Six of one, half a dozen of the other…I think not

(Sorry, this is a long one! and caveat: I am not claiming that Wikipedia is the be all and end of definitions!)

So according to Wikipedia, PBL means two things (well, three if you count Premier Basketball League, but that’s neither here nor there). If you look up PBL on Wikipedia, the first hit is, in fact, Problem-Based Learning. Why, you may ask? I believe that this is because Problem-Based Learning has been around in various forms longer than Project-Based Learning, but the term itself was coined in the 1960s by Howard Barrows at McMaster University in Ontario, Canada ( You can read more about Barrows’ specific definition of Problem-Based Learning at this site and in my tab at the top of my website that says “Problem-Based Learning.” However, what I feel is one of the biggest parts of Barrows’ definition is the fact that “the problem is a vehicle for the development of problem solving skills” – that is it, that it is the problem – hopefully well developed and set in a context that is interesting, challenging and meaningful to students – through which the students will development and learn the problem solving skills.

Wikipedia names Project-Based Learning ( Learning) almost the same thing, however, they connect it more to Greeno’s theory of situated learning – “learning by doing” and “teaching by engaging students in investigation.” However, all of these theorists ideas range from about 1991-2006, so it would seem that a PBL by any other name…is not really the same?

One of the most important distinctions in Project-Based Learning (which I will write as PjBL, because you know, it came 2nd, for the record) is the authenticity of the task that is motivated by a larger “driving question” – students learn by creating a project and investigating what they need to do in order to organize or structure their presentation for the project.

So what does this mean in mathematics? A few weeks ago, there was quite a discussion going down on Twitter about what constituted Problem-Based Learning.




Dan Meyer seemed to be criticizing Problem-Based Learning saying that it was discussed by others as “too much curriculum and too little time for PrBL” and he says that he saw “a lot of fluff in PrBL” – to which I would say, there’s much more to Problem-Based Learning than just doing problems, Mr. Meyer. Nat Banting also asked for a clarification of what the difference was between a project and a problem in math education.

In response to this discussion and Mr. Banting’s question, I posted an image of a table I created listing differences that I saw between project- and problem-based learning in mathematics education and had hoped for some feedback.


Then I thought that maybe my description on the image of the differences wasn’t clear enough, so I thought I’d try one more time to make the distinction between the authenticity clear.

If we revisit the idea that Problem-Based Learning has at its core, problems as the vehicle for learning and constructing knowledge, I think this is at the heart of the difference of the learning/teaching experience. When I look at purposefully scaffolded and written curriculum for problem-based learning, yes there are outcomes that need to be met, there are topics that are discussed that are set by the teacher (or curriculum writer). Mathematics has within it many very interesting abstract concepts that are worthwhile to teach even though the “real-life” applications don’t have much context for students (will a student appreciate the logic and problem solving skills that are learned in factoring a polynomial, for example, or do they just accept and take it as a skill they need, or use technology and not use that part of their brain – who knows?) The point I’m trying to make is that in problem-based learning the problems would lead students to figuring out (through discussion and with prior knowledge and experience) the skills needed to create a process of factoring and perhaps “unfactoring” an algebraic expression and hopefully what that expression might represent. This is what’s known as “preauthentication” – when the curriculum writers or teachers try to come up with some kind of experience for the students that simulates the mathematicians authentic experience or “ah-ha” moment of understanding or realization. Then there are other problems that allow the students to dig deeper and apply those ideas to other areas of mathematics, and yes, real-life problems too.

In Project-Based Learning, “emergent authenticity” allows the mathematics to emerge from the ideas of the project (or driving question at hand) which is up to the students to then find out what they need to know. This is where Project-Based Learning can fall short in the area of secondary mathematics – in a world of standards. Where does the learning take place and how does it happen? Once students understand a concept, having gone through rather traditional instruction and some type of formative assessment, the project can then be given. Yes, Mr. Meyer, I believe that there is a hard balance to make there between traditional instruction and when to do the projects – quite a dilemma of time (although schools do it with very nice interdisciplinary time schedules).

However, I do believe that in problem-based learning the rigor, content, problem-solving and all the other “4C’s” skills that project-based learning also promote end up happening in the discussion and presentation of problem ideas and solutions. So I would have to argue with those who say “it doesn’t matter what you call it” and “they’re all the same thing” because the learning process in mathematics is so very different in these two methods. Hopefully, I can shed some light on the differences between PBL and what I hope will be called PjBL soon!

