An article made the rounds a while ago: Colleges Reinvent Classes to Keep More Students in Science

A lot of smart twitter folks had the same reaction I had: "duh."

Now, to be fair, at no point in my career have my techniques met with strong critique. No one has accused me of doing harm or doing less than what's required. But there's a stigma. You can feel it in professional development sessions. You're showing some veterans about how great it is to have kids talking to each other and figuring stuff out and they kind of scowl or find a way to tune out. They play along to be polite but you know 0% of what you discussed will make it back to their classroom.

And in these instances it comes back to the eternal straw man: we've GOT to prepare them for college.

No one stops to consider that college teaching is sub-par. Or that the environment is completely foreign to high school. Other than desks and books it's not the same thing. Around the time I saw this article a teacher from another campus was remarking about some college professors who were blown away by Bloom's Taxonomy, something engrained in high school PD for like, ever.

Why do most people lecture eternally? Because they do it in college. Why do most people give mountains of homework? Because they do it in college.

You know what else happens in college? Class sizes of 150. Three hours of "face" time a week. Learning for the sake of learning disappears. GPA becomes the only thing that matters. You give a student two opportunities to perform in 18 weeks and they're going to get desperate. You make a student a number on a list and they won't care about deceiving you. You find the right kids on a college campus and you can earn a master's degree in cheating. No one likes talking about that either. Now, the students I knew who relied on that sort of thing didn't last long, but essay mills are a thing.

You all had the bad professor. Sixth semester fluid mechanics for me. Fellow walks in and says what are your questions. None? Ok. Ninety minutes of reading overheads. Scolding us for not asking questions. Threatening to go faster. Whipping overheads every 15 seconds. Getting more disgruntled each time. A week later he told us that merely sitting there complacently for an entire semester will earn us a C. I got a B in the class, and I know nothing about fluid mechanics (my exam grades were all less than 50%). Nothing he did inspired me to care one iota about his material. Said professor had several prestigious teaching awards to his name because, of course.

It was anomaly, and I had plenty of good instructors. But c'mon, THAT's allowed to happen? That's what everyone was trying to prepare me for?

Why in the world should I do anything like a college professor? I know all of my students by name. I see them five hours a week, every day of the week. I ask them to perform dozens of times and watch them grow from feedback. They produce something every day. There's no need to offload the instruction into a daily pile of homework. Final exams are meaningless to me, because I've watched a student every day for months, I don't need some high stakes 80 question test to tell me if they learned anything.

If a university lecture hall forms the basis for your high school classroom culture, go visit a lecture hall and update your memories.

AuthorJonathan Claydon

I ran a seating experiment in Calculus last semester. A couple weeks in I had them request a buddy that would follow them around the room. I came up with four configurations and we rotated every couple of weeks. The kids had minimal protests. Now that I've had a chance to observe them for a while and see who really works well together, I've simplified the idea for second semester.

Before we left again they gave me some preferences. I took that and made two configurations. In the case of each kid, one of the rotations represent their "dream team" scenario.

I also spread them out a little more. Previously they only used five of the six groups available. Now we'll use them all just to provide some legroom. No one stays in the same seat, but if they had a condition like "the front works best" at least one of their rotations will put them there.

AuthorJonathan Claydon
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We have a bit of an AP retention problem at my school. Kids sign up and give up. As newly installed Calculus teacher this was a problem I wanted to solve. It started with some heavy handed talk at the beginning of the year about how no one's letting them out anymore. That worked. Some kids asked about leaving and were denied. And then, accidentally, I found another way. Thanks to something stupid I did when I was 18.

And then it just kind of escalated.

And now, coming in a week or so, we will have official Varsity Math t-shirts. I styled them up to look like a letter jacket, patches and everything. For $20 kids get a shirt, contribute to an ad in the yearbook (you better believe we're taking a team photo), and the rest goes towards having some food on the Friday before the AP Test. And then we might go play laser tag. Like, for real.

It's the dumbest and best thing.

AuthorJonathan Claydon
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I, like you probably, have to pick out some goals for any given school year. Lately, part of our goals have had to be under some technology sub-domains. One of them happens to mention online collaboration and you do some self rating, etc, etc. I'm not sure if the experience I had (and have frequently) on Twitter last week can be neatly summarized into "never/sometimes/always."

A week ago I was sitting around getting some things together during my planning period and had an impromptu discussion on the necessity of trig identities:


  • This happened in the middle of the school day in about 20 minutes
  • Bowman got a series of opinions about his question
  • More people than Bowman got something out of this

Most Pre-Cal teachers probably struggle with this question. And most likely, people teaching higher level math topics don't have a lot of local resources on the issue. Having that second opinion to validate a teaching move or direction is so key.

My local math department is great. My global math department is fannnnnntastic.

