Mistakes I Made Creating a Skills Checklist

I made some rookie mistakes with my Algebra 2 skills checklist in semester one this year.  I have invented a word to describe each problem.


In my enthusiasm for separating skills, I gave determinants of 2×2 matrices and determinants of 3×3 matrices each their own spot on the list.  They should have been combined.  I also separated multiplication of polynomials from their division and even, astonishing in the euphoric clarity of hindsight, addition of matrices from their subtraction.  This overzealous separation led to inflated grades (students got a 5/5 on matrix addition, matrix subtraction, both kinds of matrix multiplication, AND twice for finding determinants) and tests that felt kind of… stupid.


A more subtle mistake: some skills on the checklist did not have discernible differences between intro-level problems and master-level problems.  For instance, an intro simplification test might look like “3x+2p-x+2x,” and a master might look like “3x+2p-x+2x-p+2p+r,” but for some students it felt silly to bother giving the intro level test first.  If a student can calculate “3+2+8+9,” do you really need to check to make sure they can calculate “3+2+8+9+1?”  In contrast, my favorite skills had some fundamental difference between the intro level and master level tests.  For example, the “dividing polynomials” intro test asked for a division that would have no remainder, and the master test involved remainders.  This was nice because a student could pass the class with a basic understanding of the concept, but would need a more advanced understanding before getting the 100%.

Bad Planning

I did not have an effective way to deliver master-level tests to the students who wanted them.  Early in the year I promised that students would be able to attempt a master-level test on any skill once per day, and early in the year this worked great.  Late in the year, when two thirds of my students wanted six tests a day, it was harder.  I need a better system.  Luckily, this won’t be a problem again until the end of this semester, so I’ll start thinking about it then.  😉

So, now the task is to create the lists for next semester (my school doesn’t start until January 10!), hoping to avoid these problems and keep the amazing benefits I got from the system last semester.  Stay tuned!

Here’s your exponential data!

A while ago k8nowak was looking for exponential data besides credit card loans and bacteria growth.  Take a look at the data describing visits to my (very new) blog:


Looks a lot like exponential decay to me.  In retrospect, I’m almost sad to have posted and ruined the trends.  But, for those of you with very popular blogs, I think you might have an interesting option here.  Have students create a webpage, right, for any old reason – to review multiplication tables or explain the distributive property or whatever.  They could even just post their drawings, a math joke, short stories, whatever.  When they’re done, link to it from your blog!  Do it early in the morning so you don’t get the weird bump at the front edge like I did.  They’re not likely to get a lot of other traffic, so I think the data might be cleaner than mine.

This is risky if you are banking on the graph being exponential, but if you’re open to a class discussion about modeling traffic, where exponential curves will probably, but not definitely, come up, I think this would be fascinating.  You know, you could guess how many visits will come the next day… and then see how close you are!  Would this be more interesting than loans and radioactivity?

I’d be interested in hearing from other blogs who notice peaks in traffic when someone famous gives you a reference.  Is the curve always exponential decay?

I want to have some contest with you guys.

It’s not just that I want to prove myself the world’s fastest math teacher (one estimate of my speed in  http://larkolicio.us/blog/?p=42 was 18 mph!).  It’s that I think a video competition amongst teachers would be interesting for my students to judge.  So, want to race?

My idea is that several teachers would submit a video of themselves doing something.  Running is an easy idea.  Videos would be at different resolutions and different zoom levels, with different settings, so kids would have to use multiple points of reference to determine who is actually moving the fastest.  Then, we’d have our kids make their calculations.  We’d need some sort of structure to submit work from our multiple classes, and some averaging scheme to be fair when one class says their teacher was going a suspicious 50 mph (honest mistake, of course!).

Clearly it doesn’t need to be a footrace.  We could have any sort of timed event.  Is there an untimed event that we could compare?  Building a pile of sand with the best height/base ratio?  Making a function with the best (length)/(area of enclosing rectangle)?  Best record at Settlers of Catan?

Maybe you’re getting the idea that this is not crystal clear in my mind yet.  But here’s my motivation: I want to show kids that I am part of a learning network, and that other people are working on getting better at things.  I think it will motivate them.  Also, I think they’d think that interacting with students from another class would be cool, and would show them a way to interact with others on the internet besides facebook.  How can we do this?

Leave your wave username in the comments or email it to me and we could collaborate with it.  I’m “rileylark.”  I have a bunch of invites if you don’t have an account.

