Posted by: lboudreau | September 30, 2011

I wonder

So much for the goal of blogging every week…  Seems I can either live and teach, or write about it.  You’d never know I was on my 18th journal at home.

It’s been a re-energizing week.  Between catching up with Jill Gough, and reading new blogs (see blog roll to the side), and a great session with Heidi Hayes Jacobs at Trinity School, I feel engaged in a larger world again.  (Although not as tech-savvy as some– how do you link a Twitter conversation on a blog?)  Lost much sleep thinking about the classroom and larger educational world.

Two questions.

1.  What’s up with tech use and math classes?  In so many professional development experiences I’ve had, the extent of tech use in math is largely limited to the graphing calculator.  Not that I’m complaining.  It’s an elegant, powerful device, and has done just about anything students have needed, especially since CAS crunches variables, and we can go into some deep, d— thinking with it.  Wolfram-Alpha crunches the same (and then some) without having to spend $150, but it’s not quite as portable, depending on the hardware used, and you can’t bring it into an SAT room.  I like the TI CAS software, but again, not so portable.  (I do have issues with the lock TI has on the market, but that’s beyond my control at this point, so I let it go.)

So, is it the pervasiveness of a useful device that’s led to less use of other tech in the math classroom?  I think of my humanities colleagues and all the web 2.0 applications I’ve seen them use.  And the Skyping.  And polling with cell phones.  And Ning and blogs to communicate an build community.  I’ve seen far less of that with math, including in my own classrooms.  Don’t you think math classrooms would lead the way with technology?

It’s a pedagogy thing, rather than a subject thing.  The way we teach math has been so skill-centered that a calculator has sufficed.  When we change our focus, I suppose we’ll change our tools.  I suppose I need to do that, too…

2.  I’m teaching Geometry this year with College Preparatory Math (CPM).  My jury is out on it so far, although I’m willing to see where the year leads before the jury comes in.  Part of my problem at this point is judging when to let students continue and when to intercede.  When do I need to interject a comment or stop the class to discuss answers as a class?  Yesterday, as the class looked at angles (finally), they had to develop a conjecture about vertical angles.  Almost all the groups determined that vertical angles are congruent, and some even used the word (although we’re not picky about the vocabulary quite yet).  One group conjectured that vertical angles are opposite each other.  True, but rather misses the more important point.  Should I have led them into another understanding, or is it okay to let that slide for now?

It’s hard to gauge the level of my participation in this learning.  I don’t like the idea of misconceptions rattling around in heads, never addressed, or addressed so late that students struggle to re-learn.  I’m all about the benefit of the struggle to learn in the first place, and I love student-centered learning, but I don’t want to add confusion–math can be confusing enough.  So, do I step in?  And how will I catch when I need to step in?

When I’ve crafted my own curriculum I’ve had a better sense of the journey.  CPM is planned and scripted for the year, and I’m told you just need to get through the year to see the full effect.  I’m not particularly patient about it.


Posted by: lboudreau | August 20, 2011

Fractal Day

I’ve wanted to do this for years.  For the first day of school, no syllabus.  Instead, we do math.  We’re making a big old fractal.

In each of five classes, we start by looking at images of all sorts of fractals, and how complex and lovely they can be.  Specifically, we look at the Koch Curve, and how each iteration builds toward infinite length bounded by a finite distance (which seems a metaphor for a person’s life, and rather profound, but that’s a blog for another day).  We guess what’ll happen next.  We explore self-similarity a while.  We apply the word “iteration” to sports practice.  Some students mention that it looked like the side of a snowflake, which moves us into the Koch Snowflake (smart students).

Next, we move to Sierpinski’s Carpet (related to the Triangle, but easier to draw on graph paper).  We guess what comes next.  We imagine end behaviors.  We admire it’s filigree state.

Finally, we break out the markers.  Life is so often about the arts and crafts. (Sorry for the fuzziness– camera’s acting up.)


Tables of students get large sheets of graph paper, off the presentation pad.  They choose their colors, and duplicate the Carpet on the third iteration.  That level is essentially a 27X27 block square and the paper is (conveniently enough) a 27X34 block sheet (something I didn’t count out until the morning we started, which caused a momentary panic flash, but all was well).  No table finished its fractal, so we’ll resume Monday.

But we won’t be done even then.  I teach girls, who are generally relational learners, so why not connect all my classes together in some sort of uber project?  Each sheet will be connected in a larger fractal; I think we’ll have three 3X3 sheet fractals when all is completed, and we’ll have to find some place to hang them.  Can’t wait.

