Friday, February 19, 2016

Rethinking How We See Mistakes

I had a flashlight moment recently.

I was helping a student to learn how to balance chemical equations. I had done a few examples, and then I had her try one. Part of the procedure calls for a certain amount of trial and error; you try a number in one part of the equation, then you track how it fans out in the rest of the equation. She hesitated for such a long time, that suddenly I realized what was paralyzing her. She thought she was already supposed to know what the right “guess” was. I told her, put any number, fully expect it to be wrong, and then we’ll use it to get the right one. Her response was immediate, she just put down a number, and it was beautifully wrong, because even though I think she had intended it as a “Here you go I told you how stupid I am” moment of self-pity, she immediately said “Oh wait, no, this would be better.” And it was. It wasn’t right, but it was better. Her wrong answer pointed her toward a better one. It seemed like she had needed permission to jump in with a mistake, before she could even experience the unlocking that happened microseconds later.

This has led me to believe that, at least sometimes, we should be calling those “mistakes” something else. Anything that leads to illumination isn’t a “mis”-anything, it’s progress. So from now on, we’re changing our attitude towards mistakes in my class, starting with what we’re calling them - flashlights. If the growth mindset movement is correct, then what we call things affects how we see them and react to them, both on an emotional level and a cognitive one. I think “flashlight” is more positive, and, more importantly, it’s more accurate. Those flashlights are our guides. They show us where the gaps are between what was taught and what was learned. (read more about these gaps in Dylan Wiliam's “Embedded Formative Assessment”.)

Here are a few other ideas I’ve had about changing my own class's attitude toward mistakes.


When a student makes a mistake, I’m going to praise them at least as much as when they don’t make a mistake. I want them to know that at that moment, they are straight up legit teaching someone something. I’ve been saying “I’m so glad you said that!” or “That’s the best mistake I’ve seen today!” or “I was hoping someone would make that mistake!” That last one I hope makes them feel like they’re my secret accomplices in teaching. It also creates a sort of suspense in my class, like everyone’s waiting for that magic mistake to happen, the same as if it’s a jackpot they’re all trying to hit. Because that’s what it’s going to feel like when they hit it.

We can legitimize mistakes as learning opportunities if we not only talk about what the mistake was, but where it came from. Because mistakes almost always have some truth in them. For example, when kids distribute incorrectly like this: 3(2×4x) = 3×2×3×4x, it helps to say to them – I know why you did that, you were thinking 3(2 + 4x), which would be 3×2 + 3×4x. I think it’s a relief for them to know that the way they think has some grain of logic to it, at least enough so that another person can backtrack with them to where attention is needed.


If I HAVE to use the word mistake, then I’ll use an adjective like beautiful, glorious, or brilliant before it, because I don’t want mistake to be a bad word – I want it to be a sign that thinking is happening, neurons are firing, lost souls are finding their way. Those are all beautiful and glorious things to happen in class, and I want as many of them as I can get.


I’ve also been thanking my students for their mistakes, because they're doing some heavy lifting for us all. For example, the other day I asked if log 3 + log 5 could be replaced with a single logarithm using a log property. One student said no, because they didn’t have the same base. Flashlight! She thought the 3 and the 5 were the bases. Not only did this show me that at least one person was looking for the base in the wrong place, but was also not aware that the unwritten base was 10. Two things learned in one shot because of her, so this was a double flashlight, and I thanked her. Later, another student thought that:
would lead to xz = y. Flashlight! At least one person was cross multiplying instead of doing fraction multiplication. They learned when that works, and when it doesn’t, and I learned that I am so not ever going to use cross-multiplication ever again. Thanks, kid!

I’m thinking it would be nice if the other kids thanked them too. I’m not sure if I’ll actually get them to, because that would probably be a bit forced. But the way I see it, the kid making a glorious mistake right away in class, as soon as we’ve done something new, is doing everyone a favour. Everyone else can now avoid making it later, when they’re all alone. If that happens, they’ll either not have any idea that they’ve made one, or they won’t have anyone to help them straighten it out. Much better that it happens when we're all there.

