## Developing Conceptual Mathematical Understanding

In Students With GeoGebra

(The original GeoGebra file used for the dynamic image above was created by Pamela McGillivray, Broome Senior High School, WA, course participant, December 2012)

Teaching for conceptual understanding - in conjunction with teaching procedures - is critically important when teaching mathematics. There is a world of difference between students being able to ‘do mathematics’ and students understanding the mathematics they are 'doing'. As is evidenced by research, simply practising mathematical routines is not enough - developing conceptual understanding requires higher-level cognitive processes such as connecting ideas, reasoning, and solving complex problems (Silver et al., 2009; Stein & Lane, 1996)
We could refer to an approach that enables students to understand what they are working on in math lessons (for the majority of lesson time) as an Understanding-first; Procedures-second Approach - because the understanding needs to form first, as its own entity, and not as a byproduct of repeating a procedure multiple times. We can also refer to this as a hybrid approach. The article Teaching Mathematics Using An Approach That Is Both Conceptual And Procedural - Nine Keys To The Hybrid Conceptual-Procedural Approach sheds some light on a hybrid conceptual approach to teaching mathematics, one that also acknowledges the importance of teaching procedures. The article states that mathematical understanding is fostered through the use of activities and strategies specifically designed to engineer aha moments within students and that through using a conceptual approach, metacognition and other higher-order thinking processes are encouraged. To quote J.E.Schwartz from his article: |

In a conceptually oriented mathematics class, the bulk of time is spent helping the students develop insight. Activities and tasks are presented to provide learners with experiences that provide opportunities for new understandings.

## Introducing GeoGebra

GeoGebra is a fantastic tool for fostering understanding in mathematics students. However, it is not a substitute for practice. Rather, when used well, GeoGebra allows students to gain a conceptual understanding of the mathematics at hand so that when they engage in mathematical practice they understand what it is they are doing.

Consider now a very simple demonstration of developing conceptual understanding with GeoGebra. Cast your eyes on the dynamic image (gif) at the top of this article. Observe the image for 30 seconds or so. The power of this file should be self-evident. Of course, students still need to crunch the numbers underpinning the principle. However, to watch the two area sums change dynamically with changes in ‘n’ – and to witness the summed area triangles approach the area of the circle as ‘n’ increases – is stunningly powerful.

Consider now a very simple demonstration of developing conceptual understanding with GeoGebra. Cast your eyes on the dynamic image (gif) at the top of this article. Observe the image for 30 seconds or so. The power of this file should be self-evident. Of course, students still need to crunch the numbers underpinning the principle. However, to watch the two area sums change dynamically with changes in ‘n’ – and to witness the summed area triangles approach the area of the circle as ‘n’ increases – is stunningly powerful.

## Increase The Occurrence Of Aha Moments In Students Without Changing Your Pedagogy!

The following is an important point ... In the 21st Century, to teach students such a principle without giving them a dynamic, visual representation is, in my view, unacceptable. This is because files like the one above not only increase the occurrence of aha moments in students, they are readily available. However, here's the kicker - demonstrating such a dynamic file, as powerful as it is, does not require a change in pedagogy. The only thing that is required is for the teacher to plug in a data projector, manipulate the file and ask leading questions.

There is an almost limitless number of mathematical situations to which GeoGebra can be applied – even when only used as a demonstration tool.

There is an almost limitless number of mathematical situations to which GeoGebra can be applied – even when only used as a demonstration tool.

## One Teacher Had This To Say:

"I had (in the past) found Geogebra to be a not very intuitive piece of software, so learning some advanced applications was great. Using GeoGebra in class has greatly improved students' ability to visualise tricky concepts - like locus, linear and non-linear functions, trig relationships, etc. The online course was fantastic." Kathy Howard, Bathurst High Campus, 15.9.14 (2010 participant)

## Now to see GeoGebra in action - 3 ways teachers have generated aha moments by using GeoGebra in their lessons

1) Parabola As A Locus File

A well constructed GeoGebra file has the ability to enable the viewer to gain insight into mathematical principles prior to teacher intervention. I think you’ll agree that the 'Parabola as a locus' file below achieves this well. When looking at the file, ask your self:

“What can you see going on here?"

"What principle can you see at work”?

(The original GeoGebra file used for the dynamic image above is one of the 50+ files which participants create)

## 2) Using A File As A Demonstration Experience

Files such as the one above offer an excellent way to summarise or revise mathematical principles. However, an arguably more powerful application of such a file is to project the file PRIOR to teaching the associated mathematics. In this way, the file provides a stimulus to pique students’ interest and to foster inquiry. Questions can be posed, such as “What can you see happening here? What principles can you see at work”? Students can then discuss their ideas in pairs prior to sharing with the whole class.

Using a well-constructed file to foster enquiry BEFORE teaching the associated mathematics offers advantages over the more traditional skill-and-drill approach, including superior student engagement, an increase in aha moments and promoting higher-order thinking in students leading to an increase in understanding.

Of course, there are superior applications of GeoGebra which enable students to investigate mathematical systems. (Make sure you read the Pythagoras investigation article to see how powerful GeoGebra can be in leading a user into new-to-them mathematical discoveries).

However, for teachers who do not want to disrupt their regular approach to teaching, they can reap powerful benefits by simply utilising GeoGebra as a demonstration tool. All that is required is 1) plug in a data projector and 2) demonstrate the file!

Using a well-constructed file to foster enquiry BEFORE teaching the associated mathematics offers advantages over the more traditional skill-and-drill approach, including superior student engagement, an increase in aha moments and promoting higher-order thinking in students leading to an increase in understanding.

Of course, there are superior applications of GeoGebra which enable students to investigate mathematical systems. (Make sure you read the Pythagoras investigation article to see how powerful GeoGebra can be in leading a user into new-to-them mathematical discoveries).

However, for teachers who do not want to disrupt their regular approach to teaching, they can reap powerful benefits by simply utilising GeoGebra as a demonstration tool. All that is required is 1) plug in a data projector and 2) demonstrate the file!

## 3) The Surfboard File

Another excellent demonstration of a file designed to foster conceptual understanding is the Surfboard file, originally created by December 2013 participant Anne Wolkowitsch. You will find it halfway down this article along with a suggested conceptually-based approach for demonstrating it. And you can also download the GeoGebra file from that page!

Do you use GeoGebra extensively? Are you able to modify and tweak files 'on the fly' as you teach? Or do you only demonstrate files created by others? We would love your thoughts below! (Your email address will not be required)