How Individuals Develop Advanced Mathematical Thinking

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How Individuals DevelopAdvanced Mathematical
Thinking

• David Tall
• Emeritus Professor in Mathematical Thinking

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Undergraduate mathematics education includes the
transition from the geometric and symbolic
mathematics at school to the formal constructions
of axiomatic systems in mathematics research.
To new concepts correspond, necessarily, new
signs. These we choose in such a way that they
remind us of the phenomena which were the
occasion for the formation of the new concepts.
So the geometrical figures are signs or mnemonic
symbols of space intuition and are used as such
by all mathematicians. Who does not always use
along with the double inequality agtbgtc the
picture of three points following one another on
a straight line as the geometrical picture of the
idea between? Hilbert 1900 ICM lecture.
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A simple framework that begins with the young
child and extends to the research mathematician
From the embodiment and symbolism of school to
the formal mathematics of axioms and proof
conceptual-embodied (based on perception of and
reflection on properties of objects)
proceptual-symbolic that grows out of the
embodied world through action (such as counting)
and symbolization into thinkable concepts such as
number, developing symbols that function both as
processes to do and concepts to think about
(called procepts)
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axiomatic-formal (based on formal definitions
and proof) which reverses the sequence of
construction of meaning from definitions based on
known concepts to formal concepts based on
set-theoretic definitions.
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Three Worlds of Mathematics
axiomatic
formal objects based on definitions
formal
Limits Manipulation Algebra Evaluation Fractions N
egatives Arithmetic Number Counting
definitions based on known objects
Euclidean Proof Deduction Definition Construction
Description Perception
Van Hiele
proceptual-
symbolic
conceptual-
embodied

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Three Worlds of Mathematics
axiomatic-
formal
formal objects based on definitions
Concept definitions - formal proof
Axiomatic Geometries
Formal Axiomatic Systems
EmbodiedNon-euclidean Geometries
Local Linearity
definitions based on known objects
Local Straightness
Euclidean Proof Deduction Definition Construction
Description Perception
Limits Manipulation Algebra Evaluation Fractions N
egatives Arithmetic Number Counting
cognitive development
blending embodiment symbolism
APOS
Van Hiele
proceptual-
symbolic
conceptual-
embodied

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Three Worlds of Mathematics
formal objects based on definitions
formal
embodied formal
symbolic formal
fully integrated
definitions based on known objects
embodied symbolic
symbolic
embodied

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Compression into thinkable concepts
Categories, Prototypes (Lakoff)
Typical in Embodiment (also arithmetic ? algebra)
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xy
Process-object (Piaget, Dubinsky, Sfard etc)
Typical in Symbolism (also embodied operation ?
effect)
e.g. addition ? sum
translation
free vector
Axioms-structures (Hilbert, Bourbaki)
Typical in Formalism
list of axioms
formal structure
Structure of Observed Learning Outcomes (Biggs,
Collis) unistructural, multi-structural,
relational, extended abstract
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Symbolic Compression
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Symbolic Compression
procedural
flexible
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Embodied Symbolic Compression
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Embodied Symbolic Compression
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Embodied Symbolic Compression
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Set-befores Met-befores
Long-term human learning is based on a
combination of facilities set-before birth in the
genes and builds on successive constructions
based on conceptions met-before in development.
set-before twoness set in visual apparatus.
consistent met-before 224 from whole numbers
is a consistent when complex numbers are
encountered.
inconsistent met-before take away makes
smaller from everyday objects and whole numbers
is an inconsistent when negative numbers and
infinite cardinals are encountered.
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Using language to blend knowledge structures
Margaret Thatcher, The Iron Lady was much
admired in the USA in the 1980s. Many would like
someone like her as a President, provoking the
following counter-argument
If Mrs Thatcher stood for President she wouldnt
get elected because the unions would oppose her.
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Using language to blend knowledge structures
If Mrs Thatcher stood for President she wouldnt
get elected because the unions would oppose her.
Generic Frame
President
President
Frame 1
Frame 2
American Voters
American Voters
American Unions
American Unions
Emergent concepts
Blend
e.g. Mrs Thatcher campaigning in Michigan
Creativity, conflicts
Mrs Thatcher never campaigned in Michigan!
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Complex numbers as a blend
a cognitive science analysis by Fauconnier
Turner, p. 273.
Generic Frame
elements in a set
field operations
President
points in the plane
President
positive negative numbers
Frame 1
Frame 2
American Voters
American Voters
geometric transformations
addition multiplication
American Unions
American Unions
complex numbers
Emergent concepts
e.g. real and imaginary parts, modulus, argument,
complex arithmetic
Blend
complex arithmetic
Creativity, conflicts
e.g. a square can be negative
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Complex numbers as a blend
Is this an intellectual top-down analysis? Is it
a good model for historical development? ... for
the cognitive development of the individual?
Generic Frame
elements in a set
field operations
President
points in the plane
President
positive negative numbers
Frame 2
American Voters
American Voters
geometric transformations
addition multiplication
American Unions
American Unions
complex numbers
Blend
complex arithmetic
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Is this an intellectual top-down analysis? Is it
a good model for historical development? ... for
the cognitive development of the individual?
My interest here is in the development of the
individual within the framework of mathematical
society

