## Thursday, December 19, 2013

### Digital Dust

This year has brought big changes for my classroom. It's the year I started getting my students to create their own geogebras, in an effort to deepen their learning and put them on the path of self-directed learning. I've had them use geogebra to explore various functions, and then I collaborated with our Physics teachers to have the kids create virtual projectiles. At first, I felt super excited about it all. And I still do, but now sometimes it feels like....so what? It feels like once they're done, that's it, the kids don't think about them or look at them or use them again. And if that's the case, was the work worth doing in the first place?

Moreover, am I just doing the digital version of those school projects my own kids did? The ones that took hours and tons of glue, clay, and papier mache to do, and which were displayed for a few days, then either immediately became landfill or collected dust in the house for a few years, and THEN became landfill. In either version, whatever learning happened seemed awfully short-lived.

But the last thing I want to happen is for all this year's fabulous student-created geogebras to start collecting digital dust. Not only because of the amount of work and real learning that went into creating them, but also because there's still so much learning that they can facilitate. I think the trick is to get them to USE their own work for deeper, self-directed, and self-powered learning. How to do that......

A few ideas I've come up with:

What's next for Physics:
• Idea #1: Once they've created their projectile projects, it might be time to, in the words of our Physics Guy Andy Ross, "take the engine apart." For example, investigate what happens to the components as time varies by moving the t-slider in this: (here's the actual ggb):
Maybe the teacher could demo this first, then have the kids unpack their own projectiles in the same way. It would illuminate some very tricky concepts that I know they have trouble with:
• that the horizontal velocity is constant (that pink text doesn't change as you move the t-slider)
• that it's gravity (ie the -4.9t²) that causes the vertical position to first increase, then decrease
• that gravity subtracts at first a little, then a lot as time goes on, from the vertical position
• that at the apex, the vertical velocity is 0, even though that puppy's still moving
• Idea #2: Have students use their own projectile ggbs to solve problems. As an experiment, I took a look at some of their assigned questions, and since I'm not familiar with Physics, I was looking at it kind of as a student. One of the questions was: "A metal ball is thrown horizontally at 44.4 m/s from a height of 2.2 m. What horizontal distance does it travel before hitting the ground?" In order to just picture the situation, I set it all up on my own ggb with those initial conditions:
 α = 0°, y1 = 2.2 m, v = 44.4 m/s
...then moved the t slider, and watched for the moment when the projectile's y-coordinate was 0:
At this point, I could just read the answer to the question, that the horizontal distance at that time is 28.86 m. But of course that's not the way we want them to get the answer! However, doing this directed my attention on the time variable, and then I saw that the time was the key - that once I knew that, I could find the horizontal distance. Learning enhancement opportunity provided by this ggb: It helped me own the problem - I understood what I had to do. I had to find the time needed for it to hit the ground, then use that to find horizontal position.
Of course, using ggb on the test isn't possible, but maybe using it before would make these questions clearer and easier to solve. Visualization is so important, and key to owning a problem.

What's next for Math:
• The awesome John Golden gave me this idea: Give them this, and instead of asking them to use it to find the solution set to an inequality, give them a solution set (ie x e -∞, 6] u [10, ∞) and ask them to find two different inequalities for which it is a solution.
• Along the same lines: once they've made their own function explorer, have them set the sliders so that a & b aren't 1, and h & k aren't 0. Then type in a rule in the input bar that is equivalent to it but with different parameters.
• Use the t slider to demonstrate domain. For example, for the square root function Why doesn't the point P show up until t reaches a certain value? Or why does P disappear when t reaches a certain value? What is that value? For the rational function, reinforce concepts like asymptotes - shouldn't the t-slider have a hole in it? (which is probably not possible yet in ggb. I had hoped that the t-slider would stop working when we hit an asymptote, but instead, it jumps over it. Wrong.)
• Use their formulas for zeros to talk about equivalent expressions, and simplified expressions. They all may have had it right, but they didn't all look the same.
What's next for me:

If I want them to use this tool to explore, I need to model it. So I'm going to start changing my "lessons" into something else altogether. They're going to look more like this from now on:

Introduction to Operations on Functions from Audrey McLaren on Vimeo.

