Lesson Plan Version 5.0: Final Revision

Over the course of the last week and a half I’ve taken my old lesson plan over the Fundamental Theorem of Calculus and made several revisions to it as it was examined through different frameworks. The lesson plan was originally very traditional (direct instruction to start, modeling with guided instruction, independent practice, and follow up the next day). However, as I looked at it through different lenses I made several modifications to the original lesson plan that made a better use of technology, made the learning more accessible and engaging, and leveraged networks in an effective way. You can check out my original lesson plan and my revised lesson plan directly below it, here.

Major Revisions

I want to first highlight some of the major revisions I implemented and my justification for them. I started with the beginning of the lesson. I wanted to start with some sort of inquiry style activity to get students familiar with the concepts on their own terms. I did this because often when students are faced with tasks lacking apparent meaning or logic, it will be “difficult for them (students) to learn with understanding at the start; they may need to take time to explore underlying concepts and to generate connections” (Bransford, Brown, Cocking, 1999, p. 58). You can check out the activity I developed and the Wolfram Alpha animation it’s centered around.

In addition to making a shift towards inquiry, I wanted to leverage technology in a more effective way. To do that I decided that each student would do the activity mentioned above, on a Google doc. This will allow me to easily follow along and provide feedback as they work through the activity. Frequent and timely feedback is incredibly important to the learning process (Bradsford et al, 1999, p. 59). During the proof stage of the lesson, I will have them participate in a backchannel via Google Docs, providing me with questions they still have and a summary of their understanding of the proof. I can then send this out to a few teachers in my network and get feedback on how to approach whatever student misconceptions still exist. I will still be using “low tech” methods in the collaborative whiteboarding, but will be having them share out their solutions with the class in a more structured way. I will be pushing them to verbally explain their thinking process as they worked through each problem. This gives students another means by which to express their understanding (beyond writing) which breaks down barriers to learning by allowing multiple means of expression (Rose and Gravel, 2011).

One of my last revisions was to create a more focused prompt for students focus on in there weekly blog reflection. My research on Gifted and Talented Learners suggested that it’s good for students to consider how they used inductive and deductive learning so I built that into the learning prompt (Sheffield, 1994, p. xvi). In addition to the blog post post they will also have to give constructive feedback on their blog posts to each other. They will look at a peer’s post through a critical lens which will help students further explore their own understanding of the concept.

Thoughts on the Revision Process

This process has allowed me to see assessment and evaluation differently. Some of the technology I’ve implemented will allow me to assess and provide feedback during and after the lesson in a much more effective way. In other lessons I want to build in a better continuous feedback loop to help students understand where they’re at in the learning process. I tried to do this before, but I think I have some techniques that will allow me to do a better job of it in the future.

More broadly speaking I’ve grown as a professional in this process. Now that I’ve studied the constructivist approach to learning, Universal Design for Learning, the TPACK framework, and network learning I will be able to better utilize these frameworks in my other lessons. I won’t do it in such a formal way, but as I revise in the future I will look through each one of these lenses to create effective lessons that integrate technology and reach more learners. These are powerful tools that I didn’t have prior to going through those revisions. I think being a quality educator means being able to evaluate lessons from different perspectives and I think I’m closer to that standard now.


Bransford, J. D., Brown, A. L., & Cocking, R. R. (1999). How people learn: Brain, mind, experience, and school. Washington, D.C.: National Academy Press.

Rose, D.H. & Gravel, J. (2011). Universal Design for Learning Guidelines (V.2.0).Wakefield, MA: CAST.org. Retrieved from http://www.udlcenter.org/aboutudl/udlguidelines

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.


Lesson Plan: Version 2.0, TPACK Revision

Throughout the next couple weeks we will take the lesson we have chosen (see version 1.0 here) and analyze/revise it in the context of various “lenses”. The first is the TPACK framework, which I have outlined below.

My goal for this lesson is to take it from it’s current state, very dry and not based in constructivist philosophies, to a more engaging and inquiry based lesson. I will be viewing this lesson through the Technology Pedagogy and Content Knowledge (TPACK) framework. The first context this framework focuses on is technology. I will determine what kinds of technology can best help my students achieve the learning objective. Pedagogy is the the various methods I will employ to help my students learn my objective. Content knowledge is the well of knowledge that I have about my content area that I will draw from as I design and implement my lesson. The intersection of these three contexts is the focus of the TPACK framework (Mishra and Koehler, 2006).

