I spent this week with a group of about 60 teachers at a California Math Science Partnership grant in Kings County. This is our third summer together. In the past I’ve always shared resources through Dropbox and/or Bitly. This year I’ve decided to share links, handouts, and a bit of commentary with a blog post instead.

On to the resources!

# Slides, Slides, and More Slides!

The slides for the entire week are here.

# Problem Solving

We started each day with 30 minutes of “problem solving” (really just my excuse to share some fun things I’ve discovered or created over the past few months).

### Monday: Visual Patterns

Created by @fawnpnguyen

My landscape version of the student handout is here.
My old (two-column, portrait) version of the handout is here.
Here are the slides I use to introduce Visual Patterns to my students (over the course of multiple days) in PDF, Keynote, and PowerPoint.

### Tuesday: The Running Game

This is a work in progress, but I’m happy with how things are moving along. I’ll probably write a blog post in the next few weeks describing the project. At that point I’ll add a page to the blog with a catalog of all the challenges.

For now you can find the first two challenges here. Look around in the images folder for the solutions.

### Wednesday: Estimation 180

Scroll down to find the handout. The latest version (including the space for reasoning) should be on Estimation180.com soon. If it’s not, you can get it here.

Update: The latest version of the handout (including space for reasoning) is posted here on Estimation180.com. A post by Andrew Stadel describing how to use the handout is here.

### Thursday: Numblurs

Back in May Niko Rowinsky tweaked this game to create an excellent, logic-rich challenge for students. I love it. My students love it, too. How to play is in the full slide deck (and here for those who don’t like hunting for needles in slidestacks).

### Friday: Daily Desmos

Created by @dandersod, @j_lanier, and @mjfenton

If you enjoy solving the challenges, consider submitting your own. Details on how to contribute are here. In most cases, creating your own challenge is easier than solving someone else’s!

We spent the mornings looking at various tasks from mathpractices.edc.org and discussing how they aligned to the CCSSM Standards for Mathematical Practice.

Here are the goods (hopefully with appropriate credit given where due):

# Planning

Teachers had 90 minutes each day after lunch to design units and lessons. I wanted to share some awesome ideas I’ve picked up from the #MTBoS recently, so Monday and Tuesday I gave brief presentations to kick off the planning time.

On Monday I nearly ran out of breath trying to share all of the awesomeness contained in Fawn Nguyen’s blog posts on Deconstructing a Lesson Activity (Part 1 and Part 2). If you haven’t read the full posts… Go. Read. Unless you just don’t care. (In which case, shame on you!)

On Tuesday we looked at Dan Meyer’s Makeover Monday series. Too much awesome to describe. I will say that several of the teachers have really taken the makeover model and run with it. Fun to watch!

# Five Practices

I gave a brief presentation each afternoon on the key ideas from Five Practices for Orchestrating Productive Mathematical Discussions. My talking points and the discussion questions are in the full slide deck.

For those who didn’t win one of the free copies of the book, I highly recommend you pick it up to add some meat to our daily discussions. Drop a line in the comments if you try these ideas out in your classroom. I’d love to hear how things are going.

# Assessment

I’ll just drop some links here and hold off on the commentary.

SBAC Pilot Test (on the Smarter Balanced website)

# Blogs

If you don’t read math blogs, you should. If you don’t know where to start, here are a few ideas.

### If you teach any kind of math…

This is just the tip of the math blogging iceberg, but it’s a great place to start. Enjoy!

## Flubmaster

Have you ever seen someone take a potentially excellent mathematical task and destroy it by flubbing the presentation? Have you ever done that yourself? I’m 2 for 2 so far (with a heavy emphasis on the second offense), so it’s with some excitement and a little bit of nervousness that I share my first Three Act task.

## Running with Scissors Smartphones

About two years ago I began running with a smartphone to track my distance, pace, etc.. Initially, this on-the-run-phone-death-grip was a result of the fact that I was too lazy (cheap?) to purchase an armband case. However, after a while I found I liked running with my phone in hand. Several months ago I looked down and thought, “Hey, I could take screenshots while I run and…”

The result of that brainstorm, and much marinating and tinkering afterwards, is this, my first real attempt at a Three Act task.

## Request for Critique

I’m fairly certain there is an interesting task contained within the screenshots I’ve grabbed, but (as hinted at in the introduction) I’m afraid I may have bungled it away.

