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?

Painting with Functions

I recently read (parts of) Paul Lockhart’s 2002 essay about math education.  His critique is essentially that we are taking the art out of math by forcing students to focus on every little mechanical subtlety before ever letting them create their own mathematics.  Faced with an upcoming unit on polynomials, which I am frankly dreading because I can’t think of a way to make dividing the things interesting, I decided to spend 30 minutes of a class letting the students play at math, to make creations of their own.  I started them off with the following ggb file (actually I started them with a  set of such files, one with two factors, one with three, one with four, and this one with five).  Drag the orange lines around.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

I gave them this list of possible questions to get started, but I emphasized that if they found anything fun or interesting, they should feel free to explore that instead, to branch off.  Then, for the next 30 minutes, I walked around the classroom admiring what the students were doing, reassuring kids that they really didn’t have to follow any particular instructions, and giving geogebra tips to kids that wanted to move beyond my initial setup.

Results were mixed.  At the end of the class period I had some students who were extremely happy with their creations.  Some students were upset that they were not able to create anything that they thought was cool.  Several students approached real mathematics.  One boy noticed that if he put an even number of lines on top of each other, the curve didn’t cross the x-axis, but an odd number of lines would cause the curve to cross, and furthermore that the more lines were on top of each other, the bigger and flatter the flat part would be (e.g. the vertex of a quadratic vs. a quartic).  A girl in the corner of the classroom, after changing some of the linear factors to quadratic factors (and later to a sine factor), and changing my original curve from a product of functions to a quotient, noticed that when the dividing function was allowed to reach zero all sorts of crazy stuff happened.  Zero students created a formal theorem and proved anything about their observations.

This glimpse of a radically different kind of math education was startling.  Some students were really just finger painting, dragging lines around and randomly changing stuff to see what happens.  Other kids were trying to create particular effects.  Some kids would have continued for another hour, and others were bored and frustrated after 15 minutes.  All of this behavior seems a lot like an art class in 1st grade.  If these kids had math once or twice a week since elementary school, and it was taught like an art class, do you think they would be up to proving theorems of their own by now?  I don’t mean anything revolutionary – I don’t expect we could ever turn every student into a new branch of mathematics – but don’t you think they might be interested in dividing polynomials by 10th grade just to see what happens?

Click here for the GGB file.

A footrace: slope and systems of equations

I taught a lesson about slope and systems of linear equations after the style of Dan Meyer’s “What Can You Do With This” series.  I videotaped a footrace between myself and some of my students, and showed it in class.

I tried to follow several of Dan’s ideas about a good video segment like this.  There is no clear focal point specified by the video, and no imposed frame of reference.  I decided not to show the beginning or end of the race in the initial video with the hopes that the omission would incite interesting questions.

The startup question was simply, “who wins?”  Some students immediately identified the runners that hit the right edge of the screen as the winners, and a few asked where the finish line was.  My plan at this point was to let some controversy mount and then steer the conversation a little, but to my delight, in the midst of the conversation, a student in the back suggested we just figure out who is going faster.  So, our question was narrowed.  I took the opportunity to point out that this is what problem solving is about – figuring out what questions to ask.  I somewhat artificially changed the student’s question to “how fast are the different runners going?” and challenged the class to figure it out.

Which is actually a pretty complicated task, given only the video above.  I was using the quicktime player to show the video, which has a pretty great slider control that can practically move between individual frames.  But when some students suggested that we need a timer, and another group said that we’d also need a frame of reference, I pulled up a new version.

I also had a video with just a timer and another with just the line ready in case they only asked for one of them.

From here, the conversation led itself at what felt like a fantastic pace.  There was some controversy about whether or not we could really find a speed by just finding the slope between two points, and about what points to use, and whether or not it mattered.  We checked with two different kinds of calculations, and got similar speeds, so the kids were convinced (though this sort of proof makes me feel a little dirty).

