The “Wounded Healer” Archetype in the PBL Teacher

I have been doing a lot more work with teachers this year as I am not in the classroom.  I love watching people teach and talking to them about their teaching.  It is clearly a passion for so many people and the modeling of lifelong learning has been so inspirational for me and their students.

One issue that seems to arise in all PBL classrooms, no matter how progressive the teacher, is this feeling that they need to somehow, someday really just not allow the students to be frustrated.  Even those who buy into the whole PBL, student-centered, productive struggle pedagogy – deep inside they understand the belief from their own education, that math is black-and-white there needs to be some resolution that is acknolwedged and /or provided by the teacher.

I was talking to a friend about this dilemma a while ago (thanks @phiggiston!) and saying how interesting it is to me that a teacher’s belief from their past can, in the moment, while teaching, often override their beliefs in the current pedagogy.  In other words, if a teacher has not experienced independent learning as needed in PBL, it is extremely difficult to not give into the impulse to “save” the students from that feeling of struggle or unease.

Well, coincidentally, @phiggiston has a background in both religious work and in psychotherapy training, so the first the he says to me is, “it’s kind of like the patient-therapist relationship in a way.” And I’m thinking, my teaching is nothing like being a therapist, but of course, I listened intently.  I guess there is a Jungian theory that says that “sometimes a disease is the best training for a physician.”  In fact, Jung goes as far as to say that

“a good half of every treatment that probes at all deeply consists in the doctor examining himself, for only what he can put right in himself can he hope to put right in the patient.”


So what does this mean for PBL teaching?  I had to think of this for a while and also read some Jung as I am not up on the psychological theories that connect to education.  I wasn’t quite sure that this “Wounded Healter” achetype paralleled the PBL teacher as much as I originally thought.  Here are some points:

  • Jung says that for the wounded healer the therapeutic encounter should be regarded as a dialectical process  It’s not just I’m going to the doctor and she’s going to tell me what wrong with me.  There needs to be some kind of dialogue in order for a real healing to happen.  In the classroom, I would argue that this is true about the teacher-student relationship.  Traditionally, it has been that not having dialogue would result in learning that was not as long-lasting, effective and/or connected to the students own ideas.  It is pretty clear that the PBL teacher needs to create the dilectical process in order for the best learning to happen.
  • Jung argues that the physician must help create a safe space where the “patient’s “inner healer” is made available to her unconsciously.” At the same time the physician, should let go of the way she is activiated by the same wounds. This idea is extremely relevant in the PBL classroom.  Why do we want to make students comfortable and relieve their anxiety about mathematical learning?  My take would be because we hate the way it makes us feel. Knowing that struggle is all to close in our memory can actually help us hand over the power to “heal themselves.”  If we can get over that feeling, it will become more of the norm in the classroom.
  • There are risks to this type of teaching – the risk of being vulnerable because you are looking at your own wounds, and also looking fragile to the patient (or student).  This is a very common concern of teachers who are beginning PBL teaching.

“The experience of being wounded does not make him/her less capable of taking care of the patient’s disease; on the contrary, it makes him/her a companion to the patient, no longer acting as his/her superior.”

In other words, it is worth the experience of creating that open relationship.  I go back to Hawkins’ theory of learning (I-thou-It) in which the relationships that exist form a triangle between teacher-student-material.

Hawkins (1974)

Hawkins (1974)

All of these relationships must be nurtured in order for the best learning environment to exist. (For more on this check out Carol Rodgers presentation slides here.)


So does this mean if you did not have this type of experience learning math that you can’t learn to empower your own students in this way?  I think not.  When I ilook back on my own mathematical experiences many of them were extremely traditionally taught.  However, I think what you need to have inside you is both the belief that students are capable of owning and constructing their own knowledge and the ability to create a space that allows them to remain uncomfortable.  You have to be willing to let go of your own insecurities and anxieties about learning math and realize that the more you do that, the more the students will feel it as well.

