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.


Networked Learning Project: Planning Phase

For my networked learning project I want to learn how to use HTML and CSS to code a website. By the end of this four weeks I want to have coded and launched a “landing page” for all of my digital “stuff” (my blog, twitter, class sites, etc.). I’ve wanted to learn HTML for a long time as a hobby, but I also think there’s value in understanding how the internet works. When I run into problems with my sites I don’t want to be at the mercy of whatever site is hosting my webpages. I want to have the knowledge and control to change things and trouble shoot as they come up. I also like the idea of having countless customization options that many sites (like Weebly or WordPress) don’t offer or allow.

I currently have basically no knowledge of how HTML works. Because of this, to start my learning, I’m going to take a course on codeacademy.com. (I wanted to use p2pu.org which is a more social site for learning to code, but it would not send me an email confirmation so that I could comment within the site, making it basically unusable from the social aspect of the site.) Within Code Academy you get to pick your goals. It then leads you through the steps to that goal. I really like that from each step you can access user forums and engage with people that have gone through that step. I’ve used these forums on multiple occasions already.

As I get further into learning to code I will undoubtedly be relying on forums to support my learning. The great thing about forums is that if I have a question, there is a community of people that want to answer it (and usually can!). I then hope to be able to contribute to these forums once I understand HTML better. StackExchange looks like an awesome community that I will likely rely on as a resource. I will be documenting my learning via this blog, so follow along if you’re interested!

Here I am learning about headings.

Zach learns headings


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.

Lesson Plan Revision 3.0 (UDL Revision)

In this revision of my original lesson plan I want to evaluate my lesson in the context of the Universal Design for Learning (UDL) framework. I will first reflect on how my lesson fits within the UDL framework in it’s current state. I will then identify a set of revisions that will make it better aligned with the UDL framework, paying special attention to accommodating gifted and talented learners. UDL is an ever evolving framework that attempts to make learning as accessible as possible to as many students as possible.

Lesson Plan in the Context of UDL (Current Form)

The UDL framework is broken into 3 majors principles: (1) provide multiple means of representation, (2) provide multiple means of action and expression, (3) provide multiple means of engagement (Rose and Gravel, 2011).

Multiple means of representation

In it’s initial form my lesson only uses one or two means of representation. The main lesson is me talking at the whiteboard. This is fairly one dimensional and only provides representation through the visual means of accessibility. In addition, this leverages only one medium (text). This is problematic since the text medium creates barriers for some students (Rose and Gravel, 2011). I will consider ways to alleviate this shortcoming below. I also need to look at the ways in which I’m supporting transfer of knowledge to new problems in the context of calculus. Ultimately my goal is not to simply have this concept exist in isolation, but to make it so that students can effectively apply it outside of the lesson.

Even though this lesson only provides for primarily visual/text accessibility, there are other aspects of the principle that I do effectively. From the front of the room I do a very solid job of being explicit about the meaning of notation. It’s important to make sure everyone understands the symbols and notation that we’ll be using throughout the lesson. I also build on prior knowledge at the beginning of my lesson in order to help bring context to the concept. Barriers to learning occur whenever students either lack background knowledge, or have background knowledge that they don’t know is valuable to learning a particular concept (Rose and Gravel, 2011).

Multiple Means of Action and Expression

One aspect of this principle that I want to focus on improving is providing options for expression and communication. I currently have students expressing their knowledge collaboratively through the mega whiteboard activities and through their daily assignment. These are largely written means and some students can better express their knowledge through different means (Rose and Gravel, 2011). I will outline ideas for improving this aspect of my lesson below.

Another aspect of this principle that I do well at certain points during the lesson, but worse at other points is enhancing students capacity for monitoring progress. I think the beginning of the lesson needs a new technique for providing feedback on students immediate understanding and for helping students provide themselves with feedback. When students break into group work I do a much better job at listening to conversations and providing quick feedback. Students will also get feedback from their peers during this time.

Multiple Means of Engagement 

This final principle of the UDL model, in the context of my original lesson, needs to be greatly improved. I currently provide very little to motivate the lesson, besides emphasizing the importance of the fundamental theorem in the context of calculus. If the information I’m presenting does not engage students in some meaningful way it “is in fact inaccessible” (Rose and Gravel, 2011). The lesson does not currently provide students much choice in terms of assessment, timing of tasks, or tools for gathering and contracting information.

