Detecting strengths and weaknesses in learning mathematics through a model classifying mathematical skills

References

• American Psychiatric Association. (2013). Diagnostic and statistical manual of mental disorders (5th ed.). DSM-5. Author.
   Google Scholar
• Andersson, U. (2008). Working memory as a predictor of written arithmetical skills in children: The importance of central executive functions. British Journal of Educational Psychology, 78, 181–203.10.1348/000709907X209854
   Google Scholar
• Andersson, U., & Östergren, R. (2012). Number magnitude processing and basic cognitive functions in children with mathematical learning disabilities. Learning and Individual Differences, 22, 701–714.10.1016/j.lindif.2012.05.004
   Google Scholar
• Ansari, D., & Lyons, I. M. (2016). Cognitive neuroscience and mathematics learning: How far have we come? Where do we need to go? ZDM Mathematics Education, 48, 379–383.10.1007/s11858-016-0782-z
   Google Scholar
• Arbuckle, J. L. (2014). Amos 23.0 User’s Guide. Chicago, IL: IBM SPSS.
   Google Scholar
• Ashcraft, M. H. (1992). Cognitive arithmetic: A review of data and theory. Cognition, 44, 75–106.10.1016/0010-0277(92)90051-I
   Google Scholar
• Augustyniak, K., Murphy, J., & Phillips, D. K. (2005). Psychological perspectives in assessing mathematics learning needs. Journal of Instructional Psychology, 32, 277–286.
   Google Scholar
• Bartelet, D., Ansari, D., Vaessen, A., & Blomert, L. (2014). Research in developmental disabilities dognitive subtypes of mathematics learning difficulties in primary education. Research in Developmental Disabilities, 35, 657–670.10.1016/j.ridd.2013.12.010
   Google Scholar
• Brainerd, J. (1979). The origins of the number concept. New York, NY: Praeger Publishers.
   Google Scholar
• Browne, M. W., & Cudeck, R. (1993). Alternative ways of assessing model fit. In K. A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 136–162). Beverly Hills, CA: Sage.
   Google Scholar
• Butterworth, B. (2010). Foundational numerical capacities and the origins of dyscalculia. Trends in Cognitive Sciences, 14, 534–541.10.1016/j.tics.2010.09.007
   Google Scholar
• Campbell, J. I. D. (1987a). Network interference and mental multiplication. Journal of Experimental Psychology: Learning, Memory, and Cognition, 13, 109–123.
   Google Scholar
• Campbell, J. I. D. (1987b). Production, verification, and priming of multiplication facts. Memory & Cognition, 15, 349–364.10.3758/BF03197037
   Google Scholar
• Campbell, J. I. D. (1991). Conditions of error priming in number-fact retrieval. Memory & Cognition, 19, 197–209.10.3758/BF03197119
   Google Scholar
• Carpenter, T. P., & Moser, J. M. (1982). The development of addition and subtraction problem – solving skills. In T. P. Carpenter, J. M. Moser, & T. P. Romberg (Eds.), Addition and subtraction. A cognitive perspective (pp. 9–25). Hillsdale, NJ: Lawrence Erlbaum Associates.
   Google Scholar
• Coles, A. (2014). Ordinality, neuro-science and the early learning of number. In C. Nichol, S. Oesterle, P. Liljedahl, & D. Allen (Eds.), Proceedings of the joint PME 38 and PME-NA 36 conference (Vol. 2, pp. 329–336). Vancouver, BC: PME.
   Google Scholar
• Cooper, R. G. (1984). Early number development: Discovering number space with addition and subtraction. In C. Sophian (Ed.), Origins of cognitive skills (pp. 157–192). Hillsdale, NJ: Erlbaum.
