Reasoning about Representations: Effects of an Early Math Intervention

References

• Ahlberg, A. (2011). Communicating mathematics in primary school. In J. Emanuelsson, L. Fainsilber, J. Häggström, A. Kullberg, B. Lindström, & M. Löwing (Eds.), Voices on learning and instruction in mathematics (pp. 79–95). Gothenburg: National Centre for Mathematics Education, University of Gothenburg.
   Google Scholar
• Andersson, U., & Lyxell, B. (2007). Working memory deficit in children with mathematical difficulties: A general or specific deficit? Journal of Experimental Child Psychology, 96(3), 197–228.
   PubMed Web of Science ®Google Scholar
• Aubrey, C., Dahl, S., & Godfrey, R. (2006). Early mathematics development and later achievement: Further evidence. Mathematics Education Research Journal, 18(1), 27–46.
   Google Scholar
• Aunio, P., Hautamäki, J., Sajaniemi, N., & Van Luit, J. E. H. ((2010)). Early numeracy in low-performing young children. British Educational Research Journal, 35(1), 25–46. doi: 10.1080/01411920802041822
   Web of Science ®Google Scholar
• Aunio, P., & Niemivirta, M. (2010). Predicting children’s mathematical performance in grade one by early numeracy. Learning and Individual Differences, 20(5), 427–435. doi: 10.1016/j.lindif.2010.06.003
   Web of Science ®Google Scholar
• Aunio, P., & Räsänen, P. (2015). Core numerical skills for learning mathematics in children aged five to eight years – a working model for educators. European Early Childhood Education Research Journal, 24(5), 684–704. doi: 10.1080/1350293X.2014.996424
   Web of Science ®Google Scholar
• Aunola, K., Leskinen, E., Lerkkanen, M.-K., & Nurmi, J.-E. (2004). Developmental dynamics of math performance from preschool to Grade 2. Journal of Educational Psychology, 96(4), 699–713. doi: 10.1037/0022-0663.96.4.699
   Web of Science ®Google Scholar
• Baddeley, A. D. (2000). Short-term and working memory. In E. Tulving, & F. I. M. Craik (Eds.), The Oxford Handbook of memory (pp. 77–92). New York: Oxford University Press.
   Google Scholar
• Bailey, D. H., Siegler, R., & Geary, D. C. (2014). Early predictors of middle school fraction knowledge. Developmental Science, 17(5), 775–785.
   PubMed Web of Science ®Google Scholar
• Baroody, A. J., Cibulskis, M., Lai, L., & Li, X. (2004). Comments on the use of learning trajectories in curriculum development and research. Mathematical Thinking and Learning, 6(2), 227–260. doi: 10.1207/s15327833mtl0602_8
   Google Scholar
• Baroody, A. J., Eiland, M., & Thompson, B. (2009). Fostering at-risk preschoolers’ number sense. Early Education and Development, 20(1), 80–128. doi: 10.1111/desc.12155
   Web of Science ®Google Scholar
• Bisanz, J., Sherman, J., Rasmussen, C., & Ho, E. (2005). Development of arithmetic skills and knowledge in preschool children. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 143–162). New York: Psychology Press.
   Google Scholar
• Bishop, D. V. M. (2003). Test for Reception of Grammar Version 2, TROG-2.Manual. Department of Experimental Psychology University of Oxford. London: Person Inc.
   Google Scholar
• Browne, M. W., & Cudeck, R. (1993). Alternative ways of assessing model fit. In K. Bollen, & J. Long (Eds.), Testing structured equation models (pp. 136–162). Newbury Park, CA: Sage.
   Google Scholar
• Claessens, A., Duncan, G., & Engel, M. (2009). Kindergarten skills and fifth-grade achievement: Evidence from the ECLS-K. Economics of Education Review, 28(4), 415–427.