AuthorJonathan Claydon
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Seen non that long ago as a Pre-Cal idea. This variant for Calculus is a little closer to the original described by Kate. Students are given two stacks of cards. A judge is appointed. The judge flips over one card from each stack. Students must draw a graph that meets both conditions. Here are the cards (Word doc).


The white stack focused on things like critical numbers and extrema. The colored stack dealt with the continuity of the function. And of course there were a couple goofy ones thrown in there. Rather than white boards we used a pile of typo-laden tests I had sitting around for drawing.

As far as playing the game goes, I had them pass out chips if someone got it right. Some time later when we did this in Pre-Cal, I found the chips unnecessary. In fact the "who did it best" angle didn't work out so well. Not enough people were sketching it correctly to create debates. So if we play this again in the future, you get credit if you get it correct, and multiple people can get a point. One group eschewed the game entirely and would just flip cards and try to get it right.

As a game with a definitive winner it didn't work out so well. As an activity to foster discussion about graph behavior, it succeeded. I loved that one group just wanted to understand.

AuthorJonathan Claydon

As we kick off the long and winding second semester, I have some additions to share.

First, the debut of a sparkling new Calculus section. As I've only taught it for one semester, there isn't much yet. Second, Pre-Cal is updated with assessments from last semester complete with final exams. Any activity I thought was clever has been bookmarked in those sections too.

AuthorJonathan Claydon

I have never been challenged so much by students. I have never enjoyed a semester so thoroughly start to finish. And, I have never taught at such scale.

This semester was nuts. My students were amazing, and we haven't even gotten to the movies yet. So many times this year I had to step back and come to grips with how much of everything was going on. When we return for the spring, it will be an adjustment remembering what it's like to present to a noisy audience of 36.

I like playing with numbers. The sheer quantity of things got me thinking. Fiddling with Illustrator got me this (full size PDF):

Some personal observations:

  • No one tell first year me about this
  • I interact daily with about 15% of the junior and senior classes
  • I interact daily with about 9% of the school
  • There are more Pre-Cal students in 3 sections than I had in 4 last year
  • Some months ago I estimated I've taught 800 students, it's 759
  • 90 of them had me more than once
  • That number excludes about 200 who know me via athletics or otherwise
  • My Pre-Cal students are amazing
  • Calc leaves me cautiously optimistic

Exciting. Excited.

AuthorJonathan Claydon

Previously, my Pre Cal students were messing around with the graphs of sine and cosine. Some common errors rose to the surface. The assessment that came a couple days later revealed that students were having trouble calculating periods and getting the subtle details of cosine correct. To top it off, now I'm throwing tangent graphs at them and they just found it weird.

Luckily, Frank says it best.

Some days before we had started any of this trig graph stuff, Kate Nowak shared an excellent curve sketching game for Algebra II. Instantly I knew I needed to try it. It was too good not to. In fact, it spawned activities for Pre-Cal AND Calculus. More on the Calculus one later.


To summarize Kate's game: distribute decks of cards with different features on them, require students to draw something that has those features, and judge one another for accuracy.

I made a few modifications for my purposes.

  • Deck of 22 half index cards
  • One six sided die
  • Whiteboards
  • Markers

On the cards were sets of features that could apply to a sine, cosine, or tangent graph. Students pick a judge, the judge flips a card and rolls the die. A 1/2 means you draw a sine with those features, 3/4 cosine, and 5/6 a tangent. Student groaning when it comes up tangent is optional.

I printed this off on Avery 5160 labels (just like Log War and Inverse Trig War):

I threw a couple goofy ones in there because that's how I roll.

Let's Play

I told someone to keep score and to make sure they rotated the judges. For two classes we awarded a point for the "best" interpretation of the graph. I noticed the same kids were getting the point every time so I changed the rules for class three. Anyone who does the graph correctly gets a point.

I thought they'd get through about 15 cards in the 40-45 minutes. In reality it might have been closer to ten. A few cards caused some confusion. In a few cases the judge wasn't sure if anyone had done the graph correctly. I suspect if we played this towards the end of our time with trig graphs I would've had fewer debates to settle. There were still lots of good things going on here. If anything it hammered out the differences in sine and cosine for lots of them. And we solved the period calculation problem. The goofy cards produced some excellent results.

It was a lot of good graphing practice and discussion without feeling like graphing practice and discussion. It kept their attention for quite a while. I think we could easily do this again for some other stuff. The Calculus version went equally well.

AuthorJonathan Claydon

More in my desire to help the students less this year.

I've had a good schtick going for a few years. I take something that ordinarily would be classwork and pass out colored paper and markers and have the students do a display version that goes in the hallway. In my life as an academic teacher, these assignments were pretty scripted. In this instance we're discussing trig graphs.

