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ZDM

, Volume 49, Issue 5, pp 701–716 | Cite as

Design of tasks for online assessment that supports understanding of students’ conceptions

  • Michal Yerushalmy
  • Galit Nagari-Haddif
  • Shai Olsher
Original Article
First Online:

Abstract

In the present study, we ask whether and how online assessment can inform teaching about students’ understanding of advanced concepts. Our main goal is to illustrate how we study design of tasks that support reliable online formative assessment by automatically analyzing the objects and relations that characterize the students’ submissions. We aim to develop design principles for e-tasks that have the potential to support the creation of rich and varied response spaces that reflect convincingly the students’ perceptions. We focus on studying the design principles of such an interactive rich task, and the representations and tools that support automatic real-time analysis of submitted answers, primarily in the form of free-hand sketches. Using a design research methodology, we describe a two-cycle study focusing on one e-task concerning tangency to the graph of a function. We characterize the mathematical attributes of examples constructed by students in an interactive environment. Checking the correctness of answers is only one of the functions of the STEP platform that we use. That platform can identify submissions that contain partial information (e.g., missing or misplaced tangency points), can categorize these answers, and allow the teacher to analyze them further. The reported analysis demonstrates how the design of the tasks and the characterization of mathematical attributes can help to expand our understanding of different ways of conceptualizing.

Keywords

Calculus Tangent Example Assessment Formative Technology 
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Notes

Acknowledgements

This research was supported by the Israel Science Foundation (Grant 522/13) and the Trump Foundation (Grant 191).