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.

PBL & James and the Giant Peach: Try looking at it a different way

James Henry Trotter: “When I had a problem, my mum and dad would tell me to look at it another way.” (Roald Dahl)

I’ve always thought that PBL fostered creative problem solving as opposed to memorization of pneumonic devices.  One of my students today proved me right when I gave a “quick quiz” on the use of the idea of tangent.  We had discussed tangent in class for only two days and in two ways – one as a slope of a line with a given angle and from that idea we discussed how it could be interpreted as the ratio of the sides of a right triangle (if you put a right triangle under the line).

Of course, during this conversation some student who had studies the ever popular SOHCAHTOA before mentioned this in class and told everyone that they had just memorized this and that’s how they knew it.  I said that’s fine but I’d like them to try to think about the context of the problems and see if this helps make any sense of it for them.

So today on the quiz one student was attempting this problem – very basic, very procedural, not at all something that I would call atypical of a textbook-like problem on tangent.

A bird is sitting on top of the Main School Building and looks down at the end of the baseball field with an angle of depression of 4 degrees.  If the MSB is 87 feet tall, how far away is the end of the baseball field?

So the student attempts to create a ratio with the sides of the triangle and even sets it up correctly.  However, because she does the algebra incorrectly, she gets an answer that is extremely small 8.037 x 10^-4.  In fact, during the quiz, she calls me over and asks what it means, she doesn’t remember scientific notation and starts getting all anxious because we didn’t do anything like this in the problems in the previous two days?  How can the answer be that small?  I said well, you better go back and think of something else.

In most classes, a student in this situation might stress out, try to do the problem over again with the limited perspective of “TOA” or of just viewing the right triangle in one way.  However, because this students had also learned other students’ perspectives of tangent as slope of a line what this girl did at this point was to see it from a different way.  Interestingly, this is what she did.  In an alternative, albeit confusing way of writing the equation of the x-axis, she wrote y=0x to represent the ground.  Then she found the tangent of 4 degrees and used that as the slope of a line.  She put the bird at the point (0,87)

She writes the equation y= – (tan4)x + 87 and explains that this is the equation of a line that makes a 4 degree angle with the x axis and has a y-intercept of 87.  Then she realizes that if she finds the intersection of that line and the x-axis, she would find how far the building is from the baseball field.  This is what she does and uses her graphing calculator to get the right answer.

When she hands in this quiz to me, I half expected that tiny little answer as her distance to the baseball field.  But what I got was an amazingly inventive solution and a correct answer.  With a problem that didn’t make sense, she looked at it a different way and ended up getting the right answer.  It was amazing what changing your perspective could do and this was great evidence that even under pressure, the habits of creativity and connection were paying off.

Using Journal Writing in PBL

Over the years, especially in PBL with mathematics, I have found that students greatly appreciate the authorship and ownership that comes with keeping a journal in my classroom.  In fact when I asked my students earlier this year, “When do you feel most confident in this class?” and here are some of the feedback responses they gave me:

“When I am about to hand in a journal.”
“When I am writing a journal entry because there are various concepts that initially don’t understand, and after discussions I make big discoveries and therefore it makes writing about it easier for me.”

There is something that I have come to appreciate about the way students grow to be able to show how they understand a concept.  Recently, I read a student’s journal entry and thought it was so amazing that I asked him if I could blog about it.  I thought that it really showed how he moved through his understanding of the concept and how he struggled with it to the end. In fact, he presented this problem in class thinking that he had gotten it right and wonderfully, kept going and learned something in the process.

The student – let’s call him Pete- was dealing with a problem that was towards the end of a thread that dealt with the concept of distance – distance between two points, distance between a point and a line, distance between two lines, etc.  This question was asking students to think about two different types of distance.  Here’s the problem:

“Plot all of the points that are 3 units away from the x axis and write an algebraic expression for those points.  Then plot all the points that are 3 units away from the point (5,4) and write an algebraic expression for those points.”

Up until now all we had discussed was writing expressions for Pythagorean distance between two points and writing equations for equations of lines.  We had also talked about the fact that the closest distance from a point to a line is the perpendicular distance.  So Pete was easily able to answer the first part of the question seeing that the set of all points that were 3 units away from the x axis were both the line y=3 and the line y=-3.  He drew a diagram discussing his concept of distance from a point to a line and how he visually (in his mind and physically on paper and at the board that day in class – connected them together).

