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:

  1. Your district’s intended approach (traditional vs. integrated)
  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!

Difference of Squares Game

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?

If you’re interested in reading responses, and even contributing your own, head to the Google Doc!

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.
  3. Write some commentary about the assessment in a Google Doc.
  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.”

Me: “Awesome. Get ready. (Pause.) Ready?”

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?”

Me: “Ready?”

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!”

binder clips

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.


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.


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.

Screen Shot 2013-04-05 at 10.53.11 PM

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.


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.


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?


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.

A Day In… Honors Precalculus with Trigonometry

A Little Introduction for “A Day In…”

Today was the first day back from spring break. I decided it’s time to take the subtitle of my blog (better through reflection) seriously. How so? By writing a recap of one period for each of my classes, including things that went well, things that didn’t, things I can do to get better, and any other takeaways (or questions to consider) that come to mind.

This is the first installment. More will follow as time and energy allow.

The Setting

4th Period, Tuesday, April 2, 2013

Honors Precalculus with Trigonometry

How Things Went Down

As students walked in the room I welcomed them back from spring break. This semester I tweaked how I enter things in PowerSchool (assignments get grouped by chapter now, rather than entered individually), so I’ve been brainstorming how to help students keep track of their assignments without the same level of detail online. My most recent (and I hope final) attempt is this handout. I spent two minutes explaining how it works and how I expect them to use it. It’s pretty easy. They just copy down whatever it says under “Do @ Home” on the Daily Plan. As an example, here’s today’s:

P4 Daily Plan.095

I suppose that slide won’t make any sense unless I share the Course Outline. (I used to call it the Assignment Schedule). Students get one of these on the first day of the year. It’s my wonderfully lazy way of communicating assigned homework.

Alright, back to today. We then played SET for two or three minutes. For the first round students had 60 seconds to work quietly in small groups to find as many sets as possible in the Advanced Puzzle Mode (12 cards, exactly six sets are present, no cards are removed when a set is found). When the 60 seconds were up, each group had an opportunity to share one of the sets they found by recounting the card positions (e.g., “1-5-12” or “3-4-9”) of the. For the second round students had 60 seconds to find as many sets as possible in the Basic Classic mode (12 cards, three cards removed and replaced when a sets is found). The students did a great job, finding 17 sets in 60 seconds, a record for the week. The record for the year—held by 7th period AP Calculus AB—is 20 sets found in 60 seconds.

I was excited for today for a number of reasons. One, I typically love my job (even though I’ve had some rough stretches this school year) and I’ve missed the students after 10 days of no school (honestly). Two, I was a little bit excited about today’s lesson because I took what I thought was a rather lackluster notes handout and spiced it up tried to spice it up with some Desmos graphing action. (As it turns out, the “supplementary” handout was garbage. More on that in a moment.)

After playing SET, I grouped students in pairs, gave every student a copy of the supplementary handout, and had each pair grab a laptop (either of their own or from the laptop cart I checked out for the period). I expected this part of the lesson to last about 5 minutes, but with some typical tech-related delays many students took closer to 10 minutes to finish. (And it didn’t help that the handout lacked a clear goal. More on that later.) While waiting for the last few groups to wrap things up I invited other students to write their responses to questions 1, 2, and 3 on the board.

After the supplementary handout, we turned our attention to the notes handout. (I’ll share my frustrations with and potential fixes for the handout below.)

As we finished Example 3, the bell rang. I would have liked another 1-2 minutes to debrief, summarize, etc., but I didn’t manage class time particularly well today.

What I Liked

It was good to see the kids again. They did a great job playing SET. Some students made important connections in spite of my poor sequence of activities. Did I mention they found a lot of sets?

What I Didn’t Like

This could drag on for a while if I’m not careful. Time for bullets!

  • Kids were bored (not everyone, but definitely some; and not just bored, but bored!!!)
  • The school laptops take multiple minutes to start up (those who used their own laptops finished the entire “activity” in half the time)
  • The first half of the supplementary handout didn’t have a clear purpose; the second half didn’t provide students with any engaging tasks (kids need a longer leash for healthy/rich exploration, not 60-second tasks that need teacher intervention before students can move on to the next mini-task)
  • The original notes handout starts with two big ugly definitions/properties boxes (Ugh! don’t lead with this kind of thing, Michael!)
  • I should point out that students had already discovered the content of the first box in a previous investigation, so I’m not totally opposed to including something like this as a summary in a later handout… But to lead with it? More ugh!
  • I blew right by the second box (“These are not the properties you’re looking for…”), deciding to introduce/develop these properties with the help of a right triangle in a two-layer Cartesian/polar coordinate plane (is that even mathematically sound?), but in a very “lecture-y” manner, where my students could have developed these properties on their own with a well-written sequence of questions
  • Example 1 could be improved (ideas below)
  • Example 2 requires completing the square, the kids were super shaky on this, and I wasn’t prepared for their confusion (in every other year that I’ve taught Precalculus, I had taught most of the students the year before in Algebra 2 or Honors Algebra 2, so I usually have a good read on their algebra skills; I didn’t teach any of this year’s Precalculus students last year, so I’m not as in tune with their strengths and weaknesses as I have been in years past)
  • Example 3 was rushed (by me, to finish before the bell rang)