• through
• compression of complex situationsinto
  thinkable concepts using language
• connection between thinkable concepts in
  coherent knowledge structures
• blending knowledge structures in new ways,
• to build theories and solve novel problems.

President
President
American Voters
American Voters
American Unions
American Unions
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Illustrations of the Framework in Action
College Algebra
Calculus
Proof
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College Algebra
For some, audits and root canals hurt less than
algebra. Brian White hated it. It made Julie
Beall cry. Tim Broneck got an F-minus. Tina
Casale failed seven times. And Mollie Burrows
just never saw the point. This is not a
collection of wayward students, of unproductive
losers in life. They are regular people with
jobs and families, hobbies and homes. And a
common nightmare in their past. Deb
Kollar, Sacramento Bee (California), December 11,
2000.
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College Algebra
Arithmetic builds on conceptual
embodiment collecting into sets to count Putting
sets together to add Taking away elements to
subtract Sharing into equal subsets to divide
all arithmetic expressions can be calculated
Algebraic expressions cannot be calculated. What
are the embodiments for
Algebra can be a minefield of dysfunctional
met-befores
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College Algebra
5x3 13 as a process of evaluation 5x3 9x-5
as two processes? The didactic cut (Filloy
Rojano) As a balance? (a conceptual
embodiment) Makes sense for positive values
(Vlassis 2002) Negative values? Do the same
thing to both sides change sides, change
signs put the 3 in 3x6 over the other side and
put it underneath. A functional embodiment (in
the sense of Lakoff?) moving terms around without
conceptual meaning.
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College Algebra
A study of students taught to do the same thing
to both sides (Lima Tall 2006) The students
performed similarly on 5t3 8, 3x1
3x. They did not see the first equation as a
process equals number. They did not imagine the
equation as a balance. Instead, they moved the
terms around using rules which were a combination
of symbol shifting and magic through a sequence
of actions to get the answer. Consider this in
APOS theory, Lakoffs theory of embodiment,
(conceptual/functional), met-befores in
conceptual embodiment symbolism.
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Calculus
Reform Calculus focuses on 3 (or 4) different
representations graphic, symbolic, analytic,
(verbal). A three worlds view embodied
locally straight symbolic locally linear formal
epsilon-delta limit Harvard calculus locally
straight locally linear. But there is a huge
difference between the two...
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Calculus
Local straightness is embodied
You can see why the derivative of cos is minus
sine
slope zero
slope
slope
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Calculus
Local straightness is embodied
You can see why the derivative of cos is minus
sine
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Calculus
Local straightness is embodied
You can see why the derivative of cos is minus
sine
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Calculus
Local straightness is embodied
It gives non-examples of differentiability of
great insight
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Calculus
Local straightness is embodied
It gives reasons to question naïve ideas
Let n(x)bl(1000x)/1000
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Calculus
Local straightness is embodied
It makes sense of differential equations ...
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Calculus

• Hahkiöniëmi (2006) studied his own teaching in a
  framework of embodiment and symbolism.
• He found
• The embodied world offers powerful thinking tools
  for students.
• The derivative is considered as an object at an
  early stage.
• students followed different cognitive paths,
  including an embodied route, a symbolic route and
  a combination of the two.
• He questioned the use of a symbolic approach
  using a process-object theory and a too-simple
  application of Sfards notion that operational
  precedes structural.
• A structural embodiment gives a meaning for the
  object that is to be constructed accurately using
  symbols.

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Calculus
Hahkiöniëmi (2006)
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Proof
focusing on given definitions and rules of
deduction
focusing on met-befores in the embodied and
symbolic worlds

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Proof
formal
formal objects based on definitions
embodied formal
symbolic formal
fully integrated
definitions based on known objects
embodied symbolic
Verbalised Geometry
Algebraic Geometry
General Algebra
Generic Arithmetic
Generic Pictures
Specific Arithmetic
Specific Pictures
symbolic
embodied

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Proof
formal
Marcia Pinto 1998
formal objects based on definitions
embodied formal
symbolic formal
fully integrated
definitions based on known objects
embodied symbolic
Verbalised Geometry
Algebraic Geometry
General Algebra
Graphs
Generic Arithmetic
Generic Pictures
Generic Pictures
Specific Arithmetic
Specific Pictures
Keith Weber 2003
symbolic
embodied

