Transcript Working memory and executive functions

Embodied cognition, verbal
redescription, and conceptual
metaphors in mathematics and
science education
Paul Leseman
Utrecht Summer School
August 22, 2013
Affection
Important
The problem of (linguistic) meaning
(and intention)
• Semantics (and pragmatics) is in cognitivist
approaches a neglected topic.
– Tautological systems, like dictionaries, focus on Ilanguage, like in intensional semantics.
• The quale problem in (dualistic) cognitive
theories.
– Can artificial intelligences feel pain, have emotions
and real desires, why not?
• What grounds or anchors meaning?
The philosophical problem
• Grice:
– Communication relies on conventions, that is, on agreements
between language users.
– How do we reach these agreements – by what kinds of
communication means?
• Tomasello:
– Basic ‘trust’ in the other and a fundamental social interest.
– Joint attention & awareness of intention to share information.
– But also a primary shared system of meanings, based in our
perception organs, bodies, and the activities we undertake in
our environments.
– Referential word (sentence, discourse) meaning.
Language and action
• Cognitivist – Chomskian - view: language is an
independent module (or set of modules),
needs to be triggered, ‘contains’ grammar, not
connected to action and perception.
– Evidence from patients.
• Functional – Vygotskian, Tomasello’s – view:
language is part of bodily action and
perception, serving certain (social) purposes.
– Evidence from language development studies.
Young children’s gesturing: interface between
embodied cognition and conventional language
“By tying action and sound, parents ground language
in the same multimodal learning processes that undergird
all of cognition” (Lesson 5: Be social)
Teachers, scientists and infants do
not differ that much
Explaining gestures in language
Explaining mathematics
Explaining origins of biomolecules
Explaining grammar
Experiment by Núñez et al.
• Let f be a strictly increasing function from the interval [0,
1] to the interval [0, 1]. There exists a number a in the
interval [0, 1] such that f(a)=a.
• Groups of graduate students in mathematics discussing
the proof of the theorem.
– Dynamic and static concepts expressed in language.
– Observed gestures (smooth gross arm-hand motion, staccato fine
hand-fingers motion).
• Strong correlations between expression and type of
motion:
• Discussing concepts like increase, continuity, intersection, nearing limit
correlated with smooth gross motor motion.
• Discussing concepts like containment and small enough region with
staccato type of fine motor motion.
Language & body: some evidence
• Many verb meanings are understood in terms
of body and movement; interference studies.
• Semantic induction of emotional states.
• Use of space / visuo-spatial memory to retrieve
verbally represented memories.
• Visuo-spatial working memory in text
comprehension.
Brain evidence of actionperception based language
• Somatotopic organization of (pre)motor cortex.
• Similarities between motor processing and
lexical processing of words with action
(=motor) meaning.
• Activation of motor cortex upon processing
action words: lexical (parallel) or post-lexical?
• Semantic binding and concepts – work of
mirror & canonical neurons in frontal cortex?
Somatotopy of Action Observation / Action
Word understanding
Observing
an action
Foot Action
Hand Action
Action word
processing
Mouth Action
Mirror & canonical neurons
• A neuron (or circuit of neurons) that is activated in
parallel to a person’s actions and when the person
observes similar actions of others.
• A neuron (or circuit of neurons) that is activated in
perceiving a/o acting upon invariant properties of
objects (‘graspability’) and actions (‘reaching’).
• Higher order mirror / canonical neurons.
• Mental simulation and prediction; new assemblages?
• Essential for joint attention, ‘mind reading’ and
cognitive co-construction in play and work.
PPC: Posterior Parietal Cortex
STS: Superior Temporal Sulcus
BA44: Brodmann 44, premotor, posterior part of Broca’s in left
hemisphere
BA 6: Premotor
Joint operation of mirror &
canonical neurons
• Mirror neurons are active in both conditions (i) and
(ii):
– (i) the production of a specific motor action (e.g. grasping with fingers)
by an agent;
– (ii) the observation of a conspecific performing the motor action.
• Canonical neurons are active in both condition (i)
and condition (iii):
– (iii) the observation of an object which provides the affordance for the
motor action in (i), when that object is not being acted on by a
conspecific.
• Human mirror-system enables mental simulation
and language-regulated mental simulation.
Primary and conceptual metaphors
• Metaphors: mappings of a source domain (with
particular automatically available inferences) to a
target domain (while preserving part, most or all of
the inference structure).
– Perception-motor or image schema’s, or elementary ‘cogs’
– Primary metaphors: cognitive structures built out of
elementary cogs
• Conceptual metaphors: cognitive structures built out
of primary metaphors, blends with other conceptual
metaphors.
• Conceptual systems (like in mathematics, science).
Exploratory play
What are these children doing?
• Constructing the basic building blocks of cognition and
later conceptual systems.
– Containment, close-contact, support, piling-up, verticality, far-near,
behind-in-front, force (several types), agency, source-path-goal, motion
(especially when the milestones of self-locomotion are passed), and
many hundreds of elementary cognitions more.
– Moreover, many other embodied experiences with bodily states,
emotions, social interaction, …
• What happens next?
– Emergence of primary metaphors = connection with language.
– Emergence of conceptual metaphors (within language, across domains
while preserving the anchoring in embodied cognition).
– Emergence/acquisition of abstract conceptual systems (like in
mathematics, science).
Predicting spatial language from 18 (t1) to
26 months (t3)
Spatial
cognition (1)
Prepositions
Productive (3)
++
Breadth & Depth
Exploration (1)
++
+++
General
Vocabulary (1)
Spatial
Language 3
Spatial-action
verbs
Receptive(3)
Spatial-action
verbs
Productive(3)
Oudgenoeg et al., in prep.
