Getting Better by Nailing the “Why”

Last weekend I presented at the CMC North conference in Asilomar, CA. I’ve attended this conference for most of the last eight years (though I took the last two years off after my wife and I had twins). It’s always been one of my favorite conferences, and this year was no exception. In fact, this may have been my favorite math-related trip to Asilomar yet!

I gave a 90-minute presentation on Desmos, the “free and fantastically beautiful online graphing calculator.”


I had a blast during the session and received a lot of positive feedback, but something felt a bit off. This is more or less the same Desmos presentation I gave twice in CMC South, but after those sessions (particularly the one on Saturday morning) I felt like I was on top of the world.

After this session at CMC North, I had a nagging sense that I need to make some core improvements. I spent some time reflecting over the rest of the weekend and the first half of this week, and (with the help of a few online math friends) have settled on three things to address:

  1. Do a better job nailing the “why” of Desmos. Let’s say someone attends my session. And another half dozen sessions over the course of the weekend. Then rapid reintegration into family life on Sunday afternoon/evening, and rude awakening/come-down-from-the-mountain re-entry into school life on Monday. Why would anyone—even those who enjoyed the session and were momentarily impressed or inspired—deviate from their longstanding routine to spend time and energy changing the way they teach and their students learn? That’s the question I need to answer, whether in relation to Desmos, Posers + Solvers, Twitter as My PLC, Technology and the Curious Mind, or any of the other sessions I’ve been about lately. And whatever the answer is, it needs to bleed through in every part of the session.
  2. Less is more. I think I crammed too much into this one. My goal was not to provide an everyone-watch-me-and-try-to-do-the-exact-same-thing experience. Those almost always explode into frustration within minutes (at least for me as a presenter) when working with a group of that size. Instead, my aim was to inspire participants with some new ideas for how Desmos could be used. I assumed some previous Desmos experience, which was a safe assumption for many (though not helpful for those few who were new to the tool). However, I think everyone wins if I slow down, take a few more breaths, and invite people to try a few things along the way instead of just near the end.
  3. Bump regression to the middle or end. Regression was launched literally the day before CMC South. It was like candy, or a new toy, or… I don’t know, something entirely awesome. So what did I do? I crammed it into the beginning of my session, since that’s the piece of I was most excited about. But I derailed an otherwise fun and engaging lesson—Hundred (Thousand) Meter Dash—with some rapid-fire regression keystrokes and a few rounds of “don’t worry about how to do this, just watch me do it.” The result? More than a few lost (and/or bored) participants. The fix? Push that piece to the end, and/or modify it considerably. In particular, I could “fast forward” through the keystrokes and instead load a pre-made graph to show off some already-built slider-based models, then some regression-based models. I think I’ll better understand how to revamp this portion of the session after I nail down the “why” of my presentation.

Speaking of the why, I’m debating between one big takeaway, or three key ideas. Whether the list includes one item or three, this will definitely be on it:

The whole point is to get to the math conversation.

Desmos paves the way to that conversation better than any tool I know.

Your Turn

With that as the backdrop, I have two questions for you:

  • What is the “why” of a session you’ve given recently? If you haven’t nailed it down, how could you?
  • What is your “why” for Desmos? If you had 15 seconds to convince someone that they cannot live without Desmos in their classroom moving forward, what would you say?


I decided to pull my own session out of my two-part CMC North Sessions Recap series because it was less a recap of my session and more a reflection of how to improve the material for a future conference or workshop. If you’re interested in reading more about my experience at CMC North 2014, check out these posts:

CMC North 2014 Recap: The Sessions, Part 2

Part 1 of this session recap is available here.

Session 5

Video Games and Making Math More Like Things Students Like
(Dan Meyer • Saturday, 11:00-12:00 pm)

If you’ve never seen Dan Meyer present, you’re missing out. Great content, great delivery. His slides are polished, there’s always something worth pondering, and the audience usually ends up in stitches once or twice. This session was no different.

I sat next to a woman who was a little skeptical at the beginning. Based on the title, I think she expected to hear about math-based video games (i.e., a thin veneer of video game painted on top of drill-and-kill practice). This was my first time seeing Dan give this particular talk, but after reading a few related posts from his blog I was anticipating another message, and suggested as much. Turns out my suspicions were correct, and my neighbor (and I) thoroughly enjoyed the session.