   Web of Science ®Google Scholar
• Claessens, A., & Engel, M. (2013). How important is where you start? Early mathematics knowledge and later school success. Teachers College Record, 115, 1–29. doi: 10.1016/j.econedurev.2008.09.003
   Web of Science ®Google Scholar
• Clarke, B., Baker, S., Smolkowski, K., Doabler, C., Strand Cary, M., & Fien, H. (2015). Investigating the efficacy of a core kindergarten mathematics curriculum to improve student mathematics learning outcomes. Journal of Research on Educational Effectiveness, 8(3), 303–324. doi: 10.1080/19345747.2014.980021
   Web of Science ®Google Scholar
• Clarke, B., Smolkowski, K., Baker, S., Fien, H., & Chard, D. (2011). The Impact of a comprehensive tier 1 kindergarten curriculum on the achievement of students at-risk in mathematics. The Elementary School Journal, 111(4), 561–584. doi: 10.1086/659033
   Web of Science ®Google Scholar
• Clements, D. H., & Sarama, J. (2007). Effects of a preschool mathematics curriculum: Summative research on the Building Blocks project. Journal for Research in Mathematics Education, 38(2), 136–163. doi: 10.2307/30034954
   Web of Science ®Google Scholar
• Clements, D. H., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York: Routledge.
   Google Scholar
• Clements, D. H., & Sarama, J. (2011). Early mathematics intervention. Science, 333(6045), 968–970.
   PubMed Web of Science ®Google Scholar
• Clements, D. H., Sarama, J., Spitler, M. E., Lange, A. A., & Wolfe, C. B. (2011). Mathematics learned by young children in an intervention based on learning trajectories: A large-scale cluster randomized trial. Journal for Research in Mathematics Education, 42(2), 127–166.
   Web of Science ®Google Scholar
• Cross, C. T., Woods, T. A., & Schweingruber, H. (2009). Mathematics learning in early childhood: Paths toward excellence and equity. Washington, DC: The National Academic Press.
   Google Scholar
• Dehaene, S. (1997). The number sense: How the mind creates mathematics. Oxford: Oxford University Press.
   Google Scholar
• De Smedt, B., Janssen, R., Bouwens, K., Verschaffel, L., Boets, B., & Ghesquière, P. (2009). Working memory and individual differences in mathematics achievement: A longitudinal study from first grade to second grade. Journal of Experimental Child Psychology, 103, 186–201.
   PubMed Web of Science ®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.
   Google Scholar
• Desoete, A., Ceulemans, A., De Weerdt, F., & Pieters, S. (2012). Can we predict mathematical learning disabilities from symbolic and non-symbolic comparison tasks in kindergarten? Findings from a longitudinal study. British Journal of Educational Psychology, 82, 64–81.
   PubMed Web of Science ®Google Scholar
• Duncan, G. J., Claessens, A., Huston, A. C., Pagani, L. S., Engel, M., et al. (2007). School readiness and later achievement. Developmental Psychology, 43(6), 1428–1446. doi: 10.1037/0012-1649.43.6.1428
   PubMed Web of Science ®Google Scholar
• Durham, R. E., Farkas, G., Scheffner Hammer, C., Tomblin, J. B., & Catts, H. W. (2007). Kindergarten oral language skill: A key variable in the intergenerational transmission of socioeconomic status. Research in Social Stratification and Mobility, 25(4), 294–305.
   Google Scholar
• Dyson, N., Jordan, N. C., Beliakoff, A., & Hassinger-Das, A. (2015). A kindergarten number-sense intervention with contrasting practice conditions for low achieving children. Journal for Research in Mathematics Education, 46(3), 331–370. doi: 10.5951/jresematheduc.46.3.0331
   PubMed Web of Science ®Google Scholar
• Dyson, N. I., Jordan, N. C., & Glutting, J. A. (2011). A number sense intervention for low-income kindergarteners at risk for mathematics difficulties. Journal of Learning Disabilities, 46(2), 166–181. doi: 10.1177/0022219411410233
   PubMed Web of Science ®Google Scholar
• Fälth, L., Gustafson, S., Tjus, T., Heimann, M., & Svensson, I. (2013). Computer-assisted interventions targeting reading skills of children with reading disabilities - A longitudinal study. Dyslexia, 19(1), 37–53.
   PubMed Web of Science ®Google Scholar
• Farrell Pagulayan, K., Busch, R., Medina, K., Bartok, J., & & Krikorian, R. (2007). Developmental normative data for the Corsi Block-Tapping task. Journal of Clinical and Experimental Neurophsychology, 28(6), 1043–1052.
   Web of Science ®Google Scholar
• Fosnot, C. T. (2007). Investigating number sense, addition, and subtraction. Grades K-3. Portsmouth, NH: Firsthand Heinemann.