First and foremost, I’d love to receive your general feedback. What works, what doesn’t, what could be improved? Is there an interesting task buried in there, and have I done it any justice?

I also have a few specific questions in mind. If you’re interesting in reading and/or responding to those, head over here. I expect I’ll want/need feedback on most (all?) of my Three Act tasks, so I threw something together to keep a running tally of my Three Act uncertainties, should anyone be inclined to weigh in on specifics.

I know it’ll require a bit of browser-tab-juggling,  but please leave any feedback in the comments below, or hit me up directly on Twitter (@mjfenton).

Thanks in advance for sharing your thoughts. I look forward to getting better at this with your help!

## Which Run? (a.k.a. Now I’m Just Rambling)

I’ve captured screenshots of seven or eight runs over the past few months. Depending on the run, I’ve taken screenshots at every 1/2, 1/3, 1/4, or 1/5 of the total distance (or sometimes every 0.25, 0.5, or 1 mi), plus the “countdown” at the end (every 0.01 mi for the last 0.13 mi of the run).

With the various total distances and screenshot “splits” I’m considering creating a series of problems of varying difficulty, all of which require students to think proportionally, interpolate, extrapolate, and explain their reasoning. I think a series of these problems might exist best as simple stills of three screenshots, maybe like this:

Sequels would include, “When was Mr. Fenton at the 1 mile mark? How far after 23 minutes? 37 minutes?” And so on.

## Getting On With It

Okay, ramble over. Time to hit “Publish” and see what the world thinks of what I have created, not what I might create.

## Postscript

This afternoon was my first experience adding more than a single image to Dan Meyer’s 101qs.com. It really is a Three Act task paradise. Thanks, Dan (and everyone else who contributed to the site’s quality by using it and asking for Dan to make it better).

UPDATE: Okay, so my warning about messing up the presentation was apparently quite warranted. I never bothered to check if the distance meter in the middle of the screenshots was accurate. Thanks for nothing, Nike… it’s not even close. That essentially kills a major strategy I intended students to use in solving the problem.

There were a few suggestions on Twitter for how to use this not-to-scale-ness as part of the lesson, one of which seems particularly worth exploring.

For now, my solution was to re-do the Three Act task to offer students enough information to find the solution along another path.

The results are Nike Running 2A (given distance, find time) and Nike Running 2B (given time, find distance). Again, I covet your feedback.

## Woohoo!

After an inexcusably long delay, the Better Assessments blog is now up and running. Head on over to have a look at Stephanie Reilly’s Algebra 2 quiz on exponents and adding polynomials.

Add your voice to the conversation by asking questions and providing feedback in the comments, and consider submitting your own assessments while you’re at it! Details (including alternative ways to play) are over at the blog.

## Pathways Through the Common Core

I recently joined a conversation on Twitter about pathways through the Common Core State Standards and potentially-shifting opportunities for advanced students. It seems I’m not alone in wondering how a transition to the CCSSM will play out in our actual classrooms and departments.

I teach in a very small math department (two members for the entire 7-12 program), so I am particularly curious to know how debates are unfolding and plans are taking shape in other school districts (like yours!).

For districts both large and small, I imagine it would be helpful to know the questions others are grappling with, as well as the solutions they’re proposing to the many challenges that will arise as we make this transition. If you’re interested in adding your voice to the conversation, drop a line in the comments describing as many of the following as you please:

2. Timeline (and other relevant details) for your transition
3. A link to a course sequence/pathway (if you have one), or a list of the options students have at each grade level
4. Plans for acceleration (i.e., what to do with/for your students who want/need/deserve to be challenged)
5. Plans for remediation (i.e., what to do with/for your students who struggle to the point of failure in one or more classes)
6. Concerns and challenges
7. Other random insights
8. Lingering questions
9. Whatever else comes to mind

Since I’m not interested in highlighting our approach as anything worthy of emulation, I’ll share my school’s plans, questions, and so forth, in the comments.

Thanks in advance to all who chime in!

## A Game!

Finals week often has awkward down time for students. With that in mind, I made a game for my middle school students to play next week.

I’d love some feedback on the directions, the scoring system, and the game itself.

And if you play with your students, let me know how it goes!

## A Few Words About Points

The decision to award more points for even values was arbitrary (I could just as easily have chosen odds) but also intentional (I want to motivate students to observe patterns/behavior and use their observations to target certain values or types of numbers).

