Category: Agency

Shared Failure as a PBL Experience

“A self-compassionate attitude could help us feel comforted when we witness the fallibility of other humans.”

Newman, 9/4/19 Greater Good Magazine

This is the conclusion of a research study that was done by researchers at the University of Waterloo, when they asked 100 recruited students to record a video about themselves to be rated on a scale from 1-9 on how “great” they were (see study for more details).

Unbeknowst to them, there were no objective observers, the study actually began when students were shown peers’ scores sometimes mediocre and sometimes outstanding. The question really was “how do you react when you have a sense of failure in the face of others’ success?” Of course, the findings were interesting. It depended on the person. If they were the type of person that had a great sense of how life is wrought with failure and learning from our mistakes, they were more likely to have an adaptable, compassionate, and sympathetic reaction to others’ failure. However, if the student had the habit of being hard on themselves for failing, that is often what they reflected onto others and had a hard time being sympathetic.

As I read this research study, I could not help but put all of this into the context of the PBL classroom. I immediately thought “this is why whole-class discussion with sharing authority is so important.” In order for the failure of the presenter to be shared by the class and experienced as the shared experienced of failure, students should hear ideas, discuss the pros and cons and loopholes and catch the mistakes together. This could not have happened in isolation or in pairs (caveat: not to say that whole class discussion is the only way it can happen or is the only PBL should happen). The shared experience of confusion, being sympathetic to the confusion, the risk-taking of embarassment or being wrong, needs to happen for everyone to say “I know what that feels like.”

Of course, this needs to be done a classroom that fosters the idea that this is a good thing – we need to make mistakes and learn from them – in a safe, non-hierarchical environment where status and positioning are being closely reflected upon by the teacher. What this study shows is that the other mindset (e.g. fixed, as we’ve come to call it) won’t necessarily embrace the shared failure and could make the shared failure counterproductive. This is a hugely important part of the dynamics of a PBL classroom.

I can’t help but recall a student that I interviewed for my dissertation research and how she captured her feeling about shared failure, in that by contributing ideas, the failures or mistakes become a shared success:

You could kind of add in your own perspective, and kind of give you this sense like, “Ooooh, I helped with this problem” and then another person comes in and they helped with the problem, and by the end, no one knows who 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 only see 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.”

Kaley, Schettino Dissertation 2013

Earning Your Status…and Eating it too.

When I was in elementary school, I was lucky enough to have a teacher named Mrs. Bayles who believed that what it meant to be “cool” was enjoy solving really interesting problems. I remember one time she gave everyone in class a piece of pie and asked us all “What’s the best way to start eating this piece of pie?” and everyone else in the classroom immediately took their fork and stabbed it right in that pointy corner, where, they argued, they would get the most of the juicy center of whatever type of pie they had. I was sitting with my group of friends (mostly girls) who were self-defined math “geeks” (although I think back in 1976, that’s not what they were called). We all kept thinking about that and eventually came to the conclusion that we wanted to start with the crust because thought saving the middle for last was a great idea.

Mrs. Bayles thought that was so awesome and asked the four us to come up to the front of the room, draw a diagram on the board and give evidence as to why eating pie from the back of the piece of pie was somehow better than eating it from the tip.  We thought we were the Albert Einsteins of pie-eating.  We just loved it. Even though the other kids in the class thought it was kind of weird, since we could justify our choice with a good argument, we stood together and most importantly, Mrs. Bayles respected our evidence and let us have our authority in our say.

One of the things in recent years that has become a passion of mine in the mathematics classroom, and more precisely, the mathematics probem-based learning classroom, is the idea of status and positioning of students in the discourse and learning that occurs.  This has become such an important issue that I invited one of our keynote speakers this year at the PBL Math Teaching Summit to speak on this very subject.  After many, many years of hearing teachers’ concerns about how to handle a student who tends to dominate a conversation, or who doesn’t speak enough, or what happens when kids get off on the wrong the track when discussing a problem – it is about time that this socio-emotional topic (which includes race, gender, privilege, equity, and all things relational that made me start studying this pedagogy in the first place years ago) be moved to forefront of the mathematics classroom once and for all. (Aside – huge thank you to Teresa Dunleavy who gave an awesome talk on this BTW!)

