I teach integrals early in Calculus, super early really. Roughly two months ago I introduced the concept. It sounds a little crazy, but it's very handy when it comes time to do curve sketching. Somewhere along this way I coined the term "the stack" for the relationship between f, f', and f'' with derivatives and integrals serving at the means of transport between layers in the stack.

Previously, I've had students graph f, f', and f'' on separate axes, this year I changed it up. I don't really know if it's different or better, but it is more space efficient. Super bonus fact, I introduced curve sketching with a review of sketching polynomials (of the factored variety).

Here was the arts and crafts project of the week, given a graph of varying points in the stack, finish the stack:

Introducing integrals prior to this is helpful when having students determine if their stack is reasonable: if I start with a linear function of f'', I know integrating that function should yield a quadratic, and integrating again should give me a cubic. The followup activity involves interpreting the sketches: validating minimums, maximums, and points of inflection. A common narrative in Calculus is starting at an arbitrary point in the stack and asking students to interpret information about a different function entirely (given f', what's up with f?).

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AuthorJonathan Claydon

One of those "teaching is such a grind" moments. I don't know how common this is, but I'm constantly thinking about my courses in the long term and super immediate short term. The march toward May constantly at odds with what we're able to accomplish in any given 50 minutes. Generally speaking I think I know how to help students learn something, but what gets internalized and how students make personal breakthroughs is just such a mystery.

Compounding the problem is an issue I think teachers fail to recognize sometimes, the ever growing gap between my fluency in a subject, and where students are when they get to me:

The longer you teach a subject, the greater your comfort level. At a certain point, you can recite the entire curriculum without reference, and walk someone through the connections to past and future courses. You can Calculus in your sleep.

Students don't have the benefit of running through Calculus over and over and over again. They're seeing it for the first time, expecting you to be their faithful guide. Their base fluency in math will be about the same year after year. And that's where it can become frustrating. You forget what it's like to learn derivatives for the first time, and what do you mean I have to explain integration again, don't you get it yet!?

I've never ranted out loud, but we've all been there, explaining something that just feels so simple for the 5th time to someone who isn't there yet. Constantly reminding myself of the fluency gap keeps me sane. You can't teach 15 years of math grind in 8 months.

And yet, the struggle remains real. Calculus has moments where we make immense progress, and then I set them loose on an assignment and I seem to be answering very basic questions over and over again. I have options. I can push forward because darn it we have a curriculum to get through, get mad and rant about their dedication, or find ways to give them time. In most cases the students are asking the questions from a genuine place. They really want to make the connection, they know the explanation was good but gosh darn it Mister why isn't this making sense to me?

I've run through the AP ringer twice now. I know the simple things are the enemy. It's not worth bothering with the complex scenarios and phrasing if they can't do the simple things. What good is recognizing an integral is necessary if you have no idea how to integrate?

So we stop. I give them a small mountain of derivatives and integrals to chew on for a week. They get better, and we slowly start moving forward again. Despite having to take a couple of pauses, we're still on track pretty well. But there's a balance here. You can't do this all the time. Kids need a push. It's very easy for them to say "we're good" and bring the pace to a crawl. 

The data seems to indicate this is mutually beneficial. We've done two benchmarks in Calculus so far, and the students are doing about 15% better year over year. But it's easy to get excited about that data. Sure, averages are up, but what does that mean? Last year I felt super confident with my data, and got burned. I now read the numbers with a more skeptical eye. If only 10% of my students scraped together a decent performance last year, what does 15% better mean? At the same time, I know I've pushed this group harder and placed more emphasis on vocabulary. So what does 15% better on harder material mean? I have no idea.

The point is to say that it's very easy to obsess over the end goal, whether that's a district-level or campus-level goal, state testing or national testing watermark. Those end goals are good external motivators. But the struggle will remain. You've still got inexperienced learners who are going to take exactly the amount of time they need before they start improving their fluency. Kids just take forever sometimes, it's what they do. There are tools to speed this up, there are more efficient ways to teach a subject the next year, but if they just need a few days to work things out, are you willing to give it to them?

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AuthorJonathan Claydon

Wendy was looking for help on assessment the other day:

My thoughts on assessing in Calculus are ever changing. I attempted to adapt it to two-attempt SBG with a colleague, it didn't work super great. Then we tried an SBG-ish hybrid system. Then I went A/B/Not Yet, and now their assessments are graded on an A/B/Not Yet scale but I don't do a lot of the grading.

Gradebook

For the purposes of reporting, I keep track of grades. The assessments are about once a week, sometimes longer. It takes a while to cover enough unique material in Calculus for it to be worth assessing, part of the reason double SBG clunked. Each one has two or three sections. These assessments are purely for mechanical stuff. I cover all the phrasing and conceptual stuff through AP style benchmarks. These sections are recorded individually and students can earn an A (95), B (85), or Not Yet (0 or 50). All the sections added together are worth 50% of the grade. I shoot for 5 or 6 in a grading period (six weeks).

Practice

Early on I realized that little grades like this are a big pile of whatever. Since there are two and a half topics in Calculus we are constantly addressing the same things over and over, just refining our applications of them. We don't really have the full picture until April. I use these graded assessments as little checks along the way. What are we doing well? What could we do differently? What topics can students comfortably explain to one another?

