At long last it's AP Test Day. What started with a hurricane and some kids on the couch has finally finished. When I started teaching Calculus, I had a personal mandate to improve the results on the AP Exam. Though it is just a test, there was no reason in my mind our students couldn't be successful at it. They're more than capable of piecing that thing together.

I started trying to be very specific about predictions and found that in general that back fired. This year, the thought was, let's just look for promise. Who shows the indication they'll be able to make it work on May 15?

Shortcomings

While I was able to do a significant amount of catch up despite the 12-day delay, a few things just never happened. In AB I still sacrifice things like related rates, in depth discussions of the mean value theorem, and an in depth look at solving differential equations. As a program we weren't grasping the basics anyway, so those fringe things weren't making a big difference. On top of that I just didn't have the time. I needed it for other things.

In BC we pretty much got it all, though I'd say they have some other fringe weaknesses. Mean value theorem, fiddling with Lagrange error bounds, and the more obscure convergence tests.

Despite the shortcomings, in our final review sessions there was very little practice exam material that was not within my students' grasp.

AB's Long Prep

A majority of my focus was on getting the AB group ready to go. Historically results have shown me that students struggle with free response. They lack some of the exacting finesse you need to make sufficient answers. Starting in March, I removed their standard assessment system and replaced it with released free response questions. In some cases students were given the use of a picture or calculator where normally it wasn't allowed because of where we were at in the material. Students completed 4 of these sets, composed of 9 full length FRQ on the subjects of particle motion, function/rate analysis, volume, and data tables. I tracked their results and some questions were used as follow ups once we had discussed some pitfalls.

If you're interested, Set 1 was 2012 #6, 2008 #4, 2009 #1; Set 2 was 2016 #3, 2017 #2, 2013 #1, Set 3 was 2013 #5, 2014 #2; and Set 4 was 2012 #2.

The goal was to expose them to the language and requirements of a free response questions early and often so that they were less intimated by them when the exam rolled around. In addition to these assessment items, they were given a pack of "ultimate" free response questions that I wrote myself a couple years ago. For College Board reasons I can't post them, but if you are a credentialed AP teacher, please get in touch.

We had many conversations about showing mathematical thought in their answers. There should be a clear setup (what information am I going to use?), work (what am I going to do with the information I chose?), and conclusion (how do I interpret my work?). Students who put the effort in improved dramatically at these as we got closer to the end. In some cases they exclaimed how easy the question was, something I had never heard before.

BC's Prep Integration

I exposed BC to released material early and often. We were doing free response questions since October. They worked through a full AB exam in December. My challenge with this group wasn't going to be language or organization, but grasping the sheer amount of material we had to cover.

When we got to their unique topics in the spring, I integrated AP-level material at every step of the way. Almost every topic discussion was ended with a look at relevant multiple choice and free response questions. They took a number of Desmos assessments that were structured like free response questions. These kids were very familiar with the requirements of a free response question.

Final Stages

Each version of Calc wrapped up new material in early April. On April 16, I started doing after school tutorials. My students have a number of commitments, so I made these flexible. AB students needed to see me once a week over the course of 4 weeks. For BC I reserved two Wednesday afternoons for us to discuss whatever they needed.

I posted sign up sheets for AB students.

Each week had a theme. In some cases I covered material we didn't have time for and the last two weeks we talked about exam strategies. My students have limited experience with big exams like this, and time management is something that needs improved. In addition, they will think themselves in circles for 10 minutes on a question they should probably skip. We discussed doing a full reading of the exam before starting any work, helping them to prioritize their time. Finally, we talked about what was really necessary for a passing score. Many many many students are unaware that 45% or so represents a passing standard on this exam.

Out of 62 AB exam takers, they showed up after school an average of 2.64 times. I can't make these things mandatory (I know the sign says mandatory, but that's for effect) or grade them for showing up or anything, so that's pretty good considering it was all voluntary. 40 students did what was requested and showed up 3 or 4 times.

All the BC students attended both of their sessions.

Results

When we concluded new material I gave both groups a significant number of assignments to work through. Class time was work time for several weeks, with tutorials in the afternoons. Each group got several skills-based assignments and then separate AP-level material. BC worked through another full length exam. AB had a selection of multiple choice and their "utlimate" free response. We went over the AP-level material either in class or after school last week. Students were given access to answer keys.

