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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 ).

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Embodied cognition, verbal redescription, and conceptual metaphors in mathematics and science education

Paul Leseman

Utrecht Summer School

August 22, 2013



The problem of linguistic meaning and intention
The problem of (linguistic) meaning (and intention)

  • Semantics (and pragmatics) is in cognitivistapproaches a neglected topic.

    • Tautologicalsystems, likedictionaries, focus onI-language, like in intensionalsemantics.

  • The qualeproblem in (dualistic) cognitivetheories.

    • Canartificialintelligencesfeel pain, have emotions and realdesires, whynot?

  • Whatgroundsoranchorsmeaning?

The philosophical problem
The philosophical problem

  • Grice:

    • Communicationreliesonconventions, that is, onagreementsbetweenlanguageusers.

    • How do we reach these agreements – bywhat kinds of communicationmeans?

  • Tomasello:

    • Basic ‘trust’ in the other and a fundamentalsocial interest.

    • Joint attention & awareness of intention to shareinformation.

    • Butalso a primaryshared system of meanings, based in ourperceptionorgans, bodies, and the activities we undertake in our environments.

    • Referential word (sentence, discourse) meaning.

Language and action
Language and action

  • Cognitivist – Chomskian - view: language is an independent module (orset of modules), needs to betriggered, ‘contains’ grammar, notconnected to action and perception.

    • Evidencefrompatients.

  • Functional – Vygotskian, Tomasello’s – view: language is part of bodilyaction and perception, servingcertain (social) purposes.

    • Evidencefromlanguagedevelopment studies.

Young children s gesturing interface between embodied cognition and conventional language
Young children’sgesturing: interface betweenembodiedcognition and conventionallanguage

“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
Teachers, scientists and infants do notdifferthatmuch

Explaininggestures in language


Explainingorigins of biomolecules


Experiment by n ez et al
Experiment byNúñez 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
Language & body: someevidence

  • Manyverbmeanings are understood in terms of body and movement; interference studies.

  • Semanticinduction of emotionalstates.

  • Use of space / visuo-spatialmemory to retrieveverballyrepresented memories.

  • Visuo-spatialworkingmemory in textcomprehension.

Brain evidence of action perception based language
Brainevidence of action-perceptionbasedlanguage

  • Somatotopicorganization of (pre)motor cortex.

  • Similaritiesbetween motor processing and lexical processing of wordswithaction (=motor) meaning.

  • Activation of motor cortex upon processing actionwords: lexical (parallel) orpost-lexical?

  • Semantic binding and concepts – work of mirror & canonical neurons in frontal cortex?

SomatotopyofActionObservation / Action

Word understanding





Action word



Mirror canonical neurons
Mirror & canonical neurons

  • A neuron (or circuit of neurons) that is activated in parallel to a person’sactionsandwhen the personobservessimilaractions of others.

  • A neuron (or circuit of neurons) that is activated in perceiving a/o actingupon invariant properties of objects (‘graspability’) and actions (‘reaching’).

  • Higher order mirror / canonical neurons.

  • Mental simulation and prediction; new assemblages?

  • Essentialfor joint attention, ‘mind reading’ and cognitiveco-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
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) byan 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
Primary and conceptualmetaphors

  • Metaphors: mappings of a source domain (withparticularautomaticallyavailableinferences) to a target domain (while preserving part, most or all of the inferencestructure).

    • Perception-motoror image schema’s, orelementary ‘cogs’

    • Primarymetaphors: cognitivestructures built out of elementarycogs

  • Conceptualmetaphors: cognitivestructures built out of primarymetaphors, blendswithotherconceptualmetaphors.

  • Conceptualsystems (like in mathematics, science).

What are these children doing
What are these childrendoing?

  • Constructing the basic building blocks of cognition and later conceptualsystems.

    • Containment, close-contact, support, piling-up, verticality, far-near, behind-in-front, force (several types), agency, source-path-goal, motion (especiallywhen the milestones of self-locomotion are passed), and manyhundreds of elementarycognitions more.

    • Moreover, manyotherembodiedexperienceswithbodilystates, emotions, socialinteraction, …

  • What happens next?

    • Emergence of primarymetaphors = connectionwithlanguage.

    • Emergence of conceptualmetaphors (withinlanguage, acrossdomainswhile preserving the anchoring in embodiedcognition).

    • Emergence/acquisition of abstract conceptualsystems (like in mathematics, science).

Predicting spatial language from 18 t1 to 26 months t3
Predictingspatiallanguagefrom 18 (t1) to 26 months (t3)


cognition (1)






Breadth & Depth











Vocabulary (1)

Oudgenoeg et al., in prep.