However, in his journal he wrote about how the second part of the question seemed just as easy to him at first.  “I assumed I needed to do a straight line. I then saw ‘three units’, so I put a point on (5,1), and drew the line y=1.  If (5,1) was 3 away, I thought, shouldn’t all the points on the line be 3 away?” Here’s what his first diagram looked like:

Pete’s First attempt

Pete had tried to use his understanding of distance being “three units” away from a point in the same way that being “three units” away from a line in the previous problem.  However, when he was at the board, another student told him they had thought of it another way and shared with Pete something Pete realized very soon…. “Only 1 point on each of the lines was actually 3.  The rest of the points were actually all further than 3 units from the point.”  Here’s his diagram of his realization of that:

Second attempt

So now Pete is discussing how he is using his knowledge of Pythagorean distance and seeing that only the vertical and horizontal points are actually the required 3 units away.  Huh, how does he move forward now?  At the moment when he was in class, it took another classmate to say that ALL the points were supposed to be 3 units away and that he (that classmate) thought it would look more like a circle.  Pete was still determined to correct his own work (which I just love) then attempted this drawing:


Third attempt

Pete writes:

“This, I thought, would cause all points on the line to be 3 units away from point (5,4).  However, I was again wrong.  The blue line on the diagram shows a point on one of my lines that was more than 3 units from (5,4). The red line shows a point on one of the lines that is less than three units from (5,4) [it would’ve been great if he went into more detail, but at this point I’m so psyched that he’s going into this much detail!]  The green lines are points that are 3 units away from point (5,4).  I have effectively created a range of lengths from (5,4) opposed to what the question was asking for which was 3 units from (5,4).”

This is some of the most insightful journal writing I’ve ever seen from a high school student.  Pete is moving through his understanding of what it means for a point to be 3 units away from another point (even when another student has imposed their understanding on him) and is trying to show me how he came to understand his classmate’s argument that it is a circle.  Realizing that there are points that are more than 3 units away (farther out than this diamond shape), 3 away (at the vertices) and closer than 3 (along the sides of the diamond)….well, that there was a range and not all constantly 3, showed him that his ideas was not correct.

Pete then draws this diagram:

Getting the concept

After understanding this, Pete writes:

“It made perfect sense!…Any point from the centerpoint of a circle to any point on the circle was the same length (the radius).  I immediately drew the connection.  3 was the radius and (5,4) was the center.  the distance between the middle point and any point on the circle was 3!”

Although Pete didn’t write about the discussion that ensued about the algebraic expression, I still felt like the goals of metacognitive journaling were reached with this entry.  Pete has articulated why he chose this problem to me, he started with his misunderstanding, how another student or the class discussion/experience had helped move him through his understanding and he could clearly write about how he now has a good understanding of the mathematical concept.  I was so proud of Pete.  Yes, it’s true, not every student gets to this point of writing mathematically by January of the school year, but the growth that occurs in each individual student is what is important – not necessarily the level of maturity in the writing.  However, I have learned not to prejudge or dismiss any student who starts off at a lower level because I believe they are all capable of growth.

If you are interested in looking at my grading rubric for journals or asking me questions about how I use them in the course, don’t hesitate to get in touch. (If you are looking for the grading rubric, make sure you scroll to the third page of that pdf).

Top 5 Recommended Readings for PBL Teachers Part 2

So, I finally got this done and I’ll continue with the top three readings that I just found extremely useful in my teaching last year.

3. The Innovators’ DNA: by J. Dyer, H. Gregersen and C. Christensen

I rarely recommend books that I have not read yet, but this one is actually on my list to read next so I am recommending it because everything about it just feels right to me.  Again, this is not an education book, but a book that is really for business people.  The research that was done in preparation for writing this book was looking to see what characteristics people who are viewed as transformative innovators in the business world all share.  The authors have come up with five major traits or behaviors that innovators share –

  1. associating
  2. questioning
  3. observing
  4. experimenting
  5. networking

You can read a wonderful summary of this book at this link, but I would highly recommend the book as well.  It is our job as progressive educators and teachers of PBL to teach these skills.  If it isn’t obvious to us already, as PBL teachers, I’ll say it again – that PBL is custom-made for teaching these types of skills which clearly is what this book is stating employers are now looking for.

One thing that I do not read enough of is how PBL encourages the skill of associating.  I write a lot about this in my blog and researched it in my dissertation.  In fact, connection is one of the main themes that came out in my research that students enjoyed about PBL.  The skill of associating is a major skill that is extremely important to innovation and in fact, Steve Jobs in quoted as saying, “Creativity is connecting things.”  Allowing students to practice making those connections themselves is key in order for students to practice their own creativity, especially in mathematics.