What will happen in July? I don't really know. I have some good feelings based on my observations the last month. I tried to gauge things based on how students asked me questions. Most knew what to do and were seeking confirmation, with others it was apparent they were at step 0. Fortunately, I think that group was minimal.

I would be very amazed if any BC student got below a 2, they've all shown strong levels of comprehension all year. The BC cut lines are also a little more forgiving because of the content demands. A number (in my wildest dreams, all) of them will pass without question. AB feels like ~70% of them should register on the scale (2+). I scratched in some predictions for each kid, but I'm prepared to be surprised in either direction. Ultimately, I want the 1 to become the minority score in my AB results. How many can make the jump from 2 to 3+ will just have to wait. Fingers crossed for about 70 days.

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AuthorJonathan Claydon
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Last week at NCTM 2018, I presented some thoughts from a radical Algebra II experiment I ran some years ago. While Algebra II was the center piece, the topic is broader. What story are you trying to tell with your curriculum? How do you integrate pieces so that students can see a theme in the skills and concepts they learn throughout the year? Why should what you teach in September matter in April?

This is a summary of those ideas from my talk.

The Script

The textbook models of Algebra II are pretty consistent. A whole parade of function types, done in isolation, over the course of a school year. Your experience probably looks something like this:

I dutifully followed a script similar to this for four years of Algebra II. But there were lingering problems. In the Fall of 2012, I included this as part of my final exam:

I made what I thought was a fair assumption: we had covered linear, linear systems, absolute value, and radical functions throughout the fall system and students should be able to work with them when mixed to together.

I was completely wrong. Students had forgotten fundamental algebra mechanics and were thrown trying to remember not only how the non-linear ones were dealt with differently, but even what they were in the first place. The topics we discussed had been so spaced out that students had made no valid connection that equation solving is a universal thought, different functions merely introduce new mechanics.

Second, we had this section in the book on parent functions. There was an activity that accompanied it:

Students spent a class period matching equations, words, and graphs together to form a parent function library of sorts. Then we never talked about this activity again. It served no purpose other than to say "ooh, activity!"

The Flip

I spent the summer of 2013 frustrated with Algebra II and determined to find a better way. There was good material in this course, but it was being handled in such an awful way. The messaging was all wrong. If you take the Texas State Standards for Algebra II and make a word cloud, you get something like this:

Clearly, the individual functions themselves are not the most important players here. Equations and Inequalities are the most frequently used terms. Incidentally, you see how small "real world" is there? Probably could stand to fix that too.

I revisited the parent function activity and thought: what if this was the entire course? What if we talked about parent functions at the very beginning of school and referenced it throughout the year? What if we focused on not just solving equations with one function at a time, but with many functions at time?

Pivot Algebra Two was born. Let's have students see how a set of skills play out with many different kinds of functions all at once. I divided the curriculum into two parts. Part 1: linear, radical, quadratic, and absolute value. Part 2: log, exponential, rational, polynomial. In each part, we would do overviews of the skills in play and slowly spiral up the difficulty.

In addition to taking tons of times to play with equation mechanics (roughly 6 weeks on linear, quadratic, absolute value, and root functions with increasingly higher difficulty), students got to see connections between their work and graphs. I don't know about you, but prior to this experiment, my kids hated graphing. It was a tedious process that seemed unrelated to anything. In this vision of Algebra II, I was going to give graphs a purpose.

The Motivation

When it was time to start a month of graphing, they groaned. As the ones that came before, they hated graphing and didn't understand the point. But then I took some classwork we had done previously. At this point they all agreed that solving equations was super easy. I graphed one side of the equation, and then graphed the other. I clicked the intersection points. MINDS BLOWN. The graph and the equations are related! I can check my answer easily! I don't need your help anymore!

It was a beautiful moment. I used it as a jumping off point to more complex algebra. In both our look at advanced quadratics and logarithms, we started with the graphs. We know that the intersection point of two graphs is equivalent to the solution of the equation. What algebra do we need to prove that?

Kids saw the power of the quadratic formula. They saw transformations for what they were, ways of manipulating various parent functions into equations.

Then we summarized what we learned about our first four functions:

The Payoff

The efficiencies in this system were apparent pretty quickly. I was no longer reteaching simple algebra moves. Kids had them down. We could tackle mechanics previously unheard of in Algebra II. We even put logarithm properties to good use with problems like this:

There is a lot of overhead in this problem. You have to understand how logarithms combine, how they are undone, that the resulting quadratic has two solutions, and that it's possible not all of those solutions are valid. And, to my sheer astonishment, the kids were telling me this was easy. Absolutely bonkers to think that this problem, which take a good 5-6 minutes of kid time to work, was easy! I'd ask if they need any help and they'd say, nope, I can just graph it.

