No power: exponential expressions are not processed automatically as such.
Journal
Psychological research
ISSN: 1430-2772
Titre abrégé: Psychol Res
Pays: Germany
ID NLM: 0435062
Informations de publication
Date de publication:
Jul 2021
Jul 2021
Historique:
received:
22
11
2019
accepted:
01
07
2020
pubmed:
25
7
2020
medline:
31
7
2021
entrez:
25
7
2020
Statut:
ppublish
Résumé
Little is known about the mental representation of exponential expressions. The present study examined the automatic processing of exponential expressions under the framework of multi-digit numbers, specifically asking which component of the expression (i.e., the base/power) is more salient during this type of processing. In a series of three experiments, participants performed a physical size comparison task. They were presented with pairs of exponential expressions that appeared in frames that differed in their physical sizes. Participants were instructed to ignore the stimuli within the frames and choose the larger frame. In all experiments, the pairs of exponential expressions varied in the numerical values of their base and/or power component. We manipulated the compatibility between the base and the power components, as well as their physical sizes to create a standard versus nonstandard syntax of exponential expressions. Experiments 1 and 3 demonstrate that the physically larger component drives the size congruity effect, which is typically the base but was manipulated here in some cases to be the power. Moreover, Experiments 2 and 3 revealed similar patterns, even when manipulating the compatibility between base and power components. Our findings support componential processing of exponents by demonstrating that participants were drawn to the physically larger component, even though in exponential expressions, the power, which is physically smaller, has the greater mathematical contribution. Thus, revealing that the syntactic structure of an exponential expression is not processed automatically. We discuss these results with regard to multi-digit numbers research.
Identifiants
pubmed: 32705335
doi: 10.1007/s00426-020-01381-6
pii: 10.1007/s00426-020-01381-6
doi:
Types de publication
Journal Article
Langues
eng
Sous-ensembles de citation
IM
Pagination
2079-2097Subventions
Organisme : Israel Science Foundation
ID : 1348/18
Informations de copyright
© 2020. Springer-Verlag GmbH Germany, part of Springer Nature.
Références
Avcu, R. (2010). Eight graders’ capabilities in exponents: making mental comparisons. Practice and Theory in System of Education, 5(1), 39–48.
Bargh, J. A. (1992). The ecology of automaticity: Toward establishing the conditions needed to produce automatic processing effects. American Journal of Psychology, 105, 181–199.
doi: 10.2307/1423027
Bonato, M., Fabbri, S., Umiltà, C., & Zorzi, M. (2007). The mental representation of numerical fractions: Real or integer? Journal of Experimental Psychology: Human Perception and Performance, 33, 1410–1419.
pubmed: 18085953
pmcid: 18085953
Brysbaert, M. (1995). Arabic number reading: On the nature of the numerical scale and the origin of phonological recoding. Journal of Experimental Psychology: General, 124, 434–452.
doi: 10.1037/0096-3445.124.4.434
Campbell, J. I. D. (1997). On the relation between skilled performance of simple division and multiplication. Journal of Experimental Psychology: Learning, Memory, and Cognition, 23, 1140–1159.
pubmed: 9293626
pmcid: 9293626
Cipora, K., Soltanlou, M., Smaczny, S., Göbel, S. M., & Nuerk, H.-C. (2019). Automatic place-value activation in magnitude-irrelevant parity judgement. Psychological Research Psychologische Forschung. https://doi.org/10.1007/s00426-019-01268-1 .
doi: 10.1007/s00426-019-01268-1
pubmed: 31734821
pmcid: 31734821
Cohen, D. J. (2010). Evidence for direct retrieval of relative quantity information in a quantity judgment task: Decimals, integers, and the role of physical similarity. Journal of Experimental Psychology: Learning, Memory, and Cognition, 36, 1389–1398.
pubmed: 20804282
pmcid: 20804282
Dehaene, S. (1989). The psychophysics of numerical comparison: A reexamination of apparently incompatible data. Perception and Psychophysics, 45, 557–566.
pubmed: 2740196
doi: 10.3758/BF03208063
pmcid: 2740196
Dehaene, S., Dupoux, E., & Mehler, J. (1990). Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison. Journal of Experimental Psychology: Human Perception and Performance, 16, 626–641.
pubmed: 2144576
pmcid: 2144576
DeWolf, M., Bassok, M., & Holyoak, K. (2013). Analogical reasoning with rational numbers: Semantic alignment based on discrete versus continuous quantities. In M. Knauf, M. Pauven, N. Sebanz, & I. Wachsmuth (Eds.), Proceedings of the 35th Annual Conference of the of the Cognitive Science Society. Austin, TX: Cognitive Science Society.