Knowledge is not mental in the
first place
• Physical environment is a rich source of
information, contains ‘cognition’ in relation to
the perceiving and acting body.
• Sensorimotor behaviors in the physical
environment (and bodily ‘pointers’) ‘bind
together’, create coherence in cognition.
• Conceptual (≈mental) knowledge is based in
the human capacity of simulation.
Elementary embodied cognitions
are essentially multimodal
• Objects and space are encountered through
sight, sound, movement, smell, haptic/touch
and proprioceptive experience, …
• Multiple (time-locked) entries to the primary
full, multimodal meaning of what is the world.
• Rich distributed (sensory-motor) meanings of
emerging concepts.
• Inferences across sensory modalities and
action systems.
Emerging knowledge of numbers
• Early (non-symbolic) number sense: specific or
general?
–
–
–
–
Small numbers (subitizing) vs. bigger numbers.
Numbers vs. quantities (magnitudes).
Size-distance effect.
SNARC effect.
• Verbal-symbolic (exact) math.
– Mapping problem.
– Role of ‘math-talk’.
• Early ‘number sense’:
Eyetracking during number sense
task with 8 months old babies
– 4 months old infants
see a difference
between 8 and 12
dots.
– Increasing accuracy,
also addition and
subtraction.
– Innate system to
which the symbolic
code needs to be
mapped?
Brain evidence
• Math and arithmetic are based in multiple
interconnected brain systems, involving visualspatial processing, verbal processing, visualsymbol processing – coordinated by prefrontal
systems (executive functions).
• Left-right Intra-Parietal Sulcus
• Ventrolateral Prefrontal Cortex.
Numberline task with eye-tracking:
Child M, normal development, normal math achievement
SNARC effect
(Spatial-Numerical Association of Response Codes effect)
• Shorter response times (even, odd) for left
hand × small numbers, right hand × big
numbers.
• ‘Mental number line’ > cultural influences.
Non-spatial (but embodied)
representation of number
• Experiment with undergraduate college students asked to
represent magnitude of non-symbolic (dots) and symbolic
numbers (count words) .
• Different response formats, spatial and non-spatial:
–
–
–
–
Number line
Dynamometer-squeezing
Bell-striking
Loudness of vocalisation
• All representations formats are used quite accurately and thus
can represent number/magnitude, but on a logarithmic scale.
• Linear representations only with count words and number
line, but showing more interindividual variation in accuracy.
Cognitive aspects of mathematics
• Definition of function in everyday language: “continuous
process proceeds without gaps or interruptions or sudden
changes” – motion-like language (in math text book).
• Concept of limit in formal definition, related to continuity
idea:
Let a function f be defined on an open interval containing a,
except possibly at a itself, and let L be a real number. The
statement limx→af(x)=L means that Aε > 0, Eδ > 0, such that if 0
< |x-a| < δ, then |f(x)- L| < ε
• No motion! Yet, mathematicians themselves often use
motion-language (approach, increase, oscillate, cross, …)
Nunez & Lakoff, 2005
Nunez, 2011
Use of motion-related metaphors
in mathematical thinking
• Reflects two basic cognitive mechanism, rooting in
embodied cognitions:
• Source-path-goal schema (or ‘cog’), with automatic
inferences such as:
– a trajectory that moves, a source location (start), a
goal/destination, a route, an actual trajectory of motion, a
position of a trajector at a certain point in time, a direction, a
final location, …
• Fictive (imagined, mentally simulated) motion, also
often expressed by gesturing.
– In addition, other metaphors and cognitions can be used for
understanding and reasoning with static notions, such as
number is a point in space, closeness, …
Time
• Expressions with time metaphors are ubiquitous:
– “The elections are ahead of us”
– “The winter is behind us”
• Conceptual metaphor: Time Events Are Things In Sagittal
Unidimensional Space (i.e., the trajectory of an arrow in
the space in front, or EGO-referenced), with important
inferences – embodied knowledge – ‘cogs’).
– Past=behind, future=in front, present=co-location,
near vs. far, transitivity, ….
• Additional way: Time-point referenced (“the day before
yesterday”).
Source: Núñez, 2008
Universal?
• In most languages, speakers associate past tense
with ‘behind’ and future tense with ‘in front’.
• The case of Aymaran language: exactly the opposite;
Ego-R or Timepoint-R?
• Detailed analysis of verbal expressions along with
gestures reveal:
– Pointing to front space along with the use of past tense =
present; pointing in front space near-far along with past
tense = past/past perfect.
– Pointing backwards = future (that can’t be known).
– Pointing from near-to-far location= timepoint reference.
Human abstraction is thus not merely “socially constructed”.
Its constructed through strong non-arbitrary biological and
cognitive constraints that play an essential role in constituting
what human abstraction is, from everyday ideas to highly
sophisticated mathematics. Human cognition is embodied,
shaped by species-specific non-arbitrary constraints.
But:
Out of the hundreds of possibilities of expressions and other
manifestations of meaning individuals have to learn to pick
the ones that are of their community and serve particular
(communicative, cultural, scientific) purposes best within the
community.
Source: Núñez, 2008
Implications for education
• Abstract concepts in mathematics and science: which
(ultimately embodied) conceptual metaphors are at
stake, how can the teacher use them (expand,
assemble, blend) in educational dialogues to foster
understanding?
• Relevance of concrete (embodied, multimodal)
experience with phenomena (or models of
phenomena).
• Using gestures and imagination.