Instead of using slides in my voicethreads/videos, I'm going to use geogebra, and instead of supplying my students with accompanying notes to fill in, they'll get the geogebra in the video to play with. This way I can model using the tool and enable the kind of exploration I want them to do.

As usual, any other suggestions or feedback are hysterically welcome!

## Monday, December 2, 2013

### Projectile Projects!

Here are the projects! I and my fellow LearnQuebec teachers, Kerry Cule and Andy Ross, have just received these joint physics-math projects.

I am beside myself. Speechless, which doesn't happen a lot. Such variety - sports, nursery rhymes, video games, abstract art, pirate ships. And such creativity!

Each caption is a link that takes you to the html5 version at geogebratube.org. They all work, so have fun!

Enjoy!

 AB - hockey

 AG - baseball

 AGR - Nursery rhymes
 BC - rocket

 CP - soccer

 CC - pacman
 JM - CoD

 KR - abstract art

 KLD - catch the ball
 LF - Jim's gym class
 RM - volleyball

 SC - soccer

 TC - archery

 VM - football

 ZB - pirate ship

 This is me now: Projectile tears of joy!

An embarrassment of riches:

Many questions and conversations arose in the course of this project. Some I plan to bring up to the whole class. And all came from the kids' own individual problems that they encountered along the way to trying to get their projectiles to fly. Some of these are ideas that I'm sure all physics teachers try to get across every time they do the unit, but I also think that some of these probably never would have even come up without this project as a backdrop. Riches beyond imagination!

"My formulas are right, but my projectile won't fly!" Launch velocity has a threshold: You need a certain velocity to overcome gravity - if the greatest velocity your slider allowed was 12, your projectile won't go up practically at all, because gravity overcomes it almost immediately: eg at 2 secs y = 12 (2) sin 50 - 4.9 (2)² = -1.2 m

Size of projectile: One student had a projectile whose diameter could be varied. Will that affect the path?

Value of g: One student had a slider for g. When would we need to vary the value of g? And should it be 9.8 m/sec² over the moon?

Dimensional analysis: What are the units of all the variables in the sliders? What must be the units along the axes?

Realistic values for variables - Why allow negative velocity, initial position, or time? Do they make any sense for your situation? Under what circumstances would those make sense?

Frame of reference - Not every student used the first quadrant as the location of their situation. Is that ok?

Angle - some allowed their angle to go up to 360°. Does that make any sense for this context? Under what circumstances would it make sense?

Other stuff: One student's projectile was an arrow, and one was a rocket. Super bonus: How to get them to move realistically along their flight pats? eg it begins with the tip pointing up and other end down, then they slowly reverse. Two different points joined by a segment? And what would be the difference between the coordinates of those two points? One has an angle that's the other one delayed, by a phase shift perhaps?

Math stuff: A couple of students defined the position of their projectile as the intersection between two lines - the vertical line x = horizontal position of the projectile at time t, and the horizontal line y = vertical position of the projectile at time t. Mathematically sound. Works. Never thought of that. Mind blown.

Next post will have student reflections about these projects - more riches. Any feedback would be hysterically appreciated, especially by these hard-working rocket scientists, literally!

## Thursday, November 28, 2013

### More Student-Created Geogebras - and some pushback

In this post:
• The ggb assignments so far
• The benefits of student-created geogebras (and the evidence)
• Student feedback
The assignments so far:

I've spent so much time trying to write this post without making it ridiculously long, so I gave up and did part of it in a video! This shows you two things at once - what they had to do, and what they actually did:

The benefits, and the evidence:

