Recently, I discussed attempts to stop explaining so much. Experiments with assessments are showing success. Skills are great, but I want to see who is thinking. Part of the reason I don't mind letting students (even PreAP, the horror!) use their notebooks on assessments is that a well done thinking question can't be anticipated. Through the course of working with the skills, the student either developed a strong working idea of the concept and therefore can answer just about anything, or they didn't.

Last week, variations of this question were posed:

Sure, you can slap that into y= and see. But it'll be obvious that's what you did when your explanation comes up short. Also, I never approached rationals that way. How do you interpret that ratio? What secrets will it give away?

There were three versions of this question. In two of them, the ratio straight up didn't work. The asymptote or discontinuity would show in the wrong place. This one though, will produce asymptotes at x = -7 and x = 3 as shown. 90% of students agreed and said, yes, this could be the correct graph because they factored the given ratio and saw the right looking numbers. Full credit was awarded. A small handful though...

While not explicitly being tested, we discussed a few days earlier how you get a general idea of a ratio's behavior by doing some testing on either side of the asymptote. Choose a test point on either side, and see if it would produce a positive or negative number. While not super accurate, it'll get you close.

It blew me away that these students saw something generally correct but still weren't satisfied. The behavior on either side needed to match too. While the top picture was inaccurate (-6 placed on the wrong side of -7), the student was suspicious, and I love it. The kid in the bottom picture nailed it exactly. I agree with the behavior for THIS asymptote, but hey man, this is the right number and all, but it's still no good. Disagree.

I mean, just wow.

Here's the bigger picture idea. It didn't occur to me to try this line of thinking with rationals until my 5th year teaching Pre-Cal (6th year overall, 4th year of listening to smart people on Twitter). When you say you want kids to really know the material, are you sure you've got a complete handle on it yourself?

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AuthorJonathan Claydon

Standards based grading has been a huge success for me. It's helping my students have conversations about success and becoming determined about showing improvement. Most of them have stopped talking about That Number. The number that I dislike. The one all the top kids in your class obsess over. 100. Ideally, it should all be about growth and feedback, but tradition dictates we give kids a number. They live and die by that number. See also: college, where sub-par teaching methods and the pursuit of 4.0 poisons just about everything.

AP culture feeds the beast. That GPA bonus is enticing. Surely extra credit must be offered. I NEED to have an A. If you can just give me two points mister, it's just two points!

I dislike it intensely. It takes a fantastic opportunity, exposure to high level curriculum, and turns it into a war for 100. How do you teach a group of Calculus kids, kids who have fought for 100 forever at this point, to forget about the stupid number and focus on learning for the sake of learning?

A few years ago I helped adapt my typical SBG approach to Calculus. Subdivide the class into topics, test each topic twice, honor the best score, and offer full replacement after school. Throw a six weeks exam in there to keep a little accountability. It didn't work. Sure, kids demonstrated improvement. But many blew off the initial attempt. TONS of them showed up after school to try and replace everything. All in pursuit of monkeys, er, 100.

This year? Reset button.

The Strategy

The first failure is trying to take Calculus and force it into nice little topics. After a two year experiment, it was pretty clear Calculus doesn't like being treated that way. It's a Big Ideas kind of class. It's primarily conceptual. Everything matters. You can't forget about September.

The second failure is removing too much accountability from the assessments. That after school thing was a crutch. Special rules had to be written.

The third failure was a disconnect between the assessment material and the design of the AP Exam. AP scores were terrible. Blowing off assessments probably didn't help. Seeing the format of the exam too late probably didn't help either.

What do we do?

Giant tests out of 100 are out of the question. Assessment needs to remain short and frequent. Reassessment should be available, but limited. Concepts matter. More accountability needs to exist. I should have the freedom to put anything and everything on them.

The Method

After a lot of discussion with a fellow Calculus teacher who wanted to do the same thing, we arrive at our current method. These are given once a week-ish.

It's a hybrid sort of thing. Each section is evaluated in a big picture sort of way. No nitpicking -1 or -2 junk, just feedback and an overall evaluation. The front page is skills. The top is for the deep catalog stuff. The bottom is fresh. The back is all conceptual. Written answers, thought questions, justifying answers and all that. Each is a separate entry in the gradebook. Kids still go right to the number, but it's not That Number. Only skills are eligible for a retry. Concepts are important. Six weeks exams are now mini-AP tests.