More points for higher numbers was un-arbitrarily intentional (I want to motivate students to tinker with larger numbers).

I want students to hunt in a clearly defined, finite space, hence the 1-100 boundaries. Good idea? Bad idea? I’m not sure yet. I’ll let you know how things go next week.

## CCSSM: Approaches to Remediation

Howdy, Internet. Thursday I asked this:

I immediately received a link to a helpful article from @reimerpaul. Shortly after that @wahedahbug and a few others expressed interest in having a larger conversation about how best to remediate for students who aren’t really ready to move on to the next course in the CCSSM sequence.

(For the record, at my school we’re going integrated in high school, but I think best practices/wise policies regarding remediation can easily apply in either pathway.)

I really want to know three things from as many people as are willing to share:

1. What are your school’s current policies and practices regarding remediation?
2. What if any changes will your school make in your transition to CCSSM?
3. Ignoring school culture, resistance to change, limited time/energy/resources, and other annoying realities, what would your ideal approach to remediation look like?

## Better Assessments: It’s Time to Begin

For background, go here and here.

Once you’re ready to play, do this:

1. Create an assessment you don’t hate (or select one you’ve already created).
2. Save/upload the assessment to Dropbox, Google Drive, Scribd, or some other tool where you can share a link to the file.
4. Complete this form.

I’m a little conflicted about whether there should be a deadline for submissions, so I’ll let you self impose one if you find that helpful. The idea is to submit something relatively soon so we can move on to the next steps (an ongoing discussion centered around the submitted assessments).

More details about the structure and expectations for the discussion will come soon. For now, get those assessments ready!

P.S. Details regarding the other major component of this project—gathering assessment-related resources and posts here—will also come soon.

P.P.S. If you have suggestions for how to improve the form (#4 above) or the submission process, drop a line in the comments or send a note to @mjfenton.

## Homework Crisis

If my WordPress stats reveal anything, it’s that the blogosphere likes me best when I’m in reflecto-panic-crisis mode. Well, thanks to a post from John Scammell over at Zero-Knowledge Proofs, I’m back at it. The subject this time: Homework.

I’ve spent a lot of this school year weighing the usefulness (or lack thereof) of my various classroom practices. I lecture too much, my students spend far more time doing more “exercises” than “problems,” my assessments need some serious work… The list goes on. In recent months I’ve thought occasionally about the effectiveness of my homework policy. John’s post has me thinking about it again, and this time I don’t believe I’ll be able to rest until I’ve sorted out what my approach should look like and how I’ll get there.

## In the Past…

In my first year of teaching, my homework sins were many. (1) I tried to grade it all myself. For about two weeks, anyway. Then I tried to grade 5 problems per assignment, all myself. Still terrible, and now for twice as many reasons. Eventually I “outsourced” homework grading to the students themselves and a TA. Better in some ways, worse in others, but still broken, because… (2) I would allow the start-of-class homework discussion to last 15 to 20 minutes (out of a 45 to 50 minute period). Horrible. Shamefully horrible. There was then never enough time to address new material, which meant I could expect widespread struggles on the next homework assignment, which meant another long homework discussion to start class the following day, and on and on. For me, it was wash, rinse, repeat at least 150 times per semester. (3) “Preparing students for the homework” became the driving force of all my instruction. Which was weird, because I was the one who selected the homework, but then once I did that I felt like I was no longer in control. The assigned homework was in control. When we weren’t prepared for the sometimes well-selected, sometimes poorly-selected assignment, I felt like the day was lost. I would occasionally send kids home to suffer through the assignment anyway (lots of frustration and tension and guilt mixed in with that approach). Other times I would postpone the assignment (which was always received with loud applause from students) and consider myself a failure—at least for the day—because the lesson didn’t “work” and we were “falling behind.”

## In the Present…

It would take thousands of words to describe all of my imperfections as a teacher. But I could probably use almost as many to describe the ways I’ve grown over the past nine years. I may not be particularly good at this teaching gig, but I’m better than I used to be.

I feel like I’ve successfully addressed the major issues outline in (1) and (2) above by streamlining our in class “homework check” routine. Of the many things I’ve tried over the years, I’ve been reasonable happy with two approaches: Hard copies of solutions (one paper per pair of students) or slides of solutions, with either method placed right at the start of class. There have been advantages and disadvantages to each approach, with each growing larger or smaller depending on the particular group of students, but the common result has been a start-of-class homework routine that usually takes between 2 and 4 minutes and provides every student with feedback on every attempted problem as “immediately” as I can manage given my available tools and my limited skill set.

