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Year: 2018

Total 23 Posts

Remainders

By • April 20, 2018 • 20 Comments

Here are three items that crawled into my head some time during the last few months and didn’t find their way out yet. This post is brain surgery.

Lesson Exploder

I have thought about this tweet from David Coffey at least once per week for the last five months.

The Song Exploder podcast interviews artists about the craft of songwriting. The artists describe their motivations for creating their songs, what they were trying to accomplish, and how they tried to accomplish it, all while the Song Exploder team teases out key elements of the song for illustration. I feel smarter about the craft of songwriting whenever I listen to it. Maybe not as smart as if I had spent a year at the Oberlin Conservatory of Music, but for how much smarter I’m feeling, it’s hard to argue with Song Exploder’s cost (free) and scale (internet-sized).

Now swap “teacher” for “artist” and “lesson” for “song.” I know what we can swap in for “Oberlin Conservatory of Music.” Classroom visits. Lesson studies. Problem solving cycles. Professional learning communities. Those are all very effective and also very expensive. I don’t know what to swap in for “Song Exploder,” though — an option that is less effective but basically free and scales with the internet.

What kind of digital media could we use to a) highlight something significant and useful about the craft of teaching b) as quickly as possible c) distributed as widely as possible d) in a form that’s replicable and episodic? (Song Exploder is up to 133 episodes right now.)

What current examples can we find? Teaching Channel videos? Blog posts? Lesson plans? Unedited classroom video? Marilyn Burns distills classroom anecdotes into really popular tweets.

What inspiration can we take from other fields? Delish videos? NFL Red Zone / Mic’d Up? Mystery Science Theater 3000? Twitch streaming?

New Jersey Turnpike

I can’t figure out the tolls on the New Jersey Turnpike.

If you don’t come from turnpike territory, how it works is you enter the turnpike somewhere and you exit the turnpike somewhere else. You pay depending on where you entered and exited.

My assumption is that the pricing would look pretty linear as a function of the miles traveled. Like this:

But it doesn’t. It looks like two linear functions with the second piece starting maybe at the Garden State Parkway. (Why?) And the Pennsylvania Turnpike exit is also way more expensive than a linear function would predict. (Why again?)

Here is the website that tells you the cost of different trips on the turnpike. Eric Berger, our CTO at Desmos, helped me type code into my browser’s Javascript console that returned all the data. Feel free to dig in. I’m looking for answers to my questions about pricing and I’m also interested in possible classroom applications of the data.

Cape Town’s Zero Day

Cape Town has a water crisis and a website that until recently calculated a “Zero Day” for their water reserves, a day when faucets will run dry and people will collect a daily allotment of water from central locations throughout the city.

That’s either terrifying or mathematically interesting, depending on which part of my brain I subdue while I’m thinking about it. How do they calculate that zero day? How can we put students in a position to appreciate, replicate, and even adapt those calculations for their own contexts?

Designing for Mathematical Surprise

By • March 30, 2018 • 18 Comments

“Surprising” probably isn’t in the top ten list of adjectives students would use to describe math class, which is too bad since surprise lends itself to learning.

Surprise occurs when the world reveals itself as more orderly or disorderly than we expected. When we’re surprised, we relax assumptions about the world we previously held tightly. When we’re surprised, we’re interested in resolving the difference between our expectations and reality.

In short, when we’re surprised we’re ready to learn.

We can design for surprise too, increasing the likelihood students experience that readiness for learning. But the Intermediate Value Theorem does not, at first glance, look like a likely site for mathematical surprise. I mean read it:

If a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

[I slam several nails through the door and the floor so you’re stuck here with me for a second.]

Nitsa Movshovits-Hadar argues in a fantastic essay that “every mathematics theorem is surprising.” She continues, “If the claim stated in the theorem were trivial it would be of no interest to establish it.”

What surprised Cauchy so much that he figured he should take a minute to write the Intermediate Value Theorem down? How can we excavate that moment of surprise from the antiseptic language of the theorem? Check out our activity and watch how it takes that formal mathematical language and converts it to a moment of surprise.

We ask students, which of these circles must cross the horizontal axis? Which of them might cross the horizontal axis? Which of them must not cross the horizontal axis?

They formulate and defend their conjectures and then we invite them to inspect the graph.

In the next round, we throw them their first surprise: functions are fickle. Do not trust them.

And then finally we throw them the surprise that led Cauchy to establish the theorem:

But you can’t expect me to spoil it. Check it out, and then let us know in the comments how you’ve integrated surprise into your own classrooms.

