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!

Need Some Help Looking Forward

So I’m trying to figure out how to reach more people and thinking about the future of my professional development plans with PBL for all levels of teachers.  I’ve gotten some great feedback from people about the PBL Math Summit so far (from the two years we’ve had it) and I have some ideas about how to create some better online resources too.  If you have the time, and are interested in helping me out, would you please fill out this short survey about PD Needs for PBL Math Professional Development.  Also, tell others who could give me insights too.  Thanks so much for reading my blog and for also being inspired to be interested in PBL math teaching!

 

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.

Why Teachers Don’t Give Feedback instead of Grades, and Why We Should

First in a series of posts about my experiences with “Feedback Before Grades”

Holy Mackerel is all I have to say – Ok, well, no I have plenty more to say – but after about a week and a half of holing myself up with my colleague, Kristen McVaugh, (big shout-out to Ms McVaugh who is not only teaching PBL for the first time but was willing to dive into this amazing journey of alternative assessment with me this year too), I am totally exhausted, almost blind as a bat, partially jaded and crazy – but mostly ready for a drink.  This little looped video of Nathaniel Rateliff and the Night Sweats pretty much sums it up…

So here was our well-intentioned plan:  we wanted to start the year off with a different type of assessment.  I put out my feelers on twitter and asked around if anyone had a rubric for grading assessments where the teacher first gave only feedback and then allowed students to do revisions and then once the revisions were done the students received a grade. Kristen and I knew a few things:

  1. we wanted to make sure the revisions were done in class
  2. we wanted to make sure the revisions were the students’ own work (tough one)
  3. we wanted to give students feedback that they needed to interpret as helpful so that we weren’t giving them the answer – so that it was still assessing their knowledge the second time around
  4. we wanted to make sure that students were actually learning during the assessment
  5. we wanted students to view the assessment as a learning experience
  6. we wanted students to be rewarded for both conceptual knowledge and their skills in the problem solving too

So we created this rubric Initial Draft of Rubric for Grading.  It allowed us to look at the initial conceptual understanding the student came to the problem set with and also the initial skill level. Kristen and I spent hours and hours writing feedback on the students’ papers regarding their errors, good work and what revisions needed to be done in a back-handed sort of way.

Here are some examples:

Student 1 Initial Work

Student 2s initial work

Student 3 initial work

 

Some kids’ work warranted more writing and some warranted less.  Of course if it was wonderful we just wrote something like, excellent work and perhaps wrote and extension question.  The hard part was filling out the rubric.  So for example, I’ll take Student 3’s work on problem 6 which is the last one above. Here is the rubric filled out for him:

Student 3’s Rubric

You will notice that I put problem 6 as a 1 for conceptual understanding and a 2 for skill level (in purple). In this problem students were asked to find a non-square quadrilateral with side lengths of sqrt(17).  Student 3 was definitely able to find vertices of a quadrilateral, but he was unable to use the PT to find common lengths of sides.  I gave him feedback that looking at sqrt(17) as a hypotenuse of a right triangle (as we had done in class) would help a bit and even wrote the PT with 17 as the hypotenuse in the hope of stimulating his memory when he did the revisions.

The day of the revisions Student 3 was only capable of producing this:

Student 3 revisions

He followed my direction and used 4 and 1 (which are two integers that give a hypotenuse of 17, but did not complete the problem by getting all side lengths the same. In fact, conceptually he kind of missed the boat on the fact that the sqrt(17) was supposed to be the side of the quadrilateral altogether.

 

One success story was Student 2.  She also did this problem incorrectly at first by realizing that you could use 4 and 1 as the sides of a right triangle with sqrt(17) as the hypotenuse but never found the coordinates of the vertices for me. I gave her feedback saying there might be an easier way to do this because she needed vertices.  However, she was able to produce this:

Student 2s revision

Student 2s revision

Although she did not give me integer-valued coordinates (which was not required) and she approximated which officially would not really give sqrt(17) lengths it came pretty darn close! I was impressed with the ingenuity and risk-taking that she used and the conceptual knowledge plus the skill-level. Yes, most other kids just used some combination of 1’s and 4’s all the way around but she followed her own thought pattern and did it this way.  Kudos to student 2 in my book.

Next time I will talk about some of the lessons we learned, other artifacts from the kids’ work and what we are changing for next time! Oh yeah and some great martini recipes!

Why Can’t We All Just Get Along?: Some Inquiry Math Classes are not Content-less

Maybe it’s just how I am, or maybe I’m just always worried about what people are going to say about me, but I am hesitant to criticize other teachers publicly in the blogosphere. I’ve always felt this camaraderie with others once I’ve learned they were a teacher even if we are very different from each other – different disciplines, different pedagogical styles, different countries – there are still fundamental commonalities that even public and private school teachers have.

