research into the use of visualization and proof in ... · david tall, natural and formal 5 an...

David Tall, Natural and Formal 1

Research into the use of visualization and proofin calculus and mathematical analysis.

David Tall

Professor in Mathematical Thinking,Mathematics Education Research Centre,

University of Warwick, CV4 7AL [email protected]

David Tall is the lead author on a new review of‘Technology and Calculus’. He will discussresearch into the effectiveness of new ways ofteaching the calculus and their relationship withthe more formal aspects of analysis. This willinclude a range of related ideas, from the brainactivity that underlies visual and formal thinkingto its manifestion in a variety of visual and formalapproaches to the subject.

Ideas are from four main sources:

Tall, D, Smith, D, and Piez C. (in preparation). A very rough draft of ‘Technology andCalculus’ (as above).

Tall, D. (2000). Biological Brain, Mathematical Mind & Computational Computers (howthe computer can support mathematical thinking and learning), Proceedings of ACTM,Chang Mai, Thailand.

David Tall & Marcia Maria Fusaro Pinto (2001). Following students’ development in atraditional university classroom, PME 25.

David Tall (in press), Natural and Formal Infinities, to appear in Educational Studies inMathematics.

All available through the link Papers for USA September 2001 at

www.warwick.ac.uk/staff/David.Tall (or via davidtall.com)


Concept definitions and deductions

Formaleg vector spaces

linear maps

Technicaleg Rn

matrices

GraphicNumeric Symbolic

Expertviewpoint

Cognitive reconstruction

Cognitive expansionwith some reconstruction

CognitiveDevelopmentof Students

Two types of advanced mathematical thinkingas exemplified in linear algebra,

(also occurring in calculus-analysis, group theory, geometry etc)


axioms/definitions

Formal ideas

Examples/images CONCEPTDEFINITION

CONCEPT IMAGE

FORMALDEDUCTIONS

proof: theorem:mental conception:

THOUGHTEXPERIMENTS

Natural experiences

Some constituents in constructing a formal theory


Natural experiences

axioms/definitions

Formal ideas

Examples/images

structure theorem

formal embodiment

CONCEPTDEFINITION

CONCEPT IMAGE

FORMALDEDUCTIONS

proof: theorem:mental conception: formal embodiment:

THOUGHTEXPERIMENTS

Building new formal embodiments from a formal theory


An example of an embodied formal concept

Define an ordered field F…

R is an ordered field that satisfies the completenessaxiom.

Consider R Ã F.

Define x Œ F to be an infinitesimal iff

–r < x < r for all positive r Œ R

and x is infinite if either x > r for all r Œ R, or x < r

for all r Œ R,

STRUCTURE THEOREM

An element a in an ordered field F … R is either

infinite or a = a+e where a Œ R and e is infinitesimal.ProofIf a is not infinite, then b < a < c for b, c Œ RLet S = { x Œ R | x < a}

S is non-empty (b Œ S) and bounded above by c. So,

by completeness of R, S has l.u.b. a Œ R. Now show

e = a – a is infinitesimal.


Embodying the structure theorem.–1/ε

(negativeinfinite

quantity)

1/ε(positiveinfinite

quantity)

c–ε c+ε

c

An ordered field F with infinitesimals and infinite elements that seem difficult to see!

Define m:FÆF by

me

( )xx c

=-

.

0

c

c–ε

µ

c+ε

µ(c–ε) µ(c+ε)µ(c)

–1 1

Visualising the function m as a mapping of a number line with infinitesimals

c

c–ε

µ

c+ε

c–ε c+εc


Natural and formal learners(Pinto 1998, Pinto and Tall, 2001)

Formal learners essentially construct the theory bydeduction, coping with the great cognitive strain asbest they can, producing a deductive formal theory.Natural learners—working from their conceptimagery—reconstruct it taking account of moregeneral ideas met in the course. They must thendevelop the formal theory from their reconstructedimagery, producing a formal theory integrating bothimagery and deduction.

Formal learning Natural learning1 Initialobstacles

Based on conceptdefinition, so may beproblematic either(a) disjoint from images,but partial procedures(b) attempt to link toimages, but weak links

Informal (based on conceptimage) so may have(a) formalism rejected,maintaining images(b) formalism embedded in

informal knowledge, withsome conflict

2 TheoryBuilding

Formal construction oftheory

Formal reconstruction (withsome conflict)(a) Thought experiments,reconstructing images(b) Deductions reconstructingformal theory

3 Formal theory Formal (deductive) Formal (integrated)

Natural and formal routes to learning formal mathematics


Sequences Series Continuity Derivative FinalInterview

1. Initial obstacles Rolf (a)Robin(a& b)

Rolf (a)Robin(b)

[Rolfwithdrew]Robin (b)

Robin Robin2. FormalConstruction Ross Ross

Ross Ross3. Formal(deductive) Ross

Students following an essentially formal route

Sequences Series Continuity Derivative FinalInterview

1. Initial obstacles Cliff (a)Colin (b)

Cliff (a)Colin (b)

Cliff (a) Cliff (a) Cliff (a)

Colin (a) Colin (b) Colin (b)2. FormalReconstruction Chris

(a&b)Chris(a&b)

Chris(a&b)

Chris3. Formal(deductive) Chris

Students following an essentially natural route


WHAT HAPPENS IN THE BRAIN?

Houdé et al (2000). Shifting from the perceptualbrain to the Logical Brain: The Neural Impact ofCognitive Inhibition Training. Journal of CognitiveNeuroscience 12:4 712–728.


