Literature: Cognition and mathematics in education

Rekenproject: Psychologie en rekenen

Ben Wilbrink

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Andreas Demetriou, Maria Platsidou, Anastasia Efklides, Yiota Metallidou & Michael Shayer (1991). The development of quantitative-relational abilities from childhood to adolescence: Structure, scaling, and individual differences. Learning & Instruction, 1, 19-43. online?

• The present article discusses the nature of the quantitative-relational system, which involves the basic arithmetic operations, proportional reasoning, algebraic ability, and the integration of these three abilities. Representative sets of tasks that assess each of these abilities were administered to 372 Greek subjects aged 9 to 16 to test hypotheses about (a) the structure and (b) the development of this cognitive system, and (c) about the role of individual differences variables, such as subjects’ gender and SES. (

Asael Y. Sklar, Nir Levy, Ariel Goldstein, Roi Mandel, Anat Maril & Ran R. Hassin (2012). Reading and doing arithmetic nonconsciously. PNAS, 109 (48) 19614-19619 abstract

Elena Salillas and Nicole Y. Y. Wicha (online first June 15 2012). Early Learning Shapes the Memory Networks for Arithmetic: Evidence From Brain Potentials in Bilinguals. Psychological Science, OnlineFirst. abstract

• The goal of this study was to determine whether LA+ establishes the quality and strength of representations for multiplication facts in adult bilinguals, and whether these representations are independent of natural language dominance. We addressed this question by measuring behavior (in a reaction time, RT, experiment) and brain electrical responses (in an ERP experiment) to multiplication-fact retrieval in adult bilinguals. From this data, we inferred the strength and interconnection of arithmetic fact representations in each language. We selected Spanish-English bilinguals who learned both languages early in childhood and who were proficient in both languages but learned arithmetic in only one. Critically, the specific language in which arithmetic was learned (Spanish or English) was independent of which language was more dominant.
• Thus, a connection between language and math is established at the time of learning and maintained into adulthood independently of natural language dominance. This finding may have important implications for teaching and testing young bilinguals, and it highlights the idea that bilinguals should not be treated as two monolinguals in one brain (Grosjean, 1989).
• ook: taal.htm#Salillas_Wicha
• Check in ieder geval ook nog de volgende items uit de literatuuropgave
  • Dehaene, S., & Cohen, L. (1995). Toward an anatomical and functional model of number processing. Mathematical Cognition, 1, 83 - 120.
  • Dehaene, S., Spelke, E., Pinel, P., Stanescu, R., & Tsivkin, S. (1999). Sources of mathematical thinking: Behavioral and brain-imaging evidence. Science, 284, 970 - 974.
  • Delazer, M., & Benke, T. (1997). Arithmetic facts without meaning. Cortex, 33, 697 - 710.
  • Frenck-Mestre, C., & Vaid, J. (1993). Activation of number facts in bilinguals. Memory & Cognition, 21, 809 - 818.
  • Galfano, G., Rusconi, E., & Umilta, C. (2003). Automatic activation of multiplication facts: Evidence from the nodes adjacent to the product. Quarterly Journal of Experimental Psychology, 56A, 31 - 61.
  • Gelman, R., & Butterworth, B. (2005). Number and language: How are they related? Trends in Cognitive Sciences, 9, 6 - 10.
  • Grosjean, F. (1989). Neurolinguists, beware! The bilingual is not two monolinguals in one person. Brain & Language, 36, 3 - 15.
  • LeFevre, J., Bisanz, J., & Mrkonjic, L. (1988). Cognitive arithmetic: Evidence for obligatory activation of arithmetic facts. Memory & Cognition, 16, 45 - 53.
  • McCloskey, M., Marcuso, P., & Whetstone, T. (1992). The nature and origins of mathematical skills. Amsterdam, The Netherlands: Elsevier.
  • Spelke, E. S., & Tsivkin, S. (2001). Language and number: A bilingual training study. Cognition, 78, 45 - 88.