Being not perfect

This is a big one, and it’s probably not going to be popular, even with me. I think our kids need us to not just SAY it’s ok to make mistakes, we need to BE okay with them.

Embracing my own mistakes: Because I’m not just talking about their mistakes, I’m talking about mine too. The ones I’m so careful not to ever let them see me doing. That’s why whenever I have to figure something out in front of them I get so totally flustered that I usually say, “Ahem, well kids, I don’t want to take class time to do this, I’ll ahem figure it out later and get back to you. Move along now, nothing to see here.” And I thereby give them the message “Mistakes are great! For you that is, not me. For me they’re embarrassing, humiliating, and scary and they never happen anyway so yeah.” I need to face it, enlist input, and maybe even get help, for example ask “Why do I doubt my answer is right? What kind of answer would make more sense? Up until where did you get the same thing?”

Problem solving on the fly: When a teacher only ever shows students how to solve a problem that they (the teacher) already know how to solve, that’s great, and of course I do that, but we’re really being disingenuous if it stops there. We’re showing them a nice sequence of rules, being fluidly followed by a calm, confident person who is already in possession of the answer, and who therefore couldn’t be any less like them when they’re solving a new problem. I think we need to give our kids the chance to watch how we handle something that is truly new, something in which we truly have no idea what to do first. And show them how we’re not afraid of that feeling, that nobody needs to be afraid of that feeling, because everybody feels that feeling! Let’s get stopped in our tracks together, try stuff together, mess up, go back, sleep on it, admit we're secretly looking for the answer know, normal life.


But if a mistake happens during a test, it’s very bad, right? We all know what marks do to kids, and how they absolutely halt all learning, whether the marks are good or bad. So yes, of course, I don’t want mistakes happening then, and nobody does. But they will happen, at any time, so I’d rather get the kids armed with strategies to detect and fix them. And most importantly, remain calm.

Friday, February 12, 2016

New Intro to Log Properties

This went well.

I had thought I'd do this via a desmos activity, and I even started to make the slides, but then decided to do it live in class with everyone totally in sync, because I wanted everyone to witness something at the same time.

Here was the sequence, and the narrative, and the results in italics:

No calculators!

What do your order of operations instincts tell you to do first here:  
multiply the 8 and 16

Okay then do it!

Great, and how much is that log?
Some discussion here. How did you get that? Some recognized 128 as a power of 2, some did not. I allowed those who weren't as familiar with the powers of 2 to use their calculator ONLY to try out different powers of 2. NO log buttons. I wanted them to COUNT how many 2's were being multiplied, in order to prepare for more counting AND tallying.

Great! How about this one then?  
Some answered 9 very quickly, some answered log base 2 of 512. We discussed both answers. Those who got the 9 quickly were very good at expressing how they did it by using the already-known exponent for 128 and increasing the total number of 2's by the additional 2. My plan worked!

And what about   ?
16. Lit up the board like a Christmas tree. This was what I wanted. Everyone to just count the total number of 2 factors. I pointed out that now they were seeing numbers through a new filter. Now 128 is 2^7, and 512 is 2^9.

Try these the same way - don't actually multiply the arguments, just see them through a filter: 
I'm happy to report that these were very handily answered and justified correctly, by more than just the usual one or two people.

Now comes the real challenge. Let's go back to that first example, and say it in English, along these lines:

After a bit of coaching:

And what's another way of saying "the exponent that 2 needs in order to get 8"?
After just a hint to use the word log......
Let's go back to the other ones you just worked out and say them in English:
This went super fast!

Now let's generalize:
And for the FIRST time in my career, my students told ME the additive property of logs, instead of the other way around. The third line here was supplied, for the first time, by my students:
That felt good.