Involving students in assessment

The norm is for the teacher to write a test, lead a few lessons, describe what will be on the test, and then administer the test, right?  Students are not involved in making the assessment that will determine their grades.  My form of summative assessment changed drastically this year, but it still does not address this issue.  From what I hear from other teachers (you?), there’s a lot of room in all of our classrooms for more student involvement in assessment creation.

The tempered radical recently wrote about the benefits of explaining to students exactly what the point of each lesson is.  It’s the kind of thing that seems so obvious when you say it like that, but this is relatively new research pointing at this stuff.  In the November 2009 issue of Educational Leadership, “The Quest for Quality” references research from 2006 and 2009 to make the radical claim that “students learn best when they monitor and take responsibility for their own learning.”  It goes on to say “This means that teachers need to write learning targets in terms that students will understand.”  Sam Shah wrote about an experience talking directly with students about what it means to think and act like a mathematician that was so powerful for him that he considers it a genesis for himself as teacher.  And meanwhile, I think I’m totally rad for talking with students about what they want out of my class.

It’s incredible that this stuff is new, right?

Some teachers are on to this already.  I read about teachers developing rubrics with their classes, and others having students write questions from which the teacher will select his favorite three, etc.  These teachers are already reaping the benefits:

  • students feel (are) respected
  • students feel (have) ownership of the assessment, which gives them a new responsibility
  • students know a lot about the assessment before the lessons are all over, which seems, you know, better.

These benefits are obvious and supported by research.

So, if you use an assessment scheme based on written tests (like I do), what are the best ways to get some of these benefits?  I want to experiment with having kids write and critique their own questions for sure, since this seems easy to implement and, at its worst, is a form of review.  I already ask them to assess their progress towards their personal goals.  What else can I do?

I am hereby declaring a new goal for the first month of next semester: I will find a way to include each student in the act and process of his or her own assessment, at least a little.  I’m aiming high – I don’t mean that I will include “the class” in creating “the assessment.”  Whew.  There’s something to think about on the 14-hour drive home for Christmas!

Please leave comments if you have ideas.  I just set this kind of big goal, and to be honest, guys, I don’t know how I’m going to meet it yet.


I’m trying out CPM’s Algebra 2 book and so far it’s pretty much fantastic.  I don’t want to give away many details of the lesson plan I used today, since they asked me not to reproduce it.  But I have to tell you about the basic mechanic: polydokus.

A complete polydoku has 4 main sections – one for each of two polynomial factors, one for the product of said factors, and another area for the work.  You can figure it out from this already-solved puzzle:


Of course, much of this data is redundant.  Try your luck at this unsolved puzzle:

polydoku_unsolvedThese puzzles are fun and satisfying.  I tried it in class today and students had the perfect amount of difficulty with them.  I explained the puzzles only as much as I explained them here, and the students seemed to enjoy figuring them out.

The payoff comes in when the puzzle looks like this:


Have your kids solve this polydoku, and then ask them, “Hey, by the way, what’s polydoku_question?”

The CPM lesson went on to have the kids discover remainders, and even connected it to the factor theorem for finding roots of an equation, but I’ll let you ask CPM about that.

You know who’s a polydoku convert?

[pointing emphatically at myself with my thumbs] This guy.

Acknowledging Student Time and Autonomy

Though I’ve always believed it on an intellectual level, I’ve recently begun to understand with new depth that kids are young people who deserve the full respect that all people deserve, and are not in some way inferior.  I have been slowly identifying instances in which I subconsciously considered my time more valuable than theirs, by changing my office hours at the last minute or assigning homework I hadn’t analyzed thoroughly.  The fact that we (adults) require students to come to school for hundreds of hours every year has become newly startling to me.

As teachers, we sometimes claim a kind of ownership over a significant percentage of our students’ lives.  I was surprised to realize that even those students that have done the least work towards the goals I set dedicated 100 hours to my class.  Do you know how many episodes of The Wire you can watch in 100 hours?  Like, a hundred.

Don’t get me wrong.  I work harder in each of my classes than any of my students ever has.  I expect them to do work for me (for them, but whatever), and I don’t feel bad about that.  But this is the first year that I’ve really acknowledged that explicitly to them.

