A few nights ago I was working on an assessment and I spat this in my twitter machine:

I do this often as a way of talking to myself while working on things late in the evening. Often it's just to be funny, sometimes it's a little more serious, but I always I figure it's late and not a lot of people are reading. Not so much this time. It happens. There were a few reactions I want to address though, as my use of "points" there was misinterpreted in a couple ways.

Don't use them!

Ideally, yes. I feel you here. Unfortunately I have a gradebook I have to maintain with a minimum number of assignments per grading period, so I've got to do something.

Let the kids decide!

I tried this once before. In 2015 I implemented A/B/Not Yet grading in Calculus. We'd take an assessment, kids would look at the solution, and then rate themselves. Generally, kids were not adept at rating themselves. I had no good system for dealing with students who rated Not Yet, I was too busy with athletics to have any kind of viable after school system. Collecting the ratings was very time consuming and I was poor at communicating how to determine what should be what. A rubric you say? At that point this work saving system has now become more work than another system would be, so no thanks.

It was interesting experiment, but one I chose not to continue. Your experience may be different.

SBG!

I never never never assume someone is familiar with my teaching journey. These responses were expected (and welcome!) and I chose not to reply to them, because it'd be too easy to come across as that guy who is all "well I wrote the book on SBG blah blah blah..." because that's not a good look. But to those who suggested SBG, yes, I love it as a system and it works super great in a lot of contexts. I have used in Algebra 2, Pre-Cal, and College Algebra with great success. If you are interested in my history with the systems, I believe I have tagged the posts appropriately.

What I do these days...

In general, most classes work great for SBG. I have an SBG system in place with College Algebra and the kids like it. It's extremely similar to the system I came up with a long time ago. However, AP Calc has really never been SBG friendly in my opinion. Implementing a built-in retry system is really the problem. And with the speed you have to move with AP Calc, eventually in class assessments just become a burden. Last spring, AB Calc shifted entirely to free response based assessment because that's what we needed to do. It didn't work, but I still liked it and have some thoughts for this year. In general, with Calculus I will break stuff into a topic, assign some general value to the category, and give a handful of questions about that standard. The points vary, the kinds of questions vary, and there is no built-in retry. It's not really SBG. It's also not a test worth 100 points.

Here's the assessment I was working on when I tweeted:

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This particular assignment was for my BC group. The complaint was about how to weight the various sections based on the time it would take to complete them and the complexity. When I grade something like this I take an overall picture of the work. I check for correctness, offer comments, and give students a chance to discuss their work with others. Each one of these sections is an entry in the gradebook. But 1 point ≠ 1 correct problem, I take the whole body of work into account based on any trends in error I may see.

Maybe that clears things up, but maybe it doesn't. Non-traditional graders of the world I'm very with you.

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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!

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AuthorJonathan Claydon

A series of seven posts on major turning points in my teaching career. A study of where I was, where I am, and where I'm headed.

ALL POSTS IN THIS SERIES

Curriculum has fast become my favorite focus in recent years. It started with an observation as I taught Algebra II, that the way I did things was inefficient, uninteresting, and lacked depth. Students were doing surface level material and never getting very far in a particular topic. There was a lack of cohesion in the school year. It felt that we were just studying things at random, hopping around on a whim. I want curriculum that's interesting to teach and that spends 9 months telling a story with identifiable payoffs.

This is different from a discussion of individual lessons. I'm talking about big picture. How do you create the tool that drives the school year?

Where It Was

I followed the textbook, roughly in order. My first year I spent a lot of time relearning the material, just trying to make sure I didn't make any major mistakes the next day. Even then I still screwed up, a natural process when teaching something for the first time. I wasn't worried about what message I was trying to send over 9 months, I was just worried about making in through to the next Friday. As I was transitioning into standards based grading, I noticed that the curriculum just wasn't satisfying. Planning this way is also just terrible. I have sat in on one two many conversations that go "well, we have 2 days for 1.1, and 1 day for 1.2, but we HAVE to finish 1.3 by Tuesday..." Ugh. Just, ugh. What generally happens 9 months later when you plan like this usually includes the phrase "...we just ran out of time." This shows a lack of vision. In August, you should have some idea of where you want to be in May, with a working knowledge of how to get there.