I have shared this specific story with so many people at this point in time, but I find it so important that I want to repeat it here for Sam Shah’s “How does your class move the needle on what your kids think about …. who can do math?” prompt for this “Virtual Conference on Mathematical Flavors.” I feel it is something I’ve worked on for almost twenty or so years and I still don’t have it down to a science, I just know that I can’t let it go anymore.

In my classroom, I allow students to use dynamic geometry apps or technology as much as they want to justify their answers or as evidence for their thinking, as long as it doesn’t go awry (and of course, as long as it is correct and they can describe their thinking).  Two years ago, I had a student (for whom I will use a pseudonym here because I’ve used his real name in the story, but not on the Internet), I’ll call Ernie.  Ernie was one of those kids who could do no wrong – very popular in his current class, very successful academically, which made him very outspoken in his ideas, very good-looking in our white, hetero-normative, social class acceptable way and to top it off – (what Dr. Dunleavy says is usually one of the highest privileges in white schooling) – an exceptional athlete.  Mix all of these privileges together and what automatically came with him into this class? Mathematical status.

Mathematical status doesn’t mean that he was not a good math student, that’s for sure.  Ernie worked very hard and had excellent intuition, as well as good retention from his past math courses (–hmm, I feel like I’m writing comments from the fall term here..) These were neither here nor there however to the rest of the class.  When students bring their own thoughts and impressions of a student into the class with them, its the class itself that priveleges that other student (in this case Ernie) the high mathematical status that he had.  There might have been other students in the class who should have had higher status but because they were not as outspoken, had different relationships with others, were messy or not as articulate about their ideas, asked “stupid” questions (you know that’s not what I mean) or whatever the behavior that was exhibited – the other students in the class would assign a low mathematical status to other students by the things they say, brushing aside questions or simply by just not listening.

So one day we were discussing a question about the congruence of two triangles that were in the different orientation, plotted with coordinates.  Students were supposed to come up with some triangle congruence criteria (I believe it was supposed to be SSS) for why the two triangles were congruent – this was at the very beginning of the concept of Triangle Congruence.  Ernie had plotted the triangles on GeoGebra and simply said, “These two triangles are not congruent because all of the correponding sides are not equal” stepped back, matter-of-factly with pride in a job well done.

There was thoughtful silence in the room as the class looked at this diagram up on the board projected from his laptop.  There was no arguing with the fact that the sides of his triangles were not all the same.  However, there was still confusion I could see in some of the students’ faces.  Some kids asked him to find the lengths of the sides.  “That’s what I got,” “Oh I see what I did wrong,” and “Thanks for clarifying” were some of the comments that Ernie received.  Under her breath, I heard one girl just whipsering to herself, “That doesn’t make sense” and I tried to follow up on the comment, but she would have none of it.  We spent maybe 5 more minutes of me trying to get anyone else to make a comment. It got to the point where I even got out my solutions because even I was doubting myself (the power of Ernie’s status) because I had sworn that those two triangles were supposed to come out congruent.  I knew some of those kids knew it too.  Why weren’t they all saying something?  It was as simple as a misplaced point.  Not a huge problem, why couldn’t anyone call him on it?  I decided to do a little experiment.  “OK, well let’s move on then, but I really think there’s a way to show that these two triangles are congruent.”  Ernie was intrigued.  He couldn’t be wrong so tried to start finding his error, but couldn’t.  I said, “no, no, I want everyone to go home tonight and try to see if we can find a way to show that these triangles are congruent.”

Jump to the next day in class and kids are sharing their solutions from the previous nights struggle problems.  Before we start discussing them, I say, “Did anyone think about the problem that Ernie presented yesterday?” Radio silence….I wasn’t sure that anyone would actually do it, so I had come prepared with a geogebra diagram of my own.  I projected it on the whiteboard and asked if they noticed anything.  Still no one said anything (outloud so everyone could hear, but I could tell that some students were at least talking to each other).  Suddenly, Ernie says, “Oh my gosh, I plotted the wrong point! It was supposed to be (6,1).” There was this huge metaphorical sigh of relief from the whole class at this moment that could be felt by everyone.  I just coudn’t understand it.  Although no one was willing to speak up that they knew Ernie had been wrong, they were all relieved that that he realized his own error.