Dubinsky, E. (1991). Constructive aspects of reflective abstraction in advanced mathematics. In L. P. Steffe (Ed.), Epistemological foundations of mathematical experience. New York: Springer-Verlag.
Ebersbach, M., & Wilkening, F. (2007). Children's intuitive mathematics: The development of knowledge about nonlinear growth. Child Development, 78, 296–308.
pubmed: 17328706
doi: 10.1111/j.1467-8624.2007.00998.x
pmcid: 17328706
Fischer, M. H. (2003). Cognitive representation of negative numbers. Psychological Science, 14, 278–282.
pubmed: 12741754
doi: 10.1111/1467-9280.03435
pmcid: 12741754
Fitousi, D., & Algom, D. (2006). Size congruity effects with two-digit numbers: Expanding the number line? Memory and Cognition, 34, 445–457.
pubmed: 16752607
doi: 10.3758/BF03193421
pmcid: 16752607
Ganor-Stern, D. (2013). Are 1/2 and 0.5 represented in the same way? Acta Psychologica, 142, 299–307.
pubmed: 23419807
doi: 10.1016/j.actpsy.2013.01.003
pmcid: 23419807
Ganor-Stern, D., Pinhas, M., Kallai, A., & Tzelgov, J. (2010). Holistic representation of negative numbers is formed when needed for the task. The Quarterly Journal of Experimental Psychology, 63, 1969–1981.
pubmed: 20437298
doi: 10.1080/17470211003721667
pmcid: 20437298
Ganor-Stern, D., Tzelgov, J., & Ellenbogen, R. (2007). Automaticity and two-digit numbers. Journal of Experimental Psychology: Human Perception and Performance, 33, 483–496.
pubmed: 17469981
pmcid: 17469981
García-Orza, J., & Damas, J. (2011). Sequential processing of two-digit numbers: Evidence of decomposition from a perceptual number matching task. Zeitschrift Für Psychologie/Journal of Psychology, 219(1), 23–29.
doi: 10.1027/2151-2604/a000042
García-Orza, J., Estudillo, A. J., Calleja, M., & Rodríguez, J. M. (2017). Is place-value processing in four-digit numbers fully automatic? Yes, but not always. Psychonomic Bulletin and Review, 24, 1906–1914.
pubmed: 28138835
doi: 10.3758/s13423-017-1231-x
pmcid: 28138835
Henik, A., & Tzelgov, J. (1982). Is three greater than five: The relation between physical and semantic size in comparison tasks. Memory & Cognition, 10, 389–395.
doi: 10.3758/BF03202431
Huber, S., Bahnmueller, J., Klein, E., & Moeller, K. (2015a). Testing a model of componential processing of multi-symbol numbers—evidence from measurement units. Psychonomic Bulletin & Review, 22, 1417–1423.
doi: 10.3758/s13423-015-0805-8
Huber, S., Cornelsen, S., Moeller, K., & Nuerk, H. C. (2015b). Toward a model framework of generalized parallel componential processing of multi-symbol numbers. Journal of Experimental Psychology: Learning, Memory, and Cognition, 41, 732.
pubmed: 25068853
pmcid: 25068853
Huber, S., Nuerk, H. C., Willmes, K., & Moeller, K. (2016). A general model framework for multi-symbol number comparison. Psychological Review, 123, 667–695.
