Category: Relational Pedagogy

Teaching + Openness = Pay It Forward

I’ve always considered myself a rational person.  I mean, I majored in math and then went on to get a masters in pure math.  I really always loved the logic and thought that went into proof and how I could think rationally about so many things in life.  Even when life threw me curve balls, I could find a way to rationalize most things and think them through in some way.

In 1996, my mother died of colon cancer at age 56 – probability of a female getting colon cancer at that age around that time was about .018%.

In 2015, my father succumbed to a number of cancers (but mainly breast cancer that metasticized in his brain) at the age of 83.  He was quite a fighter – probability of a male getting breast cancer now is about .01%.

In November 2018, my son, at age 21 has been diagnosed with Acute Myeloid Leukemia – probability of this happening is 0.2%. (all from cancerstatisticscenter.cancer.org )

All of the doctors say that none of these are related to each other at all – independent events, right? So the probability of this happening in my life is 3.6 x 10^-5.  Pretty crazy, huh? Next to impossible, probably isn’t supposed to happen to anyone.

After trying to deal with this rationally, I decided to deal with it irrationally.  I wouldn’t talk to anyone about it.  Clearly no one could help – what was the point of discussing it with anyone?  Loss, grief and sadness were always a part of my life and this was nothing new.  I just had to suck it up and get over it.  Do my best and move on.  Get through it, I always did.

My husband refused though.  We deal with things differently.  He wanted to tell the world.  He writes emails giving people updates about how my son is doing – people I haven’t even met and I felt it was a huge invasion of our privacy.  But something happened that totally surprised me.   These people, some I knew and some I didn’t know, started sending us things – cards, notes, emails, gifts cards to Panera, cookies, offers to make us dinner, and many other things I never expected.  I couldn’t fathom why they would do this.  I would say to them, “I don’t know how to repay you.” And their response would always be, “You don’t have to, just pay it forward.”

180 degree turn to the world today – students in MAGA hats harrassing indigenous peoples at a protest rally, supreme court upholding the ban on transgender peoples in the armed forces, our bullying, self-absorbed president keeping federal workers from having their paychecks, Republican elected officials losing their integrity and concept of their job to look out for their constiuents – what is happening?  Where is the kindness and care in the world today?  I was losing it quite regularly.

In all of this confusion in my brain, I felt the need to let people know that I had to cancel some public speaking and courses that I had committed to this spring.  I was nervous about tweeting about something so personal, but I know some folks who follow might have been trying to sign up for my course or looking forward to my NCTM talk.

That’s when it hit me.  I received so much love, support and kindness from the virtual PLC of math teachers that I did not expect.  I could not believe it.  People I knew, people I knew virtually, people I’ve never met – all reaching out and telling me that they cared and wished us the best in this struggle.  It was an interesting comfort at a time when my personal struggle was being exacerbated by the complex chaos of the world today.

I do believe that there is no balance between chaos and calmness.  We just learn to take both as it comes.  The peace that was given to me from the #MTBoS and #iteachmath community definitely allowed me the moment of calm that I needed and I will always remember that.  You all are surely paying it forward from someone who did something for you and I hope that I can do the same someday.  I do think that teaching has something to do with paying it forward as we do every day, honestly.  People say they teach because they love it.  But in some way, the relationality that it takes to be a good teacher, be open with people and understand our students is the way in which we pay it forward every day.

Thank you so much for giving me this moment of calm at this time in my life.

 

How do you use empathy to teach math?

1 Comment

This post is part of the Virtual Conference on Mathematical Flavors, and is part of a group thinking about different cultures within mathematics, and how those relate to teaching. Our group draws its initial inspiration from writing by mathematicians that describe different camps and cultures — from problem solvers and theorists, musicians and artists, explorers, alchemists and wrestlers, to “makers of patterns.” Are each of these cultures represented in the math curriculum? Do different teachers emphasize different aspects of mathematics? Are all of these ways of thinking about math useful when thinking about teaching, or are some of them harmful? These are the sorts of questions our group is asking.

 

One of the things that is interesting about teaching with PBL is how students often describe enjoying this type of math class more than others they have had in the past. It’s hard for students to paint a picture of what it is that produced their enjoyment.  The interesting thing is that it is often not the mathematics they enjoy, but the class itself – the interactions and relationships between the people in the class, and should they be solving some interesting problems that pertain to mathematics, that’s pretty great, too.

