How Individuals Develop Advanced Mathematical Thinking David Tall Emeritus Professor in Mathematical...

How Individuals Develop Advanced Mathematical Thinking David Tall Emeritus Professor in Mathematical Thinking

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 a>b>c the picture of three points following one another on a straight line as the geometrical picture of the idea between? Hilbert 1900 ICM lecture. Undergraduate mathematics education includes the transition from the geometric and symbolic mathematics at school to the formal constructions of axiomatic systems in mathematics research.

conceptual-embodied (based on perception of and reflection on properties of objects); 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 3+4 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. 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);

formal embodied symbolic Euclidean Proof Deduction Definition Construction Description Perception Limits Manipulation Algebra Evaluation Fractions Negatives Arithmetic Number Counting definitions based on known objects formal objects based on definitions [Van Hiele] axiomatic conceptual- proceptual- Three Worlds of Mathematics Advanced Mathematical Thinking

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

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

Compression into thinkable concepts Categories, Prototypes (Lakoff) Process-object (Piaget, Dubinsky, Sfard etc) Axioms-structures (Hilbert, Bourbaki) Structure of Observed Learning Outcomes (Biggs, Collis) unistructural, multi-structural, relational, extended abstract Typical in Embodiment (also arithmetic algebra) Typical in Symbolism (also embodied operation effect) Typical in Formalism 3+4x+yx+y e.g. addition sum free vector translation list of axiomsformal structure

Symbolic Compression SOLO unistructural multi- structural relational extended abstract (uni-structural at next level) procept theory single procedure multi- procedure processprocept APOSActionProcessObject

Proceptual Divide Symbolic Compression procedural flexible

Embodied & Symbolic Compression

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: 2+2=4 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.

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: Using language to blend knowledge structures If Mrs Thatcher stood for President she wouldnt get elected because the unions would oppose her.

President American Voters American Unions Prime Minister British Voters British Unions President American Voters American Unions President American Voters American Unions Mrs Thatcher American Voters American Unions Elected Leader Voters Organized labour Generic Frame Blend [e.g. Mrs Thatcher campaigning in Michigan] Conceptual Blending and the Minds Hidden Complexities Fauconnier & Turner 2002 Emergent concepts Creativity, conflicts Mrs Thatcher never campaigned in Michigan! Frame 1Frame 2 Using language to blend knowledge structures

Complex numbers as a blend points in the plane geometric transformations positive & negative numbers addition multiplication President American Voters American Unions President American Voters American Unions complex numbers elements in a set field operations Generic Frame Blend Emergent concepts [e.g. a square can be negative] Creativity, conflicts Conceptual Blending and the Minds Hidden Complexities Fauconnier & Turner 2002 complex arithmetic e.g. real and imaginary parts, modulus, argument, complex arithmetic a cognitive science analysis by Fauconnier & Turner, p. 273. Frame 1Frame 2

Complex numbers as a blend President American Voters American Unions President American Voters American Unions Is this an intellectual top-down analysis? Is it a good model for historical development?... for the cognitive development of the individual? points in the plane geometric transformations positive & negative numbers addition multiplication complex numbers elements in a set field operations Generic Frame Blend complex arithmetic Frame 2

President American Voters American Unions President American Voters American Unions 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 situations into 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.

Illustrations of the Framework in Action College Algebra Calculus Proof

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.

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 2a+3b+4a,3a4b,2+3x,4x2 = 2x+3 College Algebra

5x+3 = 13 as a process of evaluation 5x+3 = 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 3x=6 over the other side and put it underneath. A functional embodiment (in the sense of Lakoff?) moving terms around without conceptual meaning. 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 = 3+x. 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. College Algebra

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...

slope + slope zero slope Local straightness is embodied: You can see why the derivative of cos is minus sine Calculus

Local straightness is embodied: You can see why the derivative of cos is minus sine Calculus

Local straightness is embodied: You can see why the derivative of cos is minus sine The graph of sine upside down.... slope is sin x Calculus

Local straightness is embodied: It gives non-examples of differentiability of great insight Calculus

Local straightness is embodied: It gives reasons to question nave ideas: Let n(x)=bl(1000x)/1000 Calculus

Local straightness is embodied: It makes sense of differential equations... Calculus

