Category: tech enthusiasm

Total 104 Posts

Teach the Controversy

Here is how your unit on linear equations might look:

  1. Writing linear equations.
  2. Solving linear equations.
  3. Applying linear equations.
  4. Graphing linear equations.
  5. Special linear equations.
  6. Systems of linear equations.
  7. Etc.

On the one hand, this looks totally normal. The study of the linear functions unit should be all about linear functions.

But a few recent posts have reminded me that the linear functions unit needs also to teach not linear functions, that good instruction in [x] means helping students differentiate [x] from not [x].

Ben Orlin offers a useful analogy here:

If I were trying to teach you about animals, I might start with cats and dogs. They’re simple, furry, familiar, and lots of people have them lying around the house. But I’d have to show you some other animals first. Otherwise, the first time you meet an alligator, you’re gonna be like, “That wet green dog is so ugly I want to hate it.”

Michael Pershan then offers some fantastic prompts for helping students disentangle rules, machines, formulas, and functions, all of which seem totally interchangeable if you blur your eyes even a little.

Not all rules that we commonly talk about are functions; not all functions are rules; not all formulas have rules; not all rules have machines. Pick two: not all of one is like the other. A major goal of my functions unit to help kids separate these ideas. So the very first thing I do is poke at it.

And then I was grateful to Suzanne von Oy for tweeting the question, “Is this a line?” a question that is both rare to see in a linear functions unit (where everything is a line!) and important. Looking at not lines helps students understand lines.

So I took von Oy’s question and made this Desmos activity where students see three graphs that look linear-ish. The point here is that not everything that glitters is gold and not everything that looks straight is linear. Students first make their predictions.

Then they see the graphs again with two points that display their coordinates. Now we have a reason to check slopes to see if they’re the same on different intervals.

Finally, we zoom out to check a larger interval on the graph.

I’m sure I will need this reminder tomorrow and the next day and the next: teach the controversy.

BTW. In addition to being good for learning, controversy is also good for curiosity.

Bonus. Last week’s conversation about calculators eventually cumulated in the question:

“Calculators can perform rote calculations therefore rote calculations have no place on tests.” Yay or nay?

I’ve summarized some of the best responses – both yay and nay – at this page. (I’m a strong “nay,” FWIW.)

[Desmos Design] Why We’re Suspicious of Immediate Feedback

One of our design principles at Desmos is to “delay feedback for reflection, especially during concept development activities.” This makes us weird, frankly, in Silicon Valley where no one ever got fired for promising “immediate feedback” in their math edtech.

We get it. Computers have an enormous advantage over humans in their ability to quickly give students feedback on certain kinds of work. But just because computers can deliver immediate feedback doesn’t mean they always should.

For example, Simmons and Cope (1993) found that students were more likely to use procedural strategies like trial-and-error in a condition of immediate feedback than a condition of delayed feedback.

I think I can illustrate that for you with this activity, which has two tasks. You get immediate feedback on one and delayed feedback on the other.

I’ll ask you what I asked 500 Twitter users:

How was your brain working differently in the “Circle” challenge [delayed feedback] than the “Parabola” challenge [immediate feedback]?

Exhibit A:

The circle one was both more challenging and fun. I found myself squinting on the circle to visualize it in my head while with the parabola I mindlessly did trial and error.

Exhibit B:

With the circle, the need to submit before seeing the effect made me really think about what each part of the equation would effect the graph in each way. This resulted in a more strategic first guess rather than a guess and check approach.

Exhibit C:

I could guess & check the parabola challenge. In the circle challenge I had to concentrate more about the center of the circle and the radius. Much more in fact.

Exhibit D:

I couldn’t use trial and error. I had to visualize and estimate and then make decisions. My brain was more satisfied after the circle.

Exhibit E:

I probably worked harder on [the circle] because my answer was not shown until I submitted my answer. It was more frustrating than the parabola problem – but I probably learned more.

This wasn’t unanimous, of course, but it was the prevailing sentiment. For most people, the feedback delay provoked thoughtfulness where the immediate feedback provoked trial-and-error.

We realize that the opposite of “immediate feedback” for many students is “feedback when my teacher returns my paper after a week.” Between those two options, we side with Silicon Valley’s preference for immediate feedback. But if computers can deliver feedback immediately, they can also deliver feedback almost immediately, after a short, productive delay. That’s the kind of feedback we design into our concept development activities.

BTW. For a longer version of that activity, check out Building Conic Sections, created by Dylan Kane and edited with love by our Teaching Faculty.

[Desmos Design] Algebra Is Power, Not Punishment

This is the first of several posts where I’ll use the activities Desmos created last quarter to illustrate our design principles.

One of those principles is:

Create an intellectual need for new mathematical skills.

Nowhere is that principle more necessary, in our view, than in the instruction of algebraic expressions. Three of my least favorite words in the English language are “write an expression” because they so often mean we’re asking the student to do the difficult work of variable manipulation without experiencing any of the fruit of that work.

In both of the questions below, students are likely to experience the work of writing an expression as punishment, not power.

Given the width of the lawnmower (W) and the length of the rope (L), write an expression for the pole radius (R) that will make the lawnmower cut the lawn in a perfect spiral.

Given the width of the pool in tiles (n), write an expression for the number of tiles that will fit around the pool border.

