Interleaving and problem writing

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It’s official: my shift to problem-based learning has consumed me. It’s all I seem to be thinking and writing about for the last three months. Radical change can do that, I guess.

Like I’ve said before, it has been really fun to think about the algebra 2 curriculum in this new way. Instead of running through units like what is done traditionally, I’ve been nonlinear about my planning. One glaring example of this is how I’m teaching trigonometry and trigonometric functions. In all my previous years, I’ve had one or more isolated units in the spring. Now, instead of doing one big chunk of trig near the end of the year, my kids are learning it in bite-sized pieces over the course of the entire year. I’ve realized that there’s no reason why they can’t explore the unit circle while also learning how to factor or analyze exponential functions.

And this sort of interleaving happens through problems, which is the other reason why I’ve been so excited this year. Specifically, it has been sequencing and writing the problems that has me on the edge of my seat. This is been a challenge that I’m now addicted to.

Before this year, if someone would have asked me to write the problems that I give my students, I would have cried in fear. Why would I want to do that? My life consisted of going to jmap.org or the MTBoS search engine and that was it. The point is that I would find problems, not write them. Besides, even if I wanted to write them, it would have taken waaay too long.

That has changed.

First, I have found the time. Second, PBL requires that I write problems if I want to adequately meet the needs of my kids. That said, I’m not writing all of the problems the kids are doing, but definitely a majority of them. Thanks to Exeter and folks like Carmel Schettino, there are so many problem gems already out there to use and adapt.

As such, I’ve been thinking about problems in new ways. Specifically, I’ve never deliberately thought about the different types of problems that exist and when to throw any given one to the students based on the math that I want them to explore.

Recently, I’ve been especially interested in the student-work based problems, like algebra by example. In essence, they force the kids to analyze math work and then transfer that analysis to a new problem. The work can have mistakes. The problems usually spur some good discussion, too. (Mine haven’t, but I’ve heard they do.)

I don’t claim that they’re great problems and should be used by others, but I’ll close with a few that I’ve written and used with my students.

 

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That’s all for now.

 

bp

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Growing pains

Today a student in my class was brought to tears. 

She cried because of my teaching. Specifically, the problem-based, discussion-based learning that I’ve adopted has been troubling her. She said that she was lost and hasn’t learned anything so far this year. Knowing the student, I politely disagreed, but she was having none of my excuses. On top of this, she’s also missed several days because of an illness. She told me more – and then the tears came. She cried, hurting because of the confusion and emptiness she felt for her math class.

It was after class and I did my best to console her, but I didn’t really know how to react. It’s not every day that a kid cries in my classroom. I tried to reassure her that I’m not out to make her life miserable, that I was on her team, that she should trust the process, that she needs the teacher far less than she thinks she does, that I would never abandon her or any other student. I offered tutoring. 

Tutoring?? The girl is crying!

It’s needless to say, but my response failed miserably. 

If I wasn’t already aware, this powerful moment shed light on how my own growing pains with PBL have transferred to the students I serve. I’m learning to teach again. They’re uncomfortable and worried. All in all, it sucks. 

Afterwards, I couldn’t help but wonder whether or not overhauling my teaching is really worth it in the end. Do the perceived long-term benefits outweigh the hopelessness that may be creeping into the minds of my students? 
bp

Posted in analysis, reflection | Tagged , , , | 1 Comment

My experience at Phillips Exeter Academy

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To build upon my experiences this summer at the Exeter Mathematics Institute and to improve the newfound problem-based classroom, yesterday I paid a visit to the renowned Phillips Exeter Academy in Exeter, New Hampshire.

I observed six mathematics classrooms, had a private meeting with some students and had lunch with a few Exeter teachers. In between all of that, I also had some time to roam freely about the leafy campus, spending a good chunk of time at their library. I was on campus from 8am to 5pm.

I tend to process experiences pretty slowly. I say that because I know that I won’t be able to fully synthesize this visit for at least a few days — especially after I’m back in my own classroom. Nonetheless, I’m going to get out my immediate reactions with what else? Isolated bullet points whose main ideas are scattered and disorganized. Clearly, I still blog for myself.

