In a brief pre-lab session, we talked about sine and cosine as periodic functions - we reviewed concepts like amplitude and period, but did not discuss the how these functions are related to trigonometry. The activity was presented as a printed page of instructions without much additional background information.
The goals of the lab activity were:
- have the students explore and better understand how changes in the algebraic representations of the sine and cosine functions affected their graphs;
- understand that simple periodic functions can be combined to create more complex periodic functions;
- observe that periodic functions can be combined to create circles, spirals and other interesting curves;
- experience how mathematical ideas can be explored in a playful way.
Using desmos.com following the provided instructions, the students would create graphs of sine and cosine, and adjust parameters to change the amplitude and period.
These waves were combined into a new more complicated wave:
Combining in a different way allows the sine and cosine to form a circle:
And, with some small changes, the circle can be made into a spiral that can be wound up by adjusting parameters:
Some students went on to additional instructions that allowed them to create other spiral patterns, like this:
Four classes of fifteen students at a time participated in the lab activity, it was hosted in a small computer lab with about ten workstations. Students had various levels of English language proficiency, were in various grades (grade 9 - grade 11), and had previously completed one other Desmos-based activity.
The triple e framework helps us think through what worked in this activity and what could be improved.
Engagement Questions
Does the technology tool help students focus on the learning goals (content) with less distraction(s)?
The students were mostly focused on working through the instructions, rather than on exploring and experimenting. Because the groups of students were small, I was able to help the students focus on experimenting by going around the lab and asking questions about how the changes in the parameters affected the graphs. Once further prompted, the responsiveness of the technology (moving the sliders allowed graphs to be easily altered) helped engage the students in the learning goals.
Does the technology tool help to motivate students to begin the learning processes?
Desmos has an interesting design - unlike many similar tools, construction does not proceed by navigating through layers of menus searching for the right button to click - there are almost no menu items at all. Desmos also does not use a specialized or proprietary set of commands to carry out actions - instead it uses standard mathematics conventions: to create something, you simply begin describing what you want to create using standard mathematics. This design is similar to how Geomter's Sketchpad approached geometric constructions. This can be a little intimidating at first, but once students know how to start, there is a very low barrier to beginning to create meaningful mathematics.
Does the technology cause a shift in the behavior of the students, where they move from passive to active social learners (co-use)?
In requiring students to provide the descriptions of what they want to make, and to actually add in the "sliders" and parameters required to modify their constructions, Desmos is an active learning environment. It provides the opportunity for social interaction through sharing of graphs - this was a feature that we did not explicitly use - better use of the platform would take advantage of this social dimension.
Enhancement Questions
Does the technology tool aid students in developing or demonstrating a more sophisticated understanding of the content? (creates opportunities for creation/production over consumption)
The desmos.com graphing tool definitely creates opportunities for creation/production. Once engaged, some students were experimenting and creating their own variations on the curves and graphs outlined in the handout.
In this particular activity, it can be argued whether or not the students were achieving a more sophisticated understanding: using the technology, students were able to go beyond where they could have in exploring the combinations of functions, but we intentionally skipped over much content (the detailed behavior of the functions, the trigonometric foundations, etc.). Ultimately, explorations like this one provide a far and high level view of where we can go with periodic functions, other activities can be used to help solidify the foundations for students so that they can gain confidence in their understandings.
Does the technology create scaffolds to make it easier to understand concepts or ideas?
The slider concept definitely provides a scaffold that allows students to understand how changes in parameters affect graphs of functions. Because Desmos uses standard mathematical conventions, the representations of the functions in desmos.com look exactly like the algebraic representations that students would use with pencil and paper. This isomorphism between the representation of the functions inside and outside the tool means that the learnings from the technology can be applied without the technology: the scaffolding can be removed and the structure of the learned concepts remains intact.
Does the technology create paths for students to demonstrate their understanding of the learning goals in a way that they could not do with traditional tools?
The dynamic nature of the graphing tools provides opportunities to demonstrate understanding that simply do not exist outside the tool. This is one of the great advantages of using dynamic graphing and dynamic geometry tools, like Desmos.
Extension Questions
Does the technology create opportunities for students to learn outside of their typical school day?
As a web-based and free tool, Desmos is available anytime inside and outside school. Students were encouraged to use the tool for other classes and assignments. Some created accounts on desmos.com and were experimenting after the activity and class were over.
Does the technology create a bridge between school learning and their everyday life experiences?
Desmos does have the potential to be used for more recreational and artistic uses. Within the realm of mathematics, students can use desmos.com in many of their classes (and beyond), primarily because (1) it has many powerful mathematical features, and (2) it uses standard mathematics - which is, as we like to say, a universal language.
Does the technology allow students to build skills, that they can use in their everyday lives?
One of the fears that stalk all mathematical tools is that, in using the technology, students will not develop the skills and understandings that they would develop in doing the calculations and graphing by hand.
Because it uses standard mathematical constructs and representations, students who want to go further in their Desmos explorations end up learning more mathematics and mathematical conventions in order to do more with Desmos. Graphs in desmos.com are very close visually to (high quality) hand drawn graphs, so working in Desmos can help inform students of best-practices in drawing their own graphs by hand. Part of a comprehensive approach to learning using Desmos would be to pair online graphing with pencil and paper activities. Students can go further and do more with the technology, and the use of technology can help improve their work without the technology.
Improvements
Reflecting on the triple e questions, raises the following possible areas for improving this activity, and the use of the desmos.com graphing calculator in general:
1. Be aware of how the desmos.com interface affects student engagement. New users can be intimidated by the "blank canvas" that Desmos presents.
2. Be aware that engagement with the tool may be limited by the student's understandings of standard mathematical conventions. Use this as a learning opportunity to review/learn the standard conventions that are used in Desmos.
3. Ensure that any Desmos activity allows students to go beyond what they would be able to do without the tool, enhancing their understanding of the underling and/or associated concepts.
4. Enhance student learning by encouraging students to take the time to explore changes in their constructions by adding in sliders, and by combining constructions in various ways.
3. Take advantage of the accessibility of desmos.com to encourage students to explore mathematics outside of school and extend their learning.
4. Extend mathematics beyond the isolated experience of performing a calculation or drawing a graph by taking advantage of the sharing options on desmos.com to have students share the results of their explorations.
References
Desmos Graphing Calculator. (2019). Desmos Graphing Calculator. [online] Available at: https://www.desmos.com/calculator, accessed on February 6, 2019
Kolb, E. (2013) Engage, enhance and extend learning: Find out what these terms really mean when you integrate technologies into your lessons. Learning & Leading with Technology, 40(7), 21-27. Retrieved from http://www.learningandleading-digital.com/learning_leading/201305?pg=22#pg22
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