Issue (3) is an ongoing struggle, and something I hope to consider further. Maybe it’s related to what I discuss below, maybe not.

## Still in the Present…

For all the improvements I’ve realized, there are still some glaring weaknesses in my current approach to homework. In my honors classes, almost all of my students complete the assignment each night, but some of them spend a ridiculous amount of time on it. Is this the most effective way for them to learn? Is this the healthiest way to spend their evening? Is there a way to transform what I do in the classroom so that they end up learning and practicing at least as much as they do now, but without my stealing so much of their non-school time with school-related things? (If you think the answers are not “No, no, and yes,” you are hereby required to explain in the comments.)

## In the Future…

Should I do away with homework all together? Posts like John Scammell’s make me think I should. But then most of my classes don’t have the dismal homework completion rates that John and many other thoughtful teachers point to as one advantage of the no-homework approach. Will I use Dan Meyer’s 2007 approach? Or his 2008 approach? (Anyone know the latest thoughts out of Camp Meyer?) And while we’re linking to blog posts about assigning or not assigning homework, are there other posts I should read?

In all likelihood, my course responsibilities for next year will include AP Calculus AB, Honors Precalculus with Trigonometry, Honors Algebra 2, and Honors Algebra 1. In these classes, the homework completion rate is through the roof, plus or minus 3%, and (as I’ve shared above) I think many of the students benefit from the practice. It would be tempting to say that my homework policy is “good enough” for these classes. The students will do the work, they’ll remember to bring it to class, they’ll grade it efficiently, I’ll assume they’re getting the feedback they need to draw conclusions about their strengths and weaknesses, and that together we’ll be able to decide how to proceed from there.

One of the classes I don’t expect to teach next year is regular Algebra 1. This year’s section of Algebra 1 has been my most challenging ever. Among other things, my homework completion rate has been discouraging and not at all like the majority of my classes. I would guess that less than 50% complete the assignment each night, and for those who do only about 20% (or less) benefit from the practice. For a class like this, my current homework approach isn’t working. They need (and deserve) something better, something more thoughtful, something less frustrating, something more effective, something like what John Scammell describes in the post I mentioned above.

Does the fact that they aren’t served well by my current homework system mean that my other more compliant classes would also be better served with a different system? Or does the high completion rate mean I should leave things the way they are for these classes?

Should I have one approach to assigning homework for all of my classes, irrespective of the habits of an individual class? Or should I tailor my policies to the tendencies of my students, and define effectiveness in relation to the particular set of students?

Clearly I need some help sorting all of this out. Thanks in advance for anything you can offer in the comments.

## A Day In… Honors Algebra 1

“A Day In…” posts are averaging 1997 words per post. Holy wow! Time for a shorter one.

## The Setting

3rd Period, Wednesday, April 3, 2013

Honors Algebra 1

## How Things Went Down

Bells rang. Sets were found. Homework was checked. Estimations were made, reasons were given, the answer was shown.

And then, the lesson began. (Full disclosure: I wrote the lesson a year after reading this, and was even more influence by a jigsaw-puzzle-building activity—solo, solo, then tag-team—I heard about from some friends who work here.)

Me: “Does anyone have a magic phone with a stopwatch?”

Student R: “I do.”

Student R: “Yep.”

Me: (Walking to the front of the room with my bucket of binder clips…) “The rules are as follows: I am allowed to use my left hand only, one clip at a time. Got it?”

Everyone: “Uh… What are you talking about?”

Everyone: “Okay, we still have no idea what you’re talking about. But sure, whatever.” (This is a paraphrase.)

Me: (While dumping the binder clips on the floor…) “Student R, give me a countdown.”

Student R: “3… 2… 1… Go!”

My task then becomes clear to the students, as I proceed to pick up and toss the binder clips into the bucket as fast as my left hand will let me (one clip at a time, mind you). I’m right-handed, so this takes a while. 100 seconds to be exact. (Two years in a row, 100 seconds exactly.)

It gets a little awkward after about 30 seconds (70 seconds to go!!!) so I banter with the students for about 20 seconds, invite them to hum the Jeopardy theme music for another 30 seconds, and ask them to cheer me on for the last 20 seconds. Some oblige, some do not. (Hey, that’s not unlike the rest of my experience in teaching!)

