Monday, March 3, 2014

The Day My Brain Talked to Me, or How I Learned to Think Like a Logarithm

Honestly, what with my withering attention span and emotional ups and downs, it’s amazing that I manage to get anything done. But I do nevertheless, and this is what I managed to get done on the afternoon of this past Valentine’s Day. I needed to write about it because it was kind of awesome, and because it gave me some insight into my students' reflections in their eportfolios.

What happened is, I inadvertently went on a quest, in my head.  The journey itself was fascinating, at least in retrospect, because it seemed like a meta-cognitive experience. Since it's probably the kind of thing I'm supposed to be getting my kids to do and write about in their eportfolios, it's very fortunate that I did it myself, and apropos that I write about it too. And although they say that the journey is more important than the destination, it was pretty sweet when I got there.

What was the problem?

This little paragraph in our textbook that has tormented me for many years:
The rule of a logarithmic function can be written in the form f(x) = alogcb(x - h) + k. However, certain algebraic manipulations allow you to transform this rule and write it in the standard form f(x) = logcb(x - h).

To worsen the torment, I've said to my students "Right. Sorry. I don't know what those certain algebraic manipulations are, but it says here that they exist and they work, so let's just go with it, shall we?" And another little piece of my self-respect as a teacher snapped off.

Anyway, I had a ton of other things I was supposed to be doing, so naturally, I started playing with the expression to try to disappear the a and the k. I just kind of fell into it, just like that. Kind of like Frodo.

And so began the quest:

(There is math here, but I've tried to intersperse it with enough other stuff that anyone who's ever tried to figure something out might be able to identify with it.)

Anyway, the k part was easy, so I started with that. Easy, I just replaced the k:

At this point, I couldn't help but notice that that last term, in red, looked and sounded a lot like Louis CK, so I watched a few of his hilarious videos. Very productive.

(Approximately a half hour later): The whole reason I did that last step was in preparation for this next one, in which we...

Smoosh the logs::

Normal people would call this last step "applying the first property of logarithms" , or, "because the sum of the logs is the log of the product", but I and my students call this "smooshing the logs." (Sorry, former and current students, for using vocabulary that no one else recognizes or takes seriously. But you had fun, am I right?)

So far, I had managed to get rid of the k by combining it with the c and the b to form what's in red:

But that was very routine, baby stuff, nothing new. In fact, I'd gotten this far in previous years. The real rub was what to do with that darn "a". I was convinced that I could only get rid of it by applying another property in an equally routine and unoriginal way:


Normal people: Apply the third property of logarithms.
Me and my students: Actually, that's what we'd say. Sometimes I model normal behaviour to keep them guessing.

But the only thing I could think of to do next was this really lame exponent thing:

On Valentine's Day, for some reason, that (x - h) with the exponent of "a" sorely vexed me.

The Cycle of Distraction and Torment

I then began a cycle, which I repeated many times that afternoon,which went a little like this:

I'm afraid I became a bit crazed. I'm looking right now at all the papers I scribbled on furiously. Here's one of them: 

The Turning Point

It wasn't that I finally knuckled down, or suddenly discovered a new math thing, or gained insight from geogebra. The turning point came about when myself talked to myself. I literally heard my brain say to me "You're not thinking like a logarithm. Think like a logarithm. BE an exponent."
Then this actual logarithm idea kept popping up, something I remembered noticing a long time ago. An idea that wasn't terribly difficult, but it had been of my own making:

If I say that:
that is, they both equal 9, I'm really saying that the exponent that 3 needs to turn it into a 9 is the opposite of the exponent that 1/3 needs to turn it into a 9. In log language, that's:

Or more generally:


How to think like a logarithm:

Now this didn't solve my problem, but it gave me a hint about how I needed to think. Stop mindlessly applying routine algebraic procedures, and think about what a log is, how it behaves, and how logs with different bases can be related.

Now I saw "a" as a number multiplying an exponent, and I played around with this kind of thing:

but if I change my base to , or 3, and still want to get 729, then it's:

I'll spare you the details of my further number experiments, for there were many, and they were feverish, but the important thing is this:

At this point, there was nothing on this earth that could have distracted me.

Or even discouraged me. I was possessed. I still didn't know the answer, but I knew I was on the very verge of getting it, and now that I look back, this was the most exhilarating part. I owned the problem, and I knew in my bones that it was just a matter of time before I'd get it. I was actually shaking a bit.

I can count the number of times this has happened to me on the fingers of two hands. Maybe that's the way it's supposed to be, after all, it was pretty exhausting! (The last time it happened, I wrote about it here. Projectiles. Enough said.)

