Some thoughts on the nascent 2016-17 school year. Cross posted: SBISD VANflections

Biggest Success

I entered the year with some plans for Pre-Cal. Initially I had some grand ideas, but the result on paper was a little reordering and some and a more intensive Algebra unit. In practice, the move has been to focus on explanations with a mix of mechanics. The first round has proven to be very fruitful. Recently we had a fantastic moment where students were given a triangle like this:

In my approach to trig I'm trying not to explicitly teach methods as much as possible, instead seeing if students can piece together ideas from what we have talked about and what they've seen before. In this case, we never explicitly discussed finding two missing sides. Two students cracked the puzzle (that sin 42 is a known quantity that can be exploited), but the methods were very different. One determined it through algebra (setting sin 42 = y/37 and solving). Another went through elaborate guessing using his trig table, using the approximate value of sin 42 and then fiddling with numerators until that numerator divided by 37 came close to the same decimal value.

This "figure it out" method is slowly creeping into Calculus, but it's harder.

Biggest Challenge

Time. I mentioned it a while ago. I'm at school like 65 hours a week, plus however many to play catch up and do minor things like, you know, plan.

Summer Learning

Let's do a break down of #1TMCThings, shall we? That's where all the summer learning comes from.

TMC13 - Wide-eyed, the community is every bit as energetic and passionate as I had hoped
TMC14 - Presenting, having people confirm curriculum ideas I had and provide feedback
TMC15 - Collaborating with Michael Fenton, sharing the Varsity Math love, getting input from Lisa and Dan on how to improve my Calculus teaching
TMC16 - Bruce Cohen's great Calculus problems, helping Lisa rebuild Algebra II, remembering Julie Reulbach should be pronounced JULIE REULBACH!!!!

I never walk away from conferences with a mountain of ideas, it's more about being around energetic people who will spitball about math notebooks with you at 11pm on the floor of a dorm.

AuthorJonathan Claydon

A few days ago I pulled off a minor miracle.

First, demonstrate the power rule and a few of the trig derivatives (handy Desmos thing!). Then, as a side discussion, write the equations of motion they're probably familiar with from their algebra-based physics course [x(t) = x0+v0*t+1/2*a*t^2 etc.]

With x(t), v(t), and a(t) sitting up there, take the derivative of x(t). WHAT DID YOU DO. Take the derivative again. OMG NO WAY NO WAY.

Then, grab a tennis ball and mimic this flight path.

Questions to ask as you flail around with a tennis ball:

  • generally, is the slope at the beginning positive or negative?
  • what do they say about velocity at the peak of these flight paths?
  • what slope would a tangent line have at the peak?
  • generally, is the slope at the end positive or negative?

Confirm their findings a few times. I generalize it with something like "so, you're telling me velocity is plus, zero, then minus here?" Nods all around. However you want to word the conclusion is up to you, but make a connection between the position changes being caused by the sign changes in velocity. In the middle we start losing velocity, the ball can no longer maintain its height, etc.

Now the tricky leap. Have them visualize the balls velocity. Again, arms flailing, move the tennis ball in a negative sloping line, mirroring what they told you before with "plus, zero, minus." They'll stumble here a bit because (at least my groups anyway) aren't super comfortable with velocity graphs. They've seen them, but have they seen them?

The connection to the power rule is huge here. At some point you've established that power functions reduce by one power when taking a derivative. So somewhere in the back of their head should be "quadratics reduce to linear." A few repeats of that line and BAM, they make a small step towards curve sketching f'(x) based on knowledge of f(x).

If you jump that mental hurdle (take your time), the next move will be easier. Start talking about the velocity changes of the ball. Plus, zero, minus (lots of tennis ball flailing). Plus, zero minus (flail flail flail). What's the slope of the velocity curve? Negative? Ok...and what's the general sign convention for gravity? Nega---OMG YOU DID IT AGAIN.

In 20 minutes of tennis ball flailing you just discussed a majority of differential calculus. Refer to this metaphor as much as possible. It is the most helpful thing, especially when it's time to establish the role of integrals.

AuthorJonathan Claydon

In recent years I have become dissatisfied with the kind of assessment questions I ask on particular topics. This year I'm making an effort to increase the cognitive demand of my questions and focus less on mechanics.

As we wander through triangles, prior to inverse trig on the calculator, we cover approximating angles using a sin/cos/tan table. I want students to get a feel for the trends present as you move from 0-90º, something that's non-obvious if you just show them the calculator straight away.