At that point we record my time. I then dump the clips on the group a second time. I ask for a volunteer. (“Thanks, Student J!”) This brave volunteer then picks up the clips as fast as he can using two hands, one clip at a time (per hand). His time is 59 seconds. (60 seconds last year.)

Then the fun part, essentially stolen from the world of Three Act Math Tasks: Students make an estimate for how long they think it will take the Fenton-and-Student-J-Tag-Team to pick up the clips (same individual rules apply).

Guesses are made, clips are dumped, the stopwatch is readied, and the clip cleanup commences.

We’re an amazing team, so we finish the task in 40 seconds.

From that point the lesson is rather predictable, so I won’t bore you with the details (though we did have some great conversations in this “predictable” portion because of the seeds planted in the introduction).

## What I Liked

The lesson was fun to teach, and the kids were definitely engaged.

I love the extra buy in from students that I get simply by asking them to guess before we measure, calculate, etc..

All the guesses were reasonable! No one offered the absurd (yet tempting, for the totally lost) answer of 100 + 59 = 159 seconds. Why? Because the setting/context/problem type was set before the students in such a tangible way. “Of course the tag team will finish faster!”

## What I Didn’t Like

The lesson doesn’t do a good job of building on the reasoning students were engaged in during the introduction once we transition to a search for more efficient solutions. By no means do I dive headfirst into a “watch and mimic” approach. But the students who had no idea how to approach the problem in the first place (i.e., the students who could do no more than make an educated guess) are still unable to do more than make an educated guess.

There is a decent amount of semi-downtime for students in the first 10 minutes of class. The advantage here is that we create the data as a class. The disadvantage is that only a few of us are actually involved in generating the data. I don’t have a fix for this yet, but I would like to involve more students or decrease the downtime (or both).

## How I’ll Get Better

Immediately after teaching the lesson I began brainstorming improvements for next year. This is my attempt. My goal was to create something that would help students develop two efficient approaches that emphasize/promote understanding in the midst of finding the solution, but that didn’t require me to be a central part of the conversation while it unfolded.

I was happy with the handout and excited to use it sooner rather than later, so instead of waiting until next year I presented it to my students the day after the first lesson. I was pleased with the results, as students learned efficient methods without abandoning their reasoning. (Sadly, this abandonment-of-reason-for-the-sake-of-efficiency happens too often for many of my students, especially when we transition from estimates and arithmetic approaches to algebraic ones.) And while they didn’t develop the methods entirely on their own (to expect that of them at this point in the year would require that I’ve expected similar things all year long, which sadly I have not), there was a lot of great conversation followed by some favorable assessment results a few days later.

## Questions

Need some inspiration before you head to the comments? Consider responding to one or more of these:

1. What do you think of the first handout (Day 85 Notes)? What do you like, what would you change, and why?
2. What do you think of the second handout (Day 85 Practice)? What do you like, what would you change, and why?
3. Do you have any ideas for helping me solve the “downtime” issue described above? Or is it a non-issue, and I should just relax?
4. I want to help my students grow in their ability to develop efficient problem solving strategies on their own. What sorts of things can I do throughout the year to help them improve in this regard?

## Another Day In… Honors Precalculus with Trigonometry

Originally, I figured I would write one post per class for this “A Day In…” series. But then something strange happened in Honors Precalculus: this week.

So even though I have an Honors Algebra 1 post (or two) burning a hole in my brain, I need to process the goings-on of another day in Precalculus.

Onward!

## The Setting

4th Period, Thursday, April 4, 2013

Honors Precalculus with Trigonometry

## How Things Went Down

The bell rang. Kids graded homework (two hard copies of solutions handouts per table of four kids) while I walked throughout the room. Most students begin grading a few minutes before the bell, so we finish pretty quickly and they get detailed feedback on each assignment (and I don’t spend 2+ hours grading every day after school).

We then played SET. Next, students signed up for their CSU Fresno Math Field Day events—or wrote down why they could not attend. This took approximately 300% longer than it should have, and I have more kids opting out this year than ever before, both of which were a little frustrating. (Formerly, I’ve required my honors students to participate, unless they have an unavoidable conflict. But I’m growing tired of the tension this policy creates so I’m making it optional from here on out.)

So there we are, moving forward quick-as-molasses, finally ready to begin the lesson. Using this handout (an exploration from Paul Foerster’s Precalculus textbook) students were supposed to graph polar curves on their calculators in order to determine which of the apparent points of intersection were “true” points of intersection (and therefore solutions to the system).

