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

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.

Anja S. Greer Conference 2013

What a great time we had this week in my courses!  I am so excited by all of the folks that I met and the CwiC sessions of other leaders that I went to.  Pretty awesome stuff presented by Maria Hernandez from NCSSM, my great colleague Nils Ahbel, Tom Reardon, Ian Winokur, Dan Teague, Ken Collins and many others.  I was so busy that I didn’t get to see many other people’s sessions so I feel somewhat “out of it” unfortunately.

I want to thank everyone that came to my CwiC’s and remind them to be sure to go and pick up my materials on the server before they leave.

For my participants – here are the links to the course evaluations:

Moving Forward with PBL: Course Evaluation

Scaffolding and Developing a PBL Course:  Course Evaluation

PBL – Students making Mathematical Connections

As someone who has used Problem-Based Learning for almost 20 years and sad to say has never been part of a full-fledged Project-Based Learning curriculum, what I know best is what I call PBL (Problem-Based Learning).  I know there is a lot of confusion out there is the blogosphere about what is what, and with which acronyms people use for each type of curriculum.  I did see that some people have been trying to use PrBL for one and PBL for the other, but I guess I don’t see how that clarifies – sorry.

So when I use the acronym PBL in my writing I mean Problem-Based Learning and my definition of Problem-Based Learning is very specific because it not only implies a type of curriculum but an intentional relational pedagogy that I believe is needed to support learning:

Problem-Based Learning (Schettino, 2011) – An approach to curriculum and pedagogy where student learning and content material are (co)-constructed by students and teachers through mostly contextually-based problems in a discussion-based classroom where student voice, experience, and prior knowledge are valued in a non-hierarchical environment utilizing a relational pedagogy.

Educational Psychologist and Cognitive Psychologists like Hmelo-Silver at Rutgers University have done a lot of research on how students learn through this type of scaffolded problem-based curriculum dependent on tapping into and accessing prior knowledge in order to move on and construct new knowledge.  There was a great pair of articles back in 2006/2007 where Kirschner, Sweller & Clark spoke out against problem- and inquiry-based methods of instruction and Hmelo, Duncan and Chinn responded in favor.  I highly recommend reading these research reports for anyone who is thinking of using PBL or any type of inquiry-based instruction (in math or any discipline).  It really helps you to understand the pros and cons and parent and administrator concerns.

However, after you are prepared and know the score, teachers always go back to their gut and know what works for their intuitive feeling on student learning as well.  For me, in PBL, I look at how their prior knowledge connects with how, why and what they are currently learning.  One of the best examples of this for me is a sequence of problems in the curriculum that I use which is an adaption from the Phillips Exeter Academy Math 2 materials.  I’ve added a few more scaffolding problems (see revised materials) in there in order to make some of the topics a bit fuller, but they did a wonderful job (which I was lucky enough to help with)and keep adding and editing every year. The sequence starts with a problem that could be any circumcenter problem in any textbook where students use their prior knowledge of how to find a circumcenter using perpendicular bisectors.

“Find the center of the circumscribed circle of the triangle with vertices (3,1), (1,3) and (-1,-3).”

Students can actually use any method they like – they can use the old reliable algebra by finding midpoints, opposite reciprocal slopes and write equations of lines and find the intersection points.  However, I’ve had some students just plot the points on GeoGebra and use the circumcenter tool.  The point of this problem is for them to just review the idea and recall what makes it the circumcenter.  In the discussion of this problem at least one students (usually more than one) notices that the triangle is a right triangle and says something like “oh yeah, when we did this before we said that when it’s an acute triangle the circumcenter is inside and when it’s an obtuse triangle the circumcenter is outside.  But when it’s a right triangle, the circumcenter is on the hypotenuse.”

Of course then the kid of did the problem on geogebra will say something like, “well it’s not just on the hypotenuse it’s at the midpoint.”


Dicussion will ensue about how we proved that the circumcenter of a right triangle has to be at the midpoint of the hypotenuse.

A day or so later, maybe on the next page there will be a problem that says something like

“Find the radius of the smallest circle that surrounds a 5 by 12 rectangle?”

Here the kids are puzzled because there is no mention of a circumcenter or triangle or coordinates, but many kids start by drawing a picture and thinking out loud about putting a circle around the rectangle and seeing they can find out how small a circle they can make and where the radius would be.  When working together oftentimes a student see a right triangle in the rectangle and makes the connection with the circumcenter.

A further scaffolded problem then follows:

“The line y=x+2 intersects the circle  in two points.  Call the third quadrant point R and the first quadrant point E and find their coordinates.  Let D be the point where the line through R and the center of the circle intersects the circle again.  The chord DR is an example of a diameter.   Show that RED is a right triangle.”

Inevitably students use their prior knowledge of opposite reciprocal slope or the Pythagorean theorem.  However, there may be one or two students who remember the circumcenter concept and say, “Hey the center of the circle is on one of the sides of the triangle.  Doesn’t that mean that it has to be a right triangle?”  and the creates quite a stir (and an awesome “light bulb” affect if I may say so myself).

A few pages later, we discuss what I like to call the “Star Trek Theorem” a.k.a. the Inscribed angle theorem (I have a little extra affection for those kids who know right away why I call it the Star Trek Theorem…)

I will always attempt to revisit the “RED” triangle problem after we discuss this theorem.  If I’m lucky a student will notice and say, “Hey that’s another reason it’s a right triangle – that angle opens up to a 180 degree arc, so it has to be 90.”  and then some kid will say “whoa, there’s so many reasons why that triangle has to be a right triangle”  and I will usually ask something like, “yeah, which one do you like the best?” and we’ll have a great debate about which of the justifications of why a triangle inscribed in a circle with a side that’s a diameter has to be right.  So who are the bigger geeks, their teacher who names a theorem after Star Trek or them?


Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential and inquiry-based teaching. Educational Psychologist, 41(2), 75-86.


Hmelo, C. E., Duncan., R.G., & Chinn, C. A. (2007). Scaffolding and achievement in problem-based and inquiry learning: A response to Kirschner, Sweller and Clark (2006). Educational Psychologist, 42(2), 99-107.