The week to week stuff just stopped cutting it for me.

At the same time, I also realized that students do NOT care what chapter you're in, or what the section number is, or any of that textbook organization type stuff. They are, however, slightly more interested when you talk in topic names.

The use of SBG and topic names had an effect on the way I discussed curriculum. I stopped caring about chapter numbers and section numbers and instead focused on the material itself. It wasn't Chapter 5, Section 2 anymore, it was Motivating the Quadratic Formula. That was an important step I think. When you view a course as a collection of topics, you're more inclined to organize them into something that makes better sense to the student.

Algebra II, a course I will always defend, offers a great case study. Is it necessary to deal with the start up cost of solving equations 5 disparate times through the school year? Could you cover all the mechanics up front?

Where It Is

It was those Algebra II questions that lead me to a very intense project, my Pivot Algebra Two idea from 2013. Rather than think about the course as a list of functions where you work on the same skills weeks apart, what if you focused on the skills and iterated through their applications for various function types?

This skeleton lead me to rewrite the entirety of Algebra II around the major skills I wanted students to develop. I wanted the basic operations we learned in August to help us with more challenging situations in May. Along the way we could take a minute to summarize what we had applied to a subset of functions.

The end result was the most fun I've had teaching. We went really deep into topics that were unthinkable before. We had time for awesome projects. It was a great group of kids. All of it driven by a cohesive narrative. The project paid great dividends down the road.

My initial run at Calculus was challenging. I spent the following summer grinding away at the curriculum, looking for inefficiencies. I was searching for my narrative. After a lot of work I found it: the integral and the derivative need each other, let's explore the many contexts of their relationship. The result was a road map that gives students a basic idea of the course in about 5 weeks.

No saving things until later because they "weren't ready" or something. The more a student knows about a course up front, the more you can with the material later, the more you can communicate the story of Pre-Cal, or Geometry, or whatever.

Where It Is Going

Thinking about the story I want to tell with a course has been a huge breakthrough. Textbooks are of little concern to me. I use them as reference to get an idea of a course's topics and building a model document from there. For example, next year I will be teaching Calculus BC for the first time. I will consult a textbook but the actual rhythm of the course will find its way into some document like the one I use for AB. There's a possibility I'll be doing College Algebra as well, meaning a return to the idea I started in Algebra II. As an instructor, manipulating curriculum in this way has made me incredibly familiar with a course. I could write pages and pages about a logical progression of Pre-Cal from memory. I've been thinking about the interconnection of its topics for years.

Every year I find it easier and easier to fine tune a curriculum. Delete some wasted days here, find a new connection here, and carve out room to go deeper.

One of these days I'll figure out a way to take these personal notes and develop them into real guides for people who want to do the same thing. It's been several years, but that was the concluding step for my Algebra II project.

Takeaway

Many of you are probably in the same position. I know lots of teachers who have little regard for the order preferred by a textbook publisher.

When you sit down this summer to think about your courses, consider the story you want to tell. What connections do you want to establish? How can you spiral back to information as much as possible? How do you want August to influence May?

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AuthorJonathan Claydon

A couple years ago at TMC15 I got a sneak peek at Desmos Activity Builder. At the time it was fairly limited but there was a lot of promise. At the time of its introduction I wasn't sure I could make use of it and that proved true. I didn't have access to enough devices and the iPads I did have were aging quickly and becoming a pain to manage. I attempted one of the first Marbleslide activities in early 2016 and the hardware just croaked.

Fast forward a bit and now I have access to a fleet of Chromebooks. The number of students who can bring a device from home has increased dramatically. iPad hardware in particularly has accelerated so rapidly in recent years that the struggle I saw before is gone.

I experimented with a few use cases this year, just to see what there was to see.

Match Me

Started simple. I took an activity I had done previously where students had various graphs on paper and had to recreate the pictures in Desmos. It looked like this:

Not bad, worked pretty well for a couple years. With Activity Builder I could work through the same idea but get students to add more detail and learn a bit more about the functions of the calculator.

