A while ago I shared some thoughts about my iPad workflow (singular). Maybe you read it? A non-descript tall man did and then it got retweeted to the moon and back.

I didn't touch on the logistics much. It is not a 1:1 environment. The classes that used these the most had 36, 35, and 34 kids in them. Sharing was required.

Here are the ingredients of the tasks:

  • iPad/iPhone sitting on the same wireless network as my teaching computer
  • Desmos
  • Pages
  • Brother HL-3170CDW
  • handyPrint ($5), makes USB connected printers visible to iOS devices

Printing wasn't always involved, but in the case where the kids were designing something, the idea was the print it out and add some details. Any sort of like on high mandated Chromebook thing would need to answer the printing question for me. Desmos is exponentially better than spending time graphing things by hand. The Pages portion isn't totally necessary, but if you want to be efficient with toner and paper, Pages lets you place multiple photos on a page. Printing from the iOS Photos app gets you just one.

The sharing thing went better than I had hoped. For graph matching tasks like the one in this photo, usually two per kid was fine and I wouldn't pass out all the devices. Some kids elected to use their phone, which is totally cool with me. Sharing in this context offered a chance for more discussion. I've done this in academic and it lead to a lot of one kid just waiting for the other to figure it out. My PreAP group seems to be more eager and less likely to let that happen.

In instances where a printout was required, I would pass out all 25 and there was less sharing, but it still happened. Students working together were allowed to design whatever it was together and share output, so long as they printed two copies and annotated whatever additional information was required. Each kid had to hand in something. Often they'd sit there and share the device but create enough unique work so that they were NOT the same, which was cool.

I never felt constrained by having fewer devices than students. And I'm having a hard time convincing myself it's time to purchase more. I had several pairs who preferred sharing to working on their own. I dare say this is almost the perfect ratio. As always, I will continue to see if there's something better, but this was hard to beat.

AuthorJonathan Claydon

I'm concluding my first year teaching Caculus AB. The exam is two weeks from today.

It was a bit weird start to finish, there is a lot I want to fix.

The Bad

Teaching a course for the first time is overwhelming. Teaching something so trenched in tradition as an AP course even more so. In July I attended a week long orientation for new AP teachers and it was decidedly old school. Lots of fiddly TI-84 knowledge, handouts, and homework. And it's tough to argue with this approach. The leader of our workshop is incredibly well respected in the AP community and demonstrates results year after year.

Throughout the year I felt like I lectured way too much. But at the same time there is just so much fiddly stuff in this curriculum that it felt difficult not doing so. I also screwed up a lot or found myself overlooking details that became important later. For example, I had a big misfire with Riemann sums that I didn't notice until like, last week.

I had very little in the way of creative presentation. I had a couple of interesting ideas in the fall, but there was just nothing in the spring really. My current Calculus resources aren't much more than the assessments I gave. I felt rushed every day of the second semester and wrapped up the content over a week later than I wanted to. Our hybrid block bell schedule was a hinderance most of the time, but that's changing next year.

Assessment could have been better. Calculus is so much more conceptual than I thought and after January I felt like their grades in the course were completely irrelevant. I assigned homework 15 times. I have no clue if it was much use. I found the textbook of no use to the students other than for homework problems.

The big nagging issue is that AP Exam hanging over our heads. My school has no success in recent history. And I dislike teaching to a test as my main goal.

The Good

There are some things to celebrate. Low AP scores were one challenge, quitting was another. Varsity Math was a huge success. My Pre-Cal students are capital J Jealous. We have t-shirts, stickers, and 90 minutes of laser tag in the school cafeteria to prove it.

Speaking of Pre Cal, taking on PreAP Pre Cal offered dividends. Calculus numbers are higher for next year, and I know 95% of the kids on the list.

The idea of Throwback Thursday worked well. I kept that up (sometimes accidentally) all year long. A lot of the Calculus curriculum is putting new vocabulary to old math.

On the assessment front, increasing their exposure to College Board material seems like it's working. I gave a series of practice exams at the end of each grading period. Prior to Spring Break, they had a big one that helped me determine who could succeed on the real exam. The 25 of 50 who cleared that hurdle stand a fighting chance.

While I find teaching to the test a negative, I don't find the AP Exam to be a bad test. It is very concerned with concepts and vocabulary. I think there is a lot I can do differently to get the conceptual ideas across without overtly invoking the test as much. A lot of the high volume algebra that most instructors try to scare kids with really isn't present here. In fact, as a strategy that seems bad, it is way too easy to overthink what's required of you with this material, and overloading students with mountains of algebra and trig identities just makes them paranoid.

