B.G.: (no, not the Stayin' Alive guys):
Before geogebra, I was attached to a wonderful software called Efofex, which I used mainly to make beautiful graphs, algebraic expressions, and diagrams for my slides, tests, worksheets etc. But I always wished that my students could use it as well. I could see the potential of the visualization, instant feedback, or trying out a theory about a function. Unfortunately, Efofex was not free, and at the time, I was in a school in which students only got computer access in the computer lab, which was always booked to the limit anyway.
Sometime in 2010, I heard Dan Meyer mention Geogebra during one of his talks, and I immediately downloaded this free miracle to my computer for the first time. After playing with it a bit and getting my students to download it to their computers, I started making geogebras with questions in them for my students, questions that they would answer by typing in a function, or constructing a triangle. Fun, paperless, cool.
Since then, my geogebras have evolved into tools for my students to explore, predict, experiment, & manipulate in order to answer their own questions instead of mine. I still have a long way to go to make it all work in a truly Inquiry-Based way, but that's not the point I want to make here.
It was during the creation of those exploratory geogebras that I experienced rich learning that truly belonged to me.
Every time I created something that had to behave a certain way, to respond to changing conditions using actual math, I learned something. What that was depended on what I was doing, what I was missing, & what I happened upon, in other words, it depended on who and where I was at that moment.
What I learned, and what I want my students to learn:
Math: When I created the virtual ferris wheel, I learned that the b in the equation y = a sin bt + k was there to change seconds into degrees. I had never really understood that until I had to make the ferris wheel turn with the angle slider. I remember a student asking me about b many years ago, and I just said, "Well, b is the frequency." "But," she persisted, "what is it really? Where can I see it?" Not only did I not know what she meant, I didn't know the answer. Sorry, students of my past.
About self-organization: When I made the absolute value inequation solver, which I thought would take 10 minutes but actually took an entire weekend, I learned that I needed to be way more systematic in order for it to work. I had to make a list, on paper, and check things off as I put the conditions into the "conditions to show object" field.
About physics: When I was struggling with making the virtual basketball below, I already knew the formulas for projectile motion, but I was so stuck on finding the rule of the quadratic function that I didn't even consider using those formulas. I thought "What good does votcos Θ do me here? I need a rule! I was all about plugging away at finding the a, h, and k for y = a(x - h)² + k. The math I'd been teaching for years was interfering!
About geogebra, For that same basketball example, I learned that in order to make a point move around, I could use a time slider, and make each of that point's coordinates depend on time, rather than punching in an entire function rule. The slider variable can belong to anything, not just a function rule but a coordinate as well.
About my own brain: I don't know how many times I woke up in the morning to discover that my brain had been figuring things out while I was asleep. Audrey, it said, look what I made for you. Again, I won't go into detail about what it solved because the point is this:
I think that only happens when you are truly engaged, when you care about what you're doing, and when you believe that it's within your grasp.
I'll never forget how good it felt watching that basketball move in a parabolic fashion as I moved the time slider. It's not the most beautiful basketball net, I know, but it's mine, and so is the new understanding that I have of all these things.
The thing is, the miracle with geogebra is that I can't learn something in it without also learning something about the math or the physics or my brain. And vice-versa. And that's just me - what would this look like for my students? I have no idea but it'll be a heck of an interesting experiment.
So how do I get this to happen with my students?
I and my LearnQuebec colleague Kerry Cule have decided to continue our Physics-Math collaboration, and have our students create a virtual manipulative. Something that they have to get to behave in the geogebra the way it does in real life, based on something they're learning in Physics. I'll be teaching them how to use geogebra, they will choose a topic from Physics (eg projectile motion, or potential energy, or vectors, etc), and using whatever math they need, they will create the geogebra.
I know this means more time and more work, but I'm convinced it will engage them the way it did me. Maybe not all, but more than just the few who are already strong, motivated, and disciplined. I want to get the creative kids to sink their teeth into this, and get the math right so that it measures up to their artistic standards.
Here's how their paths have unrolled so far this year, starting in the second week of September:
1. See the interactivity: I gave them a few simple old-fashioned geogebra worksheets in which to answer questions. I snuck in some checkboxes, which revealed questions one at a time. So their first exposure to geogebra was not to create, but to use, and see the interactive nature of it.
2. Experience the interactivity: Next, I gave them some geogebras in which they could experiment with functions: in one case by directly editing the rules of 2 functions and seeing how that impacted the graphs of their sums, quotients etc., and in the other case using sliders to change a single function's parameters.
3. Familiarization: They watched my two videos : "The Basics" and "Dynamic" & did the accompanying practice files. This got them familiar with the drawing buttons, the input bar, and how to change object properties, like colour and style.
4. Making: They created a file which included any 3 different types of functions (using input bar), 3 different types of shapes (using buttons), and 4 different types of links between those (eg segment between points, or midpoint between points etc). Everything had to be a different colour and style. They were really pretty! Here's a snapshot of one:
5. (Just today): Connect ggb to what we're studying: We worked out formulas for the zeros of an absolute value function, which are:
Then we looked at this, which was about Important points in the absolute value function:
I asked how geogebra is always showing the correct intercepts no matter where the sliders are? What formula might geogebra be using to do this? Then I showed the formula that was already right there in ggb, which matched the one we had just come up with:
One student's comment" "It's a miracle!" Kind of , yes! They then predicted the formulas that ggb was using for the other intercepts and the vertex. Message: Give ggb the right formula and those points will actually be where you want them to be, no matter where the slider is.
6. They will watch the next video "Sliders" and do the practice that goes with it.
7. Then they will do this assignment. I am fairly shaking with anticipation.
My students' A.G. eras are about to begin!