Limiti e possibili sviluppi

Concludiamo questa tesi analizzando i limiti del nostro studio e suggerendo, di conseguenza, alcuni possibili sviluppi.

Un primo limite è dato dal grado (II elementare) degli intervistati. A tale livello scolare, non in tutte le classi vengono introdotti i numeri oltre il centinaio, e nei casi in cui succede ci troviamo comunque di fronte ad un primissimo approccio a tale fascia numerica. Possiamo pertanto supporre che le strategie impiegate e le prestazioni conseguite si evolvano con il progredire del livello scolare e con l'aumento dell'esposizione a numeri maggiori. Un possibile sviluppo, dunque potrebbe prevedere un allargamento dello studio a più livelli scolari, con una concentrazione su di un numero maggiore di studenti.

Rimanendo su questo lone di analisi dello sviluppo cognitivo per la linea dei numeri, potrebbe risultare interessante anche uno studio di tipo longitudinale, che coinvolga un numero anche ridotto di studenti intervistati a più riprese, e che mostri come l'idea di linea dei numeri venga modicata per ciascuno studente con il passare del tempo e con il progredire del livello scolare.

Un altro limite dello studio è costituito dalla dicoltà di riconoscere le strategie adottate dagli studenti. Un esempio è costituito dall'etichetta benchmark, che è emersa solamente per uno studente ma che verosimilmente è stata adottata anche da altri intervistati. Attraverso domande e item dedicati, un possibile studio successivo potrebbe esaminare più da vicino un ristretto campionario delle strategie più spesso utilizzate. Prendendo in considerazioni anche livelli scolari superiori, inoltre, è plausibile che il grado di introspezione degli studenti aumenti e che quindi si riescano a raccogliere più indizi espliciti sulle strategie messe in atto.

Inne, le strategie osservate sulle consegne 3 e 4 (posizione-a-numero) hanno oerto solo uno sguardo parziale sui processi cognitivi messi in atto. Abbiamo riscontrato alcune tendenze (sottostimare o sovrastimare sistemati- camente, schiacciare i numeri al centro o sugli estremi, concentrare le stime corrette in determinate fasce numeriche) senza però riuscire a comprenderne completamente la natura. Un possibile studio di tipo quantitativo, su un campione sucientemente largo di studenti, potrebbe prendere in conside- razione delle analisi multivariate delle varianze che leghino tra loro fenome- ni come quelli descritti (o altri analoghi riscontrati sperimentalmente) con fattori come il grado scolare, la valutazione dell'insegnante, il genere, le pre-

3.4. CONCLUSIONI 95 stazioni su altre consegne sulle linee dei numeri o in altri ambiti matematici specicamente designati.

Bibliograa

Ashcraft, Mark H. e John Battaglia (set. 1978). Cognitive arithmetic: Evi- dence for retrieval and decision processes in mental addition. In: Journal of Experimental Psychology: Human Learning and Memory 4.5, pp. 527 538. doi: 10.1037/0278-7393.4.5.527.

Ashcraft, Mark H. e Kelly S. Christy (nov. 1995). The Frequency of Arith- metic Facts in Elementary Texts: Addition and Multiplication in Grades 1-6. In: Journal for Research in Mathematics Education 26.5, pp. 396 421. doi: 10.2307/749430.

Ashcraft, Mark H., Rick D. Donley et al. (1992). Working Memory, Auto- maticity, And Problem Diculty. In: The Nature and Origins of Ma- thematical Skills. A cura di Jamie I. D. Campbell. Vol. 91. Advances in psychology. Elsevier B.V. Cap. 8, pp. 301329. isbn: 978-0-444-89014-6. doi: 10.1016/S0166-4115(08)60890-0.

Aubrey, Carol, Ray Godfrey e Sarah Dahl (mag. 2006). Early mathematics development and later achievement: Further evidence. In: Mathematics Education Research Journal 18.1, pp. 2746. doi: 10.1007/BF03217428. Augustyniak, Kristine, Jacqueline Murphy e Donna Kester Phillips (2005). Psychological Perspectives in Assessing Mathematics Learning Needs. In: Journal of Instructional Psychology 32.4, pp. 277286.