   Google Scholar
• Cragg, L., & Gilmore, C. (2014). Skills underlying mathematics: The role of executive function in the development of mathematics proficiency. Trends in Neuroscience and Education, 3, 63–68.10.1016/j.tine.2013.12.001
   Google Scholar
• De Smedt, B., Noël, M.-P., Gilmore, C., & Ansari, D. (2013). How do symbolic and non-symbolic numerical magnitude processing skills relate to individual differences in children’s mathematical skills? A review of evidence from brain and behavior. Trends in Neuroscience and Education., 2, 48–55.10.1016/j.tine.2013.06.001
   Google Scholar
• Dehaene, S. (1997). The number sense. How the mind creates mathematics. New York, NY: Oxford University Press.
   Google Scholar
• Dehaene, S. (2001). Precis of the number sense. Mind and Language, 16, 16–36.10.1111/mila.2001.16.issue-1
   Google Scholar
• Dehaene, S., & Cohen, L. (1997). Cerebral pathways for calculation: Double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex, 33, 219–250.10.1016/S0010-9452(08)70002-9
   Google Scholar
• Desoete, A., & Roeyers, H. (2006). Metacognitive macroevaluations in mathematical problem solving. Learning and Instruction, 16, 12–25.10.1016/j.learninstruc.2005.12.003
   Google Scholar
• Dowker, A. D. (2005). Individual differences in arithmetic: Implications for psychology, neuroscience and education. Hove: Psychology.
   Google Scholar
• Field, A. (2009). Discovering statistics using SPSS. London: Sage.
   Google Scholar
• Fletcher, J. M., Lyon, G. R., Fuchs, L. S., & Barnes, M. A. (2007). Learning disabilities: From identification to intervention. New York, NY: Guilford Press.
   Google Scholar
• Fuchs, L. S., & Fuchs, D. (2002). Mathematical problem-solving profiles of students with mathematics disabilities with and without comorbid reading disabilities. Journal of Learning Disabilities, 35, 564–574.10.1177/00222194020350060701
   Google Scholar
• Fuchs, L. S., Fuchs, D., Compton, D. L., Bryant, J. D., Hamlett, C. L., & Seethaler, P. M. (2007). Mathematics screening and progress monitoring at first grade: Implications for responsiveness to intervention. Exceptional Children, 73, 311–330.10.1177/001440290707300303
   Google Scholar
• Galton, F. (1880). Visualised numerals. Nature, 21, 252–256.10.1038/021252a0
   Google Scholar
• Geary, D. C. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114, 345–362.10.1037/0033-2909.114.2.345
   Google Scholar
• Geary, D. C. (2004). Mathematics and learning disabilities. Journal of Learning Disabilities, 37, 4–15.10.1177/00222194040370010201
   Google Scholar
• Gebuis, T., & Reynvoet, B. (2011). Generating nonsymbolic number stimuli. Behavior Research Methods, 43, 981–986.
   Google Scholar
• Halberda, J., & Feigenson, L. (2008). Developmental change in the acuity of the “number sense”: The approximate number system in 3-, 4-, 5-, and 6-year-olds and adults. Developmental Psychology, 44, 1457–1465.10.1037/a0012682
   Google Scholar
• Hanich, L. B., Jordan, N. C., Kaplan, D., & Dick, J. (2001). Performance across different areas of mathematical cognition in children with learning difficulties. Journal of Educational Psychology, 93, 615–626.10.1037/0022-0663.93.3.615
   Google Scholar
• Hecht, S. A., Torgesen, J. K., Wagner, R., & Rashotte, C. (2001). The relationship between phonological processing abilities and emerging individual differences in mathematical computation skills: A longitudinal study of second to fifth grades. Journal of Experimental Child Psychology, 79, 192–227.10.1006/jecp.2000.2586
   Google Scholar
• Hedges, L. V., & Olkin, I. (1985). Statistical methods for meta-analysis. New York, NY: Academic Press.
   Google Scholar
• Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6(1), 1–55.
   Google Scholar
• Hubbard, E. M., Piazza, M., Pinel, P., & Dehaene, S. (2005). Interactions between number and space in parietal cortex. Nature Reviews Neuroscience, 6, 435–448.10.1038/nrn1684
   Google Scholar
• Jonas, C., & Jarick, M. (2013). Synesthesia, sequences and space. In J. Simner & E. M. Hubbard (Eds.), The Oxford handbook of synesthesia (pp. 123–148). Oxford: Oxford University Press.
