Activity and Diagnosis

A couple years ago, I sat against the wall in my sons’ bedroom, taking screenshots of my smartphone once every four minutes for an entire evening. (Why, you ask? For math!)

The result: Charge! (quite likely my only decent entry into the world of three act tasks).

About a year ago, I created an Activity Builder version of the task. And I’ve never been satisfied with it.

I think it’s better than nothing at all, but undeniably worse than the slide-driven, conversation-rich, paper-and-pencil version I posted on my blog a while back.

When it comes to the Desmos activity building code, this activity—even in its AB-powered form—succeeds on several fronts, including “create problematic activities” and “connect representations” (among others).

However, it struggles in ways that overshadow its strengths. Most notably:

#5 – Give students opportunities to be right and wrong in different, interesting ways. The AB version of Charge! feels too scripted. Too narrow. Rather uninteresting. “Do this. Now this. Next, this. Now do this.” And so on, all the way through the activity. There’s really just one path, and the activity leads students along it with minimal opportunity (or even need) for careful reflection or critical thought.

#8 – Create objects that promote mathematical conversations between teachers and students. This is a tricky one. I believe the activity could generate classroom discussion, but that the sheer number of screens works against that possibility, rather than in support of it. Let’s assume a 50-minute class period, with 45-minutes dedicated to this activity. That’s just 2.5 minutes per screen, which isn’t terribly conducive to classroom discussions. It might be wise to trim the number of screens so that there’s room for deeper discussions on a smaller set of screens.

#13 – Ask proxy questions. Would I recommend this activity? Nope. Not in its current form. I’d be much more comfortable recommending the original slide-based version. With the activity parsed into 18 step-by-step style screens, there’s no one screen with anything really interesting happening on it. One of my colleagues likes to consider the quality of an activity by asking whether any of the work students do on a given screen could be considered fridge-worthy. In other words, if they could print it out and take it home to show mom and dad, would it end up on the fridge as a proud display of something deep or delightful? Again, because I’ve chopped the interesting work in Charge! into 18 tiny bits, the answer is no. Nothing fridge-worthy in this approach.

Invitation

• Will you offer your own diagnosis? A second opinion of sorts? What do you think is wrong with this activity?
• Better yet, will you offer suggestions for your own treatment? How would you make this activity better?
• Better still, will you build and share a new, better activity that addresses the shortcomings identified in one of the diagnoses?

Drop a line (or two) in the comments, or let me know what you think on Twitter (@mjfenton).

I’ll be back Friday with a new treatment of my own.

Cheers!

On Monday, I shared an activity addressing slopes of parallel lines. Here’s the diagnosis. Here’s the original activity.

As a quick recap, this activity does a poor job with at least two principles from the Desmos Activity Building Code:

#4 – Create problematic activities.

#5 – Give students opportunities to be right and wrong in different, interesting ways.

There were three comments on Monday’s post (which is approximately three comments more than I usually get!). Go team!

Sherrina Clark suggested inquiring about students’ prior knowledge, including a concrete context, and asking students to make (and test) a prediction about what makes a pair of lines parallel.

Sam Hoggard invited me to consider the form of the equation, and whether ax + by = c might lend itself to the kind of noticing and conjecturing I’m looking for here.

Paul Jorgens reimagined the activity as a series of “finish my parallelogram” challenges. In fact, this link could be the most valuable part of this entire post. Go check it out. Right now. Seriously. I’ll hang out right here.

Okay, you’re back? Nice. Note how Paul incorporated these principles:

• Incorporate a variety of verbs and nouns. (Sketch, explain, convince, write equations, notice, imagine, etc.)
• Ask for informal analysis before formal analysis. (He asks for a sketch on Screen 1 with no grid, then adds some structure and asks for another sketch on Screen 2.)
• Give students opportunities to be right and wrong in different, interesting ways. (I am in love with Screen 8. Holy cow.)

Paul also identifies some areas for improvement, and mentions that this is a work in progress. But there’s a lot to love already.

And while I think there’s still room to grow in problematizing this task, it’s a big step up from my original activity.

My Upgrade

The best laid plans, etc. My original intention was to design and build a new version of this activity. I made it through the design stage, but only part way through building. Here goes:

Screen 1. Show students a graph of two lines. Ask them to adjust one line (using a movable point) so the lines are parallel. Then click “test it!” If the lines are parallel, they’ll never intersect, right? So I’ll find out where the lines do intersect, and zoom the window to show that point. “Bummer! Your lines intersect 103 units away from the origin,” or something like that.

