Rekenproject: Getalbegrip

Ben Wilbrink

rekenproject thuis
rekendidactiek
    algoritmen
    getalbegrip
    basale rekenvaardigheden‘cijferen’
    optellenaftrekkenvermenigvuldigendelenbreukenmeten
    meetkundealgebra en rekenenkans (en combinaties)
    materialen
    woordproblemen




A big problem in discourse on math education is its shallowness. An extreme yet important example is the notion that basic arithmetic is not that crucial anymore since the arrival of the Japanese pocket calculator. This notion denies the importance of a good number sense in daily life, in taking care of one’s health, also one’s financial health, as well as in participating in one’s community (vocational as well as in other communities) (see research by Reyna, below). Yes, it is more difficult to imagine the predicaments of people with a lack numeracy, yet the problem is the same as that with straigh or functional analphabetism: people feel and really are shut out from participating fully in their communities. The economic costs of lack of number sense might easily be staggering (as percentage of GNP).


Most of the research publications (mentioned on this page) are on the development of number sense in youg children. That is not exactly the notion of number sense that I am aiming at in the exposition above, yet they might be relevant in developing a broader conception of number sense, number sense as one important goal of math education, number sense in adult life.






Valerie F. Reyna, Wendy L. Nelson, Paul K. Han & Nathan F. Dieckmann (2009). How Numeracy Influences Risk Comprehension and Medical Decision Making. Psychological Bulletin, 135, 943-973. Open access http://goo.gl/JuJ1zD


Anders dan de goeroes van de 21e eeuw ons willen doen geloven, gaat het dus niet om creativiteit en probleemoplossen, maar om zoiets eenvoudigs als getalbegrip. Dan ligt het ook voor de hand om te opperen dat wat slecht is voor de eigen gezondheid, dat ook is voor de eigen economie, en voor de economie van ons allen.

Waar gaat het over, over contextproblemen? Jawel, en tamelijk ingewikkelde ook nog (wie het laatste boek van Kahneman [Thinking fast and slow] heeft gelezen, zal een en ander herkennen), maar daarnaast ook over eenvoudiger zaken, bijvoorbeeld of 0,001 kleiner is dan een kans van een op honderd.

Een probleem dat ik in dit artikel zie is het volgende: er worden veel onderzoekresultaten aangehaald die laten zien dat volwassenen met gering getalbegrip kwetsbaarder zijn voor allerlei vormen van misduiding van situaties en getallen. Hier is een andere verklaring uiteraard dat die grotere kwetsbaarheid samengaat met geringere intellectuele capaciteiten; ik heb bij het diagonaal doornemen niet gezien dat de auteurs dit probleem bespreken. Hoe dat ook zij: geringere intellectuele capaciteiten hoeven het onderwijs niet te beletten ook deze leerlingen tot een behoorlijk niveau van getalbegrip te brengen.

Een ander probleem dat ik bij herhaling tegenkom: de testjes en toetsen op getalbegrip gebruiken vooral contextopgaven, of op zijn minst benoemde getallen. Ik ben bang dat in de meeste van deze tests geen onderscheid is gemaakt tussen het testen op getalbegrip als zodanig, en het aanpakken van contextopgaven. Ik ben benieuwd of ik in de literatuur bij dit artikel daar uitzonderingen op tegenkom. Ik wil dolgraag weten of onderwijs dat gericht is op getalbegrip als zodanig, als nevenopbrangst heeft dat dan ook contextopgaven op eenvoudige wijze kunnen worden aangepakt omdat het werken met de getallen zelf geen bijzondere aandacht behoeft.

Bij lezing van dit artikel weten we weer waar we het allemaal voor doen. Dit lijkt me absoluut een sleutelpublicatie voor de vraag waar het makkelijk om kunnen gaan met getallen en hun eenvoudige bewerkingen eigenlijk nuttig voor is. Want deze auteurs halen het echt niet in hun hoofd om de rekenmachine als oplossing in stelling te brengen!

Het is hiermee nog een open vraag of basis- en voortgezet onderwijs dat is ingericht naar de dogmatiek van het realistisch rekenen, de leerlingen beter toerust op hun volwassen leven dan rekenonderwijs dat is gericht op rekenvaardigheid als zodanig, los van toepassingen. Dit is natuurlijk het transfer-probleem. In de filosofie van het constructivisme, en van Adri Treffers, zou het transferprobleem kwijt zijn te spelen door van meet af aan het rekenonderwijs in contexten te plaatsen. Wie daar ook maar even verder op doordenkt, zal moeten constateren dat het onzin is. Waarschijnlijk is het transferprobleem onoplosbaar zolang het als transferprobleem, en niet als leerprobleem wordt beschouwd. Dat klinkt raadselachtig, maar er is niet meer mee bedoeld dan dat leren bijna per definitie leren in nieuwe situaties is. Om transferproblemen te begrijpen, is inzicht in leerprocessen nodig. Zie verder het boek van Stellan Ohlsson (2011) ‘Deep Learning’ aantekeningen daarbij.