I am currently working on a quasi-research project about this and when/how PBL teachers choose to intervene in class discussion.  If there is anyone who is interested in helping me out with this, I’d really appreciate it.

Yours, Mine and Ours

Yesterday we had a speaker in our faculty meeting who came to talk to us about decision-making process in our school.  He spoke about the way some colleges, universities, independent schools are very different from businesses, the military, and other governing bodies that have to make decisions because we are made up of “loosely-coupled systems.” These are relationships that are not well-defined and don’t necessarily have a “chain of command” or know where the top or bottom may be.  They also don’t necessarily have a “go-to” person where, when a problem arises, the solution resides in that location.  The speaker said that this actually allows for more creativity and generally more interesting solution methods.

About mid-way through his presentation he said something that just resonated with me fully as he was talking about the way these systems come to a decision cooperatively.

“The difference between mine and ours is the difference between the absence and presence of process.”

Wow, I thought, he’s talking about PBL.  Right here in faculty meeting.  I wonder if anyone else can see this.  He’s talking about the difference between ownership of knowledge in PBL and the passive acceptance of the material in a direct instruction classroom.

Part of my own research had to do with how girls felt empowered by the ownership that occurred through the process of sharing ideas, becoming a community of learners and allowing themselves to see others’ vulnerability in the risk-taking that occurred in the problem solving.  The presence of the process in the learning for these students was a huge part of their enjoyment, empowerment and increase in their own agency in learning.

I think it was Tim Rowland who wrote about pronoun use in mathematics class (I think Pimm originally called it the Mathematics Register). The idea of using the inclusive “our” instead of “your” might seem like a good idea, but instead students sometimes think that “our” implies the people who wrote the textbook, or the “our” who are the people who are allowed to use mathematics – not “your” the actual kids in the room.  If the kids use “our” then they are including themselves.  If the teacher is talking, the teacher should talk about the mathematics like the are including the students with “your” or including the students and the teacher with “our”, but making sure to use “our” by making a hand gesture around the classroom.  These might seem like silly actions, but could really make a difference in the process.

Anyway,  I really liked that quote and made me feel like somehow making the process present was validated in a huge way!

Considering Inclusion in PBL

It’s always refreshing when someone can put into words so eloquently what you have been thinking inside your head and believing for so long.  That’s what Darryl Yong did in his recent blogpost entitled Explanatory Power of the Hierarchy of Student Needs.  I feel like while I was reading that blogpost I was reading everything that I had been thinking for so long but had been unable to articulate (probably because of being a full time secondary teacher, living in a dorm with 16 teenage boys, being a mother of two teenagers of my own and all the other things I’m doing, I guess I just didn’t have the time, but no excuses).  Darryl had already been my “inclusive math idol” from a previous post he wrote about radical inclusivity in the math classroom, but this one really spoke to a specific framework for inclusion in the classroom and how in math it is necessary.


In my dissertation research, I took this idea from the perspective of adolescent girls (which, as I think towards further research could perhaps be generalized to many marginalized groups in mathematics education) and how they may feel excluded in the math classroom.  These girls were in a PBL classroom that was being taught with a relational pedagogy which focuses on the many types of relationships in the classroom (relationship between ideas, people, concepts, etc.)  – I did not look at it from the perspective of Maslow’s Hierarchy of Student Needs and this is really a great tool.

Interestingly,  I came up with many of the same results. My RPBL framework includes the following (full article in press):

  1. Connected Curriculum– a curriculum with scaffolded problems that are decompartmentalized such that students can appreciate the connected nature of mathematics
  2. Ownership of Knowledge – encouragement of individual and group ownership by use of journals, student presentation, teacher wait time, revoicing and other discourse moves
  3. Justification not Prescription– focus on the “why” in solutions, foster inquiry with interesting questions, value curiosity, assess creativity
  4. Shared Authority – dissolution of authoritarian hierarchy with deliberate discourse moves to improve equity, send message of valuing risk-taking and all students’ ideas

These four main tenets were what came out of the girls’ stories.  Sure many classrooms have one or two of these ideas.  Many teachers try to do these in student-centered or inquiry-based classrooms.  But it was the combination of all four that made them feel safe enough and valued enough to actually enjoy learning mathematics and that their voice was heard. These four are just a mere outline and there is so much more to go into detail about like the types of assessment (like Darryl was talking about in his post and have lots of blogposts about) the ways in which you have students work and speak to each other – how do you get them to share that authority when they want to work on a problem together or when one kid thinks they are always right?

