# Cornerstone Mathematics: designing digital technology for teacher adaptation and scaling

## Abstract

We report the results of a design-based research project in England that embeds digital technology. The research followed from two phases in the USA: (1) a design phase that used dynamic representations to foster conceptual understanding of hard-to-teach mathematical ideas, and (2) a research phase that measured the efficacy of the resulting technology-based curriculum units as implemented in Texas schools. The goal of the third phase in England was initially to “scale up” the US approach. We determined, however, that the materials had to be re-designed for adaptability by English teachers. We report how the features of the innovation—particularly its technological infrastructure—could be leveraged, not only to achieve positive learning outcomes, but also to lay the foundations for change in pedagogy and learning at scale. We identify an emergent framework of design affordances for teacher adaptability that are particularly salient when technology is a critical element.

## Keywords

Professional Development Digital Technology Learning Gain Teacher Adaptation Professional Development Session## Notes

### Acknowledgments

We gratefully acknowledge funding by the Li Ka Shing Foundation. The research has entailed collaboration among, in LKL, Philip Kent; and in SRI International, Jennifer Knudsen, Ken Rafanan, Teresa Lara-Meloy, Anna Werner, Gucci Estrella, and Nicole Shechtman.

## References

- Artigue, M. (2002). Learning Mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work.
*International Journal of Computers for Mathematical Learning,**7*(3), 245–274.CrossRefGoogle Scholar - Baker-Doyle, K. J. (2011).
*The networked teacher: How new teachers build social networks for professional support*. New York, NY: Teachers College Press.Google Scholar - Blumenfeld, P., Fishman, B. J., Krajcik, J., Marx, R. W., & Soloway, E. (2000). Creating usable innovations in systemic reform: scaling up technology-embedded project-based science in urban schools.
*Educational Psychologist,**35*(3), 149–164.CrossRefGoogle Scholar - Cheung, A. C. K., & Slavin, R. E. (2013). The effectiveness of educational technology applications for enhancing mathematics achievement in K-12 classrooms: A meta-analysis.
*Educational Research Review,**9*, 88–113.CrossRefGoogle Scholar - Coburn, C. E. (2003). Rethinking scale: Moving beyond numbers to deep and lasting change.
*Educational Researcher,**32*(6), 3–12.CrossRefGoogle Scholar - Confrey, J., Hoyles, C., Jones, K., Kahn, K., Maloney, A., Nguyen, K., et al. (2009). Designing software for mathematical engagement through modelling. In C. Hoyles & J.-B. Lagrange (Eds.),
*Digital technologies and mathematics teaching and learning: Rethinking the terrain*(pp. 19–46). New York, NY: Springer.CrossRefGoogle Scholar - Drijvers, P., Kieran, C., & Mariotti, M. A. (2010). Integrating technology into mathematics education: Theoretical perspectives. In C. Hoyles & J.-B. Lagrange (Eds.),
*Mathematics education and technology—rethinking the terrain*(pp. 89–132). New York, NY/Berlin, Germany: Springer.Google Scholar - Drijvers, P., & Trouche, L. (2008). From artefacts to instruments: A theoretical framework behind the orchestra metaphor. In G. W. Blume & M. K. Heid (Eds.),
*Research on technology and the teaching and learning of mathematics*(Vol. 2, pp. 363–392)., Cases and perspectives Charlotte, NC: Information Age.Google Scholar - Dynarski, M., Agodini, R., Heaviside, S., Novak, T., Carey, N., Campuzano, L., et al. (2007). Effectiveness of reading and mathematics software products: Findings from the first student cohort. Washington DC: National Center for Educational Evaluation.Google Scholar
- Heid, M. K., & Blume, G. W. (2008). Algebra and function development. In M. K. Heid & G. W. Blume (Eds.),
*Research on technology and the teaching and learning of mathematics: Research syntheses*(Vol. 1, pp. 55–108). Charlotte, NC: Information Age.Google Scholar - Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., et al. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study.
*Cognition and Instruction,**26*(4), 430–511.CrossRefGoogle Scholar - Hoyles, C., & Lagrange, J.-B. (2009).
*Mathematics education and technology—rethinking the terrain*. New York, NY: Springer.Google Scholar - Hoyles, C., & Noss, R. (2003). What can digital technologies take from and bring to research in mathematics education? In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.),
