Every so often I stop and realize that I have been at this for a decade. I have never been actively working on such a particular idea for so long. I think it’s safe to say the crisis of career I faced a long time ago has been settled. This is what I’m supposed to do. In recognition of this “holy crap 10 years” and the fact that I like making charts, I made a little stress diagram of those 10 years.

The taller the bar, the more work I felt like I was doing. Underneath is the variety and quantity of preps I had. Soccer balls and volleyballs represent coaching years. Trophies are major awards. Years where I took on a new prep or big responsibility spiked the work load, as I figured out how to do something for the first time. I put a LOT of effort into things when I’m doing them for the first time. REALLY quickly I figured out I like being thoughtful about my assignments and not just taking them from a binder, and that took a lot of time. Now I reap the rewards of that investment constantly. In the case of a new prep, there’s curriculum to map, assignments to make, and unknowns to solve. With a new responsibility, the time management needs a rebalance.

Along the way my confidence grew. You start to see that kids are buying what you’re selling, and that you can sell it really well. You get comfortable in the space, adapting good ideas to any old prep. College Algebra (Algebra 3 in local parlance) is this self-paced little wonderland because of all the grind that came before.

The first five years I felt I had something to prove. I was an outsider to education, a random guy with an alternative certification who did not know what he was doing. My first group of kids were very kind and said I did a good job, but I really did not know what I was doing. I wanted to show my school that I belonged, and that I could be trusted. Earning trust in the workplace is the hardest thing to do, and is so valuable once you have it.

Now though? Man, this is just the best. Yes I have three preps. Yes I coach a sport. Yes I’m co-running our National Honor Society. But it’s just so smooth. I’m not really sure what the shift was, but it’s an enjoyable place.

I want to deep dive some more into some other trends, so prepare your self for 9 more emoji-laden charts.

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AuthorJonathan Claydon

Some of my biggest experiments have been with assessment. It started with an SBG adoption in Pre-Cal and Algebra 2 seven years ago and it really changed the way I view assessing students. Scores out of 100 are silly and arbitrary, so I don’t bother. These days Calculus takes assessments that are segmented by a particular topic, usually integrating a variety of skills into a short set of questions. Normally when you have an assessment, tradition says you should review.

Long ago before SBG I would write up reviews that students would complete the day before an assessment, you’ve probably done the same. In practice creating a review is almost as much work as writing the assessment. I quit doing stand alone reviews years ago because I think they send a signal that classwork isn’t as important, this review is all you should care about. I want students to be diligent about completing classwork and seeing its purpose, so I’ve designed my “review” around that idea.

Both flavors of Calculus are taking an assessment today, here was their “review:”

I put the list on the screen and kids can take a picture. That’s it. We spend zero class time on this because everything on the list is represented in some piece of classwork we did in the days before. Students who were diligent about organizing their classwork should be able to find anything. The only thing that takes time is if a student has a question about what I mean by a topic. For example, they may want to clarify what I mean by “factor a polynomial into a sketch-able equation.”

I’ve found the practice effective. Previously students have said I’m too vague about what might be on an assessment, so they can’t focus their time. In fact, I’ve been too vague because usually the assessment isn’t written until later. I think these simple lists give them the focus they want without spending a ton of time on a purpose-built review. More importantly, it helps me focus when writing the assessment, to make sure I stick with whatever ideas I had when writing this list. I am incredibly bad about changing my mind constantly about approaching things. This has brought some much needed focus to my work as well.

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AuthorJonathan Claydon

Every so often the stars align and we have a real casual experience all day long in my room. Yesterday was the end of a marking period and every kid was in “finish this stuff” mode.

In Calculus we were wrapping up a function analysis project:

BC Calculus was up to something similar, and we talked about a benchmark they took recently.

Over in College Algebra kids had two tasks to accomplish, wrap up a project on polynomials:

…and take a test. After submitting their project, they grabbed their weekly (ish) assessment and completed it on their own time. College Algebra is a very casual environment. Whole group instruction isn’t really a thing in there. In any given week kids have a to do list (new lesson, assignment, assessment) and have to complete it by the end of the week. There are 23 kids in this class and they work at 23 different paces. Often that means a handful are done early. Most of the time they’ll get a bonus activity (via Desmos or something else), or sometimes I’ll bust out the puzzles:

So, in yesterday’s 50 minute period, lots of kids were wrapping up their assessment, a few handed in their projects and started/finished the assessment, and others were finished, all with some Weezer in the background. 23 kids, all hanging out, taking care of whatever they needed to get done.

