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

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## 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## Notes

### Acknowledgements

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

## References

- Ainsworth, S. (1999). The functions of multiple representations.
*Computers & Education, 33*(2), 131–152.CrossRefGoogle Scholar - 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 - 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 - 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 - Black, P., & Wiliam, D. (1998). Assessment and classroom learning.
*Assessment in Education, 5*, 7–74.CrossRefGoogle Scholar - Black, P., & Wiliam, D. (2014). Assessment and the design of educational materials.
*Educational Designer, 2*(7). http://www.educationaldesigner.org/ed/volume2/issue7/article23/. - 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 - 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 - 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 - Common Core State Standards in the USA (2010). http://www.corestandards.org/Math. Accessed 11 June 2017.
- 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 - 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 - Even, R. (1990). Subject matter knowledge for teaching and the case of functions.
*Educational Studies in Mathematics, 21*(6), 521–544.CrossRefGoogle Scholar - Goldenberg, P., & Mason, J. (2008). Shedding light on and with example spaces.
*Educational Studies in Mathematics, 69*(2), 183–194.CrossRefGoogle Scholar - Janvier, C. (1987).
*Problems of representation in the teaching and learning of mathematics*. Hillsdale, NJ: Erlbaum.Google Scholar - 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 - 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 - 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 - 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 - 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 - 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 - OECD (2012). https://www.oecd.org/pisa/pisaproducts/pisa2012-2006-rel-items-maths-ENG.pdf. Accessed 11 June 2017.
- 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 - 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 - Sangwin, C. J., & Köcher, N. (2016). Automation of mathematics examinations.
*Computers & Education, 94*(C), 215–227.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - Watson, A., & Mason, J. (2005).
*Mathematics as a constructive activity. Learners generating examples*. Mahwah: Lawrence Erlbaum Associates.Google Scholar - 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 - Yerushalmy, M. (2005). Challenging known transitions: Learning and teaching algebra with technology.
*For the Learning of Mathematics, 25*(3), 37–42.Google Scholar - 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 - 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 - 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