PBL – Page 2 – lazy 0ch0

#blackbrillance + social justice + problem-based learning

One of my summer reads has been The Brilliance of Black Children in Mathematics by Jacqueline Leonard and Danny B. Martin (inspired by Annie Perkins). I’m almost three-quarters of the way through it. It is rather dense because it’s packed with research, but I’ve been enjoying it.

Chapter 6 has stood out. It focused on the development of culturally relevant, cognitively demanding (CRCD) mathematical tasks. The authors of the gave this definition of CRCD:

Culturally relevant, cognitively demanding tasks should be mathematically demanding tasks and embedded in activities that provide opportunities for students to experience personal and social change. The context of the task may be drawn from students’ cultural knowledge and their local communities. But, the use of context goes beyond content modification and explicitly requires students to inquire (at times problematically) about themselves, their communities, and the world around them. In doing so, the task features an empowerment (versus deficit or color-blind orientation) toward students’ culture, drawing on connections to other subjects and issues. CRCD tasks ask students to engage in and overcome the discontinuity and divide between school, their own lives, community and society, explicitly through mathematical activity. The tasks are real-world focused, requiring students to make sense of the world, and explicitly critique society — that is, make empowered decisions about themselves, communities, and world. (p. 132)

The authors go on to caution the reader that finding/creating a CRCD task isn’t enough:

It should be reiterated here that task creation is by far only the beginning. Culturally relevant pedagogy necessitates that teachers learn about students, their culture, and their backgrounds. Ladson-Billings (1994) indicates that the teacher must be the driving force to creating a culturally relevant classroom. The contexts of the tasks alone will not necessarily make for the culturally relevant environment. It is the thinking behind the tasks and the actions during the implementation that make them culturally relevant. Without the appropriate set up of the task and the accompanying discussion and connection to the students and/or their communities, the task although created as culturally relevant, will lose its relevance. (p. 134-135)

This all got me thinking about all of the problem-based learning that I did last year with my kiddos. Our focus all year was thinking about, discussing, and solving problems that built on each other. As such, the big ideas of the algebra 2 curriculum were slowly uncovered through the problems. I used a range of pedagogical approaches but mainly leaned on whiteboarding (VRG and VNPS) to foster small and whole group discussions. On top of all this, back in June, I learned of Brian Lawler, who has done work around how teaching mathematics equitably requires problem-based learning. It’s an interesting take and learning from him provided even more incentive for me to improve my PBL approaches. Here are the slides to a presentation that he gave at the PBL Summitt in 2016.

So reading through chapter 6, it hit me that the PBL setting that I’m constantly improving affords my kids frequent, bite-sized opportunities to have meaningful discussions about relevant, empowering mathematics — exactly what I didn’t do last year. I centered all of the problems in contexts typically found on the Regents exams, which surely has its place, but when considering that 90% of my students are either Black or Latinx, it is an issue. The bottom line was that there was a strong disconnect between the problems I curated and my students’ lived realities. Here’s an example from last year’s problems (I could have chosen many more):

While fairly procedural, it’s a pretty standard Regents problem. Most algebra 2 teachers in New York wouldn’t complain too much about it.

Other than the unrealistic nature of the problem, what I’m coming to grips with is that the discussion we have a problem like this involves just mathematics, not the implications of the mathematics and how it directly affects how my students view themselves and/or society. The challenge I’m setting forth to myself now is to find ways to change the narratives that my problems present to my students that will help us have more meaningful, transformative conversations.

For instance, after combing through the website Radical Math, I found myself thinking about all those payday loan joints that are everywhere in the city, especially in Black and Latinx communities like where my school is located (and where I myself live). With interest rates as high as 400 percent, they help create a wicked cycle of debt that cripples many folks who are struggling to make ends meet — some of whom are quite possibly parents of my students. In addition, they target people of color. I’m thinking that instead of focusing on Bella, Ella, and their mythical interest rates, I could help my students explore about the damaging impact these lenders have our communities through introducing data from the above sources and through a series of problems that they grapple with. It’s not perfect, but here’s an example:

I’m pretty bad at using math to generate discussions about broader social issues like race. But then again, apart from beyond the white dudes, I’ve never had math problems to catalyze such discussions. I hope I’m better with facilitating discussions about problems like this, to help students see how they can better identify with math. If so, the result could be something important, relevant, and empowering.

This is a long post.

Last thing. The authors shared some examples of these sorts of problems that were created by graduate students who were also teachers. What was interesting was that, after studying the problems, the authors found that “very few of the teachers used race as a basis for their culturally relevant tasks.” Instead, the primary culture the teachers relied on was age. For me, it’s easy to get excited about some other aspect of problem set and get swept away in White culture, so this is a reminder to deliberately seek to address race in the problems and activities I use.