   Google Scholar
• Jordan, N. C., Hanich, L. B., & Kaplan, D. (2003). A longitudinal study of mathematical competencies in children with specific mathematics difficulties versus children with comorbid mathematics and reading difficulties. Child Development, 74, 834–850.
   Google Scholar
• Karagiannakis, G., Baccaglini-Frank, A., & Papadatos, Y. (2014). Mathematical learning difficulties subtypes classification. Frontiers in Human Neuroscience, 8, 57. Retrieved from  http://doi.org/10.3389/fnhum.2014.00057
   Google Scholar
• Karagiannakis, G., & Cooreman, A. (2014). Focused intervention based on a classification MLD model. In S. Chinn (Ed.), The Routledge international handbook of dyscalculia and mathematical learning difficulties (pp. 265–276). London: Routledge.
   Google Scholar
• Kosc, L. (1974). Developmental dyscalculia. Journal of Learning Disabilities, 7, 164–177.10.1177/002221947400700309
   Google Scholar
• Kouba, V. L. (1989). Children’s solution strategies for equivalent set multiplication and division word problems. Journal for Research in Mathematics Education, 20, 147–158.10.2307/749279
   Google Scholar
• Koumoula, A., Tsironi, V., Stamouli, V., Bardani, I., Siapati, S., Annika, G., … von Aster, M. (2004). An epidemiological study of number processing and mental calculation in Greek schoolchildren. Journal of Learning Disabilities, 37, 377–388.10.1177/00222194040370050201
   Google Scholar
• Krueger, L. E., & Hallford, E. W. (1984). Why 2 + 2 = 5 looks so wrong: On the odd-even rule in sum verification. Memory & Cognition, 12, 171–180.10.3758/BF03198431
   Google Scholar
• Lewis, K. E., & Fisher, M. B. (2016). Taking stock of 40 years of research on mathematical learning disability: Methodological issues and future directions. Journal for Research in Mathematics Education, 47, 338–371.10.5951/jresematheduc.47.4.0338
   Google Scholar
• Lyons, I. M., & Ansari, D. (2015). Numerical order processing in children: From reversing the distance-effect to predicting arithmetic. Mind, Brain, and Education, 9, 207–221.10.1111/mbe.2015.9.issue-4
   Google Scholar
• Lyons, I., & Beilock, S. (2013). Ordinality and the nature of symbolic numbers. Journal of Neuroscience, 33, 17052–17061.10.1523/JNEUROSCI.1775-13.2013
   Google Scholar
• Mammarella, I. C., & Cornoldi, C. (2005). Difficulties in the control of irrelevant visuospatial information in children with visuospatial learning disabilities. Acta Psychologica, 118, 211–228.10.1016/j.actpsy.2004.08.004
   Google Scholar
• Mammarella, I. C., Giofrè, D., Ferrara, R., & Cornoldi, C. (2013). Intuitive geometry and visuospatial working memory in children showing symptoms of nonverbal learning disabilities. Child Neuropsychology, 19, 235–249.10.1080/09297049.2011.640931
   Google Scholar
• Mammarella, I. C., Lucangeli, D., & Cornoldi, C. (2010). Spatial working memory and arithmetic deficits in children with nonverbal learning difficulties. Journal of Learning Disabilities, 43, 455–468.10.1177/0022219409355482
   Google Scholar
• Mazzocco, M. M. M. (2005). Challenges in identifying target skills for math disability screening and intervention. Journal of Learning Disabilities, 38, 318–323.10.1177/00222194050380040701
   Google Scholar
• Mazzocco, M. M. M., & Myers, G. F. (2003). Complexities in identifying and defining mathematics learning disability in the primary school-age years. Annals of Dyslexia, 53, 218–253.10.1007/s11881-003-0011-7
   Google Scholar