First the benefits associated with the specific tasks, in other words, the benefits I foresaw. I've also included the kids' actual word-for-word reflections (in colour), which I think provide evidence of great learning:
• Formulas:
• Finding them: Benefit: To find those formulas, they had to move up a step on the ladder of abstraction. They were manipulating equations with no actual numbers in them. Up until now, someone else has done this for them.
• Today I figured out the coordinates for my zero. I also edited my y-int and zeros conditions.
• I feel so proud of myself for figuring out the rule of the zero on my own. I dont know why but that was something i was having a hard time with and i got it!
• Entering them properly: Benefit: To get them to work, they had to be very careful about where to put brackets, about using only variables that were already defined, and of course to not make any typos.
• I had trouble with the y-intercept and the zero. Turns out, in both cases, I wasn't putting the brackets at the good place!
• OH MY GOSH!!!! It works!!! FINALLY!! Okay, so my mistake was sillly, I had written my P like this: P = (t, a sqrt(b(x-h)))+k I had put my last bracket in the wrong place. It is now: P = (t, asqrt(b(x-h))+k).
• With today's class though, I was able to know what to do and come with this product! What bothers me with this one though, is that the y-intercept doesn't seem right. The rule is what we usually use I guess, but the number geogebra gives me in the text doesn't seem correct! I'll try to find what the simplified rule is and write that instead of the big thing ( y= a*sqrt(b(x-h))+k )

• Conditions:
• Finding them: Benefit: This was very challenging for everyone. They've never had to do this before - systematically list all the possibilities for a certain math situation.  I used to give out all this information for free, but no more. This stretched their algebra minds, no doubt.
• .I have to say one of the toughest part was figuring out when my text box should appear or not appear. It can get very complicated. I noticed though that when the theres a y intercept text box need's to show up it's the same as when a theres a zero textbox; the only difference being the parameters used.
• I made domain, range, function is increasing/decreasing, y int, and now I am working on the zeros but I am having trouble figuring out what to write as conditions for zeros to appear.
• The longest part about the texts was actually finding WHAT the conditions were. This is why making students use geogebra helps them understand how the function actually works and it's like having animated notes you can use to study with.
• I am SO SO SO SO exited with this last version:) I have my increasing and decreasing in however i also made the function change colour according to whether or not its decreasing! i did look at the vt for help but i must say that i did learn a few things on my own! like at first i accidentally made the line only appear when it was decreasing so i needed to play around with it until i was able to make it change for both:) so exited!
• Entering them properly: Benefit: Logic! Boolean operators! Truth! I never even got to teach this before!
• I was really happy to have figured out how to get my conditions right for my text that I did a happy dance in my head. So I figured that a>0, b>0 and a<0,b<0 means that it will increase and if a<0,b>0 and a>0, b<0 it will decrease. I was really happy.
• So for the y textbox in this example it was (h≥0&&b≤0)||(h≤0&&b≥0). The zero text box was the same thing except with k and a.

• Expression for the moving point P
• Finding it: Another step up the ladder of abstraction. That t-slider may well be their first opportunity to see a letter as a number whose value is varying, because that's literally what's happening as they move that beautiful little dot along. And that's not the same thing as a variable. Some letters are more variable than others.  Not to mention that that moving point P was really a preparation for the next big project for physics about projectiles.
• Because of our last ggb assig nment i was able to figure out that point P was x as t and then the entire rule represented y (t as x).
•  i had no problem with entering the rule and the 4 parameters but when i entered point P with t-slider the point P does not always stay on that functions line segment, im not sure what could be the problem. So far everything seems to be simple since we have done similar to this in the linear function except slider-t.
• Entering it properly: Benefit: Again, being careful with the brackets, and also getting the big picture after 3 or 4 functions.
• like i had said i have no problems with the parameters but i did have a problem with correctly adding point P with slider t and making it follow the line but with help from you we have noticed that it was tiny mistake in the way i had written my rule when i was entering the coordinates of the point P

