|Website - https://www.geogebra.org|
|Subject : Mathematics, Science|
|Grade : 6-12|
|Curriculum : Can be aligned|
|Language Support : English, Hindi, Kannada and Tamil|
|Device Compatibility : Mobile, Tablet, Desktop|
|Offline Access : Yes|
|OS Compatibility : Android,iOS,Windows,Mac OS,Linux|
|Accessibility : Supports screen readers, keyboard navigation, fontsize, color & contrast controls|
|License : GeoGebra Non-Commercial License which includes GNU GPL v3 and CC-BY-NC-SA 3.0|
|Topics : Knowledge Deepening , Tool , Mathematics , Simulations , 3D Graphing|
|Curator : Saurabh Thakur, Amitabh Anand|
|Date of curation : 15 June 2020|
Do your students struggle to understand the concept of algebraic variables and how they are used in different contexts? Do they also struggle with curves, curve sketching, or coordinate geometry in general? Do they find it difficult to understand the relation between the algebraic notation of functions and their geometric equivalents?
If the answer to any of these questions is yes, here is an effective solution to these hiccups in mathematical learning!
GeoGebra is a Dynamic Geometry Environment which allows for the creation and manipulation of mathematical objects in an idealised digital environment. As the name itself suggests, GeoGebra has different views (graphics, algebra, spreadsheet etc.) such that any addition or modification made in one of the views gets automatically reflected in other views. For example, if you add a point in the graphics view, a corresponding entry is made in the algebra view with its x and y coordinates. Similarly, the properties of such objects can be altered using any of these views. This feature becomes especially useful when dealing with more complex mathematical objects like lines, circles, planes, conic sections, 3D shapes etc.
In my experience, facilitating sessions on GeoGebra with students not only builds their mathematical skills, but helps foster a host of digital literacy skills as well. For instance, importing various kinds of media and exporting final projects in various formats are some generic skills used across software products. Learners very soon pick up on the ‘alt+tab’ shortcut to move across the windows. They become familiar with how to input special mathematical symbols like caret (^), α, β, γ, fractional numbers etc.
In the initial days, students tend to miss out the function of the ‘move’ tool and struggle with accommodating the relevant portions of their drawings on the screen. With some practice, they master the grid, axes, zoom and xAxis:yAxis tools to achieve this and feel empowered. By using the polygon tool for simple activities and with some scaffolding by facilitators, students generally reach the stage where they can efficiently define polygons and other mathematical objects. The knowledge of integers (signed numbers) is implicit in the Cartesian coordinate system and learners seamlessly reach a stage where they internalise their meaning and use. With the help of coordinates of objects as simple as a point, the discussion about decimal numbers is never too far. Learners get hold of what it means for a number to be correct upto one decimal place or two decimal places.
In GeoGebra, some objects are drawn independent of other objects, for example, a standalone point or a line. Then, there are dependent objects like the intersection point which depends on already existing objects. The software maintains this difference using color codes. This distinction instills an intricate understanding of geometric construction in learners.
Angle and angle rotation are some concepts students are found struggling with, partly due to the nature of involvedness of the concepts themselves, and partly due to the non-explicitation of concepts and conventions used. GeoGebra gives opportunities to learners to understand the fact that angles measure rotation and anti-clockwise rotation is conventionally positive. Using the ‘sliders’ feature with ‘Segment with Given Length’ and ‘Angle with Given Size’ helps highlight these features and generate discussions around the same. See the dancing figure activity.
Active Knowledge Making: GeoGebra is a digital tool for creating mathematical visualisations based on the inputs and commands given by a user. To be able to use particular inputs and commands, the users continuously need to try different sets of options available in the form of math objects, their relations and combinations, see the effects of their manipulation and adjust according to the output. Educators can create dynamic worksheets for their students for demonstrations or to help them discover mathematical concepts on their own through the process of guided discovery. All actions on the virtual platform provides a safe environment to learners for hit and trial and learn from mistakes. See this example of a dynamic worksheet concerning derivative of a quadratic function.
Multimodal Meaning: GeoGebra is a visual delight for mathematics enthusiasts as it supports the visualisation of several kinds of mathematical objects, provides the facility to import images, create animations and export them as pictures and gifs. The software supports the construction of ideal 1D, 2D and 3D mathematical objects. It is also possible to add audio files using the ‘play sound’ command but it is not convinient for beginners.See this example of a simple animation project using the ‘sliders’ feature in GeoGebra. Sliders are basically variables.
Collaboration: One of the ways GeoGebra fosters collaboration is through its online portal. Apart from being able to download the software for various OS platforms, users can create an account on geogebra.org and go to the ‘resources’ section to find a plethora of existing classroom resources contributed by the GeoGebra community. Projects and activities can be saved and marked as ‘favorites’ for future use. The downside of the resources on the GeoGebra website is that they do not have a comment, feedback or discussion feature. However, people can collaborate on the GeoGebra help page. In the offline version, since GeoGebra projects (.ggb) are portable, they can be reused on different computers. Learners can also collaborate by doing projects collectively on the same computer.
Accessibility: While creating worksheets in Geogebra, users have the flexibility to control the font size, color and contrast of math objects, sliders, texts used etc. By unchecking the ‘Selection Allowed’ in Object Properties -> Advanced option, one can allow accessibility using keyboard shortcuts. All individual tools in GeoGebra are self-explanatory in the sense that they have ‘how to use’ tooltips attached to them. These tooltips are also customisable. On the web version, GeoGebra also allows various Operating System (OS) specific Screen Readers to automatically read the text in the Graphics view.
Adaptability: In the formal mathematics curriculum, introduction to geometric objects, basics of coordinate geometry are introduced only in the upper primary classes (6-8). Hence, my take is that GeoGebra is meaningful for students from class 6-12. Through open licensing (GNU GPL, CC-BY-NC-SA), GeoGebra allows all kind of mixing, adaptation and reusing of resources for non-commercial purposes. As the latest offering on this online portal, ‘GeoGebra Classroom’ provides the facility to create classrooms, add members, create assignments, monitor student progress, administer polls, all in real time. It can also be integrated with other Learning Management Systems.