• Mazzocco, M. M. M., & Räsänen, P. (2013). Contributions of longitudinal studies to evolving definitions and knowledge of developmental dyscalculia. Trends in Neuroscience and Education, 2, 65–73.10.1016/j.tine.2013.05.001
   Google Scholar
• Mulligan, J. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21, 33–49.10.1007/BF03217544
   Google Scholar
• Mulligan, J. (2011). Towards understanding the origins of children’s difficulties in mathematics learning. Australian Journal of Learning Difficulties, 16, 19–39.10.1080/19404158.2011.563476
   Google Scholar
• Mulligan, J. T., & Mitchelmore, M. C. (1997). Young children’s intuitive models of multiplication and division. Journal for Research in Mathematics Education, 28, 309–330.10.2307/749783
   Google Scholar
• Mulligan, J. T., & Mitchelmore, M. C. (2013). Early awareness of mathematical pattern and structure. In L. English & J. Mulligan (Eds.), Reconceptualizing early mathematics learning (pp. 29–45). Dordrecht: Springer Science-Business Media.10.1007/978-94-007-6440-8
   Google Scholar
• Mulligan, J. T., Mitchelmore, M. C., & Stephanou, A. (2015). Pattern and structure assessment (PASA): An assessment program for early mathematics (Years F-2) teacher guide. Australian Council for Educational Research. Melbourne: ACER Press.
   Google Scholar
• Nieder, A. (2005). Counting on neurons: The neurobiology of numerical competence. Nature Reviews Neuroscience, 6, 177–190.10.1038/nrn1626
   Google Scholar
• Núñez, R., & Lakoff, G. (2005). The cognitive foundations of mathematics: The role of conceptual metaphor. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 109–125). New York, NY: Psychology Press.
   Google Scholar
• Passolunghi, M. C., & Siegel, L. S. (2004). Working memory and access to numerical information in children with disability in mathematics. Journal of Experimental Child Psychology, 88, 348–367.10.1016/j.jecp.2004.04.002
   Google Scholar
• Piazza, M. (2010). Neurocognitive start-up tools for symbolic number representations. Trends in Cognitive Sciences, 14, 542–551.10.1016/j.tics.2010.09.008
   Google Scholar
• Piazza, M., Facoetti, A., Trussardi, A. N., Berteletti, I., Conte, S., Lucangeli, D., … Zorzi, M. (2010). Developmental trajectory of number acuity reveals a severe impairment in developmental dyscalculia. Cognition, 116, 33–41.10.1016/j.cognition.2010.03.012
   Google Scholar
• Pinel, P., Piazza, M., Le Bihan, D., & Dehaene, S. (2004). Distributed and overlapping cerebral representations of number, size, and luminance during comparative judgments. Neuron, 41, 983–993.10.1016/S0896-6273(04)00107-2
   Google Scholar
• Raghubar, K., Cirino, P., Barnes, M., Ewing-Cobbs, L., Fletcher, J., & Fuchs, L. (2009). Errors in multi-digit arithmetic and behavioral inattention in children with math difficulties. Journal of Learning Disabilities, 42, 356–371.10.1177/0022219409335211
   Google Scholar
• Raven, J., Court, J. H., & Raven, J. C. (1995). Coloured progressive matrices. Oxford: Oxford Psychologists Press.
   Google Scholar
• Reigosa-Crespo, V., Valdés-Sosa, M., Butterworth, B., Estévez, N., Rodríguez, M., Santos, E., … Lage, A. (2011). Basic numerical capacities and prevalence of developmental dyscalculia: The Havana Survey. Developmental Psychology, 48, 123–135.
   Google Scholar
• Rousselle, L., & Noël, M. P. (2007). Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs. non-symbolic number magnitude processing. Cognition, 102, 361–395.10.1016/j.cognition.2006.01.005
   Google Scholar
• Santi, G., & Baccaglini-Frank, A. (2015). Possible forms of generalization we can expect from students experiencing mathematical learning difficulties. PNA, 9, 217–243.
