In the year 10 subject Mathematics B, students are required to demonstrate an understanding of a variety of functions (linear, quadratic and circular) including the effect of varying specific coefficients when written in standard form.
The time given to this topic is substantial (5 weeks with 5 periods per week) considering it forms the foundational understanding required for all VCE mathematics subjects, Further Mathematics, Mathematical Methods and Specialist Mathematics
The past/current delivery primarily consists of direct instruction with few inquiry tasks. It closely follows that described by Zakaria, Ibrahim, & Maat (2014) and Bossé & Nandakumar (Bossé & Nandakumar, 2005) where we focus on factorisations skills avoiding fractional and radical work. It has been common to observe students had limited understanding of what a ‘solution’ actually was and general approached the topic as rote learned knowledge and process rather than deep understanding of the concepts
O’Conner & Norton’s (2016) showed, especially for quadratics, that developing relational understanding and functional reasoning was critical to deeper understanding. They propose the use of a multiple representational digital approach warranted considering the affordances supplied by GDS.
Desmos – providing effective multiple representations and inquiry-based learning tasks
Not all GDS tools are equal, Montijo (2018) showed that Desmos was statistically beneficial to students mathematical understanding, and importantly, problem solving skills and confidence.
The inherent complexity of devices hinders students ability to access it successfully (Hillman, 2014). Desmos is regarded as one of the most intuitive and user friendly packages available to generate multiple representations (Banting, McCormick, Twitchell, & Harvey, 2017)
Weekly Implementations Schedule
- Tool familiarisation: Build students understanding of the GUI, interface and output by generating simple graphs and experimentation. Repeated modelling of the multiple representations available and how they link.
- Introduction to learning activities: Linear relationship activities, focus on collaboration (within and outside of Desmos) and continued experimentation (there are no wrong answers, just different options).
- Continued learning activities: Advanced linear activities, introduction to quadratic functions and parabolas using inquiry-based task.
- Continued learning activities and introduction to creation: Advanced Quadratic and circular function activities. Student driven image design draft which offers an atypical product to demonstrate learning
- Desmos art final image: combines symbolic manipulation and computation with the skills of creation, planning, experimentation and others. Most students should be using Desmos and functions in alignment with higher order thinking skills by this stage of the program
Concerns and Reflections
Gray & Thomas (2001) found lack of time and consistent use of GDS tool led to students still approaching the topic procedurally rather than conceptually.
Pierce, Ball & Stacey (2009) found that while GDS and CAS provide affordances around algebraic manipulation to focus on graphical representations this may be at the expense of future assessment outcomes (eg non calculator active examinations in VCE)
Disconnect between tools used during learning and assessment may impact end of year exam results if students are not properly transitioned (Stacey & Wiliam, 2013).
Cognitive load associated with transitioning from Desmos to handheld CAS calculator notation and a barrier to students demonstrating product during assessment task (Pierce, Stacey, Wander, & Ball, 2011)
Montijo (2018) found students easily adapted to the clean interface Desmos provides
Desmos provides short and medium cycle assessment and feedback immediately which is pivotal to student progression (Hattie, 2012)
Desmos is available at any time allowing students practice time anytime at school and home helping students access the learning in a comfortable environment.
Desmos does not have fixed timelines for completing tasks, all tasks are repeatable and can be done at students desired pace, furthermore, students can still access tasks if they have been absent from class.
Banting, N., McCormick, K. K., Twitchell, G., & Harvey, S. (2017). Desmos Art. The Variable, 2(4), 25–28.
Bossé, M. J., & Nandakumar, N. R. (2005). The factorability of quadratics: Motivation for more techniques. Teaching Mathematics and Its Applications. https://doi.org/10.1093/teamat/hrh018
Gray, R., & Thomas, M. O. J. (2001). Quadratic equation representations and graphic calculators: Procedural and conceptual interactions. In Proceedings of the 24th Mathematics Education Research Group of Australasia Conference (pp. 257–264).
Hattie, J. (2012). Visible learning for teachers: Maximizing impact on learning. Routledge.
Hillman, T. (2014). Tracing the construction of mathematical activity with an advanced graphing calculator to understand the roles of technology developers, teachers and students. International Journal for Technology in Mathematics Education, 21(2), 37–47.
Montijo, E. (2018). The effects of Desmos and TI-83 Plus graphing calculators on the problem-solving confidence of middle and high school mathematics students. Dissertation Abstracts International Section A: Humanities and Social Sciences. ProQuest Information & Learning. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&db=psyh&AN=2017-29361-010&site=ehost-live
O’Connor, B. R., & Norton, S. (2016). Investigating Students’ Mathematical Difficulties with Quadratic Equations. Mathematics Education Research Group of Australasia.
Pierce, R., Ball, L., & Stacey, K. (2009). Is it worth using CAS for symbolic algebra manipulation in the middle secondary years? some teachers’ views. International Journal of Science and Mathematics Education. https://doi.org/10.1007/s10763-009-9160-4
Pierce, R., Stacey, K., Wander, R., & Ball, L. (2011). The design of lessons using mathematics analysis software to support multiple representations in secondary school mathematics. Technology, Pedagogy and Education, 20(1), 95–112.
Stacey, K., & Wiliam, D. (2013). Technology and Assessment in Mathematics. In M. A. (Ken) Clements, A. J. Bishop, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Third International Handbook of Mathematics Education (pp. 721–751). New York, NY: Springer New York. https://doi.org/10.1007/978-1-4614-4684-2_23
Zakaria, E., –, I., & Maat, S. M. (2014). Analysis of Students’ Error in Learning of Quadratic Equations. International Education Studies. https://doi.org/10.5539/ies.v3n3p105