Kinga Morsanyi, Chiara Busdraghi and Caterina Primi (2014). Mathematical anxiety is linked to reduced cognitive reflection: a potential road from discomfort in the mathematics classroom to susceptibility to biases. Behavioral and Brain Functions 2014, 10:31 doi:10.1186/1744-9081-10-31 open access





Sleutelpublicaties zijn o.a. van Susan Carey, zie haar pdf.







Carmen Brankaer , Pol Ghesquière & Bert DeSmedt (2014). Numerical magnitude processing deficits in children with mathematical difficulties are independent of intelligence. Research in Developmental Abilities, 35, 2603-2613. abstract [researchgate.net for the article itself]




M. G. Tosto cs. (2014). Why do we differ in number sense? Evidence from a genetically sensitive investigation. Intelligence, 42, 35-46. free access




Magda Praet & Annemie Desoete (2014). Enhancing young children’s arithmetic skills through non-intensive, computerised kindergarten interventions: A randomised controlled study. Teaching and Teacher Education, 39, 56-65. open access en abstract




Lance J. Rips (2013). How many is a zillion? Sources of number distortion. Journal of Experimental Psychology: Learning, Memory, and Cognition, 39, 1257-1264. concept




Pui-Wa Lei, Qiong Wu, James C. DiPerna and Paul L. Morgan (2009). Developing Short Forms of the EARLI Numeracy Measures : Comparison of Item Selection Methods. Educational and Psychological Measurement, 69, 825-842. abstract


Altijd interessant om te bestuderen hoe getalbegrip wordt geoperationaliseerd en testbaar gemaakt. Ik moet dit wel eens bestuderen, maar nu even niet.



Purpura, D. J., Baroody, A. J., & Lonigan, C. J. (2013, March 18). The Transition From Informal to Formal Mathematical Knowledge: Mediation by Numeral Knowledge. Journal of Educational Psychology. Advance online publication. doi: 10.1037/a0031753 researchgate.net



Andreas Obersteiner, Kristina Reiss & Stefan Ufer (2013). How training on exact or approximate mental representations of number can enhance first-grade students' basic number processing and arithmetic skills. Learning and Instruction, 23, 125-135. abstract




Meijke E. Kolkman, Evelyn H. Kroesbergen & Paul P.M. Leseman (2013). Early numerical development and the role of non-symbolic and symbolic skills. Learning and Instruction, 25, 95-103. abstract


Dit onderzoek is van belang voor de didactiek van het rekenonderwijs in groep twee en drie. Mijn werkhypothese is dat het rekenonderwijs in groep drie bepalend is voor de kwaliteit van het rekenonderwijs, en vooral ook voor de eerlijkheid van dat rekenonderwijs. Krijgen alle leerlingen de gelegenheid hun rekenvaardigheid goed te ontwikkelen, of maakt de school al vroeg onderscheid tussen goudhaantjes en lelijke eendjes (als de school dat doet, dan valt te vrezen dat het niets wordt met die lelijke eendjes, die anders prachtige zwanen hadden worden).



Meijke E. Kolkman (2013). Numerical development and the role of working memory. Reken erop! Ontwikkeling van numerieke vaardigheden en de rol van werkgeheugen (met een samenvatting in het Nederlands). Proefschrift Utrecht. ophalen pdf



Booth, J. L., & Siegler, R. S. (2008). Numerical magnitude representations influence arithmetic learning. Child Development, 79, 1016e1031. http://dx.doi.org/ 10.1111/j.1467-8624.2008.01173.x researchgate.net Grade 1. "Thus, representations of numerical magnitude are both correlationally and causally related to arithmetic learning."



Gilmore, C. K., McCarthy, S. E., & Spelke, E. S. (2010). Non-symbolic arithmetic abilities and mathematics achievement in the first year of formal schooling. Cognition, 115, 394e406. http://dx.doi.org/10.1016/j.cognition.2010.02.002 info



Daniel C. Hyde, Saeeda Khanum & Elizabeth S. Spelke (2014). Brief non-symbolic, approximate number practice enhances subsequent exact symbolic arithmetic in children. Cognition, 131, 92-107. doi:10.1016/j.cognition.2013.12.007 free access



Jordan, N. C., Glutting, J., & Ramineni, C. (2010). The importance of number sense to mathematics achievement in first and third grade. Learning and Individual Differences, 20, 82e88. http://dx.doi.org/10.1016/j.lindif.2009.07.004.