Somebody said, “OK, so who wins here?” and another student responded, “It depends on where the finish line was.”  Again, I pointed out the great problem solving going on here (I like to emphasize that coming up with a question is just as important as being able to answer it).  You can guess where this is going – we found the point at which it would be a tie using a system of linear equations.

I gave the video to the kids to play with themselves on individual computers.   For an honors assignment I asked them to figure out how much of a head start I gave the other pack of racers, in feet, and several of them chose to tackle the problem.  I considered the class a success because the students liked it, and I had everyone practicing solving systems of equations with interest and a purpose.  And for those students that steered the conversation with narrowing questions (and many more with viable alternate questions) it was good practice problem solving.  The lesson took 30 minutes.

There are several ways to improve this lesson.  I didn’t have video of the finish or the start, to give the class closure, the proof that they were right.  That’s not always bad, but it would have rounded this lesson off better, and I think it’s important to have kids leaving with as much satisfaction as possible.  Also, some of the students in the race were also in my class.  They knew the outcome of the race beforehand (to a finish line not specified in the video), and they knew that I had given them a head start, etc.  I can’t wait to use this video next year (although I made the mistake of including a freshman who will probably be in my class eventually).  This definitely affected our conversation in class a couple of times.  Then again, maybe it’s worth it to have actual class members running in the video on screen.

This lesson inspired by Dan Meyer, Mr. Sweeney, and a newfound category of interesting situations for class.  Please leave comments, criticisms, comparisons, and suggestions for improvement!

Interesting questions aren’t enough without scaffolding.

For one thing, you might be wrong about interest levels.  Today I stretched a 10-minute conversation into a 40-minute bore-fest by failing to anticipate low interest levels.  I was not prepared to scaffold what I thought would be a self-directing conversation.

I started the class with a warmup of some pretty easy questions.  “If a car moves at 50 mph for an hour, how far does it go?”  And also “for one minute, for one second, for 0.1 seconds?”  The students could solve these problems with little difficulty – they handled unit conversions like pros.  Then I asked, “how far does this car go?”

I wanted to give them a crisis, that old methods couldn’t solve.  I thought this would motivate at least the usual level of excitement in our class discussion, but the kids didn’t care.  Immediately, eyes started wandering around the room, and wrists came up to support bored heads.  Really?  But I was so sure this would be cool.  190 mph is pretty fast, guys!  And our old methods don’t work!  Cool!  Right!?

I prodded.  “Can anyone think of any way to estimate how far this car went?”  I got a few answers: vaguely half-formed references to a physics class, maybe (vf-vi)/t, oh maybe that’s not right, I don’t know, you can’t figure it out, why would you want to figure this out?

At one point someone said it would be nice if we knew how long this took, so I switched to the video with the overlaid timer.  Attention spiked at this, but we spent almost 40 minutes struggling with this question, and attention faded again quickly.  My hope for the period was that we would come up with some version of numerical integration, or approximation by differentials, etc.  Something like “we can approximate how far the car went between seconds 5 and 15 by using an average speed.”  Our actual product was pretty good – the students eventually made a stab at the average velocity of the car over the whole 50-second period, and multiplied by 50 seconds, to find 1.4 miles.  But it took so long that we didn’t have time to refine or generalize the concept.  I could have made this lesson much better.

Ways to Improve

  1. Prior discussion of some tools to help us with this sort of problem.  I wanted to go from this problem to using differentials to approximate square roots, but that ordering might be backwards.
  2. When the students were just staring blankly, I should have asked, “Well, did the car go a thousand miles?”  They would have been able to see that the car did NOT go a thousand miles, and I could ask them how they knew.  There are any number of focusing questions in this vein that do not give away the answer or the process.
  3. I could have just asked, “how far did it go between the 5- and 6-second mark?”  From there the students would have an opportunity to generalize from a smaller problem to a bigger one.
  4. Video that includes terrain outside the vehicle would no doubt be much more impressive.
  5. Any others?  What intermediate steps could I interpose to lead my students to a numerical integration?  Please leave more ideas in the comments!

