The Gift of Being Wrong

All week I've been watching my students' geogebra assignments progress by watching successive versions of them pop up in their epearl portfolios. Fascinating, truly fascinating to find out that I was wrong about so many things. Wrong about what I thought would be difficult and easy. And I'm trying to see these as gifts.

• I thought the easy part would be understanding what I wanted them to do, because, as I always tell people, I am so good at communicating. WRONG:
• A lot of kids interpreted "Create a geogebra file about the linear function that displays the graph and equation of any linear function y = ax + k for any values of a and k."  as "Draw one particular linear function, using whatever value of a and k that you feel like at the moment." What I wanted was sliders for the slope and the initial value. So maybe, just maybe, I should have said that in the first place.

Gift #1: A wake-up call. Get over yourself.

• I thought the hard part would be figuring out how to get the zero and y intercept to always be in the right place, regardless of which linear function is currently set by the sliders. WRONG:
• Once the sliders were in and working, many did this just by using the "Draw a point" button and placing a point right on the axis, no algebra needed. Which is not what I wanted.
• Darn geogebra is too nice - it assumes that when you put a point on the x-axis, that you always want it to stay there, even as the function changes. SO I had to edit the assignment description to say that the intercepts have to be done WITHOUT using a drawing button - use the input bar only.

Gift #2: A reminder that there's more than one way to skin a cat.

• That feels lame somehow, I mean if there's an easier way to do something, who wouldn't choose that? But too bad, there it is, this is an opportunity for them to gain conviction in algebraic formulas. Deal with it.

• I thought the easy part would be figuring out the formulas for the x and y intercepts of the linear function - in fact, I thought they'd already know them, considering these are gr 11 kids, and strong students, who have already studied the linear function for 2 years. WRONG:
• They didn't know those formulas, or they didn't remember them. So okay, fine, we spent some time solving equations like  a|x - h| + k = 0, so they could use the same method to solve the linear equivalent. Well no one could! It was no problem for them to solve 2x + 3 = 0, but it was another thing entirely when they had to treat the a and k as if they were known numbers.

Gift # 3: A surprise - I found a huge gaping hole in their algebraic toolboxes! Let the mending begin....

• I know that many people would say "Why get them to use formulas when it's more important that they understand and be able to figure it out from first principles?" But at this point, I think it's important for them to be able to generalize using algebra, and to use it to save time and cognitive load.
• Besides, if they have to derive and then type in Z = (-k/a, 0) for the zero, and then immediately see that it works, then they get to own that formula, and believe it. And lo and behold, once that happened, I got a lot of "Oh! Cool! It works!"  It seemed like the idea that algebra always tells you the truth was new to them!

• I thought that very few would try the bonus points, and I predicted who those few would be. I don't have the final versions yet, they're due tomorrow, but so far, RIGHT:
• One student put in almost all of the bonus features PLUS checkboxes
• One student wrote a text that contains, instead of inert letters, an object that changes with the sliders. I only just figured that one out last weekend.
• One student couldn't figure something out so she went online and read the geogebra manual! I wept when I read that in her reflections.
• The rest are doing the basic stuff, which is fine. It still feels like they're learning about the linear function in a whole new way.

I'll share their work and reflections here, once I get their permissions of course. For now, I plan to upload their work to geogebratube, or embed them right here, but once they have their own blogs going, they'll be doing all that themselves. Hmmmm.....I wonder if there are already any geogebras on geogebratube that came from students instead of teachers?

Life B.G. and A.G. - Before Geogebra and After Geogebra

I'm almost at the point where I see my teaching life divided into two eras: Before Geogebra and After Geogebra. It's been such a fascinating journey, and before it goes on, I need to document the major milestones thus far:

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.

A.G.:

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!

Followup on Adjusting my EFA Dials

This post is a followup to the lesson I foretold here.

How did it go? Great!

Here's how it all went down - outline here and observations after, of course colour-coded. I do that.

1. I had them open this geogebra while I screen-shared to the whole class, and we went through the questions/instructions (the ones embedded in the ggb file) together:
2. Once they were done with gasmin, then on their own, they did the same with: tosna, phyxyx, and drin, all the names of which came from this hilarious article. By now, they knew what each of those words really meant - addition, subtraction, multiplication, and division. 
3. Now it was time to play - they changed the blue and green function rules to see what function they ended up with.
4. Once they made a really cool-looking one, they uploaded it to this padlet wall:




Created with Padlet

Observations:

1:

• First question was answered very quickly, and unamimously: How is the red dot obtained from the blue and green dots? (Adding the blue y to the green y, keep x's the same.)
• Next question was almost unanimous: What type of curve is the red dot forming? (linear)
• Now I had hoped for a bit of discussion: Why is it linear? (constant slope did come up, maybe next time I'll ask, will a linear plus a linear always be a linear? What kind of functions would you have to add to get a quadratic?)

2:

• Using words like drin instead of the real ones was fun, but one student observed that at first those words intimidated her, so I'll have to make sure next time that it's clear at the outset they are nonsensical.

3. 

• This was definitely fun. I could tell because I only asked them to upload one, and many did more than that. I had trouble getting them to stop. Problems you want.
• Once someone figured out that the trig functions made waves, everybody jumped onto the band wagon. I'm not sure how or when, but I think they got the idea that the goal was to cover as much of the graph as possible. If it were possible to break geogebra, these functions would have done it.
• It just so happens that this week I am also having them start to learn how to use geogebra on their own, and during the course of doing this, I had lots of opportunities to reinforce some of the main ideas. It was a real hand-in-glove happening.

4.

• OMG I love padlet! No sign-in, just double-click and upload whatever you want. So easy. I think I only had to explain that to one person, it's just so intuitive.
• And look at it! I of course had to pick a background from my garden.

Now for the algebra segue:

With 5 minutes left, I wrote this on the board and asked what is the y coordinate for the red point? No problem, all knew to find the blue y and the green y and then add them. One student said something about 11x - 6. Where'd you get that from, I asked? And lo and behold it was perfectly explained that one could add the polynomials together, then use that to find the red y.

What's next:

Tonight they're watching the greatly reduced voicethread, and tomorrow we'll do the application problem. Last year, I remember having to work out the entire thing for them, so this year could not possibly be any worse. I'm pretty sure it'll be better!