Mundy, E., & Gilmore, C. K. (2009). Children’s mapping between symbolic and nonsymbolic representations of number. Journal of Experimental Child Psychology, 103, 490e502. http://dx.doi.org/10.1016/j.jecp.2009.02.003



Inglis, M., Attridge, N., Batchelor, S., & Gilmore, C. (2011). Non-verbal number acuity correlates with symbolic mathematics achievement: but only in children. Psychonomic Bulletin & Review, 18, 1222e1229. http://dx.doi.org/10.1016/j.neuron.2006.11.022.



Alex Hebra (2003). Measure for measure. The story of imperial, metric, and other units. The Johns Hopkins University Press. books.google isbn 0801870720


Benoemde getallen! Ik heb nog geen tijd genomen om er wat beter in te kijken. Bladerend viel me de behandeling op van dimensieloze formules, zeg maar werken met benoemde getallen zonder ze noodzakelijk ook te hoeven benoemen. Voor het rekenonderwijs niet oninteressant. Een gedachte die mij ook al snel bekroop bij het doorbladeren: er zijn nogal wat onderscheiden eenheden, dus wat is eigenlijk nut en noodzaak van het behandelen van de meest voorkomende eenheden in het rekenonderwijs? Dat is dus de klassieke vraag of leerlingen niet eerst het rekenen tot op grote hoogte zouden moeten beheersen, voordat zij worden ingeleid in het werken met benoemde getallen? Zie overigens voor het thema van meten mijn webpagina hier.



Ellen Peters & Par Bjalkebring (2014). Multiple Numeric Competencies: When a Number Is Not Just a Number. Journal of Personality and Social Psychology, Oct 6 , 2014, No Pagination Specified. abstract



Sonia L. J. White, Dénes Szücs & Fruzsina Soltész (2012). Symbolic number: The integration of magnitude and spatial representations in children agd 6 to 8 years. Frontiers in Psychology, pdf.



A. M. Leslie, Rochel Gelman & C. R. Gallistel (2008). The generative basis of natural number concepts Trends in Cognitive Sciences, 12, 213-218. pdf



C. R. Gallistel & Rochel Gelman (2005) Mathematical Cognition In K Holyoak & R. Morrison The Cambridge handbook of thinking and reasoning (559-588). Cambridge University Press. pdf



Steven T. Piantadosi, Joshua B. Tenenbaum & Noah D. Goodman (2012). Bootstrapping in a language of thought: A formal model of numerical concept learning. Cognition, 123 199-217. pdf



Kathryn Davidson, Kortney Eng & David Barner (2012). Does learning to count involve a semantic induction? Cognition, 123 162-173. pdf



Sashank Varma & Daniel L. Schwartz (2011). The mental representation of integers: An abstract-to-concrete shift in the understanding of mathematical concepts. Cognition, 121 363-385. abstract



Robert S. Siegler, Clarissa A. Thompson & Michael Schneider (2011 accepted). An integrated theory of whole number and fractions development. Cognitive Psychology. pdf



Ian M. Lyons & Sian L. Beilock (2011). Numerical ordering ability mediates the relation between number-sense and arithmetic competence. Cognition, 121, 256-261. pdf



Julie Sarama & Douglas H. Clements (2009). Early Childhood Mathematics Education Research. Learning Trajectories for Young Children. Routledge.



Nancy C. Jordan, Joseph Glutting & Chaitanya Ramineni (2010). The importance of number sense to mathematics achievement in first and third grades. Learning and Individual Differences, 20, 82-88.

Susan D. Nickerson & Ian Whitacre (2010): A Local Instruction Theory for the Development of Number Sense, Mathematical Thinking and Learning, 12:3, 227-252. abstract

Mary K. Hoard, David C. Geary & Carmen O. Hamson (1999). Numerical and arithmetical cognition: Performance of low- and average-IQ children. Mathematical Coognition, 5, 65-91. pdf



Ian M. Lyons & Sian L. Beilock (2011). Numerical ordering ability mediates the relation between number-sense and arithmetic competence. Cognition, 121, 256-261 pdf



David Barner & Jesse Snedeker (2005). Cognition, 97, 41-66. pdf



Jennifer S. Lipton & Elizabeth Spelke (2006). Preschool children master the logic of number word meanings. Cognition, 98, B57-B66. abstract



Catherine Sophian & Yun Chu (2008). How do people apprehend large numerosities? Cognition, 197, 460-478. abstract



Alexis Palmer & Arthur J. Baroody (2011). Blake’s development of the number words “one”, “two” and “three”. Cognition and Instruction, 29, 265-296. abstract



Ian M. Lyons & Sian L. Beilock (2011 in press). Numerical ordering ability mediates the relation between number-sense and arithmetic competence. Cognition, 121. abstract



Gavin R. Price, Michèle M. M. Mazzocco & Daniel Ansari (2013). Why Mental Arithmetic Counts: Brain Activation during Single Digit Arithmetic Predicts High School Math Scores. The Journal of Neuroscience, 33, 156 –163. pdf


Might be kind of proof that losing automaticity in basic arithmetics might harm further mathematical work.