How I’ll Get Better

Alright, the point of this isn’t to stew but to reflect, and through these reflections to get better. So here goes (bullets again to avoid turning this long post into a truly gargantuan one):

  • If kids were bored, that’s (usually) my fault. Here, it definitely was. I failed to give them an engaging sequence of tasks. Too often they were just waiting for the next teacher-led portion of class. With some extra time in planning the lesson, I could design something where students work primarily in groups and we only do the whole-class thing for a very brief introduction and a more detailed (student-led, teacher-facilitated) debriefing session.
  • I can’t do anything about the school laptop frustrations. However, I could plan ahead and tell students when to bring their laptops to class (or have them bring them every day just in case). More and more students are bringing laptops and tablets to school. Maybe 25 to 30% now? As that grows, it becomes easier (at least for me) to incorporate web browser-based activities into class.
  • The supplementary handout suffers from a lack of clear purpose. Was it supposed to help students learn some general graphing basics (toggling on/off, sliders, domain restrictions, etc.)? Or was it designed to have students explore a particular graph? Or to graph three on their own?
  • When I think about it some more, the entire lesson suffered from a lack of clear purpose. Are we focusing on graphs and the impact of parameters (our conversation drifted there today)? Or is the goal to convert from polar equations to Cartesian? To be honest, I need to rewrite the entire sequence of polar lessons to give students more practice graphing, making observations about the impact of parameters in polar equations, etc., before asking them to convert polar equations to Cartesian form. So next time? Here’s what I envision: A handout with 12 images (six on the front, six on the back). Students working in pairs (each with a handout, one laptop per pair). The directions: Use sliders on my pre-made graphs to match each graph. Write down the “winning” equations. Explain how you did it. Then, on the following day we can focus on converting the equations to Cartesian form to have the “a-ha” moment(s) of “Hey, that really is a vertical line/horizontal line/circle with center (#,#) and radius #!” (Oh, reflection! How you resemble rambling!)
  • Example 1 would be much better if it took over an entire page with: (1) A big fat polar coordinate plane so students can sketch what they see on Desmos or their calculators and (2) A big fat table for students to complete (theta values given, r values missing) so students can see clearly/numerically where r first becomes negative and what happens on the graph at this point. (Do I need to bring in auxiliary Cartesian graphs from the get go? Sam’s link to this applet has me wondering if that’s key early on and if I maybe wait too long to draw it in…)
  • Example 2 would be much better if we first developed the properties (rather than giving students a pathetic box o’ properties (“Hey, where did those come from?” “Pay no mind, pay no mind! Back to the examples!”). This could be done with a mini-investigation.
  • Example 3 would be decent if it followed the revamped Examples 1-2, provided that I move to the back of the room and have students wrestle as a whole class with how to make the conversion based on what we’ve done in Example 2.

Most of the rambling above, while helpful (I think), is focused on just a couple of lessons in a single chapter. But I’m similarly dissatisfied with a lot of the lessons in my Precalculus course. I don’t own it the way I do my Algebra 1 and Algebra 2 courses (despite their many and sometimes deep flaws). Algebra 1 and Algebra 2 are the classes for which I put the textbook on the shelf and wrote my own curriculum (here and here if you can bear to look). Did I mention the many and deep flaws?

At any rate, I know my Algebra 1 and Algebra 2 courses inside and out. I’ve wrestled with the sequence of topics, the sequence of lessons, the sequence of examples within those lessons, and I look forward to wrestling with how to turn these teacher-centered lecture-heavy courses into ones packed full of activities, investigations, explorations, rich problems, engaging tasks, etc.

Is it time to chuck the textbook for Precalculus? Can I even afford to make a move like that? I’m not exactly running around with heaps of free time these days, thanks to four beautiful little kiddos. Maybe my days of writing entire courses from scratch—flawed as they may be—are numbered? I don’t know. But I do know this. I want to get better, and I think know that I can. Reflection is time-consuming, but so worth it.

And the best wisdom I’ve heard so far on this issue (but the hardest thing for me to be satisfied with)? Baby steps. Just keep taking baby steps.

