Baccaglini-Frank, A., & Maracci, M. (2015). Multi-technology and preschoolers’ development of number-sense. Digital Experiences in Mathematics Education, 1–21.
Google Scholar
Clements, D. H., & Sarama, J. (2007). Early childhood mathematics learning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 461–555). Reston, VA: National Council of Teachers of Mathematics.
Google Scholar
Clements, D. H., & Sarama, J. (2010). Learning trajectories in early mathematics—sequences of acquisition and teaching. Encyclopedia of Early Childhood Development: Numeracy, 1–6.
Google Scholar
Corbin, J., & Strauss, A. (2015). Basics of qualitative research: Techniques and procedures for developing grounded theory. Thousand Oaks, CA: Sage publications.
Google Scholar
Creswell, J. W., & Plano Clark, V. L. (2011). Designing and conducting mixed methods research (2nd ed.), Thousand Oaks, CA: SAGE.
Google Scholar
Falloon, G. (2013). Young students using ipads: App design and content influences on their learning pathways. Computers & Education, 68, 505–521.
CrossRef
Google Scholar
Goldin, G. A. (2003). Representation in school mathematics: A unifying research perspective. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 275–285). Reston, VA: NCTM.
Google Scholar
Goldin, G. A., & Kaput, J. M. (1996). A joint perspective on the idea of representation in learning and doing mathematics. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 397–430). Hillsdale, NJ: Erlbaum.
Google Scholar
Holgersson, I., Barendregt, W., Rietz-Lepannen, E., Ottosson, T., & Linstrom, B. (2013). Can children enhance their arithmetic competence by playing an especially designed computer game? Proceedings from NORSMA 7: The Seventh Conference of the Nordic Research network on Special Needs Education in Mathematics. Copenhagen. Retrieved from http://www.diva-portal.org/smash/record.jsf?pid=diva2%3A693178&dswid=1175.
Inhelder, B. (2013). The early growth of logic in the child: Classification and seriation (Vol. 83). Routledge.
Google Scholar
Ladel, S., & Kortenkamp, U. (2013). An activity-theoretic approach to multi-touch tools in early maths learning. The International Journal for Technology in Mathematics Education, 20(1), 3–8.
Google Scholar
Ladel, S., & Kortenkamp, U. (2016). Artifact-centric activity theory—A framework for the analysis of the design and use of virtual manipulatives. In P. Moyer-Packenham (Ed.), International perspectives on teaching and learning mathematics with virtual manipulatives (pp. 25–40). New York: Springer.
Google Scholar
Larkin, K., & Milford, T. (2018). Mathematics apps—Stormy with the weather clearing: Using cluster analysis to enhance app use in mathematics classrooms. In N. Calder, K. Larkin, & N. Sinclair (Eds.), Using mobile technologies in the teaching and learning of mathematics. Mathematics Education in the Digital Era: Springer.
Google Scholar
Le Corre, M., & Carey, S. (2007). One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition, 105, 395–438.
CrossRef
Google Scholar
Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Lawrence Erlbaum Associates.
Google Scholar
Manches, A., & O’Malley, C. (2012). Tangibles for learning: A representational analysis of physical manipulation. Personal and Ubiquitous Computing, 16, 405–419.
CrossRef
Google Scholar
Moyer-Packenham, P. S., & Bolyard, J. J. (2016). Revisiting the definition of a virtual manipulative. In P. Moyer-Packenham (Ed.), International perspectives on teaching and learning mathematics with virtual manipulatives (pp. 5–16). New York: Springer.
Google Scholar
Moyer-Packenham, P. S., Bullock, E. P., Shumway, J. F., Tucker, S. I., Watts, C., Westenskow, A., Anderson-Pence, K. L., Maahs-Fladung, C., … Jordan, K. (2016). The role of affordances in children’s learning performance and efficiency when using virtual manipulative mathematics touch-screen apps. Mathematics Education Research Journal, 28(1), 79–105.
Google Scholar
Moyer-Packenham, P. S., Shumway, J. F., Bullock, E., Tucker, S. I., Anderson-Pence, K. L., Westenskow, A., Boyer-Thurgood, J., Maahs-Fladung, C., … Jordan, K. (2015). Young children’s learning performance and efficiency when using virtual manipulative mathematics iPad apps. Journal of Computers in Mathematics and Science Teaching, 34(1), 41–69.
