Abstract
Representational transformation is a technique for deepening our understanding of mathematical concepts and facilitating the process of solving mathematical problems. This article suggests that representational transformation can contribute to thought processes in four ways. Firstly, representational transformation helps people employ a wider range of cognitive resources. When one representation of a concept or a problem is transformed into another representation, the sensory-motor resources that are involved in the processing of the base representation are employed to process the target representation. Secondly, representational transformation helps in the process of organizing scattered information into suitable units. This gives structure to various parts of information and makes them much easier to handle. Thirdly, representational transformation helps in the process of suppressing irrelevant information and unnecessary elements of problems. This is a key step in organizing and structuring information. Fourthly, representational transformation helps in the process of discovering general rules and turning them into explicit knowledge.
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References
Bauer MI, Johnson-Laird PN (1993) How diagrams can improve reasoning. Psychol Sci 4(6):372–378. https://doi.org/10.1111/j.1467-9280.1993.tb00584.x
Borghi AM, Binkofski F, Castelfranchi C, Cimatti F, Scorolli C, Tummolini L (2017) The challenge of abstract concepts. Psychol Bull 143(3):263–292. https://doi.org/10.1037/bul0000089
Boroditsky L, Gaby A (2010) Remembrances of times east: absolute spatial representations of time in an Australian aboriginal community. Psychol Sci 21(11):1635–1639. https://doi.org/10.1177/0956797610386621
Broaders SC, Cook SW, Mitchell Z, Goldin-Meadow S (2007) Making children gesture brings out implicit knowledge and leads to learning. J Exp Psychol Gen 136(4):539–550. https://doi.org/10.1037/0096-3445.136.4.539
Butcher KR (2006) Learning from text with diagrams: Promoting mental model development and inference generation. J Educ Psychol 98(1):182–197. https://doi.org/10.1037/0022-0663.98.1.182
Casasanto D, Jasmin K (2012) The hands of time: temporal gestures in english speakers. Cognit Linguist 23(4):643–674. https://doi.org/10.1515/cog-2012-0020
Chedid G, Brambati SM, Bedetti C, Rey AE, Wilson MA, Vallet GT (2019) Visual and auditory perceptual strength norms for 3,596 French nouns and their relationship with other psycholinguistic variables. Behav Res Methods 51(5):2094–2105. https://doi.org/10.3758/s13428-019-01254-w
Chen IH, Zhao Q, Long Y, Lu Q, Huang CR (2019) Mandarin Chinese modality exclusivity norms. PLoS ONE 14:e0211336. https://doi.org/10.1371/journal.pone.0211336
Chui K (2011) Conceptual metaphors in gesture. Cognit Linguist 22(3):437–458. https://doi.org/10.1515/cogl.2011.017
Cooperrider K, Núñez R (2009) Across time, across the body: transversal temporal gestures. Gesture 9(2):181–206. https://doi.org/10.1075/gest.9.2.02coo
Crutch SJ, Jackson EC (2011) Contrasting graded effects of semantic similarity and association across the concreteness spectrum. Q J Exp Psychol 64(7):1388–1408. https://doi.org/10.1080/17470218.2010.543285
Dreyer FR, Frey D, Arana S, von Saldern S, Picht T, Vajkoczy P, Pulvermüller F (2015) Is the Motor System Necessary for Processing Action and Abstract Emotion Words? Evidence from Focal Brain Lesions. Front Psychol 6:1661. https://doi.org/10.3389/fpsyg.2015.01661
Eikmeier V, Alex-Ruf S, Maienborn C, Ulrich R (2015) How strongly linked are mental time and space along the left-right axis? J Exp Psychol Learn Mem Cogn 41(6):1878–1883
Farrugia MT (2017) On semiotics and jumping frogs: the role of gesture in the teaching of subtraction. For the Learn Math 37(2):2–7
Filipović Đurđević DF, Popović Stijačić M, Karapandžić J (2016) A quest for sources of perceptual richness: Several candidates. In: Halupka-Rešetar S, Martínez-Ferreiro S (eds) Studies in language and mind. Filozofski fakultet uNovom Sadu, Novi Sad, Serbia, pp 187–238
Fuchs T (2012) The phenomenology of body memory. In: Koch SC, Fuchs T, Summa M, Müller C (eds) Body memory, metaphor and movement, 84. John Benjamins Publishing Company, pp 9–22
Gallese V (2003) The manifold nature of interpersonal relations: the quest for a common mechanism. Philos Trans R Soc Lond B 358:517–528
Gallese G, Lakoff G (2005) The brain’s concepts: the role of the sensory-motor system in conceptual knowledge. Cogn Neuropsychol 22(3):455–479
Gemsbacher MA, Robertson RR (1999) The role of suppression in figurative language comprehension. J Pragmat 31(12):1619–1630. https://doi.org/10.1016/S0378-2166(99)00007-7
Gibbs RW (2006) Metaphor interpretation as embodied simulation. Mind Lang 21(3):434–458. https://doi.org/10.1111/j.1468-0017.2006.00285.x
Gibbs RW (2013) Walking the walk while thinking about the talk: embodied interpretation of metaphorical narratives. J Psycholinguist Res 42(4):363–378. https://doi.org/10.1007/s10936-012-9222-6
