Looking at PBL Practice from a Thematic Perspective

So I’m here down in Florida – loving it (all sing-songy like Oprah would say).  I’ve been to so many talks that have been great learning experiences so far.  The weather is beautiful – I went for a very long walk and tried to think about what my talk was missing.  I did a bunch of edits and now I think I’m ready to post it.

Here’s the powerpoint of the talk:

Here’s the document that I handed out with some “threads” of themed topics:

Three Threads Document

Please contact me with any questions, comments or concerns – I love talking to people about PBL and my work.

I’ve looked at life from both sides now…

This past July, I spent a few days at the MAA Mathfest in Chicago for the first time. The main reason I went was because the Academy of Inquiry-Based Learning was having a Conference within the Mathfest with the theme of “Diversity in IBL.”  IBL is generally what college faculty call the type of teaching and learning that many of us at the secondary level has been calling PBL for years.  I was so interested to hear many mathematics professors talking about the struggles of writing curriculum, dealing with facilitating discussion, using writing – all of the same parts of this type of teaching that we may have been talking about for so any years.  I highly suggest that we could benefit from talking to each other.  If you would like to get involved with this movement, Stan Yoshinobu, the Director of the Academy of Inquiry-Based Learning, has put forth some challenges for his community.  Check them out.

One of the most interesting talks that I attended was by a professor from Denison University, Lew Ludwig, titled, “Applying Cognitive Psychology to the Mathematics Classroom.”  As a devout social constructivist, I generally like going to talks where I can learn more about other views of education.  Seeing both sides definitely helps me understand many of the views of my colleagues and see if evidence supports my own perspective. Ludwig had published a review of another article that was titled, “Inexpensive techniques to improve education:  Applying cognitive psychology to enhance educational practice”(Roediger and Pye, 2012).

Basically, the presentation summarized three simple techniques that cognitive psychology had evidence helped student learning. The three techniques were called

  1. The distribution and interleaving of material and practice during learning.
  2. Frequent assessment of learning (test-enhanced learning, continual assessment)
  3. Explanatory questioning (elaborative interrogation and self explanation; having students ask themselves questions and provide answers or to explain to themselves why certain points are true).

In the original article, the authors write:

“Repetition of information improves learning and memory. No
surprise there. However, how information is repeated determines
the amount of improvement. If information is repeated back to back
(massed or blocked presentation), it is often learned quickly but
not very securely (i.e., the knowledge fades fast). If information is
repeated in a distributed fashion or spaced over time, it is learned
more slowly but is retained for much longer”

When this was reported, I was first in shock.  I couldn’t believe I was hearing something in a presentation about Cognitive Psychology that was actually supported by the definition of PBL that I use.  The curriculum I use takes the idea of looking at topics and teaching them over a longer time span, but distributed among other topics.  I have called this decompartmentalization of topics, which helps students see the connectedness of mathematics.

The second idea, consistent assessment, is based on the concept that testing is not really a great measure of how much a student has learned, but it actually solidifies the learning that has occurred.  So three groups of students were given different ways of learning by reading a passage of information. The first group read a passage four times. The second group read the passage three times and had test.  The third group read the passage once and was tested three times.  Their performances on tests on the information in the passage 5 minutes later and then one week later.

Diagram of retention testing research

from Roediger & Pye (2012) p.245

So if we connect the idea that testing is not the best method of seeing how much students have learned and the fact that consistent assessment actually helps students retain their knowledge, what I do in my PBL classrooms, is not only “test” but do all sort of forms of assessment (writing, oral assessment, hand-in homework with feedback, labs, quizzes, problem sets, self-assessment, etc.) alternately throughout the term.  There is probably not a week where students are not assessed in at least 2 ways. I feel that this has led students to have good retention of material and the assessments are strong measurements of their learning.