Baccaglini-Frank, Anna (37 giu. 2015). Preventing low achievement in arithmetic through the didactical materials of the PerContare project. In: Proceeding of ICMI STUDY 23: primary mathematics study on whole number. A cura di Xuhua Sun, Berinderjeet Kaur e Jarmila Novotná. Macao, CN, pp. 169176.

Baccaglini-Frank, Anna e Maria Giuseppina Bartolini Bussi (dic. 2015). Buone pratiche didattiche per prevenire falsi positivi nelle diagnosi di discalculia: il progetto PerContare. In: Form@re - Open Journal per la formazione in rete 15.3, pp. 170184. doi: 10.13128/formare-17182. Baccaglini-Frank, Anna, Pietro Di Martino et al. (5 dic. 2017). Didattica della matematica. Manuali. Mondadori Università. isbn: 9788861845503. Banks, William P. e David K. Hill (feb. 1974). The apparent magnitude of number scaled by random production. In: Journal of Experimental Psychology 102.2, pp. 353376. doi: 10.1037/h0035850.

Bender, Andrea e Sieghard Beller (1 mag. 2011). Cultural Variation in Numeration Systems and Their Mapping Onto the Mental Number Line. In: Journal of Cross-Cultural Psychology 42.4, pp. 579597. doi: 10 . 1177/0022022111406631.

Berch, Daniel (1 lug. 2005). Making Sense of Number Sense. Implications for Children With Mathematical Disabilties. In: Journal of Learning Disabilities 38.4, pp. 333339. doi: 10.1177/00222194050380040901. Berch, Daniel et al. (dic. 1999). Extracting Parity and Magnitude from

Arabic Numerals: Developmental Changes in Number Processing and Mental Representation. In: Journal of Experimental Child Psychology 74.4, pp. 286308. doi: 10.1006/jecp.1999.2518.

Brainerd, Charles J. (1979). The Origins of the Number Concept. Praeger special studies. Praeger. isbn: 9780030293061.

Brannon, Elizabeth M. e Herbert S. Terrace (gen. 2000). Representation of the numerosities 19 by rhesus macaques (Macaca mulatta). In: Journal of Experimental Psychology: Animal Behavior Processes 26.1, pp. 3149. doi: 10.1037//0097-7403.26.1.31.

Brannon, Elizabeth M., Courtney J. Wustho et al. (1 mag. 2001). Numeri- cal Subtraction in the Pigeon: Evidence for a Linear Subjective Number Scale. In: Psychological Science 12.3, pp. 238243. doi: 10.1111/1467- 9280.00342.

Butterworth, Brian (23 apr. 1999). The Mathematical Brain. Macmillan. isbn: 0333735277.

 (12 gen. 2005). The development of arithmetical abilities. In: Journal of Child Psychology and Psychiatry 46.1, pp. 318. doi: 10.1111/j.1469- 7610.2004.00374.x.

Butterworth, Brian e Robert Reeve (2008). Verbal Counting and Spatial Strategies in Numerical Tasks: Evidence from Indigenous Australia. In: 21.4: Number as a Test Case for the Role of Language in Cognition, pp. 443457. doi: 10.1080/09515080802284597.

Butterworth, Brian, Robert Reeve e Fiona Reynolds (1 mag. 2011). Using Mental Representations of Space When Words Are Unavailable: Studies of Enumeration and Arithmetic in Indigenous Australia. In: Jour- nal of Cross-Cultural Psychology 42.4, pp. 630638. doi: 10 . 1177 / 0022022111406020.

Butterworth, Brian, Robert Reeve, Fiona Reynolds e Delyth Lloyd (2 set. 2008). Numerical thought with and without words: Evidence from indi- genous Australian children. In: Proceedings of the National Academy of Sciences of the United States of America 105.35, pp. 1317913184. doi: 10.1073/pnas.0806045105.