And it played out that way all day long. Hanging out, doing some math, listening to music. It is my favorite learning environment. To spend the whole day in it is simply sublime.

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AuthorJonathan Claydon

I’m on a mission to fix all kinds of things about Calculus, especially Calculus AB. Next on deck is Curve Sketching. Previously, while working on integration of Intermediate Value Theorem and Mean Value Theorem, I integrated some curve sketching ideas. Students calculated average rates of change and created rudimentary graphs of a function’s first derivative.

A few weeks later, students now have more familiarity with the first and second derivative and we can talk about how those tables help us analyze a function.

Start with an arbitrary function and interval, and create a table of f’(x) and f’’(x) in Desmos.

At the moment we aren’t concerned with the graphs, those this will be useful later on. Have students recreate the table, but we’re going to declutter the results. Rather than worry about all the values generated, let’s look at whether the first and second derivative were positive or negative at the point.

Having discussed the Intermediate Value Theorem, we have a discussion about where values of zero should appear on our table. For the first derivative, we reestablish a connection we made before, that if the slope of a function is positive, it must be increasing. A zero should mark the transition between increasing/decreasing or decreasing/increasing and these points are important enough to have names.

Next we have a discussion about the second derivative, which is a newcomer to the party. Some days before this activity, we plotted tangent lines, computed second derivatives, and looked at whether the tangent line was an overestimate or underestimate. That opened up the idea of concavity, that the concavity of a function plays a role in how accurate a tangent line will be.

Now it’s time to define concavity a little better. We look for points where the second derivative must be zero and what that could mean. At this point I’m talking with the table and graphs in view, so students can see that something is happening to f(x) at the point where there should be a zero on f’’(x).

Going back to their horizontal table, we now annotate the table with our findings. Based solely on sign value, we can quickly determine where a function is increasing, decreasing, concave up, concave down, and the role of the various critical numbers.

The purpose of all this is to improve a HUGE weakness I’ve seen over the years. For whatever reason, while I could get students sketching f, f’, and f’’ like geniuses, there was a disconnect between how they were making their sketches and what they represented. If a non-sketching question said something about the first derivative being positive, I’d get nothing but blank stares. Very few of them were able to determine that corresponded to increasing behavior.

By building this competency with tables AND graphs, I’m hoping things improve quite a bit. By sticking with equations of tangent lines and tables as recurring themes, I’m hoping free response style questions are more comfortable. It’s way too early to tell, but I’ve really liked how this is going.

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AuthorJonathan Claydon

Since becoming a goofy idea I revealed to the public in 2015, it is FINALLY time for you to get involved in the action. I am proud to announce that Varsity Math merchandise is now on sale for a limited time. Grab a t-shirt, sticker, patch, socks, or a full matching set! If you would like to donate to the classroom needs of my students, you have that option as well.

Varsity Math is something I enjoy very very much and I think the kids like it ten times more than I do. Come join the team!

For some more background info, have a look at the highlights from last year’s Varsity Math Day.

Limited to customers in the United States only. TX residents add 8.25% sales tax on all purchases (except donations made through the button). Store open for a limited time.

VARSITYMATH.NET

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AuthorJonathan Claydon

You spend a lot of time around young people and you can’t help but do a lot of analysis of how you were as a young person. By far the biggest difference I see in myself and my students is how local their field of view seems to be, at least in my observations. When I say local field of view, I mean a lot of their time and energy is spent on the present and some handful of hours ahead of the present. What’s happening next week? month? year? You wouldn’t know they were concerned with such things.

And that’s not a bad thing, kids should get to be kids and only worry about the here and now as much as possible. Planning their next meal with friends or scraping together $10 for an impulse purchase is what youth should be about. Yes, college or post secondary plans, but kids are just so good at living right now.

Somewhere in the last 15 years, my brain got bored with right now. Or perhaps, with enough experience, you can handle the right now so well that it just doesn’t require a lot of effort?

