Summary
A range of digital technology to support mathematics learning is freely available on the internet, particularly apps offering immediate access to different representations and calculations. In this article, we analyze some learning apps (available for smartphones) with an integrated computer algebra system (CAS) that offer support when learning how to solve equations. In the context of solving quadratic equations, use of apps in an informal way to learn how to solve not only touches on learning issues in the field of algebra, but also aspects of students’ self-regulation and the use of technology. These different aspects are discussed in the theoretical background and are used to guide our methodological approach to analyze different CAS-integrated learning apps.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aldon, G., Cusi, A., Morselli, F., Panero, M., & Sabena, C. (2017). Formative assessment and technology: Reflections developed through the collaboration between teachers and researchers. In G. Aldon, F. Hitt, L. Bazzini, & U. Gellert (Eds.), Mathematics and technology: A C.I.E.A.E.M. sourcebook (pp. 551–578). Cham: Springer.
Arcavi, A. (2005). Developing and using symbol sense in mathematics. For the Learning of Mathematics,25(2), 42–47.
Bandura, A. (1986). Social foundations of thought and action: A social cognitive theory. Englewood Cliffs: Prentice-Hall.
Barzel, B. (2007). New technology? New ways of teaching—No time left for that! International Journal for Technology in Mathematics Education,14(2), 77–86.
Barzel, B., & Holzäpfel, L. (2017). Gleichungen verstehen. mathematik lehren,169, 2–7.
Beal, C., Arroyo, I., Cohen, P., & Woolf, B. (2010). Evaluation of animal watch: An intelligent tutoring system for arithmetic and fractions. Journal of Interactive Online Learning,9(1), 64–77.
Bell, B., & Cowie, B. (2001). The characteristics of formative assessment in science education. Science Education,85(5), 536–553.
Block, J. (2015). Flexible algebraic action on quadratic equations. In K. Krainer & N. Vondrová (Eds.), Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education (pp. 391–397). Prague: ERME.
Brown, A. L., Ellery, S., & Campione, J. (1998). Creating zones of proximal development electronically. In J. Greeno & S. Goldman (Eds.), Thinking practices: A symposium in mathematics and science education (pp. 341–368). Hillsdale: Lawrence Erlbaum.
Buchberger, B. (1990). Should students learn integration rules? ACM SIGSAM Bulletin,24(1), 10–17.
de Lima, R. N., & Tall, D. (2006). The concept of equations: What have students met before? In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.). Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 233–240). Prague: PME.
Dehaene, S. (1997). The number sense: How the mind creates mathematics. Oxford: Oxford University Press.
Ehret, C. (2017). Mathematisches Schreiben: Modellierung einer fachbezogenen Prozesskompetenz. Wiesbaden: Springer Spektrum.
Feierabend, S., Plankenhorn, T., & Rathgeb, T. (2017). JIM-Studie 2017: Jugend, Information, (Multi-) Media – Basisuntersuchung zum Medienumgang 12- bis 19-Jähriger. Stuttgart: Medienpädagogischer Forschungsverbund Südwest.
Graesser, A. C., McNamara, D. S., & VanLehn, K. (2005). Scaffolding deep comprehension strategies through Point & Query, AutoTutor and iSTART. Educational Psychologist,40(4), 225–234.
Heid, M. K. (1988). Resequencing skills and concepts in applied calculus using the computer as a tool. Journal for Research in Mathematics Education,19, 3–25.
Heugl, H., Klinger, W., & Lechner, J. (1996). Mathematikunterricht mit Computeralgebra-Systemen: Ein didaktisches Leherbuch mit Erfahrungen aus dem österreichischen DERIVE-Projekt. Bonn: Addison-Wesley.
Hillmayr, D., Reinhold, F., Ziernwald, L., & Reiss, K. (2017). Digitale Medien im mathematisch-naturwissenschaftlichen Unterricht der Sekundarstufe: Einsatzmöglichkeiten, Umsetzung und Wirksamkeit. Münster: Waxmann.
Klinger, M. (2019). “Besser als der Lehrer!” – Potenziale CAS-basierter Smartphone-Apps aus didaktischer und Lernenden-Perspektive. In G. Pinkernell, & F. Schacht (Eds.), Digitalisierung fachbezogen gestalten: Herbsttagung vom 28. bis 29. September 2018 an der Universität Duisburg-Essen (pp. 69–85). Hildesheim: Franzbecker.
Klinger, M., & Schüler-Meyer, A. (2019). Wenn die App rechnet: Smartphone-basierte Computer-Algebra-Apps brauchen eine geeignete Aufgabenkultur. mathematik lehren, 215.
Klinger, M., Thurm, D., Itsios, C., & Peters-Dasdemir, J. (2018). Technology-related beliefs and the mathematics classroom: Development of a measurement instrument for pre-service and in-service teachers. In B. Rott, G. Törner, J. Peters-Dasdemir, A. Möller, & Safrudiannur (Eds.), Views and beliefs in mathematics education: The role of beliefs in the classroom (pp. 233–244). Cham: Springer.