   Google Scholar
• Geary, D., & vanMarie, K. (2016). Young children’s core symbolic and nonsymbolic quantitative knowledge in the prediction of later mathematics achievement. Developmental Psychology, 52(12), 2130–2144. doi: 10.1037/dev0000214
   PubMed Web of Science ®Google Scholar
• Geary, D. C. (2011). Cognitive predictors of achievement growth in mathematics: A 5-year longitudinal study. Developmental Psychology, 47(6), 1539–1552. doi: 10.1037/a0025510
   PubMed Web of Science ®Google Scholar
• Geary, D. C., Hoard, M. K., Byrd-Craven, J., Nugent, L., & Numtee, C. (2007). Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability. Child Development, 78(4), 1343–1359. doi: 10.1111/j.1467-8624.2007.01069.x
   PubMed Web of Science ®Google Scholar
• Gersten, R., Chard, J. C., Jayanthi, M., Baker, S., Morphy, P., & Flojo, J. (2009). Mathematics instruction for students with learning disabilities: A meta-analysis of instructional components. Review of Educational Research, 79(3), 1202–1242. doi: 10.3102/0034654309334431
   Web of Science ®Google Scholar
• Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and intervention for students with mathematics difficulties. Journal of Learning Disabilities, 38(4), 293–304.
   PubMed Web of Science ®Google Scholar
• Gough, P., & Tunmer, W. (1986). Decoding, reading, and reading disability. Remedial and Special Education, 7(1), 6–10.
   Google Scholar
• Griffin, S. (2007). Early intervention for children at risk of developing mathematical learning disabilities. In D. B. Berch, & M. M. M. Mazzocco (Eds.), Why is math so hard for some children? (pp. 373–396). Baltimore: Paul H. Brookes.
   Google Scholar
• Gustafsson, J.-E., & Stahl, P. A. (2005). STREAMS 3.0 User’s guide. Mölndal: MultivariateWare.
   Google Scholar
• Gustafsson, J.-E., Yang Hansen, K., & Rosén, M. (2013). Effects of home background on student achievement in reading, mathematics, and science at the Fourth Grade. In M. O. Martin, & I. V. S. Mullis (Eds.), TIMSS and PIRLS 2011: Relationships among reading, mathematics, and science achievement at the fourth grade – implications for early learning (pp. 181–287). Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College.
   Google Scholar
• Hu, L., & Bentler, P. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structure Equation Modelling, 6(1), 1–55.
   Web of Science ®Google Scholar
• Johnsen Høines, M. (2000). Matematik som språk: Verksamhetsteoretiska perspektiv. Stockholm: Liber.
   Google Scholar
• Jordan, N. C. (2007). The need for number sense. The roots of many students’ math difficulties are evident as early as kindergarten. Educational Leadership, 65(2), 63–66.
   Google Scholar
• Jordan, N. C., Glutting, J., & Ramineni, C. (2010). The importance of number sense to mathematics achievement in first and third grades. Learning and Individual Differences, 20(2), 82–88. doi: 10.1016/j.lindif.2009.07.004
   PubMed Web of Science ®Google Scholar
• Jordan, N. C., Hanich, L. B., & Kaplan, D. (2003). Arithmetic fact mastery in young children: A longitudinal investigation. Journal of Experimental Child Psychology, 85(2), 103–119.
   PubMed Web of Science ®Google Scholar
• Jordan, N. C., Kaplan, D., Ramineni, C., & Locuniak, M. N. (2009). Early math matters: Kindergarten number sense and later mathematics outcomes. Developmental Psychology, 45(3), 850–867. doi: 10.1037/a0014939
   PubMed Web of Science ®Google Scholar
• Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
   Google Scholar
• Kleemans, T., Segers, E., & Verschaffel, L. (2011). Cognitive and linguistic precursors to numeracy in kindergarten: Evidence from first and second language learners. Learning and Individual Differences, 21, 555–561.
   Web of Science ®Google Scholar
• Kolkman, M. E., Kroesbergen, E. H., & Leseman, P. P. M. (2013). Early numerical development and the role of non-symbolic and symbolic skills. Learning and Instruction, 25, 95–103. doi: 10.1016/j.learninstruc.2012.12.001
   Google Scholar
• Koponen, T., Salmi, P., Eklund, K., & Aro, T. (2013). Counting and RAN: Predictors of arithmetic calculation and reading fluency. Journal of Educational Psychology, 105, 162–175.