Goldin-Meadow S (2015) From action to abstraction: gesture as a mechanism of change. Dev Rev 38:167–184. https://doi.org/10.1016/j.dr.2015.07.007
Gu Y, Mol L, Hoetjes M, Swerts M (2017) Conceptual and lexical effects on gestures: the case of vertical spatial metaphors for time in Chinese. Lang Cognit Neurosci 32(8):1048–1063. https://doi.org/10.1080/23273798.2017.1283425
Hegarty M, Kozhevnikov M (1999) Types of visual–spatial representations and mathematical problem solving. J Educ Psychol 91(4):684–689. https://doi.org/10.1037/0022-0663.91.4.684
Jirout JJ, Holmes CA, Ramsook KA, Newcombe NS (2018) Scaling up spatial development: a closer look at children’s scaling ability and its relation to number knowledge. Mind Brain Educ 12(3):110–119
Kang S, Tversky B, Black JB (2015) Coordinating gesture, word, and diagram: explanations for experts and novices. Spat Cogn Comput 15(1):1–26. https://doi.org/10.1080/13875868.2014.958837
Khatin-Zadeh O, Yazdani-Fazlabadi B, Eskandari Z (2021) The grounding of mathematical concepts through fictive motion, gesture and the motor system. For the Learn Math 41(3):19–21
Kita S (2000) How representational gestures help speaking. In: McNeill D (ed) Language and Gesture. Cambridge University Press, Cambridge, UK, pp 162–185
Kita S, Davies TS (2009) Competing conceptual representations trigger co-speech representational gestures. Lang Cognit Process 24(5):761–775. https://doi.org/10.1080/01690960802327971
Kita S, Alibali MW, Chu M (2017) How do gestures influence thinking and speaking? the gesture-for-conceptualization hypothesis. Psychol Rev 124(3):245–266. https://doi.org/10.1037/rev0000059
Lakoff G (2008) The neural theory of metaphor. In: Gibbs RW Jr (ed) The Cambridge handbook of metaphor and thought. Oxford University Press, Oxford, pp 17–38
Lakoff G (2014) Mapping the brain’s metaphor circuitry: metaphorical thought in everyday reason. Front Hum Neurosci 8:958. https://doi.org/10.3389/fnhum.2014.00958
Lakoff G, Johnson M (2003) Metaphors we live by. University of Chicago Press, London
Lynott D, Connell L, Brysbaert M, Brand J, Carney J (2019) The lancaster sensorimotor norms: multidimensional measures of perceptual and action strength for 40,000 English words. Behav Res Methods 52(3):1271–1291. https://doi.org/10.3758/s13428-019-01316-z
McNeill D (1992) Hand and mind: what gestures reveal about thought. University of Chicago Press, Chicago, IL
Meier B, Schnall S, Schwarz N, Bargh JA (2012) Embodiment in social psychology. Top Cogn Sci 4(4):705–716
Miklashevsky A (2018) Perceptual experience norms for 506 Russian nouns: modality rating, spatial localization, manipulability, imageability and other variables. J Psycholinguist Res 47(3):641–661
Miles LK, Nind LK, Macrae CN (2010) Moving through time. Psychol Sci 21(2):222–223. https://doi.org/10.1177/0956797609359333
Moseley R, Carota F, Hauk O, Mohr B, Pulvermüller F (2012) A role for the motor system in binding abstract emotional meaning. Cereb Cortex 22(7):1634–1647. https://doi.org/10.1093/cercor/bhr238
Newcombe NS, Frick A (2010) Early education for spatial intelligence: why, what, and how. Mind Brain Educ 4(3):102–111
Niedenthal PM (2007) Embodying emotion. Science 316(5727):1002–1005. https://doi.org/10.1126/science.1136930
Rolke B, Ruiz Fernández S, Schmid M, Walker M, Lachmair M, López JJR, Hervás G, Vázquez C (2013) Priming the mental time-line: effects of modality and processing mode. Cogn Process 14(3):231–244
Santiago J, Lupiáñez J, Pérez E, Funes MJ (2007) Time (also) flies from left to right. Psychon Bull Rev 14(3):512–516
Sell AJ, Kaschak MP (2012) The comprehension of sentences involving quantity information affects responses on the up-down axis. Psychon Bull Rev 19(4):708–714. https://doi.org/10.3758/s13423-012-0263-5
Speed LJ, Majid A (2017) Dutch modality exclusivity norms: simulating perceptual modality in space. Behav Res Methods 49(6):2204–2218
Staats S (2018) Coloring conjectures with sound, silence, syntax and gesture: a multimodal poetic analysis. For the Learn Math 38(2):2–8
Torralbo A, Santiago J, Lupiáñez J (2006) Flexible conceptual projection of time onto spatial frames of reference. Cogn Sci 30(4):745–757
Wiemers M, Bekkering H, Lindemann O (2014) Spatial interferences in mental arithmetic: evidence from the motion-arithmetic compatibility effect. Q J Exp Psychol 67(8):1557–1570. https://doi.org/10.1080/17470218.2014.889180
Williams LE, Bargh JA (2008) Experiencing physical warmth promotes interpersonal warmth. Science 322(5901):606–607
Zhong CB, Leonardelli GJ (2008) Cold and lonely: does social exclusion feel literally cold? Psychol Sci 19(9):838–842
Zhu R, You X, Gan S, Wang J (2019) Spatial attention shifts in addition and subtraction arithmetic: evidence of eye movement. Perception 48(9):835–849. https://doi.org/10.1177/0301006619865156
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Khatin-Zadeh, O. How does Representational Transformation Enhance Mathematical Thinking?. Axiomathes 32 (Suppl 2), 283–292 (2022). https://doi.org/10.1007/s10516-021-09602-2
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DOI: https://doi.org/10.1007/s10516-021-09602-2