Other benefits that I honestly wasn't even expecting:
• Doing one function would have been good, but doing several has allowed them to get a bigger picture. Some used their older assignments to figure out what to do in the new one, and saw patterns emerging.
• Engagement: There are some students who are definitely more engaged doing this than anything else. Some students had 10 or 11 versions before they finished. It's addictive. Even when it's not working. Especially when it's not working.
• Each time they do something they can check it right away. And that involves action, ie moving a slider, which then causes another action ie a colour change - it's like watching a movie!
• Opportunities now to talk about why different formulas work ie |t - h| and |h - t| give same result
• Opportunities for eloquence - eg they all input this for their formula for the x intercept of the square root function:$\left&space;(&space;\left&space;(&space;\frac{-k}{a}&space;\right&space;)^{2}\div&space;b+h,0&space;\right&space;)$, which is completely correct, but not very pretty. I get to ask them which formula they'd rather type in, the one above, or this one:$\left&space;(&space;\frac{k^{2}}{a^{2}b}+h,0&space;\right&space;)$, and by the way this is why we simplify algebraic expressions.
• Opportunities to talk about presentation - lining up your sliders so they're nice and neat, colour coding, using checkboxes to not crowd the screen
And here's more general feedback:

This was a good practice to learn about square root functions. When using geogebra, it gives a more indepth explination of how the function works by letting me explore its movement.

This was a great way to learn. I was able to see how the parameters affect each other and how it affects the function.
I feel that these assignments are really helpful for making sure we understand the concept. It allows us to put what we learned in class and in voice threads to use in a very creative way. I can't wait to see what else we will be doing with geogebra.
I am happy to be done and i have realised that this geogebra was similar to the other one except that there were more conditions to show objects and more rules about certain intercepts or zeros that we needed to enter. But by doing all these things i have learned more about the square root function and my understanding has been increased about what happens to the function when certain parameters are either negative or positive and so on.
i'm really happy i finished it on time ( i have an english response that has been due for a week, so you should be quite happy, i'm not very good with homework)

Pushback:

Finally: I had a bit of pushback. A group of students asked to meet with me to talk about all the geogebra they've been doing. Their concerns were that they were getting too dependent on the software and were not developing their algebra skills. They pointed out that they can't use it on their tests, so they feel unprepared for them. Really important input! Then another student approached me and mentioned that they would rather have more notes and less geogebra. Part of me says "Listen to your students, they're your eyes on the ground" and the other part says "They're not used to working this hard or doing this much independent thinking, let them get used to it."

Well, for the time being, they've got one more geogebra project to do, and that's the big one, the one I've been thinking about since last spring - the physics/math virtual manipulative project. After that, I'll give them a geoge-break. But till then, I'm full steam ahead, because there's just too much great stuff happening that's telling me this is all worth it.

## Tuesday, November 19, 2013

### Twitter is the New Staff Room

I used to work in a school that had about forty teachers. Forty teachers, that is, and one staff room. Which meant that teachers from every department shared one big, giant, open room. Now, many years after I left, I realize that, as it turns out, we shared a lot more than that. Today, I find myself missing it, and looking for a close substitute in, of all places, Twitter.

The Great Big Staff Room

Back then, the fact that there were no walls between our desks made it easy for us to know each other, at least on some level. Whether or not we taught the same subject, or even ever had a full conversation, we eventually gained a sense of the people with whom we worked. That happened a little bit everyday, in the bits and pieces that we caught in walking past someone's desk and saying good morning, or happening upon a juicy conversation amongst a gaggle of teachers, or noticing someone's new haircut. It happened whenever we watched each others' reactions to the unintentionally hilarious and exasperating intercom announcements. It happened when someone needed to vent, or to share good news, or bad news, or when something truly dreadful was happening in the world, like 9/11. That staff room gave us, over time, a sense that we were all in this together, whatever "this" was at the time.

Not everyone liked it that way. Some people found the noise made it hard to concentrate, and others felt that it was a place where everyone just whined and complained. Not me. I loved that staff room. I honestly looked forward to walking in there every morning. I didn't love everyone in it, and I'm sure there were people who didn't love me, but I loved the feeling of being a part of something. (And, okay, there were people there whom I loved.)