Küchemann, D. (1981). Algebra. In K. Hart (Ed.), Children’s understanding of mathematics: 11-16 (pp. 102–119). London: Murray.
Leuders, T., & Prediger, S. (2016). Flexibel differenzieren und fokussiert fördern im Mathematikunterricht. Berlin: Cornelsen.
Ma, W., Adesope, O., Nesbit, J., & Liu, Q. (2014). Intelligent tutoring systems and learning outcomes: A meta-analysis. Journal of Educational Psychology,106(4), 901–918.
Nattland, A., & Kerres, M. (2009). Computerbasierte Methoden im Unterricht. In K.-H. Arnold, U. Sandfuchs, & J. Wiechmann (Eds.), Handbuch Unterricht (pp. 317–324). Bad Heilbrunn: Klinkhardt.
Nydegger, A. (2018). Algebraisieren von Sachsituationen: Wechselwirkungen zwischen relationaler und operationaler Denk- und Sichtweise. Wiesbaden: Springer Spektrum.
Pane, J. F., Griffin, B. A., McCaffrey, D. F., & Karam, R. (2014). Effectiveness of cognitive tutor algebra I at scale. Educational Evaluation and Policy Analysis,36(2), 127–144.
Pierce, R., & Stacey, K. (2002). Algebraic insight: The algebra needed to use computer algebra systems. Mathematics Teacher,95(8), 622–627.
Pierce, R., & Stacey, K. (2004). Monitoring progress in algebra in a CAS active context: Symbol sense, algebraic insight and algebraic expectation. International Journal for Technology in Mathematics Education,11(1), 3–12.
Ritter, S., Anderson, J. R., Koedinger, K., & Corbett, A. (2007). Cognitive Tutor: Applied research in mathematics education. Psychonomic Bulletin & Review,14(2), 249–255.
Rittle-Johnson, B., Schneider, M., & Star, J. R. (2015). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review,27(4), 587–597.
Schoenfeld, A. (2014). What makes for powerful classrooms, and how can we support teachers in creating them? A story of research and practice, productively intertwined. Educational Researcher,43(8), 404–412.
Stacey, K. (2011). Eine Reise über die Jahrgänge: Vom Rechenausdruck zum Lösen von Gleichungen. mathematik lehren, 169, 6–12.
Stacey, K., Steinle, V., Gvozdenko, E., & Price, B. (2013). SMART online formative assessments for teaching mathematics. Curriculum & Leadership Journal, 11(20).
Thurm, D., Klinger, M., Barzel, B., & Rögler, P. (2017). Überzeugungen zum Technologieeinsatz im Mathematikunterricht: Entwicklung eines Messinstruments für Lehramtsstudierende und Lehrkräfte. mathematica didactica, 40(1), 19–35.
VanLehn, K., Lynch, C., Schulze, K., Shapiro, J. A., Shelby, R., Taylor, L., et al. (2005). The Andes physics tutoring system: Five years of evaluations. In G. McCalla, C. K. Looi, B. Bredeweg, & J. Breuker (Eds.), Artificial intelligence in education (pp. 678–685). Amsterdam: IOS Press.
Vygotsky, L. S. (1978). Mind in society: The development of higher psychology processes. Cambridge: Harvard University Press.
Walsh, M., Moss, C. M., Johnson, B. G., Holder, D. A., & Madura, J. D. (2002). Quantitative impact of a cognitive modeling intelligent tutoring system on student performance in balancing chemical equations. Chemical Educator,7, 379–383.
Webel, C., & Otten, S. (2016). Teaching in a world with Photomath. Mathematics Teacher,109(5), 368–373.
Werquin, P. (2007). Moving mountains: Will qualifications systems promote lifelong learning? European Journal of Education,42(4), 459–484.
Wiliam, D., & Thompson, M. (2008). Integrating assessment with learning: What will it take to make it work? In C. A. Dwyer (Ed.), The future of assessment: Shaping teaching and learning (pp. 53–82). Mahwah: Lawrence Erlbaum.
Winter, H. (1984). Begriff und Bedeutung des Übens im Mathematikunterricht. mathematik lehren, 2, 4–16.
Woolf, B. P. (2009). Building intelligent interactive tutors: Student-centered strategies for revolutionizing e-learning. Burlington: Morgan Kaufman.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Barzel, B., Ball, L., Klinger, M. (2019). Students’ Self-Awareness of Their Mathematical Thinking: Can Self-Assessment Be Supported Through CAS-Integrated Learning Apps on Smartphones?. In: Aldon, G., Trgalová, J. (eds) Technology in Mathematics Teaching. Mathematics Education in the Digital Era, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-19741-4_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-19741-4_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-19740-7
Online ISBN: 978-3-030-19741-4
eBook Packages: EducationEducation (R0)