The explanation part is what I want to get at. I incorporate a lot more discussion into assessment this year. We've done 5 so far and in each cases the students had a period of time where they could talk to each other about the task at hand. Sometimes the entire time, sometimes for only a few minutes. Then we'd discuss. Then I'd force them to go back and look at everything until they had a decent idea. No leaving questions blank or giving up. You aren't allowed to declare intellectual bankruptcy.

Grading is done by the students. I give them access to an answer key and they spent part of their time sifting through it. I ask for honesty in their ratings and I think for the most part I get it. Some students will ask for my opinion of their work before committing to a rating.

This is a time consuming process, a reason I minimize these assessments and stop doing them altogether at the end of January. Planning an assessment that is comprehensive, challenging, and completable in the time allotted is hard enough. Accounting for 10-15 minutes of discussion and 10-15 minutes of grading is equally difficult.

If you have a goal of assessing once a week or moving through a very long list of topics (like the way I do Pre-Cal), you will find this method rough. If you have a class that is pretty focused in scope and you have some flexibility in your time table, give it a shot.

The discussions are fascinating to listen to.

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AuthorJonathan Claydon
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Here's a silly classroom thing to brighten your mood.

Last summer I gathered a couple groups of 11th graders together in June for Summer Camp. At some point during the first week, related in no way, I was watching an episode of Casey Neistat's vlogs. In the middle of an episode, for a whopping two seconds or so, he has a moment where his daughter is watching silly kid song videos. Specifically, Baby Shark:

I wish I could remember the episode (and I spent a long time trying to find it again), and the context of the discussion that lead to this. But towards the end of Summer Camp Session 1, I played Baby Shark for the kids. The next day they demanded more. Then on the last day I asked for feedback:

This song is old. It's some traditional summer camp song that I had never heard. But it's infectious, apparently, because by the end of Session 2, kids had learned the dance, were showing their younger siblings, and were bringing me videos of little cousins dancing along to this shark song.

School year starts. Summer camp kids are like, PLAY THE BABY SHARK. The rest are befuddled. But it spreads. Soon the whole cheerleading squad knows the song. Kids in band are singing it to each other. Kids in band who have no idea about the origin are singing it to each other and doing the dance moves. Cheerleaders are CHASING ME DOWN singing the song. Heck, I'm driving the volleyball team back from an event and the older ones sing it FULL VOLUME on the bus ride home. Later on one of my 9th grade players (who would not know the origin) sings it at random in practice.

MISTER THERE'S A HALLOWEEN VERSION

MISTER THERE'S A CLUB VERSION

Now they're asking when the Thanksgiving and Christmas versions come out.

The drum major arranged the song for multiple instruments. He video taped himself and four others playing it on brass instruments. He learned to play it on the guitar. Another student learned it on the piano. The two volunteered to play it in the talent show while a bunch of goof balls dance around in shark costumes (and omg was it not hard to find volunteers for THAT).

Not done.

There was a fall festival at a local elementary school, some of my students were volunteering and MISTER THEY HAD STUFFED SHARKS AND I TOOK SOME FOR YOU.

The kids take turns with them at their table. One kid had them dancing along to Michael Jackson while I was playing music. Another boy, I swear I am not kidding, uses one as a comfort animal because it helps him concentrate (and IT WORKS).

The whole thing has just taken on a life of its own. I just don't even know what to say. Kids are great. I'm not sure what this has to do with teaching math, but it seems to help.

Don't underestimate the selling point of silly things.

Unrelated duct tape shark from long before the madness.

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AuthorJonathan Claydon

Since my Algebra II experiment years ago, I've been obsessed with students to start making connections between graphs and algebra. Fairly consistently I have students who dismiss graphing and something uninteresting because no one seems to making connections about why it's so interesting. That's it's possible the "x= " they're able to find with a calculator represents something. Pre-Cal has some algebra objectives and I've brought what I learned with that experiment along for the ride.

In addition to emphasizing graphing, I've been demanding a lot of explanations from my kids this year. A go to for a while now has been justifying a situation. Recently I gave kids a graph and three possibilities. In class we focused on the quadratic formula, it's super solving powers, and it's ability to show you the x-values of the intersection points for a pair of functions. In the setup for this question, I didn't specifically ask them to use this method, though most went with that approach. Here's a representative sample:

The brute force method. The student solved all three and made a connection to the picture with their work. Nothing wrong with it.

However, some found other ways:

An excellent observation of graph transformations. The functions used were simple enough that spotting some features was a quick way to accomplish the task.

The next student made a similar find:

Not only did they attempt a quadratic formula solution (apparently done on the calculator and not recorded) but they weren't satisfied that the mere presence of an imaginary number would do the trick. The slope of the line was important!

This last one might be my favorite:

Not only do they grind through the quadratic formula here, but they remember that if those intersection points are solutions, they are useful! I think only one other student (out of ~60) tried this out.

I was pleasantly surprised because I only had one method in mind when designing the question, forgetting that kids could wander in so many interesting directions.