Lesson Plan Version 1.0 (Through the TPACK Lens)


This lesson plan, in it’s current form, uses minimal technology. I use a whiteboard for the main instruction and to introduce the concept via a proof. Partway through the lesson students will utilize the collaborative whiteboards located at each pod to work on example problems. I think that I am currently underutilizing the technologies available to me. Even if I don’t necessarily add technology to the lesson, I think I can use the current technologies (the white boards) in a much more effective fashion.


In the lesson’s current form the pedagogy is mainly direct instruction. The proof at the beginning of the lesson is important to understanding the concept, and as I mentioned in the first blog post, I believe it needs to stay in the lesson in some way. I wonder about the location of the proof however. I’m not sure that the best place for it is at the beginning. As Bransford, Brown, and Cocking (1999) point out in their book How People Learn: Brain, Mind, Experience, and School, the authors explain that when faced with tasks lacking apparent meaning or logic, it will be “difficult for them (students) to learn with understanding at the start; they may need to take time to explore underlying concepts and to generate connections” (p. 58). This lesson currently does a poor job of taking that fact under consideration. Some of the pedagogy is okay. There is a period during the lesson when students will be working in small groups on example problems. This allows students to work collaboratively and to construct some meaning from the concepts, but only after a lot of the meaning has been given to them directly. They are not given time (or proper methods) to construct it for themselves.

Content Knowledge 

This lesson is conceptually difficult, even for me. I understand it for myself, but struggle to do an effective job of helping my students truly understand it. Understanding the Fundamental Theorem requires a solid understanding of the meaning of the derivative. Students also need to have a solid understanding of the definite integral, beyond just being able to complete the basic definite integral problems. A basic understanding of limits is also helpful. One of the reasons this concept is so difficult for students to understand is that it relies on the strong understanding of so many other concepts in calculus. Beyond the calculus concepts that underly the Fundamental Theorem, a strong understanding of the meaning of a function is also important. Many students make it all the way to calculus without a strong understanding of the meaning of a function. A misconception at any one of these concepts can make the understanding of the proof and it’s extensions difficult.

The Context

Much of the context of this lesson was explained above but I can’t stress the importance of taking this into consideration enough. There has to be a solid understanding the previous concepts. In addition to prior concepts, providing students a view of the big picture is also really important, so I need to help students see where the concept leads also (Bransford et al, 1999, p. 42). This concept helps us find antiderivatives for numerous functions that we would not be able to find otherwise. Providing students with this information should help them to better contextualize the concept.

Intersections: Technology and Pedagogy

The value in the TPACK model is in understanding that all of these pieces are connected. The pedagogy I utilize is directly affected by the technology I have available and vice versa. In it’s current form my technology (mainly the large whiteboard at the front of class and the “mega” whiteboards on each pod are being underutilized as a pedagogical tool. My lesson plan is currently very teacher centered and not learner centered. I need to spend some more time digging into the concept to develop other ways to better utilize my technology. I’m not sure yet if “new” technologies (like Wolfram Alpha, or other powerful graphing tools) will be beneficial or not.

Intersections: Content and Pedagogy

The important thing to understand about the intersection of the content and the pedagogy is that this concept is incredibly dynamic. The pedagogy utilized depends on the students’ construction of the prior knowledge leading up to the lesson, more so than many concepts. In a sense the quality instruction in the weeks leading up to this concept are as important as the lesson itself. One of my goals in this lesson revision is to spend time really deconstructing the content for myself and from this deconstruction find a more inquiry based approach.

Intersections: Technology and Content

Often there is an assumption that mathematics is married to calculators. In this lesson the calculator is almost a hinderance. Anything the calculator can do will essentially be a shortcut and will cause the students to create misconceptions. I want the technology that we use to help students reason their way through the concepts and develop meaning as they go. I want to avoid technology that will provide shortcuts but result in misconceptions.

Striking a proper balance between these three intersections should result in a quality lesson. My aim is to take a very teacher centric lesson, and turn it into a more inquiry based lesson in which students can better construct the concept of the Fundamental Theorem.


Bransford, J. D., Brown, A. L., & Cocking, R. R. (1999). How people learn: Brain, mind, experience, and school. Washington, D.C.: National Academy Press.

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