I asked a question recently that sparked some really amazing answers. I offered a triangle and a suggested value for an angle. Students had to demonstrate whether or not they agreed with that value. The trig table students created is in steps of 5º, some judgement calls are necessary when getting intermediate values.

Two students started on the notion that the value was fact and used it as a check on the other angle:

Given the same triangle with the suggested value of 23º, another student took up the disagree position, likely using a calculator to play around with sin outputs (22º is not on their table):

Yet another student decided to convince me beyond all doubt it was wrong because look bro, it doesn't matter which one of these trig functions I use, they're ALL telling me your suggestion is junk:

Finally, a separate question about triangles that generate circles and what happens when your interior angle trips on multiples of 90º:

We generated the unit circle on the notion that a series of triangles with the same hypotenuse creates a circle with a radius equal to the hypotenuse. Here's a handy Desmos illustration. The magic collapsing triangle is something I save for after students are given something like sin(90º) and give me the "wait what, how is that a triangle?"

I was super pumped reading through all these as I graded.

AuthorJonathan Claydon

I "change" a bit when kids have me for Calculus. Pre-Cal follows a pretty clear script: regular tests, SBG, and tons of grades. Relatively speaking I do all the work in the feedback department. Kids (most of the time) make use of my input to demonstrate improvement on future assessment attempts. Very slowly I realized that model didn't fit my intentions for Calculus.

It's AP, so in theory it'd be good to give them a proper college experience, or leave them with some skill they'll find useful as an adult. Obviously that means I lecture for 90 minutes twice a week and make them derive the product rule from first principles, right? Uh, not quite.

Towards the end of last year I put a lot of the grading burden on the students. We would take short little assessments, they'd open up the answer key and rate themselves on the A/B/Not Yet system. A majority were pretty honest with me. This year I decided to start that system from the get go. They had an assessment a few days ago, they spent 30 minutes working independently, and then 7 minutes having a discussion where they could make additions/corrections to their work with markers. They couldn't ask me any questions. I collected the papers and glanced at them very briefly just to have an idea how to focus the next day. The following class period I gave them access to the answer key, they rated themselves, and then I focused my part of the day on the topic they struggled with the most (in this case, forcing continuity in a piecewise function).

Some people would argue this is a very anti-college approach. It will discourage studying, or something. As far as grading methodologies go, it is very anti-college. Lots of feedback opportunities, abilities to discuss the test they just took in particular will make a lot of heads explode. I'm looking for two things: can the kids learn how to value learning for the sake of learning, and can they see how beneficial it is to explain something to or listen to an explanation from a peer. That last bit is the only reason I survived college, spending untold hours learning from and teaching peers in the library prior to exams.

But, but, but the GRADES! Who cares? I don't. They shouldn't either. In 8 months they need to be really good at this stuff. Their grade on September 2nd? Whatever. You do this long enough and you can find the kids who are putting the appropriate effort in and rate them appropriately. Very few are getting anything for free here.

I have this conversation with them and it throws them off. Their whole life has been nothing but achieving x out of 100 and here I am telling them to screw that noise. The tension in the room was thick when I passed out the first assessment (which they assumed was a test, but I never called it a test, did I children?).

Then I drop that "have a 7 minute discussion" bomb on them and they understand me. The conversations are FANTASTIC. The clearest positive I got last year is kids loved the instantaneous feedback they were getting by having to rate themselves. Maybe, just maybe, we can get somewhere here.

Now that they know the format, you would (rightly) argue that the discussion part is going to be a key to lots of them thinking that studying is not necessary. They can just hold out. I'm with you there. At the same time, short though the assessments may be, I'm trying to make them more demanding. Given the working pace of a teenager, 7 minutes isn't going to save anyone.

AuthorJonathan Claydon

Slowly I've felt myself transitioning into a mid-career mindset. I've been teaching for a long time, it's pretty clear this is what I like to do and what I should be doing, but I haven't been teaching long enough where the end is in site. It's a plateau. I've felt it coming.

Some evolutionary shifts are underway, some intentional, others not.


I'm more constrained this year. Due to an unforeseen accident with a staff member (they are fine), I have stepped in to help coach volleyball, a sport that starts August 1 in Texas. You may or may not be aware that I've coached soccer for my whole career. Being involved with a sport that's in season is demanding: an additional 25-30 hrs a week for 3 months in this case. Plus, it's not like I left soccer, so I still have that athletic period to manage and a 4 month crunch of its own that starts up in December. I could go on, but this isn't a "woe is me" kind of thing, it's a "first year me would be FREAKING THE F OUT right now" reflection of how far I've come.

