I took Algebra 1 as an 8th grader, then Algebra 2, PreCalculus, and THEN Geometry. I felt like I had a pretty good grasp on the mathematical concepts presented up until that point in my mathematical journey. Almost from the very beginning, I struggled with Geometry. I tried to reason through what was going on and come up with a reason WHY things worked the way they did, but I was completely lost. About 6 weeks into the course, I gave up. I resorted to the "memorize and repeat" method of learning Geometry--memorize the information and repeat it on the test. It worked! I finally had an A in Geometry. I was totally in the dark as to what was going on mathematically, but no one questioned me because I had an A.
I went through most of my college career with a very low understanding of Geometry.
Fast forward to graduate school... I took a math course at Auburn that had my least favorite topic in it, Geometry. I was really nervous. After a week, I actually understood WHY some things worked the way they did. It wasn't perfection level, but the lightbulb had finally come on for me. I didn't dread going to class because I knew that I was actually going to learn the reason behind why something worked, not just a "drill and kill" approach to Geometry.
Well, what was it that finally caused that Lightbulb Moment??
If you have never used Geogebra and you don't understand Geometry, meet your new favorite geometry tool!! I was actually able to explore my ideas and conclusions as to WHY something worked--not just that it did. Up until that point in time, all of my Geometry instruction was very direct: here's the rule, here's the tricks, here's a problem. No explorations. No investigations.
Using software like Geogebra, you are able to explore the properties of shapes without having to construct it all by hand. You can construct one case, move your figure, and see if it is true for another case. Geogebra is a type of "Dynamic Geometry Software".
If we use tools like Geogebra in our classrooms, we have the power to take Geometry instruction to the next level! Our students can explore constructions or difficult geometry questions/tasks without getting lost or distracted by the possibility of human error. They can use their geometric knowledge to explore difficult questions quickly instead of several class days when doing the constructions by hand.
Here are three reasons why I think that Dynamic Geometry Softwares (DGS) have the ability to transform instruction in the Geometry classroom.
1--They allow students to explore geometric constructions in a new light. "...using DGS, once a geometrical object has been constructed, the geometrical relations that were used in its construction retain their integrity when the object is dragged (Dick & Hollebrands, 2011, pg. 33)." When students use DGS, they can use what we will refer to as "The Drag Test". When using "The Drag Test", you are able to see properties/relationships that transcend a single example, but see what is true for ANY figure with a specific set of givens. This allows them to reason through WHY these relationships occur--not just memorizing their existence.
A great example that is given in FHSM is constructing an isosceles triangle so that it is always isosceles no matter how you move a vertex of the triangle. In the book, some students used the radii of a circle while others used a segment and its perpendicular bisector to construct the triangle. Both of these worked when "The Drag Test" was used after the construction (Dick & Hollebrands, 2011).
2--They allow students to explore geometric operations. After you first lean about reflections, rotations, and translations, you start exploring about how you can combine them in different ways. What happens if I had 2 reflections, or a rotation and a reflection, etc. Using Geogebra, you are able to use the tools that will reflect, rotate, and translate any figure. You can use sliders to explore what would happen if the angle was different. By using a DGS, you allow students to construct their own argument and test their theories instantaneously--as compared to drawing each theory out by hand (Dick & Hollebrands, 2011).
3--They allow students to prove or disprove generalizations in geometry. When looking at figures or properties of figures in geometry, you can create figures where some properties are true, but the key is discovering properties that are true regardless of the size & orientation of a given figure. This is where DGS comes in to play. Suppose you are exploring properties of quadrilaterals. After constructing a given quadrilateral, what is true? What isn't? If I were to change my figure in any way, is it still true? These are all questions that are very quickly answered and explored using a DGS like Geogebra (Dick & Hollebrands, 2011).
If you have never taken the time to explore a Dynamic Geometry Software, please do! If you struggle with geometry, take the time to look into these types of software. Geogebra is not the only one, there are many more! I am familiar with TI-NSpire and Geometer's Sketchpad. Look into DGS, see if there is a way you can change something you already do in your classroom to incorporate technology in this transformative way in your classroom.
Please comment and share your thoughts! :)
Resources:
Dick, T. P. & Hollebrands, K. F. (Eds.). (2011). Focus in high school mathematics: Technology to support reasoning and sense making. Reston, VA: National Council of Teachers of Mathematics.
1--They allow students to explore geometric constructions in a new light. "...using DGS, once a geometrical object has been constructed, the geometrical relations that were used in its construction retain their integrity when the object is dragged (Dick & Hollebrands, 2011, pg. 33)." When students use DGS, they can use what we will refer to as "The Drag Test". When using "The Drag Test", you are able to see properties/relationships that transcend a single example, but see what is true for ANY figure with a specific set of givens. This allows them to reason through WHY these relationships occur--not just memorizing their existence.
A great example that is given in FHSM is constructing an isosceles triangle so that it is always isosceles no matter how you move a vertex of the triangle. In the book, some students used the radii of a circle while others used a segment and its perpendicular bisector to construct the triangle. Both of these worked when "The Drag Test" was used after the construction (Dick & Hollebrands, 2011).
2--They allow students to explore geometric operations. After you first lean about reflections, rotations, and translations, you start exploring about how you can combine them in different ways. What happens if I had 2 reflections, or a rotation and a reflection, etc. Using Geogebra, you are able to use the tools that will reflect, rotate, and translate any figure. You can use sliders to explore what would happen if the angle was different. By using a DGS, you allow students to construct their own argument and test their theories instantaneously--as compared to drawing each theory out by hand (Dick & Hollebrands, 2011).
3--They allow students to prove or disprove generalizations in geometry. When looking at figures or properties of figures in geometry, you can create figures where some properties are true, but the key is discovering properties that are true regardless of the size & orientation of a given figure. This is where DGS comes in to play. Suppose you are exploring properties of quadrilaterals. After constructing a given quadrilateral, what is true? What isn't? If I were to change my figure in any way, is it still true? These are all questions that are very quickly answered and explored using a DGS like Geogebra (Dick & Hollebrands, 2011).
If you have never taken the time to explore a Dynamic Geometry Software, please do! If you struggle with geometry, take the time to look into these types of software. Geogebra is not the only one, there are many more! I am familiar with TI-NSpire and Geometer's Sketchpad. Look into DGS, see if there is a way you can change something you already do in your classroom to incorporate technology in this transformative way in your classroom.
Please comment and share your thoughts! :)
Resources:
Dick, T. P. & Hollebrands, K. F. (Eds.). (2011). Focus in high school mathematics: Technology to support reasoning and sense making. Reston, VA: National Council of Teachers of Mathematics.
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