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From formal proof back to embodiment
proceptual symbolism
Structure theorems take us back from axiomatic
formalism to conceptual embodiment and proceptual
symbolism
An equivalence relation on a set A corresponds
to a partition of A A finite dimensional
vector space over a field F is isomorphic to
Fn Every finite group is isomorphic to a group
of permutations Any two complete ordered
fields are isomorphic (to R).
In every case, the structure theorem tells us
that the formally defined concept has an embodied
meaning, and (in 3 cases) a symbolic meaning for
manipulation and calculation.
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From formal proof back to embodiment
proceptual symbolism
Formal mathematical thinking is supported by
met-befores from embodiment and symbolism. e.g.
the number line as an embodiment of R. Using
Dedekind cuts to fill in the irrationals
intimates there is no room for infinitesimals.
Formal Theorem Let K be any ordered field
extension of R, then, for x ?K, either xgtr ?r?R,
or xltr ?r ?R, or xae where a ?R and e is
infinitesimal (i.e. -rlt e ltr for all positive r ?
R).
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From formal proof back to embodiment
proceptual symbolism
Formal Theorem Let K be any ordered field
extension of R, then, for k ?K, either kgtr ?r?R,
or kltr ?r ?R, or kae where a ?R and e is
infinitesimal (i.e. -rlt e ltr for all positive r ?
R).
Proof Suppose k lies between b, c ?R, where
bltkltc. Let S r ?R rltk, then b ?S, and c is
upper bound for S. Let a be the least upper bound
for S and define e ka. Then k ae. Now show
that e is an infinitesimal by using the
properties for the definition of least upper
bound in R.
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From formal proof back to embodiment
proceptual symbolism
Formal Theorem Let K be any ordered field
extension of R, then, for k ?K, either kgtr ?r?R,
or kltr ?r ?R, or kae where a ?R and e is
infinitesimal (i.e. -rlt e ltr for all positive r ?
R).
Corollary If k is any finite element of K, i.e.
bltkltc (b, c ?R.) Then k ae where a is real and
e is infinitesimal.
Definition If k is any finite element of
K,where k ae (a real, e infinitesimal)then
the standard part of k is defined to be st(k) a.
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From formal proof back to embodiment
proceptual symbolism
The embodiment of K. For any a,d ?K, define the
(symbolic) transformation m K?K, to be m(x)
(xa)/d.
K
K
What happens to fixed x, for smaller d?
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From formal proof back to embodiment
proceptual symbolism
The embodiment of K. For any a,d ?K, define the
(symbolic) transformation m K?K, to be m(x)
(xa)/d.
K
K
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From formal proof back to embodiment
proceptual symbolism
The embodiment of K. For any a,d ?K, define the
(symbolic) transformation m K?K, to be m(x)
(xa)/d.
ae
K
K
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From formal proof back to embodiment
proceptual symbolism
The embodiment of K. For any a,d ?K, define the
(symbolic) transformation m K?K, to be m(x)
(xa)/d.
ae
K
K
m(ae)
m(a)
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From formal proof back to embodiment
proceptual symbolism
The embodiment of K. For any a,d ?K, define the
(symbolic) transformation m K?K, to be m(x)
(xa)/d.
ae
K
R
m(ae)
m(a)
st( )
st( )
Now take the standard part of the image.
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From formal proof back to embodiment
proceptual symbolism
The embodiment of K. For any a,d ?K, define the
(symbolic) transformation m K?K, to be m(x)
(xa)/d.
ae
K
R
m(ae)
m(a)
st( )
st( )
Give the original name to the image of each point.
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From formal proof back to embodiment
proceptual symbolism
The embodiment of K. For any a,d ?K, define the
(symbolic) transformation m K?K, to be m(x)
(xa)/d.
ae
K
a2e17e2
R
a2e
ae
a
Give the original name to the image of each point.
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From formal proof back to embodiment
proceptual symbolism
In this way, new embodiments are developed from
formal structure theorems. This has happened
before in history. e.g. the algebraic solutions
of equations made no embodied sense initially. A
century later the complex plane gave a meaningful
embodiment. The symbolism works in a broader
context with a new blend of formalism, symbolism
and embodiment.
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The Three Worlds of Mathematics as a Framework
for Mathematical Thinking
Advanced Mathematical Thinking
formal objects based on definitions
formal
definitions based on known objects
embodied formal
symbolic formal
fully integrated
Cognitive development blending met-befores through
compression to connection between thinkable
concepts
embodied symbolic
symbolic
embodied
Mathematicians live in a blend of the three
worlds, with individual preferences for aspects
of embodiment, symbolism and formalism.

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