The most important thing to remember in PBL is that if we do not consider inclusion in PBL then honestly, there is little benefit in it over a traditional classroom, in my view. The roles of inequity in our society can easily be perpetuated in the PBL classroom and without deliberate thought given to discussion and encouragement given to student voice and agency, students without the practice will not know what to do.  If we do consider inclusion in the PBL classroom, it opens up a wondrous world of mathematical learning with the freedom of creativity that many students have not experienced before and could truly change the way they view themselves and math in general.

Disruption in Presence: Missing PBL Math Class

What do we all do with kids who miss out on the wonderful rich discussions where the learning happens in a PBL math class? @0mod3 asks what to do about kids’ absences. (thanks for the great question!)

It’s not as simple as “get the notes from somebody who was there” is it?  What did they actually miss by not being in class? Yes, new vocabulary possibly, new concepts, whether their problems were right or wrong – these things can all be “looked up” in some ways in another students notes or with a conversation with the teacher or a tutor just like in any other mathematics class.  So what is it we are really concerned with that they missed?

It seems that DReycer is hitting the nail on the head in her second tweet here.  Of course, it’s the experience of being a part of the rich mathematical discussions.  Hearing other students’ ideas and deciding for themselves or analyzing critically in the moment what they think of those ideas – is it right? wrong? potentially right? more efficient? similar to what I did?  These experiences are very hard to re-simulate for students who are absent from the PBL classroom.

When students come to me who have missed class.  I do tell them to look at other students’ notes.  However, this is because of how I tell students in my classes to take notes.  Kids are supposed to attempt the homework problems on one side of the notebook and then on the other side take “note of” what the other student who is presenting the problem did differently from them.  Eventually when we, as a class, come to some type of consensus about how the problem connects to a new concept or to a problem we have already done, it is then that a student should take note of the new idea as we formalize it into a theorem or new idea.

Absences will always be a problem for us who teach in the PBL classroom since we can’t recreate the in-the-moment learning that happens when a student sees another’s presentation (unless you feel like having parental consent for recording every single class, and even then you can’t really have the interaction with the student that missed it) however, what you can do is make the most of the time when each kid is there. PBL is by its nature relational learning and student and teacher presence is extremely important.  Be sure that students are the ones who are talking and asking questions in order for them to actively be engaging with the presenter.  Be sure that you are present to their needs when they return from an absence.  On days when they are not there, it might be enough for them to ask questions on the next day after they have read through the vocabulary or seen someone’s complete solution.  Sometimes active learning the next day can just be enough.

I’d love to hear other people’s ideas and thoughts!

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.

Tracking, PBL and Safety in Risk-Taking

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

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

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

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

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

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

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

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

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

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

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

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

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

Thanks John!

How do we get kids to value others’ ideas in math class?

Some recent common situations:

A very gifted student comes to me (more than once) after class asking why he needs to listen to other students talk about their ideas in class when he already has his own ideas about how to do the problems.  Why do we spend so much time going over problems in class when he finished all the problems and he has to sit there and listen to others ask questions?

A parent asks if their child can study Algebra II over winter break for two weeks and take a placement test in order to “pass out” of the rest of the course and not have to take mathematics.  A college counselor supports this so that they can move forward in their learning and get to Calculus by their senior year.