Students could more in the matching realm, in this case finding a sin and cos function that matched the black line. Eventually they could create projects that included center lines, amplitude markers, etc.

Pretty good. Being able to build more complex prompts let more students know more fiddly details about the calculator. I liked that a lot.

Assessment

I used Google Forms quite a bit this year, and realized (well ok, Dan nudged me) that Activity Builder can be used to gather the same kind of information, though one screen at a time. Bonus, it understand math notation natively. I experimented on Calculus and used sketching screens for part of their final exam.

Sketching with the sub-par Chromebook trackpads isn't the best, but that can be remedied with some cheap wired mice. Pretty cool to see 89 sketches on top of each other.

I also used it for a two-fold assessment. Students were given access to a saved calculator with a bunch of regions on it, they had to determine expressions for the area or volume of that region, and then enter it in a separate Activity Builder.

Also cool. I really appreciate the detail that has gone into the teacher dashboard screen. Though there is room for improvement. Examining student responses screen by screen wound up being a little tedious here. Though I'm not sure a spreadsheet generated by a Google Form would've been any more efficient.

Going Further

I really liked what I learned using Activity Builder this year. Though Dylan Kane and others dropped some quality thought bombs on the subject. There can be a lot of silence while students work through these. Instant gratification may not lead to the most genuine student guesses. A subset of students may just hammer away at parameters until it works. I tried to counter the idea by requiring explanations after students had a chance to experiment. I think it helped a bit. Though Dylan's Conics activity is really something. You get no idea what your submission looks like until hitting a button.

There's a lot to think about here. The challenge of drawing a circle around a subset of dots is just brilliant. I need to bring more of this to the way I design activities.

Conclusion

Really excited to see how this evolves over the next year. There has been a lot of effort put into the feature and it's impressive how far it has come since I first saw it. Desmos curated activities are top notch. I think Activity of the Year should go to Jennifer Vadnais and her mini-golf game:

I know the intended audience is a younger crowd, but I had plenty of juniors and seniors cursing this thing. Bravo.

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AuthorJonathan Claydon

A few years ago I hit upon this project for polar equations of conics. Objects orbiting stars can be modeled more or less as polar ellipses, with their host star as the focus point. With a little Wikipedia finagling, you can recreate our solar system.

It's pretty cool, and with new Desmos labeling abilities, it's easier to distinguish what's what. However, the project had a case of the samsies. Every kid or set of kids was making the same thing. There was a lack of creativity. I don't know why it took me so long, but this year I changed it up. I wrote up an explanatory document and let them design a solar system of their own.

We got some very nice and orderly systems, and some whose planets wouldn't survive very long before ramming each other to pieces. I encouraged creativity with the themes.

This one took a little longer than I thought (about 2 hours or so), but I find it important to relax a bit on time requirements if kids are putting in a lot of effort, and that was definitely the case here. Lots of good discussions about how to vary their objects, what various eccentricities would do, and how to manipulate orbits just so.

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AuthorJonathan Claydon

It's a little baffling that this is our sixth adventure out on the sidewalk.

The complete archive:
Sidewalk Chalk Adventures
Return of Sidewalk Chalk
Sidewalk Chalk Three
Sidewalk Chalk, The Fourth One
Sidewalk Chalk Five

Premise

I'm a fan of big, obvious evidence that my classes have something to say. Sidewalk Chalk Day is one of the oldest ways we make that happen.

This year, about 250 students in Pre-Calculus and Calculus took to perimeter of our school and decorated the campus. For Pre-Cal, students were tasked with designing two types of polar equations. They graph the functions and write the equation responsible. Calculus has played along too, but never on the same day. Their task was to generate a slope field, region between curves, or f/f'/f'' family and immortalize it in chalk. Students use Desmos, print out their creation and bring to life like it's elementary school all over again.

Fly High

For the latest installment I brought along some new helpers:

It's always hard to capture the scope of this project as it has grown over the years. We cover about half a mile of sidewalk over the course of 5 hours and consume a small mountain of chalk. These spectacular views brought to you by my robot friends:

It's a great way to spend a spring day. Enjoy this fly-by:

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AuthorJonathan Claydon

Fun milestone a couple weeks ago:

Sounds crazy right? Lots of people would have you believe the future of computers in the classroom is those dystopian pictures of kids in cubicles with headphones on. Turns out you have other choices. So what were we doing during this magical week?

