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
... or again ....
An SNA has a geometrically complex structure (the attractor is
not a nite collection of points and it is not dierentiable), but it
exhibits no sensitivity to initial conditions.
Khovanov, Khovanova, McClintock & Anishchencko, 2000
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
Consensus?
SNA have
complicated geometry (strange)
non-positive Liapounov exponents (nonchaotic)
no sensitive dependence (nonchaotic)
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
A THEOREM (Glendinning, J
ager and Keller, 2006)
Strange nonchaotic attractors have sensitive dependence on initial
conditions.
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
Moral
Don't
it.
accept what others say unless you understand why they are saying
Personal integrity/honesty { own up to what you don't understand (you
usually nd others don't understand it either, though there's a balance...)
(And maths can catch you out if you don't do this { it has ATTITUDE
and demands RESPECT)
Paul Glendinning
Maths with attitude
University of Manchester
NCETM: Introduction
Maths with attitude
1. Jenkins and the quadratic
2. Hooke's Law
3. My research: SNA
Conclusion
Conclusion
Maths is personal
we need to be aware of this in our teaching
we expect students to achieve a `good' attitude towards being a
mathematician
(but we don't teach this explicitly).
Attitude goes beyond curriculum, technique, mode of teaching { and
central to the culture of mathematics.
... but I'm still trying to work out the resolution!
THANK YOU FOR YOUR ATTENTION.
Paul Glendinning
Maths with attitude
University of Manchester