Tweet from a fellow PBL teacher:

Over the summer, a student wants to move ahead in a math course and they watch video after video on Khan Academy and take a placement test that allows them to move ahead past geometry into an Algebra II course.  Why would they need to spend a year in a geometry course when they have all of the material they need in 5 weeks of watching videos all alone?

It is a very accepted cultural norm in the U.S. that math is an isolated educational experience.  I’m not quite sure where that comes from, but for me, it remains a rather traditionalist and damaging view of mathematical learning.  I would even go so far as to say that it could be blamed for the dichotomous view of mathematics as black or white, right or wrong, fast or slow, etc.  For many students, if they don’t fit that mold of a mathematics learner who can learn math by watching someone do it, sitting nicely and taking notes for 45 minutes while we ‘cover’ section 2.4 today, then they are ‘bad at math.’

Leone Burton once said that the process of learning mathematics is an inherently social enterprise and that coming to know mathematics depends on the active participation in the enterprises so valued and accepted in that community (Burton, 2002).  In other words, if we accept the status quo of the passivity of mathematics learning that is what we will come to believe is valued.   In her research on the work of research mathematicians and their mathematical learning she found that the opposite of the status quo was true.  The collaborative nature of their practice had many benefits that mathematicians could claim including sharing work, learning from one another, appreciating the connections to others’ disciplines and feeling less isolated (Grootenboer & Zevenbergen, 2007).  Collaboration was highly valued.

We are doing students a disservice if we allow them to remain in the status quo of being passive mathematics students or thinking that they do not have to share and/or listen to others.  The CCSS are asking (well, requiring) them to critique others’ work and give feedback on problem solving methods.  They need to be able to say what they think about others’ ideas and construct their own argument.  How are they going to learn how to express their reasoning if they don’t listen to others and attempt to make sense of it?

When working and/or learning in isolation students are not asked to do any of this or even asked to make mathematical sense oftentimes.  They are just asked to regurgitate and show that they can repeat what they have seen.  How do we know they are making any sense if they do not have to respond to anyone or interact with a group?  The importance of the social interaction becomes apparent in this context.

So what I try to explain to students is that mathematics means more to me than just being able to have a concept “transmitted” to them by someone showing them how to do something, but for them to actually do mathematics in a community of practice.  Creating that community takes a lot of work and mutual respect, but it’s something that is definitely worth it and I encourage everyone to keep inspiring me to keep doing it!  Thanks @JASauer.

Blog Challenge Day 3: Do I really practice what I preach?

So the question for today is “Discuss one observation “area” that you would like to improve upon for your teacher evaluation.”  This is a tough one for me because as a teacher at an independent school formal evaluations are done in the second and sixth years so I don’t have formal evaluation “areas” per se.  Last year, I had a colleague sit in on my classes and give me feedback over a month’s period  and it was extremely helpful to have his perspective.  I also have many teachers come from other school at different points in the year in order to learn about problem-based learning, so I am used to having people in my classroom, but I haven’t really asked for feedback in one particular area in very long time.

However, I do believe that something I wonder about when I speak to teachers learning about PBL is how well I really facilitate PBL discussions.  I know what I’m supposed to do but the time constraints and the issues of adolescent life often keep me from being the best I can be.   I know I can be hard on myself, but if I had an expert in questioning, wait time, reactions to statements, inquiry and scaffolding who could come in and watch me teach for a week or so, that would probably be the best thing for me right now.  It would be so helpful.  So if anyone is willing…please get in touch!