I spent about ten minutes of the first period of my classes this year telling each class why I choose to teach and what I get out of it.  I told them what I expect from them and why I thought it was important for them to succeed in my goals – important enough that I would be requiring them to do it, even if they didn’t like it.  Then, I spent the rest of the class giving kids time to talk and write about what their own goals for the class were.  I was very clear that I wanted them to meet my goals, but, given that those were required, what else did they want?  We spent some time talking about the importance of action steps, and they all came up with some ways to make sure they were working towards their own goals.  I left room for their goals on my syllabus.

Later in the semester we came back to these goals briefly, just to remind the students of the concept and to let them assess their progress.  I never graded any of it, and I didn’t require them to keep track of anything, but acknowledging that they were autonomous with full-member status in the humanity club got us off to a warm and fuzzy start.

I have no research to back this up, but, with this approach:

  • Students are given the opportunity to find something important in my class.  Some of them might have goals like “get better at doodling” at first, but, this year at least, those students got bored with those goals.  Later they picked goals more like “get better at taking notes,” “understand where formulas come from more,” or “get more involved in class discussions.”
  • When students have a personal goal, that is really theirs, with no supervision or grading or judgement, that goal is automatically interesting.
  • The students realize that I value their time.
  • I feel better after a really boring class because of this initial discussion.  The students know what my goals are for them, and they know I’m not just jerking them around.
  • When students do choose to share a goal with me, perhaps to ask for help in achieving or measuring success, I feel much closer to them because of that moment of unguarded honesty.  My dedication increases and my rewards multiply.

Overall, I’m not sure that this is worth the amount of time I spend on it.  Many kids just forget about their goals and go about business as usual, and I only mention them every three or five weeks.  What do you think?

How do you show your students respect?

Bag of Tricks #1 – Index Cards

In “Bag o’ Tricks” posts, I’ll give activities that require almost zero prep, but inject a shot of fun, practice, activity, assessment, remediation, or whatever in a small amount of class time.

This post’s focus is index cards.  My students like them – I think they are just nicer objects than sheets of paper.  These are perhaps my favorite no-prep activities.

Memory (20 minutes)

  1. Each student gets two index cards.
  2. On one index card, each student writes an expression of a given type (e.g. an anonymous differentiable function like “2x+sin(x)”).  Every student must use a pencil.
  3. On the other index card, each student writes a corresponding expression after a given operation (e.g. differentiation – “2 + cos(x)”).  After this step each student has two cards that are connected by the given operation, but not by name or any other property.
  4. In pairs, students swap cards and check each other’s work.
  5. Each student gets another two index cards and repeats the process.  Each student now has a total of four cards, two pairs of linked cards.
  6. Students form groups of four, shuffle their combined sixteen cards together, and lay them out upside down.  The cards are (hopefully) indistinguishable.
  7. The students play memory (in teams of two, or not).  A team flips over one card, and then another.  If they match through the operation, they keep the pair, get a point, and go again.  If the cards don’t match, the next team is up.

This activity is great, after you figure out how to make sure students write problems of the appropriate difficulty.  They need to be pretty easy.  Memory is hard when its just pictures of barnyard animals, you know?  I use it to have students practice derivatives over and over again.  Every time they see, for example, “2x,” they have to think “what is the derivative of 2x, and what might have 2x as a derivative?”  You need a problem that’s easy, but takes lots of practice.  Distributing polynomials, finding logarithms, solving linear equations, etc.  The first time I used this activity I put, like, physics word problems on one card and answers on another.  Let’s just leave it at “don’t do that” and move on, please ;).

Benefits of memory:

  1. A bunch of practice
  2. It’s reasonably fun
  3. Kids write their own problems and solve them
  4. Each student gets the advantage of knowing 2 of the 8 answers right away.  This almost guarantees some success for every student – everyone can feel engaged, even if their skill level is lower than the others’.

Write and Swap (5-7 minutes)

  1. Each student gets an index card and creates an example problem.
  2. Students swap cards at their table (I have tables of two) and confirm that the problems are in the proper form, etc.  Any questions about problem creation are resolved.
  3. The teacher moves quickly and energetically around the room, picking cards swiftly out of kids’ hands and giving them replacement cards from other kids.  This works elegantly – the teacher can move in any pattern, so as soon as problems are written they can be swapped out, but students who need more time may take it as the teacher is passing out cards.
    After this step each student has a new card in front of them, and they don’t know exactly where it came from.
  4. Each student solves the problem on his or her card.
  5. Students swap cards at their table and confirm solutions.  Any questions about problem solution are resolved.

