Category: futuretext

Total 21 Posts

What’s Next For Me: Desmos

I’m signing on with Desmos as their Chief Academic Officer. Job one is producing the best digital math curriculum in the world. We’ve started that project already.

This is an easy call. I need a question to carry me through my thirties and I can’t think of a better one than, “What does the math textbook of the future look like?”

I’ve known for awhile I need a certain set of collaborators for that project. I have worked with Eli, Eric, and Jenny for the last three years. We need each other. They need what I do (the math teaching stuff) and I need what they do (the computery stuff). They’re great at what they do and we get along great. Stay tuned.

This New York Times Article Is The Future Of Math Textbooks

I raved for a minute on Twitter last week about this New York Times article. You should read it (play it? experience it?) and then come back so I can explain why it’s what math curriculum could and should become.

The lesson asks for an imprecise sketch rather than a precise graph.

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This is so rare. More often than not, our curricula rushes past lower, imprecise, informal, concrete rungs on the ladder of abstraction straight for the highest, most precise, most formal, most abstract ones. That’s a disservice to our learners and the process of learning.

You can always ask a student to move higher but it’s difficult to ask a student to move lower, forgetting what they’ve already seen. You can always ask for precisely plotted points of a model on a coordinate plane. But once you ask for them you can’t unask for them. You can’t then ask the question, “What might the model look like?” Because they’re looking at what the model looks like. So the Times asks you to sketch the relationship before showing you the precise graph.

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Their reason is exactly right:

We asked you to take the trouble to draw a line because we think doing so makes you think carefully about the relationship, which, in turn, makes the realization that it’s a line all the more astonishing.

That isn’t just their intuition about learning. It’s Lisa Kasmer’s research. And it won’t happen in a print textbook. We eventually need students to see the answer graph and whereas the Times webpage can progressively disclose the answer graph, putting up a wall until you commit to a sketch, a paper textbook lacks a mechanism for preventing you from moving ahead and seeing the answer.

This isn’t just great digital pedagogy, it’s great pedagogy. You can and should ask students to sketch relationships without any technology at all. But the digital sketch offers some incredible advantages over the same sketch in pencil.

For instance:

The lesson builds your thinking into its instruction.

Once it has your guess – a sketch representing your best thinking about the relationship between income and college participation – it tailors its instruction to that sketch. (See the highlighted sentences.)

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The lesson is the same but it is presented differently and responsively from student to student. All the highlighted material is tailored to my graph. I watched an adult experience this lesson yesterday, and while she read the personalized paragraph with interest, she only skimmed the later prefabricated paragraphs. It should go without saying that print textbooks are entirely prefabricated.

It makes your classmates’ thinking visible.

The lesson makes my classmates’ thinking visible in ways that print textbooks and flesh-and-blood teachers cannot. At the time of this posting, 70,000 people have sketched a graph. It’s interesting for me to know how much more accurate my sketch is than my classmates. It’s interesting to see the heatmap of their sketches. And it’s interesting to see the heatmap converge around the point that the lesson gave us for free, a point where there is much less doubt.

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In a version of this article designed for the classroom, students would sketch their graphs and the textbook would adaptively pair one group of students up with another when their graph indicated disagreement. Debate it.

I’m not saying any of this is easy. (“Sure! Do that for factoring trinomials!”) But we aren’t exactly drowning in great examples of instruction enhanced by technology. Take a second and appreciate this one. Then let me know where else you think this kind of technology would be helpful to you in your teaching.

Featured Comment

Avery:

And as far as I know, even with Apple proclaiming “Textbooks that go beyond the printed page” since 2012?, there isn’t a single digital math textbook doing this yet.

[Makeover] Central Park & These Tragic “Write An Expression” Problems

Previously: [Makeover] These Tragic “Write An Expression” Problems

tl;dr. I made another digital math lesson in collaboration with Christopher Danielson and our friends at Desmos. It’s called Central Park and you should check out the Walkthrough.

Here are two large problems with the transition from arithmetic to algebra:

Variables don’t make sense to students.

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We give students variable expressions like the exponential one above, which they had no hand in developing, and ask them to evaluate the expression with a number. The student says, “Ohhh-kay,” and might do it but she doesn’t know what pianos have to do with exponential equations nor does she know where any of those parameters came from. She may regard the whole experience as one of those nonsensical rites of school math which she’ll forget about as soon as she’s legally allowed.

Variables don’t seem powerful to students.

In school, using variables is harder than using arithmetic. But what does that difficulty buy us, except a grade and our teacher’s approval? Meanwhile, in the world, variables are responsible for anything powerful you have ever done with a computer.

Students should experience some of that power.

One solution.

Our attempt at solving both of those problems is Central Park. It proceeds in three phases.

Guesses

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We ask the students to drag parking lines into a lot to make four even spaces. Students have no trouble stepping over this bar. We are making sure the main task makes sense.

Numbers

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We transition to calculation by asking the students “What measurements would you need to figure out the exact space between the dividers?” This question prepares them to use the numbers we give them next.