The major takeaways from the session:

Six Lessons from Video Games
  1. Video games get to the point.
  2. The real world is overrated.
  3. Video games have an open middle.
  4. The middle grows more challenging and interesting.
  5. Instruction is visual, embedded in practice, and only as needed.
  6. Video games lower the cost of failure.

I’m struggling a bit with how to incorporate all of these ideas into my classroom. But one that I think will make an immediate and positive impact is #3, “video games have an open middle.” As it turns out, this phrase from Dan’s talk is what inspired Robert Kaplinsky and Nanette Johnson to create Whether you take problems directly from the website, or simply draw inspiration from what you find there, I highly recommend checking it out.

A few more tweets from the session:

Session 6

Desmos: Infinite Graphing Power on Every Device. For Free.
(Me! • Saturday, 1:30-3:00 pm)

Selected resources from my session are available here:

A reflection on my session (less a recap, and more a few thoughts on how to get better) is coming soon.

[UPDATE: My session reflection post is here.]

Session 7

Implementing Real World Problem-Based Math Lessons
(Robert Kaplinsky • Saturday, 3:30-5:00 pm)

I planned my session attendance this year entirely based on presenter names. I missed Andrew Stadel, Dan Meyer, and Robert Kaplinsky at CMC South in October, so I made sure to put them at the top of my list for CMC North. The one problem—unique to Asilomar, possibly—that such an approach brings, is that the physical distance between sessions might make on-time arrival difficult. For Dan’s session, that turned out not to be a big deal. He was in Merrill Hall, the biggest room at the event, and there were (thankfully) unlocked, unguarded closets full of chairs on each side of the hall, seemingly designed for latecomers like me to select a chair and find some open ground. (Surprisingly enough, front left had room for one more chair.)

My session ended at 3 pm, about 2 miles away from Robert’s session. A short walk, a bus ride, a light jog in the wrong direction (I really should learn to read a map), and a more intense jog in the right direction, led me to a packed-out room designed for 30 or 40. I was the 55th person to walk through the doors, and I found the last few square inches of unoccupied carpet halfway beneath the water table on the side. (Oh, and I had signed up to be the “presider” for Robert’s session, so I was supposed to introduce him. “Er, sorry Robert. Can you introduce yourself? I’m having trouble breathing right now.”)

Okay, traveling nonsense aside, Robert’s session was excellent.

He led with his 100-by-100 task, and we actually lingered there for quite some time. Despite the significant time dedicated to the problem, we seemed to change gears regularly, swapping our student hats for our teacher hats, and back and forth for a few rounds. (As a teacher trainer and conference speaker, I took a few mental timeouts to note the way Robert led the group of teachers through the session. Aspiring presenters, get thee to a Kaplinsky session! And put Stadel and Meyer on your bucket list, for that matter.)

I remember telling Matt Vaudrey after attending his La Cucina Matematica presentation in Palm Springs (recap) that the highlight for me was the way he showed participants just how far/deep one of his tasks would go. Rather than seeing a rapid-fire “best of” showcase of great tasks, I really enjoyed camping on one context for an extended period of time. I’m sure some teachers would prefer to see a dozen different tasks, but I think I benefit more from delving deep into a smaller set.

All of that “delving deep” and exploring one thing from multiple angles that I enjoyed in Vaudrey and Stevens’ CMC South session was alive and well in Robert’s session here in Asilomar. Seeing him launch the task, lead us through the turn from Act 1 to Act 2, drawing connections to the Common Core practice standards (all eight, if I remember correctly), adding problem based lesson commentary, displaying student work, and responding to some problem-based lesson FAQs along the way… All of those moves—anchored to a single rich task—proved very helpful to me as a teacher who appreciates this style of lesson, but hasn’t seen too many teachers take it from start to finish.

Toward the end, Robert talked for a bit about “The Four C’s.”

I’d suggest his blog post on the subject as required reading. Here’s a teaser:

As a district math coach, my challenge has been successfully demonstrating problem-based learning in academically diverse classes. I am frequently unsure of what to expect as I go into unfamiliar classrooms to work with a variety of students. An interesting problem that achieves wonderful results in one class causes frustration and anxiety in another class that appeared similar on paper. These struggles have led me to come up with four C’s that I believe teachers should focus on to improve their success with problem-based learning: communication, curiosity, critical thinking, and content knowledge.

Session 8

(Lots of cool folks • Saturday, 7:30 pm)

Yes, the Ignite talks were recorded. No, the videos aren’t available. Late spring would be my guess based on last year’s release dates.

