Another function relationship project, this time from BC Calculus. Prior to the break we were working with functions defined as the integral of another, and how you could figure out absolute minimums and maximums of a given function. There was a lot of interest in this topic from the AP test last year. A number of items designed to show fluency in function relationships didn’t go well for students at large. The issues involved being very particular with your evidence and how you use notation. At TMC 18 while you were playing games at game night, me and Dave Cesa were sitting in the corner talking about this. We know how to party.

Now, we went through these ideas at the very end of the semester, so to simplify things, we made an assumption that our function would have an initial condition of f(0) = 0 and that all accumulations would happen left to right. First thing for the new semester is revisiting that idea and being a little more flexible.

Here were the student instructions:

Students used Desmos to graph their function and compute integrals at any milestone point, local minimums or maximums of F(x). Students calculated the area of their discrete pieces and kept running totals along the x-axis. In tending to precision, they collected their data into a table of x and F(x) and used a series of integrals to show their computations.

While I’ve covered this topic in years past, I have not been as particular as I need to be when it comes to notation. A real struggle in FRQ for my students is showing their knowledge of notation. Often they will cut corners by not including it, or it will be written incorrectly (integrals without dx, for example).

Like their AB counterparts, students didn’t shy away from tricky situations, including points that would register as “fakes” that they could ignore when rectifying their totals. Desmos allowed us to construct polynomials easily and get a feel for using integral notation with named functions. In parallel, when computing integrals with Desmos I will have students replicate the process on their TI-89 to verify they are proficient with both.

AB Calc will repeat the project in the coming weeks and all of this is a good sign for better efforts with precision and argument structure.

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AuthorJonathan Claydon

A huge problem in AB Calc has been an understanding of a function, its first derivative, and second derivative in a lot of contexts. Students have be able to infer the behavior of a function from the graph of f’, a table of f’, or just the equation of f’. Being able to construct all three elements when given one is a path to fluency.

Prior to break, students in AB Calculus had to create a polynomial that represented a first derivative. Based on that graph, they had to identify minimums, maximums, and points of inflection for the original function f. In addition, they highlighted the differences in concavity and when f could be expected to be increasing or decreasing. They translated their findings into tables of f’ and f’’ that validated their findings, gave justifications for their findings, and sketched f based on the derivative they created.

We were working on this fluency in class through a variety of prompts. Sometimes we started with a table, other times a picture, and others the equation. Here students created an equation and built out the whole process on their own. The results are a lot more polished than the version I tried last year.

Here’s the full write up students were given:

Most impressive to me is that students didn’t shy away from tricky to analyze functions. A lot of students created derivatives that would generate “fake” maximums, minimums, or points of inflection (where f’ or f’’ has a value of 0 but doesn’t complete the required sign change) and it can be tricky to do sketches based off that. But every student who attempted one was on the right track with their thinking. A few hit a common curve sketching snag of drawing an original function with the right features, but everything was upside down.

Collectively though, this group is doing a great job with the ideas.

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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

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

I’m working on community building a little more within my classes. Now that things are a little smaller I want to place a greater emphasis on the whole group being involved, rather than 5-6 kids at one table. Last year’s BC Calculus group was my first foray with a sub 20 class in a while, and I structured like it was a big class. Kids sat at three tables, and they stayed more or less confined to those tables. At the end of the year I felt like in a room of 15 people, they should’ve known each other and worked with each other better.

Year two I’m trying to fix that. This year’s group of 14 sits at two tables of 7 normally. I didn’t assign seats, they could just pick wherever. In this setup they’re with long time friends or whatever. But at least once a week I make them mingle.

I randomly assign partners for the day and make them combine the two tables into one big table. Sometimes there are snacks.

One, I want to make sure they’ve had multiple conversations with everyone in the room. As we progress through the year, I want them to seek out any kind of person for assistance, because they’ve worked with everyone in the room. Second, I want them to operate like a unit. Last year the class was all seniors. This year it’s a mix. I want the 11th graders to feel like they belong, and the 12th graders to respect their membership in the class.

There are things to improve upon, but this is a good start. They’re now used to the idea that I will randomly arrange them whenever I see fit. I want to better tailor the assignments for partner day, make them use 1 computer instead of 2, or version the task a bit so everyone isn’t doing the same thing. At the moment I’m making them confirm all work with their partner, and it’s working for the most part.

This isn’t anything special. Search for “visual random grouping” and you’ll find people who have been doing this and doing it better than me for years. I felt like the scale at which I had to operate wasn’t conducive to it. With that no longer a problem, I figure this pilot couldn’t hurt.

It also does more of what I always want in a classroom, students facing other students. My location in the room is irrelevant. When they’re sitting at the dinner table and we need to talk about our findings of the day, I’ll grab a seat with them. It is an incredibly relaxing way to teach. It’s the way my AP Government teacher always started class, everyone in a circle, discussing the current events of the day. I really looked forward to it every day.

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AuthorJonathan Claydon

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:

Screen Shot 2018-09-03 at 11.13.28 PM.png

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

AP scores came out last week. Last year I did some digging around with my numbers and was hopeful. This year, well...

This data combines kids in AB only and the AB subscores from the BC group. They were, by far, the most capable group I've had. They showed a lot of growth during the year. They did some great stuff when I made an explicit point to get better on free response. All my indications told me that they should've done fine. I even had what I thought was an extensive prep process. All told, great group. I am not mad at them (any of you random kids who might read this, I'm not mad at you, for serious kthxbye). And yet CollegeBoard replies "lol, you wish."

I've got nothing here. Apparently something went terribly wrong. When you pull out BC, it's like, kinda better, but not really.

Most likely the sheer scope of the course got to them in the end. The fact that so many at least registered on the AB scale is encouraging. I don't think there's anything wrong with how we invite kids into this course.

But It's Just...

A thousand times yes. It's one indicator. I'm still good at teaching, yadda, yadda, yadda. I get it. I'm more or less just frustrated and some of that you wouldn't necessarily understand if you don't work at my school and understand what happens at all points in our feeder patterns. There are a lot of issues out of my control. There have been some meetings with admin (who are super supportive, btw, no one's mad there either) to discuss what we can work on that's within our control. There are some ideas in the works.

That said, there is not a single thing on these exams that is outside what our students can do. If we're going to put ourselves out there as super proud nerds and everything, it would be reasonable to have some results to back it all up. Accepting that we're going to suck at these exams forever is not going to happen. I would like to quit hiding behind the "it's just one indicator" thank you.

Anyway, teaching is hard. Reading these reports sucked and now I have more work to do this summer than I thought. Blehhhhhhhhhhhhh

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AuthorJonathan Claydon
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