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AuthorJonathan Claydon

The hard part with new initiatives is keeping them going. Sometimes building and maintaining momentum is easy. Five years ago I decided to make notebooks a thing, for example. Am I as fired up about them as I was years ago when I figured everything out? Not exactly. But it's still an idea that needs my support. It continues to be a proven system. It'll last for decades. Take something different, like technology implementation. How do you keep up a strong technology presence for five years? ten years?

Ditching procedures if they no longer motivate you is easy. The thing about school years is your students move on, the knowledge of your procedures with them. My current groups are clueless about how things worked years ago.

What about bigger, organizational goals? When I took over Calculus I had some problems I wanted to solve. By accident it spawned Varsity Math.

New ideas are awesome, but then you realize they come with maintenance. Varsity Math is now a living thing that needs attention or it'll just disappear. How do we make it more permanent? A permanent reminder to myself "HEY KEEP DOING THIS THING."

There was a neglected corner down the hall that never had anything interesting on it. A random gap between unused lockers. For years and years I had visions of putting some kind of installation in place, some legacy to the work I love. Other parts of the school have great art installations from decades past, I wanted something to contribute to the line up. The Estimation Wall was a good first step, but temporary.

Presenting the Hall of Fame.

Painting begins. A day later random passersby are buzzing about the maroon wall.

Piecing together the enormous stencil created with a few hundred 8.5x11 pages.

Arguably the easiest part of stencil construction.

"They painted the wall? What are they doing?" "They" have something big in mind...

Some wall finish technicalities required improvisation to draw the 80" circle.

Hours of letter cutting, taping, and gluing we're ready for the final step.

All done!

This project was completed off and on from October 6 to October 31. It involved the effort of a couple dozen students in my Calculus classes working on random bits at a time. One crew did the maroon base coat. Another did the initial assembly of the letters into something coherent. Others cut and glued the assembly onto stencils boards. Others spent hours hand cutting each letter from the boards. And a final group jumped in for white paint and touch ups. In case you were curious, the capital letters are 2051.94 points tall.

It's awesome, and it's ours.

Class pictures of each Varsity Math crew (including Summer Camp) will go here as well as a notice board of students who pass the exam. I want decades of classes on here.

Coincidentally I saw a former student from the original crew right before this was completed and I showed her pictures of this. She couldn't believe what she help start.

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AuthorJonathan Claydon
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I adopt initiatives slowly. Specifically tech related stuff. I'm an enormous skeptic, trying really hard to avoid gimmick implementations. Recently, I found that the exciting world of iPads was really kind of bland given the expense and maintenance hurdles.

This school year I have a fresh batch of Chromebooks, which do everything I was getting out of iPads but with a little more flexibility.

A couple weeks ago I had some goals, teach some Desmos fundamentals and get the kids to create some trig function graphs and print them out. This is not, by any means, very amazing, but if you were curious about Chromebooks, here's what I've done. It was pretty simple and the tech was able to enhance the final product.

Learn Desmos

We spent a day getting familiar with the interface. I wanted this year's group to be more proficient with lots of things: axes controls, projector mode, restrictions, and line types to name a few. I pointed them towards the official Learn Desmos page and gave them a few topics to work through. At the end I had everyone go to the main Desmos calculator site and link their Google account to Desmos, enabling saving for future projects.

Activity Builder

Two years ago I started tackling graphing tasks with several rounds of how did I make this? Students were given a printed set of graphs, some guidelines, and asked to recreate them. At the end we discussed the effects of any parameters involved. I liked the concept enough that I filmed it for my PAEMST application.

I first saw Activity Builder during the Desmos morning session at TMC15. It was interesting, but at the time I was willing to wait. In the interim it has improved dramatically. The addition of a folder that can be hidden from students is the magic feature I think I was waiting for. Some real effort has been put into the teacher dashboard. I kept it open on a laptop only I could see as the students hummed along. The goal of class recently was to work through properties of trig functions and have discussions about amplitude, frequency, and translation effects. If you fancy a look, here's the two activities they worked through: Graphs of Trig Functions and Pro Trig Graphing Skills

Output

Now it was time for them to show me what they've learned. I took an old standby project and wrote more exhaustive instructions. I even increased the amount of things they would be doing. With Google Docs as a point of distribution, I was no longer worried about instructions that could fit on a single computer screen.

Trig Graphing Project

Students were able to save their work in Desmos and then work on it outside of class if they wanted. The instructions didn't include the writing that was required, but you can see an example of the final product here:

Later on, it was time to assess. In the past students would be given a few trig functions to graph. They'd be mediocre at best. With better tools available, I refocused the assessment to be more language based.

Conclusion

In general, tech needs to get out of the way for me. I want students spending their time on the task at hand, not making something work. There were struggles in this progression because we were working through the start up costs: learning Desmos, setting up printers, and the fiddly nature of Google Docs. In the end, we made a lot of progress, made some nice projects, and now I can raise the demands of other activities.

Thanks to the people at Desmos who have really been working hard to make all this so easy to deploy.

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AuthorJonathan Claydon

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.

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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.

Challenge

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?

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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.

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AuthorJonathan Claydon