Mary K. Hoard, David C. Geary, and Carmen O. Hamson (1999). Numerical and Arithmetical Cognition: Performance of Low- and Average-IQ Children. Mathematical Cognition, 5, 65-91. pdf

• abstract Neuropsychological and developmental models of number, counting, and arithmetical skills, as well as the supporting working memory and speed of articulation systems, were used as the theoretical framework for comparing groups of low- and average-IQ children. The low-IQ children, in relation to their average- IQ peers, showed an array of deficits, including difficulties in retaining information in working memory while counting, more problem solving errors, shorter memory spans, and slower articulation speeds. At the same time, the low-IQ children’s conceptual understanding of counting did not differ from that of their higher-IQ peers. Implications for the relation between IQ and mathematics achievement are discussed.
• Geary website: publications

J. Joy Cumming & John Elkins (1999). Lack of Automaticity in the Basic Addition Facts as a Characteristic of Arithmetic Learning Problems and Instructional Needs. Mathematical Cognition, 5, 149-180. abstract

• Analysis by error type showed most errors on the multidigit sums were due to fact inaccuracy, not algorithmic errors. The implication is that the cognitive demands caused by inefficient solutions of basic facts made the multidigit sums inaccessible. Suggestions for instruction of students who have problems learning basic arithmetic are made.

W. H. Winch (1910). Accuracy in school children. Does improvement in numerical accuracy ‘transfer’? Journal of Educational Psychology, 1, 557-589.

Als ik goed heb gekeken, is dit het eerste psychologische onderzoek naar rekenen dat in dit tijdschrift is gepubliceerd. Het is best grappig om te lezen hoe Winch probeert om het nauwkeurig werken (accuracy of numerical computation) te onderscheiden van het redeneren (arithmetical reasoning). Een vroeg onderscheid tussen oefenen en begrijpen? In ieder geval gaat het over sterke ideeën in die tijd over de transfer van nauwkeurig leren rekenen naar nauwkeurigheid op andere gebieden. Winch verwijst niet naar eerder onderzoek van het rekenen; daar was hij waarschijnlijk niet van op de hoogte, of dat was in de VS nog niet gedaan.

Judith T. Sowder, Nadine Bezuk & Larry K. Sowder: Using principles from cognitive psychology to guide rational number instruction for prospective teachers. In Thomas P. Carpenter, Elizabeth Fennema & Thomas A. Romberg (Eds.) (1993). Rational Numbers. An Integration of Research. Erlbaum. (239-259)

Daniel Bajic & Timothy C. Rickard (2011). Toward a generalized theory of the shift to retrieval in cognitive skill learning. Memory & Cognition, 39, 1147-1161. pdf

Jamie I. D. Campbell, Roxanne R. Dowd, Jillian M. Frick, Kendra N. McCallum & Arron W. S. Metcalfe (2011). Neighborhood consistency and memory for number facts. Memory & Cognition, 39, 884-893. abstract

Lynn S. Fuchs, Douglas Fuchs, Donald L. Compton, Sarah R. Powell, Pamela M. Seethaler, Andrea M. Capizzi, Christopher Schatschneider & Jack M. Fletcher (2006). The Cognitive Correlates of Third-Grade Skill in Arithmetic, Algorithmic Computation, and Arithmetic Word Problems. Journal of Educational Psychology, 98, 29-43. abstract

Michael W. Faust, Mark H. Ashcraft & David E. Fleck (1996). Mathematics anxiety effects in simple and complex addition. Mathematical Cognition, 2, 25-62. abstract

David A. Sousa (2008). How the brain learns mathematics. Corwin Press. p style='text-indent: 0em'>

NCTM (2006). Curriculum Focal Points. http://www2.k12albemarle.org/dept/instruction/math/Pages/NCTM-Curriculum-Focal-Points.aspx en pdf

Howard Wainer (1980). A Test of Graphicacy in Childre. Applied Psychological Measurement 4, 331-340. abstract

Magda Colberg, Mary Anne Nester, and Marvin H. Trattner (1985). Convergence of the Inductive and Deductive Models in the Measurement of Reasoning Abilities. Journal of Applied Psychology, 70, 681-694. abstract

M. H. Ashcraft, 1992. Cognitive arithmetic: A review of data and theory. Cognition, 44, 75-106.abstract

Roi Cohen Kadosh & Ann Dowker (Eds.) (2015). The Oxford Handbook of Numerical Cognition. Oxford University Press. [UB Leiden PSYCHO C6.-172] info

• • a.o.
  • Marcus Giaquinto: Philosophy of number 17-34 abstract
  • Oliver Lindemann and Martin H. Fischer: Cognitive Foundations of Human Number Representations and Mental Arithmetic 35-44 researchgate.net (proof)
  • Michael Andres and Mauro Pesenti: Finger-Based Representation of Mental Arithmetic 67-88 abstract
  • Jean-Philippe van Dijck, Véronique Ginsburg, Luisa Girelli and Wim Gevers: Linking Numbers to Space: From the Mental Number Line towards a Hybrid Account 89-105 researchgate.net
  • Hans-Christoph Nuerk, H.-C., Moeller, and Klaus Willmes: Multi-digit Number Processing: Overview, Conceptual Clarifications, and Language Influences 106-139 abstract
  • Jamie I.D. Campbell: How Abstract is Arithmetic? 140-157 abstract
    . . . . the kinds and references of problem operands are fundmentally relevant to cognitive arithmetic, despite being irrelevant to arithmetic as a formal operation.

