Digital Innovation: GeoGebra
GeoGebra is a good case of meaning-driven digital innovation. It gives insights into the radical problems of conventional math teaching and learning and redefines the problem to be solved. It transforms abstract math concepts into visible and meaningful ones through digital tools. It creates a benign and dynamic digital learning environment, changing conventional math education ways.
What’s GeoGebra?
GeoGebra is an open-source and multi-platform interactive math software. It provides users with “hands-on” interactive applets that everyone can access through the web portal or the local application. With digital manipulatives, such as operating with points, coordinates, equations, functions, drawing shapes, and making measurements, users can visualize math concepts and develop a deep understanding of a particular math topic.
Math is full of abstract conceptual knowledge, the most challenging part of teaching and learning. To address it, Geogebra makes learning in math much more meaningful and easier by visualizing conceptual knowledge.
Conceptual Knowledge and Procedure Knowledge
Conceptual knowledge is the basic knowledge of mathematical arrangement about the relationship and interconnection of mathematical ideas, which enables one to explain and bring meaning to mathematical procedures. In Conceptual and Procedural Knowledge, James Hiebert, a professor at the University of Delaware, defined conceptual knowledge of mathematics as a chain of multiple knowledge. Generally, conceptual knowledge(knowing why)can be viewed as a web of knowledge, cognitive chains where the relationship among the nodes is equally essential with the discrete discs of information nodes.
Procedure knowledge is another vital part of mathematics. In his book, Prof. James Hiebert described the Case of Mathematics had categorized procedural knowledge into two parts, (a) knowledge of mathematical symbols and (b) knowledge of algorithms or rules to be used in solving mathematical problems. To know the procedures is to be aware of the approach to be used in the process of manipulating mathematical symbols. Procedural knowledge focuses on the skills needed in problem-solving.
The ability to relate conceptual and procedural knowledge will benefit the students tremendously, especially in acquiring and applying procedural knowledge. A study conducted by Eisenhart et al. (1993) found that conceptual knowledge is about knowing mathematical structures, which involves linking ideas to explain and provide meanings to math procedures. Developing the ability to connect conceptual knowledge with mathematics symbols has a process of giving meanings to the symbols. However, relating conceptual knowledge with procedural knowledge is difficult. It is due to persistent challenges in gaining abstract concepts, and it is unease to measure since we couldn’t observe students’ understanding of abstract concepts. But procedural knowledge is related to anything the students do, which can be observed when students conduct specific steps to solve a math problem.
Conventional Math Education and Problems
As mathematics is a discipline that requires abstract operations under many sub-learning areas, it’s the most challenging art for students to learn. Teaching conceptual-based knowledge, in reality, is particularly intricate since it’s abstract. For math educators, it’s more convenient to transfer procedural expertise and teach the specific skill of solving math problems. Like solving the triangle area by a specific formula. So, Conventional education focuses on teaching procedural knowledge and giving students practice on the procedural type of questions in preparing for the math test. Also, math educators lack efficient tools to develop effective and visual teaching materials regarding conceptual knowledge, and teachers heavily leverage match textbooks which are rigorous but boring and ineffective.
On the other hand, students tend to depend solely on the learning process they have experienced in school. Consequently, students’ conceptual knowledge of math isn’t well developed. It lacks an open learning platform for students to practice math and error trials without limitation.
The Innovation of Meaning by GeoGebra
1. Transform abstract concepts into visible and meaningful ones.
The most creative part of GeoGebra is making learning abstract math much more meaningful. One example (see the pic below) is to show the basic concept of a triangle. Three interior angles plus together are always 180 degrees. Then three exterior angles plus together are always 360 — the visualization animation show this abstract knowledge intuitively. Through visuals and interactive material, GeoGebra shows how the visualization process guides the student to understand triangles better. Further, through interactive practice with the case, you will know this math concept better by interactively practicing with the figurative case.
The function is another challenging math topic for students. Students might be able to find the answer by rote memorizing formulas for mathematical functions but couldn’t truly comprehend the situation. One characteristic of function is a relation for which every input gets mapped to ONE and ONLY one output. But this abstract description is hard to thoroughly incomprehensible. GeoGebra provides intuitive visualization examples and animation-based activities as designed in the software, which easily improves our understanding of it.
It has been proven that the teaching and learning process using the GeoGebra digital tool significantly enhances and strengthens users’ conceptual knowledge of mathematics. Visual elements in math education are found attractive by students; more than that, visualization (images, interactive graphics, etc.) brings meaning to an abstract concept, helping students relate the concept with the other knowledge they have been familiar with or the actual situation in daily life. Visualizing abstract concepts allows students to tap their conceptual knowledge to give meaningful associations of mathematical symbols with the other entities. It allows them to connect conceptual knowledge with sequences, algorithms, or procedures, reducing the large process or procedures to be learned in mathematics.
When students understand the core mathematical concept better, it will be easier to apply it in other fields and solve procedural problems. According to Bu, Mumba, and Alghazo (2011), using GeoGebra software allows students to involve in mathematics modeling, exploration of problems, and exposure to open-ended questions. Thus, students can diagnose their process of solving mathematical problems.
2. A highly dynamic interactive platform
Knowledge isn’t easily transmitted from one person to others. Knowledge is actually developed by ourselves through amounts of practice within the right learning environment. So not just tools, creating a great learning environment is truly important.
GeoGebra changes the conventional way of teaching and learning math. It does not just provide practical tools; it builds a new learning community. Geogebra developed a highly dynamic online community for math educators and students to free trial and error. Over 100 million students and teachers freely access the Geogebra web community regularly.
The Geogebra community provides possible interaction between teachers and students, which is rare in conventional learning. Research revealed that Geogebra teaching makes teaching processes fun, enables students to participate in the class effectively and that the learning environment creates a positive competition between groups with in-group and out-group interaction. Those characteristics expose students and teachers to a broader teaching and learning scope inside and outside the classroom.
Teachers can ease create and share interactive match teaching samples through the web community of Geogebra. Through the community, math educators worldwide make all kinds of math simulations. They can fully leverage these rich resources or create interactive worksheets, even applets that include the simulations built by others, along with their helpful directions. It enables teachers to widen their teaching scope by exploring various approaches to content delivery. It even allows teachers to be more creative in their teaching, making it more effective, mainly through the two-way education and learning process.
Students can more easily interact with the visualized math through GeoGebra mobile tools or the web portal of Geogebra. For example, traditional methods for students to perform construction need to use all kinds of physical tools, like a compass, ruler, etc., which can be time-consuming and inconvenient. GeoGebra makes it quick, easy, and fun as long as there are clear directions.
In addition, students can use their ideas and present their works. It is very different from conventional learning, where students passively wait for the teacher to deliver information as they do not have the chance to present their ideas. Students are becoming more active and responsible for their learning process as they are personally involved in the GeoGebra module, which allows a self-learning process. It also gives students free trial-and-error opportunities when they practice a specific math topic on this platform. Let them make assumptions, test, and generalize from the results provided by seeing the student as a mathematician who discovers.
GeoGebra shows an excellent example of digital innovation. Digging out the actual problems of existing mathematics education, then transforming conventional mathematics education with the help of digitalization, gives it new meaning, thus significantly improving the interest and efficiency of people’s learning.