2. The Five Elements of Effective Thinking by Ed Burger and Michael Starbird

This little gem, published in 2012, was the focus of Ed Burger’s key note address at the 2012 NCTM Annual conference.  He actually didn’t try to sell the book too much, but focused on the idea of teaching effective thinking (so then, yeah, I went and bought the book – what can I say, he’s a great speaker).  As I was reading through it, all I could think about was how relevant it was to teaching mathematics with PBL.  If every student in a PBL classroom took to heart every one of the five elements that are put forth in this book, the classroom would be so much more effective (as would any classroom).

So Burger and Starbird but forth these five elements of effective thinking:

  1. Understand Deeply
  2. Make Mistakes
  3. Raise Questions
  4. Follow the Flow of Ideas
  5. Change (which they call the Quintessential Element)

So, you might ask – what’s so great about those?  I know this?  Well, it’s not those five that are so great – if you are a PBL teacher you probably are already telling your students these already.  What I think is so great about this book are the pieces of advice that Burger and Starbird give for each of these five elements.  In each chapter, these are not only examples from their own teaching but actual ways to promote each of these elements not only individually but in your classroom as well.  The anecdotes that are shared in the book are not only heart-warming but as a teacher you can see how you can make them useful in your own practice.

The combination of deliberately stating these five (and adding CHANGE as the most important) is really key for PBL.  Students may know that you want them to understand deeply and in order for them to do that they need to raise questions about their own understanding, but if you don’t constantly and deliberately create a culture for them and you in your classroom it is not a message they will receive seriously.

And the best book, that I would highly recommend reading:

  1. A New Culture of Learning, by Douglas Thomas and John Seely Brown

This book, in my opinion, is what PBL is all about.  Whether you teach in a school that uses a problem-based curriculum, uses text books and is trying projects, or if you are just trying to create a more student-centered approach to your teaching – this book is getting at the heart of what is creating a change in our schools nationwide.  It is why there is a huge movement going on with teachers in our nation trying to find something different to do in their classrooms.  Thomas and Brown describe this movement as a shift from a “teaching-centered culture” in our nation’s schools to a “learning-centered culture” which may be the most important shift in education since organized schooling began in the U.S. altogether.

This shift is based on the idea that knowledge is flexible (yes, the idea of Truth with the capital T does not exist – shhhh, don’t tell anyone).  Even in mathematics, the way that we solve problems and even the mathematics that we teach students – which topics are “most important” today- is changing rather regularly.  This has become so much more clear and visible because of not only the Internet itself, but our access to it.  Thomas and Brown suggest that we must be willing to admit that what is most important about education now is not what we teach in schools, but how students learn.  Can a student learn in the collective? Are they able to harness different modes of inquiry?  Do they experiment in their learning? This shift in the purpose of schooling is not really new to teachers but to our society it is major.  Teachers need to learn how to make this switch and articulate the deliberateness of what they are doing in their classroom in order to focus on the shift. (By the way, this also has major ramifications for teacher educators).

 I love the five dispositions that will help construct the new culture of learning (very applicable to a PBL environment!)

  1. Keep an eye on the bottom line (ultimate goal is to improve)
  2. Understand the power of diversity (strongest teams are rich mix of talents and abilities)
  3. Thrive on change (create, manage, seek out change)
  4. See learning as fun (reward is converting new knowledge into action)
  5. Live on the edge (explore radical alternatives and innovative strategies, discover insights)

All of this is so relatable to my own classroom and curriculum.  The more I create problems and experiences that allow my students do have these dispositions, the more I know that I am fostering the “culture of learning” instead of a traditional culture of “teaching.”

So that’s it.  My top 5 list of readings for PBL teachers – please let me know what you think and if you end up utilizing any of these authors’ ideas.  I know that I have been invigorated by these readings and hope that you will be as well!  Have a happy and fulfilling 2014!

