Mental imagery in Computer Science Alain Finkel, LSV, ENS Cachan - - PowerPoint PPT Presentation

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Mental imagery in Computer Science Alain Finkel, LSV, ENS Cachan & CNRS France ECSS'2009 Mental Imagery in CS 1 The thesis Learning, understanding, memorizing, Hence thinking is facilitated by explicit construcHon of mental

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Mental imagery in Computer Science

Alain Finkel, LSV, ENS Cachan & CNRS ‐ France

ECSS'2009 1 Mental Imagery in CS

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The thesis

  • Learning, understanding, memorizing,…
  • Hence thinking

is facilitated by explicit construcHon of mental

  • bjects/representaHons and explicit

manipulaHons through mental imagery.

ECSS'2009 Mental Imagery in CS 2

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Mental images versus real images

  • Images, photos in the external world seem

real but…they are not always (specially now):

ECSS'2009 Mental Imagery in CS 3

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ECSS'2009 4 Mental Imagery in CS

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The reality of mental imagery…

Have we really images in the mind ? In the brain ? May be, we have formula in the mind but we have the feeling to see something,… Etc…. Philosophical quesHons with no definiHve answer… How to increase the quality of teaching, the knowledge of students and why not the quality of research communicaHons ? AVract more students for scienHfic studies…

ECSS'2009 Mental Imagery in CS 5

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Plan

  • 1. Mental imagery: generaliHes
  • 2. Examples in mathemaHcs
  • 3. Examples in CS
  • 4. The MI of genious research
  • 5. Conclusions

ECSS'2009 6 Mental Imagery in CS

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Fact 1: Mental Imagery “exist” and also some laws.

ParHal proof: mental rotaHon (Vandenberg) and mental deplacements (Kosslyn)

ECSS'2009 7 Mental Imagery in CS

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Test of mental rotaHon

ECSS'2009 8 Mental Imagery in CS

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Test of mental rotaHon

ECSS'2009 9 Mental Imagery in CS

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Test of mental rotaHon

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Test of mental rotaHon

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Kosslyn’s experience, 1978

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Fact: « mental speed » is constant A B

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Fact 2: MI and percepHon share mental ressources

proofs: psychological & physiological

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Perky’s experience, 1910

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Mutual exclusion: it is difficult to see outside and inside in the same time

Mental Imagery in CS

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Mellet’s experience, 1995

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Observing the brain during MI and Perception: = and ≠

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Dog, fly and elephant in the mind

  • Imagine a dog, and then add a fly

Time = c

  • Imagine a dog and then add an elephant

Time = c + 200 ms

  • Kosslyn (1975)

ECSS'2009 Mental Imagery in CS 16

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proof: Paivio 1970 image+verbal > verbal

ECSS'2009 17 Mental Imagery in CS

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Pre‐conclusion on MI

  • 1. MI “exist”: mental rotaHon, mental speed.
  • 2. MI versus percepHon. Mutual exclusion,

mental resources, aVenHon.

  • 3. Double coding is beVer than unique one.

ECSS'2009 Mental Imagery in CS 18

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Plan

  • 1. Mental imagery: generaliHes
  • 2. Examples in mathemaHcs
  • 3. Examples in CS
  • 4. The MI of genious researchers
  • 5. Conclusions

ECSS'2009 19 Mental Imagery in CS

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Algebraic idenHHes

  • Theorem: (a + b)2 = a2 + 2ab + b2
  • Verbal proof:

(a + b)2 = (a + b) (a + b) by definiHon = a2 + ab + ba + b2 by the rules of computaHon = a2 + 2ab + b2 by commutaHvity of mulHplicaHon

  • MemorizaHon: by memorizing the proof, by

repeHHon of the melody, by seing it in the mind and outside,…

ECSS'2009 Mental Imagery in CS 20

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Visual proof

ECSS'2009 Mental Imagery in CS 21

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Concrete proposal

  • Present to students the two mental objects

(words and images) for increasing understanding and memorizaHon.

ECSS'2009 Mental Imagery in CS 22

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(a + b)3 = a3 + 3a2b + 3ab2 +b3

  • Verbal proof and verbal memorizaHon: as

usual, symbolic computaHon.

  • Not so easy to memorize.

ECSS'2009 Mental Imagery in CS 23

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It is also possible to see and only aier to compute

ECSS'2009 Mental Imagery in CS 24

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Verbal computaHon or visual manipulaHon ?

  • At the beginning: words or images ?
  • Difficult to know
  • Let us use both.

ECSS'2009 Mental Imagery in CS 25

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S(n) = 1 + 2 + 3 + … + n

  • Proof of S(n) = n(n+1)/2 is possible by recurrence

but with no intuiHon, no understanding.

  • Verbal/algebraic proof, playing with the formula

(Gauss find) S(n) = 1 + 2 + 3 + …+ (n ‐ 1) + n S(n) = n + (n ‐ 1) + …+ 3 + 2 + 1 2xS(n) = (n+1)+(n+1)+…+(n+1) = n(n + 1) = n(n+1)

ECSS'2009 26 Mental Imagery in CS

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ECSS'2009 27

Visual idea…

1 2 3 4 n

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ECSS'2009 Mental Imagery in CS 28

and proof

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S(n) = 1 + 3 + 5 +… + (2n ‐ 1)

  • Verbal: S(n) = ? Easy inducHon for S(n) = n2

but how to find S(n) = n2 ?