Several times over the past five or six years my students have worked their way through this exploration. And with some wandering about the room, listening in on conversations, offering a bit of guidance where appropriate, and so forth, my students have been successful. With that prior success in mind, I didn’t really prepare for this lesson.

That. Was. A. Mistake.

If the lesson was a train, then it pulled slowly out of the station, flew off the rails, crashed into something big and destructive and flammable, and burst into flames. At least there was no ambiguity. It was undeniably horrible.

When I realized the depravity of our situation, I called for everyone’s attention in order to make an announcement:

Hey guys, this isn’t going well, and it’s my fault. I didn’t prepare for this lesson as well as I should have. I want everyone to stop working on the handout and find something else to do. You can work on something from another class or just relax and chat with your friends. I’m going to sit down to rewrite the handout. If I can fix what’s broken in 5 or 10 minutes, we may resume. If not, we’ll pick things up tomorrow.

The subtext (which I didn’t verbalize to the kids): I value your time and effort too much to waste it with some half-baked lesson primed for disaster.

I then spent the next 20 minutes (yep, we didn’t resume the lesson) rewriting the handout. The bell rang, I invited them to have a great rest of their day, and that was it.

## What I Liked

There’s some cool stuff that is supposed to happen in that lesson, and Foerster’s handout has been great in the past at helping my students wrestle with these ideas.

Aside from those potential good things, there wasn’t a whole lot I liked from that class period. I suppose I could score my students’ response to my abandoning ship on the positive side of the ledger. They were gracious and forgiving, though probably only because they were in a good mood after 20 minutes of relaxation.

## What I Didn’t Like

I’ve already addressed most of what I didn’t like about my lesson above, but I will add more detail for why I think the handout didn’t stir up its former magic. The handout was designed for the TI-84. None of my students have TI-84s anymore. A few years ago we made the shift to TI Nspire handhelds, and the first group of kids who made the switch are now in Precalculus.

So why did the lesson come to a screeching halt? There was a total mismatch between (1) the guidance provided and the demands made by the handout, and (2) the technology students had access to. Granted, the TI Nspires are newer, shinier, and (at least in my opinion) better than the TI-84s. But a handout written for another device doesn’t care about newness or shininess.

## How I’ll Get Better

That 20 minutes (with my students sitting around, happily chatting with one another) was the most productive (and professionally enjoyable) 20 minutes I’ve had in the last three months. I can think of a few reasons why:

1. I knew what was broken, and I had some ideas for how to fix it.
2. I felt the pressure of the clock. Class was ending soon, and I wanted to at least get the lesson rewrite well on its way while my ideas were fresh.
3. I’ve been digging through dozens of amazing teachers’ lessons via Twitter and blogs, so I had a few more ideas floating around my head than I usually do.
4. I was excited to try my hand at writing a lesson in a way that would allow me to move off center-stage in order to let the students take on the most active roles.

So I wrote feverishly for 20 minutes during the last bit of fourth period. Then for another 15 minutes during lunch. Then for another 30 minutes after school. Then for another 30 to 45 minutes before I went to bed.

I ended up with this handout. And a gen-u-ine teaching buzz. I was so excited for the next day to roll around so I could bring what I created (really, what I modified; for better or for worse my new handout owes its existence to Foerster’s lesson/handout) to my students, to see how they would respond, what they would learn, what questions they would have afterwards, etc.. I haven’t had this sort of feeling for quite a while, and I quickly identified the reason why: I haven’t spent this much time thinking about and writing (or re-writing) a lesson in a number of years. It’s not that I don’t spend time preparing for my classes these days, but a lot of what I do now consists of reusing last year’s lessons, with or without some minor tweaks. In years past I would spend hours and hours getting ready for a day, sometimes just for a single class. That investment of time often led to decent returns (that is, decent lessons), which in turn led to an I-can’t-wait-until-tomorrow vibe.

In fact, while reflecting on all of this I thought back to what I now consider my favorite season of teaching: the spring of 2008. That was the semester during which I wrote and taught a trigonometry unit to my Honors Algebra 2 students as part of a masters project. The lessons were all student-centered and (as I recall them, anyway) fairly engaging.