Stephanie Bugden & Daniel Ansari (2011). Individual differences in children's mathematical competence are related to the intentional but not automatic processing of Arabic numerals. Cognition, 118, 32-44. abstract, Numerical Cognition Laboratory : publications


Misschien is er wel een verband met dit onderzoek html [via De Bruyckere, 30-1-2014].



Stanislas Dehaene (1997). The Number Sense. How the Mind Creates Mathematics. Oxford University Press. isbn 0195110048 info newer revised edition



Jean Piaget, Kurt Resag, Arnold Fricke, P. M. van Hiele & Karl Odenbach (1970/1973). Rekenonderwijs en getalbegrip. Bosch & Leuning. Vertaling en bewerking door J. Snijders. Oorspronkelijke titel: Rechenunterricht und Zahlbegriff. Westermann Verlag.


Karl Odenbach: Ter inleiding.

Kurt Resag: De vorming van het getalbegrip bij het kind.

Jean Piaget: De ontwikkeling van het getalbegrip bij het kind.

Arnold Fricke: Operatief denken in het rekenonderwijs als toepassing van de psychologie van Piaget.

P. M. van Hiele: Piagets bijdrage tot ons inzicht in de kinderlijke vorming van het getalbegrip.

Arnold Fricke: Operatief getalbegrip.



Kristin Krajewski & Wolfgang Schneider (2009). Early development of quantity to number-word linkage as a precursor of, mathematical school achievement and mathematical difficulties: Findings, from a four-year longitudinal study. Learning and Instruction, 19, 513-526 abstract



Kristen P. Blair, Miriam Rosenberg-Lee, Jessica M. Tsang, Daniel L. Schwartz & Vinod Menon (2012). Beyond Natural Numbers: Negative Number Representation in Parietal Cortex. Frontiers in Human Neuroscience, volume 6 html


Getallenlijn



Paul Ernest (1985). The number line as a teaching aid. Educational Studies in Mathematics, 16, 411-424. abstract


Over de getallenlijn. Paul Ernest is een sociaal constructivist, dat is dan weer minder. . Nuttig kader.



Michael Schnieder, Roland H. Grabner & Jennifer Paetsch (2009). Mental Number Line, Number Line Estimation, and Mathematical Achievement: Their Interrelations in Grades 5 and 6Journal of Educational Psychology, 101, 359-372. (getallenlijn)



Véronique Izard & Stanislas Dehaene (2008). Calibrating the mental number line. Cogniiton, 106, 1221-1247. pdf of online first - abstract



Rafael Núñez, D. Doan & Anastasia Nikoulina (2011). Squeezing, striking, and vocalizing: Is number representation fundamentally spatial? Cogniiton, 120, 225-235. pdf

from the Conclusion Spatial representations for number appear to be, out of the multiple possibilities, a specialized and highly useful type, but one that is only the most visible tip of the iceberg resting on this fundamental magnitude sense. While nonlinear number mappings may be more fundamental, the powerful linear number-to-line mappings are ubiquitous and culturally prominent in the modern world, as a result of a long history of cultural practices and ongoing refinement.



McCrink, K., Dehaene, S., & Dehaene-Lambertz, G. (2007) Moving along the number line: The case for operational momentum. Perception and Psychophysics, 69(8), pp. 1324-1333. pdf download


Tellen



I. Lyons & S. L. Beilock (2009). Beyond quantity: Individual differences in working memory and the ordinal understanding of numerical symbols. Cognition, 113, 189-204.



Henry Railo, Mika Koivisto, Antti Revonsuo & Minna M. Hannula (2008). The role of attention in subitizing. Cogniiton, 107, 82-104. abstract



Sharon Sui Ngan Ng & Nirmalo Rao (2010). Chinese number words, culture, and mathematics learning. Review of Educational Research, 80, 180-206. abstract




Kees Buys (1991). Telactiviteiten voor kleuters. Bekadidact.


Dit is een boek voor leerkrachten. Het wordt waarschijnlijk ook op een aanta pabo’s gebruikt. Het boek kan waarschijnlijk dienen als voorbeeld van rekenactiviteiten in de eerste twee groepen van het basisonderwijs in Nederland. Een interessant aspect van de didactische uitwerkingen van Kees Buijs (zijn naam komt alleen op kaft en titelblad voor, gespeld met Griekse y) zijn de uitbundige contexten die hij construeert.



Barbara W. Senecka & Susan Carey (2008). How counting represents number: what children must learn and when they learn it. Cognition, 108, 662-674. abstract



Catherine Sophian (2007). The origins of mathematical knowledge in childhood. Lawrence Erlbaum.


Chapter 2: Children's counting. pp. 15-41.