Now they use arithmetic to calculate the space width for a given lot. They do that three times, which means they get a sense of the parts of their arithmetic that change (the width of the lot, the width of the parking lines) and those that don’t (dividing by the four lots).

This will be very helpful as we take the next big leap.

Variables

We give students numbers and variables. They can calculate the space width arithmetically again but it’ll only work for one lot. When they make the leap to variable equations, it works for all of them.

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It works for sixteen lots at once.

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Variables should make sense and make students powerful. That’s our motto for Central Park.

2014 Jul 28. Here is Christopher Danielson’s post about Central Park on the Desmos blog.

Featured Comment

Grant Wiggins:

In thinking further about your complaint about “Write an expression” I think what is also going on in this app is a NEEDED slowing down of the learning process. The text (and too many teachers) are quick to jump to algorithms before the students understands their nature and value. Look how long it takes to get to the concept of an appropriate expression in the app: you build to it slowly and carefully. I think this is at the heart of the kind of induction needed for genuine understanding, where the learner is helped, by scaffolding, to draw thoughtful and evidence-based conclusions; test them in a transfer setting; and learn from the feedback – i.e. the essence of what we argue understanding is in UbD.

Kevin Hall:

One reason I like this activity so much is that it hits the sweet spot where “What can you do with it?” and “What does it mean?” overlap.

Waterline & Taking Textbooks Out Of Airplane Mode

tl;dr – This is about a new digital lesson I made with Christopher Danielson and our friends at Desmos. It’s called Waterline and its best feature is that it shares data from student to student rather than just from student to teacher. I’ll show you what I mean while simultaneously badgering publishers of digital textbooks. (As I do.)

Think about the stretches of time when your smartphone or tablet is in airplane mode.

Without any connection to the Internet, you can read articles you’ve saved but you can’t visit any links inside those articles. You can’t text your friends. You can’t share photos of cats wearing mittens or tweet your funny thoughts to anybody.

In airplane mode, your phone is worth less. You paid for the wireless antenna in your tablet. Perhaps you’re paying for an extra data plan. Airplane mode shuts both of them down and dials the return on those investments down to zero.

Airplane mode sucks.

Most digital textbooks are in airplane mode:

  • Textbooks authored in Apple’s iBooks Author don’t send data from the student’s iPad anywhere else. Not to her teacher and not to other students.
  • HMH Fuse includes some basic student response functionality, sending data from the student to the teacher, but not between students.
  • In the Los Angeles Unified iPad rollout, administrators were surprised to find that “300 students at three high schools almost immediately removed security filters so they could freely browse the Internet.” Well of course they did. Airplane mode sucks.

The prize I’m chasing is curriculum where students share with other students, where I see your thoughts and you see mine and we both become smarter and life becomes more interesting because of that interaction. That’s how the rest of the Internet works because the Internet is out of airplane mode.

Here’s one example. In Waterline we ask students first to draw the height of the water in a glass against time. We echo their graph back to them in the same way we did in Function Carnival.

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But then we ask the students to create their own glass.

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Once they successfully draw the graph of their own glass, they get to put it in the class cupboard.

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Now they see their glass in a cupboard right alongside glasses invented by their friends. They can click on those new glasses and graph them. The teacher sees all of this from her dashboard. Everyone can see which glasses are harder to graph and which are easier, setting up a useful conversation later about why.

We piloted this lesson in a local school and asked them what their favorite part of the lesson was. This creating and sharing feature was the consensus winner.

A selection:

  • Making my own because it was my own.
  • Trying to create your own glass because you can make it into any size you want.
  • Designing my own glass because I was able to experiment and see how different shapes of the glass affects how fast the glass filled up.
  • My favorite part of the activity was making my own glass and making my other peers and try and estimate my glass.
  • My favorite part of the activity was solving other people’s glasses because some were weird shapes and I wanted to challenge myself.

Jere Confrey claimed in her NCSM session that “students are our most underutilized resource in schools.” I’d like to know exactly what she meant by that very tweetable quotation, but I think I see it in the student who said, “I also liked trying out other’s glasses because we could see other’s glasses and see how other people solved the problem.”

I know we aren’t suffering from too many interactions like that in our digital curricula. They’re hard to create and they’re hard to find. I also know we won’t get more of them until teachers and administrators like you ask publishers more often to take their textbooks out of airplane mode.

Feedback From Computers Doesn’t Have To Be Boring

David Cox sent his students through Function Carnival where they tried to graph the motion of different carnival rides. (Try it!)

Every student’s initial graph was wrong. No one got it exactly right the first time. But Function Carnival doesn’t display a percent score or hint tokens or some kind of Bayesian probability they’ll get the next graph right. It just shows students what their graph means for that ride. Then it lets them revise.

David Cox screen-recorded the teacher view of all his students’ graphs. This is the result. I love it.

BTW. I’m hardly unbiased here, having played a supporting role in the development of Function Carnival.