Thanks for reading! Wisdom and insight welcome in the comments as usual.

A Further Call to Arms

Entering the Discussion

Two Thursdays ago I jumped in the middle of an ongoing discussion about assessment by posting this. It was essentially a call to arms, a request to join forces in strengthening the quality of our assessments.

Then, just after midnight on Saturday morning Daniel Schneider blew my mind by posting this. If you haven’t read it yet, stop messing around on my blog and get yourself over to his. Make sure you follow all of the rabbit-holes provided in the links. (He wasn’t kidding when he said he was a master aggregator.)

(Update: While I was fiddling with this draft, Daniel threw this down as well. Go ahead and put another educator on the list of people I want to be like when I grow up.)

What’s Next

So where do we go from here? What should we do with all of the interest and enthusiasm surrounding assessment? I see one thing as a no-brainer:

Let’s create a centralized location with links out to quality posts and articles about creating excellent mathematics assessments.

I’m hopeful that a certain master aggregator will lend a hand here. (Update: He will!) Over time we can add more links to resources and even invite members of the so-called better assessments movement to write articles addressing specific topics of need.

Archive or Conversation?

Beyond that, I see two lines of attack: (1) Build an archive; (2) Foster a conversation.

Let me explain.

It would be incredibly valuable to have access to a well-populated, easily-searchable database full of rich assessment questions.

It would be even more valuable—incomparably so—if we learned, as an entire community, to write such rich questions.

In the first case (the archive-building scenario) the focus is more on writing and/or gathering good questions and assessments, and then serving them up in a helpful way. A noble task, to be sure. One I hope others will take up and carry to great heights. (By the way, if that’s you, check out OpusMath, follow them on Twitter, and start uploading like crazy.)

But more than a comprehensive archive of excellent problems with a slick user interface, I think our most essential need right now is to develop an army of amazing assessment authors.

If you agree that the ongoing, teacher-developing conversation is at least as important as the creation of a fantastic archive, then I invite you to join me in the following challenge.

Write An Assessment You’re Proud Of

In a recent email exchange with Daniel, I shared a massive vision I have for creating this group of great assessment writers. He suggested starting with something smaller (something vital, yet attainable) and building from there.

With that in mind, here is my/his/our challenge to ourselves and to you:

I want to challenge the blogotwittersphere to write an assessment they’re proud of. To pick a skill/concept/objective and write a targeted assessment that measures this objective at various depths. Or, if they already have an assessment they’re proud of, to share why they’re proud of it—what is it about this question/this series of questions that makes this assessment meaningful? That finds a way to assess both procedural and conceptual understanding. That gives students an opportunity to exceed expectations. That has an ‘explain’/’justify’ component. I don’t know if these things are possible for all skills and objectives, but this is why I want others to be thinking about it too.

The end goal is: create an assessment you’re proud of in terms of format or questions or depth or all of the above, and explain why you think this is something worthwhile.

(I tried to recast his challenge in my own words, and realized I was better off stealing the thing wholesale, with his permission, of course.)

Where Do I Sign Up?

If you’re interested in playing along at home, start with a one-minute survey. Then hop on the Twitter and spread the word! The more voices we have in the conversation, the better we’re all going to get.

What You Can Expect From Me

Once this blog post goes live, I’ll do the following:

  • Set up (the “home” for the better assessments conversation)
  • Gather already-existing resources together in an Assessment Authoring Boot Camp section of the blog (Daniel Schneider has agreed to lend a hand; additional volunteers welcome)
  • Put out an official “call for assessments” over a two week (?) period in the near future, including submission guidelines
  • Feature a small number (one to three?) of these assessments each week as guest posts on the blog (including the “explain…” bit from Daniel’s challenge), and invite the “better assessments” community to offer feedback, constructive criticism, etc.

As with any group project (and I hope this turns into a massive group project), better ideas will come as soon as brains other than my own start their wheels-a-turnin’. I’m not married to any of the details sketched out above, so long as we find a way to establish an ongoing and positive conversation about assessment.

The Dailyness is the Key

I feel I should explain the last bullet under “What You Can Expect From Me.”

Let’s dream big. Imagine that 100 math and science teachers from all sorts of different grade levels and courses submit assessments when the call goes out. The assessments range from decent to amazing, and we’re all stoked because 100 people (100 people!) played along with this little experiment and we have heaps of assessments to look at and learn from.

If we’re not careful, we’ll squander most of the opportunity for conversation presented by 100 such submissions. My intention is to highlight a few assessments at a time so that busy teachers (that’s us!) have an opportunity to dig into each assessment in depth, over the course of an extended period of time. (Whether or not my suggested approach will achieve the goal is open to discussion.)

