Advertisement

ZDM

, Volume 49, Issue 5, pp 717–734 | Cite as

Scaffolding learner-centered curricular coherence using learning maps and diagnostic assessments designed around mathematics learning trajectories

  • Jere Confrey
  • Garron Gianopulos
  • William McGowan
  • Meetal Shah
  • Michael Belcher
Original Article
First Online:

Abstract

The paper describes how designers used the construct of learning trajectories to create a tool, Math-Mapper 6–8, to help scaffold curricula toward increased learner-centered coherence. It defines “learner-centered curricular coherence” as “an organizational means to promote a high likelihood that each learner traverses one of many possible paths to understand target disciplinary ideas in a curriculum.” The tool’s features, including its learning map, diagnostic assessments, and reporting system, are tied to its underlying foundation in learning trajectories. Three preliminary studies of the implemented tool’s effects are reported to provide insight to its influence on curricular sequencing, students’ patterns of performance in early algebra, and student responses to the assessments.

Keywords

Learning trajectories Curriculum Coherence Learning map Diagnostic Middle grades 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

We gratefully acknowledge support of the National Science Foundation (DRL-1118858) and the Bill and Melinda Gates Foundation (OPP1118124). Editing assistance by A. Maloney and M. Hennessey.

References

  1. Adjei, S., Selent, D., Heffernan, N., Pardos, Z., Broaddus, A., & Kingston, N. (2014). Refining learning maps with data fitting techniques: Searching for better fitting learning maps. In J. Stamper, Z. Pardos, M. Mavrikis & B. M. McLaren (Eds.) Proceedings of the 7th International Conference on Educational Data Mining (pp. 413–414). London.Google Scholar
  2. Barrett, J., Clements, D., Sarama, J., Cullen, C., McCool, J., Witkowski-Rumsey, C., & Klanderman, D. (2012). Evaluating and improving a learning trajectory for linear measurement in elementary grades 2 and 3: A longitudinal study. Mathematical Thinking and Learning, 14(1), 28–54.CrossRefGoogle Scholar
  3. Battista, M. T. (2011). Conceptualizations and issues related to learning progressions, learning trajectories, and levels of sophistication. The Mathematics Enthusiast, 8(3), 507–570.Google Scholar
  4. Black, P., Harrison, C., & Lee, C. (2004). Working inside the black box: Assessment for learning in the classroom. London: Granada Learning.Google Scholar
  5. Boester, T., & Lehrer, R. (2008). Visualizing algebraic reasoning. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 211–234). New York: Routledge.Google Scholar
  6. Briggs, D. C., & Peck, F. A. (2015). Using learning progressions to design vertical scales that support coherent inferences about student growth. Measurement: Interdisciplinary Research and Perspectives, 13(2), 75–99.Google Scholar
  7. Bruner, J. S. (1960). The Process of Education. Cambridge, MA: Harvard University Press.Google Scholar
  8. CCSSI (2011). Mathematics Standards. http://www.corestandards.org/Math. Accessed 17 July 2016.
  9. Chazan, D. & Yerushalmy, M. (2003). On appreciating the cognitive complexity of school algebra: Research on algebra learning and directions of curricular change. In J. Kilpatrick, D. Schifter, & G. Martin (Eds.), A research companion to the principles and standards for school mathematics (pp. 123–135). Reston, Virginia: NCTM.Google Scholar
  10. Chazan, D., & Yerushalmy, M. (2014). The future of mathematics textbooks: ramifications of technological change. In M. Stochetti (Ed.), Media and education in the digital age: Concepts, assessment and subversions (pp. 63–76). New York: Peter Lang.Google Scholar
  11. Confrey, J. (2015). Some possible implications of data-intensive research in education—The value of learning maps and evidence-centered design of assessment to educational data mining. In C. Dede (Ed.), Data-intensive research in education: Current work and next steps (pp. 79–87). Washington, DC: Computing Research Association.Google Scholar