Screen 2. “Try again. Adjust your line. Really focus this time!”

Screen 3. “This is rough, right! Try just one more time.”

Screen 4. At this point, offer a lifeline. Give the same challenge, but display the equations of the lines as students tinker. My hope/anticipation here is that they’ll notice, “Hey, my lines are pretty close to parallel right now, and the equations have almost the same coefficient for x. Let’s see if I can make those coefficients exactly the same to get the lines exactly parallel!” Maybe a little optimistic? We’ll see. For what it’s worth, this would be a pretty important discussion-facilitation screen.

Screen 5. Graph screen with a multiple choice question: “Which equation has a graph PARALLEL to the line shown here?” Four options, followed by “Explain your answer.”

Screen 6. Graph screen with a math input question: “Write an equation for a line PARALLEL to such-and-such equation.” When students hit “submit,” the line appears on the graph. Feedback, but with a bit of a delay to allow room for thinking.

Screens 7-8. Same game, but with perpendicular lines. The animation would involve rotating lines, instead of zooming to the point of intersection.

Screen 9. Similar to Screen 4. Students manipulate the line, but also see the equation. (I’m still thinking through this part, to be honest.)

Screens 10-11. Multiple choice (similar to Screen 5) followed by write your own equation (similar to Screen 6).

Screen 12. Time for a card sort.

Screen 13. I’m not sure if this is an “extension,” or just a good exit ticket style question. It goes above and beyond the activity by introducing equations in various forms, so it fits the “extension” mold in my mind.

My Analysis

Principle #4 (create problematic activities): Improved. The challenge is clear from the word go: make these two lines parallel (and later, perpendicular). Is that scenario contrived? Absolutely! Did I try this with some folks after lunch today and see it quickly move into a contest of who could make their intersection point farthest from the origin? Absolutely again!

Principle #5 (give students opportunities to be right and wrong in different, interesting ways): Improved. Not perfect, by any means. But better. It’s less step-by-step, everyone do the exact same thing, and a little more free-form exploration.

Your Thoughts?

What do you like here? Where does it fall short? How could I further improve the ideas I’m exploring (or is this one destined for the trash heap)?

Let me know in the comments, or drop me a line on Twitter.

Cheers!

P.S. If you read to the end of this one, you’re a champ! Well done.

He recently posted our conversation (in two parts) on his podcast, Transformative Principal.

If you’re into that sort of thing, you can listen in here: Part 1 and Part 2

Enjoy!

Dr. Brownell is the Director of Mathematics and STEM Education Programs at Fresno Pacific University—home of the AIMS Center—and also the host of the podcast Zone of Potential Construction. (He’s also a former instructor of mine.)

If you’re interested, you can listen to the podcast here.

You can also subscribe to the podcast. Just search for “zone of potential construction” in your favorite podcast app.

Here’s a case in that beautiful point.

Points, Lines, and Inequalities

Earlier this week I created an activity designed to help students make connections between individual points and graphs of linear equations and inequalities.

Here’s the first screen:

On its own, pretty simple. But imagine a class full of students doing this. And then imagine directing their attention to a graphical overlay of every student response.

Now we’re talking. Literally. At this point, I’m asking students:

1. What do you notice?
2. What do you wonder?
3. What do these points have in common?
4. What do they not have in common?
5. Is there a way we could summarize all of these points algebraically?

And then… Turn them loose on this:

My hope is that the previous discussion laid the groundwork for the idea that these are all points where x equals 3, so x = 3 is an (obvious?) way to summarize them algebraically. That’s how it plays out in my head, anyway. More on that in a moment.

Students then move on to the next (rather similar) prompt:

(And the activity continues in a similar fashion for a total of six scenarios—four lines, two inequalities. For a closer look, check out the links at the end of the post.)

Room for Improvement

Let’s circle back to Twitter. Yesterday I shared a link to the activity, invited folks to give it a test run, and hoped a few might offer some feedback.

I made a new Desmos activity: “Points, Lines, and Inequalities” Can you help me by giving it a try? https://t.co/ogMZydVViA #MTBoS #mathchat

— Michael Fenton (@mjfenton) March 9, 2016

The response proved incredibly helpful. In particular, Bowen Kerins offered not one or two, but nearly a dozen comments about what works, what doesn’t, and how I could make this activity even better.

I have a few takeaways from that conversation about how to make this activity better. I’ll share them here in order to clarify (for myself) and share (with others) some design principles that I think may prove helpful the next time we fire up the old Activity Builder.

