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? 
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The video club 

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This is the follow up to my post last summer that outlined my anticipation of a research-based, video-based professional development that I was to facilitate with the math department at my school. We coined the name of the PD “The Video Club.” So, with the extended arm of Math for America, I was lent a HD camera, omnidirectional microphone, and a tripod. The goal: analyze, interpret, and get better at responding to student thinking.

Here are some takeaways from the experience:

  • The entire department felt the PD was worthwhile and would love to do it again. Everyone was highly engaged throughout and over 80% of the group felt that this PD helped them gain key insights into student thinking. This was a relief. I believe in and value the work, I just hoped they would also find it worthwhile – which they did. Plus, I’ve never headed an initiative like this before, so I’m glad it was a source of growth…and not utter failure.
  • The department felt that the experience made us more conscious of the words and terms we use with students. For example, the use of the word “cancel” was recognized as something to rethink. We discovered that many students infer, for instance, that the common factors in the numerator and denominator of a fraction “cancel” out, when in fact they are a form of 1. “Cancel” made our students feel as if the factors just disappeared. This is subtle and a direct result of our utilization of the word in class. On a related note, a colleague mentioned that students’ mathematical abilities are a reflection of our teaching and that she witnessed her own shortcomings embedded in their thinking. Interesting.
  • The experience helped us develop better questions – or at least to habitually reassess the quality of questions that we ask our students. Questions should anticipate and clarify student thinking all the while pushing kids to make connections. There were instances where we spent all of 15 minutes debating a 7-second student discussion. This deliberate focus on the details of student thinking allowed us craft questions that addressed very specific areas of student understanding.
  • We realized that the more analysis we did of student thinking via the video club, the more we valued the process of analyzing student thinking. This lead us to create more opportunities for our students to discuss mathematics during class so that, in turn, we could analyze their thinking. This may have been the result of the tangible improvements in our planning and teaching that we made after each of the sessions.
  • Many teachers are mandated to analyze student work. It hit me early in the year that recording student discussions around a task was actually an elevated form of this. We weren’t interpreting written work to get at student thinking. Instead, we were watching and listening to them explain their thoughts, which is a much more sophisticated way of understanding student thinking.
  • This seems somewhat counterintuitive, but I learned a great deal about mathematics. Specifically, I learned more about how mathematical relationships and ideas are viewed through the eyes of my students. For example, I explored why it is so common for students to reference the Pythagorean Theorem when they see a triangle labeled with sides a, b, and c – no matter what the problem is asking. (Our dependency on those arbitrary letters may have something to do with it.) This type of perspective taking has proved to be incredibly powerful when it comes to developing impactful learning opportunities for my kids.
  • I came to embrace the openness of each session. I prepared prompts and questions beforehand, but insights from the team really led the way. Over time, instead of being a “facilitator,” I was just another member of the group who helped push the conversation forward. I learned that the uncertainly involved with this work is a good thing.
  • When I initially dove into this project, I was concerned about the amount of prep time required – especially since we were dealing with video. Anyone who has dealt with video knows that the editing process can be discouraging and straight-up unbearable. I was elated to find out that, from beginning to end, the process requires no editing. Sustainability!
  • Lastly, this experience afforded me an opportunity to lead my colleagues. I was empowered. And I’ve taken on other leadership roles in the building, but for reasons that I cannot seem to pinpoint, this one felt different. It may stem from my own personal belief about how this work provides exceptional hands-on improvement for teachers – and how rare this is.

I’m enthused to continue this work next year. MfA has been an invaluable partner and I’m pleased to know that I have their continued support!

 

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HOT v. the most appropriate

Every now and then I’m confronted with the idea of asking higher order thinking (HOT) questions. A lot of teachers are. In fact, there seems to be an unhealthy infatuation with HOT questions in education.

Don’t get me wrong, HOT questions certainly have their place. They serve to connect ideas, broaden perspectives, and deepen understanding of the content being learned. I strive for HOT questions. Even if it’s indirect, the heart of any good lesson, I believe, revolves around them. They’re a must.

That said, there’s much, much more to questioning than HOT questioning. In fact, by placing so much focus on HOT questions, we can lose sight of how questions build off one another and their dependency on current levels of student understanding. What is the goal of the question? How will it lead to the next? HOT questions can be too demanding and, consequently, create a gulf between what is currently understood and what’s expected. Any question is entirely dependent on our students – nothing else. They must meet them where they are.

What I strive for is not necessarily asking more HOT questions, but finding the most appropriate questions given the context. “What” and “when” questions should not be frowned upon if they are frequently used during a lesson. Instead, we should be critical of the sequence of any and all questions we ask and how this sequence impacts students’ abilities to answer HOT questions.