   Google Scholar
• Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws & the NCTM (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York, NY: Macmillan.
   Google Scholar
• Seron, X., Deloche, G., Ferrand, I., Cornet, J.-A., Frederix, M., & Hirsbrunner, T. (1991). Dot counting by brain damaged subjects. Brain and Cognition, 17, 116–137.10.1016/0278-2626(91)90072-G
   Google Scholar
• Shalev, R. S., Manor, O., Kerem, B., Ayali, M., Badichi, N., Friedlander, Y., & Gross-Tsur, V. (2001). Developmental dyscalculia is a familial learning disability. Journal of Learning Disabilities, 34, 59–65.10.1177/002221940103400105
   Google Scholar
• Siegler, R. S., & Booth, J. L. (2004). Development of numerical estimation in young children. Child Development, 75, 428–444.10.1111/cdev.2004.75.issue-2
   Google Scholar
• Siegler, R. S., & Opfer, J. E. (2003). The development of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science, 14, 237–250.10.1111/1467-9280.02438
   Google Scholar
• Szűcs, D. (2016). Subtypes and co-morbidity in mathematical learning disabilities: Multi-dimensional study of verbal and visual memory processes is key to understanding. Progress in Brain Research, 227, 277–304.10.1016/bs.pbr.2016.04.027
   Google Scholar
• Szűcs, D., & Goswami, U. (2013). Developmental dyscalculia: Fresh perspectives. Trends in Neuroscience and Education, 2, 33–37.10.1016/j.tine.2013.06.004
   Google Scholar
• Tan, P. N., Steibach, M., & Kumar, V. (2006). Introduction to data mining. Reading, MA: Addisson-Wesley.
   Google Scholar
• Tang, J., Ward, J., & Butterworth, B. (2008). Number forms in the brain. Journal of cognitive neuroscience, 20, 1–10.
   Google Scholar
• Thomas, N., Mulligan, J. T., & Goldin, G. A. (2002). Children’s representations and cognitive structural development of the counting sequence 1–100. The Journal of Mathematical Behavior, 21, 117–133.10.1016/S0732-3123(02)00106-2
   Google Scholar
• Thorndike, R. L. (1982). Applied Psychometrics. Boston, MA: Houghton Mifflin.
   Google Scholar
• Townsend, J. T., & Ashby, F. G. (1978). Methods of modeling capacity in simple processing systems. In J. Castellan & F. Restle (Eds.), Cognitive theory (Vol. 3, pp. 200–239). Hillsdale, NJ: Erlbaum.
   Google Scholar
• Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematical concepts and processes (pp. 127–174). New York, NY: Academic Press.
   Google Scholar
• von Aster, M. (2000). Developmental cognitive neuropsychology of number processing and calculation: Varieties of developmental dyscalculia. European Child & Adolescent Psychiatry, 9, 41–58.10.1007/s007870070008
   Google Scholar
• von Aster, M. G., & Shalev, R. S. (2007). Number development and developmental dyscalculia. Developmental Medicine & Child Neurology, 49, 868–873.10.1111/dmcn.2007.49.issue-11
   Google Scholar
• Ward, J., Sagiv, N., & Butterworth, B. (2009). The impact of visuo-spatial number forms on simple arithmetic. Cortex, 45, 1261–1265.10.1016/j.cortex.2009.03.017
   Google Scholar
• Watson, S. M. R., & Gable, R. A. (2013). Unraveling the complex nature of mathematics learning disability: Implications for research and practice. Learning Disability Quarterly, 36, 178–187.10.1177/0731948712461489
   Google Scholar
• Woodward, J., & Montague, M. (2002). Meeting the challenge of mathematics reform for students with LD. The Journal of Special Education, 36, 89–101.10.1177/00224669020360020401
   Google Scholar
• Zorzi, M., Priftis, K., & Umiltà, C. (2002). Neglect disrupts the mental number line. Nature, 417, 138–139.10.1038/417138a
   Google Scholar