Rather than the standard "here are twelve functions, everyone pick two" I came up with these specifications:

  • create a set of 3 sine functions and 3 cosine functions
  • "negative" amplitude must be demonstrated
  • a "faster" function must be demonstrated
  • a "slower" function must be demonstrated
  • a vertical translation must be demonstrated
  • create two graphs in desmos, one for each type
  • print and identify your creations

Negative amplitude means a reflected function. Faster and slower reference the length of the period. If you've ever struggled with the whole why do I calculate the period of sin(2x) by determining 2pi / 2, try asking the student how they would complete a 10s task twice as fast. It nixes a trick and builds better understanding all at once. This is a handy if you ever want to discuss models of electrical signals.

This activity is quick thanks to the infrastructure I've been building for a few years: tons of iPads and a color printer in the room. No computer lab sign up sheets necessary.

I liked the vast variety of functions. It also brought some misconceptions to the surface. A lot of students could create a faster or slower function, but weren't sure how to tell me the actual period. This created an excellent opportunity for follow up the day after this activity. I also fielded a lot of questions about whether vertical translations affect things like amplitude and period. Again, great follow up material for the next day.

Something you may notice. If you study the work closely, there are errors. It happens. While on display, it's practice. I expect there to be errors in practice. Students self-correcting their errors between the time we practice and we assess is what I care about. If I were to nitpick these as they went up, I counteract my goal of minimal help.

Start to finish this was one and a half class periods. I'm also getting better at anticipating activities that finish early and having something ready for the ones that finish early.

AuthorJonathan Claydon

Every year in Pre Cal I approach right triangles differently. This year is no different. The number of times you could say I "deliver a lesson" is starting to dwindle. I design tasks. The students spend class time doing those tasks, and we find the interesting questions within. Here's a short primer on how we progressed through right triangles.

Task 1: Make Some

I passed out protractors and some paper and told them to make a bunch of right triangles. Specifically, make a bunch of 30/45/60 right triangles. Side measurements are up to you. I had them make nine, it should probably be half that because the process took FOREVER. Useful as a reminder on how protractors work.

Task 2: Humor Me

After the long process of making triangles, we labeled them with opposite, adjacent, etc. For each triangle they computed the decimal value of opposite/hypotenuse and adjacent/hypotenuse and recorded the results. No explanation as to why, just entertain me for a minute here.

Wait, what?

I found this website helpful during my discussion of the results. In one class not every kid got this far, again because TOO MANY TRIANGLES, MAKE FEWER NEXT YEAR. But we worked through it. Lots of interesting stuff here. The ratios seem to be close no matter the triangle, no matter the kid that made them. And what's that? Repetition? Is there some kind of pattern here? But what are these goofy decimals about?

Unit Circle

At this point I did a rare moment of lesson delivery, aka talking. Recall of special right triangles from geometry and the ratios present. While making the triangles, many kids asked why we had to bother with the 60 degree ones when duh there's a 60 degree angle in my 30 degree one. This part. This part right here.

We spend roughly a week with the unit circle and the six trig functions and the ratios and all of that. Important difference here is that I made an effort to talk about the unit circle as a collection of triangles. We always defined the trig functions in terms of triangle sides (not x and y), making note that in this very special collection of triangles, the hypotenuse is 1 and organizing the adjacent and opposite sides as a series of coordinates is helpful. An hey, in all cases the adjacent side runs horizontally (x) and the opposite runs vertically (y). Weird.

So often the unit circle is treated as a stand alone thing that we must do because the book told us to. But the unit circle is everything! Vectors, polar coordinates, and trig graphs make so much more conceptual sense if you can convince students to get the unit circle.

Task 3: Measure Stuff

Now to branch out a bit. Right triangles that are not necessarily special. We wandered out into the school and found some objects that were tall. They measured a hypotenuse length and an angle.

That covers finding unknown sides when one side an angle are known. But what about unknown angles?

Task 4: Compute Stuff

Armed with new knowledge of trig functions and that oh hey, the calculators has those buttons, I gave them a table to complete. Values of sin, cos, and tan from 0 to 90 in increments of 5.

I hand out a series of triangles where the corner (non-90) angles are unknown. Two sides are given. Students determine which trig ratio works the best based on given information and approximate the value of their angles. A few remember the inverse feature of the calculator and I was totally find with them using it. However, the idea here is to reinforce the idea that trig is big patterns. An observant student noted that the sin and cos tables seem to be going in opposite directions. Weird.

Task 5: All the Things

In conclusion, I had them create some artwork for their notebook. Display the four quadrants, orient a triangle in each one, make the sides whatever you want, show the values of the six trig functions for the reference angle, and finally, approximate the value of the reference angle.

In the end we covered right triangle trig faster than ever. And other than the unit circle, the kids spent more time listening to each other than listening to me.

AuthorJonathan Claydon