Advertisement

ZDM

, Volume 49, Issue 5, pp 735–753 | Cite as

Learner-controlled scaffolding linked to open-ended problems in a digital learning environment

  • Alden Jack Edson
Original Article

Abstract

This exploratory study reports on how students activated learner-controlled scaffolding and navigated through sequences of connected problems in a digital learning environment. A design experiment was completed to (re)design, iteratively develop, test, and evaluate a digital version of an instructional unit focusing on binomial distributions and their applications for statistical inference. The developed materials are organized around open-ended problems linked to learner-controlled scaffolding. This study reports on a retrospective analysis of classroom observations, digital artifacts of student work, and interviews and surveys that document: (a) the ways students activated learner-controlled scaffolding linked to open-ended problems in the digital environment and (b) the observed student problem-solving pathways of activated scaffolding across connected sequences of problems related to binomial distributions and their applications for statistical inference. The results suggest that when students have the opportunity to control the level of access and challenge during problem solving using a digital medium, new opportunities are possible for the sequence of problems through which students can progress.

Keywords

Mathematics education Digital curriculum Scaffolding Problem-solving pathways 

Notes

Acknowledgements

This research was supported by the Transition to College Mathematics and Statistics project with funding from the National Science Foundation Grant DRL-1020312. All opinions and analysis expressed herein are those of the author and do not necessarily represent the position or policies of the Foundation. The research and its interpretation reported in this article are based on the author’s doctoral dissertation completed at Western Michigan University under the direction of Christian R. Hirsch and Steven W. Ziebarth. A previous version of this article was presented at the American Educational Research Association 2016 Annual Meeting. The author gratefully acknowledges the work of Ann Watkins, California State University–Northridge, in developing the print version of the Binomial Distributions and Statistical Inference unit and that of James Laser, Western Michigan University, in coding the initial shell for the digital prototype.