As teachers, we also shared our craft, and also in a gradual, organic way. My department head and mentor, Maureen Moore, had a rich vocabulary and no-nonsense approach to teaching that definitely left a mark on me over the years. She also supported me and all my ideas so enthusiastically that I couldn't help but grow in confidence as a teacher working next to her. Another huge influence for me was a young teacher named Christie Brown, whose brilliant innovation and early adopting of ed tech is responsible, I think, in large part, for my love of ed tech. And of course, there was Armand Laderoute, a retired principal who replaced me when I was on maternity leave and just never left. He was a master teacher AND impersonator of other people. A nod of the head from him and you knew you were doing something right.

But I don't remember a heck of a lot of intentional subject-specific or cross-curricular collaboration happening, although I'm sure it did, probably more so after I left and the QEP came into effect. Mostly I remember people working together to plan staff events or school events. Certainly the potential for collaboration was there in buckets, because you only had to walk over to someone's desk to get it going. Now that I look back, it seems like I missed out on a huge potential for co-teaching.

It wasn't all about the room...

The subtle yet abiding team spirit in that staff room didn't only come from the lack of walls. It also came from the specific combination of people who inhabited it. I know for certain that it came from those people, because when the people changed, the cohesion changed. As it turned out, not only had we shared a room, but we also shared a common affinity for working together, and including as many people as possible in that endeavour.

Well that staff room's long gone, and anyway, now I work online, so there's no actual staff room for me to bask in. And I am privileged to work with a staff of the most inspiring and supportive educators in Canada, but we almost never see each other. So fortunately, there's Twitter. Dear merciful heaven, there's Twitter.

On Twitter, I get to choose who's in my staff room. I can happen upon juicy conversations. I can hear peoples' reactions to things. I can get a sense of people based on their tweets, who's in their staff room, and how they respond, or don't, to me. I vent, share, or listen to others who need to. I get to be there for a friend who needs help carrying a burden.

I laugh at jokes (oh my goodness, so much laughing, just #saidnoteacherever and #overlyhonestmethods alone are enough).

I have people whom I could call mentors, but it's probably more accurate to say they inspire me, and not all in a math teacher way - some for teaching in general, pure and astonishing math skills, communication, ed tech, some just for how they interact with and help others tirelessly.

And now, despite the fact that we're not even in the same country, let alone the same room, I'm collaborating with all kinds of people all the time, and again, not always just about math (okay usually). But #bettertogether is my favourite hashtag, because it speaks to what I think I always believed, even way back in that big room.

And the learning, dear heaven, the learning. That's what I truly love, that I get to keep learning, and alongside other people who love to learn, which, as a teacher, is, I feel, is the first and best thing to live out in full view of my students. I don't mind if they forget the math they learned in my class, but I do so want them to follow their inspirations, do what they love, and share it with the world, so that's what I do. On Twitter.

And there are even some people on Twitter whom I love.

So even though I have fond memories of The Great Big Staff Room, and the people in it, for the time being I'll settle happily into my own little corner of the Twitter Sphere, with my pot of tea, my webcam, and my tweeps. All I'm missing is those intercom announcements. That's another blog post.

## Thursday, November 7, 2013

### Followup of Student-Created Geogebras

My kids are about to submit their second geogebra assignments, which are on the square root function, but I haven't even gotten around to writing about their first ones, which were on the linear function, so here goes:

First assignments: The Linear Function

This was the assignment, and here are a few final versions, with only student initials as identifiers:

AB's linearKR's linearCC's linearKR's animated linear,

As you can see there are variations in the presentation. Some use colours, some use text to guide the person using the ggb. Even with the same set of instructions, there was room in this assignment for individuality - and the animated one was just spectacular, especially to watch a quadratic function slowly forming from the trace of the linear one! I'm sure there's something cool to be done with that....later.

I gave a lot of opportunities for bonus points, the most time-demanding of which was to do the entire assignment again for a totally different function, either the quadratic the absolute value. About a third of my kids did it! Here are a few of those:

BC's quadraticBC's absolute value,  KR's animated absolute value

You're looking at the final versions, but equally fascinating and exciting to see were the various incarnations of each student's work as it evolved, as well as their reflections along the way. This was possible due to the digital portfolio tool I'm using, called Epearl.