Butterworth, Brian, Sashank Varma e Diana Laurillard (27 mag. 2011). Dy- scalculia: From Brain to Education. In: Science 332.6033, pp. 1049 1053. doi: 10.1126/science.1201536.

BIBLIOGRAFIA 99 Calabria, Marco e Yves Rossetti (2005). Interference between number pro- cessing and line bisection: a methodology. In: Neuropsychologia 43.5, pp. 779783. doi: 10.1016/j.neuropsychologia.2004.06.027.

Cantlon, Jessica F. e Elizabeth M. Brannon (1 gen. 2007). Adding up the eects of cultural experience on the brain. In: Trends in Cognitive Sciences 11.1, pp. 14. doi: 10.1016/j.tics.2006.10.008.

Carroll, Lewis (29 mar. 1876). The Hunting of the Snark. Macmillan Publi- shers.

Case, Robbie, Yukari Okamoto et al. (1996). The Role of Central Con- ceptual Structures in the Development of Children's Thought. In: Mo- nographs of the Society for Research in Child Development 61.1/2: The Role of Central Conceptual Structures in the Development of Children's Thought, pp. i+iiivi+1295. doi: 10.2307/1166077.

Case, Robbie e Judith T. Sowder (1990). The Development of Computatio- nal Estimation: A Neo-Piagetian Analysis. In: Cognition and Instruction 7.2, pp. 79104. doi: 10.1207/s1532690xci0702_1.

Chan, Ting Ting e Benjamin Bergen (2005). Writing Direction Inuen- ces Spatial Cognition. In: Proceedings of the Annual Meeting of the Cognitive Science Society. Vol. 27.

Chiao, Joan Y., Andrew R. Bordeaux e Nalini Ambady (set. 2004). Mental representations of social status. In: Cognition 93.2, B49B57. doi: 10. 1016/j.cognition.2003.07.008.

Cohen Kadosh, Roi e Avishai Henik (2006a). Color Congruity Eect: Where do Colors and Numbers Interact in Synesthesia? In: World Neurosurgery 42.2, pp. 259263. doi: 10.1016/S0010-9452(08)70351-4.

 (2006b). When a line is a number: Color yields magnitude information in a digit-color synesthete. In: Neuroscience 137.1, pp. 35. issn: 0306- 4522. doi: 10.1016/j.neuroscience.2005.08.057.

Cohen Kadosh, Roi, Jan Lammertyn e Véronique Izard (feb. 2008). Are numbers special? An overview of chronometric, neuroimaging, develop- mental and comparative studies of magnitude representation. In: Pro- gress in Neurobiology 84.2, pp. 132147. doi: 10.1016/j.pneurobio. 2007.11.001.

Cohen, Laurent e Stanislas Dehaene (1996). Cerebral networks for number processing: Evidence from a case of posterior callosal lesion. In: Neu- rocase 2.3: The Neural Basis of Cognition, pp. 155174. doi: 10.1080/ 13554799608402394.

Coles, Alf e Nathalie Sinclair (2018). Re-Thinking `Normal' Development in the Early Learning of Number. In: Journal of Numerical Cognition 4.1. doi: 10.5964/jnc.v4i1.101.

Cornoldi, Cesare e Daniela Lucangeli (1 gen. 2004). Arithmetic Education and Learning Disabilities in Italy. In: Journal of Learning Disabilities 37.1, pp. 4249. doi: 10.1177/00222194040370010501.

Critchley, Macdonald (1953). The parietal lobes. Oxford, UK: Williams e Wilkins.

Dehaene, Stanislas (1992). Varieties of numerical abilities. In: Cognition 44.1/2, pp. 142. doi: 10.1016/0010-0277(92)90049-N.

 (lug. 1993). Symbols and quantities in parietal cortex: elements of a ma- thematical theory of number representation and manipulation. In: Sen- sorimotor Foundations of Higher Cognition. Attention and Performance. A cura di Patrick Haggard, Yves Rossetti e Mitsuo Kawato. Vol. 22. Cap. 24, pp. 527574. isbn: 9780199231447. doi: 10 . 1093 / acprof : oso/9780199231447.003.0024.