We recognize that one reason variables give us power is that they let us complete lots of versions of the same task quickly and reliably. So in our version of both of the above problems, we asked students first to work numerically, both to acclimate them to the task, but especially to establish the feeling that, “Okay, doing a lot of these could get tedious.”

And then we use their expression to power ten pool borders.

And ten lawnmowers.

Those activities are Pool Border Problem and Lawnmower Math. In Picture Perfect, for another example of this principle, we give students the option of either a) filling in a table with 24 rows, or b) writing an algebraic expression once.

In each case, students are more likely to see algebra as power than punishment.

Here Are Ten New Desmos Activities

The Desmos quarter just ended and this was a huge one for the team of teachers I support.

First, we made substantial upgrades to our entire activity pool. Second, we released ten new activities in the same amount of time it took us to release one activity two years ago. This is all due to major improvements to our technology and our pedagogy.

Technologically, our engineers created a powerful scripting language that hums beneath our activities, enabling us to set up more meaningful interactions between teachers, students, and mathematics.

Pedagogically, my teaching team has spent the last year refining our digital mathematics pedagogy through daily conversations, lesson pitches, lesson critiques, summary blog posts, occasional lunch chats with guests like the Khan Academy research team, and frequent consultation with our Desmos Fellows.

The result: we cut an activity pool that once comprised 300 pretty good activities down to 127 great ones, and we gave each one of those 127 a serious upgrade, making sure they took advantage of our best technology and pedagogy. Then we added ten more.

I don’t think I’ve learned as much or worked as hard in a three-month span since grad school, and I owe a debt of gratitude to my team – Shelley Carranza, Christopher Danielson, and Michael Fenton – for committing the same energy. Also, it goes without saying that none of our activity ideas would have been possible without support from our engineers and designers.

In future posts, I’ll excerpt those lessons to illustrate our digital pedagogy. For today, I’ll just introduce the activities themselves.

Picture Perfect

Hang loads of pictures precisely and quickly using arithmetic sequences.

Game, Set, Flat

Your shipment of tennis balls has been contaminated. Use exponential functions to find the bad ones.

Graphing Stories

Graphing Stories comes to Desmos just in time for its tenth birthday.

Pool Border Problem

One of the oldest and best problems for exploring algebraic equivalence. We wouldn’t have touched it if we didn’t think we had something to add.

Laser Challenge

Use your intuition for angle measure to bounce lasers off mirrors and through targets.

Lawnmower Math

Use Algebra and the properties of circles to help you mow ten lawns automatically and quickly.

Land the Plane

Use linear equations to land airplanes safely and precisely.

Circle Patterns

Practice circle equations by completing artistic patterns.

Constructing Polynomials

Develop your understanding of the behavior of polynomial graphs by creating them piece by piece, factor by factor.

What’s My Transformation?

This is my favorite introduction to the concept of a transformation. “Actually, there’s really only one parabola in the world – we just move it around to make new ones.”

We are still testing these activities. They are complete, but not complete complete, if you know what I mean. You won’t find all of them in our search index yet. We welcome your feedback.

New Activity: Marcellus the Giant

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In Marcellus the Giant, the new activity from my team at Desmos, students learn what it means for one image to be a “scale” replica of another. They learn how to use scale to solve for missing dimensions in a proportional relationship. They also learn how scale relationships are represented on a graph.

There are three reasons I wanted to bring this activity to your attention today.

First

Marcellus the Giant is the kind of activity that would have taken us months to build a year ago. Our new Computation Layer technology let Eli Luberoff and me build it in a couple of weeks. We’re learning how to make better activities faster!

Second

When we offer students explicit instruction, our building code recommends: “Keep expository screens short, focused, and connected to existing student thinking.”

It’s hard for print curricula to connect to existing student thinking. Those pages may have been printed miles away from the student’s thinking and years earlier. They’re static.

In our case, we ask students to pick their own scale factor.

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Then we ask them to click and drag and try to create a scale giant on intuition alone. (“Ask for informal analysis before formal analysis.”)

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Then we teach students about proportional relationships by referring to the difference between their scale factor and the giant they created.

You made Marcellus 3.4 times as tall as Dan but you dragged Marcellus’s mouth to be 6 times wider than Dan’s mouth. A proportional giant would have the same multiple for both.

Our hypothesis is that students will find this instruction more educational and interesting than the kind of instruction that starts explaining without any kind of reference to what the student has done or already knows.

That’s possible in a digital environment like our Activity Builder. I don’t know how we’d do this on paper.

Third

Marcellus the Giant allows us to connect math back to the world in a way that print curricula can’t.

Typically, math textbooks offers students some glimpse of the world – two trains traveling towards each other, for example – and then asks them to represent that world mathematically. The curriculum asks students to turn that mathematical representation into other mathematical representations – for instance a table into a graph, or a graph into an equation – but it rarely lets students turn that math back into the world.

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If students change their equation, the world doesn’t then change to match. If the student changes the slope of the graph, the world doesn’t change with it. It’s really, really difficult for print curriculum to offer that kind of dynamic representation.

But we can. When students change the graph, we change their giant.

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There is lots of evidence that connecting representations helps students understand the representations themselves. Everyone tries to connect the mathematical representations to each other. Desmos is trying to connect those representations back to the world.