Calculus

We were wrapping up area between curves and various volume expressions. I figured it was a great opportunity to let them play with integral notation on Desmos. Super duper handy for expressions of volume/area with respect to the y-axis. TI-84s can't touch this.

Later in the week I gave them a pre-loaded set of regions with a request for a particular integral expression (area between curve, solid built with square cross-sections, etc) and had them enter their results into an activity builder.

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Student's ability to play with function notation here is awesome. Being able to collect math input through Desmos was a neat experiment.

Pre-Calculus

What haven't we done on computers in Pre-Cal this year?

To start, we finished a study of three-dimensional vectors with an opportunity to make some 3D objects in Tinkercad.

With our study of vectors and polar coordinates, I offered students a glimpse at what spreadsheets can do faster than people. Namely, kick out a bunch of polar points and convert them to x-coordinates and y-coordinates.

Some students needed helpers:

Then, super quickly, and with several "ooooooooohh"s, we dropped the x/y table into Desmos for an initial look at the graphs of polar equations.

We refined the ideas present here through the use of an exploration activity made with Activity Builder.

Conclusion

In each situation the computers came out because they can perform the task better and faster than pencil and paper. In the case of Pre-Cal, they extended math to a place they didn't know it could exist before through spreadsheet formulas. Calculus got a chance to speed through volume because of the ability to see a region, talk through the logic of how to define an integral around it, and then see that integral computed in the same window. That took us a million button presses and a stack of copies previously, with no option for functions defined in terms of y.

A+++ would teach again.

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AuthorJonathan Claydon

As a general rule, I don't like to let activities linger in a particular state. There are always subtle tweaks to make or the tough decision to cut something loose and try again. The advent of Chromebooks in my classroom has opened up a lot of opportunities we just didn't have before. Realized recently when it was time to talk about three dimensional vectors.

Historically, I turn the room into a giant coordinate system, we discuss how to orient ourselves in 3D space and I've had them build stuff out of straws. It was interesting at the time, though time consuming. Due to the nature of building things out of straws and electrical tape, the creations didn't last very long either.

Enter Tinkercad. I'd like to thank Autodesk for a monumental shift in the way they approach access to their software. Fifteen years ago, AutoCAD or the various things a college student might need were prohibitively expensive. It all came down to knowing the right person in the dorm who had license keys of a shifty nature. Since about 2010, they've done a total about face, making tons of great stuff available for free or cheap for students, teachers, or anyone who isn't a corporation. Tinkercad is a browser-based 3D modeling app that lets you mock up whatever you like. There are pre-built simple solids, and you can browse a whole library of community built objects and drop them into your design.

Rather than mess around with a bunch of straws and tape, I set my Pre-Cal kids loose on Tinkercad and gave them about 50 minutes to make something. The site's export options are intended for 3D printers, so we stuck with screenshots from a few different angles. We got some really awesome stuff.

Chromebook trackpads aren't the greatest, so this was a tad challenging for some. I do have touchscreen models, and kids with those were very happy with themselves. Wired mice can be had rather cheap in bulk, so that might be an idea for next year.

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AuthorJonathan Claydon

Here's a little reverse jinx I pulled:

I say reverse jinx because as Februarys go, this one wasn't too bad. Normally I have an increased work load from being soccer season, the push towards AP Exams, and it's all compounded by abysmal weather. With an absolute lack of winter around here, the work load felt a little more tolerable.

While it appears I've been pretty quiet, I have a lot going on:

Math Department

In October Varsity Math made a splash by taking over a blank wall in the hallway. It now features current class photos and our Hall of Fame. With a load of painting supplies on hand, I turned my attention to an old feature of our math department office.

It dates from the 80s, maybe? I took my painters and we're turning it into a unit circle:

A slow process (we have about 1 hour/week), but soon to be completed.

Summer Camp

It's almost time to start thinking about things for Varsity Math Summer Camp. Despite the random nature of the topics (from my point of view), all the kids consistently said they enjoyed themselves. One of them even made use of some things we talked about to work on a physics lab this year. I've done the initial advertisements to Pre-Cal students, and several are already convinced this is the thing for them. For $20 it's a pretty good deal.