 (1997). The Number Sense. How the Mind Creates Mathematics. Oxford University Press.

 (17 dic. 2002). Précis of The Number Sense. In: Mind & Language 16.1, pp. 1636. doi: 10.1111/1468-0017.00154.

Dehaene, Stanislas, Serge Bossini e Pascal Giraux (set. 1993). The mental representation of parity and number magnitude. In: Journal of Expe- rimental Psychology: General 122.3, pp. 371396. doi: 10.1037/0096- 3445.122.3.371.

Dehaene, Stanislas e Jean-Pierre Changeux (1993). Development of Elemen- tary Numerical Abilities: A Neuronal Model. In: Journal of Cognitive Neuroscience 5.4, pp. 390407. doi: 10.1162/jocn.1993.5.4.390. Dehaene, Stanislas e Laurent Cohen (1995). Towards an Anatomical and

Functional Model of Number Processing. In: Mathematical Cognition 1.1, pp. 83120.

 (1997). Cerebral Pathways for Calculation: Double Dissociation between Rote Verbal and Quantitative Knowledge of Arithmetic. In: Cortex 33.2, pp. 219250. doi: 10.1016/S0010-9452(08)70002-9.

Dehaene, Stanislas, Ghislaine Dehaene-Lambertz e Laurent Cohen (1 ago. 1998). Abstract representations of numbers in the animal and human brain. In: Trends in Neurosciences 21.8, pp. 355361. doi: 10.1016/ S0166-2236(98)01263-6.

Dehaene, Stanislas, Emmanuel Dupoux e Jacques Mehler (1990). Is nu- merical comparison digital? Analogical and symbolic eects in two-digit number comparison. In: Journal of Experimental Psychology: Human Perception and Performance 16.3, p. 626.

Dehaene, Stanislas, Véronique Izard et al. (2008). Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cul- tures. In: science 320.5880, pp. 12171220.

Dehaene, Stanislas, Manuela Piazza et al. (2003). Three parietal circuits for number processing. In: Cognitive Neuropsychology 20.36, pp. 487506. doi: 10.1080/02643290244000239.

Desoete, Annemie et al. (2009). Subitizing or counting as possible scree- ning variables for learning disabilities in mathematics education or lear-

BIBLIOGRAFIA 101 ning? In: Educational Research Review 4.1, pp. 5566. doi: 10.1016/ j.edurev.2008.11.003.

Dobel, Christian, Gil Deisendruck e Jens Bölte (1 giu. 2007). How Writing System and Age Inuence Spatial Representations of Actions. A Develop- mental, Cross-Linguistic Study. In: Psychological Science 18.6, pp. 487 491. doi: 10.1111/j.1467-9280.2007.01926.x.

Dormal, Valérie e Mauro Pesenti (2007). Numerosity-Length Interference. A Stroop Experiment. In: Experimental Psychology 54, pp. 289297. doi: 10.1027/1618-3169.54.4.289.

Dowker, Ann (2004). Individual Dierences in Arithmetic. Implications for Psychology, Neuroscience and Education. Taylor & Francis. isbn: 0203324897.

Ebersbach, Mirjam et al. (gen. 2008). The relationship between the shape of the mental number line and familiarity with numbers in 5- to 9-year old children: Evidence for a segmented linear model. In: Journal of Experimental Child Psychology 99.1, pp. 117. doi: 10.1016/j.jecp. 2007.08.006.

Fechner, Gustav (1860). Elemente der Psychophysik. Vol. 1. Elemente der Psychophysik. Breitkopf und Härtel.

Feigenson, Lisa, Stanislas Dehaene e Elizabeth Spelke (lug. 2004). Core systems of number. In: Trends in Cognitive Sciences. Language and Conceptual Development 8.7, pp. 307314. doi: 10.1016/j.tics.2004. 05.002.