Session 9

Erasing Mathematics Failure Through a Growth Mindset and Multi-dimensional Mathematics
(Jo Boaler • Sunday, 9:00-10:15 am)

In Part 1 of my CMC North recap, I mentioned that tweeting during a session forces me to focus, process, and remember what I’m hearing. Then there’s the added side benefit of making a dead simple recap. With that in mind, I’ll let my tweets do the recapping for Jo Boaler’s outstanding session.

You made it past all those tweets? You deserve a cookie! (Unfortunately, I’m fresh out.)

Session 10

Stepping Stones
(Phil Daro • Sunday, 10:45-12:00 pm)

Phil is outstanding, however… I’ve seen him present before, and I hadn’t yet set foot on the beach. I planned to head home as close to 12 noon as I could, so I skipped this last session and wandered down to the water. I’m sure Phil’s talk was characteristically excellent, but I have no regrets.

Everybody and their Mother Loves the MTBoS

That post title may be a bit of a stretch. But I know this much is true: I love the mathtwitterblogosphere, and I know my mother thinks it’s pretty swell also.

A little background… My mom teachers middle school math, just a few miles from where I do. When I started out teaching, she was my go-to resource for teaching questions (especially classroom management). I’ve often been a resource for her regarding conceptual development or activity ideas for a current or upcoming topic.

Here’s what she sent me last night at the end of an email:


Lacking any particular inspiration, I directed her to Robert Kaplinsky’s PrBL Search Engine, and then hopped on Twitter:

The response? Pam Wilson to the rescue!


In particular, this seemed like a really cool idea:


So what happened the next morning? This!


Now that’s a pretty cool Professional Learning Community/Family.

P.S. Here’s a glimpse at the handout my mom created.


P.P.S. And her thoughts on the lesson, including what she’ll tweak for next year:



CMC North 2014 Recap: The Sessions, Part 1

Here’s how I started my CMC South 2014 Recap:

Last weekend turned out to be one of my favorite weekends of my teaching career. I haven’t exactly been at this for decades, but 11 years is no short span either. The weekend was that enjoyable, at least for me.

CMC North (a conference I’ve attended several times before) didn’t disappoint, even in comparison to my weekend in Palm Springs back in October. My Saturday session wasn’t nearly as strong as in socal (more on that in another post), but we had some fun nonetheless, and several people shared some really positive feedback. On top of that, Saturday evening’s Ignite sessions were a crazy whirlwind of terror and joy.

After CMC South, I promised a two part recap: One focused on sessions, one focused on “everything else” (with an emphasis on people and community). My game plan for this post is to provide a quick recap of each session I attended. Later in the week (if I have any energy left), I’ll drop a combined “Everything Else” recap of CMC North and South.

On to the session recaps!

Session 1

“Does That Make Sense in the Story?”: Launching and Exploring Rich Problems
(Max Ray • Friday, 1:30-4:30 pm)

I kicked off my CMC North experience (scrambling into the session just minutes before it was about to start) with a hello/handshake combo from Max Ray. I’ve more or less turned into a Max Ray superfan over the past 12 months. From his book to his blog to his numerous Ignite talks… I’ve been soaking up as much of his writing and speaking as I can. It was a pleasure to meet him face to face, and great fun to sit in his low-key, problem-rich, thoughtful session to kick of what I hoped would be a great weekend.

I’ve discovered that Tweeting during a workshop or presentation helps me focus, process, and remember key ideas. There’s something about the 140 character limit that forces me to distill ideas I’m hearing into more memorable nuggets. At any rate, I put this into practice at a few sessions this weekend, including Max’s. You can find all of my Tweets related to Max’s session right here.

If you’re, notes I took during the session are available here.

Session 2

Math in the Movies
(Tony DeRose • Friday, 7:30-9:00 pm)

I planned on attending the keynote, but was (a) going to be late, (b) growing more nervous about my Ignite talk, (c) less-than-inspired by the session title. So I went back to the Pacific Grove Inn and spent the next couple of hours talking to myself on the porch.