Chapter 3: Quantitative comparison without numbers. pp. 42-63.

Chapter 4: Understanding units. pp. 64-83.

Sophian, C. (2004). A prospective developmental perspective on early mathematics instruction. In Clements, D. H., Sarama, J., &.DiBiase, A.-M. (Eds.) Engaging young children in mathematics: Findings of the 2000 national conference on standards for preschool and kindergarten mathematics education. Erlbaum. [nog niet gezien, googel op de tekst van het abstract om het hoofdstuk in google.books te kunnen lezen] abstract A prospective developmental perspective is concerned with the impact of early mathematics instruction on aspects of mathematics learning that become important later in development. The modes of instruction that produce the greatest immediate learning may not always be the ones that are best for the long term. Accordingly, instructional standards should be directed toward maximizing the long-term as well as the short-term effectiveness of early mathematics instruction. I will focus on the arena of fraction learning to develop this idea. I will suggest that much of the difficulty in fraction learning stems from conceptual issues that are not unique to fractions, but that have their origins in the way very young children think about counting and whole-number quantities.



Camilla K. Gilmore, Shannon E. McCarthy & Elizabeth S. Spelke (2010). Non-symbolic arithmetic abilities and mathematics achievement in the first year of formal schooling. Cognition, 115, 394-406. abstract



Bonawitz, EB, Shafto P, Gweon H, Goodman N, Spelke E, Schulz LE. In Press.  The double-edged sword of pedagogy: Teaching limits children?s spontaneous exploration and discovery. Cognition. mention



Hyde, DC, Spelke ES. 2011.  Neural signatures of number processing in human infants: Evidence for two core systems underlying numerical cognition. Developmental Science. 14(2):360-371. abstract



Jean-Philippe van Dijck, Wim Gevers & Wim Fias (2009). Cognition, 113, 248-253 abstract



Maria-Dolores de Hevia & Elizabeth S. Spelke (2009). Spontaneous mapping of number and space in adults and young children. Cognition, 110, 198-207. abstract



Jordan, N. C., Glutting, J., Dyson, N., Hassinger-Das, B., & Irwin, C. (2012, June 18). Building Kindergartners' Number Sense: A Randomized Controlled Study. Journal of Educational Psychology. Advance online publication. doi: 10.1037/a0029018



Wim Fias, Marc Brysbaert, Frank Geypens, and Géry d’Ydewalle (1996). The Importance of Magnitude Information in Numerical Processing: Evidence from the SNARC Effect. Mathematical Cognition, 1, 95-110. pdf



Tijdschrift: Numeracy. http://scholarcommons.usf.edu/numeracy/



Valérie Camos & Pierre Barrouillet (2004). Adult counting is resource demanding. British Journal of Psychology, 95, 19-30. abstract


Verschillende theorieën over de mentale belasting bij het tellen. Ik heb geen afzonderlijke webpagina voor tellen, vandaar de plaatsing bij getalbegrip. Interessante literatuurlijst. Zal ik die maar eens hier plaatsen?