  • This place is very old. Some of the classrooms looked like they hadn’t been renovated since the 1800’s (see photo above). The look and feel aren’t for everyone, but I found it charming.
  • The teachers were so welcoming. Each one mentioned my presence in the room and had every student introduce themselves. I shared the purpose of my visit and thanked them all for allowing me to share their space for the day. I got the vibe that they are accustomed to having visitors almost every day, but I still loved their transparency. One of the teachers valiantly tried all period to get my last name right until the moment I walked out of his classroom. It was a small thing, but I really appreciated that.
  • The students were highly motivated. I fully expected this. Maybe what I didn’t expect was how helpful and respectful they were. I got lost several times while on campus and each time I was politely helped and redirected. They also gave me some student-driven advice on how to encourage buy-in from my own students in this type of learning environment.
  • Most all of the students I spoke with came from a traditional learning setting and they all enthusiastically preferred the problem-based, discussion-based environment that Exeter has pioneered. Their families are also paying upwards of $50K a year for tuition, so yeah, there’s that.
  • In terms of instruction, I saw the same thing in every class. The period opens up with kids spending about 10 minutes putting up the homework problems (~7) on the boards around the room. For the rest of the period, the students present their own (or someone else’s) work and/or solution and the class discusses and draws conclusions. The onus was put on the students to push the lesson forward. This confirmed what I’m doing in my own classroom.
  • Every teacher spent a good amount of time sitting at the Harkness table with the students. I don’t have a Harkness table nor would I want one (give me couches and coffee tables instead), but actually sitting amongst the students during class has been a game changer for me.
  • With that said, just like in any class, there was some variation to how teachers enacted this structure. Some teachers assigned students to certain problems when they walked in by having their names on the board. In others, students openly chose their own problems. In some classrooms, students could not present their own work; they had to present someone else’s.
  • In a couple of the classes I visited, when the class got stuck, it felt like the teachers wanted to lecture — and sometimes they did…for like 15 minutes. Maybe it shouldn’t have, but this was surprising given the completely student-centric classroom that Exeter pushes.
  • This made me think about the problem sets. Every Exeter mathematics teacher uses them and they all did while I was there. If the need for direct instruction was as evident as I witnessed, are the problems scaffolded enough? How much flexibility do the teachers have when it comes to class time? Must it always be problems, problems, and more problems? Or can they filter in occasional days of enrichment based on the concepts learned from the problems?
  • Desmos was widely used in the class discussions around the problems. Most all of the classrooms had a slick setup with an Apple TV and Airplay where students could easily toggle between whose laptop/tablet screen was displaying on the projector. Other than that, there was no sign of using Desmos Activity Builder or any other structure to help maximize its obvious benefit. Maybe a problem requiring Activity Builder to answer it?
  • A few teachers used doc cams for student work. Nice.
  • I constantly saw kids taking photos of the boardwork with their phones. Since my kids can’t use their phones, this affirms why I now have a class iPad and a volunteer that snaps photos of the boardwork and emails it to everyone at the end of each class.
  • I only spent one day on campus, but if I’m honest, I felt a gulf between the teachers and students in the classrooms I visited. The focus at any given time (even at the onset of the period) was overwhelmingly on the standardized problems and less on the individual students in the classroom. Shouldn’t the problems be supplemented with other materials/resources for different classes based on the needs of the kids? Again, my sample size is incredibly small, so I may be way off.
  • From what I saw around campus, Exeter seems to be in touch with the revolution that is happening in our country right now around race, gender, sexual orientation, and other social issues. The library was exceptional on this front. At the same time, students of color were disappointingly scarce both on campus and in the classes I visited.

 

bp

Posted in mathematics, reflection, school | Tagged , , , | 6 Comments

Updates on my problem-based experiment in algebra 2

As a means of tracking the progress I make in my newfound problem-centered classroom, I’m posting some recent developments and thoughts. These notes are incredibly informal and far from polished.