Focus is the goal. Fewer topics and more time. I have a better idea of what you can accomplish in a 2.5 hour session now.

Pre-Calculus

A big change is I have access to Chromebooks this year, making spreadsheets a more realistic tool at my disposal. I learned in Summer Camp that there's a real desire by kids to wrap their heads around spreadsheets. Most having no idea of all the math stuff you can do with them. Both Vectors and Polar Coordinates can make use of these. Recently we've walked through combining vector components and finding magnitudes and directions. A spreadsheet can do this quite nicely:

We jump over to Desmos, and with the help of super slick things like auto-connecting points and labels, we can quickly render our interaction:

It's almost time for the 6th Sidewalk Chalk Day. There's a possibility Calculus will get involved, allowing us to cover a truly massive amount of sidewalk.

Sidewalk Chalk means it's time for polar coordinates, a unit I started teaching after Vectors because so many of the concepts and math are identical. Traditionally I start polar coordinates with some hand calculation and plotting of points:

But computers are so much better at these things. Can we teach a computer to calculate polar coordinates? Let's used what we learned from vectors to speed up the process:

And thanks to the super bananas awesome data table pasting, we can get something far more sophisticated than our markers could accomplish:

Very excited to see how this goes.

Calculus

Ugh, I don't know where to start here. It's time to register for the AP Exam, and as part of my five year plan, I said I wanted 80% (54 students) to register for the exam and have 30% pass. Well, determined to avoid the absolute fake out that happened to me last year (I had data to suggest that many many students would do well, it was wrong), I have tweaked the process. And well, ugh. But at the same time there are positives.

First, I learned I have a solid 20 kids who don't seem to have learned anything. It's late February. How did this happen? How much of that is my responsibility? At the same time, I have 25 who seemed to have learned everything. I'll spare you the details of my benchmarking calculations (the older a benchmark, the less it's weighted in a student's rating), but the data identified 8 highly proficient students last year. Using more difficult assignments, that same method has identified 16 individuals this year, with a higher average than the previous 8.

While there was a lot wrong about my methods last year, those top 8 all registered a 2+ on the exam. To have doubled that group is a positive.

The real disheartening thing is at the other end of the spectrum. 16 nailed it, another 9 did alright, and the remaining 55 are just wandering in the wilderness.

I have to make some hard decisions about what happens next and what is best for each student. My first year of teaching Calculus taught me that allowing kids to leave knowing nothing is a disservice. But slowing everyone down is a similar disservice.

Calculus BC

We identified 16 individuals willing to start the first full Calculus BC course in the history of my school. They're excited. I'm excited.

Classroom

Lastly, in 3-4 weeks I'm getting all new furniture. It will be a bit more flexible than what I have now but will still let me establishing the grouping methods I have come to like. More on that when it arrives.

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AuthorJonathan Claydon

Another updated lesson from yesteryear enriched by Desmos and years of me tinkering around figuring out what I'm doing. This time, polynomials.

When I taught Algebra II, we spent some time with polynomials: identifying real roots and coming up with possible expressions. It was pretty scripted. I came up with the pictures, distributed a set, had the students pick a couple, copy a paragraph and fill in the blanks for their picture. It looked like this (circa 2012):

Being smarter about this sort of thing now, I removed the script and put the design requirements in the hands of the students. Using Desmos, they were to create a series of 4 polynomials (3rd+ degree with at least one 5th degree) and tell me about them. Two went in their notebook and two went on the wall.

Great products and great discussion along the way. Students also got to play around with scale factors, the secret sauce for making any polynomial graph remotely useful. Prior to this activity we talked about hand sketching the graphs and tried a few. Later on the relevant assessment, I reversed the idea to see if they could work backwards:

Questions like this teach me to trust the students more. Previously I'd fret about asking this sort of thing without explicitly talking about it (the point of the old version of my activity). But if you really want an assessment to do its job, you see what a student can do on the fly with the introductory pieces you have provided.

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AuthorJonathan Claydon
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