Anderson, J. R. (1993). The rules of the mind. Hillsdale, NJ: Erlbaum. 
Anderson, J. R., & Lebiere, C. (1998). The atomic components of thought. Mahwah, NJ: Erlbaum.
Anderson, J. R., Reder, L. M., & Lebiere, C. (1996). Working memory: Activation limitations on
retrieval. Cognitive Psychology, 30, 221-256.
Baddeley, A. D. (1990). Human memory, theory and practice. Hillsdale, NJ: Erlbaum.
Beckwith, M., & Restle, F. (1966). Process of enumeration. Psychological Review, 73, 437-444.
Camos, V. (1998). La gestion d’une activité complexe: L’exemple du dénombrement. Approche
développementale [Managing a complex activity: The example of counting. Developmental
approach]. Unpublished doctoral dissertation, Université de Bourgogne.
Camos, V. (2003). Coordination process in counting. International Journal of Psychology, 38,
24-36.
Camos, V., Barrouillet, P., & Fayol, M. (2001). Does the coordination of verbal and motor
information explain the development of counting in children? Journal of Experimental
Child Psychology, 78(3), 240-262.
Camos, V., Fayol, M., & Barrouillet, P. (1999). Le dénombrement chez l’enfant: Double-tâche ou
procédure? [Counting in children: Dual-task or procedure?]. L’Année Psychologique, 99,
623-645 [in French].
Case, R. (1985). Intellectual development: Birth to adulthood. New York: Academic Press.
Case, R. (1992). The mind’s staircase: Exploring the conceptual underpinnings of children’s
thought and knowledge. Hillsdale, NJ: Erlbaum.
Case, R., Kurland, M., & Goldberg, J. (1982). Operational efficiency and the growth of short-term
memory span. Journal of Experimental Child Psychology, 33, 386-404.
Cowan, N. (1988). Evolving conceptions of memory storage, selective attention, and their mutual
constraints within the human information processing system. Psychological Bulletin, 104,
163-191.
Cowan, N. (1995). Attention and memory: An integrated framework. Oxford: Oxford University
Press.
Crannel, C.W., & Parrish, J. M. (1957). A comparison of immediate memory span for digits, letters,
and words. The Journal of Psychology, 44, 319-327.
Dempster, F. N. (1981). Memory span: Sources of individual and developmental differences.
Psychological Bulletin, 89, 63-100.
Fuson, K. C. (1988). Children’s counting and concepts of number. New York: Springer.
Halford, G. S. (1993). Children’s understanding. Hillsdale, NJ: Erlbaum.
Halford, G. S. (1999). The properties of representations used in higher cognitive processes:
Developmental implications. In Sigel, I. E. (Ed.). Development of mental representation:
Theories and applications (pp. 147-168). Mahwah, NJ: Erlbaum.
Lovett, M. C., Reder, L. M., & Lebiere, C. (1999). Modeling working memory in a united architecture: An ACT-R perspective. In A. Miyake & P. Shah (Eds.). Models of working memory: Mechanisms of active maintenance and executive control (pp. 135-182). Cambridge: Cambridge University Press.[PSYCHO C3.0.-159]
Potter, M. C., & Levy, E. I. (1968). Spatial enumeration without counting. Child Development, 39,
265-272.
Towse, J. N. (1993). Cognitive processes in object counting. Unpublished doctoral dissertation,
University of Manchester.
Towse, J. N., & Hitch, G. J. (1995). Is there a relationship between task demand and storage space
in tests of working memory capacity? Quarterly Journal of Experimental Psychology, 48A,
108-124.
Towse, J. N., & Hitch, G. J. (1997). Integrating information in object counting: A role for a central
coordination process? Cognitive Development, 12, 393-422.
Towse, J. N., Hitch, G. J., & Hutton, U. (1998). A reevaluation of working memory capacity in
children. Journal of Memory and Language, 39, 195-217.
Towse, J. N., Hitch, G. J., & Hutton, U. (2000). On the interpretation of working memory span in
adults. Memory and Cognition, 28, 341-348.
Tuholski, S. W., Engle, R. W., & Baylis, G. C. (2001). Individual differences in working memory
capacity and enumeration. Memory and Cognition, 29, 484-492.



Kevin F. Miller & James W. Stigler (1991). Meanings of Skill: Effects of Abacus Expertise on Number Representation. Cognition and Instruction, 8, 29-67. abstract: http://www.jstor.org/discover/10.2307/3233509



Kucian K, Grond U, Rotzer S, Henzi B, Schönmann C, Plangger F, Gälli M, Martin E, von Aster M (2011). Mental number line training in children with developmental dyscalculia. NeuroImage, 57, 782-795. abstract



T. Käser, K. Kucian, M. Ringwald, G. Baschera, M. von Aster, M. Gross (2011?). Therapy software for enhancing numerical cognition. pdf



Samantha Bouwmeester & Peter P. J. L. Verkoeijen (2012): Multiple Representations in Number Line Estimation: A Developmental Shift or Classes of Representations? Cognition and Instruction, 30, 246-260. abstract



Geary DC, Hoard MK, Nugent L, Bailey DH (2013) Adolescents’ Functional Numeracy Is Predicted by Their School Entry Number System Knowledge. PLoS ONE 8(1): e54651. doi:10.1371/journal.pone.0054651 article



Madhubalan Viswanathan (1993). Measurement of Individual Differences in Preference for Numerical Information. Journal of Applied Psychology, 78, 741-752. tekst




Paolos, J. A. (1988). Innumeracy: Mathematical illiteracy and its consequences. New Ybrk: Hill & Wang.




Lynn Arthur Steen (1999). Numeracy: The new literacy for a data-drenched society. Educational Leadership, 57, 8-13.pdf




Steen, L. A. (1990). Numeracy. Daedalus, 119, 211-231.abstract




International Numeracy Survey. A Comparison of the Basic Numeracy Skills of Adults 16-60 in Seven Countries. pdf [nog ophalen]. ERIV tijdelijk niet beschikbaar




James D. Eyring, Debra Steele Johnson, and David J. Francis (1993). A Cross-Level Units-of-Analysis Approach to Individual Differences in Skill Acquisition. Journal of Applied Psychology, 78, 805-814. abstract




Mark H. Ashcarft & Alex M. Moore (2012). Cognitive processes of numerical estimation in children. Journal of Experimental Child Psychology, 111, 246-267. researchgate




Elida Laski & Qingyi Yu (2014). Number line estimation and mental addition: Examining the potential roles of language and education. Journal of Experimental Child Psychology, 117, 29-44. [geen pdf tot mijn beschikking] abstract