Google Scholar
Moyer-Packenham, P. S., Tucker, S. I., Westenskow, A., & Symanzik, J. (2015b). Examining patterns in second graders’ use of virtual manipulative mathematics apps through heatmap analysis. International Journal of Educational Studies in Mathematics, 2(2), 1–16.
CrossRef
Google Scholar
Moyer-Packenham, P. S., & Westenskow, A. (2013). Effects of virtual manipulatives on student achievement and mathematics learning. International Journal of Virtual and Personal Learning Environments, 4(3), 35–50.
CrossRef
Google Scholar
Moyer-Packenham, P. S., & Westenskow, A. (2016). Revisiting the effects and affordances of virtual manipulatives for mathematics learning. In K. Terry & A. Cheney (Eds.), Utilizing Virtual and Personal Learning Environments for Optimal Learning (pp. 186–215). Hershey, PA: IGI Global.
CrossRef
Google Scholar
Pica, P., Lemer, C., Izard, V., & Dehaene, S. (2004). Exact and approximate arithmetic in an Amazonian Indigene group. Science, 306(5695), 499–503.
CrossRef
Google Scholar
Rick, J. (2012). Proportion: A tablet app for collaborative learning. In Proceedings of the 11th International Conference on Interaction Design and Children (pp. 316–319). New York, NY, USA: ACM.
Google Scholar
Sarama, J., & Clements, D. H. (2009a). “Concrete” computer manipulatives in mathematics education. Child Development Perspectives, 3(3), 145–150.
CrossRef
Google Scholar
Sarama, J., & Clements, D. H. (2009b). Early childhood mathematics education research: Learning trajectories for young children. New York, NY: Routledge.
Google Scholar
Sarama, J., Clements, D. H., Barrett, J., Van Dine, D. W., & McDonel, J. S. (2011). Evaluation of a learning trajectory for length in the early years. ZDM Mathematics Education, 43, 667–680.
CrossRef
Google Scholar
Siegler, R. S., & Booth, J. L. (2004). Development of numerical estimation in young children. Child Development, 75(2), 428–444.
CrossRef
Google Scholar
Smith, C. L., Wiser, M., Anderson, C. W., & Krajcik, J. (2006). Implications of research on children’s learning for standards and assessment: A proposed learning progression for matter and the atomic-molecular theory. Measurement, 4(1/2), 1–98.
Google Scholar
Spencer, P. (2013). iPads: Improving numeracy learning in the early years. In V. Steinle, L. Ball, & C. Bardini (Eds.), Mathematics education: Yesterday, today, and tomorrow (pp. 610–617). Melbourne, Australia: MERGA.
Google Scholar
Stebbins, R. A. (2001). Exploratory research in the social sciences (Vol. 48). Thousand Oaks, CA: Sage publications.
Google Scholar
Tucker, S. I. (2016). The modification of attributes, affordances, abilities, and distance for learning framework and its applications to interactions with mathematics virtual manipulatives. In P. S. Moyer-Packenham (Ed.), International perspectives on teaching and learning mathematics with virtual manipulatives (pp. 41–69). Springer International Publishing.
Google Scholar
Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally. Boston: Allyn & Bacon.
Google Scholar
Vidiksis, R., Jo, I. Y., Hupert, N., & Llorente, C. (2013). All hands on tech: math and media in the preschool classroom. In R. McBride, & M. Searson (Eds.), Society for Information Technology & Teacher Education International Conference 2013 (pp. 4453–4457). Chesapeake, VA: AACE.
Google Scholar
Watts, C. M., Moyer-Packenham, P. S., Tucker, S. I., Bullock, E. P., Shumway, J. F., Westenskow, A., et al. (2016). An examination of children’s learning progression shifts while using touch screen virtual manipulative mathematics apps. Computers in Human Behavior, 64, 814–828.
CrossRef
Google Scholar
Wilkinson, L., & Friendly, M. (2009). The history of the cluster heat map. The American Statistician, 63(2), 179–184.
CrossRef
Google Scholar
Zaranis, N., Kalogiannakis, M., & Papadakis, S. (2013). Using mobile devices for teaching realistic mathematics in kindergarten education. Creative Education, 4(7A1), 1–10.
Google Scholar