The Action

Discovering that focus on concepts got me thinking. Did I do enough to get them fluent in the concepts? During after school review sessions many students told me that they had no problem with the math, but couldn't get a handle on the vocabulary. The math really feels secondary in our preparations. The kids are kind of funny about it, they get mad that intimidating questions on differential equations boil down to Algebra 1 ideas.

But where do I go from here? I want to change a lot. I don't think my assessment strategy works in the second semester. I don't think we talked concepts enough. I did way too much talking. Stuff just took forever.

I intend for Calculus: Varsity Math Strikes Back to be an entirely different animal.

AuthorJonathan Claydon
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I've been at this a while, but you, dear up and coming teacher, should know that your career can take some turns. One direction is burn out and sadness. The other direction might lead you here:

I'm not sure if it's a direction you knew was possible. Is this the success of good relationship building? No idea. I hope so? There was lots of good work on this paper, and this student is a super hard worker. I really didn't know what to say. I did eat the cheeto, not that it affected anything. I can't be bought that easily.

How do you communicate the idea of building a classroom where a student feels safe taping a cheeto to their test knowing that you won't have a problem with it?

AuthorJonathan Claydon
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Now that the polar coordinate system is behind us, it's time for the end game material in Pre-Cal. That means sequences, summations, and limits. Over the years I have been spending less and less time with the sequences and summations portion. Primarily because I want to use it as a foundation for Calculus, discussing the problem of arbitrary area. A secondary goal is using the idea of a summation for some financial literacy.

In the short time I spent with sequences and summations, here is the random order of events:

  • Discussed sequences as an idea, everyone remembers them from some time in the past, no big deal. I wrote a series of random sequences on the board (linear, quadratic, etc) and had them generate terms. Then they added them together. No sigma notation yet.
  • Then I did a magic trick. Well, a Nowak trick. Students create a random linear pattern, generate the first 20 terms and add them together. Have someone verify your sum. They did this on little half index cards and I collected them and quickly called out sums like some sort of genius. This drove a few kids crazy trying to come up with theories.
  • Next we go Dan Meyer with some penny pyramids:
  • This group of kids weren't too intimidated by this. Many of them figured out the pattern to the number of stacks and were able to get the correct answers manually. Several thought the answer might be possible through the magic trick I demonstrated before.
  • At this point I show them sigma notation and how to use it on the calculator. They compute a few for practice.
  • Then it's Fawn's turn and I mix in some Visual Patterns. I picked pattern 2, 18, and 19. Two of those are simple, 19 is a fun kind of frustrating.
  • Next step is converging and diverging series. I pick two sequences: 4*(0.74)^n and 4*(1.02)^n and have them compute the sum of each for 10, 50, 100, 500, and 1000 terms. This is some good calculator practice and gets them to observe a few things. Why does the answer start repeating for the first one? Are these giant numbers I'm getting for the second one right? I give them some hints about the things possible in calculus when I mention that there is a simplification for computing the sum of an infinite diverging series. "We're going to learn that!?"
  • Then a summary assignment. Generate some terms, generate some terms and sum them manually, compute sums on the calculator, and identify convergence and divergence.

All of this builds towards approximating area under curves using left, right, midpoint and trapezoidal sums. Students who went through last year's version remembered it well now that I have some of them for Calculus.

Most Pre-Cal curriculum I see sticks this stuff in between vectors and polar coordinates and I find that a little awkward. Putting it right before whatever Calculus material I intend to cover has worked better for me.

AuthorJonathan Claydon

I don't write much about Calculus because frankly I'm not sure I know what I'm doing. I need another summer to reflect. The primary goal last summer was to find an assessment strategy that worked. What I came up with has been doing ok. Given the ultimate goal of the class, demonstrate 50% proficiency on a very context/vocabulary heavy exam, it has become less useful as we wind down. In fact, the last on these SBG-ish assessments will be given in April, about a month before the 2015 AP Exam. After that we're out of new material. Plus, at this point I kind of know what I have.

How? Mixed in with their standard assessment have been mock AP Exams. A theory about our historic awfulness on the exam has to do with exposure to AP like questions. After digging through tons of them, the AB test requires far less gritty math skill than you'd think. In December students took about half a test and could work on it together. A few weeks ago was their big trial by fire, half an AP test (30 multiple choice, 3 free response based on released College Board material) done independently.