• math phobia amongst many learners

• visual phobia amongst formal mathematicians.

e.g. Bourbaki:

Logical analysis was central. A mathematician had tobegin with solid first principles and deduce all therest from them. The group stressed the primacy ofmathematics among sciences, and also insisted upona detachment from other sciences. Mathematics wasmathematics – it could not be valued in terms of itsapplications to real physical phenomena. And aboveall, Bourbaki rejected the use of pictures. Amathematician could always be fooled by his visualapparatus. Geometry was untrustworthy. Mathematics should be pure, formal, austere.

(Gleick, 1987, p. 89)

When I came into this game, there was a totalabsence of intuition. One had to create an intuitionfrom scratch. Intuition as it was trained by the usualtools—the hand, the pencil and the ruler—foundthese shapes quite monstrous and pathological. Theold intuition was misleading. … I’ve trained myintuition to accept as obvious shapes which wereinitially rejected as absurd, and I find everyone elsecan do the same.

(Mandelbrot, quoted in Gleick, 1987, p. 102)

Gleick, J. (1987). Chaos. New York, NY: Penguin.


A few words about the Calculus Research

Approx 700,000 students enrolled in calculus,including about 100,000 in Advanced Placementprograms in high school. (In many colleges,mathematics majors are drawn largely from thosewho complete calculus before going to college.)

Kenelly and Harvey (1994)

12,820 students graduated in mathematics, < 2% ofthe calculus cohort. (NCES, 2001)

How many of this 2% become formal mathematicalthinkers?

The need for an approach to calculus suitable for the98%+ who do not become formal mathematicians.

Are the various reform approaches which use apragmatic technical approach as effective as atraditional approach for different student needs?


Embodied Local Straightness& Formal Local Linearity

‘Local straightness’ is a primitive human perceptionof the visual aspects of a graph. It has globalimplications as the individual looks along the graphand sees the changes in gradient, so that the gradientof the whole graph is seen as a global entity.

Local linearity is a symbolic linear approximation tothe slope at a single point on the graph, having alinear function approximating the graph at that point.It is a mathematical formulation of gradient, takenfirst as a limit at a point x, and only then varying x toget the formal derivative. Local straightness remainsat an embodied level and links readily to the globalview.


A graph which nowhere looks straight

It is the sum of saw-teeth

s(x)= min(d(x), 1– d(x)), where d(x) = x–INTx,

sn(x)=s(2n-1x)/ 2n-1 .

bl(x) = s1(x)+ s2(x)+ s3(x)+ …


A ‘smooth-looking curve’ that magnifies ‘rough’.

n(x)=bl(1000x)/1000

sinx is differentiable everywhere

sinx + n(x) is differentiable nowhere!


The gradient of cosx (drawn with Blokland et al (2000)

• an ‘embodied approach’.• it can be linked directly to numeric and

graphic derivatives, as required.• it fits exactly with the notion of local

straightness.• it uses enactive software to build up the

concept in an embodied form.


Local Straightnessand the solution of Differential Equations

A generic organiser to build a solution of a first order differential equation by hand, (Blokland et al, (2000)).


ContinuityThe blancmange graph with a rectangle selected to bestretched to fill the screen:


Embodied definition: A real function is continuous ifit can be pulled flat.

Draw the graph with pixels height 2e, imagine (a, f(a)) in

the middle of a pixel. Find an interval a–d to a+d in which

the graph lies inside the pixel height f(a)±e.

Example: f(x)= sinx pulled flat from .999 to 1.001:

Area under sinx from 1 to 1.001 stretched horizontally

The Fundamental Theorem of Calculus embodied.(Think about it!)


Embodied area and formalRiemann Integration

The area function under the blancmange and thederivative of the area (from Tall, 1991b)The embodied notions of ‘area’ and ‘area-so-far’ ascognitive roots can support Riemann and evenLebesgue integration. For further detail, see Tall(1985, 1991a, 1992, 1993, 1995, 1997), These maybe downloaded from the web-site:http://www.warwick.ac.uk/staff/David.Tall


The area function for the discontinuous function x–int(x)calculated from 0.

The area function magnified.


INTEGRATING HIGHLY DISCONTINUOUS FUNCTIONS

eg f(x)=x for x rational, f(x)=1–x for x irrational.

Idea: if (xn) is a sequence of rationals x n = an/bn

tending to the real number x, then if x is rational, thesequence (xn) is ultimately constant and equal to xotherwise the denominators bn grow without limit.

Definition: x is (eeee–N)-rational if the sequence ofrationals is computed by the continued fractionmethod and, as soon as |x – an/bn| < e, then bn < N.

Fix, say e=10 –8, N = 104 and define x to be pseudo-

rational if it is (e–N)-rational, otherwise it is pseudo-irrational.


The (pseudo-) rational area (rational step, midpoint)

The (pseudo-) irrational area (random step)


Reflections

• Visual and Formal Thinking

• Natural, Formal, and Natural Formalist thinkers

• The genuine needs of learners and society.

Natural experiences

axioms/definitions

Formal ideas

Examples/images

structure theorem

formal embodiment

CONCEPTDEFINITION

CONCEPT IMAGE

FORMALDEDUCTIONS

proof: theorem:mental conception: formal embodiment:

THOUGHTEXPERIMENTS

research into the use of visualization and proof in ... · david tall, natural and formal 5 an...

Documents