I was very unmoved or nervous about the first day of the school because I've been on duty for like a month now. Add in the two weeks I spent doing summer camp (an absolute blast) and I never really left.


A tough decision, but I think 180 Photos is out this year. It started as a project of reflection. I learned a lot in the years I took regular photos. I will still take pictures, but they won't be as lesson focused and not posted. Why stop? When I started (in year 4), I hadn't found my beats as an instructor, the regular moves I wanted to iterate on. I did a lot of experimenting with weird technology things and activities. In the four years since, I have found my beats. I started noticing that my photos were really similar. I kept seeing the same projects, moments, and material. A lot of my work now is on the presentation, how the information gets delivered to the students. That doesn't always come through in pictures. If you look at the 2014 and 2015 versions of a project, it wouldn't be immediately clear that I changed everything about the delivery to produce a similar (but with more burden on the learner) result.

What I focus on with my with my writing will change too. I'm in a lot of good places with my lessons. Though there's always room for important iterations. At this point, five years in to really regular reflection through writing, I've said what I've need to say on a lot of "what would you like to do tomorrow?" topics.


More and more I want to stop talking. Fawn read my mind (or stole my idea) in her discussion of terrible habits for math class. Already I'm seeing results in Pre-Calculus. This year we jumped right into trig, which includes a lot of review from geometry. Ordinarily this takes a lot of time, bringing up a concept, coaching them through examples, requiring them to talk me through something else. This time I was like, forget it, just let them figure it out. We're flying through triangle trig and they are having some great discussions because I just shut up.

It continues to be a bit of a problem in Calculus because there is just so much which isn't intuitive right away, for whatever reason. Or at least it feels that way. Given the constraints on my time, I'm limiting how much I will help them outside of the classroom. Meaning I'm not really going to do a lot of the grading. Any rifling through papers will be for feedback only. They can do the grading.

I won't drag out the explanation, but Calculus has such a limited scope and really really needs to be about how I'm doing at the end of the year. Assigning some sort of hard and fast grade in August isn't useful. They need feedback. Though the kid who's ranked #11 did ask if I could somehow sabotage #10. Nerds.

Random Thoughts

  • could I ever get Talking Points going?
  • what about double number lines?
  • I'm getting access to a subset of Calculus kids once a week, what non-Calculus things could I push them to do? How do you replicate summer camp during the school year?
  • how does one get started applying for a grant that could fund summer camp?
  • what's Laser Tag day look like with 95 in the Varsity Math crew?
  • who are the front runners to be my first batch of legit BC students next year?
  • how much am I into the Backstreet Boys this year?
  • Sam's ExploreMath blew us all away, I must do this, but when?

A long time ago a favorite college professor of mine remarked that I'd get bored doing the same old thing again year after year teaching high school. I'm glad he was wrong.

AuthorJonathan Claydon

This is new:

Out of nowhere, my epic class sizes have finally retreated. My average is down to 28. My room feels empty like something is wrong. Generally speaking, this is probably a good move long term. As much as I have enjoyed teaching as many kids as can possibly fit in there, the energy level required to watch over 36 kids at once is a little high. Especially when they all feel like making noise. That many kids even just whispering is crazy loud. This year it's like an entire class is missing.

Could this mean we cover more material? Possibly. I feel like Calculus will be a bit more efficient, though you could easily chalk that up to my increased experience with the subject.

It's just weird. I was wandering around the room on the first day and just kept wondering where all the kids went.

AuthorJonathan Claydon
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Time for another one, this time a draft of what I'm looking to do in Pre-Cal.

This is what you'd call a living document, similar to any sort of first draft I do for a course. I'll probably start working with this, mark it up like crazy and produce something a little better next year.

Major design goal was to ditch the Algebra review at the beginning of the year. Diving straight into trig feels like it will be fun. It will be some familiar ideas from geometry, sure, but my history with this course has shown that kids aren't quite grasping the why from geometry. Or at least, it's worn off by the time they get to me.

In all things I want students to understand why something works. Because my math teacher said so is not a sufficient answer.

The Algebra chunk, when it's time, will be a bit more thorough. In the past it's been some review of quadratics at that's it. A couple years with Calculus has shown me that we should go deeper here. There's a lot of topics that only got a surface glance in Algebra II that need the "why?" answered.

Reading this you probably wouldn't say this is the most amazing Pre Cal plan in the world. It's really not. But what I want to do when it's time for the material is really dive in to what makes it tick. For example:

This vector project felt so deep last year. Nearly every vector concept was hidden behind the phrase "draw me a picture." So simple. So not.