  12. Confrey, J., Jones, R. S., & Gianopulos, G. G. (2015). Challenges in modeling and measuring learning trajectories: A response to Briggs and Peck’s “Using learning progressions to design vertical scales that support coherent inferences about student growth.” Measurement: Interdisciplinary Research & Perspectives, 13(2), 100–105.Google Scholar
  13. Confrey, J., Maloney, A. P., & Corley, A. K. (2014). Learning trajectories: A framework for connecting standards with curriculum. ZDM—The International Journal on Mathematics Education, 46(5), 719–733.CrossRefGoogle Scholar
  14. Davis, J., Choppin, J., McDuffie, A. R., & Drake, C. (2013). Common core state standards for mathematics: Middle school mathematics teachers’ perceptions. Rochester, NY: The Warner Center for Professional Development and Education Reform.Google Scholar
  15. Dede, C. (2015). Data-intensive research in education: Current work and next steps. Computer Research Association. http://cra.org/wp-content/uploads/2015/10/CRAEducationReport2015.pdf. Accessed 4 April 2016.
  16. Engelhard, G. (2013). Invariant Measurement Using Rasch Models in the Social, Behavioral, and Health Sciences. New York: Routledge.Google Scholar
  17. Gueudet, G., Pepin, B., & Trouche, L. (2013, July). Textbooks design and digital resources. In C. Margolinas (Ed.), Task design in mathematics education. Proceedings of ICMI study 22 (pp. 327–337). https://hal.archives-ouvertes.fr/hal-00834054v2. Accessed 21 July 2016.
  18. Heritage, M. (2007). Formative assessment: What do teachers need to know and do? Phi Delta Kappan, 89(2), 140–145.CrossRefGoogle Scholar
  19. Heuvel-Panhuizen, M. Van den, & Buys, K. (Eds.). (2008). Young children learn measurement and geometry. Rotterdam: Sense Publishers.Google Scholar
  20. Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for research in Mathematics Education, 297–312.Google Scholar
  21. Koedinger, K. R., Alibali, M. W., & Nathan, M. J. (2008). Trade-offs between grounded and abstract representations: evidence from algebra problem solving. Cognitive Science, 32(2), 366–397.CrossRefGoogle Scholar
  22. Larson, M. (2016). Curricular coherence in the age of open educational-resources. https://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/Matt-Larson/Curricular-Coherence-in-the-Age-of-Open-Educational-Resources/. Accessed 13 August 2016.
  23. Maloney, A. P., Confrey, J., & Nguyen, K. H. (Eds.). (2014). Learning over time: Learning trajectories in mathematics education. Charlotte, NC: Information Age Publishing.Google Scholar
  24. National Mathematics Advisory Panel (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.Google Scholar
  25. NCTM (2014). Principles to actions: Ensuring mathematics success for all. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  26. Pepin, B., Gueudet, G., Yerushalmy, M., Trouche, L., & Chazan, D. (2015). E-textbooks in/for teaching and learning mathematics: A disruptive and potentially transformative educational technology. In L. D. English & D. Kirshner (Eds.), Handbook of international research in mathematics education 3rd Ed. (pp. 636–661). New York, NY: Routledge.Google Scholar
  27. Piaget, J. (1970). Genetic epistemology. Trans. E. Duckworth. New York, NY: Columbia University Press.Google Scholar
  28. Rittle-Johnson, B., Star, J. R., & Durkin, K. (2009). The importance of prior knowledge when comparing examples: Influences on conceptual and procedural knowledge of equation solving. Journal of Educational Psychology, 101(4), 836–852.CrossRefGoogle Scholar
  29. Sacristán, A. I., Calder, N., Rojano, T., Santos-Trigo, M., Friedlander, A., Meissner, H., & Perrusquía, E (2010). The influence and shaping of digital technologies on the learning–and learning trajectories–of mathematical concepts. In C. Hoyles & J.-B. Lagrange (Eds.), Mathematics education and technology—rethinking the terrain: The 17th ICMI study (pp. 179–226). Boston, MA: Springer US.Google Scholar
