## Monday, February 18, 2013

### A week without homework. Sort of.

Don't know where I'm going with this, all I know is, I am dazzled by the idea of getting enough real learning to happen in class that there is little to no need for mindless practice questions from the textbook or worksheets. I want as much as possible to happen in the 50 minutes we spend together. All I ask them to do outside of class is a blog post every single day this week. It can be a summary of what we did in class, what they learned in class, or it can be a summary of what they watched in the voicethread. Oh yes, there are also voicethreads BUT they are there mainly as a backup. If I don't get through all the class activities, or if I do but there's someone out there who needs to take another look, or insert a question, it's there for them. Is that cheating? Maybe it is. That's for next time. Baby steps.

Why the daily blog posts?

Three reasons:

First so that they can digest the information, process, and organize it in a way that makes sense to them.

Second, to get them actively learning. They'll be active in class, but I still think they have to do something on their own to pin down the concepts. But this way it's not just busy work, it's actual construction of their own knowledge, and not in a vacuum, because they can all check out each other's blogs. By now, everyone knows whose work to check out first! I'm hoping that the creativity that I experience when I'm writing happens for them, the urge to embed, link, colour, or otherwise clarify an idea takes over.

The third reason is that I want to give everyone another assessment option besides tests. I first tried this a few weeks ago, right after we had finished a unit called Optimization, which is really short, really simple, and in which most kids usually do very well. Some kids were disappointed in their test mark, though, so I gave them the option of doing a blog post on the chapter. The deal was, I'll only record the better of the two marks. I gave them very specific guidelines for it, which I put up on the classblog. and which I used to create this grid:

Seven students did it, and out of those, 5 got a better mark as a result.  Here are five of the posts. The results were pretty encouraging for many reasons, not the least of which was the quality of the work, and the enthusiasm they had for doing this type of assessment over tests. I asked for some informal feedback afterwards, and all those who answered felt that it helped them not only to learn the concepts and get the big picture, but it was a calmer, more authentic way to demonstrate their understanding.

The log blogs that didn't happen:

Well, the next unit was logarithms, which is a bloodbath every year, and as we reached the end, just before the test, I offered the option of blog posting again, but it was way too late. This unit was huge, about 4 or 5 times bigger than optimization, and it's way, way harder. No takers this time. No surprise. But I felt I had really dropped the ball for them. What if I had had them blogging all along?

Enter trigonometric functions:

So for trig, I'm getting them started right now. I'm thinking that if they start now, as the days go on, they will not only keep track of the daily learning, but they will refer to their previous posts, compare their's to other peoples', see connections, and deepen their understanding as they go. Hopefully, by the time I get to the test, those that want to can opt to do a final post that authentically demonstrates their understanding - although I'll have to think about the guidelines for that. It'll have to involve some sort of new problem to be solved, like the optimization one was (which by the way, everyone had a different one to solve). Just got an idea. Involving geogebra. More later.

Day one: Today

Today's activity was about converting between degrees and radians, as well as coterminal angles on the unit circle. I just had them drawing angles on the unit circle, and doing conversions all together. It's Monday, so nothing fancy. Hm, I wonder how many have blogged already....let me check my google reader feed....ooooh three so far - here, here, and here!

Note: I have also posted this at The Flipped Learning Journal!

## Wednesday, February 13, 2013

### Follow up on new trig activity

Yesterday I posted this about a new introductory activity for trig functions, so here's what happened:

Despite a few technology hiccups, things went well, evidenced by the few graphs on the gdoc as of now. Hopefully by tomorrow there will be more.