Now this would be fine if it were simply a hook into the math, something for our first day together.  But it was better than that.  Sure, we learned math, but I learned a great deal about my students.  It turns out that there are numerous ways to look at that fractal and to get it onto paper.  It was enlightening to see how each student created patterns within patterns, how she counted and checked work, how she placed her body around the table to orient her thinking to the projected image.  Or how she didn’t.  We broke out the White Out a few times.  It was a very kinesthetic, very spatial kind of day and I had a chance to see a number of strategies at play.  (It would have been better if I’d known all these students already and could link learning approaches to individuals.  As it is, I only have a general sense for each class.)

I also plan to come back to this day at points during the year.  These two simple fractals are rich in topics: probability, sequences, area and length.  I’m looking forward to seeing what we do.

Posted by: lboudreau | August 20, 2011

Inspirational Syllabus, My Attempt

Inspired by John Burke’s entry for the challenge and his initial inspiration from Dan Meyer, I decided to play with my syllabi.  Starting a new school, it seemed a perfect opportunity– not a whole lot developed yet for new courses and a clean slate with students.

I developed a general template for all classes, and tweaked as necessary for each of four courses.  This is the one for Honors Pre-Calculus.


John used common threads between people for his syllabus; I used images, because I find making math as visual as possible really helps it make sense.  These images link the Fibonacci Sequence, the unit circle, and conics– all topics we’ll study this year, but also linked to the idea of progression.  It seems clear that sequences progress from one term to another, with some pattern in play.  The unit circle also involves progression as the point rotates around the origin along the circle, and the angle changes magnitude (and maybe direction).  Conics, too, illustrate progression if we consider the shapes as the intersection of two cones and a plane, and the plane changes its approach angle.  (Conics also illustrate a progression of eccentricity, but I may not bring that up when we talk about this syllabus.)  I’m hoping students will find their own connections between the images, and we’ll have an interesting conversation, but at least I have something in my back pocket.

The final section on expectations may be the most important for my classes, but I find it’s most effective to bring it up last, after the nuts and bolts.  These items were suggested three years ago by Marsha Little when I was hashing out what I considered the most important elements of learning.  I’ve played with varying degrees of transparency over those years (sometimes students prefer not to know what’s going on behind the scenes, and sometimes they do).  This is is what I like.

I plan to talk about this on the second day of class.  On the first day, we do math (see following post).  I really like Dan’s idea: hook ’em right off.  Let’s see what sort of interest builds.

It may not be the Document of My Dreams (it’s still too long), but it’s a step closer, and I think we can use it as a springboard for learning this year.

Thanks, John.

Posted by: lboudreau | August 13, 2011

Fresh Canvas

I’ve always been influenced by the rooms in which I’ve worked, read, and, well, eaten (do like a good restaurant).  I’ve almost always been organized.  (Almost.)  Marry the two and you have a teacher who needs a well-ordered space, at least at the start of school.  I feel I can let students tear loose in creative ways if I know what’s on hand to help the process.

At a new school this year, I start from scratch, moving two pick-up’s worth of materials and books (into a room that’s about the size of Wyoming, fortunately).  The kites are up, books on shelves, non-tech materials to hand, and manipulatives stored for just the right moment.  Took five days, but it’s done.  It’s a pleasure to sit in here and anticipate what we’ll do this year.  Of course, once students arrive, the peace and quiet evaporates, and we get down to the business of using this space.  It makes a heck of a mess, but that’s often where learning occurs.

So for now I enjoy order and serenity.  In less than a week the tsunami rolls in and we’re awash in school.  I wonder what learning we’ll imprint on this canvass.




Posted by: lboudreau | August 2, 2011


I start to feel like a department chair.  We’ve had to hire a part-time teacher for two sections of Algebra II.  The school, bless them, hasn’t folded me into that process, instead giving me a bit more summer, but I did get to interview one of the candidates.  I enjoy interviewing candidates; I’m always interested in how they discuss education, and in hearing new ideas.  I feel, too, that even if the candidate isn’t hired (which happens), we’ve all had some sort of teaching encounter– an odd sort of professional development.

We hired the person I interviewed.  Fresh out of Agnes Scott, she’s new to teaching, although experienced in finance, and she’s going to be quite a fine teacher: eager, creative, and energetic.