Related: Recipes for Surprising Mathematics

2020 Feb 19. via Ben Blum-Smith.

Rough-Draft Talk in Front of Hundreds of Math Teachers

By • March 9, 2018 • 68 Comments

This was new. I was on a raised platform with seven middle school students to my left and six to my right and several hundred math teachers surrounding us on all sides.

This wasn’t a dream. The MidSchoolMath conference organizers had proposed the idea months ago. “Why don’t you do some actual teaching instead of just talking about teaching?” basically. They’d find the kids. I was game.

But what kind of math should we do together? I needed math with two properties:

  • The math should involve the real world in some way, by request of the organizers.
  • The math should ask students to think at different levels of formality, in concrete and abstract ways. Because these students would be working in front of hundreds of math teachers, I wanted to increase the likelihood they’d all find a comfortable access point somewhere in the math.

So we worked through a Graphing Stories vignette. We watched Adam Poetzel climb a playground structure and slide down it.

I asked the students to tell each other, and then me, some quantities in the video that were changing and some that were unchanging. I asked them to describe in words Adam’s height above the ground over time. Then I asked them to trace that relationship with their finger in the air. Only then did I ask them to graph it.

I asked the students to “take a couple of minutes and create a first draft.” The rest of this post is about that teaching move.

I want to report that asking students for a “first draft” had a number of really positive effects on me, and I think on us.

First, for me, I became less evaluative. I wasn’t looking for a correct graph. That isn’t the point of a rough draft. I was trying to interpret the sense students were making of the situation at an early stage.

Second, I wasn’t worried about finding a really precise graph so we (meaning the class, the audience, and I) could feel successful. I wanted to find a really interesting graph so we could enjoy a conversation about mathematics. I could feel a lot of my usual preoccupations melt away.

After a few minutes, I asked a pair of students if I could share their graph with everybody. I’m hesitant to speculate about students I don’t know, but my guess is that they were more willing to share their work because we had explicitly labeled it “a first draft.”

I asked other students to tell that pair “three aspects of their graph that you appreciate” and later to offer them “three questions or three pieces of advice for their next draft.”

  • I like how they show he took longer to go up than come down.
  • I like how they show he reached the bottom of the slide a little before the video ended.
  • I think they should show that he sped up on the slide.
  • Etc.

If you’ve ever participated in a writing workshop, you know that workshopping one author’s rough draft benefits everyone’s rough draft. We offered advice to two students, but every student had the opportunity to make use of that advice as well.

And then I gave everybody time for a second and final draft. Our pair of students produced this:

Notice here that correctness is a continuous variable, not a discrete one. It wasn’t as though some students had correct graphs and others had incorrect ones. (A discrete variable.) Rather, our goal was to become more correct, which is to say more observant and more precise through our drafting. (A continuous variable.)

And then the question hit me pretty hard:

Why should I limit “rough-draft talk” (as Amanda Jansen calls it — paywalled article; free video) to experiences where students are learning in front of hundreds of math teachers?

My students were likely anxious doing math in front of that audience. Naming their work a first draft, and then a second draft, seemed to ease that anxiety. But students feel anxious in math class all the time! That’s reason enough to find ways to explicitly name student work a rough draft.

That question now cascades onto my curriculum and my instruction.

How should I transform my instruction to see the benefits of “rough-draft talk”?

If I ask for a first draft but don’t make time for a second draft, students will know I really wanted a final draft.

If I ask for a first draft, I need to make sure I’m looking for work that is interesting, that will advance all of our work, rather than work that is formally correct.

How should I transform my curriculum to see the benefits of “rough-draft talk”?

“Create a first draft!” isn’t some kind of spell I can cast over just any kind of mathematical work and see student anxiety diminish and find students workshopping their thinking in productive ways.

Summative exams? Exercises? Problems with a single, correct numerical answer? I don’t think so.

What kind of mathematical work lends itself to creating and sharing rough drafts? My reflex answer is, “Well, it’s gotta be rich, low-floor-high-ceiling tasks,” the sprawling kind of experience you have time for only once every few weeks. However I suspect it’s possible to convert much more concise classroom experiences into opportunities for rough-draft talk.

To fully wrestle my question to the ground, how would you convert each of these questions to an opportunity for rough-draft talk, to a situation where you could plausibly say, “take a couple of minutes for a first draft,” then center a conversation on one of those drafts, then use that conversation to advance all of our drafts.

I think the questions each have to change.