What does “making students metacognitive” mean? – answering “why should someone learn?” in Math

So I recently tweeted a nice article that I read that discussed “12 Questions to Help Students See Themselves as Thinkers” in the classroom (not specifically the math classroom


and appropriately, Anna Blinstein tweeted in response:


So I thought I needed to respond in a post that spoke to this question. First of all, I should state the caveat that even when I am in a more “standard” classroom (i.e. not a PBL classroom) – which happened to me last year – I try as much as possible to keep my pedagogy consistent with my values of PBL which include

1) valuing student voice
2) connecting the curriculum
3) dissolving the authoritative hierarchy of the classroom
4) creating ownership of the material for students

I find that helping students to be metacognitive helps with all of this. An important aside her is also Muller’s definition of 21st century learning* which is much more than that 20th century learning and education that often comes with direct instruction in the mathematics classroom (not always).I think it’s important to note that the more fluid concept of knowledge that is ubiquitous with technology today and is no longer static in textbooks or delivered by teachers.  Students can go find out how to do anything (procedurally) nowadays, but it is the understanding of it that is more important and the true mathematical learning and sense making.

Anyway, I think I would write way too much if I responded to every one of the questions, but how would I use these questions in my direct instruction class that I taught last year?  What I tried to do was introduce a topic with some problems (and then we would do some practice with problems from the textbook so I could keep up with where my colleague was in the material).  Well, this course was Algebra II, which often referred to prior knowledge that always reminded students of something they had studied before.  I let them use computers to look things up on the internet and use the technology at hand, GeoGebra, Graphing Calculators, each other to ask questions about the functions we were studying.  They could look up topics like domain, range, asymptotes (why would there be an asymptote on a rational function)…but then the bigger questions like “what am I curious about?” had more to do with how did those asymptotes occur, what made vertical vs. horizontal asymptotes and then I would have them do journal entries about them (see my blogposts on metacognitive journaling – journaling and resilience, using journal writing, page on metacognitive journaling).

The more “big picture” questions like “Why learn?” and “What does one *do* with knowledge?” I find easier to deal with because the students ask those.  I think that all teachers find their own ways to deal with them, but I enjoy doing is asking students about a tough question they are dealing with in their life – I use the example of whether or not I should continue working when I had my two kids.  Was keeping my job worth it financially over the cost of daycare? and of course I had to weight my emotional state when I wasn’t working – this is why I enjoy learning and what I do with my knowledge.  When kids see that there’s more to do with functions than just points on a grid, it becomes so much clearer for them – but you know that!

What I really like about Dr. Muller’s list is that he lays out some nice deliberate ways in which we as math teachers can get students to think more clearly and reflectively about mathematics as a purposeful process as opposed to a just procedures that they can learn by just watching a Kahn Academy video.


*”Learning – here defined as the overall effect of incrementally acquiring, synthesizing, and applying information – changes beliefs. Awareness leads to thoughts, thoughts lead to emotions, and emotions lead to behavior. Learning, therefore, results in both personal and social change through self-knowledge and healthy interdependence.” Muller

Keeping the Dice Rolling: Questioning in PBL

Returning from a week-long conference is always invigorating for me – not for the reasons that many people think.  I do appreciate the great feedback I get from my “teacher-students” that I interact with during the week who are so extremely eager to learn about PBL – this truly invigorates me and allows me to do so much work over the summer myself.  However, what I always look forward to is how much I personally learn from the interactions with my students that week.  At this point, PBL is so popular in its use in mathematics classrooms across the country, although people see me as an expert in the field, I gain so much from the questions and process of those who are learning that it is so useful for me to move through that process with them all the time.  I believe this is why they call it “professional development”!  So I just wanted to give a HUGE shout-out of thanks to everyone who took my workshops, came to my Cwic sessions, had conversations with me or interacted in some way – it might have been one of the best professional weeks I’ve ever had!

Since that week in New Hampshire, I’ve done a lot of reading, editing of my own materials, and catching up with my own work.  I recently read a blogpost on edutopia entitled “The Importance of Asking Questions to Promote Higher-Order Competencies” which stood out to me as something that we talked a great deal about in my own PBL classes, although this blogpost was not specifically about PBL or math at all.  It was written by a professor at Rutgers University in the Psychology Department, Maurice Elias, who is part of the Rutgers Social-Emotional Learning Lab, and made me wonder if he had done any work with Cindy Hmelo-Silver, who is also at Rutgers and does work with PBL in Psychology.  The concept of asking questions is something that we discuss and practice in my workshops because Hmelo-Silver says that it is a characteristic of an experienced PBL teacher to ask probing questions that are metacognitive and at a higher-order level.  Interestingly, the four areas that Elias discuss are often not linked to higher-order thinking (for example, yes/no questions) so I thought I might take his “Goldilocks” example and try it through the lens of math PBL.  Elias’ four questioning techniques are 1)Suggest 2) Ask a Closed Question 3)Ask an open question and 4)The Two Question Rule.