Through all of this, I feel like I’m getting closer to where I need to be, but I’m still left thinking about the many ideas in algebra 2 and how I might address them in the midst of the looming Regents exam.

PBL v2

So my yearlong experiment with problem-based learning has concluded.

After attending Exeter Math Institute last summer, I decided to overthrow my units and use problems as the foundation of how my kids learned each day. Throughout the course of the year, week by week I wrote a bunch of original problems, edited others that I already had, and stole the rest. In the end, there were 349 problems which I would classify as mediocre at best. These problems (and other practice, including DeltaMath) were the vehicle that my students used to learn algebra 2…and be adequately prepared for the Regents exam on June 14. The 12 whiteboards wrapped around the walls of my classroom provided the platform for my students to dig into these problems each and every day.

It was idealistic, but this change was inspired to help my kids be more independent and interdependent problem solvers. I took a huge risk because I didn’t know what heck I was doing. Despite some early struggles, I stuck it out because I believed in the process and knew that real change would take time. I constantly adjusted to support my kids as they pushed themselves out of their comfort zones. There were tears. There were instances where I felt like I bit off more than I could chew. Despite support from my admin, I still felt alone because I was doing something so different, so radical, from the rest of my colleagues. It wasn’t their fault. I was hard to relate to. Mine was a messy, nonlinear pedagogical stance to teaching mathematics and, as such, others stayed away. In the end, although folks wished me well, I had no one to talk to about the day-to-day, nitty-gritty roadblocks that I ran into. Other than an independent trip to Exeter and an awesome visit from one of their teachers, I worked in isolation. This only intensified my struggles.

Anyhow, the result of all this was an uplifting, rejuvenating, and stressful school year. I have some major takeaways that will inspire next year’s work, PBL v2. I’ll let them breathe here.