Dimona Bartelet, Anniek Vaessen, Leo Blomert & Daniel Ansaric (2014). What basic number processing measures in kindergarten explain unique variability in first-grade arithmetic proficiency? Journal of Experimental Child Psychology, 117, 12-28. [geen pdf tot mijn beschikking] abstract




Iro Xenidou-Dervou, Bert De Smedt, Menno van der Schoot & Ernest C.D.M. van Lieshout (2013). Individual differences in kindergarten math achievement: The integrative roles of approximation skills and working memory. Learning and Individual Differences, 28, 119-129. [geen pdf tot mijn beschikking] abstract




Ilona Friso-van den Bos, Sanne H.G. van der Ven, Evelyn H. Kroesbergen & Johannes E.H. van Luit (2013). Working memory and mathematics in primary school children: A meta-analysis. Educational Research Review, 10, 29-44. [geen pdf tot mijn beschikking] abstract


Ik heb geen pdf van dit artikel, maar er is wel een lijst beschikbaar van publicaties (alleen van Elsevier) op dit thema: zie hier 84 Skills underlying mathematics: The role of executive function in the development of mathematics proficiency Review Article Trends in Neuroscience and Education, In Press, Corrected Proof, Available online 9 January 2014 Lucy Cragg, Camilla Gilmore  Open Access 85 A New Dimension to Teaching Mathematics Using iPads Procedia - Social and Behavioral Sciences, Volume 103, 26 November 2013, Pages 51-54 Diana Audi, Rim Gouia-Zarrad  Open Access     91 The Impact of Web Sites on Teaching and Learning Mathematics Procedia - Social and Behavioral Sciences, Volume 93, 21 October 2013, Pages 631-635 Jarmila Robová  Open Access 93 Domain-general mediators of the relation between kindergarten number sense and first-grade mathematics achievement Journal of Experimental Child Psychology, Volume 118, February 2014, Pages 78-92 Brenna Hassinger-Das, Nancy C. Jordan, Joseph Glutting, Casey Irwin, Nancy Dyson   96 The dual processes hypothesis in mathematics performance: Beliefs, cognitive reflection, working memory and reasoning Learning and Individual Differences, Volume 29, January 2014, Pages 67-73 Inés Mª Gómez-Chacón, Juan A. García-Madruga, José Óscar Vila, Mª Rosa Elosúa, Raquel Rodríguez   100 Measuring Efficiency of Teaching Mathematics Online: Experiences with WeBWorK Procedia - Social and Behavioral Sciences, Volume 89, 10 October 2013, Pages 276-282 Peter Fejes Toth  Open Access -->



Barbara W. Sarnecka (2014). On the relation between grammatical number and cardinal numbers in development. Frontiers in Psychology. Developmental Psychology. free access html or pdf




Wim Van Dooren, Erno Lehtinen & Lieven Verschaffel (2015). Unraveling the gap between natural and rational numbers. Learning and Instruction, in press




Sarah Linsen, Lieven Verschaffel, Bert Reynvoet, Bert De Smedt (2015). The association between numerical magnitude processing and mental versus algorithmic multi-digit subtraction in children. Learning and Instruction, 35, 42-50.




Ian M. Lyons, Daniel Ansari & Sian L. Beilock (2015). Qualitatively Different Coding of Symbolic and Nonsymbolic Numbers in the Human Brain. Human Brain Mappng, 36, 475-488. pdf




Helen C. Reed, Michelle Gemmink, Marije Broens-Paffen, Paul A. Kirschner & Jelle Jolles (2015). Improving multiplication fact fluency by choosing between competing answers. Research in Mathematics Edcation, 17, 1-19.pdf




Michael Schneider, Kassandra Beeres, Leyla Coban, Simon Merz, S. Susan Schmidt, Johannes Stricker, Bert De Smedt (2015, 2nd revision). Associations of Non-Symbolic and Symbolic Numerical Magnitude Processing with Mathematical Competence: A Meta-analysis. pdf researchgate




Birgit Knudsen, Martin H. Fischer, Anne Henning, Gisa Aschersleben (2015). The Development of Arabic Digit Knowledge in 4- to 7-Year-Old Children. Journal of Numerical Cognition, 1, 21-37. free access




Andriy Andriy Myachykov, Rob Ellis, Angelo Cangelosi, Martin H. Fischer (2016). Ocular drift along the mental number line. Psychological Research. abstract




Michael Schneider, Kassandra Beeres, Leyla Coban, Simon Merz, S. Susan Schmidt, Johannes Stricker & Bert De Smedt (2016). Associations of non-symbolic and symbolic numerical magnitudeprocessing with mathematical competence: a meta-analysis. Developmental Science researchgate.net


Everything correlational here. In this context it is cofusing to talk of ‘effect sizes’: nowhere is there any causal relation in sight.