I just finished reading a KQED blogpost entitled “Do You Have the Personality to be an Inquiry-Based Teacher?” that sort of summarizes the theoretical qualities that the author feels a teacher who would teach with IBL would need to exhibit in order to successfully run a classroom. It’s kind of interesting – I’m not sure I agree with it, but respect the author for putting his ideas out there. I’ve been an inquiry/problem-based teacher for almost 20 years and I don’t think I exhibit all of the qualities listed, so I’m not sure it’s quite true.

Anyway, that’s not the point – at the end of the blogpost there are about 11 comments from people who are educators and many of them are quite negative and even degrading to the author:

“I earned a Ph.D. in Educational Psychology, but phrases like this one still baffle me: “…the divide between a transmission model and an inquiry model…” ”

“First, we need to make sure that we have at least a rudimentary understanding of the language in which we will be teaching. Second, we need to make sure we can write.”

“That is what’s wrong with you teachers.You want to do it your way.”

“Some of us have been doing this for decades, where were you?”

Whoa, Whoa, Whoa…cowboys…hold your horses. This guy is just writing an essay about something he believes in. What kind of role model are we being for our students if this is how we are reacting to something we don’t agree with? What happened to civil communication? I totally agree that people are allowed to comment and voice their opinions on someone else’s opinion, but there has to be a way to do it with respect and decency.

So I am going to try to model what I would like to see as a response to something I actually do disagree with. Here is a blogpost by a very respectable Professor in Canada, who I have to be totally honest, I do not know at all. I tried to learn as much as possible about him before writing a response to his blogpost in order not to make any assumptions about him (and not make a fool of myself in doing so), so I may be wrong about some of this information because I garnered it from different websites. It seems he is a research mathematician who is currently studying to get a teaching degree, but who lectures for mathematics courses at the college level. I cannot ascertain if he has any experience teaching at lower levels (like elementary or secondary). From his blogpost it does seem like he takes pride in the amount of background research he does, which again is very respectable and I appreciate in bloggers. He seems to care a lot about student learning and from his opinions on his blog he seems to lean towards being a behaviorist and cognitive theorist in terms of learning theories.

His latest blogpost is titled “The Content-less Curriculum” and it is a critique of the movement towards 21st Century skills being a part of the mathematics classroom. It does sound like Prof. Penfound is implying that with the inclusion of “soft skills” of collaboration, critiquing others work, problem solving,communication, etc. (i.e. the MPS for the CCSS) there must be a loss of mathematical content. In fact, he says that

“there must be a trade-off for the inclusion of “soft skills” activities into an already packed curriculum. So what gets removed from the curriculum then? Content knowledge.”

I would respectfully, but wholeheartedly disagree with this. By teaching with the PBL curriculum that I use, I have all of the college prep geometry curriculum I desire and I also concurrently am assessing and teaching the skills of problem solving and the so-called “soft skills” that he is implying are an add-on. I still give quick quizzes to make sure that students are up on their basic skills that are so important for basic problem solving (or else they wouldn’t be able to do the open-ended problems they are given).  The mathematics that students leave my courses having experienced is rich and leaves an impression on the way they think.

Making blanket statements about teachers implying that we all make choices that are not based in research or good practice is just not true. I actually invite you Prof. Penfound to come visit my classes and see my IBL/PBL classes in practice and let me know what you think of your opinions of the rigor of the mathematics that is discussed. Although we are most likely at different ends of the spectrum in terms of learning theories, I do believe that students have different needs and try to work with kids’ learning needs individually. However, I do believe as @danieldmccabe does that there are going to be new outcomes required of our ever-growing diverse body of graduates in the near future (or even present). I also have to say that I have thought rather thoroughly about the implementation of a teaching program which includes “soft skills” and even wrote a dissertation on it.

It is possible to balance content and practice skills and it is what I and many other classroom practitioners strive for. I do not deny that there are some practitioners out there that are confused about what problem-based and project-based learning outcomes should be especially with regard to secondary mathematics, but that is a subject for another blogpost.  The balance between content and practice skills we should strive for does not mean that one is more important or less important and in fact they both need to be assessed with the ultimate goal being to create independent problem solvers. From my experience this does not necessarily happen in a classroom where the educator does not take into consideration the so-called “soft skills.” But that statement is, of course, based on my 25 years of anecdotal classroom experience.