Don’t think that students will value my perspective on learning simply because I say its valuable and worthwhile. There was going to be a natural struggle involved with learning through problems, but I did a poor job of setting them up for dealing with it. Next year, the first 1-2 weeks will be all about helping them meet my expectations. This may include modeling how they should approach the problems using prior knowledge and independent research, encouraging uncertainty, showing them how to document their thinking, and how to use classmates as resources. I also want to present the research behind how and why I’m structuring their learning experiences.
A more diverse set of instructional routines to discuss problems. This year I used student-led Harkness discussions, rotating stations (group speed dating), Desmos Activity Builder, structures unique to the specific problems, and traditional, teacher-directed lessons that focused on anchor problems. Before the year began, I was worried about having the right problems as they are so pivotal in this setting. As the year progressed, I realized that I overlooked the pedagogy behind implementing the problems. Even with a focus on small groups, uniform Harkness discussions simply won’t cut it for a class of 30 every day. While it is and will continue to be a foundation of what I do, students quickly tire of the routine. I’m also thinking that exploring the use of protocols may be worthwhile.
Better engagement during group work. On most days, I gave students lots of freedom when discussing the problems of the day. For much of the period, they were on their own to construct their own (with guidance from me) understanding of the problems and the related concepts. Trust was baked into each day’s discussions; their thinking inspired the success we had each day. Some days were great, but on plenty of occasions, they did what teenagers do: be lazy. I’m wondering what else I can do to foster more consistent engagement during these small group discussions.
More metacognitive journaling. I did one in the spring and I liked it. They chose a recent problem and analyzed their own thinking around it. They told the “story” of how they arrived and understood the solution. They were a lot to grade though. Maybe one per marking period?
Be better with parents. I need to have a much more transparent and stronger relationship with my parents. I almost got around to inviting one into my classroom. Nonetheless, I need to clearly communicate how students are learning, why it’s important, and how I will support them along the way. Some parents had reservations about my approach and they definitely didn’t hold back from sharing their thoughts.
Use standards-based grading. Because I didn’t have explicitly defined units for students, when they encountered the problems, they didn’t have the crutch of knowing they were working on “section 2-4,” for example. They needed to use the context of the problem (and work done on previous problems) to discern what to do. I really like this because it made more challenging for students, but it handcuffed me when because I couldn’t find a way to accurately identify and document their understandings on exams, other than a vague, overarching percentage like “74%.” I thought deeply about this a lot and decided I will need to sacrifice a little PBL to assess meaningfully and authentically. Next year, I still don’t see having units, but I do think I will attach concepts to problems, at least to start. At the start of the year, when I give them their problems, I will also give them an exhaustive list of concepts that the problems elicit over the course of the year. I will number the concepts (eg 1-52) and each problem will have an indicator showing which of the concepts the problem connects to. Maybe over time, I can move away from this and students can make the problem-concept connection on their own. Either way, with well-defined, itemized concepts, I should be able to assign qualitative measures to each student’s understandings (needs improvement, developing, proficient, mastery). Whew.
The above would allow for more meaningful retakes of exams. With “corrections,” this process was a joke this year. There was no meaningful learning and we were all simply going through the process of applying an informal curve to their exam grades. With SBG back in the fore, this means that my post-exam procedures will look more like last year.
A nonlinear approach to learning mathematics. A huge plus of the PBL as I implemented it was that it gave me the opportunity to interleave concepts like never before. Not only did I marry concepts together in natural ways that are harder to achieve with discrete units, but I was able to space out concepts over the course of several months when it would traditionally be crammed into a three-week unit and subsequently forgotten. The most obvious example of this is trigonometry. We did many problems over the course of four months, each being a small step that got us closer to learning all the concepts from the unit. All the while, students were learning about other concepts as well. I can definitely improve my sequencing of problems but, again, since concepts learned are nonlinear, this makes recall more challenging for students and harder to forget.
One formal group assessment per marking period. These are just too valuable to not include on a regular basis. The kids love them. Plus, real learning happens during an assessment! They include two-stage quizzes, group quizzes, and VNPS quizzes.
Assign problems that will be formally collected and graded. In addition to the daily problem sets that are worked on for homework and usually discussed the following day, I want to give one meaty problem that’s due every two weeks. I’ll expect integrity and independent solutions, but students are free to research how to solve them using whatever resources they want. This will hopefully promote deep thought and a formal write up of math on a complex problem. I would love to have students type up their responses. I foresee using the Art of Problem Solving texts to find these problems, at least to start.
Using DeltaMath as a learning resource, not just practice. I was surprised by how big of a role DeltaMath played in my students’ learning. Given the lingering Regents exam, my kids relied heavily on the ‘show example‘ feature of the site to explore and solidify key ideas brought out by problems that we discussed during class.
Check homework randomly, I think. Because I didn’t check homework at all, the majority of students didn’t do it. Since the homework consisted of problems that were the centerpiece of following day’s discussion, it was a necessary component of the class. I wanted students to internalize that if they didn’t do it, they would be lost the next day. It’s ok if they didn’t understand, but they had to try. Well, that didn’t happen. Most kids just tried the problems in class the next day and set us all back. A colleague gave me feedback that students will give priority to things that have incentives, like points. I get it, but refuse to accept giving a carrot for homework. To compromise, I may check the homework of a random set of 5-7 students each day. Any student is fair game and, by the end of the marking period, every student will have roughly the same number of homework checks. I had tested this out in May and I think it triggered some initiative amongst students to do homework. I also like the idea of possibly administering a homework quiz that’s based on the previous day’s homework. If they didn’t do the homework, they’ll struggle…and I’ll offer tutoring for them to make it up.
Deliberately teach problem-solving skills. I had a flawed expectation that students would somehow become better problem solvers by simply solving a bunch of problems and have discussions about them. While that happened for some, at the end of the year most of my students grew minimally when it comes to their actual problem-solving abilities. I’m still trying to figure out exactly how to get better with this, but I know purposeful reflection will play a big role. I will also need to help surface specific PBL skills for kids. I want to bring in the question formulation technique and problem posing. This is still up the in air…and I’m reading a lot about this right now.
Be uncomfortable. It’s a great thing. In past years, I unequivocally strived to have students that were comfortable and at ease with everything we did in the classroom. I hoped they would find what and how they learned as easy and unproblematic. If I’m frank, I did a pretty good job of that. This year, I landed on the cold realization that, in many ways, my students should be uncomfortable. How else will they grow? As this post showcases, I led by example.

That’s all I have for now.

A lingering thought. Years from now, I’ll probably look back at all this and realize that I was fighting a losing battle, that I was too idealistic, that my time with students could have been used more effectively. I’ll look back and see how foolish I was. Yes, foolish to think that I could somehow establish a subculture within my classroom of independent and interdependent problem solvers that relied more on themselves than on the teacher. A subculture that places little value of remembering a formula or procedure for a quick fix, but instead focused on the mathematical relationships, collaboration, productive struggle, and prior knowledge to own what and how they learned. I’ll laugh at myself and shrug it off as me being ignorant. I’ll recognize that my goals were too lofty and practically impossible in a day and age of teacher-driven learning, high-stakes exams, and point-hungry motivations.

With this in mind, I can’t help but quote Maya Angelou: “I did then what I knew how to do. Now that I know better, I do better.”

Example analysis from DeltaMath

With so much problem-based learning happening this year, I’ve been mixing in plenty of algebra by example-esque problems. They work really well because they get kids to analyze math work on their own and then use it to solve a similar problem.

I’ve been writing some of these problems from scratch (horribly), but DeltaMath has shown up on the scene and helped out in unexpected ways. At the beginning of the year, I originally intended for DeltaMath to be a review of the problems/topics we learned in class. I assign them one big assignment that’s due the day before the next exam and they do it over time as we explore ideas in class.