  30. Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York, NY: Routledge.Google Scholar
  31. Schliemann, A. D., Carraher, D. W., & Brizuela, B. M. (2007). Bringing out the algebraic character of arithmetic: From children’s ideas to classroom practice. New York: Routledge.Google Scholar
  32. Schmidt, W. H., Hsing Chi, W., & McKnight, C. C. (2005). Curriculum coherence: an examination of US mathematics and science content standards from an international perspective. Journal of Curriculum Studies, 37(5), 525–559.CrossRefGoogle Scholar
  33. Schoenfeld, A. H. (2009). Instructional research and the improvement of practice. In J. D. Bransford, D. J. Stipek, N. J. Vye & L. M. Gomez, & Lam, D. (2009), The role of research in educational improvement (pp. 161–188). Cambridge: Harvard Press.Google Scholar
  34. Schoenfeld, A. H., & Arcavi, A. (1988). On the meaning of variable. The Mathematics Teacher, 81(6), 420–427.Google Scholar
  35. Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 114–145.Google Scholar
  36. Stacey, K., Price, B., & Steinle, V. (2012). Identifying stages in a learning hierarchy for use in formative assessment – the example of line graphs. In J. Dindyal, L.-P. Cheng & S.-F. Ng (Eds.), Expanding horizons: Proceedings of the 35th Annual Conference of Mathematics Education Group of Australasia (pp. 393–400). Singapore: MERGA.Google Scholar
  37. Stacey, K., Price, B., Steinle, V., Chick, H., & Gvozdenko, E. (2009, September). SMART assessment for learning. Paper from Conference of the International Society for Design and Development in Education, Cairns, Australia, September 28–October 1. http://isdde.org/isdde/cairns/pdf/papers/isdde09_stacey.pdf. Accessed 8 May 2016.
  38. Stacey, K., Steinle, V., Gvozdenko, E., & Price, B. (2013). SMART online formative assessments for teaching mathematics. Curriculum and Leadership Journal, 11(20). http://www.curriculum.edu.au/leader/smart_online_formative_assessments_for_teaching_ma,36822.html?issueID=12826. Accessed 20 July 2016.
  39. Stacey, K., & Wiliam, D. (2012). Technology and assessment in mathematics. In M. A. Clements, A. J. Bishop, C. Keitel, J. Kilpatrick & F. K. S. Leung (Eds.) Third international handbook of mathematics education (pp. 721–751). Springer New York.Google Scholar
  40. Stark, J. S. (1986). On defining coherence and integrity in the curriculum. Research in Higher Education, 24(4), 433–436.CrossRefGoogle Scholar
  41. Stiggins, R. (2005). From formative assessment to assessment for learning: A path to success in standards-based schools. The Phi Delta Kappan, 87(4), 324–328.CrossRefGoogle Scholar
  42. Suppes, P. (1961). A comparison of the meaning and uses of models in mathematics and the empirical sciences. In H. Freudenthal (Ed.), The concept and the role of the model in mathematics and natural and social sciences (pp. 163–177). Dordrecht: Reidel Publishers.CrossRefGoogle Scholar
  43. Thrall, T., & Tingey, B. (2003). SuccessMaker® Motion: A research summary. Pearson Digital Learning Technical Note TN030409). Upper Saddle River, NJ: Pearson Education.Google Scholar
  44. Trigueros, M., & Ursini, S. (1999). Does the understanding of variable evolve through schooling? In O. Zaslavsky (Ed.), Proceedings of the 23rd International Conference of PME, vol. 4 (pp. 273–280). Haifa. Israel: Israel Institute of Technology.Google Scholar
  45. Webel, C., Krupa, E. E., & McManus, J. (2015). Teachers evaluations and use of web-based curriculum resources to support their teaching of the Common Core State Standards for Mathematics. Middle Grades Research Journal, 10(2), 49–64.Google Scholar

Copyright information

© FIZ Karlsruhe 2017

Authors and Affiliations

  1. 1.North Carolina State University, SUDDSRaleighUSA

Personalised recommendations

Cite article

Buy options