Spring has Sprung – and so has the French Garden!

So the spring term means two things for my Honors Geometry kids – the technology inquiry project and looking at the French Garden Problem.  So for those of you who are not familiar with both of those I’ll try to quickly fill you in while I talk about how they just happen to so coolly (is that an adverb?  if not I just made it up) overlapped this week.

My Spring Term Technology Inquiry Project is something I came up with three years ago when I really wanted a way to push my honors geometry students into thinking originally while at the same time assessing their knowledge of using technology.  I did a presentation last year at the Anja S. Greer Conference on Math, Science and Technology and the audience loved it.  Basically, I give students an inquiry question (one that I attribute to my good friend Tom Reardon) that they have to work on with technology and then they have to come up with their own inquiry question (which is, of course, the fun part) and explore that with technology and/or any other methods they wish.  I have received some pretty awesome projects in the past two years and I don’t think I am going to be disappointed this year either.

The French Gardener Problem is famously used in my PBL courses at the MST Conference as well.  Everyone who has taken my course knows the fun and interesting conversations we have had about the many ways to solve it and the extensions that have been created by many of my friends – an ongoing conversation exists somewhere in the Blogosphere about the numerous solutions – In fact Tom sent me a link just last fall to a more technological solution at Chris Harrow’s blog. (We’re such geeks).  Great math people like Phillip Mallinson and Ron Lancaster have also been drawn in by the attractive guile of the The French Gardener Problem.  In this problem, the main question is what fraction of the area of the whole square is the octagon that is formed inside (what is the patio for the garden)?

So the other night, after we had worked on this question in class for a couple of days and the students had meet with me in order for me to approve their original inquiry question, a student stops by to discuss his question.  John starts off with, “I can’t think of anything really. What I had wanted to do, someone else already claimed.” (I’m not letting them do a question that someone else has already decided to look into.  So John sits in my study and thinks for a while. I told him that this part of the project was supposed to be the fun part.  I gave him some thoughts about extending some problems that he liked.  He said he had liked the French Garden Problem and thought it was really cool.  So I went back to some of my work and he started playing with GeoGebra.  Before I knew it he starts murmuring to himself, “Cool, cool….Cool! It’s an octagon too!”  I’m thinking to myself, what has he done now?  I go over to his computer and he’s created this diagram:

John's Original Inquiry Question

John’s Original Inquiry Question

I’m asking him, “What did you do? How did you get that?”  He says that he just started playing with the square and doing different things to it and ended up reflecting equilateral triangles into the square instead of connecting the vertices with the midpoints as in the original French Garden Problem.  Then he started seeing how much of the area this octagon was and it ended up that it was……you don’t think I’m going to tell you, do you?

Anyway, it just made my night, to see the difference in John when he came by and the by the time he left.  He was elated – like he had discovered the Pythagorean Theorem or something.  I just love this project and I would encourage anyone else to do the same thing.  Leave a comment if you end up doing it because I love to hear about any improvements I could make.

Looking for the Teacher of Grit

I’m in the middle of working on organizing my courses for the Exeter conference in about a week and something I’m really struggling with is trying to articulate to teachers how they can impart to their students this idea of grit in the PBL classroom.  So I started doing a little research online (besides looking through all of the books I have read on the subject).  I took Angela Duckworth’s Grit Test at her lab’s website (got a 3.63 grit score- grittier than 60% of other U.S. citizen’s my age…hmmm).  Then I started reading some blog posts of other PBL teachers and writers like here on the MAA’s blog which is trying to encourage math students to tinker with problems or here which is more of an all-purpose index of resources to teaching grit. There was this wonderful video of a teacher in NH who created a neat grit curriculum for her 5th grade class (with Angela Duckworth too)

John Larmer of the Buck Institute wrote a really nice blog entry on how project-based learning fosters grit in students. I even found a nice video of Po Bronson, author of Nurture Shock (the book about how parents have failed kids because we don’t let them fail).  This is a short video of how Mr. Bronson believes we should be allowing kids to fail these days.