I may be her official mentor, but I’m new to the school, and they may assign her to someone who, say, already knows Haiku.  Regardless, I will be a mentor to some degree, and I take it seriously.  While there, I had the opportunity to help Lovett develop its mentor teacher program, and it’s become a viable, useful program.  It’s also portable; I freely admit that my best ideas are stolen, and I plan to “borrow” from the Lovett program as we welcome a new math teacher to AGS.

I find this terribly rewarding.  After a summer as a mentor teacher for Breakthrough Atlanta, I am more than ever convinced that good teaching starts with good mentors– from a walking tour of the People and Places You Really Want to Know, to suggestions about dealing with students, to brainstorming creative approaches.  Teaching is at heart a human endeavor, not done best in a vacuum, and everyone benefits– everyone learns– when mentoring works.  I think this is especially true with new teachers; it was certainly true for me– I feel nothing but gratitude for colleagues who showed me the ropes, pushed me to another level, or simply paid attention to what I was trying to do.  (And due to their efforts, I have a library of good ideas– see earlier comment about theft.)

So, the year starts with teaching.  We meet tomorrow for that walking tour and to share curriculum.  And start talking about students and classes.  Makes me happy.

Posted by: lboudreau | July 21, 2011

Lessons Learned

We’re in the final teaching week of Breakthrough Atlanta.  Eight math interns gave post-tests to students yesterday, and graded them all within an hour.  The results are in: 7th graders saw a fairly significant improvement from the pre-test, while 8th graders saw a much more modest one.  In fact, there was some question for a while as to whether 8th graders had seen any improvement at all.  This, after all the energy and good teaching in and out of classrooms.  All very discouraging.

Now, Breakthrough is in large part a preview of concepts students will see this coming school year, so having a number of students who haven’t yet mastered slope or simplifying radicals (or even adding negative numbers) may not be a surprise.  And these interns/ infant educators haven’t mastered teaching.  They are very good, though, and students showed they knew a great deal in class, so we all had high expectations for post-test results, and found ourselves dismayed.

The next day, though, I think there are lessons to learn in this.

One, although our pre- and post-tests are not as long and rigorous as the CRCT, they are standardized tests, and you just can’t measure everything learned by using a standardized test.  Just as important are the eagerness to answer questions in the classroom, or the statement directed at one intern: “You make math so easy,” or the feedback at conferences that students love math now.  What students share in a class discussion illustrates just as deep an understanding as the paper and pencil snapshot, perhaps taken on an off day.  But these other indicators are harder to quantify than a test, so they often get discarded.

Two, it’s hard to know, really, what a student has learned.  It’s the bottom line in my own classroom, but I never feel like I have an accurate measure of it.  How much more difficult is it for younger teachers?  It’s like I have to look at a portfolio of indicators: tests, questions, tutorial time, projects, engagement in class.  (See lesson #1 above.)  So, how do you know you’ve prepared a student for the next level?  How do you know if that student is truly prepared?  If that’s the case, a poor showing on a test might just have to be a surprise.

Three, on the flip side, you have to take some things on faith.  These interns have middle school students, for whom it’s a miracle that learning takes place on some days.  Interns have planted seeds that they will never see sprout, and it’s important to acknowledge that.  Otherwise, it’s easy to lose all hope of making a difference.

Four, about the test itself.  I made the tests based on input from interns, but even so, it didn’t totally reflect what students had done in class.  Did that matter?  I’ve grappled with this idea that all four interns in each grade level should be in lock step during these weeks.  They haven’t always given the same lesson, or even covered the same topics, on the same day, and that’s worked well.  Each intern had the flexibility to go at the right pace for their students or to interject an activity.  In the end, they got to all of the standards.  (Part of the issue is that the standards are general in places, and not all interns got to, say,  simplifying the square root of 50.)  So, this test of those standards is relatively fair, but not totally accurate.  But isn’t that the nature of the beast?  A test that’s totally fair to every student may be too simple for some.  I think there’s something inherently underhanded– in a good way– about asking kids to combine concepts in new ways on a test.  “Thinking on your feet” isn’t on the list of NCTM Standards, but it could be (maybe they just label it “Critical Thinking”).

Five, this leads to the question of common assessments.  If I were department chair (and soon I will be), I think I’d look harder at purposes for the tests within the context of a class than at how tests match from teacher to teacher.  I would be concerned that, once we all agreed on a test, then we’d all agree on how to teach to it (and only teach to it), and leave a lot off the table.  I acknowledge the importance of having some degree of common experience for students within a course, but there needs to be room for teachers to go to their strengths, as these interns have.  In the end, we all met the goals of the program.

