Written Proofs and Dynamic Geometry

The question has been asked, "If we have software like Geogebra, is there a need to have formal proofs in Geometry class when I can simply construct a figure in Geogebra that maintains its properties under the 'drag test?"

The first thing that I did was go to Wolfram Alpha and look up the mathematical definition of "Proof" and it said 

The definition of mathematical proof does not specify written or otherwise.  

I think that written proofs go hand in hand with a dynamic sketch in a program like Geogebra.  In order to construct an appropriate figure, a few things must first happen: 

1-You must have an understanding of the mathematics.  You can not just guess to draw a perpendicular bisector, circle, or parallel line.  Most students would not go to those constructions without some indication as to why we would need them.  

2-By using the elements of a construction, you are also using the reasoning that goes with the written theorems that have already been established. 

By using the Dynamic Geometry Software (DGS), we allow our students to have a visual as well as a written proof. Many students learn visually and allowing them to use DGS to construct formal proofs, we help them through the reasoning process by seeing the theorems at work.  

I am currently working on a project for a mathematical technology class at Auburn, and in this class, we are to show different properties of a problem using DGS.  DGS has properties that allow you to show "step-by-step" processes.  So students could use the advanced features of DGS to illustrate their formal proofs in class.  

In NCTM's Principles to Actions(2014), they talk about Mathematical Action Technology (like DGS) and how it allows students to see the same information in a variety of ways.  "The ability to shift between different representations of a problem (e.g. visual/graphical, symbolic, numerical) can help students develop a deeper understanding of mathematical concepts. Further research suggests that the effect of working with virtual manipulatives on a computer screen is equivalent to using physical materials (Sarama and Clements, 2009)" (NCTM, 2014, p. 84). 

So, is there a need for formal proofs, YES! It is another way of representing the geometric sketch.  I think that both forms of the proof are necessary and allows students to reason through and see the written proof in action.  

References:

National Council of Teachers of Mathematics. (2014). Principles ot Actions: Ensuring mathematical success for all. Reston, VA: National Council of Teachers of Mathematics.

Sarama, J., & Clements, D. H. (2009). 'Concrete' Computer Manipulatives in Mathematics Education. Child Development Perspectives , 3 (3), 145-150.

The Technology "Crutch"

Over the past few weeks, I have been working on a few projects for a Technology in the Secondary Mathematics Classroom course that I am taking at Auburn.  I have learned a lot of new things, and we definitely aren’t done yet! I am looking forward to all the new things that I will learn about over the rest of the summer. 

So…why is this “blog relevant”??

I have realized something over the last few weeks that I never really consciously realized before.  If you are using Dynamic Geometry Software, Graphing Calculator, or some sort of Mathematical Action Technology, you MUST still understand the mathematics behind what you are doing (NCTM, 2009).  You can’t complete an activity just because you are using technology because the technology can’t do the work for you.  You have to also have an understanding

Here is how I came to this realization: I am working on a project for my technology class using Dynamic Geometry Software: specifically Geogebra and a TI-Nspire CAS.  My geometry skills aren’t perfect by any means, but in order to complete this project, I have to have some understanding of vocabulary and an understanding of how to use the technology to accomplish the goal. 

A lot of times, people who are against the use of technology think that by using technology we are giving students a crutch to help them “get through” a math course (NCTM, 2014).  I personally disagree.  I have a degree in mathematics education, and despite my degree, I still have to have some grasp on the concepts that we are using. 

Isn’t our goal as teachers to prepare our students for things that will come next? By using technology in our classrooms, we are providing students the opportunity to learn to use tools—in a supportive environment—that will help them in college and future careers.

Please share your thoughts on technology in the classroom below!

The "Lightbulb" Moment

Anyone who knows me will tell you very quickly that I am definitely a math geek!  It's true! I LOVE math!!  I enjoy solving problems just because I enjoy the "thrill of the chase" that comes with solving any math problem.  I have been this way since as long as I can remember.  I have just always LOVED math! I did well in high school math... at least until I got to Geometry.  

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?? 

GEOGEBRA!

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.