References

  1. Abdu, R., Schwarz, B., & Mavrikis, M. (2015). Whole-class scaffolding for learning to solve mathematics problems together in a computer-supported environment. ZDM, 47(7), 1163–1178.CrossRefGoogle Scholar
  2. Bakker, A., Smit, J., & Wegerif, R. (2015). Scaffolding and dialogic teaching in mathematics education: Introduction and review. ZDM, 47(7), 1047–1065.CrossRefGoogle Scholar
  3. Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. Elementary School Journal, 93(4), 373–397.CrossRefGoogle Scholar
  4. Barab, S. (2014). Design-based research: A methodological toolkit for engineering change. In R. K. Sawyer (Ed.), The Cambridge handbook of the learning sciences (2nd ed.) (pp. 151–170). New York, NY: Cambridge University Press.Google Scholar
  5. Bell, C. V., & Pepe, S. J. (2012). Scaffolding students’ opportunities to learn mathematics through social interactions. Mathematics Education Research Journal, 24(4), 423–445.CrossRefGoogle Scholar
  6. Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for Research in Mathematics Education, 29(1), 41–62.CrossRefGoogle Scholar
  7. Boaler, J., & Staples, M. (2008). Creating mathematical futures through an equitable teaching approach: The case of railside school. Teachers College Record, 110(3), 608–645.Google Scholar
  8. Brown, A. L. (1992). Design experiments: Theoretical and methodological challenges in creating complex interventions in classroom settings. The Journal of the Learning Sciences, 2(2), 141–178.CrossRefGoogle Scholar
  9. Chapin, S., & O’Connor, C. (2007). Academically productive talk: Supporting student learning in mathematics. In W. G. Martin, M. Strutchens & P. Elliott (Eds.), The learning of mathematics (pp. 113–128). Reston: National Council of Teachers of Mathematics.Google Scholar
  10. Chen, W., Rovegno, I., Cone, S. L., & Cone, T. P. (2012). An accomplished teacher’s use of scaffolding during a second-grade unit on designing games. Research Quarterly for Exercise and Sport, 83(2), 221–234.CrossRefGoogle Scholar
  11. Choppin, J. (2011). Learned adaptations: Teachers’ understanding and use of curriculum resources. Journal of Mathematics Teacher Education, 14(5), 331–353.CrossRefGoogle Scholar
  12. Choppin, J., Carson, C., Borys, Z., Cerosaletti, C., & Gillis, R. (2014). A typology for analyzing digital curricula in mathematics education. International Journal of Education in Mathematics, Science, and Technology, 2(1), 11–25.CrossRefGoogle Scholar
  13. Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81–89.CrossRefGoogle Scholar
  14. Cobb, P., Gravemeijer, K., Yackel, E., McClain, K., & Whitenack, J. (1997). Mathematizing and symbolizing: The emergence of chains of signification in one first-grade classroom. In IN D. Kirshner & J. Whitson (Eds.), Situated cognition: Social, semiotic, and psychological perspectives (pp. 151–233). Mahwah: Erlbaum.Google Scholar
  15. Cognition and Technology Group at Vanderbilt (1990). Anchored instruction and its relationship to situated cognition. Educational Researcher, 19(5), 2–10.CrossRefGoogle Scholar
  16. Collins, A. (1992). Toward a design science of education. In E. Scanlon & T. O’Shea (Eds.), New directions in educational technology (pp. 15–22). New York: Springer.CrossRefGoogle Scholar
  17. Collins, A. (1999). The changing infrastructure of education research. In E. C. Lagemann & L. S. Shulman (Eds.), Issues in education research: Problems and possibilities (pp. 289–298). San Francisco: Jossey-Bass.Google Scholar
  18. Coxford, A. F., Fey, J. T., Hirsch, C. R., Schoen, H. L., Hart, E. W., Keller, B. A., et al. (2003). Contemporary mathematics in context: A unified approach, courses 1–4. Columbus: McGraw-Hill.Google Scholar
  19. Davis, J., Choppin, J., Roth McDuffie, A., & Drake, C. (2013). Common Core State Standards for Mathematics: Middle school teachers’ perceptions. Retrieved March 15, 2014, from http://www.warner.rochester.edu/files/warnercenter/docs/commoncoremathreport.pdf.
  20. Dyer, M., & Moynihan, C. (2000). Open-ended questions in elementary mathematics: Instruction and assessment. Larchmont: Eye on Education.Google Scholar
  21. Edelson, D. C. (2002). Design research: What we learn when we engage in design. Journal of the Learning Sciences, 11(1), 105–121.CrossRefGoogle Scholar
  22. Edson, A. J. (2014). A study on the iterative development and efficacy of a deeply digital instructional unit on binomial distributions and statistical inference. Kalamazoo: Western Michigan University.Google Scholar
  23. Edson, A. J. (2016). A design experiment of a deeply digital instructional unit and its impact in high school classrooms. In M. Bates & Z. Usiskin (Eds.), Digital curricula in school mathematics (pp. 177–193). Charlotte: Information Age Publishing.Google Scholar