Geometry

Arithmetic

Algebra

[photo by Devin Rossiter]

Must Read: Larry Berger’s Confession & Question About Personalized Learning

By • February 14, 2018 • 13 Comments

Larry Berger, CEO of Amplify, offers a fantastic distillation of the promises of digital personalized learning and how they are undone by the reality of learning:

We also don’t have the assessments to place kids with any precision on the map. The existing measures are not high enough resolution to detect the thing that a kid should learn tomorrow. Our current precision would be like Google Maps trying to steer you home tonight using a GPS system that knows only that your location correlates highly with either Maryland or Virginia.

If you’re anywhere adjacent to digital personalized learning — working at an edtech company, teaching in a personalized learning school, in a romantic relationship with anyone in those two categories — you should read this piece.

Berger closes with an excellent question to guide the next generation of personalized learning:

What did your best teachers and coaches do for you–without the benefit of maps, algorithms, or data–to personalize your learning?

My best teachers knew what I knew. They understood what I understood about whatever I was learning in a way that algorithms in 2018 cannot touch. And they used their knowledge not to suggest the next “learning object” in a sequence but to challenge me in whatever I was learning then.

“Okay you think you know this pretty well. Let me ask you this.”

What’s your answer to Berger’s question?

BTW. It’s always the right time to quote Begle’s Second Law:

Mathematics education is much more complicated than you expected even though you expected it to be more complicated than you expected.

Featured Comment

SueH:

I have come to believe that all learning is personalized not because of what the teacher does but because of what’s happening inside the learner’s brain. Whatever pedagogical choices a teacher makes, it’s the student’s work that causes new neural networks to be created and pre-existing ones to be augmented or strengthened or broken or pruned.

Scott Farrand:

I’ll accept the risk of stating the obvious: my best teachers cared about me, and I felt that. Teaching is an act of love. A teacher who cares about each student is much more likely to, in that instant after a student responds to a question, find the positive value in the response and communicate encouragement to the student, verbally and nonverbally. And students who feel cared for are more likely to have good things going on in their brains, as described by SueH.

Desmos + Two Truths and a Lie

By • February 2, 2018 • 57 Comments

I’m absolute junk in the kitchen but I’m trying to improve. I marvel at the folks who go off recipe, creating delicious dishes by sight and feel. That’s not me right now. But I’m also not content simply to chop vegetables for somebody else.

I love the processes in the middle — like seasoning and sautéing. I can use that process in lots of different recipes, extending it in lots of different ways. It’s the right level of technical challenge for me right now.

In the same way, I’m enamored lately of instructional routines. These routines are sized somewhere between the routine administrative work of taking attendance and the non-routine instructional work of facilitating an investigation or novel problem. Just like seasoning and sautéing, they’re broadly useful techniques, so every minute I spend learning them is a minute very well spent.

For example, Estimation 180 is an instructional routine that helps students develop their number sense in the world. Contemplate then Calculate helps students understand the structure of a pattern before calculating its quantities. Which One Doesn’t Belong helps students understand how to name and argue about the names of mathematical objects.

(Aside: it’s been one of greatest professional pleasures of my life to watch so many of these routines begin and develop online, in our weirdo tweeting and blogging communities, before leaping to more mainstream practice.)

I first encountered the routine “Two Truths and a Lie” in college when new, nervous freshmen would share two truths about themselves and one lie, and other freshmen would try to guess the lie.

Marian Small and Amy Lin adapted that icebreaker into an instructional routine in their book More Good Questions. I heard about it from Jon Orr and yesterday we adapted that routine into our Challenge Creator technology at Desmos.

We invite each student to create their own object — a circle graph design in primary; a parabola in secondary.

We ask the student to write three statements about their object — two that are true, and one that is a lie. They describe why it’s a lie.

Here are three interesting statements from David Petro’s circle graph design. Which is the lie?

  • The shaded part is the same area as the non shaded part.
  • If these were pizzas, there is a way for three people to get the same amount when divided.
  • If you double the image you could make a total of 5 shaded circles.

And three from Sharee Herbert’s interesting parabola. Which is the lie?

  • The axis of symmetry is y=-2.
  • The y-intercept is negative.
  • The roots are real.

Then we put that thinking in a box, tie a bow around it, and slide it into your class gallery.

The teacher encourages the students to use the rest of their time to check out their classmates’ parabolas and circle graphs, separate lies from truth, and see if everybody agrees.

Our experience with Challenge Creator is that the class gets noisy, that students react to one another’s challenges verbally, starting and settling mathematical arguments at will. It’s beautiful.

So feel free to create a class and use these with your own students:

2018 Feb 6. I added eight more Two Truths & a Lie activities on suggestions from y’all!

BTW. Unfortunately, Challenge Creator doesn’t have enough polish for us to release it publicly yet. But I’d be happy to make a few more TTL activities if y’all wanted to propose some in the comments.

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