Investigating the circle property with GeoGebra
GeoGebra - Ideal for junior high mathematics!
Most of the GeoGebra articles on this site demonstrate the power of GeoGebra for junior high school mathematics. The focus on GeoGebra for junior high mathematics is, in part, because there is a strong misconception in the math education world that GeoGebra is a tool (almost entirely) for high-level, calculus-based mathematics and high-level geometry.
The Learn Implement Share blog articles aim to encourage those math teachers who are not utilising GeoGebra (or similar dynamic geometry software) as a regular demonstration tool, to start doing so. I have reason to suspect the portion of mathematics teachers not yet regularly utilising GeoGebra (or similar) to be significant. I suspect the common reason for this is a lack of awareness. It is logical to assume that the following misconceptions are held by these teachers:
The Learn Implement Share blog articles aim to encourage those math teachers who are not utilising GeoGebra (or similar dynamic geometry software) as a regular demonstration tool, to start doing so. I have reason to suspect the portion of mathematics teachers not yet regularly utilising GeoGebra (or similar) to be significant. I suspect the common reason for this is a lack of awareness. It is logical to assume that the following misconceptions are held by these teachers:
- GeoGebra is difficult to use.
- Utilising GeoGebra will require a significant change in my teaching approach.
- GeoGebra is inappropriate for the level of mathematics I teach.
GeoGebra - powerful even only as a demonstration tool
For many teachers the reason will be simpler – they simply do not yet realise how powerful GeoGebra is, even when used as a simple demonstration tool.
In the article Let GeoGebra transform your math teaching I show how using GeoGebra as a demonstration tool can revolutionise 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 revolutionise 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-with-GeoGbera' 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-and-show-the-file' method into an even 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 'uninitiated' teacher.
Two categories of student-led investigations which relate to the 5 tips:
Category 1)
Beginner investigations: for students who are new to investigations and/or less likely to stay on-task.
Category 2)
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.
Category 2)
Advanced investigations: for students who are self-starters and enjoy investigating mathematical systems in a self-directed manner.
Five tips for student-led GeoGebra tasks
- Engage.
- Teach basic skills using self-directed notes or videos.
- Create open-ended investigations.
- Scaffold the investigation.
- Encourage collaboration.
NOTE: When introducing student-led activities, the potential positives are high as are the chances of glorious, dismal failure!! Therefore, you will need to consider several factors, the first of which is the engagement level of your students.
1. Engagement
Generally speaking, how engaged are your students? Are they familiar with student-centred activities? How do they respond to them? 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 centred 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 problem. 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 being a success, as suggested below.
Those who teach functional, generally motivated students who are prepared to explore mathematical systems are clearly ready to proceed.
Where disengagement is the norm it is important to understand that introducing technology per se won’t fix the problem. 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 being a success, as suggested below.
Those who teach functional, generally motivated students who are prepared to explore mathematical systems are clearly ready to proceed.
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. In other words, don’t plan for any structured investigations at the same time as students are learning the skills. Allow the initial investigation to be ‘play time’. Students will naturally want to play anyway so avoid causing an unnecessary war - make their play 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 20 ways see how many closed spaces you can make.”
In regard to the skill instructions, deliver them via student centred printed handouts or videos. Stress to students that they must READ (or WATCH) the instructions and follow them. Avoid allowing students to ask you or anyone else any questions relating to the instructions unless they have read/watched the instructions twice. And/or use the 'Three before me' strategy - "You must ask 3 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 that’s all they know. Your task is to wean them off the spoon feeding. It won’t happen overnight!
In regard to the skill instructions, deliver them via student centred printed handouts or videos. Stress to students that they must READ (or WATCH) the instructions and follow them. Avoid allowing students to ask you or anyone else any questions relating to the instructions unless they have read/watched the instructions twice. And/or use the 'Three before me' strategy - "You must ask 3 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 that’s all they know. Your task is to wean them off the 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 steer 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 (Category 1) 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 Category 1 investigations, is to build ‘construct files’. A construct file is a GeoGebra file with the instructions written inside the file (see the gif examples below and above). 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 another instruction source. I realise we just established that investigations need to be open-ended. However, as category 1 investigations, highly scaffolded files - as per the files on display here - could be ideal. But ideally, once you move beyond Category 1 into more advanced investigations you will want to make the investigations as open-ended as possible. Advanced construct files could be created for this purpose.
One option, ideal for Category 1 investigations, is to build ‘construct files’. A construct file is a GeoGebra file with the instructions written inside the file (see the gif examples below and above). 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 another instruction source. I realise we just established that investigations need to be open-ended. However, as category 1 investigations, highly scaffolded files - as per the files on display here - could be ideal. But ideally, once you move beyond Category 1 into more advanced investigations you will want to make the investigations as open-ended as possible. Advanced construct files could be created 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.
Encourage collaboration
Ideally, investigations are collaborative. Therefore engineer collaboration to occur, especially for the advanced (Category 2) investigations.
Learn Implement Share offer 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:
Learn Implement Share offer 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:
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)
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