Session 3

Get Students to Argue Through Number Sense Activities
(Andrew Stadel • Saturday, 8:00-9:00 am)

I’ll take the lazy way out and summarize my experience in Andrew’s session with a few tweets:

That certainly doesn’t do it justice, but it’s a start. My favorite things  from Andrew’s session were seeing him in action for the first time (I’ve been stealing his estimation and three act tasks online for about two years) and his framework for number sense activities:

  • A framework for number sense activities, and constructing viable arguments
  • Simple visual and/or question
  • Competition or guessable answer
  • Students create a representation
  • Students assess each other (accountability)
  • Students define what is important, vocabulary

I scribbled a few other notes down during the session if you want to have a look.

Session 4

Calculus Adjacent: Designing Math Electives Accessible To All
(Bree Pickford-Murray • Saturday, 9:30-10:30 am)

After Andrew’s session I hopped on a yellow school bus and rode over to the Middle School for a session from a fellow Igniter (@btwnthenumbers):

I walked in the door a minute before the session was supposed to start, a little out of breath. (Did anyone else run from the bus to a session or two this weekend? Just me? Oh well…) Luckily, I found an open seat—next to @cheesemonkeysf, no less!—and we were off. (As the “presider” I had the honor of introducing Bree, something made a little more awkward by my out-of-breath-ness).

For years I’ve thought about enriching our math department with some non-calculus electives, but I’ve never pulled the trigger. Sitting around a group of people who were in a similar position, and quite a few who have already journeyed down that road with courses like Cryptography, History of Math, Topics in…, Topology, and Game Theory, was a great way to spend the middle of the morning. Bree did an excellent job leading the discussion. There were about 25 people in the room, and I believe almost every chimed in at least once during the session. I left with some peer-inspired inspiration, quite a few ideas to explore, and a slight increase to the number of math titles on my Amazon wishlist.

Also, there was a woman in the session (her name was Laura Hawkins… and I’m thinking this might be here) with a handful of amazing comments/quotes. I wrote them down in my notes, which are here. I also created a Google Document to collect “Courses Offered” and “Courses Considered” ideas from everyone in the session. I never ended up sharing it since it wasn’t my session. (It seemed like it might have been rude to jump in like that.) Anyway, this is my blog, so there’s no need for restraint. If you’d be interested in contributing to it, have at it!


Well, that’s enough typing for a bit. I’ll share some thoughts/notes from Dan Meyer, Robert Kaplinsky, and Jo Boaler’s sessions very soon.

[UPDATE: Part 2 is available here.]

CMC North 2014 Ignite

An Extra Jolt of Fear

I gave an Ignite talk (5 minutes, 20 slides, ready-or-not auto-advance every 15 seconds) at this year’s CMC North conference in Asilomar. It was quite possibly the most fun I’ve had at any math conference. In the hours leading up to the event, I was more than a little nervous. Then I saw the speaking order:



Just before we went on, one of my co-igniters asked whether it was possible to ruin one’s career in just five minutes. We hoped not. As it turned out, we did alright. In fact, from where I sat, the other nine presenters were all amazing.

To Script or Not To Script

After preparing my slides, I wrote out a script of what I intended to say. I’ve never scripted anything before, but then again I’ve never done anything with such an unforgiving format. I imagined myself falling silent for an entire slide or two. That make-believe mental picture wasn’t pretty, and scripting seemed like the answer. I struggled to get past 80% memorized, so I abandoned the “word for word” approach, didn’t look at my notes at all on Saturday, and made it my goal simply to drive home the main idea of each slide. For anyone who skipped the Ignite talks (shame on you!), or for those who attended but are curious how my live presentation (and occasional bumbling) compared to what I originally had in mind, here are my slides along with what I would have said if I had been able to stay 100% on target.

Slides and Script


[Suzanne Alejandre presses the space bar, and we’re off!]



Here’s my goal for the next five minutes: To convince you that some of what passes for innovative use of technology in the classroom is actually dehumanizing and therefore destined to be ineffective. And what better way to kick things off than to waste two slides on a blue police box.



For those not addicted to Doctor Who, that’s the TARDIS. An amazing box of infinite potential. It’ll take you anywhere in space, and any time in history. It’s basically a time-machine-powered promise of total freedom.



Here’s another set of boxes with a similar promise. Bigger on the inside? Check. Ability to take you anywhere? Well, sort of. Time travel? Not so much. Anyway, a few years ago one of the box-makers spent billions of dollars developing a shiny new box.



Then they announced that they were going to change the world. In particular? Education. Their game plan? To revolutionize the textbook. Well I watched the seven minute infomercial. And I’m not going to lie. I was stoked. The future looked amazing. And seemed like it was just around the corner.