Looking back at the lesson, I’m a little appalled that I didn’t think of 2 and 3 during the lesson.  I was so worried about saying too much and allowing the students to get through without really processing anything, that I was paralyzed and led this boring class.  Ugh!  Have you ever had a class like this?

The Videos

Assessment, One Skill At a Time

I recently realized that I was destroying some of the information that my tests collect.  I was averaging scores of multiple questions together, blending a student’s performance in different areas into a single, summative score. Instead of keeping the information that Johnny could multiply matrices perfectly (100%) but couldn’t really find inverses (50%), I was telling Johnny, “Johnny, you’re at about 75% in this class!”

And so I hit my first personal inflection point of the year.  I stopped averaging scores, and started telling students (and parents) about their strengths and weaknesses in specific skills.  For the most part I am following Dan Meyer’s example, described briefly at, and working with a vague idea of what Hans, a logic teacher here, does.   Instead of getting back a single big score for a month of class, the students get 10 or 12 separate scores.  Check out Dan’s blog for more details.

The new system immediately started helping in three important ways:

  1. Student motivation increased (dramatically in some cases),
  2. Remediation became more informed, and
  3. I started getting feedback that helps me streamline lessons before I give them and assess their efficacy after I give them!

1. Student Motivation

After switching to this new grading system, I have seen an increase in motivation in almost all of my students.  I don’t know if they like the check list, if they like to be recognized as masters of skills (and they are masters!), or if the simple act of breaking the course down into manageable chunks is what is doing it.  But it’s great.

A student that works to improve a single skill, and gets a higher grade in that skill, feels a sense of accomplishment immediately, even if he has five other skills to improve over the next week.  The change in the way I see students catching up is actually astonishing.  They can more easily see that they can do it, and they love it!

Furthermore, students that have earned 80% or 90% already seem motivated to earn the “master” designation in all of the skills on their list.  My experience with students in prior years has almost never been “I have a 95% already – can’t I please take the test again to get a 100%?,” but now it’s practically across the board.

In my old, averaging ways, I wasn’t giving my students this kind of specificity and manageability to work with.  Students with failing grades simply got a big fat “60%” on the top of a unit test.  Now they get “if you work on matrix multiplication, you’ll be at a passing level,” or “you are a master of linear equations – what did you do to get so good with those?”

2. Informed and Focused Remediation

When a student comes in to my office hours now, I can pull out my grade book and see that they aren’t yet passing in skills 13 or 18.  Since these skills are more or less independent of any other skill (an important feature of this program), we can get down to the students’ misunderstanding much faster.  Also, it’s natural to focus on skill 13, and then 18.  There’s no pressure to do everything at once for a single makeup exam that will re-test every skill simultaneously.  Especially for kids who perceive themselves as bad at math, I’ve seen an increased willingness to come to my office hours for help.

3. Formative Assessment

When I give a test, there are a few intro-level questions and a few master-level questions, and also a pretest question.  These are tests of skills that I have perhaps never mentioned before, or mentioned only briefly.  I let the students know that if they score well on these pretest questions they won’t have to take them in the future, but I don’t expect them to know how to do them (how could I?) and that they will absolutely not be penalized for doing them wrong, or just leaving them blank.  I estimate that it takes between 3 and 7 minutes to give a pretest question.  For that price I get a preview of my students’ current knowledge, before I plan a whole class about something they already know or plan to skip over something that they don’t know at all.  I also get a measurement of my skill as a teacher when the students take a test on the same concept after my lesson.  Knowing the difference between “my kids all aced this skill (but they learned it last year)” and “my kids all aced this skill (and they never even thought about it before this class)” helps me rate my lessons.

I recommend it, guys.  The switch is pretty easy, especially at the beginning of a grading period.  Writing tests is easier.  All you have to do is separate the skills you most care about (ok, this is hard), and then stop averaging!