 

Handouts – Front and Center

I always try to make it easy for people to find both my slides and handouts when I give a talk – so Here’s my powerpoint presentation from my talk entitled, “Change the Classroom, Not the Students: Creating Equity with PBL”  which I’m giving today at the NCTM Annual Conference in New Orleans – great to be here.  I also have 2 handouts which include my framework for a relational PBL class and the results of my qualitative dissertation – I’d love to hear any comments and questions and start a discussion with PBL teachers. (I do not include the videos I used in this public version of the powerpoint, sorry)


Schettino Framework Handout

Schettino Sample Problems Handout NCTM2014

There are actually a few talks here today that I would highly recommend and seem to be related to this topic of creating a classroom that allows for discussion and interaction at the level of creating equity.  One of them is on Friday, and is entitled “The Hidden Message: Micromessaging and Mathematics” and it seems to be about managing the way we talk to each other in the classroom and making sure all voices are heard.  I’m definitely going to that one!  Unfortunately, Jo Boaler is presenting at the exact same time as me!  I don’t know if I should take that as a compliment that I was put as the same time or not :(

Well, hope everyone has a great time!  Enjoy the conference!

Does PBL teach Resilience?

I just read a great blogpost by a business writer, Gwen Moran, entitled, “SIx Habits of Resilient People.” When I think of people that I admire in my life for their resilience there was usually some circumstance in their life that led them to learn the quality of resilience because they had to. Even the examples that the author uses in this blogpost – being diagnosed with breast cancer, almost being murdered by a mugger, the inability to find a job – these tragedies that people have had to deal with can be turned into positive experiences by seeing them as ways in which we can learn and grow and find strength within ourselves.

But wouldn’t it be great if it didn’t take a negative experience like that to teach us how to be resilient? What if the small things that we did every day slowly taught us resilience instead of one huge experience that we had no choice but to face? Having to deal with small, undesirable circumstances on a daily basis, with the help and support of a caring learning community would be much more preferable, in my opinion, than surviving a mugging. (Not that one is more valuable than the other). But I just wonder – and I’m truly ruminating here, I have no idea – if it is possible to simulate the same type of learning experience on a slower, deeper scale by asking students to learn in a way that they might not like, that might make them uncomfortable, that asks more of them, on a regular basis.

I think you know what I’m getting at. Does PBL actually teach resilience (while also teaching so much more)? In my experience teaching with PBL the feedback I’ve received from students has been overwhelmingly positive in the end. But initially the comments are like this:

“This is so much harder.”
“Why don’t you just tell us what we need to know.”
“I need more practice of the same problems.”
“This type of learning just doesn’t work for me.”

Having students face learning in an uncomfortable atmosphere and face what is hard and unknown is difficult. Thinking for themselves and working together to find answers to problems that they pose as well as their peers pose is very different and unfamiliar. But does it teach the habits that Gwen Moran claim create resilient people? Let’s see. She claims that resilient people….

1.Build relationships – I think I can speak to this one with some expertise and say that at least if the PBL classroom is run with a relational pedagogy then it is very true that PBL teaches to build relationship. My dissertation research concluded nothing less. In discussing and sharing your ideas, it is almost impossible not to – you need relational trust and authority in order to share your knowledge with your classmates and teacher and this will only grow the more the system works for each student.

2. Reframe past hurts. – If we assume that real-life “hurts” are analogous to classroom mistakes, then I would say most definitely. PBL teaches you to reframe your mistakes. PBL is a constant cycle of attempting a problem->observing the flaw in your solution ->trying something else and starting all over again. This process of “reframing” the original method is the means by which students learn the the PBL classroom.

3. Accept failure – This may be the #1 thing that PBL teaches. I am constantly telling my students about how great it is to be wrong and make mistakes. You cannot have success without failing in this class. In fact, it is an essential part of learning. However, students in the US have been conditioned not to fail, so that reconditioning takes a very long time and is a difficult process.

4. Have multiple identities – In a traditional classroom, certain students fulfill  certain roles – there’s the class clown, the teacher’s pet, the “Hermione Grainger” who is constantly answering the teacher’s questions, etc. But what I’ve found happens in the PBL classroom is that even the student who finds him/herself always answering questions, will also find him/herself learning something from the person s/he thought didn’t know anything the next day. Those roles get broken down because the authority that once belonged only to certain people in the room has been dissolved and the assumption is that all voices have authority. All ideas are heard and discussed. PBL definitely teaches a student to have multiple identities while also teaching them a lot about themselves, and possibly humility, if done right.

5. Practice forgiveness – This might take some reinterpreting in terms of learning, but I do believe there are lessons of forgiveness in the PBL classroom. Students who expect themselves to learn everything the first time and when they don’t, feel stupid, need to forgive themselves and realize that learning is an ongoing process. Learning takes time and maybe needs more than one experience with a topic to see what the deeper meanings and understandings really are. Since PBL is not just a repetitive, rote teaching method, students need to learn how to be patient and forgiving of their own weaknesses as a learner and take time to see themselves as big picture learners.