These connections also help improve their ideas of equivalence. We could take any line in this process, graph it, and see that the intersection points had the same x-value. We were manipulating a log into an equivalent parabola.

The Longterm Lesson

This line of thinking, finding the story in my curriculum, continues to pay off. Because of this exercise, I have improved my ability to weave concepts together in other courses. I no longer teach Algebra II, now I'm primarily focused on Calculus, but the fundamental thought behind this experiment remains. Calculus has a story, and students should be able to see it.

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Calculus is even better suited to these ideas. Limits, Derivatives, and Integrals interplay with each other in all kinds of ways. What if students saw that relationship from the very beginning of the course? What if they didn't have to wait 4 months before they even heard the word "integral?" How much better could they be at seeing those relationships?

The Conclusion

I encourage you to take a hard look at what you teach. What are the big ideas? What should students be able to do when it's all over? How can you give them an overview of their year in the first month of school?

For more detailed information, I chronicled everything about my radical journey through Algebra II. Please contact me if you need anything.

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AuthorJonathan Claydon

Every spring, during a brief period of time when it's very pleasant outside before the ravages of summer, a few hundred students venture outside and make over a half mile of public art. We call it Sidewalk Chalk Day.

Sidewalk Chalk Adventures
Return of Sidewalk Chalk
Sidewalk Chalk Three
Sidewalk Chalk, the Fourth One
Sidewalk Chalk Five
Sidewalk Chalk Six

Now etched into school tradition, Sidewalk Chalk Day features students from all kinds of math classes displaying graphs of whatever it is they have been learning recently. As one faculty member who was strolling outside as we worked put it, "I always know it's spring when the chalk goes down." The first iteration involved two sections of Pre-Cal. This year 11 classes (AB Calculus, BC Calculus, Pre-Calculus PreAP, Pre-Calculus, and College Algebra) went outside throughout the school day.

Setup

Myself and a colleague pick the day a couple weeks in advance. It's trickier to pick a day than you might think. Because of our bell schedule, only Tuesdays and Fridays work, and, as the subject that started the movement, it has to coincide with Pre-Calculus classes wrapping up Polar Equations. Go to early and the material isn't covered thoroughly enough. Wait too long and suddenly it's testing season and nothing works.

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Day in mind, we incorporate a unit that will culminate with students graphing something. Pre-Calculus students create a pair of polar equations (rose curves/cardioids or something in between); AB Calc creates regions between curves with stated integrals for area and volume using that region; BC Calc creates polar equations with integrals of area for them; and College Algebra showed off logarithms and their corresponding inverses. Each student created their own set of equations/graphs that fit the requirements. They were given two panels (or one double panel) of sidewalk to show off their work. If complete, extra panels could be decorated however they like.

While students worked I flew around a drone and took pictures. Last year there I made a video compilation. The weather didn't cooperate this year, so I had to settle for pictures. Enjoy!

Posted
AuthorJonathan Claydon

I have always liked introducing games into the classroom. Often I take trips to Target just to see what kind of new games they have that might work with kids. Some are relevant to math, and some are just fun to play in big groups. With the advent of internet connected devices everywhere, a new kind of genre has opened up, trivia games played on a console or computer that lets students enter a room code and participate in front of everyone for fame and fabulous prizes (well, not so much the prizes). You know, like Kahoot, but better. WAY better.

I have gotten endless value from the Jackbox Party Pack, specifically Party Pack 3. For $25 you get 5 games, and two of them make FANTASTIC classroom games. Right before Spring Break, with a confluence of a blood drive, field trips, and general maybe-lets-not-introduce-something-new-right-before-a-giant-holiday, I played Guesspionage with all my classes. Participants take turns approximating answers to survey questions, the other players get an opportunity to decide whether the guess is much higher, higher, lower, or much lower. Points are awarded based on the accuracy of the guesser, and who correctly said higher or lower, with a bonus for much higher and much lower. In the final round, everyone is given the same question and has to choose the three most popular answers from a set of 9 (for example: which decade would people like to live in the most?).

Thanks to The Array™, I brought in my Nintendo Switch and we were off.

