The mess of non-thematic units and why they excite me

For the last two years, I’ve adopted a problem-based, discussed-based approach for algebra 2. The whole curriculum is interleaved, meaning that big ideas are parsed and revisited over long periods of time (weeks or months) to improve retention. At any given time, students are learning small parts of a few different units. This allows for extended exposure to the topics that my kids learn. This is not how a curriculum is commonly viewed because, with this model, there are no traditional units. By “traditional” I mean thematic (e.g. Unit 6: Logarithmic Functions). Instead, these thematic units are broken down and served piecemeal to students over long stretches — mainly through problems. The sequencing of this model is indiscrete and quite messy.

As evidenced by that bewildering opening paragraph, I find all this terribly hard to communicate with others. I made my best effort to describe it here. It is often referred to as spiraling. I think Henri Picciotto does a good job of articulating it.

Thematic units have the advantage of being simpler…and easier too, I think. They are a slow-moving mass of closely-related topics that stays for a little while and then leaves when the next one comes along. Everything in them is directly linked and, therefore, these units make it easier for students to draw connections between mathematical concepts. At the same time, they encourage the isolation of facts and skills. Because related ideas are all lumped together, these units offer an easier pathway to a deep understanding in a short amount of time. Or at least the illusion of deep understanding.

These units make everything easier for the teacher, too. Thematic units and their associated lessons are far easier to plan and execute. The whole process linear; the focus of each lesson is based on the previous. There’s no untidy looping in and out of concepts, no systematic revisiting of big ideas over time. The concepts march in a clean, single-file line.

This is my guess as to why textbooks and traditional forms of curriculum have adopted thematic units. Seen in this way, they make the most sense both for the student and teacher.

But easier doesn’t make it better, right? When learning is hard, when it places a higher cognitive demand on the learner, isn’t it more meaningful? By helping students learn something small, then forget it, and then recall it after a reasonable amount of time — and iterating this process again and again over the course of the school year — can’t we help ideas cement? By not blocking out content, and instead spacing out practice and frequently assessing on the same topics at greater depth, do we help students better retain it? There’s research that says yes. Make it Stick by Paul Brown really helped me understand this.

There’s no denying the challenge that this creates for teachers. Tracing how concepts mature over the course of weeks or months is not easy. Adjustments to the sequencing can be tricky, too, because concepts are so tightly intertwined. I’ve been personally building the lessons and sequencing for two years and its still not right. Granted, I only work on it during the school year — and pretty much on the fly. Nonetheless, at least compared to traditional units, I’ve found it far more demanding and unusual to plan. And I haven’t even mentioned the loneliness — I have met no teachers during this time who are doing similar work with their curriculum. Who knows, maybe one day I’ll be disciplined enough to sit down and formally document my sequencing for other teachers to understand — and to start a conversation — but, I’m not worried about that.

I begin another school year in two weeks. I have realized that coupled with the fact my students retaining more information than ever, the messiness of interleaving has awakened and excited me these last few years. (I could do without the loneliness, though.) After years of marching in a single-file line, interleaving has made my curriculum work more of a dance. It’s interesting and lively. It moves. It sways. Fortunately, I work under an assistant principal and principal that given me the autonomy to do this. They decided to accept the consequences of the risks that I inherently took on when I decided to throw my units out the window. They trusted me even though none of us fully knew what I was doing. I think that I was at the right place at the right time because I’m not sure many other schools or departments would be on board with such a break from the norm of traditional units. My students and I have learned so much.



One graph. Ten minutes. An important conversation.

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

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:

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:

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.”



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.



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