GeoGebra is a dynamic mathematics software for teaching and learning mathematics. As tool for the teaching part is pretty easy to do. But the learning part, well, that’s always been the one that is problematic, GeoGebra or not GeoGebra.
Studies about integration of technology in teaching and learning have always acknowledged that despite the availability of the technology, teaching and learning tools like GeoGebra is still not widely used in many classes even with the availability of computers for students. If ever, it’s the teacher who uses it and more often, for demonstration and sometimes ‘staged’ discovery of concepts and for visual effects for all the students to enjoy but not to learn. I discuss my thoughts about it in my first post about GeoGebra and Mathematics.
To date, the calculator is still the undisputed teaching and learning tool in many mathematics classes. And for calculators, I can confidently claim that it is indeed both a teaching and learning tool. Students use it and can use it to investigate mathematical relationships, depending how lucky they are to have a math teacher that makes it possible. I think students use calculator not just because they know how to use it but because they understand the mathematical ideas represented by the keys. Now, if we can do the same for GeoGebra then maybe, just maybe, we can maximize its potential for facilitating mathematics learning.
GeoGebra is a great software for teaching and learning mathematics. It offers geometry, algebra and calculus tools in one environment, a great support indeed for linking mathematical concepts. On top of that it is free and an open-code software. Click here to download the latest version of the software.
Is it easy to use? Yes and No. Yes, for math teachers because they know the mathematics and can therefore easily understand the ideas and logic behind the tools. Yes, for students who have been instructed on how to use the tools and understand the mathematics and logic behind it. They can use it in solving problems and for investigating mathematical relationships. But, for the majority of students, especially those who have not learned the basic of graphing, equations, and geometric relationships, the use of GeoGebra is limited to manipulating ready-made GeoGebra applets. (Click here for my posts on solving problems about quadrilaterals or here for introducing function using Geogebra applets.) Well, yes, GeoGebra applets are easy to use but most of the time if you do not know the mathematics behind the construction or can’t construct it yourself, then the learning of the mathematics may be superficial.
Construction of math models using the software is not accessible for many younger students just starting to learn basic Algebra and Geometry. In order for them to construct a model, they will have to follow a set procedure (constructed by the teacher) without really understanding why they do what they do. So I thought why not teach GeoGebra tools and mathematics at the same time? This is a challenge I set for myself and I have no idea if it will work or not. I am thinking of doing a research of it later in the year. GeoGebra is free and faithful to mathematics so for countries like us that can’t afford to buy licensed softwares, we get the same quality teaching tool with Geogebra. I think all students need to know how to learn mathematics with it.
‘To understand mathematics is to make connections.’ This is one of the central ideas in current reforms in mathematics teaching. Every question, every task a teacher prepares in his/her math classes should contribute towards strengthening the connections among concepts. There are many ways of doing this. In this post I will share one of the ways this can be done: Use the same context for different problems.
The following are some of the problems that can be formulated based on quadrilateral BADF. You can pose these problems to your class but the best way is to simply show the diagram to the students then ask them to formulate the problems themselves.
quadrilateral
Problem #1. What is the area of the quadrilateral? Show different methods.
Mathematics is not just about the study of numbers and shapes but also about the study of patterns and relationships. Function, which can define some of these relationships, is an indispensable tool in its study. Function is the central underlying concept in calculus. It is also one of the key concepts of mathematics that can model many quantitative relationships.
Textbooks and teachers usually introduce function via a situation with the related quantities already identified. What is required of the students is to learn how to set up and represent the relationships in tables, graphs, and equation and analyze the properties. In the real world, when function is used as a model, the first thing that needs to be done is to identify the varying quantities. So, it is important to let students identify the quantities and let them determine which of these quantities may be related. This way they get a sense of what function really is and what it is for. The function is not the graph, not the table of values, and not the equation. The function is the relationship between the variables represented by these. The study of function is the study of these relationships and their properties, not finding y or f(x) given x and vice versa, not reading graphs, and not translations among the representations. These are important knowledge and skills, yes, but only in the context for investigating or learning more about the relationships between the quantities, that is, the function. Thus, for an introductory lesson for function, I find it useful to use a situation where students themselves will:
identify the changing and unchanging quantities;
determine the effect of the change of one quantity over the others;
describe the properties of the relationship; and,
think of ways for describing and representing these relationships.
These are the ‘big ideas’ students should learn about function. Of course, there are others like looking or dealing with function as a mathematical object and not only as a process or procedure for generating or predicting values. However, for an introductory lesson on function, teachers need not focus on this yet.
Sample introductory activity:
What are the quantifiable attributes or quantities can you see in the figure below? Which of these quantities will change and remain unchanged if GC is increased or decreased? Click the figure and move point C. Are there ways of predicting the values of these changing quantities?
Identifying related quantities
Click here or the image above to go to dynamic window for the worksheet.
I like this particular activity because it gives students the opportunity to link geometry/measurement concepts to algebra and learn mathematics through solving problems.
GeoGebra is a dynamic mathematics software for teaching and learning mathematics. As tool for the teaching part is pretty easy to do. But the learning part, well, that’s always been the one that is problematic, GeoGebra or not GeoGebra.