Vanbinst K, Ansari D, Ghesquière P, De Smedt B (2016) Symbolic Numerical Magnitude Processing Is as Important to Arithmetic as Phonological Awareness Is to Reading. PLoS ONE 11(3): e0151045. doi:10.1371/journal.pone.0151045 online




Joke Torbeyns, Greet Peters, Bert De Smedt, Pol Ghesquiere and Lieven Verschaffel (2016). Children's understanding of the addition/subtraction complement principle. British Journal of Educational Psychology [researched.net] abstract




Anna A. Matejko & Daniel Ansari (2016). Trajectories of Symbolic and Nonsymbolic Magnitude Processing in the First Year of Formal Schooling. PLoS ONE (Impact Factor: 3.23). 03/2016; 11(3):e0149863. DOI: 10.1371/journal.pone.0149863 researchgate




Rebecca Merkley, Daniel Ansari (2016). Why numerical symbols count in the development of mathematical skills: evidence from brain and behavior. Current Opinion in Behavioral Sciences, 10, 14–20. open access




K. Vanbinst, B. De Smedt (2016). Individual differences in children’s mathematics achievement: The roles of symbolic numerical magnitude processing and domain-general cognitive functions




Jessica M. Tsang, Kristen P. Blair, Laura Bofferding & Daniel L. Schwartz (2015). Learning to 'See' Less Than Nothing: Putting Perceptual Skills to Work for Learning Numerical Structure. Cognition and Instruction, 33, 154-197 pdf




Matthias Hartmann, Martin H. Fischer (2016). Exploring the numerical mind by eye-tracking: a special issue. Psychological Research, 80, Issue 3, May 2016 page




J. Mock, S. Huber, E. Klein, K. Moeller (2016). Insights into numerical cognition: considering eye-fixations in number processing and arithmetic Psychological Research (2016) 80:334–359 DOI 10.1007/s00426-015-0739-9 abstract




Titia Gebuis, Roi Cohen Kadosh, Wim Gevers (2016). Sensory-integration system rather than approximate number system underlies numerosity processing: A critical review Acta Psychologica researchgate




Michael Schneidera, Angela Heine, Verena Thaler, Joke Torbeyns, Bert De Smedt, Lieven Verschaffel, Arthur M. Jacobs, Elsbeth Stern (2008). A validation of eye movements as a measure of elementary school children's developing number sense. Cognitive Development, 23, 424-437.




Ben Wilbrink (6 Jan 2017) Chief economist of Bank of England Andrew Haldane, Jan 5 2017, on numeracy and economy ‪https://www.theguardian.com/business/2017/jan/05/chief-economist-of-bank-of-england-admits-errors?CMP=share_btn_tw ‬ tweet




(MAR. 17, 2017 AT 11:07 AM). Why Do We Count? By Craig Fehrman book review




Groot, W. & Maassen van den Brink, H. (2006). Stil Vermogen, een onderzoek naar de maatschappelijke kosten van laaggeletterdheid. Den Haag: Stichting Lezen en Schrijven


De analogie met laaggecijferdheid ligt voor de hand, niet?



Ellen Peters, Brittany Shoots-Reinhard, Mary Kate Tompkins, Louise Meilleur, Aleksander Sinayev, Martin Tusler, Laura Wagner and Jennifer Crocker (2017). Improving numeracy through values affirmation enhances decision and STEM outcomes. PLoS ONE 12(7): e0180674. https://doi.org/10.1371/journal.pone.0180674 open access




Anna Shusterman, Pierina Cheung, Jessica Taggart, Ilona Bass, Talia Berkowitz, Julia A. Leonard, Ariel Schwartz (2017). Conceptual Correlates of Counting: Children’s Spontaneous Matching and Tracking of Large Sets Reflects Their Knowledge of the Cardinal Principle Journal of Numerical Cognition open access




Schneider, M., Merz, S., Stricker, J., De Smedt, B., Torbeyns, J., Verschaffel, L., & Luwel, K. (in press 2018). Associations of number line estimation with mathematical competence: A meta-analysis. Child Development. researchgate.net




On the relation between the mental number line and arithmetic competencies Tanja Link, Hans-Christoph Nuerk & Korbinian Moeller . The Quarterly Journal of Experimental Psychology Volume 67, 2014 - Issue 8: Spatial biases in Mental Arithmetic abstract




G. B. Chapman & J. Liu (2009) ‘Numeracy, frequency, and Bayesian reasoning. Judgment and decision making, 4, 34-40 open


Bayesian reasoning is very difficult!



Parallel and serial processes in number-to-quantity conversion. Dror DotanStanislas Dehaene (2020). preprint
















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On the relation between the mental number line and arithmetic competencies Tanja Link,Hans-Christoph Nuerk &Korbinian Moeller