GeoGebra - An ideal tool for middle school and junior high mathematics!
There is a strong misconception in the math education world that GeoGebra is a tool (almost entirely) for calculus-based mathematics and senior high school-level geometry.
This article aims to encourage math teachers who are not yet utilizing GeoGebra as a regular demonstration tool across all classes and most topics, to start doing so. Despite its popularity, GeoGebra remains an under-utilized tool by math teachers generally. This is due to the lack of awareness of the power of GeoGebra available to teachers once they become proficient as well as some common misconceptions, namely:
This article aims to encourage math teachers who are not yet utilizing GeoGebra as a regular demonstration tool across all classes and most topics, to start doing so. Despite its popularity, GeoGebra remains an under-utilized tool by math teachers generally. This is due to the lack of awareness of the power of GeoGebra available to teachers once they become proficient as well as some common misconceptions, namely:
- That GeoGebra is difficult to use.
- That utilizing GeoGebra will require a significant change in teaching approach.
- That GeoGebra is inappropriate for 'lower' levels of mathematics.
GeoGebra - powerful even as a demonstration-only tool
The reason for many teachers not utilizing GeoGebra is simple – teachers simply do not yet realise how powerful GeoGebra is, even when used as a demonstration tool.
In the article Let GeoGebra transform your math teaching I show how using GeoGebra, as a demonstration tool, can revolutionize the impact a teacher has on their students’ engagement and understanding. Importantly, this can be achieved simply by employing the 'Plug-in-the-data-projector-and-show-the-file' method.
In the article Let GeoGebra transform your math teaching I show how using GeoGebra, as a demonstration tool, can revolutionize the impact a teacher has on their students’ engagement and understanding. Importantly, this can be achieved simply by employing the 'Plug-in-the-data-projector-and-show-the-file' method.
The 'Plug-in-and-demonstrate-a-GeoGebra-file' method is the one opportunity a math teacher has to make an immediate and significant impact on the engagement of and understanding by his/her students without having to make wholesale pedagogical changes.
In this article, however, we will look beyond the Plug-in-the-data-projector method into a more powerful use of GeoGebra, that of student-led GeoGebra investigations. Unlike simple, teacher-directed demonstrations, student-led GeoGebra investigations do require a different pedagogy to the traditional procedural approach. In addition, GeoGebra investigations require considerable planning for the teacher unfamiliar with student-centered activities..
Two types of student-led investigations to which to the 5 tips apply:
Beginner Investigations
Beginner investigations: for students who are new to investigations and/or less likely to stay on-task.
(More) Advanced Investigations
Advanced investigations: for students who are self-starters and enjoy investigating mathematical systems in a self-directed manner.
Beginner investigations: for students who are new to investigations and/or less likely to stay on-task.
(More) Advanced Investigations
Advanced investigations: for students who are self-starters and enjoy investigating mathematical systems in a self-directed manner.
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The Five tips for student-led GeoGebra tasks
- Engage.
- Teach basic skills using self-directed notes or videos.
- Create open-ended investigations.
- Scaffold the investigations.
- Encourage collaboration.
NOTE: When introducing student-led activities, the potential positives are high as are the chances of glorious, dismal failures! Therefore, you will need to consider several factors, the first of which is the engagement level of your students.
1. Student Engagement
Generally speaking, how engaged are your students? Are they familiar with student-centered activities? How do they respond to student-centered activities? Do some students tend to veer off task at the first opportunity? Is the class difficult to manage? If yes, are you hoping that a student-centered GeoGebra investigation might help win them to your cause?
Where disengagement is the norm, it is important to understand that introducing technology per se won’t fix the issue. Disengagement can be turned into engagement using a range of strategies. However, the information required to turn disengagement into engagement is way too extensive for a short article. I will say this though – teachers with classes where students are not particularly engaged but where the situation is manageable are encouraged to trial a GeoGebra investigation as long as steps are followed to maximise the chance of the activity succeeding, as suggested below.
Obviously, those who teach functional, generally motivated students who are prepared to explore mathematical systems are clearly ready to proceed with student-centered investigations.
Where disengagement is the norm, it is important to understand that introducing technology per se won’t fix the issue. Disengagement can be turned into engagement using a range of strategies. However, the information required to turn disengagement into engagement is way too extensive for a short article. I will say this though – teachers with classes where students are not particularly engaged but where the situation is manageable are encouraged to trial a GeoGebra investigation as long as steps are followed to maximise the chance of the activity succeeding, as suggested below.
Obviously, those who teach functional, generally motivated students who are prepared to explore mathematical systems are clearly ready to proceed with student-centered investigations.