  24. Edson, A. J., & Thomas, A. (2016). Transforming preserviced mathematics teacher knowledge for and with the enacted curriculum: The case of digital instructional materials. In M. L. Niess, S. Driskell & K. Hollenbrands (Eds.), Handbook of research on transforming mathematics teacher education in the digital age (pp. 215–240). Hershey: IGI Global.CrossRefGoogle Scholar
  25. Engle, R. A., & Conant, F. R. (2002). Guiding principles for fostering productive disciplinary engagement: Explanining an emergent argument in a community of learners classroom. Cognition and Instruction, 20(4), 399–483.CrossRefGoogle Scholar
  26. Fey, J. (2009). Considering the future of K-12 STEM curricula and instructional materials: Stimulating and supporting new developments. The Center for the Study of Mathematics Curriculum Workshop Series on STEM Curriculum and Instructional Design. Retrieved March 15, 2014 from http://mathcurriculumcenter.org/conferences/stem/index.php.
  27. Heaton, R. M. (2000). Teaching mathematics to the new standards: Relearning the dance. New York: Teachers College Press.Google Scholar
  28. Hiebert, J., & Wearne, D. (1993). Instructional tasks, classroom discourse, and students’ learning in second grade arithmetic. American Educational Research Journal, 30(2), 393–425.CrossRefGoogle Scholar
  29. Hirsch, C. R. (Ed.). (2007). Perspectives on the design and development of school mathematics curriculum. Reston: National Council of Teachers of Mathematics.Google Scholar
  30. Hirsch, C. R., Fey, J. T., Hart, E. W., Schoen, H. L., Watkins, A. E., Ritsema, B. E., et al. (2008). Core-plus mathematics, courses 1–4, 2nd edition. Columbus: Glencoe/McGraw-Hill.Google Scholar
  31. Hirsch, C. R., Hart, E., Watkins, A., Fey, J., Ritsema, B., Walker, R., et al (2015). Transition to college mathematics and statistics. Columbus: McGraw-Hill School Solutions.Google Scholar
  32. Holton, D., & Clarke, D. (2006). Scaffolding and metacognition. International Journal of Mathematics Education in Science and Technology, 37(2), 127–143.CrossRefGoogle Scholar
  33. Hufferd-Ackles, K., Fuson, K. C., & Sherin, M. G. (2004). Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education, 35(2), 81–116.CrossRefGoogle Scholar
  34. Jaworski, B. (1994). Investigating mathematics teaching: A constructivist inquiry. London: Falmer Press.Google Scholar
  35. Kapur, M. (2010). Productive failure in mathematical problem solving. Instructional Science, 38(6), 523–550.CrossRefGoogle Scholar
  36. Keller, B. (2014). TCMS-tools. Kalamazoo: The Transition to College Mathematics and Statistics Project, Western Michigan University.Google Scholar
  37. Lampert, M. (1990a). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27(1), 29–63.CrossRefGoogle Scholar
  38. Lampert, M. (1990b). Connecting inventions with conventions. In L. P. Steffe & T. Wood (Eds.), Transforming children’s mathematics education (pp. 253–265). Hillsdale: Erlbaum.Google Scholar
  39. Mercer, N., & Littleton, K. (2007). Dialogue and the development of children’s thinking: A sociocultural approach. London: Routledge.Google Scholar
  40. Michaels, S., O’Connor, C., & Resnick, L. (2008). Reasoned participation: Accountable talk in the classroom and in civic life. Studies in Philosophy and Education, 27(4), 283–297.CrossRefGoogle Scholar
  41. Miyazaki, M., Fujita, T., & Jones, K. (2015). Flow-chart proofs with open problems as scaffolds for learning about geometrical proof. ZDM, 47(7), 1211–1224.CrossRefGoogle Scholar
  42. Noss, R., & Hoyles, C. (1996). Windows on mathematical meanings learning cultures and computers. Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
  43. Pea, R. (2004). The social and technological dimensions of scaffolding and related theoretical concepts for learning, education, and human activity. Journal of the Learning Sciences, 13(3), 423–451.CrossRefGoogle Scholar
  44. Puntambekar, S., & Hubscher, R. (2005). Tools for scaffolding students in a complex learning environment: What have we gained and what have we missed? Educational Psychologist, 40(1), 1–12.CrossRefGoogle Scholar
  45. Reigeluth, C. M., & Frick, T. W. (1999). Formative research: A methodology for creating and improving design theories. In C. M. Reigeluth (Ed.), Instructional-design theories and models: A new paradigm of instructional theory (Vol. 2, pp. 633–651). Mahwah: Lawrence Erlbaum.Google Scholar
  46. Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246.CrossRefGoogle Scholar
  47. Remillard, J. T., Herbel-Eisenmann, B. A., & Lloyd, G. M. (Eds.). (2009). Mathematics teachers at work: Connecting curriculum materials and classroom instruction. New York: Routledge.Google Scholar
  48. Reys, B. J. (Ed.). (2006). The intended mathematics curriculum as represented in state-level curriculum standards: Consensus or confusion? Charlotte: Information Age.Google Scholar