I think we’ll experience the most growth as a community if we employ a slow-and-steady approach, rather than go after this all at once. It’s the reason many of us find blogs and Twitter more powerful tools for sustained professional growth than fantastic-but-isolated conference experiences once every year or two. The dailyness of our practice is the key to our growth.

What To Do With a Head of Steam

If this project isn’t dead in a few months, then I’ll share some details of the bigger vision I have for this assessment conversation. It has to do with recruiting and organizing people at various levels and in various subjects into assessment-writing cohorts. They’re exciting plans (at least to me), but possibly unrealistic. In the coming weeks I’ll invite some of you to tell me what has potential, what’s a waste of time, and what needs tweaking to become realistic.

Some Closing Thoughts

I think multiple-choice questions are generally inferior to free-response questions. I also think that both styles of “one-off” question are completely inferior to well-crafted performance assessments. However, I also believe that poorly-written MC and FR questions are inferior to well-written MC and FR questions. With that in mind, I think it’s entirely appropriate to allow MC and FR questions into the discussion at

With that said, I think the MC/FR/Performance Assessment classification of assessment questions isn’t as helpful as the conceptual/procedural/synthesis approach described by Daniel here and here. I certainly like this latter three-part structure better than then conceptual/procedural/application framework I’d been mulling over prior to reading all these great blog posts on assessment.

Comment Time

Heart beating with uncontrollable excitement? Bored out of your mind and wondering how you made it to the end of another lackluster post? Have a suggestion? A critique? An idea? Drop a line in the comments and keep the conversation rolling.

Expressions Challenge

The Four Fours

In my first year of teaching I came across the Four Fours challenge. I loved it. I’ve always considered puzzles, brainteasers, and trivial challenges of that sort to be rather entertaining, so I went to town on the Four Fours right away. The problem gets better as you go: easier target numbers are out of the way and the real mental effort begins as you look to fill in the gaps.

Beyond the personal puzzle perspective, as a brand new teacher (my course load then was split 50/50 between 7-8 and 9-12) I considered this challenge to be a gem for my middle school students. Here’s a few reasons why:

  • Mundane-yet-necessary practice is made more engaging in the context of a challenge (just like in this problem).
  • Shifting the task from evaluating expressions to writing expressions ramps up the critical thinking component of the task. Students have to look for and make use of structure in order to bend the expressions to their will.
  • The problem is flexible and can be presented to individuals, small groups, entire classes (class vs. class challenge, anyone?), or with a twist (find as many expressions as possible with a value of, say, 23).

And Then It Happened: The Internet Failed Me

I’m sure there are more reasons why the Four Fours is awesome. If you think of any, share them in the comments. But I know of one reason why the Four Fours is not awesome. In fact, I know of one reason why the Four Fours is worthless. That’s right, useless. Totally devoid of any value.

Wait? Just a minute ago I was singing the problem’s praises. Why the dramatic turn? One reason:

The Internet

I typically love the Internet. Defender of the integrity and usefulness of Wikipedia and all that. But not this time. Here, the Internet failed me.

See, someone decided it woud be a good idea to ruin the Four Fours by posting entire solutions all over the place. Now any time I give this problem to students, they’re one Google search away from Four Fours glory.

My Solution

So what do you do if the Internet robs you and your students of a great problem? Make another one. Keep the good stuff (see the bullets above) and tweak the parameters.

Flash back to my first year. When someone told me the full solution to the Four Fours was online, I rewrote the rules and called it Expressions Challenge. Once I got going, it was easy to make additional versions. I’ve included the directions (and handouts I mocked up this year) for the first three versions, as well as some comments on how to tweak Internet-proof the problem further.


Here are the directions for the first three versions of the Expressions Challenge. Note that there is only one difference between the three: Use the numbers in (1) any order, (2) increasing order, or (3) decreasing order. Obviously, versions 2 and 3 are more challenging.

expressions.1 expressions.2 expressions.3


Just to be clear on the order element of the directions, here are examples for each version (1, 2, and 3, respectively):



And the handouts (nothing special):

Further Defense Against the Evils of the Internet

If the Internet destroys any of the challenges above, no problem. Just tweak the problem again. Use the numbers 1, 2, 3, and 4. Or 1 through 6. Or the first four odds (or evens). You get the idea.

Also, you could change the word “numbers”  in the directions to “digits” to open up some additional possibilities. And for the record, whether your students should be allowed to use decimals, percent signs, radicals, ceiling/floor/rounding functions, etc., is totally up to you.

Let me know in the comments if you find any of this useful, or if you have a similar challenge you use with your students.