Fias, Wim et al. (15 ago. 2007). Processing of Abstract Ordinal Knowledge in the Horizontal Segment of the Intraparietal Sulcus. In: Journal of Neuroscience 27.33, pp. 89528956. doi: 10.1523/JNEUROSCI.2076- 07.2007.

Fischer, Martin H. (11 set. 2001). Number processing induces spatial per- formance biases. In: Neurology 57.5, pp. 822826. doi: 10.1212/WNL. 57.5.822.

Fischer, Martin H., Alan D. Castel et al. (2003). Perceiving numbers causes spatial shifts of attention. In: Nature Neuroscience 6, pp. 555556. doi: 10.1038/nn1066.

Fischer, Martin H., Richard A. Mills e Samuel Shaki (apr. 2010). How to cook a SNARC: Number placement in text rapidly changes spa- tialnumerical associations. In: Brain and Cognition 72.3, pp. 333336. doi: 10.1016/j.bandc.2009.10.010.

Fischer, Martin H. e Julia Rottmann (2005). Do negative numbers have a place on the mental number line? In: Psychology Science 47.1, pp. 22 32.

Fischer, Martin H., Samuel Shaki e Alexander Cruise (2009). It Takes Ju- st One Word to Quash a SNARC. In: Experimental Psychology 56, pp. 361366. doi: 10.1027/1618-3169.56.5.361.

Fletcher, Jack M. et al. (1 mar. 2011). Learning Disabilities. From Identi-

cation to Intervention. 1a _{ed. Guilford Press. isbn: 9781593853709.}

Fuchs, Lynn S. e Douglas Fuchs (1 nov. 2002). Mathematical Problem- Solving Proles of Students with Mathematics Disabilities With and Wi- thout Comorbid Reading Disabilities. In: Journal of Learning Disabili- ties 35.6, pp. 564574. doi: 10.1177/00222194020350060701.

Fuson, Karen C. (1987). Children's Counting and Concepts of Number. Three Islands Press. isbn: 1461237556.

Galton, Francis (15 gen. 1880). Visualised Numerals. doi: 10 . 1038 / 021252a0.

Geary, David C. (1994). Children's mathematical development: Research and practical applications. Washington, DC, US: American Psychological As- sociation. doi: 10.1037/10163-000.

 (1996). The Problem-size Eect in Mental Addition: Developmental and Cross-national Trends. In: Mathematical Cognition 2.1, pp. 6394. doi: 10.1080/135467996387543.

 (1 gen. 2004). Mathematics and Learning Disabilities. In: 37.1, pp. 4 15. doi: 10.1177/00222194040370010201.

 (apr. 2010). Mathematical disabilities: Reections on cognitive, neu- ropsychological, and genetic components. In: Learning and Individual Dierences 20.2, pp. 130133. doi: 10.1016/j.lindif.2009.10.008. Geary, David C., C. Christine Bow-Thomas et al. (ott. 1996). Develop-

ment of Arithmetical Competencies in Chinese and American Children: Inuence of Age, Language, and Schooling. In: Child Development 67.5, pp. 20222044. doi: 10.1111/j.1467-8624.1996.tb01841.x.

Geary, David C. e Mary K. Hoard (29 nov. 2004). Learning disabilities in arithmetic and mathematics: Theoretical and empirical perspectives. In: Handbook of mathematical cognition. A cura di Jamie I. D. Campbell.

1a _{ed. New York City, NY, US: Psychology Press, pp. 253267. isbn:}

1841694118.

Geary, David C., Mary K. Hoard et al. (2008). Development of Number Line Representations in Children With Mathematical Learning Disability. In: Developmental Neuropsychology 33.3: Mathematics Ability, Performance, and Achievement, pp. 277299. doi: 10.1080/87565640801982361. Gebuis, Titia, Roi Cohen Kadosh et al. (mag. 2009). Automatic quanti-

ty processing in 5-year olds and adults. In: Cognitive Processing 10.2, pp. 133142. doi: 10.1007/s10339-008-0219-x.