pubmed: 27617347
doi: 10.1037/rev0000040
pmcid: 27617347
Işık, C., Kar, T., Yalçın, T., & Zehir, K. (2011). Prospective teachers’ skills in problem posing with regard to different problem posing models. Procedia-Social and Behavioral Sciences, 15, 485–489.
doi: 10.1016/j.sbspro.2011.03.127
Iuculano, T., & Butterworth, B. (2011). Rapid communication: Understanding the real value of fractions and decimals. The Quarterly Journal of Experimental Psychology, 64, 2088–2098.
pubmed: 21929473
doi: 10.1080/17470218.2011.604785
pmcid: 21929473
İymen, E., & Duatepe-Paksu, A. (2015). Analysis of 8th grade students' number sense related to the exponents in terms of number sense components. Education & Science/Egitim Ve Bilim, 40(177), 109–125.
Jeffreys, H. (1961). Theory of probability. Oxford, UK: Oxford University Press.
Kallai, A. Y., & Tzelgov, J. (2009). A generalized fraction: An entity smaller than one on the mental number line. Journal of Experimental Psychology: Human Perception and Performance, 35, 1845–1864.
pubmed: 19968440
pmcid: 19968440
Kallai, A. Y., & Tzelgov, J. (2012). The place-value of a digit in multi-digit numbers is processed automatically. Journal of Experimental Psychology: Learning, Memory, and Cognition, 38, 1221–1233.
pubmed: 22449132
pmcid: 22449132
Keren, G. (1983). Cultural differences in the misperception of exponential growth. Perception and Psychophysics, 34, 289–293.
pubmed: 6646972
doi: 10.3758/BF03202958
pmcid: 6646972
Korvorst, M., & Damian, M. F. (2008). The differential influence of decades and units on multidigit number comparison. The Quarterly Journal of Experimental Psychology, 61, 1250–1264.
pubmed: 18938764
doi: 10.1080/17470210701503286
pmcid: 18938764
LeFevre, J.-A., Shanahan, T., & DeStefano, D. (2004). The tie effect in simple arithmetic: An access-based account. Memory & Cognition, 32, 1019–1031.
doi: 10.3758/BF03196878
Logan, G. D. (1988). Toward an instance theory of automatization. Psychological Review, 95, 492–527.
doi: 10.1037/0033-295X.95.4.492
Meert, G., Grégoire, J., & Noël, M. (2009). Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task. The Quarterly Journal of Experimental Psychology, 62, 1598–1616.
pubmed: 19123119
doi: 10.1080/17470210802511162
pmcid: 19123119
Menon, R. (2004). Preservice teachers' number sense. Focus on Learning Problems in Mathematics, 26(2), 49–61.
Meert, G., Gregoire, J., & Noel, M. P. (2010a). Comparing the magnitude of two fractions with common components: Which representations are used by 10- and 12-year-olds. Journal of Experimental Child Psychology, 107, 244–259.
pubmed: 20627317
doi: 10.1016/j.jecp.2010.04.008
pmcid: 20627317
Meert, G., Gregoire, J., & Noel, M. P. (2010b). Comparing 5/7 and 2/9: Adults can do it by accessing the magnitude of the whole fraction. Acta Psychologica, 135, 284–292.
pubmed: 20797686
doi: 10.1016/j.actpsy.2010.07.014
pmcid: 20797686
Meyerhoff, H. S., Moeller, K., Debus, K., & Nuerk, H. C. (2012). Multi-digit number processing beyond the two-digit number range: A combination of sequential and parallel processes. Acta Psychologica, 140, 81–90.
pubmed: 22469562
doi: 10.1016/j.actpsy.2011.11.005
pmcid: 22469562
Mohamed, M., & Johnny, J. (2010). Investigating number sense among students. Procedia-Social and Behavioral Sciences, 8, 317–324.
doi: 10.1016/j.sbspro.2010.12.044
Moyer, R. S., & Landauer, T. K. (1967). Time required for judgements of numerical inequality. Nature, 215, 1519–1520.
pubmed: 6052760
doi: 10.1038/2151519a0
Mullet, E., & Cheminat, Y. (1995). Estimation of exponential expressions by high school students. Contemporary Educational Psychology, 20(4), 451–456.