Results are not spectacular, but I'm feeling ok:

As it's about time to hand in your AP registrations, I gave an official opinion to each student based on their performance. I had to be mean and tell a lot of ambitious kids no. Two kids with 3s were two points shy of a 4 which was stupendous. Now, I have no authority to prohibit them from taking the exam, but they should know if they're going to wind up wasting their time. Or this has all been a long con on their part and they'll do great, who knows.

That healthy chunk of 2s gives me something to work with. Many of them were not far from a 3. I advised most of these kids to register. Another subset I left it up to them, some of them were in that group of 1s.

You can look at that and think, well those scores do suck. And they kinda do. With six weeks to go I feel like everyone is capable of jumping a level. All of them (that weren't just utterly lost) said it really came down to vocabulary, the math was not difficult.

I'm trying to change expectations for students in this class, and we might just slowly be getting somewhere.

AuthorJonathan Claydon

There's not much to explain that hasn't been said already:

Sidewalk Chalk Adventures
Return of Sidewalk Chalk
Sidewalk Chalk Three

A few tweaks this year. Previously the equations were scripted. I handed out a list and kids made the picture and then drew it outside. This year they designed their own: one limaçon and one rose curve with appropriate equation or table. I showed them how to do all sorts of odd things with rose curves. The scope expanded as well, enrolling my co-worker and her troupe. In total it was about 230 students who produced two graphs each. No joke, all of that put together covers about 2000 linear feet. Powered by Desmos. Because, duh.

Selected images:

There were a lot of pokéballs. Tons of these were fantastic. The size of this installation is ridiculous. Unlike previous years, it's not scheduled to rain right away so this will stay put for a while.

Three years ago I had a stupid idea walking to my car. You never know, man.

AuthorJonathan Claydon

Second semester can be rough. It is a busy time of year for me being "on" for 12 hours most days, and 16 hours on others. By Feburary I'm a little spent. And that's when I'm not trying to teach something for the first time. This February I was a little hashtag overworked, for real. I've always found student opinion to be valuable feedback. I figured prior to Spring Break it was time to take the temperature of the room. I asked my Pre Cal kids if there's anything in particular they liked, whether they found the class difficult, if it's different than prior classes and whether or not they get attention. The pile was large:

Call it fishing for complements, but these are some nice validation after a 10 week grind. It's also a chance to see what the quiet kids have to say. I am always curious about what quiet kids have to say about the hot mess noisy environments I've been known to cultivate.

Here's a selection (click for bigger):

Try it next time you're feeling a little down or if you're worried that something may not be working as intended. I don't spend a lot of time with these, I scan them kind of quickly. I'm looking for a couple of indicators: the student feels comfortable and supported. If most of them tell me that, things are going ok no matter how strung out I may feel.

AuthorJonathan Claydon

For several years I have students explore polar equations with a little puzzle solving. Eventually, students stumble upon some observations about how to build the graphs. As an example:

For something like this, students will point out that there is something awfully coincidental about the presence of 3 and 5 in the equation and the values of the various intercepts on the graph. That leads to a discussion where I have ignorantly offered them the shorthand that "the two parameters add together or subtract" to get you various places. It doesn't tell the entire truth. The presence of the + in the equation confuses a few. Why would you subtract these? Same thing happens if it was r = 3 - 5 cos theta. It still reaches out to 8, and 3 + 5 still equals 8 but that's weird mister, there's no + sign! And in the end you're memorizing a lot of rules about one integer being larger than the other, etc.

What I should have done is given a proper explanation, something I didn't figure out until a week too late last year. Now I was prepared.

Started with the same activity, students made the same observation about the weird additive property thing going on. But rather than let them hold onto a shortcut, we looked at why.

After the initial analysis, I gave them a new set of graphs with the same goal: determine the equation. After they succeeded using whatever patterns they had internalized, I had them do a breakdown on the corners (0, 90, 180, and 270). If a loop is present, there will be a negative value at one of the corners. The largest dimension happens even in the presence of a - because the value of the sin or cos is -1 there. This had an added benefit of helping students understand loops better. What does it mean to go -3 in the direction of 90? It helps sell the idea that polar coordinates have a different concept of positive and negative. It also gives them a more universal idea about how to graph a polar equation. The corners will tell me what I need to know.

I feel much better about this unit now.

AuthorJonathan Claydon
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It's polar season. You have some options: squint at poor TI reproductions or find yourself a highly efficient graphing tool.