But then I watched the video a second time. And a third time. The shine wore off. The promise disintegrated into a few gimmicks. “Textbooks got you down? You need some pinch-to-zoom action. And flash cards. And sorting activities where students don’t actually sort anything.”



Needless to say, I was bummed out. “You and me, Apple, we were going to change the world. And all you did was tech-wash the idea that teaching consists mainly of sending the right sequence of letters and images, or in this case, 1s and 0s.”



The problem with this approach is that it treats students as passive. As consumers. Worse than that, it actually trains them to be that way. And what we end up with is more of what we don’t want. Indifference. Apathy. Isolation.



But there’s one part in that video that still resonates with me. They said that if we can stimulate curiosity, we’ve got the spark for learning. Amen to that. But what is it that stimulates curiosity? Digital flash cards? Using a stylus instead of a pen?



Or maybe this: A grid of more than 900 skills begging to be mastered—with on-demand help from Uncle Sal whenever you need it. If you weren’t inspired the first time, just pause, rewind, and play it again. That should do the trick. Did you know you can play YouTube videos at quarter speed? As if normal speed wasn’t painful enough.



I’ve seen one version of our tech-saturated future, and it terrifies me, because it looks like this. All headphones and no heart. Everything that makes us most human just put on pause. This isn’t the revolution I want for my students.



It certainly isn’t what I want for my own kiddos. When they come home from school I want to ask questions like: “What made you curious today? What did you create? What inspired you?”



So here’s what I propose: Let’s use technology in ways that foster—rather than stifle—what is most valuable and most human in us and our students. Here are four ways we can do that. #1, let’s replace indifference



…with curiosity. Let’s break out our smartphones and capture strange, thought-provoking things we find in the world. In the kitchen, in the checkout line, a dying battery on your phone, or just aimlessly wandering the Internet… Keep your eyes peeled for the things that hook and engage and provoke.



#2, let’s replace consumption with creativity. I have never had a student email me to say “Hey, I just spent a week binge-watching Khan Academy videos. And it was awesome.” But out of the blue, a student of mine did send me this boat he made with an online graphing calculator. He was so proud of what he’d done.


And not because he earned some boat badge. He was stoked because he made something. And making things is awesome. You know paint by number? How about paint-a-minion by equation. All 424 of them!



#3, let’s replace competition with collaboration. I’m less interested in energy points and clicker quizzes, and more excited to see students using things like Google Drive to create and revise and collaborate.



#4… If I ever write my own young adult dystopian novel, it’ll include a scene from a K-12 classroom with 109 students. At one point the hero (an amiable hacker named Eli) will redirect every kid’s browser to…



In other words, let’s replace isolation with conversation. Headphones and computer cubicles isn’t progress. We need more discourse and more arguments in math class, not less.



If you want to flip the classroom, here’s my model. Let’s flip tech-induced indifference, consumption, competition, and isolation, and replace them with tech-inspired curiosity, creativity, collaboration, and conversation.



In the end, technology is just one tool among many. It’s as powerful or as useless (or worse, as damaging) as we make it. Let’s wield it well. Thank you.

Image Credit

Slide 2Slide 3, Slides 4-6, Slide 7, Slide 8, Slide 9, Slide 10, Slide 11, Slide 12, Slide 15a, Slide 15b, Slide 16, Slide 17, Slide 18, Slide 19


Problem of the Week Gold

I saw a question on Twitter earlier tonight asking for good Problem of the Week resources. A light brain-wracking produced four ideas. A search through old links revealed a fifth. Here’s what I suggested:

I’ll admit, mixed in there I’ve got two definitely-not-problem-of-the-week resources, and one problem of the month website. But still, they’re all fantastic resources, and go-to places for me when I’m looking for a rich problem.

What are your favorite resources (online or in print) for rich problems that might serve teachers and students well in a POW or POM context? Have a thought? Drop a line in the comments!

Gearing Up for CMC North 2014

This Friday a host of passionate teachers will converge on Pacific Grove, CA for the California Mathematics Council’s northern conference. I’m presenting a session on Desmos (almost identical to the one I gave at CMC South in October). I’m also set to give an Ignite talk Saturday evening. (Super excited for that, and more than a little nervous.)