I’ll share the first takeaway here, and a couple more in the next few days.

Takeaway #1

The cycle in screens 1-3 (place a point, imagine all the points, summarize algebraically) is missing a crucial step. It begins simply and somewhat informally, which is helpful. And then it invites students to imagine/predict, which I also think is helpful. But then it jumps straight to the summary, without pausing long enough on the “discussion” that I described above.

My current fix. Insert this between screens 1 and 2:

By asking students to name these points (note: not just their own, but also several others), I think they’ll be more likely to see (or “hear”) the repetition that leads to our algebraic summary later on:

• “x is 3, y is mumble-mumble…”
• “x is 3, y is something-something…”
• “x is 3, y is whatever…”
• “Hey, x is always 3! So then, x = 3!”

My future fix. This activity uses an existing (and lovely) feature called copy previous, where whatever a student did on one graph screen can be pushed ahead to (or “served up” on) a later one. You’ll see this in action between the screens 1 and 2 in my original activity.

What I would love here is another feature (even more lovely?) where the graphical responses from every student are served up to kiddos on a later screen. In other words, bring the graphical overlay feature that we have in the teacher dashboard into Activity Builder itself. Apply that to my original screen 3, and I think we’re in much better shape.

And the good news? Something like this is already in development. (Side note: I continue to be amazed at the engineering skill—wizardry?—of my colleagues.)

Closing Thoughts (For Now)

I think these proposed changes will prove helpful. But I also think they stretch the activity out a bit. So maybe it’s not realistic (or even helpful) to work through vertical lines, horizontal lines, linear functions, and linear inequalities all in the same activity. Maybe this is two separate activities. Or even three.

I take some comfort knowing I’m not the only one who thinks a narrower focus here could be helpful.

Like the way eq of vertical & horizontal lines are addressed here – I can see making an activity solely about that. https://t.co/MJiK15att4

— Cathy Yenca  (@mathycathy) March 10, 2016

With that in mind, here is my new-and-hopefully-improved activity. Give it a whirl, and let me know what you think in the comments.

Next Time

I’ll share some thoughts on making the middle section (linear functions) stronger.

Links

I almost forgot! You can get the original activity here. Or try it out as a student here.

In fact, I’ve created a number of Activity Builder screens like that.

However, I’ve also seen quite a few screens like this:

Notice the difference? It’s subtle, but powerful.

The ingredients are largely the same, but the vague objective is replaced by a clear target.

In each case, my followup screen probably looks something like this:

In other words, my ultimate goal here is to invite students to observe and then describe a parameter’s impact on a given graph. But my path to that end has shifted from “poke around and see what you see” to “complete this specific task, now reflect on how you made it happen.”

Here are a couple more scenarios where I’ve found this approach to be helpful:

So, what would you do with this?

P.S. For a closer look at the screens above, go here.

Dan Meyer wrote about the activity on his blog, and our resident wordsmith dropped a marbleslides post on the official Desmos blog.

[Psst: If you haven’t played yet, and want to… Go here.]

In reflecting on what I love about marbleslides, my thoughts go back to the session Katie Reneau and I co-presented at this year’s CMC North and South conferences. We presented four principles that we’ve found effective in task selection and implementation with our own students.

1. Use tasks that build capacity for SMP 3.
2. Use tasks that are accessible and extendible.
3. Use tasks that encourage iteration.
4. Use tasks that have room for multiple approaches.

There’s not much in the way of SMP 3 (constructing viable arguments…), but marbleslides is an absolute gold mine when it comes to the other three principles.

Accessible and Extendible

The activity kicks off with a challenge that “even your baby cousin” could tackle. Provided he or she is willing to hit the “launch” button.

From there, the challenges grow. Marbleslides adds one layer at a time, providing students with digital sandboxes in which to explore constants and coefficients, translations and dilations, concavity and restrictions.

As students build their skills, Marbleslides pulls back the supports. Students are asked to find (and reflect on) solutions to increasingly complex challenges.

Iteration

In our conference session, Katie and I used an Open Middle problem to illustrate the meaning (and value) of iteration in math tasks. We both love problems that offer students some early reward, and them immediately invite them to dig deeper, work smarter, etc.