Top 5 Recommended Readings for PBL Teachers of 2013 Part 1

Happy New Year!  It’s been a busy end of 2013 for me.  I’ve been doing a lot of reading and catching up with some writing.  So, the New York Times came out with their top 75 Best-Selling Education Books of 2013 and some of them are really great reads and some are just books that are commercially hyped education jargon.  I’ll let you read it for yourself and see which you think are which.  But this inspired me to think about what I would recommend as great reading for PBL teachers in terms of mathematics.  It’s not always easy to get inspired to continue with PBL so I am always on the look-out for good reads and things that might help me to find ways to motivate students in the classroom.  I also hate those lists from articles that seem to have all the answers but then when you read them nothing is ever really black and white like “To Flip or Not to Flip: that is the Question” or “5 Resolutions to Modernize Your Teaching For 2014” or “Top 100 Tools for Learning in 2014” – geez, does anyone just write about one thing anymore?  Or even give critical analysis of why these are the reasons to flip, or an argument as to the top 100 tools – anyone can make a list.

Including me!  So here goes nothing – well, I mean something.  I tried to put together some good reading that emphasizes the skills that are needed for working with students in a problem-based classroom.  One of the things I hear most from teachers is not necessarily how to work with the curriculum, but how to get students working with each other and how to foster the type of classroom community (curiosity, openness and risk-taking) that is needed in order for students to want to be engaged.

5. The Mistake Manifesto: How Making Mistakes Can Make Us Better by Alina Tugend, 2011.

I first came across Tugend’s writing when I read her Op-Ed piece in the NY Times while ago, but this essay on making mistakes says so much about Tugend’s great attitude towards how mistakes are not only helpful, but are a wiser and more powerful way of learning.  She says that “we do single-loop learning when we need to do double-loop learning.”  I love that and I believe that PBL’s  method of returning to ideas in its scaffolded and multi-topic approach often allows students to revisit ideas multiple times.  Tugend talks about how most of our society creates a fear of making mistakes because we have this idea that we aren’t supposed to make mistakes.  This is in turn makes us all risk-averse unfortunately and only allows the most unstructured students and learners to be creative innovators.  This is what we have to turn around.  Her manifesto doesn’t necessarily tell us how to do this, but it’s a wonderful argument for why we should.

4. Flow, by Mihaly Csikszentmihalyi, 1990

This book’s original intent was to investigate the psychological experience of happiness, however this past year it became connected for me to the process of problem-based learning.  OK, so this book is not from 2013 – or even from the past few years, but what happened in 2013, is that I read an article that sent me to this book.  The article was called “The Problem-Based Learning Process as finding and being in Flow” by Terry Barrett and it discussed the concept of ‘flow’ (from Csikszentmihalyi’s book) and compared the PBL process (the discourse that occurs, the exchange of ideas and that learning process itself) to the optimization of creativity that occurs in the ‘flow’ process.  In this book, Csikszentmihalyi defines ‘flow’ as “the state in which people are so involved in an activity that nothing else seems to matter.  The experience itself is so enjoyable that people will do it even at great cost, for the sheer sake of doing it.”(Csikszentmihalyi, p.4).  Wouldn’t that be great if that’s the way students viewed learning?  One way to see it is like this:


(Barrett, 2013)

The idea being that the state of flow in learning comes when the optimal problem or activity is presented to students such that the difficulty and time or skills given keeps their interest long enough to minimize anxiety and maximize love of learning and the return on their learning (reinforcement of confidence, efficacy, enjoyment, agency, etc.).   A lot of the book is based on the idea of the state of flow helping to create the optimal state of happiness so it might not relate directly to teaching, but I highly recommend the last two chapters which are entitled “Creating Chaos” and “The Making of Meaning” which can be directly translated to the PBL classroom and are highly useful for the PBL teacher looking to see how you can create the state of flow for your students.

Tomorrow I will catch up with numbers two and three! (hopefully get you #1 as well)

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’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.

Teaching Students to Become Better “Dancers”

So the other day I read a tweet by Justin Lanier that really sparked my interest.

 We all know the scenario in classroom discourse where a student asks a question – a really great question – and you know the answer, but you hedge and you say something like, “That’s a great question! I wonder what would happen if…”  So you reflect it back to the students so that they have something to think about for a little while longer, or maybe even ask a question like “Why would it be that way?” or “Why did you think or it like that?”  to try to get the student to think a bit more.  But what Justin, and the person who coined the phrase “authentic unhelpfulness” Jasmine Walker (@jaz_math), I believe were talking about was hedging because you really don’t know the answer – sincere interest in the uniqueness of the question – not because you’re so excited that student has helped you move the conversation forward, but because of your own excitement about the possibilities of the problem solving or the extension of the mathematics.