The Hook

Students should keep track. Students should get a pat on the back for success. There should be a goal unrelated to 100.

Since each test is a unique set of semi-permanent scores, the tracking is a little different. Each test is recorded on a line with the results from the three categories. If the total is 16+, you get a shiny, Calculus-only silver star.

It's the dumbest thing in the world. But an 18 year old will commit crimes for a sticker. And now that's how they talk. Did I get a star? I starred this one! I'm all ABOUT those stars! NOOOOOO 15, I HAD IT!!!!!!

Take a step back for a second. There are 20 points total. A 16 represents 80% (a comfortable 5 on the AP test, the reason it was chosen). A 15 is 75%. Think about that kid with the 15. What happens if they got a paper back with a 75? I probably can't do this. This is as hard as everyone said. Now I'm behind in math too.

When would a student with a 75 be ENCOURAGED by that result? Be willing to stop by after school to make sure that darn it they get a sticker next time? In the fight for 100, 75 is pretty unacceptable. And yet, my students have no problem with it, because the dreaded 100 has been abstracted away.

That's all I want. If you are motivated by getting As, ok. But I want you in tutorials because you're eager to learn, not because you're going to beg for some extra credit. And in class where the real evaluation isn't until May? Your number out of 100 is the least important number there is.

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AuthorJonathan Claydon
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Open ended activities have never been my strong suit. In the past it devolves into answering a million questions and a clear sense that the students aren't doing the heavy lifting. Many things blew up in my face in the my first couple years because of this. I moved to pseudo open-ended tasks. Students weren't following a recipe of steps, but they weren't wholly responsible for creating the work at hand.

With a slightly different population, it turns out my students can be easily bored with that approach. Very slowly, I've forced myself to get them to invest more effort into their work, and I have stopped explaining every little scenario.

A couple recent examples.

Composite Functions

The last little while we were discussing adding, subtracting, and multiplying functions together. Previously, I treated it as a wholly mechanical lesson. I wanted more context this year. We discussed perimeter, area, and volume as parallels to the various operations we study. On the relevant assessment, I defined length(x) and width(x) and asked for this:

We never did these as class work, but we had discussed their relevance to composite functions. Reading the answers to this gave me a really good idea of who gets function combination on a conceptual level. I've noticed that my little group of nerds is really good at rote mechanical work, making differentiators like this important. The mind blowing part is that no one gave me that "what do I do?" look that I've seen before. Win.

Rational Functions

Division of functions is an excellent segue to rationals. I had a progression last year that was ok, but clunky. After various discussions about holes, asymptotes, and how to identify each, I gave them a closing task. Design some. If they really get how an asymptote or hole is created, they can design a ratio that meets whatever specs I give.

Kids had questions, and tons of other kids were eager to share the answers. I didn't do much other than stick an errant iPad back on the wifi so they could print. Watching it work made it very apparent how and when direct instruction fails. For many students it was obvious that they were absorbing all of this passively and nothing "clicked" until it was time to make something. Assuming direct instruction is working for your students is foolish.

Conclusion

There is still work to do. But I'm so happy to discover this early, so many other things I have planned can become that much more interesting as a result. I always tell people to challenge their students, that you'll be surprised what they can do, and something forget to do it myself. It's paying dividends in Calculus as well, where you really don't have the time to explain every minor scenario.

Neither of these activities involve "real world" problems, which may bother you. But reality is in the eye of the student. If a student can create a host of things that validate their understanding of how an asymptote is created and defend their method, is that not real? Has that math become something more than words in a textbook?

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AuthorJonathan Claydon

I think a lot about room design. Nowadays, everyone in my room sits in a 6-kid pod like this:

The groups at the front of the room are a tad different, but you get the idea. This design facilitates what I want most, kids talking. To optimize it, I need to have a handle on who sits with who. Developing group chemistry is half of what makes the table system successful. In previous years, the kids are in a random arrangement for six weeks. I observe who works well with who, who seems to be displaced from their friends, and adjust it. At the semester, I give them options: choose a person or choose a table. I take their suggestions and make a third chart that lasts the rest of the year. This also gives them an anonymous way of communicating big time conflicts, if they exist.