The last one was the one I was most excited to hear about – explanatory questioning.  This seemed to give students so much more responsibility for their own learning than traditional cognitive psychology as I had understood it.  The authors of this study claim that explanatory questioning can be broken in to two areas:

Elaborative Interrogation – students generating plausible explanations to statements while they are studying and learning.  This speaks directly to the idea of mathematical discussion and how students generate explanations when they ask themselves “why?”

Self-Explanation – students monitoring their learning and describing, either aloud or silently some features of their learning.  This idea can be found all over the PBL classroom but in mine, it’s generally found most in metacognitive journaling where students use self-explanation the most.

“Obviously, the elaborative interrogation and self explanation are related because both strategies encourage or even require students to be active learners, explaining the information to themselves (perhaps rephrasing in language they understand better) or asking themselves why the information is true.”

I honestly couldn’t believe what I was reading – this is an article on educational methods based on cognitive psychology that is suggesting that we require students to be active learners and “explain the infomation to themselves”?  This is lunacy.  I have been teaching for 25 years where students have been complaining to their parents that they have had to explain things to themselves – who would’ve known that I was applying cognitive psychology?

My guess is that these ideas are only enhanced by the social aspect of the classroom and other constructivist ideas – clearly the constructivitst classroom in enhanced by or agrees with some of these cognitive psychology methods as well.  Listening to both sides of the theories is actually helpful and I’m seriously going to continue doing this! Although I never thought that there might be strong connections between cognitive psychology theories and PBL, I do know that it’s life’s illusions I recall and I “really don’t know life, at all.”

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

Connections Between IBL and PBL

At the PBL Summit a few weeks ago, we had two wonderful speakers, Julian Fleron and Phil Hotchkiss from Westfield State University who are founding members of the Discovering the Art of Mathematics Project.  They gave a great key note address on Friday night about Inquiry-Based Learning and motivating students in an IBL classroom.  You can find their talk at our Summit Resources website if you are interested.  I wrote a blogpost a few years ago about my interest in IBL and the commonalities between PBL and IBL and I thought I’d reshare in honor of them.  Enjoy!

A number of years ago, I needed some kind of suport text for a Number Theory tutorial that I was doing with two rather advanced students who had gone through the curriculum at the school where I was teaching.  These two girls were advanced enough that I knew that if I used my notes and problems from my wonderful Number Theory course from college (some many years ago) we would have a great time.  I looked online and found a great book called “Number Theory Through Inquiry” published by the MAA which came with an instructor’s supplement including pedagogical discussion and some solutions.  It sounded so much like what I was doing with my other classes that I couldn’t turn down the opportunity to see what it was like.  So I ordered the book and while I was reading the instructor’s supplement I came across something that I had not heard about before (and now I am so embarrassed to admit this). The authors described what they called the “Modified Moore Method” of instruction or Inquiry-Based Learning and went on to describe what sounded interestingly so much like what I was doing in my classroom.  I had to learn about this Moore Method.

I ended up researching R.L. Moore online and it seems that he was one of the first math teachers – ever – to think about and act on this idea of not teaching mathematics with direct instruction.  He did it all the way back in 1948, but at the college level – and it was radical there!  The idea of Inquiry-Based Learning has expanded from there, but it has really only stayed at the college level in mathematics for a very long time.  There are many initiatives at the college level, including the folks at Westfield State University who are writing a wonderful curriculum project funded by the NSF called Discovering the Art of Mathematics with is a math for liberal arts curriculum at the college level.  I think it could be used at the secondary level as well for an alternative elective in the senior year for those students who still want to take a college-level math course but aren’t ready for or interested in an AP course in Calculus or Stats.  If there are any secondary teachers interested in beta-testing this unique curriculum please contact me and let me know.  I am on the advisory board for this project.

What made me think about the connection between IBL and PBL was this wonderful blogpost I just read by Dana Ernst, of Northern Arizona University in which he describes, in such wonderful ways, the pedagogy and nature of IBL.  The similarities between the definition of IBL he cites (by E. Lee May) and my definition of PBL are eerie – and it is one of the only ones that I’ve seen that stresses a reference to teacher authority being diminished.  Many wonderful resources are given by Ernst at the end of his post as well.