Last week I mentioned dull workflows. Here's a great example. We spent a lot of time on limacons, cardioids, and roses, your usual suspects. Where polar gets really interesting is conic sections and the idea of eccentricity. Manipulating these kind of equations is tough with pencil and paper. Normally I introduce the idea by having students build a solar system first. It's a fun introduction but there's a major problem. Students understand much about what they're doing if a bunch of planetary research is their introduction. They produce nice work and everything, but I should have let them play around with general ellipses first.

This year I almost didn't learn my lesson but some weirdly timed field trips changed my mind. Basically, there wasn't enough time to get the solar system thing in before Spring Break. I had a day to play with so I figured, why don't we play around.


  • access a pre-built desmos array
  • reproduce some shapes
  • make connections between the physical size of the shapes and the parameters
  • attempt to verbalize the possible effects
  • discuss eccentricity with little idea what that word might mean

The Slider Issue

Previously I've had a desmos set up on the board and had them copy it. This time I was like, duh, make it in advance and so I had them access this:

Screen Shot 2015-03-11 at 1.39.08 PM.png

Normally, I wouldn't really go the slider route. There's a lot of room for them to just kind of fiddle at random until something worked. The problem with polar conics is that you have to make adjustments to two variables and the relationship between those variables in the equation is not trivial. I want success to require some effort, but to come quickly. The primary point of this is to have a discussion about what they did and how to classify these shapes.


I gave very few instructions. I pointed them to the ellipse generator and told them the first idea was to reproduce the graphs and record how it happened. Nearly everyone got them all within 10 minutes. After that, we were done with the iPads. Told you, dull.

I put the 8 ellipses up on the screen. I had them tell me what a and k would reproduce them. I asked them if they could tell me what they thought the two parameters did. Main theory was a controlled size and k controls how "squishy" it is. We talked about how wide or tall they thought each shape was. I stood back for a sec and let them look at those numbers versus the a values (a is half the value of the longest dimension).

Then we talked about k. What is this weird number? Why is it a decimal? Why did the shape disappear with values of 1 and 0?

And now the big question. Look at the set of shapes. Which would classify as the most eccentric? Least eccentric? At no point did I ever hint at what "eccentric" might even mean in this case.

That's the part of the workflow I care about. The iPads were the most efficient way to play with the idea. We barely used them. But it let me have a discussion with them and hammer out theories about eccentricity without ever having to tell them what that word means.

At the end of discussion I teased the solar system idea. I think we are better prepared for the concepts involved there. The thought that their elementary school teacher might have mislead them was extremely concerning.

AuthorJonathan Claydon

I've had iPads in the classroom since the fall of 2012. It started with 4 and now I have a pile of 25. What world changing things are we doing these days?

Honestly, no one would build an ad campaign around it. My primary workflow is rather dull.

I've played around with exporting student work and sharing photos. It served a purpose but at my current scale isn't practical. I tried collecting data in spreadsheets and sharing it, too many kids had nothing to do. I tried have them analyze a video. I tried having them model equations to pictures. In each case the time investment ratio was just out of whack. There were simpler ways to accomplish the same thing. Despite how easy it was to gather stuff via Google Drive, I could never find other uses for it. Despite how easy it was to have them try and teach themselves, it was just glorified handouts. I like for them to keep records of everything we do. I still love paper and pencil, marker and glue. Can we make that better?


It began last year with an Algebra II experiment. About midway through the first semester I wanted to spend a long time on graphing, and verifying that algebra you do with your pencil can be proven with a graph. It became a recurring theme in that class to take the most efficient graphing utility available and produce great work without spending forever plotting points by hand.

It was so easy. Graph it in Desmos. Arrange screenshots in Pages. Print. Glue. Repeat.

Could this enhance Pre-Cal projects? Also definitely. And nowadays they don't always need my iPads.

Entrusted with a different type of student, I wanted to give them more control of their display work. Let them experiment inside some defined parameters rather than follow a script. It has worked really well. When they don't need to graph and print, Desmos makes the best tool for trying to answer the question "how do you reproduce this?"

So no, we aren't taking photos (for math purposes anyway #hundredsofselfies), we aren't making movies, we aren't watching videos, and we aren't collaborating in Google Docs. But we are graphing stuff. We are adjusting parameters on the fly, plotting polar equations with ease, making our custom piecewise functions just so, and pretty soon, designing an absolutely giant collection of sidewalk drawings.

It's simple. It's versatile. It's out of the way. And that's cool with me.

AuthorJonathan Claydon