Once I was sure of the numbers, I put it all down algebraically to summarize. It occurs to me that this is the most boring part, because it's like a synopsis of a story, rather than the actual story. And usually this is the part we show our students, the boring synopsis!

The math behind the "a", if you're interested:

, or, said as a log:

But then:

which means that

which, put logarithmically:

See above in red, by which I can replace n:

which told me I can get rid of the "a" by combining it with the c to form a new base, c ^(1/a).

After certain algebraic manipulations

 and much checking and re-checking with geogebra, this thing of beauty finally revealed itself to me:

which I immediately tweeted, and to which my wonderful brilliant friend Jennifer Silverman immediately replied::

What happened then?

Well, I shared it with my students the following week, just so that I could have the satisfaction of NOT losing another little piece of my self-respect as a teacher for once. Their reactions could be summarized as "Do we need to know this?". But I also shared with them the journey that had lead to it, and I think that made way more of an impression. I hope what they'll remember is:
  • No one can ever say they know all there is to know about something.
  • You can never call anything you know useless, because you never know when you'll need it.
  • Your brains are miracles, not because of the facts they can hold, although that is pretty amazing, but because it can solve problems while you're not even aware of it.
  • You can actually influence how you think!
  • Everything that I ask you to do I'll do too. Including and especially, keep learning.

Saturday, February 8, 2014

Our First All-School Twitter Chat

....was this week! It was an idea that was hatched just a couple of weeks ago, and due to the enthusiasm of our principal, our staff, and our students, it happened, it was great, and it'll happen again soon.

A few facts about us that I hope WON'T make you say "Oh well, that's fine for you but it won't work for us" and then move on:
  • We're online synchronous classroom teachers. Our students are in brick and mortar schools all day but when it's time for Math, Science, Physics, or Chemistry, they get online with us. They are from all over Quebec, many in remote areas. We're all pretty used to interacting live online, in fact, we pretty much crave it due to the lack of f2f time.
  • By "all-school", I mean all 5 teachers and all 77 students. Yes, we're not your typical "school" in that sense.
  • 3/5 of the teachers (myself, Peggy Drolet, and Kerry Cule) already use Twitter with their students, so most of them already had accounts and were comfortable using it, although the chat was a first for almost everyone.
  • At the beginning of each year, we get permission from our parents for our kids to be online in many sites - google drive, twitter, blogs, geogebratube....and the list just keeps growing every year. So that part was already taken care of.
Still here? Great!
Why we wanted to do this:

Cohesion. We're scattered all over the place, admin, teachers and students alike. This would be an opportunity to feel like we all belong to something, a school, an organizations, a community. It's something that you take for granted in the brick and mortar school. We also thought it would be great for the kids to meet other online students - they typically only know those in their own class.

How it all unfolded:

Our first staff meeting after the idea was hatched was all about brainstorming for the chat. When to do it, duration, number of questions, how to engage the kids, what to talk about, the hashtag, guidelines for participants. (Our principal, Dianne Conrod, flips the weekly staff meetings so that we can spend our time doing just this kind of thing.) (Take THAT anti-flippers.) We used a googledoc to put our ideas down.

Selling the idea

The teachers agreed to talk it up in class, and Peggy made and tweeted this powtoon:

Peggy's kids were ready to go on day one, for she and her students are the über-tweeters! Mine and Kerry's, were, for the most part, intrigued but not jumping-up-and-down excited.


We settled on a half-hour chat, with 5 questions. The guidelines would be given during class but reinforced at the beginning of the chat:
  • Be honest but respectful in your answers.
  • Use #lqchat
  • Follow the Q1-A1 format
  • Have fun!

Kerry put together instructions for the kids on how to use tweetchat. I thought to myself "Why the heck have I never used tweetchat?" for it is SO much easier than trying to keep up with a chat on twitter. We all spent a little time going over that in class the day of the chat, and also how to follow the Q1-A1 format so people can keep track of the many threads that typically develop during a chat.


We decided first and foremost to make it about the students, not about math or science. For one thing, not all of them are taking the same course, but mostly we wanted it to be a social activity.

We settled on these questions:

Q1 What is your favourite Valentine's treat?
Q2: What are the pros/cons of online classes?
Q3: What is the number one piece of advice you would give next year's new Learn students?
Q4: What is the one thing about online learning that has surprised you?
Q5: What topics would you suggest for future Twitter chats?

So what happened?

Just before it was time, twitter was alight with #lqchat. Kids were testing out tweetchat, and saying hello:

Then it was go time! I moderated the chat - a first for me!