I expressed my concern with this dynamic in my classroom.  Simply asking them why didn’t anyone help Ernie with the problem yesterday in class? or what kept anyone from speaking up when they thought the triangles were congruent? wasn’t getting us anywhere.  So what I did was tried to let them know how much I wanted to hear their ideas – similar to what Mrs. Bayles did with the pie.  If students can see and hear evidence that the teacher values all voices equally, not just those that the students have given high status, can truly make a difference in how they start placing their status beliefs.

What I saw change in the class slowly, wasn’t the status that the kids all gave Ernie.  In fact, if anything he got even more from finding his own error – but what happened was that girl who had spoken under her breath, spoke a little more loudly.  Students who presumed that Ernie was correct, asked an interesting question that Ernie had to justify.  These other students were growing in the status that the others were giving them. I believe that it is very hard for us as teachers to control what the students come into the classroom believing about each other, but we can have an impact on what they believe is valuable and meaningful about what they do in the classroom.

 

 

 

The “Wounded Healer” Archetype in the PBL Teacher

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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.”

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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.)

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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.

Reducing Cognitive Load in PBL

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One of the things that I have been thinking about for a very long time is the idea of those who oppose PBL.  Namely those who prescribe to behaviorist and cognitive scientist theories of learning, which I know a great deal about because of my doctoral work.  So many teachers, parents and others have asked me about this over the past 25 years that you’d think I would have an answer.  I know I have thoughts but I do want to do more research in this area.

I do not pay lip-service to the ideas of cognitive load theory for sure and definitely respect those who follow these ideas.  I do think there is a place for thinking about this theory in PBL, but not an argument for why NOT to do it.  At its heart though, I believe the learning outcomes that are important in the different types of theory (CLT vs. constructivist learning for example) is what ends up differentiating them and also the way the knowledge is constructed.  I do believe in the importance of reducing the Cognitive Load for students so that their long-term memory can be triggered and practiced.

So I do believe there is a place for this in PBL – it just hasn’t been discussed a great deal.  There is always this us vs. them notion that one is right and the other is wrong – it comes from very strong belief systems and I totally understand where they are coming from.  However, if PBL is done well in a scaffolded, structured way, I believe that you can both reduce cognitive load and also ask students to think creatively.

Here is an image I saw from an article in the Guardian recently entitled Teachers: Your Guide to Learning Strategies that Really Work by Carl Hendrick. This graphic is describing the six ways to make your classroom best-ready for learning.

 

Positive Class room climate

 

When I was looking at this, the first thing I thought of is “This is my PBL classroom.”  However, I could tell there would have to be some discussion of the “reducing cognitive load” part.  All the other aspects, I believe you can find in some other blogpost of mine somewhere.  In a PBL classroom, the way that students get timely feedback is in so many ways (see my rubrics, journals, etc.).  The nightly homework is the scaffolding of learning and monitoring of independent practice – again when done well.  I won’t go through every one of these, but would love your takes (in the comments below) on each of them.

So then, how can we talk about reducing cognitive load in PBL – where is lecture and worked problems that the teacher does?  I would argue that the cognitive load is reduced by the scaffolding of the problems in the curriculum.  In other words, by triggering students’ prior knowledge the cognitive load is reduced in such a way that they are remembering something they have learned from the past, and then being asked to look at something new.  The “something new” goes through many phases of problems – concrete, multiple representations, all the way through to abstract – in order to slightly build up the cognitive load.  Again, this is all if it is done well and very deliberately with the idea of not to overload students’ thinking but to help to build the schemas that are needed for constructing knowledge both individually and socially.

The problems are worked by the students, yes – I will give you that.  But it is the teacher’s responsibility to make sure that the steps are correct, students get feedback on their thoughts and ideas, that on the board at the end of the discussion is a correct solution and so much more.  What this type of teaching does, in my view is both reduce cognitive load to a point, yet also allow students to gain agency and ownership of the material through their prior knowledge and experience.

Something else that Mr. Hendrick says in his article is:

“Getting students to a place where they can work independently is a hugely desired outcome, but perhaps not the best vehicle to get there. Providing worked examples and scaffolding in the short-term is a vital part of enabling students to succeed in the long-term.”

And I would ask, what does students’ success mean in this framework?  Some studies have shown that worked examples are beneficial in only some cases for student learning.  Others have shown that students that are taught with worked examples out-perform those who received individual instruction.  I could go on and on with the studies contradicting each other.  But what if they weren’t in contradiction?  What if there was a way that they could work together – both reducing cognitive load and also giving students agency and voice in the classroom?  Allowing students the freedom to become independent problem solvers but also scaffolding the learning in such a way that their cognition was not overloaded?  Maybe I’m an optimist, but I do believe there is a way to do both.