6. Have a sense of purpose – This habit is about “big picture” purpose and looking at a plan. From the research that I did, I also found that PBL brings together many topics in mathematics, allowing students to see the “big picture” connections between topics much better than traditional teaching does. The decompartmentalization that occurs (as opposed to compartmentalizing topics into chapters in a textbook) is confusing at first because they are not used to it, but eventually students see how topics thread together. Just the other day in my geometry class we were doing a problem where they were asking to find as many points as possible that were 3 units away from (5,4) on the coordinate plane. A student in the class asked, “is this how we are going to get into circles?” The whole class was like “Oh my gosh, it is, isn’t it?” Bam, sense of purpose.

All in all, I feel that PBL meets Moran’s criteria of “resilience characteristics” in ways in which it allows students to practice these habits on a regular basis.  So not only does PBL help students learn collaboration, communication and creativity, but perhaps they will see the benefits over time in learning how to move forward from a setback – just a little.

 

Sharing in Chicago! PME-NA 2013

So tomorrow I’m off to PME-NA 2013 in Chicago which is one of my most favorite conferences for mathematics education research.  I will be presenting my research findings from my dissertation on Saturday morning and I’m so lucky to be going.  I’ve posted my PMENA handout  for anyone interested in having it.  I’m also posting  the powerpoint on my slideshare site.

Teaching Students to Become Better “Dancers”

So the other day I read a tweet by Justin Lanier that really sparked my interest.

 We all know the scenario in classroom discourse where a student asks a question – a really great question – and you know the answer, but you hedge and you say something like, “That’s a great question! I wonder what would happen if…”  So you reflect it back to the students so that they have something to think about for a little while longer, or maybe even ask a question like “Why would it be that way?” or “Why did you think or it like that?”  to try to get the student to think a bit more.  But what Justin, and the person who coined the phrase “authentic unhelpfulness” Jasmine Walker (@jaz_math), I believe were talking about was hedging because you really don’t know the answer – sincere interest in the uniqueness of the question – not because you’re so excited that student has helped you move the conversation forward, but because of your own excitement about the possibilities of the problem solving or the extension of the mathematics.

I think what got me so excited about this idea was how it connected to something that I was discussing earlier this summer with a group of teachers in my scaffolding in PBL workshop in late June.  In a PBL curriculum, the need to make sure that students have the right balance of scaffolded problems and their own agency is part of what Jo Boaler called the “Dance of Agency” in a paper she wrote in 2005 (see reference).  My understanding of this balance goes something like this:

(c) Schettino 2013

So initially, the student is confused (or frustrated) that the teacher refuses to answer the question although you are giving lots of support, advice and encouragement to follow their instincts.  The student has no choice but to accept the agency for his or her learning at that point because the teacher is not moving forward with any information.  But at that point usually what happens is that a student doesn’t feel like she has the authority (mathematical or otherwise) to be the agent of her own learning, so she deflects the authority to some other place.  She looks around in the classroom and uses her resources to invoke some other form of authority in problem solving.  What are her choices?

She’s got the discipline of mathematics – all of her prior knowledge from past experiences, she’s got textbooks, the Internet, her peers who know some math, other problems that the class has just done perhaps that she might be able to connect to the question at hand with previous methods that she might or might know how they work or when they were relevant – that discipline has had ways in which it has worked for her in the past and lots of resources that can help even if it may not be immediately obvious.

But she’s also got her own human agency which is most often expressed in the form of asking questions, seeing connections, drawing conclusions, thinking of new ideas, finding similarities and differences between experiences and thinking about what is relevant and what is not.  These pieces of the puzzle are not only important but a truly necessary function of the “dance of agency” and imperative to problem solving.

Interweaving both of these types of agency (and teaching kids to do this) have become more important than ever.  Yes, being able to use mathematical procedures is still important, but more important is the skill for students to be able to apply their own human agency to problem and know how and when to use which mathematical procedure, right?  This “dance” is so much more important to have every day in the classroom and if what initiates it is that deflection of authority then by all means deflect away – but the more we can “dance” with them, with “authentic unhelpfulness” and sincere deflection because we need to practice our own human agency, the more we are creating a true community of practice.

Boaler, J. (2005). Studying and Capturing the complexity of practice – the Case of the ‘Dance of Agency’

Anja S. Greer Conference 2013

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

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

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

Moving Forward with PBL: Course Evaluation

Scaffolding and Developing a PBL Course:  Course Evaluation