Observations: (Followed by deeper thoughts):
• Thank goodness I had them print up the blank graphs just in case, because that turned out to be the only reason we were able to proceed when there were tech problems. Always have a Plan B that doesn't involve tech.
• Not everyone was able to open the ggb file, (miss I don't have java) but every group had at least one person who could. Always group when you're using a fancy-schmancy tool.
• Needed to walk through it a bit at first, emphasizing that the graph they are building is of time vs height, not of the little green ordered pairs that follow the car around in the ggb file. Did I miss an opportunity to do another hint-troduction?
• In the last class, I started out by getting them to mark the various times out on the circle - where would 15 seconds be....etc. But I feel like that was sort of spoon-feeding. Although this group needs more pushing, plus NO ONE could open geogebra, so I had to screen share it and move the car around for them. I forgive myself.
• Didn't get to the second part (distance from wall) in most cases. That's fine, they can do that on their own tonight. GACK - Homework.
• In one class, no one used the checkbox that revealed the angles, probably because no one got that far, but also because this group very early on decided that it would be linear. And none of the points they graphed disproved that - they didn't have enough of them to see the curving part. But that's okay, part of learning what something is, is learning what it isn't.
• In the other class, everyone ended up clicking the checkbox to reveal the angle lines. Got the thrill of my life when I saw "OHHHH!! THAT HELPS A LOT!"
• Nevertheless, I don't think anyone saw the angle as being significant - to them it was more about cutting the circle up into sectors. But maybe once they sleep on it, that will be more obvious. Maybe I need to change that ggb so that instead of sector lines popping up, it's an arm that rotates, and they make it rotate via a slider. And it changes colour when it hits 30, 45, 60, etc. Hmm. I wonder how to do that. Great. How long is THAT going to take me to do?
• Most of them predicted that the curve would be linear, or piece-wise linear, or an absolute value function. I heard some talk of quadratic too. I neither confirmed nor denied. It's not about getting the right answer at this point, it's about seeing the circular motion, and about how they should check out their assumptions, and how a few points can make all the difference.
Questions that I hoped would happen and did:
• Do we have to find the height for all the times you have in the table?
• Aren't the four compass points enough?
• Will it be a line?
• How can we find 5 seconds on the wheel?
• Why did you give us such a wide graph?
• Is there a rule for this type of graph?
Comments that I hoped would happen and did:
• It's just going to repeat
• It looks like hills
• It looks like teeth
• This was fun!
Tomorrow:

We're going to look at all the graphs together. At this moment there is really only one that is correct and complete for the first cycle. They are going to tell me which one(s) are right. Then and only then will I show them this! And I will bring up angle as the quantity that interlocks with seconds. Then we'll start talking about the periodicity, etc, all those things I thought we'd get to today.

## Tuesday, February 12, 2013

### New intro to trig functions

Big Trig

Just a quick post before I try this out tomorrow. Just starting the big trig unit, and I hate the way I've been doing it all these years, so I'm going to try it differently, in reverse order almost, by starting with something that I used to save for the end. At the same time, I want to weave an important concept into the mix right at the get-go, and that is the relationship between time and angle.

The activity:

I'm giving them this virtual ferris wheel, and this pdf to fill in.

They will move the little green car around the ferris wheel, and geogebra will give them its height above the ground as they do so. I expect they will make it go to the easiest places at first, that is, the points of the compass, which correspond to 15 seconds, 30 seconds, and 45 seconds, where it's easy to get the car in the right place just by eyeballing it. They'll be able to fill in the table with these ordered pairs pretty fast:
(0, 1), (15, 6), (30, 11), (45, 6), and (60, 1).

My evil plan:

But then they will have to fill in the times in between, like 5 seconds, 35 seconds, and what I'm hoping is that someone will at some point ask me, "Mrs., wouldn't it be easier if we could see the angles around the circle?'

"WHAT?!?!?" I shall say. "Well, if you insist, I suppose there's no harm in it. Click on the tiny little checkbox at the lower right corner, if you really think that knowing where the angles are will help...."

This is what they will then see:

Now they will know exactly where to put the little green car for all those times in between North, South, East, and West.

And BOOM! They will, for the rest of this unit, know that time is glued to angle. I hope.

I'd like them to share and compare their graphs, so I've made a gdoc for them to upload to. Hopefully by this time tomorrow, there will be a few there.

I will also get them to talk about other things they notice about the function, like the periodicity, the symmetry, the total absence of an asymptote, the restricted range, whatever!

I already made this for the answers, but I had so much fun doing it, maybe I should make them do it themselves, at the end, when they'll know a thing or two about trig functions.

Ideas for the future:

I'm also thinking that as I add new parameters to the basic functions, I'll keep coming back to this and changing, for example, the radius of the wheel, or the speed at which it's turning, or the position at which the car begins. If you can think of anything else, please feel free. I'm sure someone else has already thought of this and done it for years, but oh well, better late than never. Fingers crossed!

## Sunday, February 3, 2013

### Jump-starting collaboration

This year, about 3 years into flipping my math class, one thought that has consumed me is - What would a student miss out on by not coming to my class? What advantage is there to meeting with me and the other students everyday, why not just watch the recorded lessons, do the assigned work, write the tests, collect the mark, and be on their way? The answer has to be more than just that they would miss out on getting help. True. But there are kids who would do fine even without that. And I want there to be more of an advantage to coming to class than the individual tutoring.