Whether you’re attending and want to keep an eye out for others in the MTBoS, or you’re not attending and you’re interested in stoking your math conference jealousy, here’s where you’ll probably find me:

Friday, 1:30-4:30 pm [Pre-session]

Max Ray • “Does That Make Sense in the Story?”: Launching and Exploring Rich Problems

Friday, 7:30-9:00 pm [Keynote]

Tony DeRose • Math in the Movies

Saturday, 8:00-9:00 am

Andrew Stadel • Get Students to Argue Through Number Sense Activities
Steven Leinwand • Shift Our Mindsets from Remembering How to Understanding Why
Jennifer North Morris, John Berray • Do the Math: Like Your Life Depends On It

Saturday, 9:30-10:30 am

Bree Pickford-Murray • Calculus Adjacent: Designing Math Electives Accessible To All
David Foster
• The Decisions and Shifts Required by the CCSS
Michael Serra • Martin Gardner and the Mathematical Practices
Patrick Callahan • Mathematical Reasoning: Why We Are Bad at It

Saturday, 11:00-12:00 pm

Dan MeyerVideo Games and Making Math More Like Things Students Like
Brad Fulton
• Designing and Implementing Performance Tasks

Saturday, 1:30-3:00 pm

Michael Fenton • Desmos: Infinite Graphing Power on Every Device. For Free.

Saturday, 3:30-5:00 pm [The stacked hour]

Robert KaplinskyImplementing Real World Problem-Based Math Lessons
Andrew Stadel
• Modeling Mathematics Using Problem-Solving Tasks
Annie Fetter • Noticing and Wondering, a Vehicle to Understanding a Problem
Nanette Johnson • Fostering Perseverance with Interesting Math Problems
Elizabeth Statmore • Talk Moves & Task Structures for Productive Mistake Analysis

Saturday, 7:30 pm [Ignite]

My talk is called Technology and the Curious Mind. Here’s the flyer for the event:


Sunday, 9:00-10:15 am

Jo Boaler • Erasing Mathematics Failure Through a Growth Mindset and Multi-dimensional Mathematics

Sunday, 10:45-12:00 pm

Phil Daro • Stepping Stones

P.S. Hey Twitter folks… Whether you’re at the conference or not, keep an eye on #CMCN14.

Counting Fish

Last night my oldest son (Caleb, 5) asked if we could play Monopoly before he went to bed. I said, “Uh, no thanks.” (It was after 8 pm, and I still had aspirations to be in bed while it was still called Thursday.) He then asked if we could play the “fish game” instead.


“That, my friend, is a great idea.” And with that, it was on.

Round 1

We played our first round; I was thoroughly destroyed. (I’m not sure how. I should at least be able to put up a good fight. He’s only five. And I’m not that uncoordinated.) Then—while channeling my inner Christopher Danielson and Andrew Stadel—I asked Caleb how many fish he thought we had each caught.

He made a guess about mine (7), and then we counted. I decided to line them up:


“Off by one. Not bad.”

Then we looked at Caleb’s catch. His guess? 16.


He then counted his fish. Not surprisingly, he lined his up (just as I had). But he made an interesting move:

Was that a coincidence? Or was that move inspired by what he’s learned and learning about our number system?

Round 2

I find these TMWYK conversations wonderfully interesting, and Caleb is happy to play along, provided that we don’t linger for too long. With that in mind, we moved on to another round. I decided to capture a video of our second battle, and the counting that would follow:

Another interesting move! Arranging in fives. Coincidence? Or is he starting to wrestle with 5 (half of 10) as another friendly number at his disposal?



I shared the video with my wife before heading off to bed last night. She was similarly intrigued by the way he arranged the fish while counting. And though my curiosity has yet to be quenched (it will take some followup conversations to figure out his level of intentionality in arranging the fish in that way), I noticed on this second viewing that Caleb arranged the fish by color. In the first case (10 + 3), he ran out of room, and decided to put the last color (red, with three fish) on the next row.

As for the second round, where Caleb arranged things into fives? There were five different colors of fish, three per color (except for that last one).

Next Time

One of the things I love about sharing these conversations is that in writing them down I almost always think of another question or two I might have asked along the way. The arrangements of fish (complete or otherwise) now look to me like fertile soil for rich mathematical conversations about addition, subtraction, multiplication, and factors. In making estimates and checking them by counting, we have a great opportunity to discuss about “more than” and “less than,” and could easily reflect on whether we tend to over- or underestimate in our guessing.

Granted, those are things I thought of only after the fact, while writing down the less-interesting version of things (reality). But for me, that’s the value. In the same way that reflecting on my teaching practice helps me grow as a teacher, reflecting on my conversations with my kids will help me grow in my ability to challenge and encourage and excite them through our father-son or father-daughter discourse.