I do remember back in 2003, when I published my first article on my experiences at Emma Willard, after I left Exeter (where they called in Harkness teaching because of the table), in attempting to teach the way I wished to.  I had no idea what to call what I was doing.  I believe in my first article I called it teaching with a Problem-Solving Curriculum (PSC).  After I started my doctoral work, I found PBL and I realized that’s what it was.  Then I read more and more and realized that others thought PBL was project-based learning and called what I did discovery learning.  After reading about R.L. Moore, it sounds like he was doing it all along since 1948 and called it IBL.  In whatever branch of the pedagogical family tree you find yourself, if you are asking students to look at mathematics with wonder and question what they know – you should know that you are supported, know that you are doing good work and know that there is someone out there who has done it before and wants to discuss it with you.

PS – I’m hoping to attend the Legacy of R.L. Moore Conference next year in Austin, if anyone is interested!

Think about where the learning happens in PBL

After a few weeks of recovery, I wanted to write about having a BLAST of a time at our first attempt of putting together the PBL Summit my friend Nils Ahbel and I organized from July 16-19.  I wanted to thank all of those who came and participated in the discussions and talks and who shared their ideas so freely.  It’s such a great reminder of the huge resources we all are to each other as math teachers.  I know that I at least tripled my Professional Learning Network and hope that all of the participants did too.

I’d also like to thank everyone who gave feedback and the amazing ideas for next year – including a pre-conference session for those of you who might have been PBL “newbies” and might have needed more of an intro, topic-level groups, more in-depth SIGs for people who want to dive deeply into writing or assessment writing too.  The ideas just kept flowing and I think we will have a wonderful plan for next year too.

One of the take-aways that I left the PBL Summit with was how differently people view what “learning” means in PBL.  From my long career both teaching and studying PBL, I have had a lot of time to form my own frameworks for student knowledge construction and pedagogical theory and often take for granted that all of us are on the same page. As I have traveled and talked to many other math teachers and heard others who are experts in PBL (both PjBL and PrBL) speak, I realize more and more that we are often NOT on the same page.  This does not mean that any one of us is more right or wrong.  We just need to understand each other more.

My big question to everyone I talk to is “where/when does the learning happen?” or “where/when do the students construct their knowledge and understanding of the mathematics?”  If students are presented with a problem, for example they watch a wonderful interesting video of a basketball player shooting at a basket or watching someone fill a water tank and they come up with their own question based on a real-life phenomenon from the video, how do those students know the mathematics to answer those interesting questions?  If students are sitting through direct instruction lessons to be exposed to the mathematics but using them to answer their own questions, this is definitely an improvement than passive mathematics classes of the past.  Having students take ownership of the material in this way is is a powerful method of creating agency for mathematics learning.  The problems that they are solving and from where they are posed are extremely relevant to the motivation and agency in learning.

I would posit that PBL can be more and mean more and in more ways to student learning. Even when posed with a good problem (one they did not come up with themselves).  In PBL, students can:

  • see the need for a new method without the teacher introducing it
  • see the need for discussing other students’ ideas
  • find their own organizational strategies for problem solving
  • access prior knowledge that they did not realize they needed before
  • use their resources to discuss the problem with each other
  • use resources to find new solutions and follow their own thinking
  • make connections between topics in mathematics that they might not have realized before
  • create community in the mathematics classroom (like in other disciplines – humanities, fine arts and science)
  • realize that reflection is one of the most important parts of the learning process
  • learn to relate to others in math class
  • see mathematics as a creative endeavor

and so much more. I’d love to hear from people some that I have left out.  In my mind, even the mathematical learning happens in these contexts and students are the shapers of where and when this happens.  Robert Kaplinsky is one of those amazing PBL teacher/speakers who has a somewhat different approach than I do, but is very similar in many ways and I heard him say this April, “Don’t teach what students need to know before they do a problem-based lesson.” In that way, we are all on the same page, for sure.