*Second international handbook of research in mathematics education*(pp. 323–349). Dordrecht, The Netherlands: Kluwer.CrossRefGoogle Scholar - Kaput, J. (1992). Technology and mathematics education. In D. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 515–556). New York, NY: Simon and Schuster Macmillan.Google Scholar - Kaput, J., Hegedus, S., & Lesh, R. (2007). Technology becoming infrastructural in mathematics education. In R. Lesh, E. Hamilton, & J. Kaput (Eds.),
*Foundations for the future in mathematics education*(pp. 173–192). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Kaput, J., & Roschelle, J. (1998). The mathematics of change and variation from a millennial perspective: New content, new context. In C. Hoyles, C. Morgan, & G. Woodhouse (Eds.),
*Rethinking the mathematics curriculum*. London, England: Falmer Press.Google Scholar - Kynigos, C., Clayson, E., & Yainnoutso, N. (2012).
*Constructionism 2012, theory, practice and impact: Conference proceedings*. Athens, Greece: Educational Technology Lab.Google Scholar - Laborde, C. (1995). Designing tasks for learning geometry in a computer-based environment. In L. Burton & B. Jaworski (Eds.),
*Technology in mathematics teaching—a bridge between teaching and learning*(pp. 35–68). London: Chartwell-Bratt.Google Scholar - Noss, R., & Hoyles, C. (1996).
*Windows on mathematical meanings: Learning Cultures and Computers*. Dordrecht, The Netherlands: Kluwer Academic.CrossRefGoogle Scholar - Noss, R., & Hoyles, C. (2013). Constructionism & microworlds. In R. Duval, R. Sutherland, & M. Sharples (Eds.),
*The technology-enhanced learning reader*. New York, NY: Springer.Google Scholar - Papert, S. (1980).
*Mindstorms: Children, computers and powerful ideas*. London, England: Harvester Press.Google Scholar - Penuel, W. R., Fishman, B. J., Cheng, B., & Sabelli, N. (2011). Organizing research and development at the intersection of learning, implementation, and design.
*Educational Researcher,**40*(7), 331–337.CrossRefGoogle Scholar - Raudenbush, S. W., & Bryk, A. S. (2002).
*Hierarchical linear models: Applications and data analysis methods*(2nd ed.). Newbury Park, CA: Sage Publications.Google Scholar - Roschelle, J., & Jackiw, N. (2000). Technology design as educational research: Interweaving imagination, inquiry & impact. In A. Kelly & R. Lesh (Eds.),
*Research design in mathematics & science education*(pp. 777–797). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Roschelle, J., & Shechtman, N. (2013). SimCalc at scale: Three studies examine the integration of technology, curriculum and professional development for advancing middle school mathematics. In J. Roschelle & S. Hegedus (Eds.),
*The SimCalc vision and contributions: Democratizing access to important mathematics*(pp. 125–144). Berlin, Germany: Springer.CrossRefGoogle Scholar - Roschelle, J., Shechtman, N., Tatar, D., Hegedus, S., Hopkins, B., Empson, S., et al. (2010). Integration of technology, curriculum, and professional development for advancing middle school mathematics.
*American Educational Research Journal,**47*(4), 833–878.CrossRefGoogle Scholar - Roschelle, J., Tatar, D., & Kaput, J. (2008a). Getting to scale with innovations that deeply restructure how students come to know mathematics. In A. E. Kelly, R. Lesh, & J. Y. Baek (Eds.),
*Handbook of design research methods in education*(pp. 369–395). New York: Routledge.Google Scholar - Roschelle, J., Tatar, D., Shechtman, N., & Knudsen, J. (2008b). The role of scaling up research in designing for and evaluating robustness.
*Educational Studies in Mathematics,**68*(2), 149–170.CrossRefGoogle Scholar - Schneider, B., & McDonald, S. K. (Eds.). (2007).
*Scale-up in education: issues in practice (Vol. 2)*. New York: Rowman & Littlefield Publishers, Inc.Google Scholar - Sinclair, N., Arzarello, F., Gaisman, M., Lozano, M., Dagiene, M., Behrooz, E., & Jackiw, N. (2010). In C. Hoyles & J-B. Lagrange (Eds.)
*Mathematics education and technology*—*Rethinking the terrain*(pp. 61–78). New York, NY: Springer.Google Scholar - Sturman, L., & Cooper, L. (2012).
*Evaluation of Cornerstone Mathematics pilot in England: Unit 1, linear functions*. Slough, England: National Foundation for Educational Research.Google Scholar - Vahey, P., Knudsen, J., Rafanan, K., & Lara-Meloy, T. (2013a). Curricular activity systems supporting the use of dynamic representations to foster students’ deep understanding of mathematics. In C. Mouza & N. Lavigne (Eds.),
*Emerging technologies for the classroom: A learning sciences perspective*(pp. 15–30). New York, NY: Springer.CrossRefGoogle Scholar - Vahey, P., Roy, G., & Fueyo, V. (2013b). Sustainable use of dynamic representational environments: Toward a district-wide adoption of SimCalc-based materials. In S. Hegedus & J. Roschelle (Eds.),
*The SimCalc visions and contributions: Democratizing access to important mathematics*(pp. 183–202). New York, NY: Springer.CrossRefGoogle Scholar