I showed the video to Caleb. (“Look, little man! You’re on the Internet!”)

Then, regarding the first round: “Why did you arrange the fish like that?”

Caleb: “Because I ran out of room.”

And for the second round: “Why did you line them up that way?”

Caleb: “Because I love rainbows, and I made them like a rainbow.”

Well, numerical motivation for (10 + 3) and (5 + 5 + 4) arrangements may be a few months off. But he does have a nice eye for design. :)

And if I can pull another classroom takeaway from this conversation, it would be this: The best way to know what they’re thinking? Ask! In a parent-child exchange, this happens naturally and easily in conversation. In the classroom, we’ll have our fair share of individual conversations like this, but also a great number of whole-class-all-at-once interactions. The manner in which we ask for their thinking changes (e.g., a written response instead of a spoken one), but the importance of inquiring remains.

Where’s the emphasis? Right where it belongs.

One of the things I love about Desmos is that it allows students and teachers to keep the focus where it should be. Working on a linear approximation problem in Calculus? It’s easy to get caught up in algebraic and numerical details and lose sight of the big (somewhat amazing) picture: We can approximate a crazy curve with a simple line! (Provided we stay in the neighborhood, of course.)

And if you make a mistake, it can be an absolute bear to track it down. Is your issue differentiation? Evaluating a function? Or is your weak spot related to what’s happening visually in linear approximation?

Here’s a problem from last year’s AP Calculus review workbook:


There’s a lot of great work on the page.

  • Take the derivative? Check.
  • Find the slope of the tangent line? Check?
  • Find the equation of the tangent line? Check.

But that’s where things fall apart.

Now, imagine you’re a calculus student. You’ve been hammering away at this thing for several minutes. Maybe you don’t even remember what the problem’s asking for in the first place. You get to this point, you scan the original problem, see an input value of 4.2, and presto-whammo, plug it in and out comes 0.4. “Great news, everyone! That’s on the list! Well done, folks. On to the next problem!”

Only, that answer is entirely wrong. So how do you debrief this problem with a student, small group, or class, so they can see the source of the error? For problems with even a sliver of something graphical, Desmos has become my go-to tool for helping students find and fix their errors.

Here’s what I built with a pair of students last year (with a link to the live graph here):

Screen Shot 2014-11-12 at 6.43.51 PM

Students can’t use Desmos on the AP exam (for now, anyway), so I’m not trying to permanently sidestep what they ultimately must be able to do sans technology (or with a device from that “other” graphing calculator company). But what we can do in class with Desmos is build a better visual/conceptual sense of what’s happening in this problem so they’ll be more prepared for something similar in the future.

Here’s a short list of what this Desmos graph did for us in this scenario:

  1. We offloaded the algebraic and numerical work of finding the derivative (and evaluating it for a particular x-value) and built the tangent line in a matter of seconds (rather than minutes). Mentally, we’re still fresh, and ready to focus on what the problem is really asking us to do: compare the function and the tangent line.
  2. We gave things specific names so we could call on them in our time of need. Okay, that may sound a little dramatic. But think about why we even bother with function notation. Why give a function a name? Well, why did your parents give you a name? So they could call on you! (“Alfred! Get down here and pick up your comic books!”) So then, why do we give functions names? Because function notation is on the Chapter 8 test in Algebra 2? No! We give functions names so we can call on them. So they can do our bidding. If you don’t name it, it’s difficult to put a function to work for you. Give it a name? Now our wish is its command. (It sounds a little bit like we’re going to take over the world, with math as our trusty sidekick.)
  3. We gave things specific names so we could keep clear in our own minds the various moving pieces in the problem. There’s a function. We called that f. There’s a tangent line. We called that l (or t, or whatever). As we approach the end of the problem, and we start looking for the error, it’s easier to avoid simply evaluating f(4.2) or l(4.2), because we know we’re dealing with both f and and l. (How could we forget?! We named them! We practically gave birth to them.)
  4. We visualized the error with a beautiful little orange bar, and in doing so imprinted on our minds (for future problems) what linear approximation error looks like.
  5. We dropped a slider in so we could answer a hundred related problems in a matter of seconds, further clarifying for the confused student (or teacher; I was terrified of linear approximation my first two years of teaching Calculus) what the problem is really about, and all with a nifty, dynamic burst of compare-and-contrast.