Knowledge is not mental in the first place
Knowledge is not mental in the first place

  • Physical environment is a richsource of information, contains ‘cognition’ in relation to the perceiving and acting body.

  • Sensorimotorbehaviors in the physical environment (and bodily ‘pointers’) ‘bind together’, createcoherence in cognition.

  • Conceptual (≈mental) knowledge is based in the humancapacity of simulation.

Elementary embodied cognitions are essentially multimodal
Elementaryembodiedcognitions are essentiallymultimodal

  • Objects and space are encounteredthroughsight, sound, movement, smell, haptic/touch and proprioceptiveexperience, …

  • Multiple (time-locked) entries to the primaryfull,multimodalmeaning of what is the world.

  • Richdistributed (sensory-motor) meanings of emergingconcepts.

  • Inferencesacrosssensorymodalities and actionsystems.

Emerging knowledge of numbers
Emergingknowledge of numbers

  • Early (non-symbolic) numbersense: specificorgeneral?

    • Smallnumbers (subitizing) vs. biggernumbers.

    • Numbers vs. quantities (magnitudes).

    • Size-distance effect.

    • SNARC effect.

  • Verbal-symbolic (exact) math.

    • Mappingproblem.

    • Role of ‘math-talk’.


taskwith 8 monthsoldbabies

  • Early ‘numbersense’:

    • 4 monthsoldinfantssee a differencebetween 8 and 12 dots.

    • Increasingaccuracy, alsoaddition and subtraction.

    • Innate system to which the symbolic code needs to bemapped?

Brain evidence
Brain evidence

  • Math and arithmetic are based in multiple interconnected brain systems, involving visual-spatial processing, verbal processing, visual-symbol 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
Numberlinetaskwitheye-tracking:Child M, normaldevelopment, normalmathachievement

Snarc effect spatial numerical association of response codes effect
SNARC effect(Spatial-NumericalAssociation of Response Codes effect)

  • Shorter response times (even, odd) forleft hand × smallnumbers, right hand × big numbers.

  • ‘Mental numberline’ > culturalinfluences.

Non spatial but embodied representation of number
Non-spatial (butembodied) representation of number

  • Experiment withundergraduate college studentsasked to represent magnitude of non-symbolic (dots) and symbolicnumbers (countwords) .

  • Different response formats, spatial and non-spatial:

    • Numberline

    • Dynamometer-squeezing

    • Bell-striking

    • Loudness of vocalisation

  • All representationsformats are usedquiteaccurately and thuscanrepresentnumber/magnitude, buton a logarithmicscale.

  • Linearrepresentationsonlywithcountwords and numberline, butshowing more interindividualvariation in accuracy.

Cognitive aspects of mathematics
Cognitiveaspects 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
Use of motion-relatedmetaphors in mathematical thinking

  • Reflectstwobasiccognitivemechanism, rooting in embodiedcognitions:

  • Source-path-goal schema (or ‘cog’), withautomaticinferencessuch as:

    • a trajectorythat moves, a sourcelocation (start), a goal/destination, a route, anactualtrajectory of motion, a position of a trajector at a certain point in time, a direction, a finallocation, …

  • Fictive (imagined, mentallysimulated) motion, alsooftenexpressedbygesturing.

    • In addition, othermetaphors and cognitionscanbeusedforunderstanding and reasoningwith static notions, such as number is a point in space, closeness, …


  • Expressionswith time metaphors are ubiquitous:

    • “The elections are ahead of us”

    • “The winter is behindus”

  • Conceptualmetaphor: Time Events Are Things In SagittalUnidimensionalSpace(i.e., the trajectory of anarrow in the space in front, orEGO-referenced), with important inferences – embodiedknowledge – ‘cogs’).

    • Past=behind, future=in front, present=co-location, near vs. far, transitivity, ….

  • Additionalway: Time-pointreferenced (“the daybeforeyesterday”).

Source: Núñez, 2008


  • In most languages, speakers associate past tensewith ‘behind’ and futuretensewith ‘in front’.

  • The case of Aymaranlanguage: exactly the opposite; Ego-RorTimepoint-R?

  • Detailedanalysis of verbal expressionsalongwithgesturesreveal:

    • Pointing to front spacealongwith the use of past tense = present; pointing in front spacenear-faralongwith past tense = past/past perfect.

    • Pointingbackwards = future (thatcan’tbeknown).

    • Pointingfromnear-to-farlocation= timepointreference.

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.


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
Implications 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. foreducation

  • Abstract concepts in mathematics and science: which (ultimatelyembodied) conceptualmetaphors are at stake, howcan the teacher usethem (expand, assemble, blend) in educationaldialogues to fosterunderstanding?

  • Relevance of concrete (embodied, multimodal) experiencewithphenomena (or models of phenomena).

  • Usinggestures and imagination.