  • Image:

ECSS'2009 29 Mental Imagery in CS

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Types of definiHon: a funcHon f is a …

  • Verbal (≠types):

– Triple f=(X,Y,x‐‐>y) such that

for all x,y,z (f(x)=y and f(x)=z) implies y=z

– Subset f of XxY saHsfying:

1. for all x, there exists at most an y s.t. (x,y) ∈ f 2. for all x,y,z, if (x,y) ∈ f and (x,z) ∈ f then y=z

– Formula f(x,y) saHsfying: for all x,y,z, (f(x,y) and f(x,z)) implies y=z

  • Image:

– Curb… but not every funcHon is representable by a (visible) curb and some curbs are not funcHons – Graph: every node in X has at most a successor in Y (ok for finite graphs and if X=Y).

ECSS'2009 30 Mental Imagery in CS

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A funcHon is…

  • Mixte: verbal+image+movements

– All the previous examples: words + images + feelings – Process which takes input x, works on x and produces output y – Algorithm compuHng f(x) – Other….

  • Examples
  • Difficult to see: f(real\raHonal)=1 et f(raHonal)=0
  • Verbal, sequenHal view: g(n)=0 si Tn stops (in n steps) on input n.
  • InteresHng to see: f(x)=x, f(x)=x2, f’(x)=x,…
  • Use mixted representaHons !

ECSS'2009 31 Mental Imagery in CS

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Plan

  • 1. Mental imagery: generaliHes
  • 2. Examples in mathemaHcs
  • 3. Examples in CS
  • 4. The MI of genious researchers
  • 5. Conclusions

ECSS'2009 32 Mental Imagery in CS

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Graphs and finite automata

Graph:

– relaHon (edges) between a set E of verHces. – adjacency matrix – picture of edges linking a finite number of verHces – Abstract or concrete graph, labyrinth – examples (metro,…)

Finite automaton:

– 5‐uple (E,A,δ,e0,F) – (Abstract) labelled graph – (Concrete) labelled graph: labyrinth with rooms, named one‐way corridors, entrance and exit rooms – A machine which produces outputs, which reads inputs, both.

ECSS'2009 33 Mental Imagery in CS

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Regular langages

  • Recognized by an automaton: image‐movements for seing

words…

  • Generated by a regular grammar : verbal‐sequenHal, possible

to add a tree (image)

  • Described by a regular expression: verbal, staHc.

ECSS'2009 34 Mental Imagery in CS

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Pumping lemma

Lemma: Let L be a regular language. There exists kL ∈ N such that any word w ∈ L, of length larger than kL, can be factored w = xuy , with u non empty and xuny ∈ L for all n ∈ N. Proof:

  • from the regular expression ?
  • From the regular grammar ?
  • BeVer by using the Kleene Theorem for the bridge

between algebra and machines and then by using the visual/kinesthesic representaHon of an automaton as a graph.

ECSS'2009 35 Mental Imagery in CS

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Bridges

  • Kleene theorem: FA = REG, a bridge between

– machines (automata) and algebra (languages) – concrete objects (graphs) and abstract objects (subsets) – images (graphs, machines) and words (formula)

  • Büchi theorem: MSO = REG, a bridge between

– Logics (verbal) and automata (graphs, machines)

ECSS'2009 Mental Imagery in CS 36

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Other bridges

  • Ginsburg & Spanier: SL = Presburger logics

– Geometry (visual), algebra (verbal), numbers (concrete objects) and logics (formula, equaHons, abstract words)

  • Rabin, Comon, Wolper: Presburger Logics ⊆ FA

– Logics and automata (hence algorithmics for logics).

  • REG(Np)= Presburger logics

– Algebra and logics and more or less complex bridges.

ECSS'2009 Mental Imagery in CS 37

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Plan

  • 1. Mental imagery: generaliHes
  • 2. Examples in mathemaHcs
  • 3. Examples in CS
  • 4. The MI of genious researchers
  • 5. Conclusions

ECSS'2009 38 Mental Imagery in CS

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Mental images from 2000 years

  • Avant JC: Platon, Aristote, Epictète…
  • 1600: Descartes
  • 1700‐1800: Locke, Hume, Berkeley, Kant,…
  • 1890‐1950: Husserl, Heidegger, Merleau‐Ponty, Proust, Freud
  • 1900‐1930: Tichener, Binet (psycho. IntrospecHve) : profils visuels, audiHfs, moteurs
  • 1910: Perky (conflit entre percepHon et évocaHon)
  • 1930: Pavlov, Skinner, Watson (comportementalisme)
  • 1933: SémanHque générale (Korzibsky)
  • 1936: Turing, thèse de Church
  • 1940‐60: Ecole de Piaget
  • 1956: Naissance des sciences cogniHves, Miller (7+2), Galanter, Pribram
  • 1956‐60: Naissance des thérapies cogniHves: Beck et Ellis (USA)
  • 1970‐1980: Paivio, Kosslyn, Pinker, Denis (double codage V/A)
  • 1985: Gardner (intelligences mulHples)
  • 1986: Baddeley (Calepin visuo‐spaHal et boucle phonologique, aVenHon)
  • 1995: Damasio, Goleman
  • 2009: Berthoz, Dehaene, Mellet,…

ECSS'2009 39 Mental Imagery in CS

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Aristote

« Jamais l’âme ne pense sans image » « Never the mind thinks without image »

ECSS'2009 40 Mental Imagery in CS

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Descartes

Prefers words than image and created the analyHc geometry, i.e. solving geometric problems by solving equaHons !