2. Teach basic skills first using self-directed notes or videos.
First thing's first - you will need to teach students some GeoGebra skills before expecting any full-scale investigation. Allow the initial investigation to be ‘playtime’. Students will naturally want to play anyway so avoid causing an unnecessary 'war'. Rather, allow their play to meet your need. Teach the basics and ask them to play (explore, investigate) but don’t cloak this in mathematical investigation language. As an example, after teaching students how to create lines, segments, perpendiculars, circles and arcs you might ask them to create an ‘award-winning’ design. Or set them a challenge, for example, “Using these 5 objects in a total of 15-20 ways see how many closed spaces you can make.”
In regard to the skill instructions, deliver them via student-centered printed handouts or videos. Stress to students that they must READ (or WATCH) the instructions and follow them before asking anyone for help. Avoid allowing students to ask you or anyone else any questions relating to the instructions unless they have read/watched the instructions twice. Then apply the 'Three before me' strategy - "You must ask three students for help on an issue before asking me:" When they do ask you, have them read the instructions back to you. Lead them to find the answer. Be aware that many students will want you to spoon-feed them because they are familiar with spoon-feeding. Your task is to wean them off spoon feeding. It won’t happen overnight!
In regard to the skill instructions, deliver them via student-centered printed handouts or videos. Stress to students that they must READ (or WATCH) the instructions and follow them before asking anyone for help. Avoid allowing students to ask you or anyone else any questions relating to the instructions unless they have read/watched the instructions twice. Then apply the 'Three before me' strategy - "You must ask three students for help on an issue before asking me:" When they do ask you, have them read the instructions back to you. Lead them to find the answer. Be aware that many students will want you to spoon-feed them because they are familiar with spoon-feeding. Your task is to wean them off spoon feeding. It won’t happen overnight!
3. Make the investigations open-ended
Beware of the temptation to make the investigations highly prescriptive by overtly steering students into discovering what it is that you want them to discover. Prescriptive and complex instructions detract from the investigative experience. However, prescriptive investigations have their place – they are appropriate as beginning investigations.
4. Scaffold the investigation
The number one reason for student investigations failing, especially with students who are unfamiliar with such tasks, is a lack of scaffolding. The form the scaffolding takes is up to you.
One option, ideal for beginner investigations, is to build ‘construct files’. A construct file is a GeoGebra file with the instructions written inside the file (see the two gif examples on this page). Assuming students know how to execute the specific skills, construct files are excellent because they guide students into creating a file without requiring them to refer to an external instruction source. I realize we just established that investigations need to be open-ended, however, as beginner investigations, highly scaffolded construct files - as per the files featured here - can be ideal. Once you move beyond the beginner stage and onto more advanced investigations you will want to make the investigations as open-ended as possible. However, some use of 'advanced construct files' may be appropriate for this purpose.
One option, ideal for beginner investigations, is to build ‘construct files’. A construct file is a GeoGebra file with the instructions written inside the file (see the two gif examples on this page). Assuming students know how to execute the specific skills, construct files are excellent because they guide students into creating a file without requiring them to refer to an external instruction source. I realize we just established that investigations need to be open-ended, however, as beginner investigations, highly scaffolded construct files - as per the files featured here - can be ideal. Once you move beyond the beginner stage and onto more advanced investigations you will want to make the investigations as open-ended as possible. However, some use of 'advanced construct files' may be appropriate for this purpose.
Scaffolding saves you time!
Another reason to scaffold an investigation is to reduce the amount of time the investigation will take. An example of such a scaffold would be to deliver a partly built file to students. Rather than expecting each student to spend 20+ minutes building the same file, construct the basis for the file yourself and allow students to explore. Using a 'similar figures' investigation as an example, the initial file could contain similar figures which move in tandem. The investigation could then be for students to explore the ratios of side lengths, areas and volumes of the similar figures. Obviously, students will need to first learn how to create dynamic formulas (formulas where the numbers change as the elements change in the file).
Encourage collaboration
Investigations need to be collaborative. Therefore engineer collaboration to occur, especially for the advanced investigations. This does not mean that students need to work in formal pairs or groups, rather, 'organic' peer teaching can be encouraged.
Yes, there's a GeoGebra-Proficiency course for teachers!
Learn Implement Share offers an engaging pathway for teachers to become proficient users of GeoGebra. Many schools navigate this pathway as a TEAM.
Here’s what one participant wrote after completing the course:
Here’s what one participant wrote after completing the course:
GeoGebra is a tool which I use to improve student engagement and understanding. It is very rewarding to see students independently exploring a topic in order to discover mathematical concepts. GeoGebra is a wonderful tool to enable students to create, manipulate and visualize, thereby gaining a better understanding of various topics. The depth and value of discussions resulting from lessons involving GeoGebra have been amazing. Rosemary Jacobitz, Northside Christian College 12.9.14 (2010 participant)
Do you use GeoGebra extensively? Do you run student-led investigations? Having read the article, are you now tempted? Would love your thoughts below! (Your email address will not be required)