The idea of “suggestion” is one that I always tried to stay away from since student voice and experience is first and foremost in my mind as a pillar of the PBL pedagogy.  Allowing students to make first attempts at making those connections on their own I believe takes precedence over critical thinking skills of choosing from alternatives.  However, that concept of making a choice between alternatives is important as well and might be a very good skill to have them practice every now and then deliberately.  I think I will begin to try this in class.  The next time when it seems like no one has an idea or when the student at the board is going in the wrong direction, I may decide to say something like “Should Joe go with the method of completing the square or factoring here?”

The second idea of asking the closed question (yes or no) is also one that I have always tried to stay away from.  In my experience it’s kind of a conversation staller, but the way it’s explained by Elias in his blogpost is actually a very interesting twist on the closed question.  It takes a yes or no question but embeds an opinion in it, so almost forces a justification of the closed question with the yes or no.  It makes the teacher find a way for the student to continue (well, the teacher must make sure the student follows up).  So for example, if the teacher asks asks, “Do you think the quadrilateral is a rhombus?” it might seem very obvious that a student could just say yes or no and the conversation could just end there.  Everything I’ve read about closed questions say that you should not phrase the question that way but be sure that the question has within it some interest in the student’s opinion. “Why do you think it’s a good idea to argue that this quadrilateral is a rhombus?” (Which is a closed question in disguise but opens up the conversation).

Then there’s the Open-Ended Question (or what Bingham calls a True Question) which I have written about before.  I talk about this in my workshops as well and real open-ended questions are questions that the teacher doesn’t really know the answer to.  I love Bingham’s analogy of trying to predict with your students what the sum of two dice will be (the answer)  but trying to keep the dice rolling for as long as possible without knowing the answer.

Dice Metaphor

What’s an example of this type of question in mathematics?  This is a tough one because as we know so well, there are definitely right and wrong answers in mathematics.  However, we can ask questions like “Why did you chose that method?” and “What do you think of Sara’s argument? Do you agree with her?” These types of questions can make mathematics teachers very uncomfortable but we can keep the box wiggling for great deal longer than we could before with these questions and they allow us to work towards the CCSS Mathematics Practice Standards of persevering and critiquing other students’ work.

Elias’ Two Question Rule isn’t just as simple as asking a follow-up question, but makes the assumption that students want to see if when you ask a question the first time, you really wanted to know what they wanted to say.  For example, in most mathematics classrooms, students are accustomed to the I-R-E form of dialogue which is short for Initiation-Response-Evaluation (Teacher-Student-Teacher) where the teacher generally knows that answer that they want for the question they have asked (kids know this, they’re not dumb).  So when the same old kids do the response part of this, instead of just doing the evaluation part, why not blindside them and actually rephrase the question and ask it again in a different way, or ask one kid themselves individually in order for them to know that you really want to hear from them?  I think that’s what Elias is talking about.  (or even better don’t use IRE, break that darn habit, I know I’m still trying to!)

We had some great fun during my workshops role modeling and just trying out different ways of questioning the mock student who was at the board – it’s hard to break old habits.  But the more we are aware of what we are trying to do and do it deliberately, the more important it becomes and bigger agents of change we can be as well. If you have any thoughts on these questioning techniques in math PBL classroom – please let me know

Hmelo-Silver & Barrows (2006). Goals and strategies of a PBL Facilitator. Interdisciplinary Journal of Problem-Based Learning , 1(1), 21-39