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Heatmap and Hierarchical Clustering Analysis to Highlight Changes in Young Children’s Developmental Progressions Using Virtual Manipulative Mathematics Apps

Part of the Mathematics Education in the Digital Era book series (MEDE,volume 12)

Abstract

The purpose of this study was to examine what patterns were revealed using heatmaps with hierarchical clustering to examine preschooler’s performance , speed , and developmental progressions in counting and seriation . The chapter describes a study conducted with 35 preschoolers who used six touchscreen virtual manipulative mathematics apps in two different learning sequences: counting and seriation . The analysis employed heatmaps coupled with hierarchical clustering to highlight changes in children’s performance , speed , and developmental progressions, between a pre- and post- assessment app after using two learning apps. This method allowed for analysis of individual and whole group data examining several tasks within each app and also several apps within each learning sequence. The analysis revealed different clusters of children grouped according to their developmental progressions which were related to incremental changes in performance and speed from the Pre to Post App use.

Keywords

  • Virtual manipulative
  • Touchscreen app
  • Developmental progression
  • Performance
  • Speed
  • Heatmap
  • Hierarchical clustering
  • Preschool
  • Mathematics app
  • Seriation
  • Counting

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References

  • 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 

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Correspondence to Patricia S. Moyer-Packenham .

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Lommatsch, C.W., Tucker, S.I., Moyer-Packenham, P.S., Symanzik, J. (2018). Heatmap and Hierarchical Clustering Analysis to Highlight Changes in Young Children’s Developmental Progressions Using Virtual Manipulative Mathematics Apps. In: Calder, N., Larkin, K., Sinclair, N. (eds) Using Mobile Technologies in the Teaching and Learning of Mathematics. Mathematics Education in the Digital Era, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-90179-4_10

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  • DOI: https://doi.org/10.1007/978-3-319-90179-4_10

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