doi: 10.1006/ceps.1995.1031
Nuerk, H., Moeller, K., Klein, E., Willmes, K., & Fischer, M. H. (2011). Extending the mental number line: A review of multi-digit number processing. Zeitschrift Für Psychologie/Journal of Psychology, 219(1), 3–22.
doi: 10.1027/2151-2604/a000041
Nuerk, H., Moeller, K., & Willmes, K. (2015). Multi-digit number processing: Overview, conceptual clarifications, and language influences. In: R. Cohen Kadosh, and A. Dowker (Eds), The Oxford Handbook of Numerical Cognition. New York, NY: Oxford University Press.
Nuerk, H., Weger, U., & Willmes, K. (2001). Decade breaks in the mental number line? Putting the tens and units back in different bins. Cognition, 82, B25–33.
pubmed: 11672709
doi: 10.1016/S0010-0277(01)00142-1
pmcid: 11672709
Nuerk, H., & Willmes, K. (2005). On the magnitude representations of two-digit numbers. Psychology Science, 47, 52–72.
Pansky, A., & Algom, D. (1999). Stroop and Garner effects in comparative judgment of numerals: The role of attention. Journal of Experimental Psychology: Human Perception and Performance, 25, 39–58.
Parnes, M., Berger, A., & Tzelgov, J. (2012). Brain representations of negative numbers. Canadian Journal of Experimental Psychology/Revue Canadienne De Psychologie Expérimentale, 66, 251–258.
pubmed: 22774801
doi: 10.1037/a0028989
pmcid: 22774801
Perruchet, P., & Vinter, A. (2002). The self-organizing consciousness: A framework for implicit learning. In R. French & A. Cleeremans (Eds.), Implicit Learning and Consciousness: An Empirical, Philosophical, and Computational Consensus in the Making. New York, NY: Psychology.
Pinhas, M., & Tzelgov, J. (2012). Expanding on the mental number line: Zero is perceived as the “smallest”. Journal of Experimental Psychology: Learning, Memory, and Cognition, 38, 1187–1205.
pubmed: 22369255
pmcid: 22369255
Pinhas, M., Tzelgov, J., & Guata-Yaakobi, I. (2010). Exploring the mental number line via the size congruity effect. Canadian Journal of Experimental Psychology/Revue Canadienne De Psychologie Expérimentale, 64, 221–225.
pubmed: 20873919
doi: 10.1037/a0020464
pmcid: 20873919
Pitta-Pantazi, D., Christou, C., & Zachariades, T. (2007). Secondary school students’ levels of understanding in computing exponents. The Journal of Mathematical Behavior, 26(4), 301–311.
doi: 10.1016/j.jmathb.2007.11.003
Reynvoet, B., & Brysbaert, M. (1999). Single-digit and two-digit Arabic numerals address the same semantic number line. Cognition, 72, 191–201.
pubmed: 10553671
doi: 10.1016/S0010-0277(99)00048-7
pmcid: 10553671
Reys, R., Reys, B., Emanuelsson, G., Johansson, B., McIntosh, A., & Yang, D. C. (1999). Assessing number sense of students in Australia, Sweden, Taiwan, and the United States. School Science and Mathematics, 99(2), 61–70.
doi: 10.1111/j.1949-8594.1999.tb17449.x
Robson, D. S. (1959). A simple method for constructing orthogonal polynomials when the independent variable is unequally spaced. Biometrics, 15, 187–191.
doi: 10.2307/2527668
Rouder, J. N., Speckman, P. L., Sun, D., Morey, R. D., & Iverson, G. (2009). Bayesian t tests for accepting and rejecting the null hypothesis. Psychonomic Bulletin and Review, 16, 225–237.
pubmed: 19293088
doi: 10.3758/PBR.16.2.225
pmcid: 19293088
Sastre, M. T. M., & Mullet, E. (1998). Evolution of the intuitive mastery of the relationship between base, exponent, and number magnitude in high-school students. Mathematical Cognition, 4(1), 67–77.