He says (in so many words) that if kids grow up without learning how to fail, they will become risk-averse.  This is what I am finding in my classroom at times.  The risk-averse kid combined with the fixed mindset kid, combined with the “I-have-to-get-into-college-and-make-my-parents-happy” kid makes the PBL classroom very difficult when you are trying to get them to take risks and be creative.  Add that to the classroom culture that they have been used to for the first 9 years of their education in the U.S. and sadly, it makes for a tough place to foster the teaching of grit.

In fact, on my most recent course evaluations I asked students what they found most challenging about the class and the two pieces that tied for first place were journal writing and

“having to be vulnerable and make mistakes in front of my peers.”

I so want to change that and I always thought that I created a classroom atmosphere where students were comfortable.  I did all of these things that the professionals are suggesting on these websites:

1. modeling risk-taking and making mistakes myself
2. talking about growth mindset regularly
3. ask them to write about positive experiences when they are proud of themselves
4. using class contribution feedback forms (self-report and analysis of class contribution sheets)
5. using strategies where students think of a wrong way before we talk about the correct solution method together.

But somehow, even at the end of the year, their fear of being wrong in front of each other (and me, some commented) is still predominantly what they say challenged them.  So I would say to Po Bronson, where is the teacher of Grit?  What is the secret?  How do I make it so?  Is there a time when it’s too late for some kids?  Most of what I’ve seen on the internet is teaching grit to elementary school children – does the fact that I am teaching high school kids make it any harder?

I finally found this great Prezi created by a teacher named Kristen Goulet which, I know, is geared towards elementary school kids, but I think I could find a way to direct it towards older students.  The idea of having them ask themselves whether their self-talk is “because of me” or “because of other” and whether it is “permanent (i.e. fixed mindset)” or “temporary (i.e. growth mindset)” definitely would help them realize how much of the way the deal with adversity is flexible.  It also helps with seeing how to have a more realistic and optimistic view of a certain situation (and is kind of hard to argue with).

So, I’m still in search for the best practices to teach grit (and apparently so is Angela Duckworth – she admits this in her TED talk), but now I know that it is way more complex than just following a certain number of steps – it has so much more to do with a student’s socio-emotional state of mind. Vicki Zakrzewski’s article “What’s wrong with Grit?” is probably the closest I got to agreeing with someone’s assessment of grit and how to teach it.  I know that I am really good at letting kids know what is important to me and doing that modeling that is important as well.  Undoing what has happened to them before they got to me is a tall order, but I’m not going to stop trying.

Creating a Conspiracy in the PBL Classroom

Any Mad Men fans out there?  I just love some of the characters and the struggles they put themselves through.  In one episode from season 5, called “Signal 30,” Lane Pryce needs to take some clients out to dinner and Roger Sterling is giving him some advice on how to woo them to sign a contract with Sterling Cooper Draper Pryce.  Since he is not an account man, Lane is nervous about landing the account.

Roger:  And then it’s kind of like being on a date.

Lane:  Flattery, I suppose…

Roger:  Within reason, but I find it best to smile and sit there like you’ve got no place to go and just let ‘em talk.  Somewhere in the middle of the entrée, they’ll throw out something revealing and you want to wait ‘til dessert to pounce on it. You know, let him know you’ve got the same problem he has.  Whatever it is, and then you’re in a conspiracy – the basis of a quote “friendship”.  Then you whip out the form.

Lane:  What if I don’t have the same problem?

Roger:  It’ll probably be something like he drinks too much, he gambles…I once went on a five minute tear about how my mother loved my father more than me and I can assure you, that’s impossible.

Lane:  Very good then, and if for some reason he’s more reserved?

Roger:  You just reverse it – feed him your own personal morsel.

Lane:  Oh I see.

Roger: (getting up to leave) That’s it, get your answers, be nice to the waiter and don’t let him near the check.

My husband and I watched this episode about the same time I was having a great deal of resistance in my class to PBL.  I was talking to my husband about how to get students to buy into the notion of learning for the sake of learning where everywhere else in their lives what measures their learning are their grades.  Why would I expect anything different from them if this is the culture they were brought up in?  They depend on their grades to get them into a good college and if their grades are not up to a certain standard, they will not “measure up.”