The questions are great. Every kid has an opinion and some of the answers will surprise you. There's also the collective FREAK OUT if a participant manages to get the question exactly right. Guesspionage allows 8 players to compete in the main game, with room for an audience who are also able to answer the questions. You can also set a family filter if you have a younger audience.

Here's a gameplay video:

Once kids understand the rules, the game runs itself. I just sit back and enjoy the arguing. When prompted to play a second round, every class was a unanimous "YES MORE GUESS NOW."

Posted
AuthorJonathan Claydon

My most elaborate creation has got to be what I affectionately call The Array™. It takes a Mac mini, my dedicated teaching computer, and duplicates its screen all around the room to six little pods. It was constructed one piece at a time starting in 2011 and might be nearing its final form, well until fancier TVs get cheaper or school becomes a VR head set experience anyway.

Here's a schematic

Each TV serves as a focus point for discussion in each little pod. I can have something on display and reference it anywhere in the room. My ability to turn College Algebra into a no lecture zone works great thanks to The Array™. While working on computers, kids have access to the room's printer and can reference the screens for examples or instructions or whatever. I can't count the number of times I've caught kids teaching each other using their local screen. It makes my big classes tiny.

Key modification this year:

These screens used to sit on their own tables, limiting my mobility at the back of the room. I found wall mounts that use picture nails (a LOT of picture nails) to do the job of cumbersome bolts, allowing these TVs to hang in place without having to damage studs or do any significant hacking away at the walls. Though it took forever, routing the cables along the wall was the smartest thing I think I've ever done.

I'm not saying go out and build your own array, but finding ways to decentralize your classroom opens you up to all kinds of cool possibilities. Some of which I will elaborate on soon...

Posted
AuthorJonathan Claydon

Last school year was very demanding. Through a lot of circumstances I had a full teaching load plus coaching duties for two very time consuming sports. Prior to that school year, I had already made the decision to step back from my school responsibilities and call the 16-17 my last year coaching. Overall, it has proven to be a very good decision. Not only have I been able to focus on my teaching more, but I have been able to incorporate a lot more time off into my schedule. This is something that gets talked about a lot and remains the most important thing, you need to have set times where you say "no" and do something, anything, else other than work.

To co-op a discussion from the fall, here's my work load #yearinmath measured in Stress Units™

While teaching still remains demanding, mostly because I have to make all kinds of stuff, it has been incredibly more manageable this year. Two nights a week (Wednesday and Friday) I don't do anything school related after hours. Saturdays are generally off limits too. As much as I like to think about school constantly (and I'm usually thinking about school constantly), I don't act on any of it when it's a night off. Secondary to this, I have become comfortable in my ability to do the job. As I told a panel of aspiring student teachers a little while ago, eventually you can plan 3 completely different 90 minutes blocks in about 20 minutes.

Regardless of experience level, make sure you're forcing yourself to take time off. Whatever it is can wait, and whatever it is can be done faster than you think.

Posted
AuthorJonathan Claydon

I have three periods of Calc AB, one of them first thing in the morning. I had the same situation last year and that group was incredibly enthusiastic and ready to start the day. This year...not so much. Lots of tired faces and it takes a while to get them going. Plus, it's the smallest group, only 21 (compared to 25 and 28 in the others).

For the first semester (when it was only 20), they sat like this (the back tables seat 6, the front 7):

Conversations were limited, and performance was poor, scoring about 10% less than the other periods on our first benchmark. In general purpose assessments they were about 5-10% lower than the other classes as well. The table at the top left in particular did a lot of staring their paper, uninterested in finding assistance around them or even asking me.

It was clear things weren't working, so when January rolled around (and we gained 1 kid from another period), I forced them to do this:

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It's a little cramped but it has made a world of difference. The low performing table is gone, and there is far more conversation happening when it's time for an assignment. When it came time for benchmark two, this class is now more in line with the others, off by a rounding error.

They always had access to each other, but forcing the issue and making them be more friendly has had the intended effect. Never be afraid to change it up when it's obvious something isn't working.

Posted
AuthorJonathan Claydon

As we headed into Spring Break, it was time for a project in AB Calc. I have been trying to stress curve relationships throughout the year. Early on we talked generics, how to indicate a minimum, maximum, and point of inflection. Then came integrals, absolute minimums, and absolute maximums. When we moved into position, velocity, and accleration, I hit those contexts again and again. By far, this is the biggest thing AB students need, how are the interactions between a function and its derivatives the same regardless of context.