This week, as a result of becoming part of the infamous CoFlipCollaborative at the Flipped Learning Journal, one answer has come to me loud and clear. Collaboration! I have been reading and hearing so much about how teachers planning their lessons together gives much better results than any one person could have achieved alone. I haven't yet collaborated like that, but I at least wanted my students to.

But you can't get people to collaborate just by telling them to. I know this for a fact, because I've tried to get my kids to do it. There are too many obstacles to it happening that way: shyness, lack of confidence, overconfidence, mismatched abilities, to name just a few. But in my case, it was also poorly chosen, ill-defined tasks. (Eg "Hey kids, in your groups, write down everything you can about....) I think this week I finally found a good balance between introducing a task, setting the stage, and letting them run with the ball, so that my students would be inclined AND able to learn and create something together.

Now then, people:

There will be math in this post  WAIT COME BACK, IT'S NOT ABOUT THE MATH I PROMISE!! I want to show you the collaboration, conversation, socializing, evolution, and discovery that happened, and all without me being involved! These are things that I think any teacher can appreciate. But if you'd really rather not hear the math part, then ok, geez, it's a shame, but skip the video and go straight to the slideshow. Math people, just so you know, they had already spent a couple of weeks getting familiar with logarithms:

I really think this was the clincher. It was just enough math info and just enough of the big ideas so that everyone would have a place to jump in once they were in their groups:

I didn't actually tell them what they'd be doing, not a word about that, until after this gentle hint-troduction. I liken this to giving someone a running start when they're learning to ride a bicycle - here you go, wheeee, isn't this easy - never mind where you're going, just keep your balance yay you're doing it! .

Setting the stage:
I then put them into groups of 2 or 3 and told them to see what they could figure out about a brand new function, one they'd never seen before, that is, $y = log_{2}x$. That's it. I didn't even say Work together! Talk! Write stuff!

Running with the ball:
What you'll see in this slideshow is kind of a time-lapse sequence. I took snapshots of what emerged in each group, along with the conversations that took place at the same time - which were easy to capture, thanks to the kids using the live chat. As I said, even if you're not a math person, I think you will be able to see elements of collaboration that every teacher would appreciate. Everything you see is in sequence, straight from the class, with a few enlargements and enhancements so you can read it more easily. I take no responsibility for the spelling mistakes, though they do vex me:

Did you see what I saw?
• every kid participating spontaneously (you probably didn't know that because their names had to be covered up, please trust me on this, everybody did!)
• kids talking to each other, being polite, having fun, questioning each other, answering each other, correcting each other, showing their strengths, pooling their knowledge, coming to consensus
• not one kid complaining, being rude, fooling around, letting everyone else do all the work, or doing all the work
• so many ongoing conversations - every time I came back to a group, there was more chat, and more progress in their work
• kids participating who had previously done almost nothing, in or out of class. Some who usually keep quiet because they think they don' t know anything, and some who are very strong mathematically, but have never had a reason or inclination to speak up before.
• two kids in particular who are very strong, and who naturally took on the role of teacher. One was very good at explaining things, and the other needed coaching. (I was able to help a bit via private message. That was REALLY cool for me.)
• not everything written was correct, but there was debate, and discussion about many things, and there was progress, via discussion, toward greater accuracy
• as the conversations progressed, so did the depth. They started comparing this function to others they had studied this year, how were the different, how were they the same
• for the math people - in two groups, a member used another technology to verify their findings, one used her graphing calculator, and another used geogebra. (Sniff!)
What I think now:
• I think the hint-troduction got enough critical math mass in each group so that they hit the ground running. They had to make a few connections on their own to get going, like what does this have to do with what we just did, but each group had enough collective ability to do it.
• this was not a PBL task, not real-life, it didn't come from their own interests, but it worked really well anyway. Imagine what might happen if I had done any of those things.
• the difference may have been because of the hint-troduction, or it may have been because at this point in the year, they are naturally more at ease with each other, and more likely to be themselves
• then again, what if I had tried something like this earlier in the year - where would we be now?
• true, fruitful, meaningful collaboration does need to happen organically, things have to just click, but like any relationship, it needs help, nurturing, and intentional effort to stay alive. In my class, as the teacher, that's where I come in
• I got many hints for my own journey into teacher collaboration. Maybe the starting point isn't a particular unit, but a common understanding of an issue, or a common problem we're facing in our classes
I need to ask my students what THEY think - is there anything that happens in class that they wouldn't want to miss out on? And did it happen this time? What do you think your students would say?