  49. Richey, R. C., Klein, J., & Nelson, W. (2004). Developmental research: Studies of instructional design and development. In D. Jonassen (Ed.), Handbook of research for educational communications and technology (2nd edn., pp. 1099–1130). Mahwah: Lawrence Erlbaum.Google Scholar
  50. Schoenfeld, A. H. (2002). Making mathematics work for all children: Issues of standards, testing, and equity. Educational Researcher, 31(1), 13–25.CrossRefGoogle Scholar
  51. Schukajlow, S., Kolter, J., & Blum, W. (2015). Scaffolding mathematical modelling with a solution plan. ZDM, 47(7), 1241–1254.CrossRefGoogle Scholar
  52. Senk, S. L., & Thompson, D. R. (Eds.). (2003). Standards-based school mathematics curricula: What are they? What do students learn? Hillsdale: Erlbaum.Google Scholar
  53. Sherin, M. G. (2002a). A balancing act: Developing a discourse community in a mathematics classroom. Journal of Mathematics Teacher Education, 5(3), 205–233.CrossRefGoogle Scholar
  54. Sherin, M. G. (2002b). When teaching becomes learning. Cognition and Instruction, 20(2), 119–150.CrossRefGoogle Scholar
  55. Sherin, M. G., & Drake, C. (2009). Curriculum strategy framework: Investigating patterns in teachers’ use of a reform-based elementary mathematics curriculum. Journal of Curriculum Studies, 41(4), 467–500.CrossRefGoogle Scholar
  56. Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.Google Scholar
  57. Silver, E. A., & Smith, M. (1996). Building discourse communities in mathematics classrooms: A worthwhile but challenging journey. In P. C. Elliott & M. J. Kenney (Eds.), Communication in mathematics: K-12 and beyond (pp. 20–28). Reston: National Council of Teachers of Mathematics.Google Scholar
  58. Smit, J., van Eerde, H. A. A., & Bakker, A. (2013). A conceptualisation of whole-class scaffolding. British Educational Research Journal, 39(5), 817–834.CrossRefGoogle Scholar
  59. Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455–488.CrossRefGoogle Scholar
  60. Stein, M. K., & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project. Educational Research and Evaluation, 2(1), 50–80.CrossRefGoogle Scholar
  61. Stein, M. K., Smith, M. S., Henningsen, M., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York: Teachers College Press.Google Scholar
  62. Stein, M. K., Remillard, J., & Smith, M. S. (2007). How curriculum influences student learning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 319–370). Charlotte, NC: Information Age Publishing.Google Scholar
  63. Stender, S., & Kaiser, G. (2015). Scaffolding in complex modelling situations. ZDM, 47(7), 1255–1267.CrossRefGoogle Scholar
  64. Stigler, J. W., & Hiebert, J. (2004). Improving mathematics teaching. Educational Leadership, 61(5), 12–16.Google Scholar
  65. Stone, C. A. (1998). The metaphor of scaffolding: Its utility for the field of learning disabilities. Journal of Learning Disabilities, 31(4), 344–364.CrossRefGoogle Scholar
  66. Sztajn, P., Confrey, J., Wilson, P. H., & Edgington, C. (2012). Learning trajectory based instruction: Toward a theory of teaching. Educational Researcher, 41(5), 147–156.CrossRefGoogle Scholar
  67. Tarr, J. E., Reys, R. E., Reys, B. J., Chávez, O., Shih, J., & Osterlind, S. J. (2008). The impact of middle-grades mathematics curricula and the classroom learning environment on student achievement. Journal for Research in Mathematics Education, 39(3), 247–280.Google Scholar
  68. The Design-Based Research Collective (2003). Design-based research: An emerging paradigm for educational inquiry. Educational Researcher, 32(1), 5–8.CrossRefGoogle Scholar
  69. Tropper, N., Leiss, D., & Hänze, M. (2015). Teachers’ temporary support and worked-out examples as elements of scaffolding in mathematical modeling. ZDM, 47(7), 1225–1240.CrossRefGoogle Scholar
  70. Usdan, J., & Gottheimer, J. (2012). FCC Chairman: Digital textbooks to all students in five years. Retrieved March 15, 2014, from http://www.fcc.gov/blog/fcc-chairman-digital-textbooks-all-students-five-years.
  71. Usiskin, Z., & Willmore, E. (Eds.). (2008). Mathematics curriculum in Pacific Rim countries: China, Japan, Korea, and Singapore. Charlotte: Information Age.Google Scholar
  72. Van de Pol, J., Volman, M., & Beishuizen, J. (2010). Scaffolding in teacher-student interaction: A decade of research. Educational Psychology Review, 22(3), 271–296.CrossRefGoogle Scholar
  73. Wang, F., & Hannafin, M. J. (2005). Design-based research and technology-enhanced learning environments. Educational Technology Research and Development, 53(4), 5–23.CrossRefGoogle Scholar
  74. Wood, D., Bruner, J. S., & Ross, G. (1976). The role of tutoring in problem solving. Journal of Child Psychology and Psychiatry, 17(2), 89–100.CrossRefGoogle Scholar
  75. Yee, F. P. (2000). Open-ended problems for higher-order thinking in mathematics. Teaching and Learning, 20(2), 49–57.Google Scholar

Copyright information

© FIZ Karlsruhe 2017

Authors and Affiliations

  1. 1.Michigan State UniversityEast LansingUSA

Personalised recommendations