Getting Kids to Drive the Learning

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It doesn’t always work this way, but it would be awesome if it did.  When PBL is perfect or ideal, the students are the ones who make the natural connections or at least see the need or motivation for the problems that we are doing.  Yeah, some of them are just really interesting problems and the get pulled in by their own curiosity, but as all math teachers know, we have a responsibility to make sure that students learn a certain amount of topics, it is quite that simple.  If students from my geometry class are going into an algebra II class with trigonometry the next year where their teacher will expect them to know certain topics, I better do my job and make sure they have learned it.

So how do I, as a PBL teacher, foster the values for the problem-based learning that I have while at the same being true to the curriculum that I know I have a responsibility to?  This is probably one of the biggest dilemmas I face on a daily basis.  Where’s the balance between the time that I can spend allowing the students to struggle, explore, enjoy, move through difficulty, etc. – all that stuff that I know is good for them – while at the same being sure that that darn “coverage” is also happening?

So here’s a little story – I have a colleague sitting in on my classes just to see how I teach – because he is interested in creating an atmosphere like I have in my classes in his.  We have just introduced and worked on problems relating to the tangent function in right triangle trigonometry in the past week and now it was time to introduce inverse tangent.  I do this with a problem from our curriculum that hopefully allows students to realize that the tangent function only is useful when you know the angle.FullSizeRender (3)

So as students realize they can’t get the angle from their calculator nor can they get it exactly from the measurement on their protractor (students had values ranging from 35 to 38 degrees when we compared), one of the students in my class says, “Ms. Schettino, wouldn’t it be great if there was a way to undo the tangent?” and the other kids are kind of interested in what she said. She continues, “Yeah, like if the calculator could just give us the angle if we put in the slope.  That’s what we want.”  I stood there in amazement because that was exactly what I wanted someone to crave or see the need for.  It was one of those “holy crap, this is working” moments where you can see that the kids are taking over the learning.  I turned to the kids and just said, “yeah, that would be awesome, wouldn’t it?  Why don’t you keep working on the next problem?” and that had them try to figure out what the inverse tangent button did on their calculator.  They ended up pressing this magical button and taking inverse tangent of 0.75 (without telling them why they were using 0.75 from the previous problem) to see if they could recognize the connection between what they had just done and what they were doing.

At the end of the class, the colleague who was observing came up to me and said, “How did you do that?” and I said, “What do you mean?” and he said, “How did you get the kids to want to learn about inverse tangent? I mean they just fell right into the thing you wanted them to learn about.  That was crazy.”  I really had to think about that.  I didn’t feel like I did anything honestly, the kids did it all.  I mean what made them all of a sudden care about getting the angle?  Why were they invested?   It doesn’t always happen in my classroom that’s for sure.  This is not a perfect science – there’s no recipe for it to work – take a great curriculum, interested kids, an open, respectful learning environment and mix well?

I do think however that a huge part of it is the culture that has been created throughout the year and the investment that they have made in their ownership and authorship in their own learning. We have valued their ideas so much that they have come to realize that it is their ideas and not mine that can end up driving the learning – and yes, I do end up feeling a little guilty because I do have a plan.  I do have something that I want them to learn, but somehow have created enough interest, excitement and curiosity that they feel like they did it.  It is pretty crazy.

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.

Revisiting Journals: Getting Kids to Look Back

I have been using metacognitive journaling in my PBL classroom since 1995.  I first learned about it the Summer Klingenstein Institute when I was a third year teacher and just fell in love with it.  At that time, the colleagues at my school thought I was crazy trying to make kids write in my classes – it was just “something else for them to do” and didn’t really help them learn but I did more reading on it and there was clearly more and  more research as time went on that showed that writing-to-learn programs especially those that prompt for metacognitive skills really do help in learning mathematics (see my metacognitive journaling link under the Research tab for more info and sample journal entries).

Every once in a while a student will write a journal entry that I think is so thoughtful that I will write about it like this one a few years ago that just impressed me with his insight into his learning process of a particular problem. But other times kids write about their understanding of their learning overall like one I’ll write about today and I am also blown away.