Do you need Desmos to teach this stuff? Maybe not. But given the option, I’ll use it every time. Last year in my classroom, we got a lot more out of a five minute Desmos-supported conversation than we did in a whole day of business-as-usual notes and examples.

So this year? We went straight to Desmos and put the emphasis right where it belongs.

P.S. I love GIFs.

Screen Recording 2014-11-12 at 07.12 PM

A Good Day in Precalculus

After struggling a bit earlier in the week, it was nice to end with a positive experience  in Precalculus on Friday. There’s nothing profound in the sentences below, but I still think it’s worth writing down, if for no other reason than increasing the likelihood that I recall some of this as I plan future lessons.

Context (for the Year)

We made the move to block schedule this year. After years of being opposed to the idea, I recently softened my stance and even began to think a block schedule approach might be helpful. I’m still learning the ropes, but the “early returns” in my own classroom (from me as well as my students) are positive.

Context (for the week)

We’ve been working on trigonometric properties and identities. I had a rather disappointing experience in Precalculus on Wednesday, and though I can’t recall the details of last Monday, I’m pretty sure it was far from amazing.

The Daily Plan

As students shuffle into class each day, I throw “The Daily Plan” on the projector. Friday’s looked like this:


Mini Exploration

Some of what I disliked about my lessons from earlier in the week related to the all-eyes-on-me approach I took. So I decided to start Friday with a miniature exploration. Again, nothing profound, but something that would:

  • Provide students with a brief review of the properties they would need to have at-the-ready for today
  • Give each group (or pair) of students an opportunity to proceed as quickly/slowly as they needed to
  • Challenge students to express their thinking in writing and in discussion
  • Allow me to wander through the room, checking progress, lingering with students who needed extra support (which I tried to supply via questions, not statements)
  • Require students to do some individual and small group thinking and wrestling before our whole class discussion/recap

Here’s what the handout looked like:

page-one page-two

And here’s a link to the two-page PDF (in all of its non-glory).

Reordering Task

After debriefing the mini exploration (which included a whole-class conversation and a “puppet volunteers” work-through of Problem 6), we moved on to a reordering task. It’s an idea I had been thinking about for a few months (years?), but one I had never put into action. After using this on Friday, I think I’m sold on its quality—at least for some problem types.

Here’s a look at the student page:


And a link to the handout, for closer inspection.

Properties Quiz

I should have done this each day for the last three days, but my moments of genius are few and far between. On several occasions in class we’ve described the properties as “puzzle pieces” and proving identities as “solving a puzzle.” If you don’t have the basic properties memorized, it’s like trying to do a puzzle with pieces missing. Apparently that metaphor was insufficiently inspiring, as many of my students spent little to no additional time at home committing the reciprocal, quotient, and Pythagorean properties to memory. And this lack of recall presents a problem for the work at hand.

An older version of me would have tried to remedy that problem by lecturing the class about the importance of blah blah blah. On Friday, I decided to skip over that part and instead gave students a few minutes of class time to work on committing these bad boys to memory. “5/5/5″ on the daily agenda meant five minutes of silent and individual study, five minutes of partners quizzing one another verbally and/or in writing, and five minutes of do-the-best-you-can-on-your-own quizzing. I may have only given them 3 or 4 minutes for each stage, but the results were great. Most students now have the “puzzle pieces” in hand, and those that do not know exactly where their weaknesses lie.

Visual Patterns

In a block schedule setting, I’m finding that focusing on trig properties and identities for the entire class period is just too much of the same thing. To mix things up—and to plant some seeds for an upcoming functions- and graphing-heavy chapter—we worked through a Visual Pattern (our first one in months). As an aside, I wish I did a better job of sticking with my start-of-the-year resolutions (e.g., “Visual Patterns will be a regular feature in such-and-such class this year.”) I suppose it’s not too late to bring it back into the mix…

Personal Takeaway

If my second sentence in this post is going to be true, I need to nail down why I think Friday was better than the other days in Precalculus last week. I think it boils down to two things:

  • In designing the lesson, I endeavored to make my students the key do-ers throughout the lesson (whether via thinking, writing, arguing, sorting, explaining, or defending)
  • In an effort to maintain student focus and fight off the feeling of the class “dragging on and on,” I provided students with several distinct (though still related) tasks

As I create my next set of daily plans, I’ll try to keep these little victories from Friday in mind.