Lesson Plan Version 4.0: Networked Learning Revision

For the next revision of my original lesson plan I want to look at how networks (both my own and my students’) can be leveraged to create a higher quality lesson. I want to quickly recap my lesson with it’s revisions. First, students will engage in an inquiry activity where they will do an exploration using this Wolfram Alpha widget. We will then have a group discussion looking at the patterns students noticed in exploring different functions with the widget. I will then transition into the proof of the Fundamental Theorem of Calculus. During this, or immediately following, I will ask students to backchannel, explaining the questions they still have with the proof, a part they understood the best, and how it fits with the activity they just did. I will then move into modeling a couple problems. They will then try some problems in small groups using the mega whiteboards, sharing out solutions with the class when they’re done. Finally, they will have independent work time. The following day we will follow this system for clearing up misconceptions on the assignment. At the end of the week they will write a blog post with the prompt “What kind of inductive and deductive reasoning did you utilize in constructing your understanding of the fundamental theorem of calculus?”


Image credit: http://innovatribe.com/tag/connected-workplace/ 

How I Currently Utilize Networks

The biggest way that my lesson currently uses networks is through their blogs. I can do a better job of making this an effective use of networks (see below), but I will often tweet out quality blog posts to my network and will occasionally get feedback from people in my network. In addition, I knew Wolfram Alpha was a great math and science resource so I explored that and (surprisingly quickly) found a simulation that increased the quality of the lesson. Although I use networks a small amount in this lesson, I think that they can be implemented in a much more effective way that will further enhance the quality of the lesson.

How Networks Could be Better Utilized

I want to focus on two specific aspects of using networks: how can I leverage my network to increase the quality of the lesson, and how can my students use their networks to gain a better understanding of the concept.

One way that I can use my network is to have them look at the backchannel the students do during/after the proof. Let me explain. The backchannel will happen on a Google doc. I won’t change anything in the Google doc (I may leave students comments but I won’t change what they originally wrote). I will then ask specific math teachers that I’ve connected with previously to scan the Google doc and give me feedback on students’ misconceptions. What do they think I need to go back and reteach? Do they have ideas for extending the concepts? What trends do they notice that I should address? I really think this would be a powerful use of my network that would certainly help me increase the quality of follow up instruction on the topic.

Another idea I’d like to explore is connecting with the physics teacher to discuss overlap in our lessons. I know the fundamental theorem has implications in science and I’d like to look at how to leverage that overlap to bring a more real world context to the concept. It might be worth my time to develop a project for the end of the unit in collaboration with him.

I also think that students could leverage their network in creative ways to increase their learning. First, I’m going to have students comment on other students blogs while considering the following questions. How does that student’s understanding of the concept differ from yours? What did he/she leave out that you would put in? What did they explain that you missed? Can you help to give that student a better understanding of the concept and if so, how? This should help each student better construct the knowledge in their own mind as well as help the person whose blog they are commenting on. This idea of explaining and discussing mathematics is especially important for gifted and talented learners to extend their learning beyond a surface level understanding of a topic (Sheffield, 1994, p. xx).

I also want them to tweet out their article using both the hashtag #mathchat and #calcchat asking for feedback on their ideas. Many of them probably won’t get feedback, but the potential for a random person to actually read their post and give feedback will motivate them to do better work.

Last, as an extension for the motivated learner, I’d like them to find a video online over the concept and critically analyze it with questions like “What did the creator do effectively and what did he/she miss?” They will then post the link to their analysis in the comments. This gives students the opportunity to participate and contribute to the conversation in mathematics. This is authentic, motivating (for some students) and will help them deepen their understanding of the Fundamental Theorem of Calculus.


Sheffield, L. J. (1994). The Development of Gifted and Talented Mathematics Students and the National Council of Teachers of Mathematics Standards. Storrs, CT: The National Research on the Gifted and Talented.

“Cooking with TPACK” Reflection

This morning we did an activity that was analogous to the Technology Pedagogy and Content Knowledge (TPACK) framework to help us draw connections between it and our teaching practices. Essentially we were given a kitchen tool at the beginning of class (a spatula for instance) and then randomly divided into groups and instructed to “make” something (fruit salad for example) using only the tools we were given at the beginning of class.

I took a few things away from this activity. First, the ability to be flexible was incredibly important. Just because you’ve never used an olive spoon to make a sandwich before doesn’t mean that it might not have a use (using the handle to spread peanut butter). Sometimes the tools that we have can be used in ways that we didn’t think possible. Our “content” in the context of the activity was sandwich making. The “technology” was an olive spoon. The pedagogy was how the spoon was used to contribute to the making of the sandwich. As teachers we are always trying to balance these three things. An appropriate intersection of the three places a teacher in a position to deliver a quality lesson (or make a quality sandwich).

Beyond flexibility, it became clear to me that a deep knowledge of all three contexts, technology, content, and pedagogy is vital to success in this model. Without the deep knowledge of each you can’t be flexible. If you are well versed in your content and various technology, but only know one avenue to delivery of the content then the quality of your lesson will not increase. The article, “Using the TPACK Framework: You can have hot tools and teach with them, too,” cites an example of a math teacher utilizing open sourced DJ software to teach about ratios (Mishra and Koehler, 2009, p.17). Without deep knowledge of his/her content knowledge, that teacher wouldn’t be able to recognize the connection between DJ software and mathematics. Likewise, without the knowledge of various pedagogies, the teacher wouldn’t be able to recognize the value that technology would have in the context of that particular concept. Without a deep understanding of each context, the overlap is lopsided and results in instruction that is not optimum.

Deep knowledge of the contexts and flexibility is important, but my biggest takeaway today was that teachers have always been doing this. They’ve always been balancing these three contexts. But in the last 50 years the technology context evolved incredibly rapidly in comparison to the other contexts. Prior to the development of computers, technology didn’t evolve terribly quickly. It was safe to assume that whatever technology you had at the beginning of your career would change relatively little by the end of your career. Or if it did change, it would change rather slowly. What we’ve seen in the last 20 years is that students now carry around more information in their pocket than entire universities contained only 20 years ago. This has the potential to fundamentally change our pedagogy for the better if we decide that there is value in developing a deep knowledge of the technological context. I would argue that there has always been value in developing that context, however at this point in history it’s much more intimidating for many educators.


Mishra, P., & Koehler, M. (2004). Using the TPACK Framework: You Can Have Your Hot Tools and Teach with Them, Too. Learning & Leading with Technology, 14-18.

TPACK Diagram