One of the Original “Makers”

Apologies to any faithful readers out there – I have had a heck of a summer – way too much going on.  Usually during the summer, I keep up with my blog much more because I am doing such interesting readings and teaching conferences, etc. (although I’m running a conference for the first time in my life!) However, this summer I was dealing with one of my biggest losses – the passing of my father after his 8 year battle with breast cancer.  I thought I would honor him by writing a post talking about a problem that I wrote a few years ago, well actually a series of problems that utilized his work when teachers of algebra I asked me how I taught the concept of slope.  So dad, this one’s for you.

In 1986, my dad, Francesco (Frank) Schettino, was asked to work on the renovations for the centennial project for the Statue of Liberty.  He was a structural steel detailer (also known as a draftsman) but he was really good at his job.  Everywhere we went with my dad when I was younger, he would stop and comment about the way buildings were built or if the structure of some stairs, windows or door frames was out of wack.  He could tell you if something was going to fall down in 10 years, just by looking at it.  At his wake last week, one of the project managers from a steel construction company that he worked on jobs for told me that they would save the interesting, most challenging jobs for him because they knew he would love it and do it right.
photo (1)I remember sitting with my dad at his huge drafting desk and seeing the drawings of the spiral stairs in the Statue of Liberty.  He talked to me about the trigonometry and the geometry of the circles that were necessary for the widths that were regulated for the number of people that they needed to walk up and down the stairs.  This all blew my mind at the time – that he needed to consider all of this.  So to be able to write problems that introduce slope to students about this was just a bit simpler to me.

If you take a look at my motivational problems on slope and equations of lines I believe it’s numbers 2 and 3 that refer to his work (excuse the small typo).  Over the years I’ve meant to go back and edit these a number of times.  If you are someone who has taken my course at the Anja S. Greer Math, Science and Technology Conference at Exeter, you are probably familiar with this series of questions because we have discussed these at length and talked about how students have reacted to them (and how different adult teacher-students have as well).  We have assumed no prior knowledge of slope (especially the formula) or the terminology at all.

Some questions that have come up: (with both students and the teacher-students I’ve worked with)

1. What does a graphical representation of “stairs” mean to students?
2. What does “steeper” mean and what causes stairs to be steep?
3.  Why are we given the “average” horizontal run for the spiral stairs? Would another measurement be better?
4. Why does the problem ask for the rise/run ratios?  Is there a better way to measure steepness?
5. (from a teacher perspective) why introduce the term “slope” in #3? can we just keep calling it steepness?

These are such rich and interesting questions. The questions of scaffolding terminology and when and how to introduce concepts are always the most difficult.  Those we grapple with specifically for our own students.  I always err on the side of allowing them to keep calling it steepness as long as they want, but as soon as we need to start generalizing to the abstract idea of the equation of the line or coming up with how to calculate that “steepness” a common language of mathematics will be necessary.  This is also where I take a lesson from my dad in terms of my teaching.  His great parenting style was to listen to me and my sisters and see where we were at – how much did we know about a certain situation and how we were going to handle it.  If he felt like we knew what we were doing, he might wait and see how it turned out instead of jumping in and giving advice.  However, if he was really worried about what was going to happen, he wouldn’t hesitate to say something like “Well, I don’t know…”  His subtle concern but growing wisdom always let us know that there was something wrong in our logic but that he also trusted us to think things through – but we knew that he was always there to support and guide.  There’s definitely been a bit of his influence in my career and maybe now in yours too.