    p. 154

  • Catherine Thevenot and Pierre Barrouillet: Arithmetic Word Problem Solving and Mental Representations 158-179 abstract
  • Kinga Morsanyi and Denes Szucs: Intuition in Mathematical and Probabilistic Reasoning 180-200 academia.edu
  • Elizabeth M. Brannon and Joonkoo Park: Phylogeny and Ontogeny of Mathematical and Numerical Understanding 203-213
  • Koleen McCrink and Wesley Birdsall: Numerical Abilities and Arithmetic in Infancy 258-274
  • Minna M. Hannula-Sormunen: Spontaneous Focusing on Numerosity and Its Relation to Counting and Arithmetic 275-290
  • Barbara W. Sarnecka, Meghan C. Goldman, Emily B. Slusser: How Counting Leads to Children's First Representations of Exact, Large Numbers 291-309 [in the line of research by Susan Carey]
  • Camilla Gilmore: Approximate Arithmetic Abilities in Childhood 310-330
  • Titia Gebuis and Bert Reynvoet: Number Representations and Their Relation with Mathematical Ability 331-344
  • Kim Uittenhove and Patrick Lemaire: Numerical Cognition during Cognitive Aging 345-364
  • Geoffrey B. Saxe: Culture, Language, and Number 367-376 Here they are, prominently: Vygotsky and Piaget. Skip the chapter, unless you are in the habit of debunking the theories of these getlemen. Problem: this chapter is the intrduction to the 'Culture and Language' section.
  • Rafael Núñez and Tyler Marghetis: Cognitive Linguistics and the Concept(s) of Number 377-401 Embodied learning
  • John N. Towse, Kevin Muldoon, Victoria Simms: Figuring Out Children's Number Representations: Lessons from Cross-Cultural Work 402-414
  • Yukari Okamoto: Mathematics Learning in the USA and East Asia; Influences of Language 415-429
  • Linda Sturman: What is there to Learn from International Surveys of Mathematical Achievement? 430-443. Amazingly, Linda Sturman does not discuss the question of exacty what it is that Pisa or TIMSS are in fact measuring. Is it proficiency in mathmatcs (arithemtics, really)? Or is it an unknown amalgam of intellectual capabilities?
  • Roi Cohen Kadosh: From single-cell neuropathology to mathematis education (Neuroscience of mathematics navigator chapter) 445455
  • Liane Kaufmann, Karin Kucian, and Michael von Aster: Development of the Numerical Brain 485-501 abstract & researchgate
  • Vinod Menon: Arithmetic in the Child and Adult Brain 502-530 pdf
  • Ian D. Holloway and Daniel Ansari: Numerical Symbols: An Overview of Their Cognitive and Neural Underpinnings 531-551 abstract. Publications (many for free download) by this research group see here
  • Vincent Walsh: A Theory of Magnitude: The Parts that Sum to Number 551-565 abstract
  • Chantal Roggeman, Wim Fias, and Rom Verguts: Basic Number Representations and Beyond: Neuroimaging and Computational Modeling 566-582abstract
  • Elena Salillas and Carlo Semenza: Mapping the Brain for Math: Reversible Inactivation by Direct Cortical Electrostimulation and Transcranial Magnetic Stimulation 583-611 concept pdf
  • Bert De Smedt and Roland H. Grabner: Applications of Neuroscience to Mathematics Education
  • Marie-Pascale Noël: When Number Processing and Calculation Is Not Your Cup of Tea
  • Brian Butterworth, Sashank Varma, and Diana Laurillard: Dyscalculia: From Brain to Education
  • Avishai Henik, Orly Rubinsten, and Sarit Ashkenazi: Developmental Dyscalculia as a Heterogeneous Disability
  • Michèle Mazzocco: Mathematical difficulties in children with and without specific genetic syndromes
  • Silke M. Göbel: Number Processing and Arithmetic in Children and Adults with Reading Difficulties
  • Jo Van Herwegen and Annette Karmiloff-Smith: Genetic Developmental Disorders and Numerical Competence Across the Lifespan
  • Karin Kucian, Liane Kaufmann, and Michael von Aster: Brain Correlates of Numerical Disabilities
  • Pekka Räsänen: Computer-assisted Interventions on Basic Number Skills
  • David C. Geary: The Classification and Cognitive Characteristics of Mathematical Disabilities in Children
  • Julie Castronovo: Numbers in the Dark: Numerical Cognition and Blindness
  • Marinella Cappelletti: The Neuropsychology of Acquired Number and Calculation Disorders
  • L. Zamarian and Margarete Delazer: Arithmetic Learning in Adults - Evidence from Brain Imaging
  • Chris Donlan: Individual Differences
  • Lee de-Wit and Johan Wagemans: Individual Differences in Local and Global Perceptual Organization
  • Ann Dowker: Individual Differences in Arithmetical Abilities: The Componential Nature of Arithmetic
  • Jo-Anne LeFevre, Emma Wells, and Carla Sowinski: Individual Differences in Basic Arithmetical Processes in Children and Adults
  • Annemie Desoete: Cognitive Predicators of Mathematical Abilities and Disabilities
  • Alex M. Moore, Nathan O. Rudig, and Mark H. Ashcraft: Affect, Motivation, Working Memory, and Mathematics
  • L. Verschaffel, F. Depaepe, and W. Van Dooren: Individual Differences in Word Problem Solving 953-974
  • Julie Ann Jordan: Individual Differences in Children's Paths to Arithmetical Development
  • Maria G. Tosto, Claire M.A. Haworth, and Yulia Kovas: Behavioral Genomics of Mathematics
  • Richard Cowan: Education 1021-1035
  • Karen C. Fuson, Aki Murata, and Dor Abrahamson: Using Learning Path Research to Balance Mathematics Education: Teaching/Learning for Understanding and Fluency 1035-1054
  • Herbert P. Ginsburg, Rachael Labrecque, Kara Carpenter, and Dana Pagar: New Possibilities for Early Mathematics Education: Cognitive Guidelines for Designing High-Quality Software to Promote Young Children's Meaningful Mathematics Learning 1055-1078
  • Nancy C. Jordan, Lynn S. Fuchs, Nancy Dyson: Early Number Competencies and Mathematical Learning: Individual Variation, Screening, and Intervention 1079-1098
  • Nick Dowrick: Numbers Count: A Large-scale Intervention for Young Children Who Struggle with Mathematics 1099-1117
  • Bethany Rittle-Johnson and Michael Schneider: Developing Conceptual and Procedural Knowledge of Mathematics 1118-1134
  • Geetha B. Ramani, Robert S. Siegler: How Informal Learning Activities Can Promote Children's Numerical Knowledge 1135-1154

