Modeling with Soap Bubbles

I am so very lucky to have a guest teacher with me this year at my school.  Maria Hernandez (from the North Carolina School of Science and Math) is probably one of the most energetic and knowledgeable teachers, speakers and mathematicians you could ever find – and we got her for the whole year!  We are so excited.  I am working with her and she is so much fun to work with.  I have been teaching calculus with PBL for almost 20 years now and thought I had all the fun I could but no!  Maria is bringing modeling into my curriculum and I’m enjoying every minute of it.

As we started teaching optimization this week, Maria had this wonderful idea that she had done before where we want to find the shortest path that connects four houses.


I let the kids play with this for about 10 minutes and then did this wonderful demonstration with some liquid soap bubbles and glycerin.  We had two pieces of plastic and four screws that represented the houses.  As the kids watched, I dipped the plastic frame into the liquid and voila-file_000

Right away the students saw what they were looking for in the shortest path.  Now they had to come up with the function and do some calculus. As they talked and worked in groups, It was clear that using a variable or one that would help them create the right function was not as easy as they thought.  However,  I was requiring them to write up what they were doing and find a solution so they were working hard.


We have been doing a lot of writing in Calculus this fall so far and they are getting used to being deliberate about their words and articulating their ideas in mathematical ways.

Here is the outline of the work they did in class: Shortest Path Lab

and here is the rubric that I will be using to grade it.


The engagement of students and the buzz of the classroom was enough to let me know that this type of problem was interesting enough to them – more than the traditional “fold up the sides of the box.”  The experience they had in conjecturing, viewing, writing the algebra and solving with calculus was a true modeling experience.

If you decide to do this problem or have done something like it before, please share – I’d love to do more like this.  I am very lucky to have a live-in PD person with me this year and am grateful every day for Maria!


Documents for CwiC Sessions at Anja Greer MST Conference 2016

Instead of passing out photocopies, I tried to think of a way that participants could access the “hand-outs” virtually while attending a session.  What I’ve done in the past a conferences is have them just access them on their tablet devices.  You can also go and access copies on the Conference Server if you do not have a device with you (you should be able to use your phone too).

These link to This is an Adobe Acrobat Documentpdf documents that I will refer to in the presentation about “Assessment in PBL”

Information on Spring Term Project and Spring Term Project Varignon 2015 (this document includes rubric)
Keeping a Journal for Math Class
Revised Problem Set Grading Rubric new
Rubric for Sliceform project and Sliceforms Information Packet

Page at my website with Rubrics and other guides for Assessment

Someday I’ll get this assessment thing right… (Part 2 of giving feedback before grades)

So, all assessments are back to the students, tears have been dried and we are now onto our next problem set (what we are calling these assessments).  What we’ve learned is that the rubric allowed us to easily see when a student had good conceptual understanding but perhaps lower skill levels (what we are used to calling “careless mistakes” or worse). We could also quickly see which problems many students had issue with once we compared the rubrics because, for example, problem number 6 was showing up quite often in the 1 row of the conceptual column.  This information was really valuable to us.  However, one thing we didn’t do was take pictures of all of this information to see if we could have a record of the student growth over the whole year. Perhaps an electronic method of grading – a shared google sheet for each student or something to that effect  might be helpful in the future – but not this day (as Aragorn says) – way too much going on right now.

We also changed the rubric a bit for a few reasons.  First, we found that when students completed the problem to our expectations on the initial attempt we felt that they should just receive 3’s for the other two categories automatically.  We considered not scoring them in this category but numerically felt that it was actually putting students who correctly completed a problem at a disadvantage (giving them fewer overall points in the end). Second, we also changed the idea that if you did not write anything on the revisions you earned 0 points for the revisions columns.  Many students told me afterwards that they felt like they just ran out of time on the revisions and actually had read the feedback.  This was unfortunate to me since we had spent so long writing up the feedback in the hope that the learning experience would continue while doing revisions.

Here is the new version of the rubric: Revised Problem Set Grading Rubric new

What we decided to do was to try the revisions this time without the “explanation” part of writing.  I think it will keep the students focused on reading the comments and attempting a new solution.  I was frankly surprised at how many students stuck to the honor pledge and really did not talk to each other (as they still got the problem wrong the second time around – with feedback).  Truly impressive self-control from the students in my classes and how they were sincerely trying to use the experience as a learning opportunity.

I do think the second assessment will go more smoothly as I am better at doing the feedback and the rubric grading.  The students are now familiar with what we are looking for and how we will count the revisions and their work during that time.  Overall, I am excited about the response we’ve received from the kids and hope that this second time is a little less time-consuming.  If not, maybe I’ll just pull my hair out but I’ll probably keep doing this!