Gebuis, Titia e Bert Reynvoet (nov. 2012a). The interplay between non- symbolic number and its continuous visual properties. In: Journal of Experimental Psychology: General 141.4, pp. 642648. doi: 10.1037/ a0026218.

 (17 mag. 2012b). The Role of Visual Information in Numerosity Esti- mation. In: PLoS One 7.5. doi: 10.1371/journal.pone.0037426.

BIBLIOGRAFIA 103 Gelman, Rochel e Elizabeth Meck (mag. 1983). Preschoolers' counting: Principles before skill. In: Cognition 13, pp. 343359. doi: 10.1016/ 0010-0277(83)90014-8.

Gerstmann, Josef (ago. 1940). Syndrome of nger agnosia, disorientation for right and left, agraphia and acalculia: local diagnostic value. In: Archives of Neurology & Psychiatry 44.2, pp. 398408. doi: 10.1001/ archneurpsyc.1940.02280080158009.

Gevers, Wim, Ineke Imbo et al. (2010). Bidirectionality in Synesthesia. Evidence From a Multiplication Verication Task. In: Experimental Psychology 57, pp. 178184. doi: 10.1027/1618-3169/a000022.

Gevers, Wim, Bert Reynvoet e Wim Fias (apr. 2003). The mental repre- sentation of ordinal sequences is spatially organized. In: Cognition 87.3, B87B95. doi: 10.1016/S0010-0277(02)00234-2.

Giaquinto, Marcus (5 lug. 2007). Visual Thinking in Mathematics. Ristampa. Clarendon Press. isbn: 0199285942.

Gibbon, John e Russell M. Church (apr. 1981). Time left: Linear versus logarithmic subjective time. In: Journal of Experimental Psychology: Animal Behavior Processes 7.2, pp. 87108. doi: 10.1037//0097-7403. 7.2.87.

Gilmore, Camilla, Shannon E. McCarthy e Elizabeth Spelke (31 mag. 2007). Symbolic arithmetic knowledge without instruction. In: Nature 447, pp. 589591. doi: 10.1038/nature05850.

 (giu. 2010). Non-symbolic arithmetic abilities and mathematics achieve- ment in the rst year of formal schooling. In: Cognition 115.3, pp. 394 406. doi: 10.1016/j.cognition.2010.02.002.

Göbel, Silke M., Samuel Shaki e Martin H. Fischer (1 mag. 2011). The Cultural Number Line: A Review of Cultural and Linguistic Inuences on the Development of Number Processing. In: Journal of Cross-Cultural Psychology 42.4, pp. 543565. doi: 10.1177/0022022111406251.

Göbel, Silke M., Vincent Walsh e Matthew F. S. Rushworth (dic. 2001). The Mental Number Line and the Human Angular Gyrus. In: NeuroImage 14.6, pp. 12781289. doi: 10.1006/nimg.2001.0927.

Goldin-Meadow, Susan (2001). Giving the mind a hand: The role of gesture in cognitive change. In: Mechanisms of cognitive development: Behavio- ral and neural perspectives. A cura di James McClelland e Robert S. Siegler. Mahwah, NJ, US: Earlbaum Associates, pp. 531.

Gottfredson, Lisa S. (1997). Foreword. In: Intelligence. A Multidiscipli- nary Journal 24.1: Intelligence and Social Policy. A cura di Douglas K. Detterman e Linda S. Gottfredson, pp. 112.

Halberda, Justin e Lisa Feigenson (set. 2008). Developmental change in the acuity of the number sense: The approximate number system in 3-, 4-, 5-, and 6-year-olds and adults. In: Developmental Psychology 44.5, pp. 14571465. doi: 10.1037/a0012682.

Halberda, Justin, Michèle Mazzocco e Lisa Feigenson (2 ott. 2008). In- dividual dierences in non-verbal number acuity correlate with maths achievement. In: Nature 455, pp. 665668. doi: 10.1038/nature07246. Hamann, Mary Sue e Mark H. Ashcraft (1986). Textbook Presentations of the Basic Addition Facts. In: Cognition and Instruction 3.3, pp. 173 202. doi: 10.1207/s1532690xci0303_2.

