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!