Reflections:
Here are a few reflections (I know, there's a lot, believe me this is just a tiny fraction) - word for word, including spelling mistakes. I've highlighted things I found particularly interesting. Also, know that I'm not only showing the positives, although the overwhelming majority were. And a change in colour means a different student's voice.

I am hardly motivated at all to do this assignment; geogabra does not help me learn and only confuses me.

"The Absolute Value Function" is uploaded!!! It's amazing how perseverance pays off!!! After hours, literally, of trying how to make the zeroes and the initial value show up proprely on my absolute value function it works. The feeling you get after it finally works is unbelievable. The excitement to see it work proprely is amazing.

Overall, I enjoyed the assignment. It had its ups and downs, but in the end, it got done. I liked working with it because it wasn't too difficult and gave me a better vision when working with functions (oh, Geogebra). Projects like these are a great learning experience while also giving students creative control, which makes learning easier.

I have to admit, Geogebra is pretty cool....

I don't think it's helping me much with understanding this, as i already don't really understand much of what to do ......i haven't figured out how to add the t into the equation to have it move along my linear line....Figuring this whole thing out is pretty frustrating because i'm not sure how to add the t into the equation and how to make it move...Well now that i sort of figured out how to use the sliders, they work! and the whole project id sort of coming together. ...I did have troubles with this project... Learning to user geogebra to do this was frustrating me a lot. But i did learn from this.

Today i finally figured out what the y-intercept was! Its very simple after you realise what it really is, simply the k! i figured it out by accidentally putting in the formula (0,-k) and realized that was a reflection of the actual y-intercept and realised it will have to be (0,k). Feeling progressive! :)...

Throughout the whole process of making this geogebra I was able to get better at using the sliders. This was demanding work, but once you know what to do its easy!

I messed around with things and added animations to them and I got point P to leave a trace (: I'm happy to have figured out these things on my own....I'm finaly done, with 10 minutes untill the due date lol.

I kind of understand abs val functions more, but i deffinalty understand ggb more. It really took me until i had to compare my linear functiona nd my abs val function ggb's to understand all the formulas and for the most part i undersnat them now.

I am not really a person that is good in technology but I was proud of me when I was able to complete this assignment . This is probably not the prettiest/ funnest/ coolest geogebra in the world but I still like it because it made me discover a little bit of technology but also I learned a lot on the math involved. Looking at my function change direction while I was playing with the sliders made me see a whole new image of how this function is workingI may not really be good with geogebras but actually working with this program is a big help me understand better the math I am learning.

It was difficult figuring out how to get the point to move along the line but i looked at the
online manual of geogebra and figured it out

i am happy with myself for not slacking off as much as i usually would, i admit i was lazy
by not doing the last thing in the bonus section, but i am still impressed with my performance.

I learned a few things about geogebra throughout this assignment at first i was only doing the geogebra for one function but then with help from you i put in the general rule and from there if i had any problems i just looked at the geogebra file in sakai on important points of an absolute value function.  It was a great project and i am excited to learn more about geogebra because it can help so much.

In my 3rd geogebra you can see that I animated both my a and k slider. I really love
that it changes it and can show how it changes by itself. I'm pretty sure it would look really
cool to show how something changes over time. The speed was at 1 at the begenning.
I found that to fast so I changed it to 0.1. I went to obeject properties for that specific
slider, went to the option slider and changed the speed (Refer to the screen shot's I
attached below).

I figured out so far how to make the colour of the line change depending if the slope (a)
is either positivenegative, or equals 0When I had finally gotten it right I was so happy
I screamed. I think I might of scared my mom.

I'm almost done. Yay! I only have the little details left. Making everything pretty!

Wow. That's what I have to say. It's a lot of work, but a lot of fun. Challenging.
It makes your brain work to figure everything out and that's great. I have always
loved anything that put my brain to work, even if that meant that I got really
fustrated. In the end though it's worth it.

