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.

Tracking, PBL and Safety in Risk-Taking

I’ve been giving a lot of thought recently to the idea of “tracking” in PBL, mostly at the prodding of the teaching fellow I’m working with this year – which is so awesome, of course.  Having a young teacher give you a fresh outlook on the practices that your school has come to know and accept (even if I don’t love them personally) is always refreshing to me.

I have taught with PBL in three different schools – two that tracked at Algebra II (or third year) point in the four-year curriculum and now one that tracks right from the start.  Anyone who has done Jo Boaler’s “How to Learn Math” course has seen the research about tracking.  So the question that my teaching fellow asked me, is why do we do it.  The answers I had for him were way too cynical for a first year student teacher to hear – “Because it’s easier for the teachers to plan lessons and assessments.” “Because the class will be easier to manage, as well as parents.”, etc.

In fact, I would have to say that in a PBL math classroom the experiences that I had with the heterogeneous groupings ended up being really advantageous for both strong and weak math students.  Here’s a great quote from a weaker student in a heterogeneously grouped math class, who was part of my dissertation research (that I have used before in presentations) when asked what the PBL math classroom was like for her:

“You could, kind of, add in your perspective and it kind of gives this sense like, “Oooh, I helped with this problem.” and then another person comes in and they helped with that problem, and by the end, no one knows who solved the problem.  It was everyone that solved the problem.  LIke, everyone contributed their ideas to this problem and you can look at this problem on the board and you can maybe see only one person’s handwriting, but behind their handwriting is everyone’s ideas.  So yeah, it’s a sense of “our problem” – it’s not just Karen’s problem, it’s not just whoever’s problem, it’s “our problem”.

This shared sense of work, I believe, rubs off on both the strong and weak students and allows for mutual respect more often than not.  Even my teaching fellow shared an anecdote from his class wherein a stronger student had gotten up to take a picture with his iPad of a solution a weaker student had just been in charge of discussing.  The presenter seemed outwardly pleased at this and said ,”He’s taking a picture of what I did? that’s weird.”

This mutual respect then leads to a shared sense of safety in the classroom for taking risks.  Today I read this tweet from MindShift:

I don’t really read that much about coding, but when something talks about risk-taking, I’m right there.  In this article, the student that decided to go to Cambodia and teach coding to teenage orphans makes a really keen observation:

“Everybody was a beginner, and that creates a much more safe environment to make mistakes.”

So interestingly, when the students in a classroom environment have the sense that they are all at the same level, it allows them to accept that everyone will have the same questions and opens up the potential that all will be willing to help.  I don’t think this has to be done with actual tracking though – I think it can happen with deliberate classroom culture moves.

I got more insight into this when asking some students in my Honors Geometry class why they don’t like asking questions in class.

“It seems to not help that much because it shows others how much I don’t know.”
“It only allows others to feel good about themselves instead of make me feel better that my question was answered.”
“If someone else can answer my question then they end up getting a big head about it instead of really helping me understand.”

I was starting to see a trend.  Now, this was not all kids, don’t get me wrong, but it was enough to get me concerned – This reminded me of a great blogpost I read by John Spencer (@edrethink) called The Courage of Creativity in which he write about how much courage it takes to put something creative out there and fail.  In mathematics, many students don’t see it as being creative, so hopefully John won’t mind if I change his quote a little bit (since I am citing him here, I hope this is alright!)

“All of this has me thinking that there’s a certain amount of courage required in [risk-taking in problem solving]. The more we care about the work [and are invested in the learning or what people think of our outcomes], the scarier it is. We walk into a mystery, never knowing how it will turn out. I mention this, because so many of the visuals I see about creativity treat creative work like it’s a prancing walk through dandelions when often it’s more like a shaky scaffold up to a mountain to face a dragon.”

Thanks John!

Minimizing Shame in the PBL Classroom…and maybe Daring Greatly?

I recently read a blogpost by one of my favorite authors, Brene Brown, of TED talk fame, and the author of a great book about vulnerability called Daring GreatlyIn her blogpost Brene wrote about some reactions to a comment she made on Oprah Winfrey’s Super Soul Sunday show where she talked about shame in schools about which she received a great deal of criticism in the blogosphere and on twitter.

I kept reading as I was shocked that anyone would be offended by anything that Brene Brown could say – especially teachers.  She has always been extremely inspiring and very supportive of teachers – as a teacher herself, her book, Daring Greatly, has a whole chapter on how schools can support a community to come together around vulnerability and become closer and foster creativity and innovation in this way.

However, she talks about the research that she has done about learning and teaching.  She says,

“As a researcher, I do believe that shame is present in every school and in every classroom. As long as people are hardwired for connection, the fear of disconnection (aka shame) will always be a reality. ..Based on my work, I do believe that shame is still one of the most popular classroom management tools.”