I’m convinced that this season was enjoyable for a number of reasons, but foremost among these is the fact that during that time I was creating content like a madman. Saving and reusing curriculum is healthy. In fact, for many of us (myself included with four to seven preps and four kids under four) it’s 100% life-saving-necessary. But if I want to remain satisfied in this profession, I know this: I have to continue creating. If I don’t, my interest will vanish like wind-driven mist.

So whether it’s the revamping of a single lesson, an entire chapter, or a whole course… Late at night, on a weekend, or over the summer… I know the key to keeping my heart in the classroom: Create. And create some more.

## Epilogue

I’m writing this post on Friday night. (No time to blog last night; I was too busy drawing up a new lesson/handout.) I won’t go into a lot of detail, but I will say that fourth period was a lot of fun today. Because of my extra hard work the day before, I got to step aside during class and let the kids do the heavy lifting of thinking, arguing, and drawing conclusions. Students also got to work through the lesson at different speeds, which is totally appropriate considering that students think at different speeds.

## Questions

I’d be very interested to know what you think of the my experience in Precalculus this week, as well as my semi-newfangled handout. In particular:

1. Is there too much scaffolding? Too much hand-holding via handout?
2. Are the lesson goals (solving polar systems via graphing, learning about auxiliary Cartesian graphs) worth exploring? I became so focused on making this old lesson work that I didn’t stop to think until I was done: Is this something we should even be studying? I think it’s cool stuff, and certainly was healthy exercise of the brain for my students, but is it essential or trivial, useful or useless? (I obviously need to rethink why I teach anything in Precalculus—or any course, for that matter. I have some serious work to do over the next couple of years in making my courses stronger, more well thought out, etc.)
3. Are the directions and questions clear?
4. Does the format help or hinder the lesson goals?

Thanks!

## Update 1

Joshua Zucker shared some great thoughts in two comments almost immediately after my post when up last night. (Check ’em out below.)

His first comment inspired me to tweak the handout a bit further (namely, the coordinate planes provided). The latest version of the handout is here.

Let me know what you think of the changes to the coordinate planes. (Hooray for Adobe Illustrator!)

Also, while making the polar grid for the handout I decided it wouldn’t be too much trouble to throw six small copies on a sheet and one large copy on a second sheet to share with my students for other activities. You’re welcome to use whatever you want from this Dropbox folder. The initial inspiration for the graph paper came from this, though the final version was improved (in my opinion) by a student comment that “It would be swell if every fifth circle used a heavier line stroke.”

## Update 2

I blog to reflect on my teaching. That alone makes it all worth it. However, more often than not someone asks a followup question that forces me to think even more critically about my teaching experiences. And it’s not at all uncommon for this person to be named Michael Pershan. Exhibit A:

I decided to respond to Michael with an update here rather than on Twitter or in the comments because I think it’s incredibly relevant to my entire reflection. With that said, here’s my reply:

I don’t think the original worksheet has any deficiencies. I love Paul Foerster’s materials, especially his explorations for Precalculus and Calculus. (In fact, Foerster was one of the first people on the list.)

The reason I revamped the handout was that it no longer worked in my classroom with my students (with the technology we’re using). My lack of planning that led to the fiery train wreck was about 90% not accounting for the changes needed in light of out shift from the TI-84 to the TI Nspire. Beyond that, I ramped up the “wordiness” and “handholding” of the lesson/handout because, frankly, my students needed it this year. Some of the wordiness is due to my poor skill as a writer, and some of it is entirely by design.

So what did I improve? Mathematically, I would say nothing. But I created a handout that worked with my students and the technology available to us. The original handout (again, see train wreck), despite its quality in other settings, was no longer functional in mine.

## Update 3

Joshua Zucker again, this time in response to my question of whether the activity included too much scaffolding:

There’s some handholding and some room for discovery, so thinking about some of the schools I’ve taught at (among the top in the country, where my honors precalc class is not unlikely to have some 10th grader who is taking the USAMO or something) it would be a bit too handholdy, but overall it seems reasonably balanced. You know your students better than I do.

Talk about a classy and affirming response that still dishes some helpful critique. Here’s how I read it: “Yeah, there is some handholding there. But depending on your situation/students, that may be entirely appropriate, especially if it allows for the discovery to happen.”

So I began wondering what this handout would look like if I was designing it for a group of students who were mathematically more proficient or more familiar with open ended questions (or both). Here’s my answer. It would be another “train wreck day” with my current practice and students, but maybe one day… Let me know what you think.