ECSS'2009 Mental Imagery in CS 41

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Einstein

No words during his thinking And a lot of MI in his texts

« les mots et le langage, écrits ou parlés, ne semblent pas jouer le moindre rôle dans le mécanisme de ma pensée »

ECSS'2009 42 Mental Imagery in CS

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Poincaré

The converse !

ECSS'2009 Mental Imagery in CS 43

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Bourbaki

Words > images AxiomaHsaHon, … No image in the books !

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Benoît Mandelbrot

Images > words He leaved the ENS because he felt the place too much verbal and he created the geometry of fractals

ECSS'2009 Mental Imagery in CS 45

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Plan

  • 1. Mental imagery: generaliHes
  • 2. Examples in mathemaHcs
  • 3. Examples in CS
  • 4. The MI of genious researchers
  • 5. Conclusions

ECSS'2009 46 Mental Imagery in CS

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What is a mental representaHon?

RepeHHon is necessary…

  • An image, a movement, a sound, linked to a

mathemaHcal concept or proof

  • Several types of representaHons, adapted to various

people

– Visual, visio spaHal: mental images or designs – AudiHve, verbal, phonologic: a sound, a poetry, a sentence,... – KinestheHc: a movement, feeling, emoHon,…

ECSS'2009 47 Mental Imagery in CS

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I understanding when I am able…

– To manipulate symbols and words (computaHon, rewriHng rules, apply knowledge,…)

And

– To create and manipulate mental objects through MI.

And

– To make connecHons between both !

ECSS'2009 Mental Imagery in CS 48

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An researcher

  • Is able to create MR and to manipulate them
  • Finds pleasure with/in his mental world
  • Mental world may become very important

then he someHmes ignores external world (mutex)

  • May ignore that other people are not like her.

ECSS'2009 49 Mental Imagery in CS

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What is the use of a mental representaHon?

  • Very important to get an intuiHon, to understand,

to remember

  • Good representaHons are hard to find so share

them !

  • One representaHon is only useful for a part of

students; one needs several representaHons of various types.

  • Good representaHons oien need some effort to

be used

  • RepresentaHons do NOT replace algorithms and

methods; they help to learn and use them

ECSS'2009 50 Mental Imagery in CS

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« ObservaHons »

  • Understand ≠ memorise
  • Understand a definition = to build a well-adapted

representation, to manipulate it (Create, test, complete our representations)

  • well-adapted = multiples codings: verbal, visual,

auditive, feeling, movement, kinesthesic, parameters

  • f attention.
  • Understand a proof ≠ follow the proof line after line +

build a well-adapted MR

  • Prove = create the proof
  • Teach = give few MR

ECSS'2009 51 Mental Imagery in CS

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Training

  • 2h: a seminar
  • 2 days to train a parHcular subject: MI, pedagogical communicaHon,

emoHons, memory, moHvaHon,…

  • 20 days (janv. 09 ‐ sept. 09): ACTA

ECSS'2009 52 Mental Imagery in CS

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Thank you

Conference: 11th ICME’2008, InternaHonal Congress on MathemaHcal EducaHon, Mexico. Journal: IJMEST’2009. Texts in hVp://plato.stanford.edu/ – hVp://plato.stanford.edu/entries/mental‐imagery/ – hVp://plato.stanford.edu/entries/mental‐representaHon/ Jacques Hadamard, a pre‐cogniHve psychologist « essai sur la psychologie de l’invenHon dans le domaine mathémaHque » – 1945, Princeton.

ECSS'2009 53 Mental Imagery in CS

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Numberous connecHons between CS and psycho

  • Virtual percepHon may replace/allow to develop MI for students

without good MR.

  • Turing created TM parHally with introspecHon (mulHplicaHon ?) and

cogniHve psychology used CS (in 1960).

  • Extending the Vision: Images in History, Colson, F. and Hall, W.

(1990)

  • Bases neurocogniHves des stratégies de navigaHon chez l'Homme,

PhD thesis under supervision of Berthoz

– Allocentré: hypocampe droit, space – Egocentrée: hypo gauche, Hme sequence – Gps: good collaboraHon between words and images

  • Virtual therapy….
  • Brain‐computer…

ECSS'2009 Mental Imagery in CS 54

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Verbal Logic Symbolic Visual Auditive Kinesthesic

Concrete

Mental Représentation

Action Conceptual

ECSS'2009 55 Mental Imagery in CS