References

  1. Ainsworth, S. (1999). The functions of multiple representations. Computers & Education, 33(2), 131–152.CrossRefGoogle Scholar
  2. Berry, J. S., & Nyman, M. A. (2003). Promoting students’ graphical understanding of the calculus. The Journal of Mathematical Behavior, 22(4), 479–495.CrossRefGoogle Scholar
  3. Bills, L., Dreyfus, T., Mason, J., Tsamir, P., Watson, A., & Zaslavsky, O. (2006). Exemplification in mathematics education. Proceedings of the 30th conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 126–154). Prague: Czech Republic.Google Scholar
  4. Biza, I., Christou, C., & Zachariades, T. (2008). Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis. Research in Mathematics Education, 10(1), 53–70.CrossRefGoogle Scholar
  5. Black, P., & Wiliam, D. (1998). Assessment and classroom learning. Assessment in Education, 5, 7–74.CrossRefGoogle Scholar
  6. Black, P., & Wiliam, D. (2014). Assessment and the design of educational materials. Educational Designer, 2(7). http://www.educationaldesigner.org/ed/volume2/issue7/article23/.
  7. Botzer, G., & Yerushalmy, M. (2008). Embodied semiotic activities and their role in the construction of mathematical meaning of motion graphs. International Journal of Computers for Mathematical Learning, 13(2), 111–134.CrossRefGoogle Scholar
  8. Buchbinder, O., & Zaslavsky, O. (2013). A holistic approach for designing tasks that capture and enhance mathematical understanding of a particular topic: The case of the interplay between examples and proof task. In C. Margolinas (Ed.), Task design in mathematics education: Proceedings of ICMI Study 22 (pp. 25–34). UK: Oxford.Google Scholar
  9. Burkhardt, H., & Swan, M. (2013). Task design for systemic improvement: Principles and frameworks. Task design in mathematics education, Proceedings of ICMI Study 22 (pp. 431–439). UK: Oxford.Google Scholar
  10. Common Core State Standards in the USA (2010). http://www.corestandards.org/Math. Accessed 11 June 2017.
  11. Clark-Wilson, A. (2010). Emergent pedagogies and the changing role of the teacher in the TI-Nspire Navigator-networked mathematics classroom. ZDM—The International Journal of Mathematics Education, 42(7), 747–761.CrossRefGoogle Scholar
  12. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131. doi: 10.1007/s10649-006-0400-z.CrossRefGoogle Scholar
  13. Even, R. (1990). Subject matter knowledge for teaching and the case of functions. Educational Studies in Mathematics, 21(6), 521–544.CrossRefGoogle Scholar
  14. Goldenberg, P., & Mason, J. (2008). Shedding light on and with example spaces. Educational Studies in Mathematics, 69(2), 183–194.CrossRefGoogle Scholar
  15. Janvier, C. (1987). Problems of representation in the teaching and learning of mathematics. Hillsdale, NJ: Erlbaum.Google Scholar
  16. Kaput, J. J. (2002). Notations and representations as mediators of constructive processes. In E. Von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. 53–74). Dordrecht: Springer.Google Scholar
  17. Koedinger, K. R., McLaughlin, E. A., & Heffernan, N. T. (2010). A quasi-experimental evaluation of an on-line formative assessment and tutoring system. Journal of Educational Computing Research, 43(4), 489–510. doi: 10.2190/EC.43.4.d.CrossRefGoogle Scholar
  18. Lockwood, E., Ellis, A. B., & Lynch, A. G. (2016). Mathematicians’ example-related activity when exploring and proving conjectures. International Journal of Research in Undergraduate Mathematics Education, 2(2), 165–196.CrossRefGoogle Scholar
  19. Naftaliev, E., & Yerushalmy, M. (2013). Guiding explorations: Design principles and functions of interactive diagrams. Journal of Computers in the Schools, 30(1–2), 61–75.CrossRefGoogle Scholar
  20. Nagari-Haddif, G., & Yerushalmy, M. (2015). Digital interactive assessment in mathematics: The case of construction e-tasks. In CERME 9-Ninth Congress of the European Society for Research in Mathematics Education (pp. 2501–2508).Google Scholar
  21. Nguyen, D. M., Hsieh, Y. C. J., & Allen, G. F. (2006). The impact of web-based assessment and practice on students’ mathematics learning attitudes. The Journal of Computers in Mathematics and Science Teaching, 25(3), 251–279.Google Scholar
  23. Olsher, S., Yerushalmy, M., & Chazan, D. (2016). How might the use of technology in formative assessment support changes in mathematics teaching? For the Learning of Mathematics, 36(3), 11–18.Google Scholar
  24. Paiva, R. C., Ferreira, M. S., Mendes, A. G., & Eusebio, A. M. J. (2015). Interactive and multimedia contents associated with a system for computer aided assessment. The Journal of Educational Computing Research, 52(2), 224–256.CrossRefGoogle Scholar
  25. Sangwin, C. J., & Köcher, N. (2016). Automation of mathematics examinations. Computers & Education, 94(C), 215–227.CrossRefGoogle Scholar
  26. Stacey, K., & Wiliam, D. (2012). Technology and assessment in mathematics. In M. A. K. Clements, A. Bishop, C. Keitel-Kreidt, J. Kilpatrick & F. Koon-Shing Leung (Eds.), Third international handbook of mathematics education (pp. 721–751). New York, NY: Springer Science & Business Media.CrossRefGoogle Scholar
  27. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.CrossRefGoogle Scholar
  28. Ubuz, B. (2007). Interpreting a graph and constructing its derivative graph: stability and change in students’ conceptions. International Journal of Mathematical Education in Science and Technology, 38(5), 609–637.CrossRefGoogle Scholar
  29. Van Dormolen, J., & Zaslavsky, O. (2003). The many facets of a definition: The case of periodicity. The Journal of Mathematical Behavior, 22(1), 91–106.CrossRefGoogle Scholar
  30. Watson, A., & Mason, J. (2005). Mathematics as a constructive activity. Learners generating examples. Mahwah: Lawrence Erlbaum Associates.Google Scholar
  31. Winicki-Landman, G., & Leikin, R. (2000). On equivalent and non-equivalent definitions: Part 1. For the Learning of Mathematics, 20(1), 17–21.Google Scholar
  32. Yerushalmy, M. (2005). Challenging known transitions: Learning and teaching algebra with technology. For the Learning of Mathematics, 25(3), 37–42.Google Scholar
  33. Yerushalmy, M. (2006). Slower algebra students meet faster tools: Solving algebra word problems with graphing software. Journal for Research in Mathematics Education, 37(5), 356–387.Google Scholar
  34. Yerushalmy, M., & Gafni, R. (1992). Syntactic manipulations and semantic interpretations in algebra: The effect of graphic representation. Learning and Instruction, 2(4), 303–319.CrossRefGoogle Scholar
  35. Zaslavsky, O., & Zodik, I. (2014). Example-generation as indicator and catalyst of mathematical and pedagogical understandings. In Y. Li, A. E. Silver & S. Li (Eds.), Transforming mathematics instruction: Multiple approaches and practices (pp. 525–546). Cham: Springer.Google Scholar

Copyright information

© FIZ Karlsruhe 2017

Authors and Affiliations

  • Michal Yerushalmy
    • 1
  • Galit Nagari-Haddif
    • 1
  • Shai Olsher
    • 1
  1. 1.Faculty of EducationUniversity of HaifaHaifaIsrael

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