Category: tech enthusiasm

Total 113 Posts

Putting the “Use” in “Look for and Make Use of Structure.”

This is beautiful, right? Put enough straight lines in the right places and your eyes see a curve.

How many linear equations did the student use to create it? You might start counting lines and assume it required dozens. For some students, you’d be right. They typed 40 linear equations and corrected a handful of typos along the way.

But other students created it using only four linear equations and many fewer errors!

The seventh mathematical practice in the Common Core State Standards asks students to “look for and make use of structure.” The second half of that standard is a heavier lift than the first by several hundred pounds.

Because it’s easy enough for me to ask students, “What structures do you notice?” It’s much more difficult for me to put them in a situation where noticing a mathematical structure is more useful than not noticing that structure.

Enter Match My Picture, my favorite activity for illustrating my favorite feature in the entire Desmos Graphing Calculator and for helping students see the use in mathematical structures.

First, we ask students to write the linear equations for a couple of parallel lines.

Then four lines. Then nine lines.

It’s getting boring, but also easy, which are perfect conditions for this particular work. A boring, easy task gives students lots of mental room to notice structure.

Next we ask students, “If you could write them all at once – as one equation, in a form you made up – what would that look like?” Check out their mathematical invention!

Next we show students how Desmos uses lists to write those equations all at once, and then students put those lists to work, creating patterns much faster and with many fewer errors than they did before. With lists, you can create nine lines just as fast as ninety lines.

What are the four equations that created this graph? Personally, I find it almost impossible to discern by just looking at the graph. I have to write the equation of one of the lines. Then another. Then another. Then another, until that task becomes boring and easy. Only then am I able to notice and make use of the structure.

Pomegraphit & How Desmos Designs Activities

Eight years ago, this XKCD comic crossed my desk, then into my classes, onto my blog, and through my professional development workshops.

That single comic has put thousands of students in a position to encounter the power and delight of the coordinate plane. But I’ve never been happier with those experiences than I was when my team at Desmos partnered with the team at CPM to create a lesson we call Pomegraphit.

It’s yours to use.

Here is how Pomegraphit reflects some of the core design principles of the teaching team at Desmos.

Ask for informal analysis before formal analysis.

Flip open your textbook to the chapter that introduces the coordinate plane. I’ll wager $5 that the first coordinate plane students see includes a grid. Here’s the top Google result for “coordinate plane explanation” for example.

A gridded plane is the formal sibling of the gridless plane. The gridded plane allows for more power and precision, but a student’s earliest experience plotting two dimensions simultaneously shouldn’t involve precision or even numerical measurement. That can come later. Students should first ask themselves what it means when a point moves up, down, left, right, and, especially, diagonally.

So there isn’t a single numerical coordinate or gridline in Pomegraphit.

Delay feedback for reflection, especially during concept development activities.

It seemed impossible for us to offer students any automatic feedback here. After a student graphs her fruit, we have no way of telling her, “Your understanding of the coordinate plane is incomplete.” This is because there is no right way to place a fruit. Every answer could be correct. Maybe this student really thinks grapes are gross and difficult to eat. We can’t assume here.

So watch this! We first asked students to signal tastiness and difficulty using checkboxes, a more familiar representation.

Now we know the quadrants where we should find each student’s fruit. So when the student then graphs her fruit, on the next screen we don’t say, “Your opinions are wrong.” We say, “Your graph and your checkboxes disagree.”

Then it’s up to students to bring those two representations into alignment, bringing their understanding of both representations up to the same level.

Create objects that promote mathematical conversations between teachers and students.

Until now, it’s been impossible for me to have one particular conversation about the tasty-easy graph. It’s been impossible for me to ask one particular question about everyone’s graphs, because the answer has been scattered in pieces across everyone’s papers. But when all of your students are using networked devices using some of the best math edtech available, we can collect all of those answers and ask the question I’ve wanted to ask for years:

“What’s the most controversial fruit in the room? How can we find out?”

Is it the lemon?

Or is it the strawberry?