doi: 10.1080/135467998387415
Schwarz, W., & Ischebeck, A. (2003). On the relative speed account of number-size interference in comparative judgments of numerals. Journal of Experimental Psychology: Human Perception and Performance, 29, 507–522.
pubmed: 12848323
pmcid: 12848323
Shaki, S., & Petrusic, W. M. (2005). On the mental representation of negative numbers: Context-dependent SNARC effects with comparative judgments. Psychonomic Bulletin and Review, 12, 931–937.
pubmed: 16524013
doi: 10.3758/BF03196788
pmcid: 16524013
Singh, P. (2009). An assessment of number sense among secondary school students. International Journal for Mathematics Teaching and Learning, 1–27. Retrieved from http:// www.cimt.plymouth.ac.uk/journal/singh.pdf .
Stango, V., & Zinman, J. (2009). Exponential growth bias and household finance. Journal of Finance, 64, 2807–2849.
doi: 10.1111/j.1540-6261.2009.01518.x
Timmers, H., & Wagenaar, W. A. (1977). Inverse statistics and misperception of exponential growth. Perception and Psychophysics, 21, 558–562.
doi: 10.3758/BF03198737
Tzelgov, J. (1997). Automatic but conscious: That is how we act most of the time. Advances in Social Cognition, 10, 217–230.
Tzelgov, J., & Ganor-Stern, D. (2005). Automaticity in processing ordinal information. In: Campbell, J. I. D. (Ed.), Handbook of Mathematical Cognition. New York, NY: Psychology Press.
Tzelgov, J., Ganor-Stern, D., & Maymon-Schreiber, K. (2009). The representation of negative numbers: Exploring the effects of mode of processing and notation. The Quarterly Journal of Experimental Psychology, 62, 605–624.
pubmed: 18609405
doi: 10.1080/17470210802034751
pmcid: 18609405
Tzelgov, J., Meyer, J., & Henik, A. (1992). Automatic and intentional processing of numerical information. Journal of Experimental Psychology: Learning, Memory, and Cognition, 18, 166–179.
Varma, S., & Karl, S. R. (2013). Understanding decimal proportions: Discrete representations, parallel access, and privileged processing of zero. Cognitive Psychology, 66, 283–301.
pubmed: 23416180
doi: 10.1016/j.cogpsych.2013.01.002
pmcid: 23416180
Varma, S., & Schwartz, D. L. (2011). The mental representation of integers: An abstract-to-concrete shift in the understanding of mathematical concepts. Cognition, 121, 363–385.
pubmed: 21939966
doi: 10.1016/j.cognition.2011.08.005
pmcid: 21939966
Verguts, T., & De Moor, W. (2005). Two-digit comparison: Decomposed, holistic, or hybrid? Experimental Psychology, 52(3), 195–200.
pubmed: 16076067
doi: 10.1027/1618-3169.52.3.195
pmcid: 16076067
Wagenaar, W. A. (1982). Misperception of exponential growth and the psychological magnitude of numbers. In B. Wegener (Ed.), Social attitudes and psychophysical measurement. Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
Wagenaar, W. A., & Sagaria, S. D. (1975). Misperception of exponential growth. Perception and Psychophysics, 18, 416–422.
doi: 10.3758/BF03204114
Wagenaar, W. (1978) Intuitive Prediction of Growth. In Burkhardt, D, and W. Ittelson (eds.), Environmental Assessment of Socioeconomic Systems. New York: Plenum.
Wagenaar, W. A., & Timmers, H. (1979). The pond-and-duckweed problem; three experiments on the misperception of exponential growth. Acta Psychologica, 43, 239–251.
doi: 10.1016/0001-6918(79)90028-3
Yang, D. (2005). Number sense strategies used by 6th-grade students in taiwan. Educational Studies, 31, 317–333.
doi: 10.1080/03055690500236845
Zhang, J., Feng, W., & Zhang, Z. (2019). Holistic representation of negative numbers: Evidence from duration comparison tasks. Acta Psychologica, 193, 123–131.
pubmed: 30622021
doi: 10.1016/j.actpsy.2018.12.012
pmcid: 30622021