Tuesday, November 24, 2015

Why I Desmosified One of My Favourite Geogebras

I'm a certified GeoGebraphile, and have been for years. All that time, I kept hearing about Desmos, and thinking - it's lovely but I can do so much more with GeoGebra. But now that Desmos has introduced its Activity Builder, I'm torn between two dynamic geometry thingies. Here's why:


First of all, that means "to transform into a Desmos activity using the Desmos activity builder", or at least that's what it seems to mean to Dan Meyer, so it might as well go into the new dictionary.

Dan was talking about paper worksheets getting desmosified, but I did it to a GeoGebra of mine. I've used this GeoGebra now for a good 3-4 years, and it's always been one of my favourites, perhaps for sentimental reasons. It was one of the first ones I made that I felt had an actual impact on my students' understanding of a concept. Rather, it intervened to correct a common mistake that many of them made. Never mind what the mistake was, the point is, for me to do this to this particular GeoGebra was kind of a big deal.

What used to happen

Typically, my students would open the applet, which would look like this, to them AND to me:

And they'd work on it on their own, or if they were so inclined, with someone sitting near them. But I could never see what they were doing at my end (I teach long distance). And if someone was stuck they'd ask, of course, but then there'd be kids who wouldn't ask, for the usual variety of reasons - too shy, no idea what to even ask, or something else had their attention. Then those who were so inclined would put their answers in, save it on their hard drive, then upload it to me on our LMS. I'd open it up, and it would look like this, to them AND to me:

And I'd assign a mark, send it back, and go over it the next day, and I felt that they'd had a productive struggle that was well handled by the visual and interactive nature of the software.

What happens now

But this week, at the last minute, when I desmosified it, it became all that and something else. The activity had all of the goodness it always had, plus it came ALIVE.

How do I show you how it looks? Not easily, because how it looks changes all the time! It's a shared experience in real time, not to mention that it doesn't even look the same to me as it does to them.

What students see:

This is what my students see when they go to and type in the class code I gave them:

Slide 1/3
(If you'd like to really see it through their eyes, go to, and type in 9Y5F where it says class code.)

This first "slide" I made is very similar to the GeoGebra version.

What I see:

One big difference in the live version though, is I knew the moment each student logged into the activity from my teacher view:

(If you'd like to really see it through my eyes, here's the link.)

On the left, I see their names as they enter. Once everyone's in, and working on slide 1, I can click on the first rectangle that's titled "Radical rules", and see this:

I can see what each student is doing LIVE in that first slide of the activity. What you see in this image is of course static, and the end result, but during the activity I sat and watched their graphs appear, disappear, adjust etc. I can also zoom in on one kid and see what they problem might be. I could tell when someone needed help, and offer it. I couldn't do that in the moment in the GeoGebra version.

Another huge difference, as you can see, is that there are 3 slides. Once they're done slide 1, they go to slide 2 and see this:

The real gold to be mined:

And I see, again, LIVE, their typed answers appear like this:

I can set it so that they can see each others' replies, so naturally I'd ask higher-order questions in that case. Or get them to.

So this isn't just a math activity to get the right answer, it's now a starting point for a discussion.

Or it's a place for them to reflect, focus, notice, wonder - to step back for a moment before moving on - together! Before we all move on for that matter! I need to know if they got the point too!

It's also a way to do on-the-spot formative assessment, because I can set it so that they can't see each others' answers.

And as the amazing people at Desmos keep working with the equally amazing #mtbos (and really there is no distinction between those entities) it will be more and more customizable, so it can become whatever you want it to be - but the LIVE part is the gold for me. I want classes in which I am surprised too!

So who do I love?

Does this mean it's over for me and GeoGebra? HELL NO! I'm still a geogebra-holic, but I'm finding that I'm using it differently now. It has features that aren't available (yet) in Desmos, or that would only be doable with an onerous cognitive load for students to handle AND learn math at the same time. Longer-term, more complex, and individual assignments like this.

In fact, I've also used Desmos activities sometimes to lead up to the major geogebras, as I wrote about here. I think the interface for desmos is less intimidating for most students. Its immediate response, the fact that there's no need to even hit enter, makes it friendlier for them.