Hannula-Sormunen, Minna M. e Erno Lehtinen (giu. 2005). Spontaneous focusing on numerosity and mathematical skills of young children. In: Learning and Instruction 15.3, pp. 237256. doi: 10 . 1016 / j . learninstruc.2005.04.005.

Helmreich, Iris et al. (1 mag. 2011). Language Eects on Children's Nonver- bal Number Line Estimations. In: Journal of Cross-Cultural Psychology 42.4, pp. 598613. doi: 10.1177/0022022111406026.

Hermelin, Beate e Niel O'Connor (giu. 1986). Spatial representations in mathematically and in artistically gifted children. In: British Journal of Educational Psychology 56.2, pp. 150157. doi: 10.1111/j.2044- 8279.1986.tb02656.x.

Holloway, Ian D. e Daniel Ansari (12 ago. 2008). Domain-specic and domain-general changes in children's development of number compari- son. In: Developmental Science 11.5. doi: 10 . 1111 / j . 1467 - 7687 . 2008.00712.x.

 (mag. 2009). Mapping numerical magnitudes onto symbols: The nu- merical distance eect and individual dierences in children's mathema- tics achievement. In: Journal of Experimental Child Psychology 103.1, pp. 1729. doi: 10.1016/j.jecp.2008.04.001.

Hubbard, Edward M., A. Cyrus Arman et al. (24 mar. 2005). Individual Dierences among Grapheme-Color Synesthetes: Brain-Behavior Corre- lations. In: Neuron 45.6, pp. 975985. doi: 10.1016/j.neuron.2005. 02.008.

Hubbard, Edward M., Ilka Diester et al. (12 nov. 2008). The Evolution of Numerical Cognition: From Number Neurons to Linguistic Quanti- ers. In: Journal of Neuroscience 28.46, pp. 1181911824. doi: 10 . 1523/JNEUROSCI.3808-08.2008.

Hubbard, Edward M., Manuela Piazza et al. (1 giu. 2005). Interactions between number and space in parietal cortex. In: Nature Reviews Neu- roscience 6, pp. 435448. doi: 10.1038/nrn1684.

Huntley-Fenner, Gavin (mar. 2001). Children's understanding of number is similar to adults' and rats': numerical estimation by 57-year-olds. In: Cognition 78.3, B27B40. doi: 10.1016/S0010-0277(00)00122-0. Hurewitz, Felicia, Rochel Gelman e Brian Schnitzer (19 dic. 2006). Some-

times area counts more than number. In: Proceedings of the National Academy of Sciences of the United States of America 103.51, pp. 19599 19604. doi: 10.1073/pnas.0609485103.

BIBLIOGRAFIA 105 Inhelder, Bärbel, Hermine Sinclair e Magali Bovet (1974). Learning and the development of cognition. Ristampa. Harvard University Press. isbn: 0674519256.

INVALSI, Istituto Nazionale per la Valutazione del Sistema educativo di Istruzione e di formazione (2017). Il Quadro di Riferimento delle Prove di Matematica del Sistema Nazionale di Valutazione.

Jarick, Michelle et al. (nov.dic. 2009). A dierent outlook on time: Visual and auditory month names elicit dierent mental vantage points for a time-space synaesthete. In: Cortex 45.10, pp. 12171228. doi: 10.1016/ j.cortex.2009.05.014.

Jonas, Clare e Michelle Jarick (1 dic. 2013). Synaesthesia, Sequences, and Space. In: Oxford Handbook of Synaesthesia. A cura di Julia Simner e Edward M. Hubbard. Oxford University Press. doi: 10.1093/oxfordhb/ 9780199603329.013.0007.

Jonas, Clare, Alisdair Taylor et al. (16 set. 2011). Visuo-spatial representa- tions of the alphabet in synaesthetes and non-synaesthetes. In: Journal of Neuropsychology 5.2, pp. 302322. doi: 10.1111/j.1748-6653.2011. 02010.x.