Think about it.  When you talk to adults about their memories of school, and specifically math classrooms, many people will tell stories of being embarrassed or humiliated about getting something wrong, about feeling less than adequate or unworthy of being in the class they were in.  Even if the teacher was not doing anything deliberate, if a student has the courage to answer a teacher initiated question and get it wrong, the response that is given can make or break their self-worth that day.

I’ve been giving this a lot of thought in the context of the PBL Classroom – How are we supposed to be teaching students how to take risks and not be afraid to be wrong and make mistakes in their learning if they have this fear of shame that is so deeply entrenched in our culture?  Especially in mathematics classrooms, how are we supposed to undo so many negative experiences that may have affected a student’s ability to allow themselves to be vulnerable and learn in this way?

PBL relies on the fact that a student is willing and able to make connections and conjecture regularly – numerous times in a class and on their own during “homework” time.  Being wrong and uncertain is really the norm and not the anomaly in this classroom.  As October rolls around and I hear more from students (and parents) about the discomfort they are feeling, I really do understand how different this is for everyone.  However, I do think we need to rely on the fact that students can be resilient and strong when pushed to try new things and to learn in a way that is good for them.  It is just that resilience that will make them better leaders, learners and more creative in the work force later on in life.

In talking to some students recently, I asked them where they thought they would learn more, in a classroom where it was laid out for them what they had to do or where they had to make choices about methods and sometimes it would be unclear.  I could tell that one girl was really struggling with that question.  She knew that it would be easier in the other classroom, but also knew that she would learn more and wanted to stay where her learning would be more effective.

What can I do to help this process go more smoothly?  Make sure that they know that I am working hard NOT to use shame as a classroom management tool.  That I am sincerely interested in the mistakes that they are making and how it is helping their learning.  I want them to grow from their errors and misconceptions and find ways to use those to their advantage.  I want to add to their self-worth not only as a math student, but as a problem solver in every way.

As Brene Brown says:

“I don’t believe shame-free exists but I do believe shame-resilience exists and that there are teachers creating worthiness-validating, daring classrooms every single today.”

I can be truly aware of the language that I use and the questions that I ask in order to make sure that everyone’s voice is heard and that my students know that I want to hear their ideas.  It’s really the only way to get them to Dare Greatly!

PS – Check out the wonderful quote by Teddy Roosevelt that I use in my PBL classes about Daring Greatly that Brene Brown used for the title of her book.

Get Comfortable with Uncertainty: A Short Dialogue

And so it begins.   The students are flustered. The emails are coming at night.  The faces stare at me, scared to death.  Although I repeat numerous times, “You do not have to come to class with each problem done and correct” students are totally freaking out about the fact that they can’t “do their homework” or they can’t “get” a certain problem on the homework.  No matter how many times I attempt to send the message the first few weeks about how unnecessary it is to come to class with a problem complete or an answer to show, students feel the need.

Tomorrow I am writing on my large post-it notes in HUGE capital letters, “Get comfortable with uncertainty because it’s not going anywhere.”  Every year about this time, I give the speech about how my homework is extremely different from any homework they have probably encountered in math class.  These are not problems that you are supposed to read, recognize and repeat.  They are there to motivate your thinking, stimulate your brain and trigger prior knowledge.  In other words,  you need to be patient with yourself and truly create mathematics.

Today I met with a young woman who I thought was about to cry.  She came and said, “I can’t do this problem that was assigned for tomorrow.”  Here’s how the conversation went:

Me: Why don’t you read the problem for me?

Girl:  Find points on the line y=2 that are 13 units from the point (2,14)

Me:  Ok, so show me what you did. (she takes out her graph paper notebook and shows that she graphed the line y=2, plotted the point (2,14)).  Great, that’s a great diagram.

Girl:  But it didn’t make sense because in order for it to be 13 units away, it had to be like, diagonal.

Me: Huh, what would that look like?

Girl: (drawing on her diagram) There’d be like two of them here and here.

Me; yeah?

Girl: But it can’t be like that….

Me: yeah? Why not?

Girl: Um…cause it wouldn’t be a straight distance.  I think..

Me: Is it 13 units away from (2,14)?

Girl: yeah, I think so…

Me: Hmmm….how far is (2,14) from the line y=2?

Girl:  Oh that’s easier – it’s like 12. ..Oh My gosh..it’s like a hypotenuse….and the other side that I don’t know is like the a and the 12 is like the b.  I can just find it.  Oh my gosh that’s so easy.  And the other one is on the other side.    Why didn’t I see that?

Me:  Well, you did…actually….

Girl: well, after you asked me that question…

Me: yeah, but eventually you’ll learn how to ask yourself those questions.