It basically came down to the fact that I was sattisfied with my work personnally
and just felt that I had tripled checke deverything enough times. I felt peaceful. I'm
not even joking. It was like everything was perfect.

When I would finish part of the activity I would be happy it worked, this made the
activity extremely satisfying and even fun. Sometimes just stepping away from my
computer and coming back would help me see my problem and then I would be
able to solve it without too much difficulty.

This was amazing. I feel like I now know a lot more about geogebra. It's a great way of
learning. I didn't think that it would be a style of learning that I would like, but I loved it.
Even though the goal was to make a linear function with many things, I learned so much
about geogabra while I was doing my assignment. I hope that we get to do something like this
again. It's fun, challenging, and creative. Emphesis on creative. Anything that is creative
I will fall in love with. I thought I was more orriented towards drawing on paper and things
like that, but being creative technology wise (using a program like geogebra) is absolutely
amazing. I didn't only learn about math and geogebra I also weirdly learned about myself
as I stated above.The universe has weird way's to teach us things about ourselves
This was absolutly AMAZING!

What I see:

• Actual enthusiasm, not only from the kids who are always enthusiastic, and not only from the kids who have an easy time with math.
• The sheer variety of what they learned - be it tech or math - is mind-boggling.
• How many things they learned by accident, or just by playing around. Reminds me of me.
• Conviction in algebraic formulas - not only getting them right, but being able to believe they tell you the truth.
• Sophistication - seeing that some letters represent a variable and others represent a known value, and solving an equation that has no actual numbers in it, only variables and parameters.
• Soft skill development - walking away to get persepctive, organizing, taking initiative, persistance, using past successes to move ahead, metacognition.
• Resourcefulness - if you don't know something look it up online, or look at another example
• Student PLN's - one student found Steve Phelps' online geogebra manual and told me about how wonderful and helpful it was! I got to tell her that I ACTUALLY MET HIM! So now she and I are officially learning from the same person - how's that for a PLN?
Nuts and bolts:
• This takes them a lot of time. And I haven't exactly removed anything from the list of things I gave my students to do last year, except I've backed way off on the weekly checklists. The in-class activities have been replaced with ones that revolve around the geogebra assignment, as opposed to whatever voicethread they're currently supposed to listen to. More about that in a future post.
• This takes me a lot of time - I have been giving feedback on each and every version, so that by the time it's due, they can decide if it's done, not me. But Epearl makes it really easy to do that - it's like an interactive dropbox.
• Epearl also makes it really easy for them to share things with other members of the class, but I didn't have them do that for this assignment. Because I didn't know how to. Next time.
• I felt I had to give them a mark for this, because they worked so hard on it.
• I got the distinct impression that some of them would have done this for no marks at all.
What I wonder:
• What are the advantages of this? I mean really? I know it looks great and it feels great, but what's happening that I wouldn't happen without something like this?
• Are they doing deeper learning, or just different tasks superficially?
• How can I get them to make use of these tools that they have created? If they never look at them again, it's pointless. Or is it? Was the learning they did to make it worthwhile on its own?
On to the square root function geogebras, due tomorrow! Some were already in yesterday!

## Saturday, October 19, 2013

### 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?

## Wednesday, October 16, 2013

### 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!

## Tuesday, October 1, 2013

### 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.

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!

## Sunday, September 29, 2013

If you are one of my current students STOP READING NOW! You will ruin the surprise for this week.

Maybe I'm just flattering myself, but some students have read my stuff.....okay, now that they're gone:

I am continuing in my quest to use Ramsay Musallam's EFA in my class. I looked at last year's voicethread about operations on functions and once I was able to get a grip, for it was so very bad, I asked myself two things:

1. Which part of this can I move out of the direct instruction and into the class activity?
2. What activity would do a better job than that part did?