In Calculus, I want these students exposed to more of their class. When I took Calculus, there were 12 of us. I sat in the same seat the entire year. Yes, I knew the names of everyone. But it would be tough to tell you when I got to interact with them. My groups are in a similar situation. The overwhelming majority have had classes together for years and years. They all know who is who. But could they learn more than that?

Enter moving buddies. You, dear Calculus student, get to choose a permanent partner. They will travel with you on your adventures through the room, meeting and working with exciting new people. I had them write down partner choices on an index card, a few had no preference, and I had to settle a couple disputes.

Each rotation lasts two weeks. This will continue for the rest of the semester. We'll do it again next semester, but some fiddling may be required if someone drops. Without getting too crazy with combinations, I did a bunch of random sorts and then made some edits until every pair of buddies visited at least 3 unique locations. I tried to make sure a set of 4 didn't travel together either. If you track a kids' movement, they'll sit with over half the class in the cycle.

I expect any payoff will take a long time to develop. But it beats sitting still.

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AuthorJonathan Claydon

One of the most frequent questions I get in person or via e-mail is how I structure my class. There's a lurking frustration about the way people present material, Pre-Cal in particular. There seems to be a connection between higher level material and the need to explain explain explain. I find it to be the least successful method for conveying information, or at least a LOT of information. In talking with my students while doing classwork, it's pretty evident that they are highly likely to tune out during lengthy explanations. Students I've tutored share the same experience. Math is 50 minutes of note taking and that's that.

I need more from my students. I experienced math classes like that. Hundreds of hours of math classes like that. I was in cruise control and never figured much out until I got a chance to talk to someone during "homework time" in high school or meeting with my study groups in college.

This is my goal:

As frequently as possible, I want YOU doing something. I get bored talking. You get bored with me talking. I have a new concept for you, here's how it's going to go:

Day 1

  • introductory question or scenario
  • questions about prior knowledge
  • how can we extend your prior knowledge?
  • new skill
  • example
  • randomly assembled boardwork
  • 10 minutes of work

Day 2

  • review prior introduction
  • offer any new insights (special cases, errors to watch for)
  • pass out classwork
  • 20-25 minutes of work
  • questions

That's about it. I was introducing function compositions the other day. I demonstrated some algebra. We rehashed expanding binomials, and they completed 3 exercises on their own. The following class day, I did some brief introductions and gave them an assignment. 25 minutes later we went over questions. On a block day with Calculus the other week, they had been prepped such that I gave them a variety of assignments and their job was to work for 80-90 minutes. I have many extension activities, but this is how I handle the nitty gritty skills.

Why is this so important? Kids will ask the questions that would stop them in their tracks at home. They'll turn to a neighbor. They'll confirm something with me. They'll offer a theory about something. They'll wonder if they're working the calculator correctly. When finished they'll find a neighbor to check with. They'll fight about stuff (I LOVE when they fight about stuff). Most importantly, I get to interact with them. Sidetrack them with a story, point out something weird, verify an assumption. I can have dozens of small moments, even if it's just to ask "are we ok?"

If I stand at the front all the time, I'll never know them.

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AuthorJonathan Claydon

My first Calculus arts and crafts! Students I had for academic Pre Cal were worried that all the fun stuff was behind them. We can still play with markers in an AP class, don't worry.

Simple idea: draw a function that is continuous over a set of intervals

Prior to this we had been discussing overall vs. local continuity, how to identify intervals of continuity, and how to formally define breaks in continuity. In this exercise, they would work in reverse. I would provide some intervals, they draw something that matches the description.

It didn't take them long, and was one of many tasks they had for the day. But they helped each other immensely with some conceptual problems.

Upon completing their picture, they had to identify an obvious discontinuity and use a three step argument to prove why it was a problem.

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AuthorJonathan Claydon

Calculus students are bad at algebra. It's not totally their fault. Algebra is hard. Algebra requires an immense out of experience to perfect. Something you may forget if you're 15 years into teaching and throw your hands up as the children yet again disappoint you.

In an effort to constantly remind my stewards that all the old stuff matters, I wanted a way to jog their memory on the regular. Thus, Throwback Thursday was born.