And they do….it’s just the beginning of the year.  We have to give them time – time to look into their prior knowledge as a habit, time to surprise themselves, time to have those moments, time to enjoy the moment and revel in the joy and courage and disappointment.  It’s all a part of the breakthrough that is needed to realize that they are creative and mathematics needs them to be.  It’s amazing and it’s worth it.

Defying Gravity as a Means to Learning from Mistakes

There’s a lot of blogging, writing and research (and anecdotal stories) out there these days about trying to foster the value in students for the appreciation in failing.   I even wrote a blog entry two years ago entitled “modeling proper mistake-making” way before I read anything or watched any videos on the Internet.  From teaching with PBL for over 17 years, I am a pro at making mistakes and watching students struggle with the concept of accepting the idea of learning from their mistakes.  This is so much easier said than done, but it is clearly something that grow to love even if only for a short time.

Last April, I had the pleasure of hearing Ed Burger at the NCTM national conference where he spoke about having students in his college-level classes required to fail before they could earn an A in his class.  In his August 2012 essay “Teaching to Fail” from Inside Higher Ed (posted at 3:00 am, which I thought was kind of funny), he talks about attempting to make a rubric for the “quality of failure” on how well a student had failed at a task.  I thought this was an interesting concept.  I mean, in order to fail well, can’t you just really screw up, like not do it at all?  Prof. Burger states that allowing students to freely reflect on their “false starts and fruitful iterations” as well as how their understanding “evolved through the failures” can be extremely beneficial.  He also states:

“To my skeptical colleagues who wonder if this grading scheme can be exploited as a loophole to reward unprepared students, I remind them that we should not create policies in the academy that police students, instead we should create policies that add pedagogical value and create educational opportunity.”

Last year for the first time, I tried a similar experiment wherein I gave students an assignment to write a paper in my honors geometry class.  They had to choose from three theorems that we were not going to prove in class.  However, it was clear that they could obviously just look up the proof on the Internet or in a textbook or somewhere, since they clearly have been proven before.  The proof was only 10 or 20% of their grade.  The majority of the paper’s grade was writing up the trials and failures in writing the proof themselves.  This proved to be one of the most exciting projects of the year and the students ate it up.  I even told them that I didn’t care if they looked up the proof as long as they cited it, but I still had kids coming to me to show my how they were failing because they wanted a hint in order to figure it out themselves.  It was amazing.

This past week I showed my classes Kathryn Schultz’ TED talk entitled “On Being Wrong” in which she talked about the ever popular dilemma of the Coyote who chases the Road Runner, usually off a cliff.

My students loved her analogy of the “feeling of being wrong” to when the Coyote runs off the cliff and then looks down and of course, has to fall in order to be in agreement with the laws of gravity.  However, I proposed a different imaginary circumstance.  Wouldn’t it be great if we could run off the cliff, i.e. take that risk, and before looking down and realizing that vulnerability and scariness, just run right back on and do something else?  No falling, no one gets hurt, no one looks stupid because you get flattened when you hit the ground?  Maybe that’s not the “feeling of being wrong” but it’s the “feeling of learning.”

Next blog entry on creating the classroom culture for “defying gravity.”

Wrong is not always bad

I recently read an article in Education Week that was proclaiming the benefits of discussing student mistakes in class. The author, Alina Tugend who has recently published a book entitled, Better by Mistake: The Unexpected Benefits of Being Wrong, cited that in some asian cultures students can be asked to work out “math problems in front of the whole class for a healthy period of time…even if [they] are doing it wrong.” She goes on to discuss that the teacher might ask the student to discuss her thought process and why they chose to do the problem that way and the decisions they made at certain points in order for the class to see the choices that were made at certain crossroads in the problem solving process. Some researchers believe that this type of discussion allows students to help create a sort of “index” of what still needs to be learned or what has already been learned.

In other words, the class is actually viewing the errors and misunderstandings as a helpful thing. They’re using the opportunity of the mistake, of being wrong, as kind of a check point to see what else they need to know. Perhaps someone else in the room might have did something differently that might have led them in a direction that was more fruitful and everyone can learn from that as well. So there is much more to be learned from this type of environment. On the one hand, the students learn the material, but on the other hand, they are learning that they can learn from each other and they learn that being wrong in the first place was actually helpful.

This also goes to the idea of how problem-based learning is ideal for this type of learning. Posing the problem in the context of a prior knowledge base, allows students to think that they have a background that is sufficient for them to do the problem, they just need to recall what that was with a little push. It also fosters was researcher Carol Dweck calls the “Growth Mindset” allowing students to believe that their intelligence and ability to succeed to flexible and not pre-determined.

I am getting excited for my course next week at the Math and Technology Conference in Exeter, NH which I believe is full. I love this conference because I always meet lots of people who are so eager to engage in mathematics and learning. It should be a great time of dialogue and I look forward to a great time!