Adjusting My Explore-Flip-Apply Dials

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.

As usual, please feel free to share your thoughts, good, bad, or ugly.

Update #2 on Coteaching Project

Team vector's story so far:

Week 1:
Content:

• the basic definition of a vector
• the different representations (arrows, components, etc)
• the various calculations involved, all of which were based on math that was really review, ie using sin and cos to find the components, using arctan to find the direction, and using pythagoras to find the magnitude.
• In math only: relationships between vectors (identical, opposite, collinear, orthogonal)

Coteachingness:

• We tried to coordinate and vary the activities in our classes, so that, of course, the kids wouldn't get bored, but more importantly so that they could see things in different depths and contexts - formally in math, and concretely in physics. For example, I showed them the polar coordinate system to help visualize vectors, particularly the way the angles are named, and Kerry and Andy had them performing various calculations involving velocity and force vectors, such as decomposing a vector. They probably paid more attention to things like units than I did. But I got to show the kids how to put their calculators in polar mode and graph pretty flowers and spirals!
• To spread out the load, we took turns making the voicethreads - Kerry made the intro to vectors one, and I made the representations one etc.
• We wanted to make the team aspect of this very visible, so we commented on all of the voicethreads, whether or not we were the author. That's another reason voicethread is a great tool for this kind of collaboration.

Week 2:
Content:

• operations on vectors - addition, subtraction, and multiplication by a scalar.
• all of these operations done in all of the various representations

Coteachingness:

• We started using a googledocs version of our joint schedule (see previous post) so we could keep each other updated.
• We tried to step it up a bit with the visible teamwork. Kerry and I had envisioned a video with all of us in it, because it would be fun for everyone, kids and teachers. We wanted to introduce vector addition with the three of us having a real tug of war, but of course time and distance got the better of us, so here's what we ended up with:

• As you can see by their comments, the kids really enjoyed this one! Just imagine if we were able to really get together and act out stuff like this - I bet the retention rate would be off the scale!
• We, the teachers, tried to meet to give feedback, but we were only able to manage 2 people at a time. All agreed that this has been great fun, and we'll try it again, if not later this year, next year for sure.

Week 3:

Content:

• Here's where our paths diverged, because this was as far as the Physics course went in terms of content. But now they would start applying their skills big time to solve vectors problems.
• In math, dot product (two different methods)
• Chasles' property
• Linear combinations
• Applications

Coteachingness:

• We're still in the same neighbourhood, but we didn't really coteach much this week.

Thoughts for next time:

I hope this isn't the end of coteaching this year, but we have to find a logical place to try it again. It may be that I'll move to chemistry - logs and exponentials maybe? Fortunately, Kerry is also the teacher for that course! Things to work on:

• We need to meet much more often, before, during, and after. More planning, coordinating, cross-feedback, followup - more goodness!
• Team videos are so happening. But we are so rarely in the same physical space that we need to know what we're doing before we meet again - which will probably be Christmas, or maybe even June.
• We need to find more ways to cross-pollinate. How can we use this strategy to get to the really deep, unexpected, authentic, real-life important learning?

Thanks for reading, any ideas or questions are hysterically welcome!

Update #1 on Physics-Math Co-teaching Project

First of all, start your weekend right, and watch this:

Don't you love that? Nothing to do with this post except that it's how I started each class today. Some of our kids are sitting in front of the computer all morning and I figured they, and everybody, could benefit from getting up, dancing and laughing for a few minutes!

The Project:

Okay here's what's happening in our first-ever co-teaching project. We had really only 3 days this week, because Monday was a holiday, and Physics doesn't happen on Friday for our kids.

Here's how we're keeping track of which kid is in whose class and when, and what they're doing there. Of course I removed the actual names of our students, but on our actual schedules, their names are highlighted as you see here:

3 class schedule with st names from amcsquared

For example, the yellow kids I teach period one are also in Kerry's period 3 Physics class. Andy has the blue kids in his period one class, then I see them period 3 etc. The black X, Y, Z, W, R, and S are kids who are only in those classes.

As you can imagine, we have to make sure that:

• Each child has access to the same content and materials, whether or not they're taking both math and physics.
• The kids who are taking both don't hear the same thing twice a day.
• If sequence is an issue, the kids who are taking both see things in the same sequence
• We don't overload them with double the work - they don't need to do the same type of practice examples for both classes

That's why we have to:

• deliver content outside of class
• keep our class activities as stand-alone as possible
• keep talking to each other

Here's this first week's schedule with my stuff filled in:

3 class schedule with audrey's activities from amcsquared

Tuesday was our first day, so we all did the usual first day stuff, getting to know you, using the tools, etc. Friday was used to cover some topics that were only part of the math program, since there's no physics class that day. I also took the opportunity to introduce them to geogebra, a tool that I hope the kids will use later this year to create a joint math-science manipulative.

Student feedback:

Only informal so far. The kids seem to be happy with the co-teaching. They've said that they really appreciate the common "homework" - so far that's the only advantage they have seen. One of my blue students arrived to class and said that he had had a question to ask me, but that during his Physics, it had been answered. Cool. I'm looking forward to when they see a deeper advantage - learning in context, or more deeply than they would normally do. That'll take a lot more meetings and experience on our parts, and of course the beginning of the year is probably the WORST time for that!

My feedback:

It has been great bouncing ideas off of each other. We just jumped in and got our feet wet, and I already feel like I'm doing a better job of teaching. For example, having to make the activities stand-alone and non-repetitive forced me to to come up with polar coordinates as another way to view vectors. I never did that before. And the kids REALLY responded to the spirals on their calculators, not to mention they were surprised that the calculators they have had for all these years could be put into polar mode.

The plan next week:

Next week, we're going to do operations on vectors, with lots of in-class practice and investigation. We had dreamed of doing a joint video, with the three of us pretending to do a tug of war....but my daughter is turning 18 this weekend so.....I don't think that video is gonna happen, time for some Camtasia magic animation techniques!

It's a start! In the meantime, Hatee-hatee-hatee-ho! (if you don't know what I mean, go back and watch The Fox. And geez, DANCE, will ya?)

Collaborative Teaching Project #1

This year, the Math and Physics online teachers at LearnQuebec are going to try co-teaching! I and two colleagues - Kerry Cule and Andy Ross, who both teach Physics, have decided to collaborate on the first unit, which is vectors. The content is pretty much the same in both of our courses, so we figured what the heck. At the very least, we'll be more efficient, and at best, we'll be able to provide richer experiences for our students.

One of the challenges is that we can never be in each other's classes, because our schedules conflict. Oh yes, and one of us lives in Montreal, one in Ottawa, and the other in New Brunswick. Another challenge is that not all of our students are in both Math and Physics. Most are, but we of course don't want those students who aren't to be at a disadvantage, so we decided to try it as follows:

Week 1: We split up the topics, and we've each created one recorded lesson, which all groups will watch asynchronously. Then, during our classes we'll do activities that put the concepts in the framework for our own subjects. 

Week 2: Similar, but we're thinking of doing videos in which we all appear, just to make things more fun. one idea is to have the teachers doing a tug of war, to illustrate vector addition. That'll also be challenging considering we don't live anywhere near each other - I think Camtasia will have to provide some magic for us there.

We'll see how it goes. So far, the most challenging thing has been finding a way to represent the joint schedules so that each teacher can immediately know which of the their students is in which other class, if any. I've made a colour-coded one that makes use of every possible colour Word's got.

At some point, I'm hoping we'll collaborate, at get our kids to collaborate also, in the spirit of my One Big Idea that I wrote about toward the end of last year.

Here goes nothing!