Today, the day after, interns all report that students are engaged in their final math projects for Museum Day.  I witnessed it myself.  Interns feel better.  I feel better.  And we keep at the task of teaching and learning math.

Posted by: lboudreau | April 12, 2011

Parsing Education

I’m going to be a mentor teacher for Breakthrough Atlanta, working with college-age students who will work with Middle School students.  The curriculum follows the standards from the Atlanta Public Schools.  Each element is rather new to me, which makes the work exciting.  I’m especially excited about training future teachers.

We’ve met to discuss the basic outlines of the program: goals, people, timing, forms.  I was struck by the detail of the learning standards; my public school colleagues have to tease apart their lesson plans in ways I don’t need to.  And I wonder.  Does that really help student learning?

I truly understand the need to have teaching/ learning goals for each lesson.  And I understand the need to standardize curriculum across a broad swath of a district– you don’t want some students having seen substantially more or less than their peers.  But from the outside,  it seems that planning so many detailed requirements consumes much time and energy.  You must be sure that you covered EXACTLY this particular type of line, or solution, or step.  Wow.  Does this really serve students?  Is it a realistic plan?  Is there time for something off the grid?  Something inventive?  Maybe something generated by students themselves?

Do students really learn all this?

Is there some sort of happy medium, in which students get a fair deal, and some space to play?  And teachers get a livable curriculum plan.

Posted by: lboudreau | April 11, 2011

Budding Architects

Today is Conics Test Day!  Woo-hoo!  Classes full of students working as architectural teams, designing a new building for the Louisiana Museum of Modern Art in Copenhagen, Denmark.  Not that I’ve been there,but it seems a likely location for fluid design.

Students already planning (overly?) elaborate designs.  Ellipses and hyperbolas and parabolas and lines, drafted on large sheets of graph paper, described in polar, parametric,and rectangular form (yay! analytic geometry).  The mind boggles.  Can’t wait to see their final work.

As an aside, I’ve lost the thrill of digital production–it’s simply been too awkward to draft and grade.  It’s probably just that we don’t have the right platform, but it’s hard to research new platforms during the school year.  So, for now we use paper, and I keep digital production on the list.  I rather like the large-format hand graphing, too.  I think engaging larger muscles may help to reinforce the mathematics.  (Maybe not, but it’s my story and I’m sticking with it.)

Here are images from the days:


The Big Work

Hard at work, or hardly working?

BIG Graphing

Ellipse or Hyperbola?

Posted by: lboudreau | March 5, 2011

Second thoughts

Or not exactly second thoughts, but re-evaluating the algebra test.

The test went very well.  I listened in on great conversation, as they talked each other through each process, and caught each others errors with negative signs, and shared factoring strategies.  For the most part, they got even the hard equations right.

Feedback from students– written notes requested on the back of the last page– tells me that they REALLY likes this assessment.  Most thought it a good balance of easy and hard (a challenging test if it had to be taken individually).  Most liked the lively nature of the test event itself.  One student wasn’t so sure about it; she thought they were making mistakes, but no one was listening to her.  She was right, there were two class-wide errors that she might have caught.  But the class average was 90. something, so I feel like that wasn’t a dreadful problem, and will actually be a conversation topic when we debrief the test after break.

But as I graded today, I started thinking about the next round of issues.  Have I fairly and accurately measured what students can do?  Could they, individually, repeat any of those problems if I gave them out again?  Was this a gauge of long-term learning and retention, or a quick and easy way to test at the end of the chapter?  Could they mix it up with even more challenging problems?

I sat in a math colleague’s class room today to proctor the SSAT.  I always peek at what’s on the walls– it’s all so interesting.  I saw a couple of algebra II problems that asked for factoring skills we haven’t used.  Not a surprise– there are so many skills, deciding what to teach is a lot like smorgasbord eating, and some days it feels like a bit of a crap shoot.  But I wondered if my students would be able to tackle them.  Probably three students, but isn’t my goal to bring everybody along?

In the end, the question is, did I choose a good strategy to help my students with deep learning?  The problem is that, things that seem good on the page and that fit innovative designs, may have lackluster results.  At least in the short term.

Posted by: lboudreau | March 3, 2011

Algebra Test

The test is made.  Testing now.  Pictures live>>>>>

Sean, the Expert

The Test


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