The stream was incredibly fast and the kids were clearly having a good time. A few memorable tweets:

We didn't trend but we did have about 700 tweets by my estimate. And by estimate I mean I counted them just now. Out of 77 students, we had 27 participating - 35% not bad! The half hour zoomed by, and I even had to think up another question on the spot -

For the full chat, here's the link to the tweetchat room, and here are the storifies that Peggy made, one for each question:

Prechat warmup
Q1:  What is your favourite Valentine's Day treat?
Q2: What are the pros/cons of online learning?
Q3: What is the number one piece of advice you would give next years’ new LEARN students?
Q4: What is one thing about online learning that has surprised you?
Q5: Do your friends who are not in online classes have a hard time understanding what it's like?
Q6: Do you have any suggestions for future chat toics?

Student feedback: I'll let the tweets do the talking:

And next day, in class, we talked more about it. Some feedback:
It was great to be able to go beyond our subjects, and get to know each other as people rather than classmates.
I loved meeting/tweeting with other students that I didn't even know existed! 
We're hoping that the kids will sell it for us now. AND maybe next time, we'll include the administrators from their schools, parents, former students - who knows? The twittersphere's the limit!

Sunday, January 19, 2014

Exponent Activity Re-visited One Year Later

Last year I did this, and afterward wrote about how it didn't have the impact I had hoped for. So immediately after that class, one year ago, I created another series of slides, thinking, this is what I'll do next time. I did it this week, and it did exactly what I had hoped for, at least for some kids.

A little background:

Kids often make the mistake of thinking that if something is doubled every hour, then it's multiplied by half of 2 every half hour. I can easily show them the faulty logic of that, but I'd rather not teach by telling them no, you're wrong, I'd much rather get them intuitively to the right place, so they can see that every half hour, it gets multiplied by root 2. The discussion about why two exponential functions are equivalent algebraically is different from the one about why they're equivalent realistically. Both are worthwhile, just different. Up till now, I feel like I've only ever dealt with the algebraic part with my students.

So here it is, my second time at bat with this lesson. I uploaded the pdf to slideshare so that it wouldn't mess up any of the formatting, but you're welcome to the editable ppt and the notes underneath this slideshow:

I also gave them a ggb that dealt with the algebraic equivalence better than I ever could by blathering on. Here are all the downloadable and editable versions if you'd like to use and tweak anything:
Slides with answers
Notes version for students

This week, I'll try another version of the equivalent exponential THINGS activity from last year, and see how it goes. It's got not only situations, but also graphs and rules. Here goes nothing!

Feedback hysterically welcome.

Sunday, January 5, 2014

Sunshine Award Post

I have been reading with great interest peoples' Sunshine Award Homework posts, and I finally got nominated myself to do one, by none other than Carolyn Durley. Thanks Carolyn!

Here are the steps we're supposed to follow once nominated:

Step 1: Acknowledge the nominating blogger:

Carolyn Durley, whose writing is just the best. She is a wordsmith, she's brilliant, and despite being really busy, she answers everyone who takes the time to write her. Always.

Step 2: 11 random facts about me:

1-I am really, really good at whistling. I sometimes do it without even thinking, at inopportune times. I often stop in the middle of a song, look up, and realize everyone's staring at me. Whether it's due to my superior skills or the inopportune-ness, I'm not sure.

2-Whenever I sing along to a song, I always sing the harmony or the backup.

3-Many years ago, I started what is now a tradition at my old school, Queen of Angels Academy (QAA), which was that for the Christmas assembly, the teachers put on a skit for the whole school. We spoofed popular shows or songs, and turned it into a story about the students of QAA. One year it was The OC, another time we did the Solja Boy dance. I heard that for Xmas 2013, they did "What Does the Fox Say?"

Top shelf of my pantry
4-I hoard Celestial Seasonings' Candy Cane Lane Tea. It's only available at Christmas, and I must have it all year.

5-I have a sister who is not my twin but who might as well be. We do look alike, have similar glasses and hair, and often show up to an event wearing identical clothes, purely coincidentally. And we both live in Dorval, a small suburb of Montreal, which makes things interesting when we shop. Once at Reitmans, I was considering a blouse when the clerk said to me "Oh you're back! Changed your mind?" It turned out my sister had just been there before me and had tried on the exact same blouse.

6-I'm not overweight, but I do very carefully watch what I eat, so I follow the Weight Watchers points system. It is the perfect combination of health, psychology, and math.

7-Biggest regret of my life - not learning to play piano when it would have been easy to. I just know my life would be even better if I could make music on a piano.

8-I have a terrible blushing problem. Even thinking about blushing makes me blush. I'm doing it now.

