Sunday, September 22, 2013

Learning GeoGebra

I am finally reaching the point at which I can put GeoGebra to good use.  I am getting to the steeper portion of the learning curve.  I am sure I have much more to learn, still.  But I have learned enough to make GeoGebra a useful tool for exploring geometry, and also for creating learning activities for my students.

My first week of trying to use GeoGebra was frustrating.  I was having a hard time making shapes that were adjustable and movable, or dynamic.  At a basic level, I did not understand the dependency of points in Geogebra.  Points can be free (completely movable), semi-dependent (only able to move along the object they are placed upon), and fully dependent (not movable, unless the object(s) they lie on move).  I didn't realize that if I placed a point at the intersection of two lines, I wouldn't then be able to move those lines by moving that point.  The point needed to come first.  In my early attempts at figuring this out, I tried to make a rectangle with adjustable dimensions.  I went back and forth between creating rectangles that were entirely rigid, to trying to use four free points and creating a quadrilateral that didn't maintain the properties of a rectangle.

A Google search for how to make a dynamic rectangle in GeoGebra led me to a YouTube video presenting a challenge to do just that.  The rectangle in the video was dynamic in three points, that is, three points could be moved and the shape would remain a rectangle.  I had imagined that four points could be adjustable, so seeing that only three could move gave some more direction to my attempts, of which there were many.  However, I still wasn't able to figure it out until I watched the longer challenge video, which provided a key hint: one diagonal of the rectangle must lie on the diameter of a circle.  Then I was able to figure out that the vertices on either end of that diagonal are the only free points on the rectangle.  Another vertex needs to lie on the circle, and a right angle is guaranteed at that vertex, according to Thales' Theorem.  Here's a link to the rectangle on GeoGebra Tube.
My dynamic rectangle with semicircle shown.
A and C are freepoints, B is semi-dependent,
and D is fully dependent.

Working through the rectangle challenge problem helped me to understand how objects and figures behave and relate to each other in GeoGebra, and how to carefully put a sketch together to allow adjustment of certain dimensions or positions.  Another thing that I particularly wanted to understand, especially for the sake of creating activities for my students, was how to show different steps in a GeoGebra worksheet.

More Google searches on this topic helped me fumble my way through using check boxes.  Several objects on a worksheet can be selected to be shown or hidden when a box is checked or unchecked.  Here's a GeoGebra worksheet that I created using check boxes.  With the help of Professor John Golden, I learned how to use a slider to achieve the same results, but with more control of what is being shown or hidden.  With a slider set to integer values, conditions to show objects could be assigned in the properties for each part of a drawing, according to the values on the slider.  This also made it easier to have objects, like text, appear for only one step, or for several steps, without the need for unchecking a box again.
A worksheet with check boxes.
worksheet with a slider to show steps.  Plus, the
rectangle's length and width can be adjusted!

Being able to use a slider will be key to making worksheets for students to use that contain several different examples.  In the near future I am hoping to make a GeoGebra worksheet to help my geometry students discover the four triangle congruence theorems (SSSSASASA, and AAS).  In a similar vein, I think a GeoGebra worksheet could be very helpful for my precalculus students learning about the Law of Sines and the SSA case, and when two triangles are possible.

Wednesday, September 11, 2013

Van Hiele's Levels of Geometric Reasoning

“School geometry that is presented in a [Euclidean] axiomatic fashion assumes that students think on a formal deductive level.  However, that is usually not the case…” (Van Hiele, 1999)

This excerpt from Van Hiele’s article, Developing Geometric Thinking through Activities that Begin with Play,  immediately made me think of one topic in particular, the bane of (most) high school geometry students’ existence: proofs.  The type of formal reasoning demanded by deductive proofs is seldom easy for students to attain, even in high school.  If students have one complaint about geometry, it is usually the proofs.

As such, I am always looking for ways to make proofs easier for students to understand.  Before we get into geometric proofs, we always play a “proof game” with simple rules for manipulating strings of letters, to get them used to justifying every statement they make.  I also try to emphasize the sequence and structure that is common among many proofs, like using definitions to change words into equations, after which properties of equality can be used to show a range of things.  We also try to scaffold heavily as students learn how to write proofs, slowly bringing them along from filling in blanks to eventually writing complete proofs on their own.

But Van Hiele’s article led me to think about students’ struggles with proof in a different light, and question whether my students have been given or are given enough opportunities to progress through Van Hiele’s levels of geometric reasoning.  Perhaps some of their difficulty with proof stems from having too little time to engage in informal reasoning.

Van Hiele’s Levels of Geometric Reasoning
1 - Visual
2 - Descriptive
3 - Informal Deductive
4 - Formal Deductive
5 - Axiomatic

I know that my students do not have many opportunities to do this, to take their time and consider what they know about geometric figures, and to discover out of their own knowledge further relationships and geometric truths.  On the first day of class, they are introduced to four postulates, and at least one or two postulates or theorems follows in each subsequent lesson.  They do come in a logical sequence, but still they are presented for students to take at face value most of the time, unless we take the time to prove them.  It would be better if my students could be presented with new ideas along the way, and allowed to explore and struggle to find new relationships on their own, before having the formalized theorems presented to them.  And they could practice justifying their discoveries, without immediately being tied to structure of a proof.  I think they would much more thoroughly internalize important concepts than they do now.

Formal proof writing will always be an adjustment, though.  It is a new style of writing for students.  But if they have had more time to explore their way through new concepts, they might be better at making the connections that tie given information to a desired conclusion.

Van Hiele’s mosaic and the activities he describes to go with it are impressive.  Its careful design provides a basis for a wide range of discussions and student discoveries.  The example of the mosaic makes me want to find more exploratory activities that can apply to the concepts my high school geometry students will work with this year.  I will certainly look for these, or try to come up with some, and look for opportune places to tie them into our curriculum.

Time might be the biggest challenge to integrating more exploratory activities.  It is at such a premium already, with all that is supposed to be covered in a term.  Giving students a day here or there to explore, rather than presenting material directly and moving on is not the most efficient way to go, time-wise.  Our math department already has moved to include days dedicated to problem solving group work near the end of each unit.  I wonder now if in geometry it might be better to use some of those days for exploratory activities at the start of a unit.  That is a suggesting I will make to our geometry team.  At the very least, I will be more apt to provide more time within lessons to have students discuss key ideas with each other.

Van Hiele, P.M. (1999). Developing geometric thinking through activities that begin with play. 
Teaching Children Mathematics, 5(6), 310-316.