One graph. Ten minutes. An important conversation.

At the beginning of class I showed this to my students:

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They came up with lots of interesting things.

  • There are three variables
  • They are functions
  • They are different colors
  • The units are millions and years
  • The scale for the millions is by 500,000’s and for years is decades
  • The domain of all 3 functions is 1920 to 2010
  • The range is 0 to 2.5 million
  • All of the functions are positive over their domain
  • The average rate of change for the red graph from 1920 to 2010 is positive
  • The average rate of change for the light purple graph from 1920 to 2010 is close to zero
  • The greatest average rate of change for all functions appears to occur from 1980 to 2000.

Then I asked them to predict what the graph was about. Most felt it detailed some sort of economic situation. Or population. Then came the reveal:

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They were shocked. We talked about possible causes for this situation, like the school-to-prison pipeline and the privatization of the prisons. More eyebrows raised. I brought up the question of what the racial breakdown of the prison system might look like. It was an important conversation.

And then we moved on to the regularly scheduled program: the lesson.

That was this week. While this class opener wasn’t directly tied to work that we’ve been doing in algebra 2 and was relatively brief, I felt compelled to have this conversation with my kids. This summer I began thinking about how to deepen the connections between social issues and math. Since I suck at projects, I thought about making these connections in smaller, bite-sized ways — comparable to problems found on a typical NYS Regents exam. In an ideal world, I would find (and write some) problems around social issues that are directly tied to the algebra 2 curriculum and discuss them with students. But this is really, really hard. Factoring by grouping doesn’t exactly lend itself to talking about racial inequities.

I was upfront with them. I said that its hard for me to relate some of the mathematics we learn to their daily lives, but we can do it in other ways. I told them that it was my responsibility to help you see how math can you uncover your world. Graphs are one way.

Through this graph of incarcerated Americans, I’ve myself learned that periodically presenting an interesting graph or data can be another way to build in time for important discussions around social justice and empowering students through math — even if the discussion isn’t wrapped up in a “problem” or directly tied to what we’re studying. This is not unlike What’s Going On in this Graph from the NY Times.



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:

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

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



I didn’t jump off the deep end and my students are better because of it

I’m not proud that I don’t know how to swim. Certainly, its a life skill, like riding a bike, that everyone should be able to do. But I don’t.

So about two years ago, I took swimming lessons at my local Y. The class was 8 weeks long and early on Sunday mornings — before any person under the age of 30 would dare show their face. That gave me some relief.

We used boogie boards, those barbell things, and practiced our breathing underwater. I struggled for all of those 8 weeks. And while I was much better than before I started the class, I just couldn’t relax in the water and let it “take me.”  I was too tense and thinking too much. My rhythm was off and I couldn’t coordinate my breathing, arm and leg movements. I was a mess.

The result: I never learned how to swim. My instructor recommended I take private lessons. YAY!

Despite my bitter disappointment, I had a chance to redeem myself on the last day of class. We were jumping into the deep end. For the duration of the class, we had stayed on the shallow end — the safe end in my eyes. But today was a chance to push ourselves farther than what we would naturally choose to on our own.

I was terrified. Of course, I waited for everyone else to go first. While everyone else was jumping, I tried to privately talk myself up to the monumental task of doing something I had never done before. I tried to think of inspirational music that would help push me over the edge.

I was up. I stepped up to the pool. As I stared into the water that was 10 feet deep, my instructor was wading a few feet away, waiting to help me if I needed it. My heart was racing. All my classmates were watching in excitement. They witnessed my struggles for 8 weeks and knew I didn’t learn how to swim. They knew this was going to be hard for me. Talk about pressure! What did I do?

I sat back down.

Yep. I sat back down. I just couldn’t push myself to do something that I had never done before, something that has avoided me my entire life. My feet were glued to the edge of the pool.

Humbly, I shared this story with my students this week. I also showed them this TEDx Talk:

It was the start of semester two, and it has been very clear that my students were still somewhat uncomfortable with the problem-based, discussion-based learning model I’ve adopted this year. They were frustrated and scared — just like I was when faced with jumping off the deep end.

After a few days away from them during Regents week, I realized that although I didn’t jump, that moment at the pool inspired me to make sure that I do everything I can to make my students uncomfortable. It made me fully embrace the disorder I seek every day. Watching the TEDx talk was great, too.

The bottom line is that I realized that am deeply responsible for helping to get my students be braver than I was on the edge of that pool. I want them to grow as mathematicians, but without pushing them out of their comfort zone, how could I ever truly achieve this? How can they ever grow if their learning of mathematics revolves around my thinking and not their own? How will they ever evolve into sophisticated thinkers if my instruction isn’t complex and push them to their intellectual limits?

They don’t realize it, but I have seen them grow immensely since September. In the past, my students had never owned the classroom the way they have this year. I’m proud of them for stepping out of their comfort zone in a big, bold way. Just because I didn’t jump off the deep end doesn’t mean that they won’t be able to.

I shared all of this with them. I think they appreciated it.



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.




Factoring trinomials by first rewriting them

So back in December, I gave this problem to my students:

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To my surprise, it got a lot of traction with the kiddos. We spent the entire period talking about it. The idea was for them to see how rewriting a trinomial with four terms helps us to factor it. I’ve used this approach, often called the “CAB method” and used with a large “X” to organize the product and sum of the A and C terms, to factor trinomials for the last several years and I really like it for two reasons:

  • It doesn’t matter if a is greater than 1.
  • It naturally integrates factoring by grouping. Traditionally, grouping is learned after factoring trinomials. But with this approach, I teach grouping before we even see trinomials. Yeah:

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So, yeah, this is all great, but as I was explaining this approach to a colleague, she asked me why it works. It was in that moment that I realized that I had no idea.

Well, it turns out that later that day she went ahead and wrote up a proof of the method.

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I read a quote somewhere or heard someone say that the real usefulness of algebra is the ability it affords us to rewrite things in order to help reveal their underlying structure. This method surely epitomizes that idea.