Due to the nature of our bell schedule, I see both sections for a long time on Thursdays. At some point during that class period, they have to complete an assignment on vintage material.

I'm not sure how effective it is yet. But, their unit circle trig is getting better. I have some other plans for this in the future. Including things like create a unit circle from scratch, calculator showdown, and equations of lines, to name a few.

As I keep saying, confidence is my enemy. I'm hoping things like Throwback Thursday will help them know some things in their sleep.

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AuthorJonathan Claydon

Piecewise Functions are a fun twist at the beginning of Pre-Cal. Graphing them however, has always been a bit annoying. The first year I taught Pre-Cal this topic was a disaster. Slowly I found a better way to do it. Start with evaluation. Have students evaluate a function over a wide range of points. Give them a grid and have them plot the results.

Some errors will surface. Overlapping pieces, boundaries drawn in the wrong location. This is natural. But, it gets better.

As a culminating move, I teach them how to graph with restrictions in Desmos. This year, I tried something new. You, dear child, are going to DESIGN a piecewise function. Normally, I suck at open ended stuff.  But, my few weeks of bizzaro world gave me the impression that my new crop of students would be up to the challenge.

Requirements:

Design a function that at a minimum

  • has 3 pieces
  • has 3 different types of behavior: increasing, decreasing, constant, or undefined
  • has 2 different types of functions: linear, constant, radical, quadratic

Present a large version of your function, the equation that generated it, and a description of the intervals that exhibit different behavior.

I set up the project with some practice on describing intervals and how to manipulate various function types (a review of transformation rules), and then set them loose.

Most went with something pretty standard, but I did get a few clever applications. Students primarily built their functions on an iPad and then made a copy. However, since I don't have a enough iPads to go around, they had the option of using their phones if they wanted. At least 3 or 4 kids in each class successful pecked out a function on the web or app version of Desmos. A few shared an iPad with a partner.

Super happy with the result. I expect this open ended thing will continue.

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AuthorJonathan Claydon

Calculus is my giant scary project this year.

For once, I'm doing something super traditional. Relying on published resources. I am not the biggest fan of textbooks, our standard issue Pre-Cal book in particular. For years and years everything I produce has been independently created. Kind of necessary if you do crazy stuff like make your own Algebra II curriculum. I made LOTS of things.

Given the formulaic nature of the AP Exam, wandering off in my own direction may not be advisable. Thankfully, I only had to buy one of the books you see here. The AP workshop I attended over the summer was free book after free book, plus the one adopted by my school.

Breaking the traditional lecture format of AP classes, that's the real goal here. These books? I can work with these books.

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AuthorJonathan Claydon

My major challenge with calculus is confidence. For whatever reason, the AP Exam causes our students to freak out. I have a number of ideas to try and fight this that will trickle out here as the year progresses. One of them is to teach them how to use resources. And second, provide them with resources that are going to mirror the instruction they're given in class. Meaning, sending them to instructional videos by some random guy on YouTube is not involved here.

To start, I want to work any homework assignments I give them. I got a special notebook for it and everything. Then I thought, well, these should be available to them. But borrowing the notebook only helps one kid at a time. Five minutes of fiddling with Dropbox later and I created my Calculus share folder:

I post the solutions around the time I give the assignment. We have several conversations in class about the right and wrong way to use solutions. Homework is such a tiny part of their grade, the lack of understanding that will come out on tests isn't worth it. Most understand that attempting it themselves is the appropriate thing to do. In fact, one kid found an error in MY work, which was cool.

That last one might scare you. Hold on, that's the TEST ANSWER KEY? Are you crazy? No. The AP Exam is the final performance. I gain nothing by giving them assessments that they can't learn from. ALL their skills have to be sharp in 8 months. If I can't get them into an analysis and growth mindset now, they'll never convince themselves they can study for something like an AP Exam next semester. Their tests are short and frequent. Material on them overlaps. Once something appears on a test, it can appear again WHENEVER. I'm asking you limit questions in March, kids. Making old assessment part of their preparation routine is something I see that can fight the confidence problem. I might have messed up, but darn it I'll be ready the next time.

As you read this they'll be taking Test 2. I'm cautiously optimistic that some of them see a value in this opportunity.

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AuthorJonathan Claydon
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