Particle motion was a huge problem last year. I found myself having to do a lot of reteach prior to the AP exam over concepts I thought we had gotten (change in direction, particle at rest in particular). In combination with my new thoughts on FRQ, I'm trying to review as we go and combine ideas as much as possible.

Student's objective in this project was to create a polynomial, restrict it, and discuss a variety of properties. Specifically, the difference between a change in position and total distance traveled, where and how a particle is said to change direction, and how to determine what's required for a speed increase/decrease.

FULL WRITE UP

The projects were all fairly similar, but here's a particularly outstanding example:

Giving students ownership of the process had really been the best move for my projects. Because there were so many opportunities for variety, students were able to have great conversations about they were or were not meeting the requirements. They also learned a lot of handy Desmos along the way (shading, integrals, absolute value). Project days are my classroom at its very best. The whole room buzzes and I just get to float around assisting as necessary.

Students did outstanding on the free response questions related to this topic. Well, not like everyone got a 9/9 outstanding, but every student got the concepts and had something to contribute when it came to finding the solutions. I'm hoping to roll our review as we go strategy into rate functions next.

Posted
AuthorJonathan Claydon

We spent about six weeks on sequences and series in Calc BC. It was an interesting struggle for me personally as I last touched on the topic in 2003 taking Calculus II. The subject appears really intimidating, just search "convergence flowchart" and see what I mean. In the end, it was not nearly as complicated as I thought it would be and the kids and I got through it without any insane flowcharts.

By far the most interesting topic is Taylor/Maclaurin/power series. It's a fascinating bit of math that causes all sorts of problems, because the tiniest variation wrecks the usability of your series completely.

As we headed into Spring Break, I had the kids do a simple exploration of Maclaurin series in particular. Primarily because the series for sin/cos are so easily modified. Their objective: modify a sin/cos function in whatever way they want, iterate a Maclaurin series to four terms, name the iterations appropriately, and take a peak at their error when x = 1.

 FULL WRITE UP

The coolest part is I don't think the kids (or myself really) was prepared for how wild their constructions could be. A few kids picked functions that worked beautifully at x = 1.

A bigger group got ok results, but had errors over 35%

And then 3 of them picked functions that just went off the rails (the joys of exponents)

450% was hardly the worst. Another was off by 22000+% and the best ever was the student whose error was infinite.

This was a nice, simple project, and I think a nice little eye-opener about how math is sometimes just a shot in the dark.

Posted
AuthorJonathan Claydon

We've finally covered enough material in AB Calculus to start tackling free response situations. In previous years I haven't done enough of this sort of thing and it's cost us. This year with a fewer days (we're probably ~1 week behind still) I wanted to integrate the language component of AP questions earlier and more often. Last year a big issue was the kinds of things kids wanted to review before the exam were concepts I thought we had nailed already. With fewer days, the less I have to reteach in late April, the better.

A few weeks ago students completed a benchmark in Desmos. We had covered enough material to handle curve sketching and data table FRQ situations. In the days since the benchmark, we have turned to position, velocity, acceleration, and general ideas about rate functions. I ended their previous testing system (short skills based items worth a max of 10 pts) and I'm now using FRQ scenarios as assessment piece.

Last Friday I gave them this:

A small hand out with a couple graphs was included. The idea was to get them used to the language and start getting used to how points are weighted in an FRQ. The class day previous to this, students were given arbitrary v(t) functions and we went through a series of 10 questions to cover the x/v/a relationship.

I gave them 40 minutes (though mentioned in reality they'd have about 30) with no notes, but open discussion. I have to say the results were night and day to previous years. I have had TONS of problems with students just leaving stuff blank, or being stumped by the phrasing. Not so this year. While not every student got 9/9 or anything, the conversations they were having were excellent and it was clear that the majority understand the concepts.

The 40 minutes ended and I collected their papers. Come Monday their papers were returned and they were given a colored pencil. Posted were all to see were some observations I made when flipping through the papers. I have made no marks on the papers. These observations served as feedback to the group at large.

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With a colored pencil, they can use this information to update or amend their work. Then we'll look at the scoring rubric for each question. Finally, I will give them a new question (2009 #1) for them to complete. The plan is to turn this into an FRQ Fridays™ kind of thing where we can minimize how much we talk about these in late April. So far I like the process.

Posted
AuthorJonathan Claydon