Why Teachers Don’t Give Feedback instead of Grades, and Why We Should

First in a series of posts about my experiences with “Feedback Before Grades”

Holy Mackerel is all I have to say – Ok, well, no I have plenty more to say – but after about a week and a half of holing myself up with my colleague, Kristen McVaugh, (big shout-out to Ms McVaugh who is not only teaching PBL for the first time but was willing to dive into this amazing journey of alternative assessment with me this year too), I am totally exhausted, almost blind as a bat, partially jaded and crazy – but mostly ready for a drink.  This little looped video of Nathaniel Rateliff and the Night Sweats pretty much sums it up…

So here was our well-intentioned plan:  we wanted to start the year off with a different type of assessment.  I put out my feelers on twitter and asked around if anyone had a rubric for grading assessments where the teacher first gave only feedback and then allowed students to do revisions and then once the revisions were done the students received a grade. Kristen and I knew a few things:

  1. we wanted to make sure the revisions were done in class
  2. we wanted to make sure the revisions were the students’ own work (tough one)
  3. we wanted to give students feedback that they needed to interpret as helpful so that we weren’t giving them the answer – so that it was still assessing their knowledge the second time around
  4. we wanted to make sure that students were actually learning during the assessment
  5. we wanted students to view the assessment as a learning experience
  6. we wanted students to be rewarded for both conceptual knowledge and their skills in the problem solving too

So we created this rubric Initial Draft of Rubric for Grading.  It allowed us to look at the initial conceptual understanding the student came to the problem set with and also the initial skill level. Kristen and I spent hours and hours writing feedback on the students’ papers regarding their errors, good work and what revisions needed to be done in a back-handed sort of way.

Here are some examples:

Student 1 Initial Work

Student 2s initial work

Student 3 initial work


Some kids’ work warranted more writing and some warranted less.  Of course if it was wonderful we just wrote something like, excellent work and perhaps wrote and extension question.  The hard part was filling out the rubric.  So for example, I’ll take Student 3’s work on problem 6 which is the last one above. Here is the rubric filled out for him:

Student 3’s Rubric

You will notice that I put problem 6 as a 1 for conceptual understanding and a 2 for skill level (in purple). In this problem students were asked to find a non-square quadrilateral with side lengths of sqrt(17).  Student 3 was definitely able to find vertices of a quadrilateral, but he was unable to use the PT to find common lengths of sides.  I gave him feedback that looking at sqrt(17) as a hypotenuse of a right triangle (as we had done in class) would help a bit and even wrote the PT with 17 as the hypotenuse in the hope of stimulating his memory when he did the revisions.

The day of the revisions Student 3 was only capable of producing this:

Student 3 revisions

He followed my direction and used 4 and 1 (which are two integers that give a hypotenuse of 17, but did not complete the problem by getting all side lengths the same. In fact, conceptually he kind of missed the boat on the fact that the sqrt(17) was supposed to be the side of the quadrilateral altogether.


One success story was Student 2.  She also did this problem incorrectly at first by realizing that you could use 4 and 1 as the sides of a right triangle with sqrt(17) as the hypotenuse but never found the coordinates of the vertices for me. I gave her feedback saying there might be an easier way to do this because she needed vertices.  However, she was able to produce this:

Student 2s revision

Student 2s revision

Although she did not give me integer-valued coordinates (which was not required) and she approximated which officially would not really give sqrt(17) lengths it came pretty darn close! I was impressed with the ingenuity and risk-taking that she used and the conceptual knowledge plus the skill-level. Yes, most other kids just used some combination of 1’s and 4’s all the way around but she followed her own thought pattern and did it this way.  Kudos to student 2 in my book.

Next time I will talk about some of the lessons we learned, other artifacts from the kids’ work and what we are changing for next time! Oh yeah and some great martini recipes!

PBL Summit News!

It’s been an extremely busy fall for me, but with the help of my friend Nils Ahbel, I have finally put together an informational flyer and schedule for the Problem-Based Learning Math Teaching Summit for next summer.  As you begin to look for professional development opportunities for yourself, please consider being a part of this great summit where like-minded math teachers can gather and share ideas.  Currently, we are making this information available and registration and final pricing will be available in January.  If you have any questions regarding the summit, please feel free to contact me.

Check out the PBL Math Summit Flyer 2015 here. For further information see the page on the PBL Summit.

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 at NCTM 2014!

One of my major goals in attending the NCTM annual conference this year was to see how widespread PBL had become in terms of mainstream education practices across the US.  I have to say that this year there were quite a few sessions that had PBL in the title or as the central theme and I was excited to see that!  Here were some of the workshops:

Problem-Based Learning (PBL) Is More Than Solving Problems – in this session the speakers were giving just a beginner’s view of what PBL is and can be in the classroom.

Change the Classroom, Not the Students – Attaining Equity Using PBL (OK, this one was mine)

Bring Back Problem-Based Learning into Methods Courses! – in this session the speaker makes an argument for using PBL methods in courses for teacher candidates and spoke about the positive experiences of preservice teachers with PBL.

Amplify the Mathematical Practices -this session focused on middle school PBL practices and how they stressed the CCSS MP standards.  This was sponsored by Amplify’s Math projects.

Making Mathematics Culturally Relevant to Students Using Problem-Based Learning – in this session, the speakers gave an example of culturally relevant pedagogy striving for equity in the classroom.  Again arguing that PBL allows for furthering equity in the practice of PBL.

Setting the Scene: Designing Your Problem-Based Classroom – in this session, the great Geoff Krall ( gave a great talk summarizing a lot of his methods relating to PBL and his protocols in getting students to work through problems in their learning.