So in summary, Desmos, and GeoGebra, I love you both. 

Monday, November 9, 2015

Reflections on Desmos Activity - Piecewise Functions

I'm finding that just about every activity I've ever done over the years is adaptable to and even improved by the Desmos Activity Builder. It's getting to the point where I'm making those instead of doing other things that are part of my actual job, like correcting,'s become a sort of guilty pleasure.

Here's the link to my latest activity:

A few reflections, some of which are based on those I've read in these posts from others.

Practical stuff:

1.Since you can't hide the equations from the students, and since those equations would be the answers I wanted them to find, I put the folder containing those waaaayyyyy down so that it would be unlikely anyone would find it. In case anyone did scroll down to line 34 for no reason other than to waste time, I rewarded them with Dan Meyer's "Nothing to see here" as a title for it. Which one kid did, and he now thinks I'm hilarious.

"That's cute Miss."
2. Echoing what Bob Lochel said here, keep the questions efficient. I put too many questions all in the same question slide. I need to exert some self-control there.


I took heed of Shelley Carranza's great post, specifically the part about pacing - letting students have time to explore, but then bringing them together for some explicit instruction before they continue. Once I saw that everyone was at slide 3 (because that's one of the things you CAN see with this tool!), I brought everyone back to slide 2 to read everyone else's answers/comments. I didn't spend a lot of time, just enough to use the "without lifting your pencil" idea to introduce the word "continuous" at this point, and to point out that something can be discontinuous but still a function. There were some other things that their comments made me want to discuss, but I knew I was going to followup next day, so I left them for next day.


I'd love to read more from others about this. Maybe if the activity is designed well enough, followup isn't an issue. Maybe my activities are too long? At any rate, the things I was concerned about were:
1. mining their comments
2. giving the answers 
3. showcasing their work.
To do this, I took screenshots of all the slides, annotated them, and made a powerpoint with those for next day.

1. Mining their comments: For the comments I saw during the activity that I really wanted to address but decided to leave until next day, here's what one of those slides looked like:

I tweaked Fawn Nguyen's use of colour-coding highlighter pens - instead of using it for assessment of student work, I used it to group their comments. I used the orange to reinforce the meaning of "continuous", the blue and green ones to discuss their "I wonder"s and my "hmmmm"s (things I wanted to address/clarify/straighten out), and I got them to answer some of the questions that had been raised. I know this has a lot of potential, but I felt like I was doing way too much talking while going through these slides.

2. Some slides had definite right answers I wanted everyone to know, so here's what that looked like:
It was also an opportunity to repeat how to specify the domain using Desmos.

3. Next slides showed everyone's custom-designed piecewise functions. Many had already tweeted theirs out, but I wanted to make sure everyone saw everyone's, and to get reactions in real time. 

You can see more of them on Twitter using the #piecewisefn hastag. Enjoy!

Thursday, October 22, 2015

Sticky Points

A sticky point is a dot that stays in the right location on a Desmos (or GeoGebra) graph no matter how the sliders are moved. Why don't I just show you?

It came out of my new and improved way of introducing my students to making their own Desmos/GeoGebras to study functions. I decided to frustrate the heck out of them so that they would beg me to show them how to do the thing I wanted them to learn in the first place. Inspired by Dan Meyer's headache-for-which-math-is-the-aspirin idea. Getting a point to stick shall be their aspirin.

The Activity:
First, as part of this desmos activity, (which I made using the amazing Desmos Activity Builder - DAB to me) they had to move the sliders for this linear function:

...then type in the coordinates of the function's y-intercept in order to get an orange dot to show up on the graph like so:

...then move the sliders around again, and repeat:

and repeat, and repeat....and of course, most of the time, the y-intercept changes, so they had to keep re-typing the orange dot's coordinates so that it was where the new y-intercept was. Frustration! Eventually (read - as soon as someone started whining) I said 

"Wouldn't it be nice if that orange dot automatically moved to the location of the y-intercept as soon as you moved the sliders?" So the next activity slide was all about what ordered pair can we type into Desmos so that that happens:

Well, because of the way the DAB is made, they could easily share their ideas with me, and each other, about how to do that:

And voila, if the orange dot moves around and is always at the y-intercept right along with the sliders, that point is sticking. It's a sticky point.