Just off the top of my head, school related things I happen to be processing and their due date:

  • logistics for laser tag (7 months)

  • approximate sales expectations for my upcoming Varsity Math merch line (1 month)

  • is summer camp happening again? how many might go? (9 months)

  • painting a wall in the school visitor entrance (3 months)

  • National Honor Society induction logistics (6 months)

  • approximate AP testers (5 months)

  • AP benchmarking ideas (2 months)

  • TMC 2019 (10 months)

  • TMC 2020 (22 months)

  • does Varsity Math have a logical conclusion (20 years)?

Stuff like what am I teaching tomorrow? what should I eat today? what game/show/movie should I get into? just aren’t interesting questions. Have I just solved those problems too many times?

Is this why I tinker with curriculum so much? To keep my own attention?

I still very much like planning what I’m teaching tomorrow and what I should eat today, but they just aren’t a source of concern. I started noticing this about 3 years ago when I passed an experience milestone.

As friend Rachel put it the other day, it’s like I’m just in a constant state of playing tug of war with the future. As one long term project ends, another steps in to take its place. The never emptying to-do list.

Does this stress me out? No. It’s bizarre really. I just really like working long term problems. I mean, I’m still in the middle of my longest problem to date, improving AP effectiveness. We’re at 4.5 years and counting with only preliminary results.

You might be tempted to call me a workaholic or something. Some have before. I just don’t see it that way. For one, I don’t see it as work. I do “non-work” things, whether you hear me talk about them or not. I haven’t gone to “work” in 10 years. I’m solving interesting problems, something it turns out I really wanted in a career. Teaching offers fascinating logistic problems that I really really really like. It’s probably the biggest thing I liked in engineering school, mapping out plans to get stuff done, forged in a (now demolished) dorm basement as I figured out what it takes to pass exams. Secretly, I don’t really care for summer break anymore (though it is still a nice respite, and I mean, duh, TMC is the greatest), as there aren’t as many problems to solve.

It’s almost a bit of a game I play with myself, “bet you can’t figure out how to start a company” being the latest.

All this to say watching kids is just fascinating stuff, it helps me try to remember what it was like before whatever switch flipped in my head.

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AuthorJonathan Claydon

Not long ago I had this simple idea for introducing the idea of an antiderivative. Turns out it wasn’t long before the idea would pay off again. In College Algebra, we were looking at radical equations. Solving stuff with square roots wasn’t new, but solving stuff with cube roots and fourth roots was. I place a huge emphasis on connecting algebra to graphs in College Algebra. If we’re solving a cube root equation, I want them to be able to graph it.

I cheat a little bit and tell them Desmos “can’t” accept a cube root as written with a radical symbol. It very much can, if you go digging around in the function buttons. However, I use it as an opportunity to introduce root and fractional exponent equivalence. It was a bit of a struggle to get kids to remember, so I thought we’d play a game with some of the extra index cards I had lying around.

As before, groups of students were given a random pile of cards. Each card had 1 match in the deck. I did not tell them how they were supposed to make matches.

Thrown into the mix were some rational expressions and their equivalent form with negative exponents. Students had no exposure to this other than from previous classes. I made the coefficients unique enough to where they were able to put things together using that as a context clue. As groups concluded their pairs, I went around and had a discussion with each group about what rules they had developed to make their matches.

Within 10 minutes, we had decoded fractional exponents and negative exponents without any boring lecture about exponent laws.

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AuthorJonathan Claydon

This moment is a little surreal. If you’ve never heard the origin story of Varsity Math, here’s the moment I revealed it at TMC 15.

At TMC 16, I spread the love to everyone in attendance, offering a free sticker to all 200 some-odd attendees. In the years since, Varsity Math has become an enshrined institution at my school. Kids are incredibly all in on the concept.

At recent conferences, I’ll sport my Varsity Math merchandise and people will ask me for some. I am only able to give out a handful of extra stock that’s left over from the school year. And I always feel bad because there’s not really a lot to go around, and I don’t want people to feel left out. Others want to get something going like this with their department but may not have the means to generate merchandise. This year, I am excited to announce that you, teachers at large, can join on the fun as well. After figuring out a lot of random paperwork, Varsity Math is now a legally recognized company in the state of Texas.