I think what got me so excited about this idea was how it connected to something that I was discussing earlier this summer with a group of teachers in my scaffolding in PBL workshop in late June.  In a PBL curriculum, the need to make sure that students have the right balance of scaffolded problems and their own agency is part of what Jo Boaler called the “Dance of Agency” in a paper she wrote in 2005 (see reference).  My understanding of this balance goes something like this:

(c) Schettino 2013

So initially, the student is confused (or frustrated) that the teacher refuses to answer the question although you are giving lots of support, advice and encouragement to follow their instincts.  The student has no choice but to accept the agency for his or her learning at that point because the teacher is not moving forward with any information.  But at that point usually what happens is that a student doesn’t feel like she has the authority (mathematical or otherwise) to be the agent of her own learning, so she deflects the authority to some other place.  She looks around in the classroom and uses her resources to invoke some other form of authority in problem solving.  What are her choices?

She’s got the discipline of mathematics – all of her prior knowledge from past experiences, she’s got textbooks, the Internet, her peers who know some math, other problems that the class has just done perhaps that she might be able to connect to the question at hand with previous methods that she might or might know how they work or when they were relevant – that discipline has had ways in which it has worked for her in the past and lots of resources that can help even if it may not be immediately obvious.

But she’s also got her own human agency which is most often expressed in the form of asking questions, seeing connections, drawing conclusions, thinking of new ideas, finding similarities and differences between experiences and thinking about what is relevant and what is not.  These pieces of the puzzle are not only important but a truly necessary function of the “dance of agency” and imperative to problem solving.

Interweaving both of these types of agency (and teaching kids to do this) have become more important than ever.  Yes, being able to use mathematical procedures is still important, but more important is the skill for students to be able to apply their own human agency to problem and know how and when to use which mathematical procedure, right?  This “dance” is so much more important to have every day in the classroom and if what initiates it is that deflection of authority then by all means deflect away – but the more we can “dance” with them, with “authentic unhelpfulness” and sincere deflection because we need to practice our own human agency, the more we are creating a true community of practice.

Boaler, J. (2005). Studying and Capturing the complexity of practice – the Case of the ‘Dance of Agency’

A Total Win…with lots of understanding

Before I left for the Anjs S. Greer Math Conference last week, I read an amazing blog entry at the Math Ed Matters website by Dana Ernst and Angie Hodge that was talking about Inquiry-Based Learning and the mantra “Try, Fail, Understand, Win.”  The idea came from one of Prof. Ernst’s student course evaluations this past spring as his student summed up his learning experience in such an IBL course.  This blog post was so meaningful to me because for each of these four words, the authors wrote how we as teachers (and teacher educators) can take this student’s perspective towards our own work.  I decided to attempt to take this attitude going off to my own conference with two courses to give and three smaller talks.  It was sure to be a busy week.

And in fact, it really was.  I had very little time to sit and listen to others’ work, which I really was quite sad about.  However, in my own classes I was so impressed with the amount of enthusiasm and excitement my participants had for PBL and their own learning.  As I sat in front of my computer this morning reading the course evaluations and their tremendously helpful input, it finally occurred to me how truly powerful the experience had been for my participants.  Many of them became independent thinkers and knowers about PBL and feel so much more knowledgeable and prepared for the fall.    Part of the class time is spent in “mock PBL class” where I am the teacher/facilitator and they are the students doing problem presentations.  We then sit and talk about specific pedagogical questions and distinctions in classroom practice.  Some of the class time is spent in challenging problem solving which is where I also learn so much from the participant’s different perspectives. “We win when we realize there’s always something we can do better in the classroom” – as Ernst and Hodge write.

The now Infamous ‘French Garden’ Problem

I want to give a huge shout out to all of my participants from last week and encourage them to keep in touch with me.  Many of you wrote in your evaluations that you still have many questions about your practice and how to integrate your vision of PBL in your classroom.  I will always be only an email away and hope that you continue to question your practice throughout the year.

My plan is to try to write some blog posts at the end of the summer/beginning of the year in order to respond to some of the remaining questioning while you plan for the beginning of the school year such as:

  1. How to plan for week one – writing up a syllabus, creating acceptable rules
  2. Helping students who are new to PBL transition to it
  3. Assessment options – when to do what?
  4. Working hard to engage students who might not have the natural curiosity we assume

If you can think of anything else that you might find helpful, please post a comment or send me a message and I’d be happy to write about it too!  Thanks again for all of your feedback from the week and I look forward to further intellectual conversation about teaching and PBL.