The next slide asked that they do the same thing for the x-intercept - to type in an ordered pair that will stick to the x-intercept, no matter how we move the sliders. This took more time, of course, which on one level is amazing to me because they spent an entire year already on the linear function - how can they not be experts at finding its zero?

BUT, on the other hand, solving 4x - 2 = 0 is very different from solving ax + k = 0. The second one requires that they see the a and the k as numbers, even though they're letters, and seeing the x as a variable, even though it, too, is a letter. I'm sure the sliders sitting right there in front of them, with numerical values showing, helps with this idea. Interestingly, one student, who happens to do a lot of coding, got it immediately.

Next day followup & new activity:
The next day, I shared the graphs of those students who had everything sticking, so that everyone could have the experience of typing in the formulas and seeing that the points stick.

On to quadratics then. I summoned the DAB and made this activity:

Which was the same idea, getting points to stick, but this time, the vertex, the y-intercept, and the zeros. The vertex was super easy and most got it right away, so now I wanted to bring up how to use the sliders to check if your point is sticking. Here were their responses to that:

After all, I want them to not only get their formulas right, I want them to be able to decide, and be their own teacher, about when they're right AND, more importantly, know when they're not.

When things got really interesting, for me anyway!
Once the vertex was sticking, it was on to the y-intercept. Here's where things got really interesting. Again, it was no problem for anyone to calculate the y-int when a, h, and k were numbers, but slow going when they were just a, h, and k. Eventually, here were their responses on the slide that prompted them to share:

They were checking with the sliders! And a few unexpected things popped up - one student mentioned the y-intercept for the general form of the quadratic, and one simplified the expression a(0-h)² + k to ah² + k. I had the opportunity to talk to those students about their particular work - the one who was thinking about the general form eventually made a whole desmos just about that, and got those same points to stick!

Which brings me to what I really love about the Desmos Activity Builder:
With a tool like this, anything is possible. It puts control, if that's the right word, in everyone's hands. 

And here's the thing - You don't get a tool like the DAB, where who learns what is all up for grabs, and use it to make something with the same single outcome for every student. It's just not possible! 

My ultimate goal....student-created GeoGebra's!

All of this was ultimately leading up to their first GeoGebras about the Absolute value function, which is their first new function for this year. They just started them yesterday, and there's plenty to add, but I already find it's going MUCH better than in previous years. They are already familiar with the sliders, AND with the idea of formulas for important points. I'll share those here soon, but in the meantime, some kids are already sharing them via the #ggbchat hashtag:

Happy DABing!

Friday, October 2, 2015

If Only I'd Used a Hinge Question 3 Weeks Ago

After correcting this week's assignments, I discovered many students are still not able to find a vector's direction, given its components. This is something that I supposedly taught 3 weeks ago, and thought I'd checked for understanding, but....oh well. A hinge question 3 weeks ago would have been awesome. I would have known who, what, why, and how bad things were, and come up with a way to straighten out their vector issues.

Haha! Get it? Straighten out the vectors? It's been a long week.

As usual, it's only when it's too late for this year's students that I have clarity on what to do, but in my defense, seeing so many possible wrong ways to do it today was what guided me to writing this hinge question.

To generate the wrong answers, I used today's mistakes. The 3 big ones I saw today: not taking absolute value of components, wrong order of ratio, wrong quadrant formula. I saw one person using the y-axis as a reference instead of the x-axis, so I'll put that in just a few answers.

Find the direction of the vector <9, -20>.