The Hidden Message: Micromessaging and Mathematics – I wanted to attend this session so badly, but I had to leave early on Saturday morning.  This session has so much to do with my own research relating to how we talk to each other in mathematics classrooms and how PBL can allow for better communication without the micromessages.  (Tujuana if you read this – get in touch with me!)

Promoting Equity through Teaching for a Growth Mindset (Jo Boaler) – in this Session Prof. Boaler reported on her work in math education with Carol Dweck’s Mindset research.  You should check out her new website if you are interested in all the resources that she has shared freely.

And that was just to name a few!  So much wonderful information out there to learn and share.  The variety and number of sessions that connected to the pedagogy, content or philosophy of PBL was overwhelming and honestly very invigorating for me as someone who has taught with PBL for over 20 years.  Seeing the interest and enthusiasm for this type of classroom practice has given me renewed energy to get me through the rest of the year!

Having Students Be More Aware of their Contributions in PBL

One of the things I do at the middle of the term to have students reflect on the way that they talk about mathematics in class, is have them evaluate their work with my Student Self-Report on Class Contribution and I give them detailed feedback on their rankings of each type of question and what I think about their work so far.  But what I’ve found in the past is that it’s hard for them to remember specific examples of when they “helped to support the point by contributing evidence” or “raised a problem in another person’s solution” weeks or days after the fact.  Students have so much to think about during a class period that it really needs to be “in the moment” for them to be deliberately thinking about what they are doing and saying to each other.

So I had this idea that I would actually have them keep track of the types of questions and comments they made to each other over a five day period at the end of the term and then hand it in to me.  This way they would check off each category of question or comment at the moment when they did it.  So I made copies of this table of Student Analysis of Contribution and had them keep it on their desks while we were discussing topics or problems in class.  They had to have something to write with while they were talking and taking notes too.

Initially the students were very concerned about how they were going to tell the difference between each of the types of contributions.  But little by little, it became easier.  These categories were not arbitrary, they happened all the time, they just hadn’t really thought about it before.   For example, when a student presented a problem but had made an error and didn’t know it, another student usually “raised a complication in another person’s soltuion” or “pointed out an unspoken assumption or misunderstanding.”  These are important contributions that they were making every day about mathematics and are important critical thinking skills they didn’t even realize they were developing.  I just wanted to help them be more specific about realizing it.

The first day we did this I heard some students say something like, “Ooh, I just…(brief pause while looking at the table)… ‘started the group discussion on a rich, productive track by posing a detailed question.” with pride and excitement (and maybe a little sarcasm).  But after three days of doing it, and students seeing that they had places that they could check off, one student actually said, “I think that this table makes our conversations more interesting.”

Here are some samples of student feedback:

Quiet student who needs to work on all types of comments and questions

This first respondent is a student is very quiet, and she knows it.  She rarely speaks in class, but is actively listening.  I’ve spoken to her about why she does not ask questions or share her ideas in class and she says that she is afraid of being wrong in front of everyone.  My hope all year has been that seeing everyone else be wrong regularly would eventually show her how acceptable it was in class.  It was clear that using this table showed her that there are many different ways to contribute to class discussion.  When she presented her problems in class (which she will do when asked) she is very capable and knows that she is contributing evidence or examples, but she rarely questions others’ work.  I do remember the time when she “built on” what was said by asking a question about someone else’s solution.  She also sometimes asks for clarification of her own understanding, but I know that she’s capable of more.  I struggle with how to encourage her to get more out of class discussion, but at least now she knows how important a role she can play.

Outgoing student who lacks interaction from other kids

This is a really interesting student who is quite inquisitive and very comfortable with sharing his ideas.  When he has a question or comment, it’s very easy to get him to go to the board and naturally start writing another idea or possible solution method.  However, I noticed was missing in his table were checks in the two rows that had to do with “inviting others” into his thought process and also making comments on others’ solutions and ideas.  It made me wonder how much time in class he spends listening to other students talking or if he is just listening to his own ideas in his own mind.  I am working on trying to get him to collaborate more – with his creative mind it would behoove him to start interacting more with others.

Outgoing student who is unaware of his effect on others

This table belongs to a student who really has overestimated his contributions to class.  He definitely spends a lot of time talking (and hence I spend a lot of time managing his unrelated talking unfortunately) and thinks that talking – any kind of talking – is useful and contributing to class discussion.  One of the things I have talked to him about is the idea of active listening and how important not speaking can be.  He has not yet caught onto the idea of being respectful while listening, and still is considering his next talking move while others are trying to make their points.  It will be a difficult discussion, but a necessary one for this young man to grow in important ways.  At least I now have this chart to refer to when I have that discussion.

So was this exercise everything I had hoped?  Not really – but it definitely had some great highlights.  Class discussion was very exciting and interesting while students were aware and deliberate knowing the different types of ways in which they could contribute.  Knowing that they had to hand this table in to me in 5 days was putting the onus on them to show that they had or had not fulfilled what I had observed of them in class.  I do believe they learned a lot about what they were capable of.  I do believe I would do it again.

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)