NCTM 2015 – Reflections

I know I’m a little late but I did want to post my own handouts and talk a little bit about my experiences at NCTM Boston this year.  I want to thank all of the great speakers  that I saw including Robert Kaplinsky, Ron Lancaster, Maria Hernandez, Dan Teague, The Young People’s Project (Bob Moses’ Group), Deborah Ball, Elham Kazemi and of course the inspiring Jo Boaler.  One of the things I thought was great about Jo Boaler’s talk on Thursday night was that even though I had heard a great deal of what she had said before, there was a different tone in the room.  I’ve been a fan of Jo’s since I first read her research in 2001 when I started my doctoral work on girls’ attitudes towards mathematics learning.  What I felt that was different that night was that she was no longer trying to convince people of anything.  There was a different message and that was “join the revolution” and the audience seemed to be on board and excited.  It made me feel very energized and empowered that a huge ballroom full of mathematics educators had bought into her ideas and were enthusiastic to make change happen.

Some of the best times I had were spent just connecting and reconnecting with people – some who I met for the first time (MTBoS folks and other Twitter folks I met F2F which was really nice) and others who were old friends who mean a great deal to me.  I forget how much the mathematical community of professionals enriches my life and makes me proud to do what I do.  Thanks to everyone who reached out to find me and say hi – or tell me a story, talk to me about what they are doing or ask a question about what I am doing.  You are all inspiring to me.

I left the conference with exciting ideas about teacher observation for PD, how teachers can share problems with each other better on the internet (awesome resources at Robert Kaplinsky’s problem-based lesson site), great ideas about agent-based models to add to courses, and ways in which teachers can talk to people about the Common Core and gain respect about the difficult work we do in teaching.  Overall, I felt like it was an amazing time.

I want to thank everyone that came to my session.  Although I had an unfortunate technological snafu and was unable to do an exercise I had planned where we were going to analyze a segment of discourse from my classroom using the framework of the MP standards (which would’ve been great), I felt that at least the resources that I shared were worthwhile for the people that came.  Here is a link to the powerpoint presentation and the handouts I gave.


Handout 1 NCTM 2015 Schettino

Weekly Learning Reflection Sheet

Handout 2 Schettino

I’ll just put in one more plug for our PBL Summit from July 16-19 this summer – we still have a few more spots and would love to have anyone interested in attending!

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.

The First Followers…how do I get them in the PBL classroom?

So I have one class this year that is rather frustrating and pretty tough to handle when it comes to buy-in with what I’m doing in the mathematics classroom.  Perhaps it’s because it’s first period, or perhaps it’s the mix of kids (quiet, shy, cynical?) – but I’m having a hard time inspiring them to speak out about their ideas or even be somewhat active in class.  This has made me wonder if I’m doing anything differently?  What’s the difference between first period and second period?  Why would this class be that much different in student make-up and personality than other challenging classes that I’ve had.

These thoughts made me remember a video I saw at a conference talk this summer and how important the “first followers” are.  This video is basically about a guy the narrator calls the “lone nut” who is dancing at a music festival (maybe you’ve seen it, it’s been around for a while) and how his leading becomes a “movement.”

It’s one of my favorites and so true.  But what I am afraid of is that the “first followers” I had in my first period class are not necessarily “followers” but students who realized they better do what I want or they won’t do well in my class.  This is not the same as “buy-in” to PBL.  This led me to think about what I needed to do in order to create first followers who would truly be inspiring and lead to more followers.  I’m not sure about this, but a couple things I tried:

  • talking about the pedagogy and how it’s different with the students
  • discussing the class contribution rubric with them and having them do a self-evaluation of their contributions to class
  • discussing listening skills when learning
  • Being deliberate about asking questions that are more open-ended (not just procedural)
  • Being less “forgiving” that it’s first period and they are tired – keeping my standards up of what I expect from them.
  • Giving praise when students take risks and learn from mistakes at the board
  • Offering a reward (like a Pez Candy) when a student is wrong but has taken a risk

So far my attempts have not been in vain, but I still don’t feel the “movement” as I do in other classes.  This has been an interesting first month with this group and I think many of them are actually learning, but don’t seem to be enjoying themselves.  I think I just need a couple more “first followers” to allow the others to see that what I am asking of them – although harder and requiring more energy and effort on their part – is actually an important part of their journey of learning.  I would love to hear from anyone who has experienced this and what steps can be taken to increase the followers in a “mob” of the whole class!