If they get 294 degrees, they're right.
If they get 426 degrees, they did 360 - arctan (-20/9), ie didn't take abs value of 9
If they get 335 degrees, they did 360 - arctan (9/20), ie wrong order of ratio, OR they did 270 + arctan (20/9)
If they get 66 degrees, they did arctan (20/9), ie wrong quadrant
If they get 384 degrees, they did 360 - arctan (9/(-20)), ie didn't take abs value of 9 AND wrong order
If they get -65 degrees, they did arctan (-20/9), ie didn't take abs value of 9 AND wrong quadrant
If they get 24 degrees they did arctan (9/20) wrong order and wrong quadrant
If they get -24 degrees, they did arctan (9/(-20)), ie wrong ord, wrong quad, no abs val
If they get 204 degrees, they did 270 + arctan(-20/9) ie used y-axis as reference AND no abs val.

Thursday, September 17, 2015

Vector Addition Goodness - and Quirkiness

This went well.

It turned out to be a quick way to generate lots of examples - simple ones and quirky ones, and to target specific examples so that we could look back and look ahead in the vector unit.

We've just started vector addition. Here's the activity I put together for my online class today:

We're all online, so anyone can easily move these arrows around by clicking and dragging them.

I split the kids into groups of 2-3, and told them to each pick one blue and one green vector, add them, then draw the resultant in red (these colours matched the ones from the previous evening's voicethread.). Since there are 6 blue and 6 green, there are many possible combinations of vectors - 36 in fact (an opportunity to talk about math that's not usually part of the vectors unit!). They didn't have to do them all, of course, just do 3 pairs, or enough to use up all the vectors.

Because they were dragging instead of drawing, it only took about 5 mins for them all to finish. I took snapshots of each group's work, then we reconvened in the main classroom space. Here were some of the results
Group 1:
First we together looked at all the examples group 1 made and decided if there were any mistakes - which happily, there weren't. Then I saw the first part of my evil plan unfold. (I had only a 1/36 chance of that combo happening, yet it happened!) The combo at the very lower left was two horizontal vectors of opposite directions being added. A quirky one that I wanted everyone to see, and since only one group did that combo, it kind of was an example that had a personality to it - it belonged to a person!

Group 2:
We repeated for group 2 - checked the answers first, then more evil plans unfolded. (Another long shot happened, geez I should buy a lottery ticket today.) In the lower left corner, someone had added a green and blue that were perpendicular to each other. I asked them to think about where they'd seen something like that before - and after a few seconds I heard - the blue and green are almost like the components of the red. Almost? OR EXACTLY?!? We talked about how all along, the components of a vector (with which they're already familiar) actually add up to that vector. When they'd been drawing a vector's components all last week, it was kind of like they started with the answer, and drew a question to go with it.

Group 3:
Weird how all my evil combos ended up in the lower left corner...anyway here we saw the same component-style example except that they're added in a different order, but still giving the same resultant.

I then showed them a combo that no one had picked, but that I wanted to address:
I had, of course, deliberately put in a blue and green that were identical to each other. We talked about what the resultant for this would look like, and it was a nice intro to something we haven't done yet - multiplication of a scalar and a vector.

What I liked about this activity:
  • the possibility of everyone seeing so many examples in a relatively short time (this all took a total of 20 minutes)
  • the possibility of interesting and unusual examples for discussion
  • the personalization of the examples - I didn't make them up, not really anyway, so that I could refer to "Susie's example", instead of a cold "number 3"
  • It gave kids who think outside of the box a chance to try something outside of the box, and feel like it's ok to do that. In fact, it was fantastic to do that!
  • It was a great intro to the geogebra that I then had them do individually, in which they were pretty much doing the same thing, manipulating vectors to add them, with a few twists, like find what vector I have to add to this one to get that one (intro to subtraction)
Next time:
  • I'll make sure to keep WAY more copies of the original arrows - easier to show combos that no one did that way
  • Find cases where two people added the exact same vectors and got the exact same resultant
  • I'll include the potential for more quirkiness, like opposite vectors adding to zero. You can never have too much quirkiness.