So this kind of thing, in which I tell the kids the questions AND the answers, is out:

And replaced with this geogebra:....in which THEY will tell ME that the functions are being added. Notice the title of the file, gasmin of functions, gives nothing away - I used one of those 27 new trig function names instead of calling it "Adding functions". They're going to watch the coordinates of the red, blue, and green points as the vertical line moves around, and THEY'RE going to say "Oh, you're adding the y's." That simple concept, that the y's get added when functions get added, is something that almost all of my students-of-the-past have missed out on, due to the crappy way I've taught it.

Next they'll predict what sort of function the red dot is forming, after which they'll check by clicking on the checkbox called "Resulting curve". Then we'll talk about why the sum of two linear functions is also linear. Maybe we'll get into the algebra at this point, we'll see how it goes, but I'm really just looking for some number-intuition here. I hid the algebra view for a reason.

Bottom line, instead of me saying "We're going to add functions together and see what happens" this will be more like "What's happening here and why?"

Dialing up the explore:

Then I'll have them play around with the actual functions involved - what happens when you gasmin a linear with a quadratic, a quadratic with a quadratic etc.

Next, I've got a whole suite of geogebras with cryptic names to follow: tosna (subtraction), phyxyx (product), and drin (quotient). I'm sure they'll catch on and already know the next operation as we go, but at that point we can shift the focus onto the wonderful functions that you can form by doing simple operations to relatively normal run-of-the-mill functions. I'll let them play around with the functions and the operations for a bit, and I really want this to be fun for them, in the doing and in the sharing, so I'll have them share their creations to a gdoc so we can all see what sort of wonderful weirdness results. Who knows, with any luck, one of them will come up with a function we'll be studying later on this year, and we shall christen it with that kid's name. Find the domain and range of the function "Susie of x".

Dialing down the flip:

This means the original voicethread will lose about 21 slides. What's wrong with me that I thought that was a good idea?!?!?.

Most of what's left is about notation and algebra - specifically the algebraic way to do an operation on two functions, which is to do the operation on their algebraic rules. For example:

if f(x) = 2x + 3 and g(x) = x²
then (f +g)(x) = (2x + 3) + (x²)
= x² + 2x + 3

Which brings me to the next improvement I need to figure out - how to get across this important idea:

(f + g)(x) = f(x) + g(x)

in other words, using the previous example

(f + g)(2) = 2² + 2(2) + 3, which comes to 11
and also f(2) + g(2) = 7 + 4, which also comes to 11

But I don't want to give them this kind of step-by-step, see-if-you-can-guess-the-point sort of way. I'd like it to be one of those cleverly orchestrated things in which they stumble onto it while they're doing something else....and I have to think of something before I give the voicethread to them to watch....not sure if I'll have time. Or an idea. But ideally, I want the voicethread to really be about pinning things down, but only if those things have already started to occupy their brains and annoy them.

 This is me now.

I hope I can think of something before the day of, which is Tuesday.

Dialing up the apply:

I may give the activity from the text, which I used last year but which that group found really hard. It involves Reginald, who has investments in two companies, company A and company B. The number of shares he has in each company is given as the rules NA = 400 - 4t, NB = 1000 + 7t. There's a table to fill in, with the column headings time, NA, NB, and total number of shares.

Isn't this the most contrived, boring application ever?!?!

But at least it doesn't tell them to add, they intuitively do that, and it then invites them to find the totals both ways - first by adding the numbers in the columns, and then by using the algebraically simplified sum of the rules, to see that it comes to the same thing.

The next part uses the same story and sequence of activities to get them to multiply two functions, company A's shares (NA = 400 - 4t) by the value of those shares, which is given as v = 0.125t + 6.25. They fill in a table, get the values by multiplying the numbers in the table, then do it using algebra, and compare.

Perfect or Tuesday?

I don't know, this application just seems dumb. That's it, that's my professional assessment. But even if it's not perfect, do I need it to be perfect, or do I need it for Tuesday? I'm betting Tuesday will, once again, win out. At least there has been some improvement over last year, and I'm going to improve the one on composite functions the same way, in fact, I've already made the geogebra for that too.