That’s happening, yes, But what I’ve found is that the kids are also using the DeltaMath to learn the new ideas by means of the examples, not just review them. They’re independently leaning on their own analysis of DeltaMath examples to learn rather than on me to hand-hold them through examples in class. Independent learners, yay!!

The result is that someone regularly comes to class saying “…on DeltaMath I learned that…,” when presenting a problem we’re discussing in class – even when its an introductory problem on a topic. And, more often than not, this opens the door for a complete student-led class discussion around the problem.

For example, take this “Factor by Grouping Six Terms” problem that I assigned earlier in the year:

When they click the “Show Example” on the top, a worked-out example appears:

Students can even filter through different types of examples of the same problem by clicking “Next Example.”

My midyear report card

So my midyear report card results are in. As always, they’re a mixed bag. Here are a few comments directly from the kiddos. First, the good:

I like the amount of time we have to explore math in the class. It’s not just sitting down listening to a teacher all period.
I like how resourceful we are and a teacher isn’t always 100% necessary.
I like how we get to put up the problems on the board and are allowed to go to other tables to compare answers or to ask help.
The way we learn from each other’s work.
I like how the students have a right in their teaching in a way.
I have the freedom to walk around and don’t have to be confined to my desk.
I like the freedom in the class and learning from the problems rather than cumbersome units.
Nobody judges others on their work.
We focus on different types of problems that all connect to each other.

And the not-so-good:

It can be improved by actually teaching a lesson so that the lesson can be more clear.
I think there should be more traditional teaching.
You can try to lead the class a bit more rather than the students teaching it.
More lessons and notes rather than just problems.
Topics can be gathered into categories by Mr. Palacios so we know what we’re dealing with.
You should talk more.
Our class should try to identify problems or topics we are confused on therefore allowing you to step in and teach the topic.
I would really want for you to take charge of the class instead of the students.
Teaching in front of the board like once a week.

Notice a theme?

Based on the comments, it’s clear to me that my students are uncomfortable with the high levels of autonomy that I have afforded them. Well, let’s talk about the structure. It doesn’t happen everyday, but usually I assign 5 problems for homework (designed as learning experiences, not traditional practice). I expect them to come in the next day, put their work to the problems up on the whiteboards and thoroughly discuss the solutions they found in small groups. While this is happening, I assess their thinking and step into their group’s conversations to help drive the learning. For the most part, they can move freely about the room, but at times I will strategically move kids to different groups, a.k.a. visible random grouping. Afterwards, I sequence the presenters for the 5 problems and a whole class discussion around the solutions to the problems closes things out.

Through this structure, I have tried to minimize the amount of direct instruction that I do all the while interleaving mathematical ideas through problems. I’ve wanted student discussion to completely direct the learning and the problems to be the vehicle that makes that happen. Damn, that sounds so good in theory. I know in September it did.

Admittedly, I probably went a little too gung-ho about the student-driven, discussion-based learning. It was just so tasty. But I could have taken baby steps. I could have tried it out for a few lessons, learned its flaws and iterated on a smaller scale. But, no, I had to go all in. And I’m drowning because of it.

But all is not lost. The kids really love working on the whiteboards and freely getting help from others in the class. This is liberating for them. They aren’t confined to their seat and they appreciate this. The whiteboards give them an outlet to collaborate, which they have been eating up. If nothing else, at least they are engaged. They just need more guidance from me. And the problem-based learning has enabled the content to be interleaved and naturally spiraled, which has been so worthwhile for long-term learning. For the most part, the kids have gotten over not having discrete units.

So where do I go from here? Well, after seeking therapy from my colleagues all day, I think I’m going to begin incorporating “anchor” problems throughout the problem sets I give students. These should take a full class period to solve and I will help guide students through them with direct instruction. I hope that they will serve as a shared experience that future problems will connect to and provide them with a basic understanding of a concept.

In addition, I want to do some problem strings with them as a whole class. Again, this will serve as another shared problem-solving experience that can allow for in-depth exploration of future problems…and more direct involvement of myself.

Every few days at the start of class, I plan on giving 5-10 minute, unannounced “checkpoints” to check for understanding on what we’ve been learning. A huge weakness of semester one was not giving the kids opportunities to validate their learning. This resulted in them feeling confused and thinking they weren’t learning. Plus, I didn’t measure where they were in their understanding of key ideas until an exam. Not good. The checkpoints will inherently result, again, in more direct intervention by me and will help me adjust how we move forward.

Lastly, we just need to have more fun in class. Things got somewhat tight and tense near the end. I hated it.

I’m going to start day 1 of semester two sharing all this with my students. I want them to hold me accountable. I’ll share my reflections and ask them to reflect on what they can do to make the second half of the year better than the first. They will write a few paragraphs and submit them to me as I’m going to hold them accountable, too. Many of them don’t do the assigned homework each night because I don’t give points for it, so I hope to pull this out of them.