 
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Slowing down student thinking…learning to notice

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I approach a group of students discussing a problem in my class. I listen. I watch. I interpret their thinking. I sense a misconception. I ask a question to clarify what and how they are thinking. Hopefully, in the end, they reach a higher level of understanding of the problem and I reach a higher level of understanding of their comprehension.

Just like other teachers, I often do this sort of complex analysis of my students in under 10 seconds. I’ve been trained to.

That said, what if I could improve this skill I have learned over the course of my career? What if I could somehow train myself to be more attuned to student thinking?

That brings me to my next project. I’m partnering with MfA this year to bring some exciting, new PD to my school. It involves using video to record student discussion and interaction around a specific task (with no focus on the teacher). Afterwards, a group of teachers gather to watch the video, brainstorm about critical moments that occurred, interpret student thinking, and formulating questions that could be asked to clarify the thought process of the students.

It’s all based on the research by Elizabeth A. van Es and Miriam Gamoran Sherin. Here’s a follow up article they wrote on selecting clips and an overview of their work.

The idea is to slow down student thinking to the point where deep analysis can happen. My hope is that teachers at my school, along with myself, are able to use this process to improve our abilities to interpret student thinking and how we address it during our lessons.

Here are some of the challenges I foresee.

  • Introducing it to teachers. You can only introduce something once and first impressions have impacts that can last until June. I must make it good.
  • Teachers accepting the idea that interprepting student thinking often contains loads of uncertainty, and that this is okay. Not everything needs a final answer.
  • Developing engaging prompts for the group when the conversation is lagging. This may depend on the quality of my preparation beforehand.
  • I don’t see overall engagement being an issue, but you never know.
  • Being able to record and edit video clips in a timely manner. Luckily, at least in the beginning, MfA will be helping with this. But how sustainable is this type of PD in the long term?

Here are a few other unrelated thoughts.

  • How will teacher analysis differ if the focus is on student understanding versus misunderstanding, if at all? Does this impact “next steps” after the session?
  • Speaking of next steps, how will those look?
  • Can I channel teachers to certain moments in the clip based on my preparation beforehand? Would this be useful?
  • I may facilitate the initial sessions, but I want to learn perspective from my colleagues about how a student may be thinking. My MfA experiences have been scintillating in this regard. There were things mentioned that I would have never thought of.
  • This PD involves using video in the classroom. When most teachers think of video, they think of the teacher being recorded as s/he teaches with best practices as the center of attention. This is not that. It should be interesting to see this dynamic play out.
  • Each session I’ve attended with MfA has focused on one group of students discussing a task. How would the session change if we examined multiple groups of students from different classes – all discussing the same task? How would this affect the analysis?
  • This type of PD hinges on teachers understanding the content, in my case math. That notwithstanding, is there a way to run something similar that focuses on student discussion, but has a more interdisciplinary approach? Perhaps CRE/advisory?


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Height Problem from Graphing Stories

Height Image from Graphing Stories

This week in my Precal class, we were working on increasing, decreasing, and constant functions. I showed them a Graphing Story. These problems are perfect for analyzing graphs of functions. I chose the one that asks the viewer to graph height vs. time based on a lady swinging on a rope in Costa Rica.

If you haven’t already, first watch it for yourself here.

I’ve done some of the other graphing stories before and I love them. We did this one, discussed various aspects of the graph, and eventually hit on increasing, decreasing and constant intervals of the function. Its a great activity, but an interesting debate arose regarding the given answer.

You notice that the answer (i.e. graph) has a relatively large increase during 7-10 seconds.

Height Graph

When I initially displayed the solution to my students, they debated whether this interval was accurate. They found it intriguing that, according to the solution, she swung roughly 32 meters higher than her original height on the platform. Some of them said yes, it was possible for her to swing 44 meters in the air. I mean, looking at the video her upswing does look VERY high – much higher than the platform she started jumped from. Some students said no, that the camera angle was playing tricks on us. It was such an awesome discussion, and to be honest, I told them I wasn’t sure about who was right, but I’d do my best to find out.

After school I spoke to a physics teacher at my school, Shane Coleman (who these students also have), about the problem to get his scientific opinion. He informed me that it is physically impossible for the lady to have swung as high as 44 meters, unless she was pushed hard, had extra momentum when she left the platform, or pumped her legs on the swing, none of which happened. He told me for the footage in the clip, her kinetic energy would not have exceeded her potential energy. In other words, she wouldn’t have been able to swing higher than her original height; it would have been less and less with every swing. I probably should have known this, because its sort of common sense, but whatever. So then I brought all this back to my Precalc kids and it made for a pretty good discussion while we ironed out the actual solution. Later that day, the students probed even more while in Shane’s class. Awesome.

Now here’s my question: have my students, Shane, and I missed something here or is the solution inaccurate? If so, my follow up question is what assumption(s) could we make about the swing that would make the given solution valid?

On a slightly different note, isn’t it also interesting that her height is 0 at two different instances, yet she never reaches the ground?

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