For Marbleslides, imagine this sequence:

• A student builds a solution, clicks launch. No stars. (Bummer.)
• But she tries again. And gets a star. (:internal-fist-pump:)
• She adjusts her graph (or graphs), and gets two more stars. (Nice!)
• So close… Just one more… A few more tweaks (and attempts), and she’s like:

• But then a classmate throws down a challenge: Five parabolas? Betchacant with just three.
• And the tweaking, the improving—the iterating—continues.

Multiple Approaches

Something else I love about marbleslides? Students can legitimately take alternate paths to the solution. Years ago I began asking students (almost like a broken record), “Great! Can you solve it another way?”

For the record, this prompting works out much better when there’s more than one mathematically meaningful approach.

And on most marbleslides levels (especially later in the activities), I’m pretty sure there are. From different paths, to different functions, to minimizing equations, to minimizing the time it takes to get all four stars… There are a lot of options for pressing deeper on any part of these tasks.

Invitation

If you haven’t tried out marbleslides yet, pick your flavor (lines, parabolas, exponentials, rationals, periodics), and let me know what you think.

If you’re already tried it, do me a favor: Save a GIF of your favorite solution (yours, a colleague’s, or a student’s) and drop it in the comments.

So I haven’t exactly been pumping out record numbers of blog posts since I shifted from the classroom to my current gig at Desmos. This lack of posts is partially related to general busyness, but it’s also related to this question that I’ve been wrestling with for the past few months:

I’m not in the classroom, so… what would I even blog about?

Over the past few weeks, it’s gradually dawned on me that my next hundred posts can be about the same things as my first hundred or so: trying and tinkering and learning and struggling and failing and succeeding, and so forth.

So here goes…

Custom Polygraph

Back in December 2014, Desmos released Polygraph, a small collection of activities designed to develop students’ informal language into formal vocabulary. If you’ve ever played Guess Who, you’ll understand the rules right away: use yes/no questions to narrow down the field until you’ve singled out the winning face (or graph).

Near the start of the summer, Desmos released Custom Polygraph so teachers build their own polygraphs to address whatever vocabulary they want.

Polygraph: Histograms

Earlier today I built a Custom Polygraph focused on histograms. My hope is that it will get students talking about shape, center, spread, and related concepts/vocabulary. I’d love some feedback, especially if you use it with students (or if you play a practice round with colleagues).

You can take a look at it here:

https://teacher.desmos.com/polygraph/custom/561fee03e104f0f5312bed70

Also, I used this graph to make the pictures:

https://www.desmos.com/calculator/arymd1ty5h

In a future post I’ll share my thoughts on what makes an effective Custom Polygraph, though that’s a work in progress, so I welcome your own insight, either on Twitter or in the comments below.

This August I’ll join the Desmos team as a teaching faculty member, where I’ll help their already-amazing crew design activities for teacher.desmos.com.

I’ll miss the classroom experience immensely. I have absolutely loved these 11 years, and I’ve been blessed by students, parents, and colleagues—including those I’ve met through Twitter and blogging—in more ways than I’ll ever be able to count.

But it’s time for a new season. As much as I’ll miss what I’ve done, I’m even more excited for what comes next.

I look forward to the opportunity to dive deep into the question of how to design meaningful and engaging digital math experiences for students. I look forward to the opportunity to rub shoulders with people at the top of my “when I grow up, I want to be like…” lists. I look forward to the challenges that are sure to come, as well as the chance to learn and grow that those challenges represent.

And not least of all, I look forward to the opportunity to make a difference well beyond the walls of my own classroom.

In my own little insignificant timeline, it’s the end of one era and the start of another. I can’t wait for this next one to begin.

On top of that, I needed an activity that would work well with a sub. The intersection of those wants and needs? A sorting activity!

Print ’em out, slice ’em up, throw ’em in a Ziploc and we’re ready for action.

Students worked in small groups on the following:

1. Plot the polar equations in Desmos.
2. Plot the Cartesian equations in Desmos.
3. Match each graph with its polar and Cartesian equations.
4. Match each graph description with its cards (polar equation/Cartesian equation/graph).

I challenged students to be on the lookout for connections between the two equation forms. This is something we’ll develop further in an upcoming lesson.

But Wait, There’s More!

Sorting activities are great because they often prompt lots of discussion within the groups. However, they’re sometimes short and sweet. With that in mind, I prepared another sub-proof task for my students: A pair of equation-converting examples, followed by several practice problems. If you’re curious to see the handout, which is seriously limited in focus/scope) click here.

Looking for the sorting activity slide deck instead?

• Original Keynote
• PDF (1 card per page)
• PDF (9 cards per page)
• PDF (16 cards per page)