How do I bring more of that to the table? Can't wait to figure it out.

AuthorJonathan Claydon

Previously in my Calculus curriculum journey, I hacked together a progression. Lots of positives from the approach. The main idea is that it's valuable to see the entire course up front and slowly iterate on the various ways to apply the two big concepts. In the end, it seemed to work. Room for improvement though, as always.

Presenting the updated version of the plan.

The second page existed in the previous version, but I don't think I shared it. There are subtle modifications throughout, mostly to reflect what actually happened as I worked my way through the original draft. The first semester was pretty jam packed in last year's version, so I altered the goals for some of the grading periods, pushing material down the line. Our pace in the second semester is much more comfortable.

Huge, huge, huge goal for me instructing this year is to focus on the abstract. Applying the chain rule, product rule, and quotient rule when only given "f(x)" and "g(x)" terms for example.

Plus a fair bit of these problems:

Bruce Cohen - TMC16 Morning Session

To the new Calculus student, this feels difficult, but is elegantly simple. Quite honestly it summarizes the AB curriculum quite nicely.

Feeling good about it.

AuthorJonathan Claydon
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The coming school year will be notable for a number of reasons, among them: no more iPads in my classroom.


It sounds like a big deal, but not really.

Use Cases

In 2012 we were given 4 iPads for classroom use by the district. My class sizes were hovering at about 30 and I knew that wouldn't be enough to have much of an impact. Slowly, through donations, personal purchases, and some additional school purchases I peaked at 33 iPads of varying vintages by spring of 2016.

In that four year span, I attempted everything. I look at distributing and collecting things via Dropbox, Google Drive, iCloud photos, you name it. I tried playing around with various apps only to find that the best workflow involved a word processor, Desmos, photos, and a printer. None of those components specifically requires an iPad. It was easy and worked every time, but it could be replicated with other methods if necessary.

Over a year ago I started to realize that there was nothing iPad-y about the way I used them, opening up questions about where my tech use goes from here. In fact, the simple use case I had was beginning to become more tedious as software pushed ever forward but the hardware aged (about 2/3 of the stock had 2011-era internals).


Managing that many iPads yourself is a pain. Mobile Device Management systems require business Apple IDs to use most of the features. Acquiring a business Apple ID requires that you own and operate an actual business, with tax ID numbers and all that. Apple makes an app called Configurator that's allegedly for this purpose, but I'm not linking to it because it never worked. Where does that leave me? Manually updating iOS thirty some odd times, retyping iCloud credentials constantly, retyping the WiFi password when the security certificate expires every week or so, and having to wipe each device at the end of each school year. Plus the cost and time associated with my charging system.

Relatively simple work, but bleh.


As cloud computing became a real, viable thing for student work (Google Docs in 2012 was uh, yeah...and it's still so-so on iOS), and as a 1:1 Chromebook pilot in AP English 4/AP Government proved useful, I started looking at Chromebooks as an alternative. Price wise it's equivalent to a refurbished iPad mini, with none of the maintenance headaches and batteries that last for-e-ver. Also, Summer Camp proved the ideal test market. It was a small group of kids, the librarian had nothing else to do with the Chromebooks for the summer, and in the years since we started tech deployment, every kid has a Google account. In the end? Totally frictionless.

A small technical hurdle was solving the Chromebook printing problem, but that wasn't terribly difficult. Some detail on that later, but the easiest way involves a printer with Ethernet (and boy did I purchase a monster).

Having a class set (which the district is providing, for that I am grateful) combined with teaching kids that will be issued one to take home (through other classes) will open up some opportunities. It was already handy for accessing Calculus solutions, and will make some of the deeper parts of Desmos more accessible.

AuthorJonathan Claydon
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On July 16 I presented a My Favorite at Twitter Math Camp. It's a follow up to a presentation I gave the year before. I discuss the ways in which the program expanded at my school. A recording (with the slides embedded) is below:

Main points: Varsity Math is now the blanket term for AP Math classes at my school; membership is up; we did some outreach at our middle schools; laser tag in the cafeteria is the best thing ever; it's a summer program now; YOU'RE INVITED!

If you attended TMC 16, I handed out Varsity Math membership stickers with instructions to 1) come up with a member number 2) write it in the white box 3) take a picture and tweet "I'm on the team! #varsitymath"

But what about you, dear reader who wasn't there and might be upset you missed out on the fun? Well, I have roughly 50 stickers remaining. If you'd like one, get in touch and we can make arrangements. If you're interested in how to start such a branded movement at your school, I'd be happy to help you with that also.

AuthorJonathan Claydon