Hahkinimi (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. Calculus

Hahkinimi (2006): Calculus

focusing on met- befores in the embodied and symbolic worlds focusing on given definitions and rules of deduction Proof

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

embodied definitions based on known objects formal objects based on definitions embodied formal symbolic formal fully integrated fully integrated symbolic embodied symbolic Generic Pictures Specific Pictures Specific Arithmetic Generic Arithmetic General Algebra Verbalised Geometry Generic Pictures Algebraic Geometry Graphs formal NATURAL based on concept image FORMAL based on concept definition Marcia Pinto 1998 PROCEDUAL learnt by rote Keith Weber 2003 Concept DefinitionFormal proof Proof

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 F n ; 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. Structure theorems take us back from axiomatic formalism to conceptual embodiment and proceptual symbolism 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. From formal proof back to embodiment & proceptual symbolism Formal Theorem Let K be any ordered field extension of R, then, for x K, either x > r r R, or x < r r R, or x=a+ where a R and is infinitesimal (i.e. -r < < r for all positive r R).

From formal proof back to embodiment & proceptual symbolism Formal Theorem Let K be any ordered field extension of R, then, for k K, either k > r r R, or k < r r R, or k=a+ where a R and is infinitesimal (i.e. -r < < r for all positive r R). Proof Suppose k lies between b, c R, where b < k < c. Let S = {r R | r < k}, then b S, and c is upper bound for S. Let a be the least upper bound for S and define = k a. Then k = a+ . Now show that is an infinitesimal by using the properties for the definition of least upper bound in R.

From formal proof back to embodiment & proceptual symbolism Formal Theorem Let K be any ordered field extension of R, then, for k K, either k > r r R, or k < r r R, or k=a+ where a R and is infinitesimal (i.e. -r < < r for all positive r R). Corollary If k is any finite element of K, i.e. b < k < c (b, c R.) Then k = a+ where a is real and is infinitesimal. Definition If k is any finite element of K, where k = a+ (a real, infinitesimal) then the standard part of k is defined to be st(k) = a.

From formal proof back to embodiment & proceptual symbolism The embodiment of K. For any a,d K, define the (symbolic) transformation m: KK, to be m(x)= (xa)/d. 01 a a+d 01 m(x)m(x) x K K What happens to fixed x, for smaller d?

From formal proof back to embodiment & proceptual symbolism The embodiment of K. For any a,d K, define the (symbolic) transformation m: KK, to be m(x)= (xa)/d. 01 a a+d 01 m(x)m(x) x K K

From formal proof back to embodiment & proceptual symbolism The embodiment of K. For any a,d K, define the (symbolic) transformation m: KK, to be m(x)= (xa)/d. 01 a a+ 01 x K K

From formal proof back to embodiment & proceptual symbolism The embodiment of K. For any a,d K, define the (symbolic) transformation m: KK, to be m(x)= (xa)/d. 01 a a+ K K m(a)m(a) m(a+ )

From formal proof back to embodiment & proceptual symbolism The embodiment of K. For any a,d K, define the (symbolic) transformation m: KK, to be m(x)= (xa)/d. 01 a a+ m(a)m(a) m(a+ ) st( ) K R Now take the standard part of the image. This is the familiar real line This is a line with infinitesimals

From formal proof back to embodiment & proceptual symbolism The embodiment of K. For any a,d K, define the (symbolic) transformation m: KK, to be m(x)= (xa)/d. 01 a a+ m(a)m(a) m(a+ ) st( ) K R This is the familiar real line This is a line with infinitesimals Give the original name to the image of each point.

From formal proof back to embodiment & proceptual symbolism The embodiment of K. For any a,d K, define the (symbolic) transformation m: KK, to be m(x)= (xa)/d. 01 a a+ a a+2 a+2 +17 2 K R This is the familiar real line This is a line with infinitesimals Give the original name to the image of each point.

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.

formal embodied definitions based on known objects formal objects based on definitions The Three Worlds of Mathematics as a Framework for Mathematical Thinking embodied formal symbolic formal fully integrated fully integrated symbolic embodied symbolic Mathematicians live in a blend of the three worlds, with individual preferences for aspects of embodiment, symbolism and formalism. Cognitive development blending met-befores through compression to & connection between thinkable concepts Advanced Mathematical Thinking

How Individuals Develop Advanced Mathematical Thinking David Tall Emeritus Professor in Mathematical...

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