Here’s a student I’ll call Meaghan reflecting on a problem that she found challenging for her.  Really, it doesn’t matter which problem it was or what topic it was, just the fact that she had a hard time with it at first, right?  The most important part was that after she wrote about how to do it correctly, she then took the time to write this: (in case you can’t read her handwriting, I will rewrite it below).

FullSizeRender
Part of Meaghan’s Journal Entry

“This problem was a challenge for me.  When I saw the question, it didn’t look that difficult but once I was trying to solve by [sic my] brain wasn’t thinking on the right track, and it was trying to use prior knowledge that was irrelevant in this case.  I wasn’t making connections to the properties of triangles that I had recently learned.”

Why is this realization so important for Meaghan?  Polya’s Fourth Principle of Problem Solving is “Look Back” – why is this fourth principle so important?  In my mind, this is where all the learning happens.  The three other principles are very clear

  1. Understand the problem
  2. Device a plan
  3. Carry out the plan

These three are all very basic – if they work, right?  But most of the time they don’t work for kids.  It’s the fourth step that we know is the most important – it’s where the critical thinking and analysis takes place.  If this part isn’t taken seriously and the right steps within it are not taken nothing happens, no moving forward, no growth.

So what did Meaghan do?  She realized that she had not made a connection between the triangle properties that we had just learned and how it applied to this problem.  She had not use the correct prior knowledge.  She  just created more openings to other knowledge that she knows- and I know what you’re thinking.  Does this mean that next time she will use the correct prior knowledge in another problem?  From my experience with kids, no, it does not.  But honestly, what I have seen is that the more they realize that there are more possibilities and also that the option of just saying “I don’t get it” or “I can’t do this” is unlikely, the more they will keep trying.

So what did Meaghan do? By just being asked to write a reflection about one problem (every two weeks) she has reinforced her own potential in problem solving on HER OWN.  That she may, in the future, weed out the irrelevant prior knowledge and possibly see the connections to the relevant prior knowledge, with more practice.  I think it’s made her feel just a little bit more confident – and they said it was just “something else for them to do.”

Think about where the learning happens in PBL

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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.

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.

Finding Inspiration In All Sorts of Places

“Kids will never understand fully if you just tell them the answer.  They have to break it down and understand it, take it piece by piece ‘cause if you get it straight on you’ll never know what happened. Like if you’re building something you’ll never understand how it’s built, you can never build it again because you don’t know what to do.”

You might think this quote comes from an experienced teacher who has worked with many students over years of seeing how they learn best.  Someone who has found that over time best practices have shown that individuals must spend time with material and grapple with their own understanding in order to learn for understanding.  However, this weekend I heard a fifth grade student named Jessica say this very quote in a video.  Words of wisdom from this young student who has experienced learning in a way that has been very meaningful to her.  Another teacher puts it this way:

“We’re asking students to do things that, at first, may be a little beyond them. But because of the way we present it, they find that they can do it. They’re not finding out how to do it by listening to the teacher explain.  They’re experiencing themselves as people who are capable of learning increasingly difficult skills. Confidence comes from knowing that “I can do it!” – Ted Swartz, Ph.D.

This may be the most controversial part of my definition of PBL.  The concept that not only the students are learning through their own inquiry and curiosity, but that they are asked to apply their own prior knowledge and to do so at an increasingly difficult rate of skills.  That they are asked to challenge themselves again and again.  Another big difference is the way that students experience themselves.  Swartz states that students experience themselves as learners who are capable of so much more than just listening to a teacher and doing what someone external to themselves tells them to do.  They are capable of their own direction in learning and of learning increasingly difficult tasks and managing those.

Also, students in these types of classrooms are concurrently practicing and learning new skills.  This is very different from the way we all learned as children and adolescents and it also goes against the culture of the math classroom in most of the U.S. today so we must set new norms and explain this to parents.  But it is something that is very rewarding.

We just had our Parents’ Weekend at my school and the two new teachers who were working on our pilot PBL shared with me stories of parents who had had negative feelings towards the curriculum at the beginning of the year.  I was nervous about how they were responding now, at the end of the school year.  However, to my surprise, my colleagues shared with me stories of how proud of these parents were of their students presenting in front of the class and articulating mathematical concepts to their classmates very well.  Are these straight A students now?  Not at all, but they are proud of their work, engaged in the discussion and enjoying math class.  These are great strides for these students.  And at the end of year where we worked very hard, this was truly inspirational.