   Web of Science ®Google Scholar
• Krajewski, K., & Schneider, W. (2009). Exploring the impact of phonological awareness, visual-spatial working memory, and preschool quantity-number competencies on mathematics achievement in elementary school: Findings from a 3-year longitudinal study. Journal of Experimental Child Psychology, 103(4), 516–531. doi: 10.1016/jecp.2009.03.009
   PubMed Web of Science ®Google Scholar
• Kyttälä, M., Aunio, P., & Hautamäki, J. (2010). Working memory resources in young children with mathematical difficulties. Scandinavian Journal of Psychology, 51(1), 1–15. doi: 10.1111/j.1467-9450.2009.00736.x
   PubMed Web of Science ®Google Scholar
• Kyttälä, M., Aunio, P., Lepola, J., & Hautamäki, J. (2014). The role of the working memory and language skills in the prediction of the word problem solving in 4- to 7-year-old children. Educational Psychology, 34(6), 674–696. doi: 10.1080/01443410.2013.814192
   Google Scholar
• Laski, E. V., & Yu, Q. (2014). Number line estimation and mental addition: Examining the potential roles of language and education. Journal of Experimental Child Psychology, 117, 29–44. doi: 10.1016/j.jecp.2013.08.007
   PubMed Web of Science ®Google Scholar
• Lerkkanen, M.-K., Rasku-Puttonen, H., Aunola, K., & Nurmi, J.-E. (2005). Mathematical performance predicts progress in reading comprehension among 7-year olds. European Journal of Psychology of Education, 20(2), 121–137. doi: 10.1007/BF03173503
   Web of Science ®Google Scholar
• Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33–40). US: Lawrence Erlbaum Associates.
   Google Scholar
• Levine, S. H., Jordan, N. C., & Huttenlocher, J. (1992). Development of calculation abilities in young children. Journal of Experimental Child Psychology, 53, 72–103.
   PubMed Web of Science ®Google Scholar
• Li, Y., & Geary, D. C. (2017). Children’s visuospatial memory predicts mathematics achievement through early adolescence. Plos One, 12(2), 1–13. doi: 10.1037/edu0000159
   Web of Science ®Google Scholar
• Lie Reikerås, E. K. (2007). Aspects of arithmetical performance related to reading performance: a comparison of children with different levels of achievement in mathematics and reading different age levels. PhD thesis. Stavanger: University of Stavanger.
   Google Scholar
• Little, T. D. (2013). Longitudinal structural equation modelling. New York: The Guilford Press.
   Google Scholar
• Lundberg, I. (1985). Longitudinal studies of reading and writing difficulties in Sweden. In G. E. McKinnon, & T. G. Waller (Eds.), Reading research: Advances in theory and practice (pp. 65–105). New York: Academic Press.
   Google Scholar
• Lundberg, I. (2001). Vilken Bild är rätt? Klassdiagnos av läsförståelse för årskurs 2 och 3. Stockholm: Natur och Kultur.
   Google Scholar
• Lundberg, I. (2007). Bornholmsmodellen: Vägen till läsning: Språklekar i förskoleklass. Stockholm: Natur och Kultur.
   Google Scholar
• Lundberg, I., Frost, J., & Petersen, O.-P. (1988). Effects of an extensive program for stimulating phonological awareness in preschool children. Reading Research Quarterly, 23(3), 263–284.
   Web of Science ®Google Scholar
• Lundberg, I., & Sterner, G. (2006). Reading, arithmetic and task orientation - how are they related? Annals of Dyslexia, 56(2), 361–377.
   PubMed Web of Science ®Google Scholar
• McCarney, R., Warner, J., Iliffe, S., van Haselen, R., Griffin, M., & Fisher, P. ((2007)). The Hawthorne effect: A randomised, controlled trial. BMC Medical Research Methodology, 7, 30. doi:10.1186/1471-2288-7-30. PMC 1936999. PMID 17608932
   PubMed Web of Science ®Google Scholar
• McIntosh, A. (2004). Mental computation: A strategies approach. Tasmania: Department of Education, Tasmania.
   Google Scholar
• McIntosh, A. (2008). Förstå och använda tal. Göteborg: Göteborgs universitet. NCM.
   Google Scholar
• McKenzie, B., Bull, R., & Gray, C. (2003). The effects of phonological and visuo-spatial interference on children’s arithmetical performance. Educational and Child Psychology, 20(3), 93–108.