  • I’ve settled on assigning 5-6 problems for homework. When they come to class, I give the groups 20-25 minutes to peer review and make sense of the problems. They show their work on the large whiteboards around the room so everyone can see. As a class we then spend the last 10-15 minutes of the period in a whole group discussion with students presenting their solutions on the large whiteboards.
  • I’m now thinking…why can’t I use visibly-random groups as they peer review the problems??
  • I need to do a better job of establishing coherence within the problems. For the first 20 problems or so, students feel like they were doing random problems covering unconnected concepts. In some ways, they were since I was trying to establish some norms and routines through the problems.
    • Admittedly, the first 20 problems lacked coherence (and therefore meaning). It’s ok to intersperse concepts, but I should have a focus (or foci) for each problem string we go through.
    • It seems around 20 problems is a fair amount for each exam to assess.
  • Duh: class size matters! Periods 1 and 8 both downsized and it made a world of difference. I now have groups of around 5-6 discussing the problems. It’s only been a few days, but this has been so much more effective than the whole-class discussions we had at the onset. As I visit groups, small group instruction is the norm. I’m doing my best to simply ask questions and avoid direct instruction on the problems. I think I need a develop a simple protocol to follow when I approach a group.
    • One thing I should get back into is asking for “group questions” only. There are too many students doing their own thing and not all students in the groups are actively discussing the same problems each day. I need to push this more.
  • After emphasizing problem-solving and group discussion ahead of answers, I started providing correct answers on the board halfway through the period. Students were uncomfortable because there was too much ambiguity in final answers (thank you high-stakes exams), especially since sometimes I can’t get around to everyone’s work.
  • I am worried about the more introverted students in the class, those not openly engaging in group discussions. At times they seem to not be engaged.
  • How should my exit slip or “closing” to each day look? Note: I need to make time for this.
  • I haven’t been surfacing problem-solving strategies as students work through problems. Related: there hasn’t been a lot of focus on the various ways and perspectives to solve these problems.
  • I need to organize a day/lesson where students purposely make connections between problems and establish big ideas for the course.
    • Makes me think of Dan Meyer’s co-authoring the class post. I’m thinking we, as a class, can create a large concept map on the wall with paper and string making connections between key concepts and problems. In this way, instead of me saying, “all these problems belong to unit 7, exponential functions,” students can surface these sorts mathematical connections for themselves and own the content. That’s the dream, anyhow.
    • Maybe start with a table with columns for problems, big ideas, key vocabulary?
  • I’m allowing for students to create a 3×5 index card for use on the exams. I don’t do review days before exams so this is my way of getting them to prepare. It also forces me to think creatively about the problems I include on exams!
  • To break up the monotony of this structure, I need to begin planning lessons that don’t revolve the same sort of group discussions. I also want students to see that class won’t always look the same.
  • I have seen whiteboards being used very effectively. Student thinking is public. At times, students are moving freely around the room to independently seek out methods and strategies.
  • With these 12 whiteboards being actively used in every part of the room, I think I have successfully defronted the room. That’s a win.
  • Because the boardwork students are doing is so important, and since students can’t use their phones in my school, at the end of the period I want a student to take photos of the boards using an iPad. They would then email it the class. This would alleviate students’ feverishly copying correct work into their notes during the whole class discussion.
  • Another thing so far that I love is that the class has been focusing on doing and actively engaging with mathematics. Plus, there’s been lots and lots of struggle with the problems. That’s great, but now I just my students to be comfortable with being uncomfortable. Hopefully in time.
  • I managed to set up a spreadsheet aligning the problems to the standards-based grading “concepts” that I used last year. Although I don’t share this with students, I’m using it to guide the problem strings that I write.
  • I’m still far away of student buy-in — which I desperately need. This is due in part because of the rough start I had in sequencing the first series of problems.

 

bp

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I threw all of my units out the window.

So I’m noticing a trend. it seems like every few years I have an epiphany that causes me to blow up my teaching and rethink what I’m doing in a major way.

Case in point, six years ago I flipped my classroom and realized what is really important when it comes to learning. Three years ago I implemented standards-based grading and learned how to be more analytical with assessment. I now find myself smack in the middle of another major shift in my practice: problem-based learning.

Attended the Exeter Mathematics Institute in August was the catalyst. Experiencing a purely problem-based classroom was new. I had known the “PBL” buzzword for a long time and thought I understood what it meant. I didn’t.

Here’s the workflow: Students explore problems for homework and we use the entire next period analyzing and discuss them. The problems are designed to enable key ideas to organically emerge during homework and class discussions. There are no units. No direct instruction. This is what they call the Harkness Method, I think.

Now I find myself thinking through and sequencing the problems I give my students like never before. This has been pretty fun. All problems need to be inherently scaffolded and since they are now a learning experience (and not just practice), they are everything. Well, I shouldn’t say everything, because the class discussions are crucial too…but without the problems, you have no meaningful discussions.

Without knowing it, I think I have been moving towards PBL for a while now. For a few years, I have been trying to think about sequencing questions/prompts to naturally guide students towards a learning objective — so many of my problems have come from handouts that I’ve developed through the years. Now I’m finding myself weaving these prompts/problems together that is problem-based and not concept-based.

And about the class discussions, that’s something that I feel I’ll be tweaking with throughout the course of this year. I’ve started out doing whole-class discussions and, with classes of 30+ students, I watched as equity quickly crashed and burned. Kids were hiding their ideas and drifting off. I’m now transitioning to smaller groups of around 6-8. I plan to move around the room to guide the group discussions. I’m still debating whether I should give solutions. Maybe towards the end of class to avoid it being a conversation killer?

If I’m honest, I’m worried. I have no idea how this will go and I’m pretty sure that I may have bitten off more than I can chew. I really believe in the process, but this is a pretty drastic change. Because I have no well-defined arrangement to the curriculum, my SBG is gone. And did I mention that I have no units?!

Patience, be with me.

 

bp

Posted in mathematics, philosophy, reflection, strategy | Tagged , , , | 4 Comments