Abstract
Co-variational reasoning has received particular attention from researchers and mathematics educators because it is considered of paramount importance for the understanding of concepts such as variable, function, rate of change, derivative, etc. Some of the critical issues that have been identified in several studies consist of the difficulty in interpreting the simultaneous variation of two quantities, particularly in overcoming coordination problems of two variables changing in tandem. A relevant question in the study of co-variational reasoning concerns representing the joint variation of quantities and performing translations between different representations. Problems of motion involving variation over time are strongly linked to the concept of co-variation and require the ability to translate a dynamic situation by means of mostly static representations. Those problems require the construction of a conceptual model that, in some way, visually contains dynamism. In taking solving and expressing as a unit of analysis and focusing on the ways in which commonly available digital technologies are used by youngsters as tools in problem-solving, we analyse the approaches used by the participants in SUB14 to a motion problem. Some surprising results of the content analysis of over 200 answers indicate that the textual/descriptive form of presenting a model of the situation had a clear dominance. The use of tabular representations along with pictorial/figurative content was also present in a high percentage of solutions. Furthermore the use of digital media was decisive in producing visuality, i.e. ways of depicting the displacement with time (quasi-dynamic representations).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Borba, M. C., & Villarreal, M. E. (2005). Humans-with-media and the reorganization of mathematical thinking: Information and communication technologies, modeling, visualization and experimentation. New York, NY: Springer.
Carlson, M. (2002). Physical enactment: A powerful representational tool for understanding the nature of covarying relationships? In F. Hitt (Ed.), Representations and mathematics visualization (pp. 63–77). Special Issue of PME-NA & Cinvestav-IPN.
Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events. Journal for Research in Mathematics Education, 33(5), 352–378.
Carlson, M., Larsen, S., & Lesh, R. (2003). Integrating a models and modeling perspective with existing research and practice. In R. Lesh & H. Doerr (Eds.), Beyond constructivism – models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 465–478). Mahwah, NJ: Erlbaum Associates.
Carraher, D. W., Martinez, M. V., & Schliemann, A. D. (2007). Early algebra and mathematical generalization. ZDM: The International Journal on Mathematics Education, 40(1), 3–22. doi:10.1007/s11858-007-0067-7.
Doorman, L. M., & Gravemeijer, K. P. E. (2008). Emergent modeling: Discrete graphs to support the understanding of change and velocity. ZDM, 41(1–2), 199–211. doi:10.1007/s11858-008-0130-z.
Greeno, J. G., & Hall, R. P. (1997). Practicing representation: Learning with and about representational forms. Phi Delta Kappan, 78(5), 361–367.
Izsák, A., & Findell, B. R. (2005). Adaptive interpretation: Building continuity between students’ experiences solving problems in arithmetic and in algebra. ZDM: The International Journal on Mathematics Education, 37(1), 60–67. doi:10.1007/BF02655898.
Matos, J., & Carreira, S. (1997). The quest for meaning in students’ mathematical modeling. In K. Houston, W. Blum, I. Huntley, & N. Neill (Eds.), Teaching and learning mathematical modeling – ICTMA 7 (pp. 63–75). Chichester, UK: Horwood Publishing.
Monaghan, J. M., & Clement, J. (2000). Algorithms, visualization, and mental models: High school students’ interactions with a relative motion simulation. Journal of Science Education and Technology, 9(4), 311–325.
Passaro, V. (2009). Obstacles à l’acquisition du concept de covariation et l’introduction de la représentation graphique en deuxième secondaire. Annales de Didactique et de Sciences Cognitives, 14, 61–77.
Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 205–235). Rotterdam, The Netherlands: Sense Publishers.
Santos-Trigo, M. (1996). An exploration of strategies used by students to solve problems with multiple ways of solution. Journal of Mathematical Behavior, 15(3), 263–284. doi:10.1016/S0732-3123(96)90006-1.
Santos-Trigo, M. (2004). The role of technology in students’ conceptual constructions in a sample case of problem solving. Focus on Learning Problems in Mathematics, 26(2), 1–17.
Sherin, B. L. (2000). How students invent representations of motion: A genetic account. Journal of Mathematical Behavior, 19, 399–441.
Zeytun, A., Çetinkaya, B., & Erbaş, A. (2010). Mathematics teachers’ covariational reasoning levels and predictions about students’ covariational reasoning. Educational Sciences: Theory and Practice, 10(3), 1601–1612.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Carreira, S., Jones, K., Amado, N., Jacinto, H., Nobre, S. (2016). Digitally Expressing Co-variation in a Motion Problem. In: Youngsters Solving Mathematical Problems with Technology. Mathematics Education in the Digital Era, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-24910-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-24910-0_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-24908-7
Online ISBN: 978-3-319-24910-0
eBook Packages: EducationEducation (R0)