Karagiannakis, Giannis e Anna Baccaglini-Frank (2014). The DeDiMa bat- tery: a tool for identifying students' mathematical learning proles. In: Health Psychology Report 2.4, pp. 291297. doi: 10.5114/hpr.2014. 46329.

Karagiannakis, Giannis, Anna Baccaglini-Frank e Yiannis Papadatos (10 feb. 2014). Mathematical learning diculties subtypes classication. In: Frontiers in Human Neuroscience 8.57. doi: 10 . 3389 / fnhum . 2014 . 00057.

Karagiannakis, Giannis, Anna Baccaglini-Frank e Petros Roussos (2016). Detecting strengths and weaknesses in learning mathematics throu- gh a model classifying mathematical skills. In: Australian Journal of Learning Diculties 21.2, pp. 115141. doi: 10.1080/19404158.2017. 1289963.

Karagiannakis, Giannis e Marie-Pascale Noel (in revisione). Mathematical Prole & Dyscalculia Test: An Online Interactive Numerical Battery for Primary School Children.

Kaufman, Edna L. et al. (ott. 1949). The Discrimination of Visual Num- ber. In: The American Journal of Psychology 62.4, pp. 498525. doi: 10.2307/1418556.

Kaufmann, Liane et al. (21 ago. 2013). Dyscalculia from a developmental and dierential perspective. In: Frontiers in Psychology 4, p. 516. doi: 10.3389/fpsyg.2013.00516.

Kazandjian, Seta et al. (9 gen. 2009). Egocentric reference in bikeidirectio- nal readers as measured by the straight-ahead pointing task. In: Brain Research 1247, pp. 133141. doi: 10.1016/j.brainres.2008.09.098.

Keil, Frank C. (1992). Concepts, Kinds, and Cognitive Development. MIT Press. isbn: 0262610760.

Koehler, Matthew e Richard Lehrer (1999). Understanding children's rea- soning about measurement: A case-based hypermedia tool for professio- nal development. Madison, WI, US: Wisconsin Center for Education Research.

Kuhn, Jörg-Tobias (2015). Developmental Dyscalculia. Neurobiological, Co- gnitive, and Developmental Perspectives. In: Zeitschrift für Psychologie 223.2, pp. 6982. doi: 10.1027/2151-2604/a000205.

Lako, George e Rafael E. Núñez (2000). Where Mathematics Comes From. How the Embodied Mind Brings Mathematics into Being. isbn: 978-0- 465-03771-1.

Lee, Kyoung-Min e So-Young Kang (apr. 2002). Arithmetic operation and working memory: dierential suppression in dual tasks. In: Cognition 83.3, B63B68. doi: 10.1016/S0010-0277(02)00010-0.

LeFevre, Jo-Anne, Gregory S Sadesky e Jerey Bisanz (gen. 1996). Selection of procedures in mental addition: Reassessing the problem size eect in adults. In: Journal of Experimental Psychology: Learning, Memory, and Cognition 22.1, pp. 216230. doi: 10.1037/0278-7393.22.1.216. Lewis, Katherine E. e Marie B. Fisher (lug. 2016). Taking Stock of 40

Years of Research on Mathematical Learning Disability: Methodological Issues and Future Directions. In: Journal for Research in Mathematics Education 47.4, pp. 338371. doi: 10.5951/jresematheduc.47.4.0338. Lourenco, Stella F. et al. (13 nov. 2012). Nonsymbolic number and cu- mulative area representations contribute shared and unique variance to symbolic math competence. In: Proceedings of the National Academy of Sciences of the United States of America 109.46, pp. 1873718742. doi: 10.1073/pnas.1207212109.

Lucangeli, Daniela (2005). National survey on learning disabilities. Roma, IT: Italian Institute of Research on Infancy.

Lyons, Ian M. e Sian L. Beilock (nov. 2011). Numerical ordering ability me-