Benefits of write and swap:

  1. Each student gets practice writing a problem, which may involve critical thinking about what is important to include.
  2. Each student thinks about four different problems in a row, but a physical interaction between each problem keeps attentions focused.
  3. Student responsibility is diffused.  Limited responsibility can help students feel safe, which can be important (though students should be fully responsible for at least some work every day).
  4. A peppy teacher can infuse the activity with energy on a slow day by zipping around the classroom in the big card swap.  Carry around a funny container instead of just holding the cards in your hand if you want.

Write and Swap is great for those times when you just want students to practice something kind of boring a few times.  It’s not great for longer problems because the phases get unsynchronized.

Most confusing part (5-7 minutes)

I got this from Science Formative Assessments, by Page Keeley.

  1. Each student gets one index card near the end of the period.
  2. Each student writes, anonymously, the thing about the class that was most confusing, least fun, whatever.
  3. The cards all go in a box and are redistributed, one card per kid.  Page Keeley recommends having the kids literally throw the cards around, but I admit to not being brave enough to try this yet.  It might make this activity really fun… or just add two minutes to its execution.
  4. Kids read their new cards aloud to the class.

The first time I tried this, I wasn’t that impressed with the results, but like any new technique I’ve gotten better at making it succinct and useful.  This activity is mostly to get a quick sense of how your lesson went, if you didn’t have any better way to do it built in.


  1. If a theme emerges, you know, that’s a great piece of information for the teacher.  Write that down on your lesson plan!
  2. You get to hear from every kid in a very low-pressure way.
  3. I imagine that kids who are embarrassed by a lack of understanding are heartened when they (inevitably(!)) hear that someone else had the same problem.

Time-independent assessment at the end of the period

This semester I started giving my students small, focused tests that attempt to isolate a single skill.  Among the many things about this method that are astoundingly great is the fact that reassessment is a snap.  A student can earn credit for a skill regardless of whether he understood it immediately or took 2 months of work to master it.  I can easily reassess his skill level in December, even if we studied the concept as a class in September.

But now there are only two weeks left in the semester, and grades will be due.  I want to be able to explore more material, and to test my students’ skill level with it.  The students want more tests, for goodness’ sake.  And if I give a test with only 3 days of class left… a student who earns a low grade at first does not get any time to improve!

It is clear that there is no way to teach new skills at the end of a grading period and give students a lot of time to become comfortable with them.  Is there a way to teach all new skills at least three weeks before the end of a grading period… and still make the last three weeks interesting and curricularly-advancing?

What do you do when a student fails a test at the very end of a grading period?

The Unfunny Valley

Why is it that I always feel the best after classes that I’ve planned the least?  Is it because less-planned classes can be more organic, following discussions more naturally, accepting tangents?  Or is it because I end up talking more in classes less planned, and don’t have to wait around bored while students practice with problems I’ve prepared?  Clearly, some theories are more complimentary than others.

Of course, when I work for eight hours on 100 minutes of a lesson, I feel great about it.  When I do a TON of work, my lessons are satisfying at reasonable rates.  But there seems to be a sort of uncanny valley that starts at about one hour of prep and doesn’t end until about four hours.  More of an uncanny crevasse.

fun vs. prep time

fun vs. prep time

My work between one and two hours goes towards tiering the lesson, or differentiating an activity.  I can invent or find an activity in this time, but it’s not enough time to make sure it’s great.  I spend this time improving the lesson at the cost of its flexibility.  Since the flexibility was the only thing making the lesson fun, and I haven’t replaced it with anything else that’s fun, I fall into the fun abyss.  After 3 hours I’ve started adding something else that’s fun – an intrinsically engaging activity or demonstration – and start clawing my way out.

Here’s the worst part: I currently spend between two and three hours preparing my classes. Previously in my career, I was hitting the easy sweet spot, at between 45 and 75 minutes.  Now I’m aiming at a harder (higher?) goal – fun and satisfaction, with big doses of student practice, understanding, and interest.

So, am I improving?  The same valley exists neither in the graph of educational content vs. prep time nor in test scores vs. prep time.  On some levels I’m more satisfied with my two- and three-hour lessons, even though they’re less fun.  However, student interest is correlated to the amount of fun I’m having.  And, dangit, so is how much I like teaching!  Will my kids this year have higher skill levels, but like math less?