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

1. Embodied cognition, verbal redescription, and conceptual metaphors in mathematics and science education Paul Leseman Utrecht Summer School August 22, 2013

2. Affection Important

3. 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?

4. 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.

5. 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.

6. Young children’sgesturing: interface betweenembodiedcognition and conventionallanguage

7. “By tying action and sound, parents ground language in the same multimodal learning processes that undergird all of cognition” (Lesson 5: Be social)

8. Teachers, scientists and infants do notdifferthatmuch Explaininggestures in language Explainingmathematics Explainingorigins of biomolecules Explaininggrammar

9. 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.

10. 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.

11. 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?

12. SomatotopyofActionObservation / Action Word understanding FootAction Observing anaction HandAction Action word processing MouthAction

13. 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.

14. PPC: Posterior Parietal Cortex STS: Superior Temporal Sulcus BA44: Brodmann 44, premotor, posterior part of Broca’s in left hemisphere BA 6: Premotor

15. 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.

16. 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).

17. Exploratory play

18. 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).

19. Predictingspatiallanguagefrom 18 (t1) to 26 months (t3) Spatial cognition (1) Prepositions Productive(3) ++ Spatial Language3 Breadth & Depth Exploration(1) Spatial-action verbs Receptive(3) ++ +++ Spatial-action verbs Productive(3) General Vocabulary (1) Oudgenoeg et al., in prep.

20. 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.

22. 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.

23. 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’.

24. Eyetrackingduringnumbersense 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?

25. 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.

26. Numberlinetaskwitheye-tracking:Child M, normaldevelopment, normalmathachievement

27. SNARC effect(Spatial-NumericalAssociation of Response Codes effect) • Shorter response times (even, odd) forleft hand × smallnumbers, right hand × big numbers. • ‘Mental numberline’ > culturalinfluences.

28. 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.

29. 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

30. 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, …

31. Time • 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

32. Universal? • 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.

33. 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

34. Implicationsforeducation • 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.