Anna A. Matejko, Daniel Ansari (2015). Drawing connections between white matter and numerical and mathematical cognition: A literature review/. Neuroscience & Biobehavioral Reviews, 48C, 35-52. download here

Edward L. Thorndike (1924). The Psychology of Arithmetic. The Macmillan Company. https://archive.org/details/psychologyofarit00thoruoft

Annotations: matheducation.htm#Thorndike; on the import and impact of Thorndike’s work see matheducation.htm#Cronbach Suppes

Miriam Rosenberg-Lee, Marsha C. Lovett & John R. Anderson (2009). Neural correlates of arithmetic calculation strategies. Cognitive, Affective, & Behavioral Neuroscience, 9, 270-285. preview; pdf download; pdf direct

ACT=R Publications in cognitive arithmetic. webpage

ACT-R Publications in Mathematical Problem Solving. webpage

ACT-R Publications in Cognitive Math. webpage

Stellan Ohlsson, Andreas M. Ernst & Ernest Rees (1992). The cognitive complexity of learning and doing arithmetic. Journal for Research in Mathematics Education, 23, 441-467 preview

Josefine Andin, Peter Fransson, Jerker Rönnberg & Mary Rudner (2015). Phonology and arithmetic in the language–calculation network. Brain and Languge, 143, 97-105. open access

Esther De Loof, Louise Poppe, Axel Cleeremans, Wim Gevers & Filip Van Opstal (2015). Different effects of executive and visuospatial working memory on visual consciousness. Attention Perception & Psychophysics pdf

Gavin R. Price1, Michèle M. M. Mazzocco and Daniel Ansari (2013). Why Mental Arithmetic Counts: Brain Activation during Single Digit Arithmetic Predicts High School Math Scores. The Journal of Neuroscience, 33abstract