I get this question all the time from other teachers – about  how to motivate students to find the love of learning and the interest in problems when they do not necessary know the solution methods to find them.  I usually tell them the same things – talking about the values of the class, grading class contribution with a viewable rubric,  grading their metacognitive journal writing, rewarding them with an interesting relationship with a great teacher…OK that might be pushing it.

However, this year is different.  I am having the hardest time trying to let them know what I want from them.  They do the homework, try their best, write down notes, but for some reason it feels different.  It’s almost as if there’s this wall between them and me and I don’t know how to get them to see my side.  I have had this problem with students in the past, but usually with a whole class.  Some of them blatantly are interrupting each other and others are obviously ignoring each other.

Then my husband says, “Maybe it’s like the conspiracy.”  I said, “What?” He said,” You know, what Roger was talking about on Mad Men.  Now, Roger Sterling is no saint (those of you who watch the show know this all too well) and I usually take what he says with a grain of salt.  I also would not ever consider taking advice from him, especially about teaching, but I allowed my husband to continue.  He said maybe what I had to do was build up the conspiracy that Roger was talking about.  I had a real problem with that because I am so committed to relational pedagogy that there’s no way I could lie to or mislead a student about their learning.  But that’s not what he really meant.

I suddenly realized that what had happened was I was teaching a curriculum that I didn’t even buy into.  I had just finished teaching them matrices and matrix operations with some problems that I had written, and it went very well.  However, in the end I did have to do Cramer’s Rule and determinants.  I tried motivating the problems about determinants with the area of a parallelogram, which kept them interested for a while, but in the end, with a 3×3 it was just here’s the way to do it.  I’m not sure that I could’ve expected them to have enough prior knowledge to derive the formula for finding a determinant of a 3×3.  As much as I tried to cover it up with problem-based learning, it was still a curriculum that is antiquated and not necessarily what I felt they should be doing and learning.  I couldn’t hide it any longer.

But we’re caught aren’t we?  Do we change the whole system – college prep curriculum, SAT required math, college expectations – and if so how do we do that? (see ahbel.com for a great article on this and a keynote address called Reflections on a 119 year old curricullum!)  Do we move beyond the required standardized testing material and allow our students to see mathematics the way we see it?  Yes, that’s the conspiracy – that’s what my husband was talking about.  When kids complain to me, I will “smile and sit there while they talk” knowing that I’m going to try to get something that we have in common.  “Do you hate solving a system of three equations with three unknowns with a determinant? Oh yeah, I did too in high school.  Wouldn’t it be great if we could do something else?  What else should we do?  Let me find some other problems that might be interesting.”  We have the same problem (literally and figuratively), now we’re on the same playing field having similar motivating factors.

And you know what?  I don’t think it would be the end of the world if they’re not revealing and you reversed it.  We are allowed to say to them that we don’t understand why we are still teaching this and these would be my reasons for taking it out of the curriculum – part of your own personal morsel.  It might actually bring you closer as a class and have you talking about how your hands are tied and we have to get through this “together.”

Yeah, there are little tricks that can be learned and carrots that can be used to get students to do what you want them to do, but in PBL, that’s not the point.  There is very little for them to mimic because it is based on their prior knowledge.  They are the ones who need to move the curriculum forward.  So in a nutshell,

  1. Take action – Get to problems in order for students to start feeling empowered and active in class.  Once they see that they are capable of a great deal on their own, it is amazing what they can accomplish.
  2. Create relationships – be sure that you are being reciprocal in your attempts at problems and valuing theirs.  The concept of Relational Trust and Authority are huge parts of a PBL pedagogy (Boaler, Bingham)
  3. But make sure that you are at least somewhat in control in the end because we are, at least for now, still responsible for making sure that some understanding of what we might consider unnecessary skills, for their next courses or future use.

As Roger said, “Get your answers, be nice to the waiter and don’t let ‘em near the check.”  Create that conspiracy.