### Mining That Face-to-face Time

When I first started flipping my math classes, class time was agony for me. I could hear the crickets chirping, and I knew there had to be better things to get my students to do than the usual text book questions. But now, about two years in, I'm having a hard time fitting things into my class time.

What am I trying to fit in you ask?

Talking time:

I don't mean lecturing, of course, and I don't mean necessarily me talking, but I do mean a whole-class discussion. I want to create some sense of group cohesion, that we are a team, on a journey, and that part of the journey is getting to know each other in this math context. I don't want to go through a whole year with a bunch of people who feel no connection to each other. And yes, it's fine if all they have in common is that they worship their math teacher, I suppose I can live with that. But if, once they're done this course, they never speak to each other again, or never think fondly of the things that they did here, I have failed them in some way.

So at the very least, we spend a few minutes chatting, maybe about math, or maybe not. I like to start the class talking about something that they have done, like their comments from the most recent voicethread, or about some of their recent blogposts. Recently we laughed about some of the words we're using in our log studies, like logalicious, smooshing, de-logging, and they came up with a new one: to "Lion King" a logarithm. More about that in another post. But in coming up with our own funny language, we all said, without words, "We are us."

Working time:

This might be just a quick checkup on the basics, for example, I often have them all work something out on the eboard at the same time.  This isn't only so that I can see if they're getting it, it's also a nice combo of individual work and group work - they're working on their own but they're checking out everyone else's work at the same time. And they love it! If they don't know what to do, they watch a peer's work unfold, and I can just get someone else to answer questions or explain as they're working. It's a great opportunity to get them talking about math in their own language. Here's how one group's work looked at the end, and by the way, not everyone started out knowing what to do, and we talked about scientific notation to boot:

Exploration time:

The goal of this time is to deepen their understanding, often doing group work. I'm not a huge fan of group work just for the sake of group work, but if it arises organically, as a result of a need to discuss, compare, gather info, then I'm all for it. There are a lot of people out there who are creating wonderful activities like this, for example, Dan Meyer, Kate Nowak, John Golden, or any of the people in the geogebratube.org community.

When I create my own, I use Malcolm Swan's great document  as a guideline. Last week, I wanted to open their eyes to some details that they may have easily overlooked, for example, that log ab² is not the same as log (ab)². Here's the activity, complete with answers, which, by the way, I did NOT make available to them until they had struggled and suffered to my satisfaction:

Individual help time:

During this time, they are working on the items I've assigned via a weekly checklist, which means this is when they are likely to need help. My goal is to speak to every single person. This is the greatest time-hog of all. I cannot honestly say that I am reaching every student everyday, à la Jon Bergmann/Aaron Sams, but I do aim for that. I try to set things up so that if I'm not available, they can ask a peer for help. (So far, in the online environment, that hasn't really happened to my satisfaction. I think that is one thing that happens more organically in the brick and mortar class. That's another post.)

Just like in any class, there are some students who always initiate a conversation with me (Mrs. I need help with...), and some who never do. For those kids, I have my checklists (info that they send me during the week via a google form) at the ready. So if I see that Susie hasn't checked off the geogebra activity.....Susie, have you uploaded your geogebra? No? You need help? Or, Tommy, you haven't checked anything off for this week, what's up? Get it together man! (I don't actually speak to my students that way. Yes I do.)

The Big Picture

I'm not saying that I can fit all of these into every single class! Most of the time, I manage to fit in the talking and the individual help, with something else sandwiched in between. And that something else is constantly evolving. My hope is for that part of the class to live up to the goal set in this brilliant post by Robert Talbert, that is, to "focus class time on the stuff where they are most likely to get stuck and need their social network and a professor to help."

But that evolution can only happen with collaboration, unless you are immortal, or don't have anything else to do in your life like, say, take care of a family or watch movies and go for walks. It takes a boatload of time to create, try out, reflect on, and improve our classroom activities, and when we share, or better yet, create things together, or EVEN BETTER YET, COFLIP, we divide and conquer that time!