   Google Scholar
• Melhuish, E. C., Sylva, K., Sammons, P., Siraj-Blatchford, I., & Taggart, B. (2008). Preschool influences on mathematics achievement. Science, 321(5893), 1161–1162. doi: 10.1126/science.1158808
   PubMed Web of Science ®Google Scholar
• Mononem, R., & Aunio, P. (2016). Counting skills intervention for low-performing first graders. South African Journal of Childhood Education, 6(1), 1–9. doi: 10.1080/08856257.2016.1141541
   Web of Science ®Google Scholar
• Mononen, R., Aunio, P., Koponen, T., & Aro, M. (2014). A review of early numeracy interventions for children at risk in mathematics. International Journal of Early Childhood Special Education, 6, 25–54. doi: 1080/08856257.2014.922794
   Google Scholar
• Morgan, P. L., Farkas, G., & Wu, Q. (2009). Five-year growth trajectories of kindergarten children with learning difficulties in mathematics. Journal of Learning Disabilities, 42(4), 306–321. doi: 10.1177/0022219408331037
   PubMed Web of Science ®Google Scholar
• Mueller, M., & Maher, C. (2009). Learning to reason in an informal math after-school program. Mathematics Education Research Journal, 21(3), 7–35.
   Google Scholar
• Mulligan, J. (2011). Towards understanding the origins of children’s difficulties in mathematics learning. Australian Journal of Learning Difficulties, 16, 19–39. doi: 10.1080/19404158.2011.563476
   Google Scholar
• Muthén, L. K., & Muthén, B. O. (1998-2012). Mplus user’s guide. Seventh edition. Los Angeles, CA: Muthén & Muthén.
   Google Scholar
• Muthén, L. K., & Muthén, B. O. (2012). Mplus user’s guide. Statistical analysis with latent variables. Version 7. Los Angeles, CA: Muthén & Muthén.
   Google Scholar
• National Agency for Education. (2017). Curriculum for compulsory school, preschool class and recreation centre 2017. Stockholm: Skolverket.
   Google Scholar
• Nunes, T., Bryant, P., Evans, D., & Barros, R. (2015). Assessing quantitative reasoning in young children. Mathematical Thinking and Learning, 17(2–3), 178–196. doi: 10.1080/10986065.2015.1016815
   Web of Science ®Google Scholar
• Nunes, T., Bryant, P., Evans, D., Bell, D., Gardner, S., et al. (2007). The contribution of logical reasoning to the learning of mathematics in primary school. British Journal of Developmental Psychology, 25(1), 147–166. doi: 10.1348/026151006X153127
   Web of Science ®Google Scholar
• Okamoto, Y., & Case, R. (1996). Exploring the microstructure of children’s central conceptual structures in the domain of number. Monographs of the Society for Research Child Development, 61(1-2), 27–59.
   PubMed Web of Science ®Google Scholar
• Papic, M. M., Mulligan, J. T., & Mitchelmore, M. C. (2011). Assessing the development of preschoolers’ mathematical patterning. Journal for Research in Mathematics Education, 42(3), 237–268.
   Web of Science ®Google Scholar
• Purpura, D. J., Baroody, A. J., & Lonigan, C. J. (2013). The transition from informal to formal mathematical knowledge: Mediation by numeral knowledge. Journal of Educational Psychology, 105(2), 453–464. doi: 10.1037/a0031753
   Web of Science ®Google Scholar
• Raghubar, K. P., Barnes, M. A., & Hecht, S. A. (2010). Working memory and mathematics: A review of developmental, individual difference, and cognitive approaches. Learning and Individual Differences, 20(2), 110–122. doi: 10.1016/j.lindif.2009.10.005
   Web of Science ®Google Scholar
• Rasmussen, C., & Bisanz, J. (2005). Representation and working memory in early arithmetic. Journal of Experimental Child Psychology, 91(2), 137–157. doi: 10.1016/j.jecp.2005.01.004
   PubMed Web of Science ®Google Scholar
• Sarama, J., & Clements, H. C. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York: Routledge.
   Google Scholar
• Schneider, M., Beeres, K., Coban, L., Merz, S., Schmidt, S. S., Stricker, J., & De Smedt, B. (2017). Associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence: A meta-analysis. Developmental Science, 20(3), 1–16.