9-I keep a feel-good file, on the advice of Kalid Azad, of It includes keepsakes from events that I organized and that went really well, like own high school's 25th reunion, which was no easy task considering the high school closed in 1979. (So did my elementary school!) It also includes things I've written and am very proud of, like an article I wrote about my mother that was published in the Montreal Gazette, and my chapter in this book.

10-Some of my worst experiences came from being a parent of kids who didn't like school. I suffered terribly from disappointment, resentment, and stress, and so did my kids. I also made so many mistakes during those years that I don't think I will ever forgive myself, or the teachers who treated my kids as if they weren't really their students, but merely irritating distractions. I can only hope my kids will forgive us all.

11-Some of my best experiences came from being a parent of kids who didn't like school. The fact that there was no safe place for these two creative, insightful, intelligent, kind, sensitive, honest, brave souls opened my eyes to the shortcomings of our schools, and myself as a teacher. I learned so much about my own students and always pictured my own kids in their place to ask myself how I would have wanted the teacher to treat them if they had been mine.

Step 3: Answer the nominating blogger's 11 questions:

Carolyn's 11 Questions:

1-Favourite quote or saying?

Coincidence? I think not!

2-If you could do your post secondary education over again what (if anything) would you do differently?

I would have participated in theatre, both performances and classes.

3-What inspired you to go into education?

I never had a choice in the matter - teaching was as natural as breathing to me. Before starting school, I nearly went out of my mind waiting to learn the things my older sister was learning. Fractions especially fascinated me. When I finally learned how to read, I taught other kids how to, including my younger brother, because I assumed they were all suffering from the same impatience I had!

4-What is your favourite educational related book you have read and why?

I Am Malala. It's not just about what happened to her, it's about her father, and his ideas about education. That man should have been nominated for the Nobel Peace Prize along with his daughter. And they both should have won.

5-If you did not have to work what would you do instead (or would you continue to work)?

I would still do my job, but someone else would sure as heck do the cooking, cleaning, laundry, etc, even the uploading and downloading part of my job that takes so much time. But I might work a bit less and travel way more!

6-What is your favourite characteristic about yourself?

I treat everyone the same, regardless of who they are, how famous they are, how old they are, what they look like, or who's watching. Because that's the kind of person I like to be with.

7-What new goals or aspirations do you have for 2014 (professional or personal)?

Same for both areas - play WAY more. Alone and in teams.

8-What one country that you have not visited, would you like to and why?

I have never been off this continent, so that's pretty much wide open. I have always wanted to go to India though. And the wine regions of France.

9-What is your favourite way to unplug and unwind?

Gardening, and drinking wine. Not at the same time though!

10-Strangest food you have ever eaten?

Sushi. I know, I don't get out much.

11-What is the scariest thing you have ever done?

I jumped off a high ledge at a swimming quarry a couple of years ago. Then I did it again right away and didn't let myself hesitate. It was so scary and exhilarating that my heart rate goes up even now when I think of it. I know, I don't get out much.

Step 4: Ask 11 questions:

1. What's the last book you read that had a profound impact on you, personally or professionally - fiction or non-fiction?

2. What is your number one most-hated-pet-peeve grammar mistake that when you hear it you want to scream? If you can't decide on one, I'd LOVE to hear them all!

3. How many careers have you had?

4. Are you the same person face to face as you are online?

5. What celebrity are you certain you could be good friends with if you ever had the chance?

6. Everyone says pedagogy first, edtech tool second, but has it ever worked the other way around for you?

7. What are your desert island foods, record albums, movies? (That only counts as one question.)

8. Who/what always makes you laugh?

9. Do you spend any time at all playing something - alone or competing with others?

10. What was the best professional conference session you ever attended, and what made it the best?

11. Have you ever watched or heard of the movie Être et Avoir? If not, what do you consider to be the best movie about teaching?

Step 5: My nominations for 11 bloggers:

My list of people was really long, but it got shortened when I eliminated people who don't actually blog, as well as those who I know have already done this sunshine award post. I hope you all don't mind being nominated (again if that's the case), and I truly look forward to reading your responses, if you decide to do this!

Kalid Azad @betterexplained
Christian Drouin @christiandrouin
Jennifer Silverman @jensilvermath
John Golden @mathhombre
Gary Strickland @SciAggie
Gregory Taylor @mathtans
Mark Sanford @hfxmark
Jason Bretzmann @jbretzmann
Daryl Bambic @dabambic
Verena Roberts @verenanz
Ines Renner @IRRenner
Susan Van Gelder @susanvg

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 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:
  • Don't ask them for answers, ask them for questions: 
    • 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 They all work, so have fun!


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

XG - basketball

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!