Summation notation, but way more

I’ve been rethinking all of my lessons this year. My hope has been to get my students to reason more. To think independently. To not be sponges. I’d like to think it’s been working. Here’s a recent lesson on summation notation that showcases this shift.

To open things up, I gave them this.

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Super accessible and relevant to summation notation. In the past, I would have chosen a bell ringer that was closely connected to a prior lesson (i.e. review) than the current one. I wanted to provide remediation. I’ve learned this year that a relevant bell ringer is pivotal to any lesson.

Here’s what came next.
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Again, very accessible. Last year Jennifer Preissel mentioned the “Stop & Jot” idea as a simple way of getting kids to write and reflect more during a lesson. Here, I gave them five minutes to express, on their own, what they wondered and noticed about the expression. After, they shared with their groups and we discussed as a class. By including “left side” and “right side,” I wanted to focus student responses. There were comments like “the +2 happens in every parenthesis” and “the number next to the +2 is going up by one.” Their observations led us to the brink of directly relating sigma notation to its expanded sum. In the past, I would jump right into defining sigma, the upper and lower limits, argument, etc. There would have been no exploring or thinking on their own.

Next, I ask them to move on to another example with the hope of finding a relationship.

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It worked like magic. They see the same pattern from the Stop & Jot and they start to generalize. They have no idea what the “E thing” is, but it’s beginning to settle in how the left and right sides relate to one another. They discuss all of this in their groups. I float around. Observing. Listening. In the past, I would show them how to find this sum and answer their questions. Again, no self-exploration and making meaning of what they see.

Now they are to dissect and interpret.

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This lacks clarity. Some students knew to write their interpretation next to the arrows, but many did not. As a checkpoint, we came back together and discussed.

Next: remove the right side.

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Things are flowing now. The scaffolds are working. They know the relationship and successfully express the sum. In the past: The students would probably be completing this problem, but instead of using their own insight to drive the work, they’d be following what I said was the correct procedure.

Finish it off.

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We come back together one more time to debrief and to address any questions the groups haven’t already. To bring things full circle, I mention the task from the bell ringer. “Ohhh!”

Lastly, on the next page, the proper names are reveled.

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We then have just enough time for an exit slip.

This lesson is heavy on notation and I didn’t want to bog them down with symbols. The goal was to find meaning first, then discuss representation. It succeeded. What I miss out on is working in reverse. Namely, using sigma notation to represent a given sum.

What I love most about this lesson has little to do with summation notation. It’s much bigger. It stems from the approach. Bottom up. Using their own insights to help them find meaning. Doing less and allowing them to put the pieces of the puzzle together. This lesson is a microcosm of how I try to teach nowadays, which is much different than in the past. It symbolizes my growth as a teacher, as a learner.

Here is the handout.

 

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Why I stopped flipping

Flip Flops

Four years ago, just before it became heavily commercialized, I flipped my classroom. I created video lessons that my students watched for homework. Class time was used for enrichment, reflection, and collaborative work. I ran with the model for a year and a half.

The other day, out of the blue, I was asked why I stopped. That made me think: back when I stopped flipping, I didn’t have this blog and never wrote about why I stopped. Here goes. Four years later.

Before flipping, I usually lectured. Sure, I disguised it with an enthusiastic and energetic delivery, but I lectured nonetheless. I wasn’t critical of my own teaching at the time, so I didn’t really think twice about it.

After I flipped, I had significantly more facetime with my kids and they had more time to reinforce new concepts. I was really happy about this. My students sat back absorbing new content like sponges, this time from a video embedded with summary questions. After all, a video lecture, however dressed up, is still a lecture.

The problem was that students weren’t discovering mathematics from my lessons. They weren’t interacting with mathematics or each other during the learning process. They weren’t debating with one another while learning something new. They weren’t being asked to find patterns and discuss them with a partner. They weren’t being challenged to make connections and develop understanding. They were using technology for learning, but not to learn. Their first impressions of so many beautiful mathematical ideas included pausing and rewinding a video that contained my face. In short, they didn’t construct their own learning. I did all of that for them.

I stopped flipping my classroom because I realized that I wasn’t flipping student learning, I was simply flipping my teaching.

I discovered that I needed them to take ownership and discover how and what they learned. What’s ironic is that I actually had to flip my classroom in order to realize this. Flipping allowed me to see my lessons through a more concentrated lens that highlighted my teacher-centered approach. More on this.

Four years later, do I regret flipping my classroom? Not a chance.

 

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Summer 2015: An Immersive Research Experience…The After Party

Earlier this summer I wrote about a research experience that I was taking part in: SMARTER, an RET instituted by NYU Polytechnic School of Engineering. The experience was profound. Here’s a recap of my takeaways.

Composite material

  • Firstly, a learned a good deal about composite materials. Going into the program, I had the slightest idea what a composite material even was. During orientation my curiosity was stoked by Nikhil Gupta, my research mentor, about his area of expertise (primarily because he mentioned that his research impacts the U.S. military). But other than that spark, I had no real foundation for where to start the research. Naturally, this equates to an information overload and, in the end, a substantial amount of learning on my end. Part of me wished my research tied in closer to what I teach everyday, but the other part of me understands that being forced into foreign territory was exactly what I needed.

Research Results Chart

  • My partner and I decided to make repurposing waste materials the centerpiece of the research. Specifically, we were interested in the impact that fly ash, human hair, and glass microspheres have when integrated with cement. What did we learn? The results were somewhat inconclusive, but we did discover an ideal proportion of fly ash to glass microspheres that would optimize the peak stress (i.e. the point at which the material begins to break down) of the composition. We expected that the human hair would have a greater impact on the overall strength of all the composites, but the results were fairly mixed.
  • A facet of the experience that surprised me was the collaboration that it involved. When you work with a perfect stranger for an extended period of time, things can get rocky. I didn’t learn to appreciate my partner’s perspective until later in the research; she taught me a ton about seeing things with a starkly different outlook and how this is necessary for the team to succeed. It also pushed me to open my mind and connect with ideas that I initially found hard to accept. I was reminded that everyone has strengths that are both unique and amazing…and that productivity sometimes hinges on the ability to focus on those strengths.
  •  The most significant impact of the experience was the uplifting inspiration it provided me. Before SMARTER, I had a deep-rooted desire to grow as a professional, but I had no ambition to further my education or seek a higher level of certification. But after a brief conversation with a Ph.D candidate who was aiding in our analysis, I left with an unexpected desire to push myself further than I ever thought I would. It was his laudable attitude coupled with the overall atmosphere at the university that left me wanting much more than just to complete my research and get back to teaching.
  • What does this mean? It means that one day I will 1) earn National Board Certification and 2) obtain a doctorate in mathematics education. Yup. I am now eagerly awaiting these immense challenges in the years to come. Thanks Eduardo.

This is my evolution thanks to SMARTER. My students, school community and myself are all much better because of it.

 

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