   Web of Science ®Google Scholar
• Sfard, A., & Lave, R. (2005). Why cannot children see as the same what grown-ups cannot see as different? – early numerical thinking revisited. Cognition and Instruction, 23(2), 237–309. doi:101207/s1532690xci2302 3
   Web of Science ®Google Scholar
• Siegler, R. S., & Booth, J. (2004). Development of numerical estimation in young children. Child Development, 75(2), 428–444. doi: 10.1111/j.1467-8624.2004.00684.x
   PubMed Web of Science ®Google Scholar
• Steiger, J. H. (2007). Understanding the limitations of global fit assessment in structural equation modeling. Personality and Individual Differences, 42(5), 893–898. doi: 10.1016/j.paid.2006.09.017
   Web of Science ®Google Scholar
• Stein, M. K., Engle, R., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340. doi: 10.1080/10986060802229675
   Google Scholar
• Sterner, G. (2014). Basic arithmetic skills. Gothenburg.
   Google Scholar
• Sterner, G., & Helenius, O. (2015). Number by reasoning and representations – the design and theory of an intervention program for preschool class in Sweden. In O. Helenius, A. Engström, T. Meaney, P. Nilsson, E. Norén, J. Sayers, & M. Österholm (Eds.), Development of mathematics teaching: Design, scale, effect (pp. 159–168). Linköping: SMDF.
   Google Scholar
• Sterner, G., Helenius, O., & Wallby, K. (2014). Tänka, resonera och räkna i förskoleklass. Gothenburg: National Centre for Mathematics Education. 
   Google Scholar
• Toll, S. W. M., & Van Luit, J. E. H. (2012). Effects of remedial numeracy instruction throughout kindergarten starting at different ages: Evidence from a large-scale longitudinal study. Learning and Instruction, 33, 39–49. doi: 10.1016/j.learninstruc.2014.03.003
   Web of Science ®Google Scholar
• Van Luit, J. E. H., & Van de Rijt, B. A. M. (2005). Early numeracy test. Doetinchem: Graviant.
   Google Scholar
• Vennberg, H., & Norqvist, M. (2018). Counting on – long term effects of an early intervention program. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.), Proceedings of the 42nd conference of the international group for the psychology of mathematics education (Vol. 4, pp. 355–362). Umeå: PME.
   Google Scholar
• Vygotsky, L. S. (1978). Mind in society. The development of higher psychological processes. Cambridge: Harward University Press.
   Google Scholar
• Vygotsky, L. S. (2004). Imagination and creativity in childhood. Journal of Russian and East European Psychology, 42, 7–97.
   Google Scholar
• Watts, T. W., Duncan, G. J., Siegler, R. S., & Davis-Kean, P. E. (2014). What’s past is prologue: Relations between early mathematics knowledge and high school achievement. Educational Researcher, 43(7), 352–360. doi: 10.3102/0013189X14553660
   PubMed Web of Science ®Google Scholar
• Wiese, H. (2003). Iconic and non-iconic stages in number development: The role of language. TRENDS in Cognitive Sciences, 7(9), 385–390. doi: 1016/S1364-6613(03)00192-X
   PubMed Web of Science ®Google Scholar
• Winsløv, C. (2004). Semiotics as an analytic tool for the didactics of mathematics. Nomad, 9, 81–100.
   Google Scholar
• Witzel, B. S., Mercer, C. D., & Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities Research & Practice, 18(2), 121–131.
   Google Scholar
• Wolff, U. (2011). Effects of a randomised reading intervention study: An application of structural equation modelling. Dyslexia, 17(4), 295–311. doi: 10.1002/dys.438
   PubMed Web of Science ®Google Scholar
• Wolff, U. (2013). MiniDUVAN. Kartläggning av fonologisk förmåga hos barn mellan 4 och 6 år. [assessment of phonological skills in children 4–6 years old]. Stockholm: Hogrefe Psykologiförlaget.
   Google Scholar
• Xenidou-Dervou, I., Van Luit, J. E. H., Kroesberger, E. H., Friso-van den Bos, I., Jonkman, L. M., et al. (2018). Cognitive predictors of children’s development in mathematics achievement: A latent growth modeling approach. Developmental Science, 1–13. doi: 10.1111/desc.12671
   Web of Science ®Google Scholar
• Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. Kilptarick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 227–236). Reston, VA: National Council of Teachers of Mathematics (NCTM).
   Google Scholar