It's the third year of Varsity Math. It's a point of pride for me and has graduated from "what's that?" to "oh, of course" status at my school.

Last year I added some embroidered patches to the apparel set. They had adhesive backs designed to stick to stuff. Turns out it was difficult to find surfaces they liked to stick to. Sewing them on was the more practical application. Kids didn't get much use of them. This year, I spent a little more money and got two-piece velcro patches. I wasn't sure what the kids might do with them, but I was pleasantly surprised to discover the backing piece made it a perfect lanyard accessory.

Typically you can't spot the Varsity Math crew unless they're wearing their shirts. Now this everyday accessory let's them show math pride everyday. It's quite a majority sporting the badges, the 12th grade counselor even sports one! It's an especially fun game of "spot the nerd" when we've had senior assemblies.

This club that's not a club is fantastic and makes me happy every day, but how do you keep the momentum going when 95-100% of your members graduate every year? It was time to make our place in the school permanent.

Coming (very slowly) to a wall near my classroom is the Varsity Math Hall of Fame. Above you can see my current crew beginning the long process of assembling what will become the stencil for the wall. Many hours of gluing and cutting later we should have a nice looking finished product.

This corner has gone unused for years and was the perfect spot for something like this. It won't be done until the end of the month (we only have a small window of time once a week to focus on it), but it will be a great place for me to display yearly photos and recognize students who pass the AP Exam.

The sudden appearance of the maroon wall has already generated buzz amongst the other students who travel through my hall.

AuthorJonathan Claydon

Since I started, I've always been forcing myself to push students further. Around Year 3 or 4 I realized that for all the kids I taught in Year 1 meant, they will forever be my worst, in a manner of speaking. I had no expectations and generally no idea what I was doing. I resisted intense instruction or tasks because I thought the kids "couldn't handle it" or would resist or because I doubted my ability to persevere when they were uncomfortable.

It took a while to silence those concerns. I spend a lot of time rethinking instruction and take my sweet time making changes, but over time I see dramatic differences.

Let's flash back to November 2012, here's a project on graphing sin/cos:

At the time I was patting myself on the back for this. Super colorful, but the demands were so simple. Take a trig function (that I made), do a 5 second frequency calculation and then graph it. No sense of scale, no comparison with the parent function, and no explanation of what the heck is going on if you're casually looking at these.

Flash forward four years to the same project:

I made nothing here. Students were required to design functions that met certain requirements in terms of frequency, amplitude, and translation. Then they had to write about everything they did. We spent some time learning Desmos and how to print, and they were off. I even required a complimentary task on the reciprocal functions:

Unthinkable in 2012 both in terms of technology available and determination on my part.


The question I want to you to answer, how do you push? What are you doing to make previous activities, concepts, and topics more demanding of your students? What would happen if a Pre-Cal student of mine from 5 years ago walked into my room today? Would they be able to hang with the young ones?

I overhear a lot of negative chatter about what kids can't do. That they're behind, or incapable of [blank] mechanic. That teaching had to slow down because the group felt like it was going by so fast. How are you pushing to fix these problems? What tiny routine or change could you incorporate to make it better?

Who's really scared of going too fast?

AuthorJonathan Claydon

Year 3 of Calculus is the refinement year. The curriculum feels good, now to improve on some major flaws and oversights. A huge part of Calculus that I underestimated is getting students to understand what notation really stands for. For example: f(2) has meaning, f'(2) has a different meaning, dy/dx can have a numerical value, that sort of thing. The real abstract concepts that demonstrate whether you get Calculus in a big picture kind of way. It's very easy to teach kids the quotient rule and have them parrot it back mechanically, but something like this will cause a major roadblock:

There are no defined functions here. Just notation. Symbolic of some meaning. If you really get Calculus, you can see right through it. You think you know the chain rule? product rule? integration? Here's a chance to prove it.

My students last year couldn't prove it. In our brief discussion after the release of this question, only one (out of 49) admitted to having any clue what to do here. The rest punted. That's a failure on my part.

There are other examples, but more abstract discussions are a feature this year. Surprisingly, our first discussion about a week ago (which included a look at this question), went really well. One of my strugglers was beside themselves with excitement over finding h'(1) if h(x) = f(x)*g(x). Blew away my expectations.

AuthorJonathan Claydon

I mused almost a year ago that I dislike the way I assess some topics in Pre Cal, particularly ones where students and myself know better tools are available to do the job.

Somewhat by accident, I decided I was going to make this year's batch of Pre-Cal students write their little fingers off. Mechanics are great and all, but can you reason? Can you explain? Can you defend an argument?

First, something rare on most math tests, some paragraph writing on the differences between the main trigonometric parent functions:

Not everyone wrote me quite the essay I was looking for, but lots of good things. I find these questions are a good way to see who pays attention to what. A lot of students will mirror your explanations (the ones citing infinity/negative infinity got that specifically from a discussion we had), or offer a new spin (vertical vs horizontal flow in the last one).

Some are more reluctant to play my game:

While it wasn't in a paragraph structure, I liked this one for the attention to detail in the diagram and the talking points breakdown. Bonus points for the use of the word "squiggly."

Earlier in the same assessment, I made them agree or disagree with some conclusions about the output of some inverse trig functions:

Let me validate your suspicions and tell you that no, not everyone churned such long discussions. Some were a little short and to the point. Others generalized a bit too much. When it comes to writing tasks, think about what you'd like the minimum to be, and add 3 sentences. There's always that subset that's going to undershoot. Providing a stretch goal will prevent you from getting frustrated when someone writes two words when you wanted a few sentences. No one was penalized for missing the 7-10 mark. I just wanted 5.

Any rumblings about having to write have disappeared. They expect it. One student even remarked after Test 1 "ok, now that I know you expect a lot of explaining, I'm going to adjust how I study."


AuthorJonathan Claydon
2 CommentsPost a comment

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
2 CommentsPost a comment

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
2 CommentsPost a comment