Starting in November, an online store will open at varsitymath.net which will redirect to a page on this site. It will be open for a couple weeks as I test out taking and fulfilling orders. Some months later it will probably open up again. You can guarantee I will do a run for TMC 19. On offer will be stickers, t-shirts, socks, and sponsorships. If you purchase a sponsorship, 100% of that money will go towards the various Varsity Math activities at my school. Students are not always able to cover the costs associated with the merchandise and activities I provide. Traditionally I am able to cover their costs, but outside help will allow me to do more and possibly, one day, lower the cost of entry for all students.

I have put out feelers and it seems like there’s a demand, now we’ll see what happens. If you would like official Varsity Math gear, keep an eye on this space and Twitter for the opening of the store. Talk to people in your math department and buy shirts as a group! I would love to see you all sporting the best in math clothing (next to Desmos swag).

Huge thanks to everyone who has shown enthusiasm for my little high school joke and got me to this day. See you soon!

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AuthorJonathan Claydon

It’s about time to start talking about antiderivatives in AB Calculus. We’ve defined a few rules so far: power rule, chain with trig, chain with e^x, and chain with ln x. For several years I’ve wanted to be early about our discussion of integrals. It has been helpful for conceptual understanding later if they know about the relationship between derivatives and integrals early. In fact, knowing about integration makes Curve Sketching a lot easier.

I was just going to do some simple introduction, but then I decided we could use a moment to get up and walk around.

Start with a 53¢ pack of index cards. Write a function on the top of the card and its derivative on the bottom. Repeat about 40 times.

Cut the cards in half. For fun, include several functions that have the same derivative.

Shuffle the cards a bunch until they’re good and scrambled.

The plan is to hand them the stack of cards and tell them that each card has a match. I’m not going to say how the cards are related to each other, only that have a match in the deck. After we spend some time sorting them, we’ll talk about the results.

How are the cards related?

Could some cards match with multiple cards?

Eventually they’ll tease out the idea that one card was the derivative and one was the original. Then bam! we hit them with the idea of antiderivative, the result of working backwards from a derivative. It is highly likely that when finding some matches they will do this, knowing they have the derivative in hand, in search of the original.

The fact that multiple functions can have the same derivative is always an interesting discussion, and just like that we’ve justified the presence of +C.

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AuthorJonathan Claydon

I have traditionally had trouble fitting in Intermediate Value Theorem and Mean Value Theorem into Calculus without them seeming arbitrary. Kind of by accident, I found a nice way to not only talk about both theorems, but introduce early idea for curve sketching. And it was super simple!

Start with a table of a continuous function:

Screen Shot 2018-10-04 at 11.26.51 PM.png

Have students calculate the average rate of change on all the intermediate intervals. In this case, 3 to 7, 7 to 11, etc. In addition, have them calculate the average rate of change for the extreme values of the table, in this case 3 to 26.

We now have a list of slopes: 1.525, 2.975, -8.6, -1.3667, 1.683 with an extreme slope of 0.1227.

Next, have students graph the values from the table.

They did this by hand, but I used Desmos here for demonstration purposes. Have them describe the behaviors of the function. Next, make a graph of the slopes, including the slope we found from the endpoints:

Great moment to talk about why we would graph the slopes this way, and the assumptions involved when taking an average rate of change. Now some questions. How many times is the slope zero? How many times is the slope equal to the overall slope? Could we sketch a function that would output these values?

Go back to their description of behaviors. When the function was increasing, what type of slope did you have? when the function was increasing? when the slope appeared to be zero?

Before you know, you’ve teased out the concepts behind the relationship between K(t) and K’(t) in addition to a good demonstration of the Mean Value Theorem at work. Finally, with a graph of the original function, a discussion of Intermediate Value Theorem comes naturally. Pick some arbitrary y-values and have students decide if they should exist. How would you know?

I don’t even know what made we think of it, but adding the graph of the slopes to the mix really made this an interesting problem. Everything I need to cover in the next few weeks is sitting right there! And students found it all intuitive, we just needed to add some formality to our justifications. I really enjoyed how this turned out.

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AuthorJonathan Claydon