What one girl, Isabelle, described enjoying about my class once, was the way in which she saw mathematics as no longer black and white – with only the teacher’s information as what counts.  In a research interview, I asked her to describe for me what that was like:

Isabelle:  Like it’s, if you have a question you can just ask it and then that can lead into, like, some conversation or [the teacher] can ask a question and then kind of leaves it out there for us, the kids, to answer it, so…

Ms. S:  OK, and why do, why do you like that better?

Isabelle:  Um, because it’s not so uptight and [laughs], like it’s not like focused, “memorize all of this stuff…”

Ms. S:  Hmm

Isabelle:  It’s more relaxed, and that helps me learn better I think.

Isabelle’s more traditional view of the mathematics classroom with its “uptight” and rigid nature reminds her of memorizing facts and formulas and she stated that she responds better to a classroom that, in her eyes, is more “relaxed” and interactive allowing her views and responses to matter.  This is extremely consistent with Frances Maher and Mary Kay Thompson’s (2001) view of the feminist classroom’s responsibility to “deliberately position students as academic authorities” in order to allow them the input for the feeling that their responses matter, but so that that they do not “dismiss their own emerging sense of themselves.”  Also, Isabelle’s feelings are consistent with what Fox Keller (1985) once called “dynamic objectivity” which she defined in terms of how we might be inclined to think about the idea of integrating student input with factual mathematical knowledge.

Dynamic objectivity is a form of knowledge that grants to the world around us its independent integrity but does so in a way that remains cognizant of, indeed relies on, our connectivity with that world.  In this, dynamic objectivity is not unlike empathy, a form of knowledge of other persons that draws explicitly on the commonality of feelings and experience in order to enrich ones’ understanding of another in his or her own right (Fox Keller, p.117).

We can view this more flexible way of viewing knowledge as necessary for including students like Isabelle who find the more rigid mathematics classroom not conducive to learning.  She would rather remain connected to the material and the persons in the classroom with her in order to facilitate learning for herself.  Many students truly enjoy the fact that students are the contributors to the knowledge and part of the authority presence in the classroom.  Because of the openness to the dynamic objectivity of the knowledge, the students are able to accept that their input is valuable.  When I asked Isabelle why she thought the students felt so compelled to participate in a PBL classroom, she had this to say:

Ms. S:  Yeah, there’s almost a guarantee that people will… I wonder why? I wonder what guarantees that everyone will have something to say.

Isabelle:  Well [both laugh] it’s probably just because geometry has like twenty… like a lot of different ways to do certain problems so there’s a lot of variations in the way that people do them, so…

Ms. S:  Hmm.

Isabelle:  That might be it, or it might just be that people feel comfortable in the situation they’re in to participate and it’s not like, “OK nobody ask questions so we can leave now.”

Ms. S: [laughs]  Yeah. Ok. So there’s a certain amount of like motivation to want to talk about it?

Isabelle:  Yeah.

Ms. S:  because it’s like interesting to hear what other people did? [pause] Um, yeah, I can’t figure that out.

Isabelle:  I think everybody like shares the same curiosity level and like when somebody… like I know in our physics class he never tells us the answer to questions and it drives everybody crazy…

Ms. S:  Huh…

Isabelle:  And then we all start talking about it to try and figure out if like we can find out the answer ourselves so and the same thing happens in my math class so…

Ms. S:  Yeah?

Isabelle:  I think it’s just the motivation to find the right answer and like, because I know everybody in my class wants to understand.

Isabelle had described a mathematics classsroom culture with a tacit understanding of the dynamic objectivity of the part that students play in the formation of knowledge.  When presented with a problem where the solution is unknown and the teacher presumes a certain lower level of authority than the students, the students take on a higher level of responsibility and curiosity in finding solutions and methods for those solutions.  Being open to a view of dynamic objectivity and the empathy that it needs, allows many students to have their comfort in this type of learning environment and fosters more equity in learning for students who have previously been disenfranchised in mathematics and science classes.

 

Fox Keller, Evelyn, (1985). Gender and Science. New Have: Yale University Press.

Maher, F. A., & Thompson Tetreault, M. K. (2001). The Feminist Classroom. New York: Rowman & Littlefield Publishers, Inc.

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

4 Comments

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.