What will it be in your classes? Find out and let us know.

2017 Jun 16. Ben Orlin adds several different graphs of his own. Delete his objects and ask your students to choose and graph their own. Then show Ben’s.

This Is My Favorite Cell Phone Policy

Schools around the world are struggling to integrate modern technology like cell phones into existing instructional routines. Their stances towards that technology range from total proscription – no cell phones allowed from first bell to last – to unlimited usage. Both of those policies seem misguided to me for the same reason: they don’t offer students help, coaching, or feedback in the complex skills of focus and self-regulation.

Enter Tony Riehl’s cell phone policy, which I love for many reasons, not least of which because it isn’t exclusively a cell phone policy. It’s a distractions policy.

What Tony’s “distraction box” does very well:

  • It makes the positive statement that “we’re in class to work with as few distractions as possible.” It isn’t a negative statement about any particular distraction. Great mission statement.
  • Specifically, it doesn’t single out cell phones. The reality is that cell phones are only one kind of technology students will bring to school, and digital technology is only one distractor out of many. Tony notes that “these items have changed over time, but include fast food toys, bouncy balls, Rubik’s cubes, bobble heads, magic cards, and the hot items now are the fidget cubes and fidget spinners.”
  • It acknowledges differences between students. What distracts you might not distract me. My cell phone distracts my learning so it goes in the box. Your cell phone helps you learn so it stays on your desk.
  • It builds rather than erodes the relationship between teachers and students. Cell phone policies often encourage teachers to become detectives and students to learn to evade them. None of this does any good for the working relationship between teachers and students. Meanwhile, Tony describes a policy that has “changed the atmosphere of my room,” a policy in which students and teachers are mutually respected and mutually invested.

Read his post. Great, right? How would you build on his work?

2017 May 26. Okay, okay, we have a bunch of font critics in the comments thread!

Featured Tweet

This is a different approach. The cell phones are in jail. But I admire the incentive for parking your phone.

Desmos Now Embedded in Year-End Assessments Across the United States

Amy X. Wang, writing in Quartz:

Enter Desmos, a San Francisco-based company that offers a free online version of TI’s graphing calculator. Users across 146 countries, most of them teachers or students, are currently logging 300,000 hours a day on the platform—and today, Desmos announced a major partnership with testing consortium Smarter Balanced, which administers academic exams in 15 US states. Beginning this spring, students in those areas will use the online tool in math classrooms and on statewide performance assessments.

When students take their year-end assessment in 15 states, they’ll see the same free calculator they’ve been using at school and at home the rest of the year. That assessment will more closely reflect what they know, rather than what they were able to express through unfamiliar or costly technology.

USA Today has the reaction quote from Texas Instruments president Peter Balyta:

Peter Balyta, president of TI Education Technology and a former math teacher, defended the purchases, saying a TI calculator “is a one-time investment in a student’s future, taking students through math and science classes from middle school through high school and into college and career.” He said TI’s technology is evolving, but that models like the TI-84 Plus come with “only the features that students need in the classroom, without the many distractions that come with smartphones, tablets and internet access.”

This is interesting. In a world where more and more assessments are delivered digitally (and pre-loaded with digital calculators) the sales pitch for hardware calculators is their lack of features, rather than their abundance.

There is clearly a market today for a calculator that lacks internet access. Around 20% of teachers in my survey said they wouldn’t let students use mobile devices on exams for reasons of “test security” and another 10% cited “distraction.”

Open, interesting questions:

  • Are those figures trending upwards or downwards?
  • Will schools and parents continue to pay Texas Instruments an estimated 50% profit margin for more test security and fewer distractions?
  • How do math coaches and instructional technologists help teachers harness the advantages of the internet while also managing concerns about security and distraction?

2017 May 12. Peter Balyta makes a longer case for hardware calculators, one which won’t surprise anyone who has followed this discussion. He mentions (1) lower cost, (2) fewer distractions, (3) greater test security, (4) more features, and (5) availability on tests.

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.)