Marie Crouzevialle, Annique Smeding, Fabrizio Butera (September 25 2015). Striving for Excellence Sometimes Hinders High Achievers: Performance-Approach Goals Deplete Arithmetical Performance in Students with High Working Memory Capacity. PLOS One free online

Frieder L. Schillinger , Bert De Smedt, Roland H. Grabner (2015). When errors count: an EEG study on numerical error monitoring under performance pressure. ZDM [ research.net ] abstract

DeCaro, M. S., Rotar, K. E., Kendra, M. S., & Beilock, S. L. (2010). Diagnosing and alleviating the impact of performance pressure on mathematical problem solving. The Quarterly Journal of Experimental Psychology: Human Experimental Psychology, 63(8), 1619-1630. abstract

Beilock, S. L., & DeCaro, M. S. (2007). From poor performance to success under stress: Working memory, strategy selection, and mathematical problem solving under pressure. Journal of Experimental Psychology: Learning, Memory, & Cognition, 33, 983-998. abstract

Daniel Ansari, Roland H. Grabner, Karl Koschutnig, Gernot Reishofer & Franz Ebner (2012). Individual differences in mathematical competence modulate brain responses to arithmetic errors: An fMRI study. Learning and Individual Differences, 21, 636-643. abstract

• "In view of the evidence associating the lateral prefrontal cortex with the implementation of cognitive control, we suggest that individuals with relatively high mathematical competence may exhibit greater awareness of calculation mistakes and implement greater control following the commission of errors."

Sian L. Beilock & Marci S. DeCaro (2007). From Poor Performance to Success Under Stress: Working Memory, Strategy Selection, and Mathematical Problem Solving Under Pressure. Journal of Experimental Psychology Learning Memory and Cognition, 33, 983-998. abstract

• Implications for High-Stakes Testing
  Our work demonstrates that the advantages individuals higher in WM have on the types of demanding math problems used in Experiment 1 and those that high-stakes tests often embody (Sternberg, 2004) are just what make them susceptible to failure when pressure is added. One might wonder how this could be the case, given that high-stakes testing has been used to gauge students’ ability for many decades. Here we show how important testing situations limit the efficacy of these evaluations. These results align with recent concerns regarding the ability of admissions tests to elicit optimal performance in underrepresented groups, especially high-achieving racial minorities and women in the math and sciences ( . . . ). Such individuals feel added pressure to perform at a high level, often in an effort to overcome wellknown and widely held stereotypes regarding the intelligence or academic skill of the social groups to which they belong ( . . . ). The finding that the more important a test is, the more likely the best performers will take up the strategies of the worst demonstrates just how perilous a strong reliance on test scores may be, especially for those most in need (and deserving) of high-level performance to advance in academics and beyond. Ironically, the conditions under which admissions tests are conducted may impact the very constructs they are attempting to measure.

Sanne H. G. van der Ven, Han L. J. van der Maas, Marthe Straatemeier, Brenda R. J. Jansen (2013). Visuospatial working memory and mathematical ability at different ages throughout primary school. Learning and Individual Differences, 27, 182-192. pdf

Brenda R. J. Jansen, Abe D. Hofman, Marthe Straatemeier, Bianca M. C. B. van Bers, Maartje E. J. Raijmakers & Han L. J. van der Maas (2014). The role of pattern recognition in children's exact enumeration of small numbers. British Journal of Developmental Psychology, 32(2), 178-194. research.net

Number comparison and number ordering as predictors of arithmetic performance in adults: Exploring the link between the two skills, and investigating the question of domain-specificity Kinga Morsanyi , Eileen O’Mahony and Teresa McCormack THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2016 http://dx.doi.org/10.1080/17470218.2016.1246577 abstract

Sloppy research?

Federico Zimmerman, Diego Shalom, Pablo A. Gonzalez, Juan Manuel Garrido, Facundo Alvarez Heduan, Stanislas Dehaene, Mariano Sigman, Andres Rieznik (2016). Arithmetic on Your Phone: A Large Scale Investigation of Simple Additions and Multiplications webpage

Tyler W. Watts, Douglas H. Clements, Julie Sarama, Christopher B. Wolfe, Mary Elaine Spitler & Drew H. Bailey (2017). Does Early Mathematics Intervention Change the Processes Underlying Children's Learning? Journal of Research on Educational Effectiveness abstract

Brain Networks Supporting Execution of Mathematical Skills versus Acquisition of New Mathematical Competence. Samuel Wintermute¤, Shawn Betts, Jennifer L. Ferris, Jon M. Fincham, John R. Anderson (2012). PLOS December 2012 | Volume 7 | Issue 12 | e50154 open access

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