The UDL framework also encourages fostering collaboration and communication. I do a fair job of this in the lesson. The group work time helps foster collaboration and communication. In addition students are asked to reflect on and communicate their ideas via a reflective blog post. The instructions for this blog post may be too vague, however.

Revisions to Current Lesson Plan, Based on UDL Framework

Multiple Means of Representation

In order to alleviate the fact that my lesson is fairly one-dimensional in terms of representation, I’d like to provide some sort of graphical representation of the concept. This will be inserted prior to the proof of the fundamental theorem. Students will be given the opportunity to explore graphs that represent the fundamental theorem and then asks to draw conclusions based on their observations. The is a pedagogical choice based on Bransford, Brown, and Cocking’s book How People Learn: Brain, Mind, Experience, and School (1999) where they cite the difficulties students have with learning “with understanding at the start” when they don’t “take time to explore underlying concepts and to generate connections” (p. 58). A digital component will be required (computers) as the exploration will utilize a Wolfram Alpha visualization.

Multiple Means of Action and Expression

I want to make sure that students have multiple ways to demonstrate and approach their learning. To do this I will have students not only do an assignment, but they will also be asked to explain their reasoning on certain problems to the class. I will also be frequently asking them during the group work to explain their reasoning verbally to me. I will also be pushing them to draw out deeper conclusions that might not be initially apparent. This is supported by Linda Sheffield, PHD, in her book The Development of Gifted and Talented Mathematics Students and the National Council of Teachers of Mathematics Standard (1994), when she says “Top students should explore topics in more depth and draw more generalizations” (p. xx). This technique adds verbal expression to the written expression already utilized (their daily assignment, reflection blogs, and lesson follow up on the next day).

In order to better monitor student learning for myself and to help students self regulate their learning I’d like to do some sort of back channeling during the proof portion of the lesson. This will force students to reflect on their learning (metacognition) during the proof and should help them evaluate their understanding. Students “need feedback to the degree to which they know when, where and how to use the knowledge they are learning” (Bransford et al, 1999, p. 59). Once the proof is finished I will be able to immediately give students feedback based on the backchannel. I think the best way to manage the backchannel is through a shared Google doc.

Multiple Means of Engagement

I want to first motivate students a bit differently. I think starting with an inquiry activity will be motivating in it’s own right. Following the inquiry activity I will discuss the implications of the fundamental theorem in the real world, just before going into the proof. This will hopefully make the lesson more relevant and therefore more engaging (Rose and Gravel, 2011).

As mentioned above, I think the blog post could use some more structure. Sheffield points out that gifted and talented students should “use and explain logical inductive and deductive reasoning” (1994, p. xvi). A good prompt for them to use in their reflection would be “What kind of inductive and deductive reasoning did you utilize in constructing your understanding of the fundamental theorem of calculus?” This will force students to reflect on how they came to their understanding


As you can see, my revision didn’t focus so much on breaking down barriers to learning, but methods for extending and deepening learning. I have a fairly homogeneous group of students and many are gifted and talented. This lead me to focus on revisions that could better support that group.


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.


“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


Lesson Plan (Version 1.0)

The lesson I’ve chose to focus on covers the concept of the Fundamental Theorem of Calculus. It is possibly the most important concept that is taught in high school mathematics and it comes shortly after students learn about definite integrals, usually a little past half way through the school year (in a typical calculus course). The last two years I’ve done this very traditionally. I begin the lesson with a proof of the Fundamental Theorem. I then do a few example problems that are similar to what they’ll have to do on their assignment. I follow up and formatively assess the next day following this process. They are also assessed on the chapter test, and through their reflective learning blogs. Here is the lesson and below is the objective.

Objective: At the end of this lesson students will be able to explain and articulate the concepts within the fundamental theorem of calculus, apply them to appropriate problem sets and use the concept in the context of more complex problems.

I’ve chosen to revise this for two reasons. First, it is incredibly dry. Regardless of how animated I am as I lecture it is a difficult concept to stay engaged with. It is a powerful concept and it deserves a lesson that is equally as powerful . Second, it is incredibly teacher centered. I need to find a way to get students to play or engage with the concept first, taking into account their preexisting knowledge and it’s affect on how they will be able to learn this concept. I want to build in a way for students to explore or tinker before moving onto the proof. In this circumstance I do think the proof is important and should stay in the lesson in some way (be it on video or in person). In the past students come away with only a basic understanding and real difficulties applying the concept to more complex problems. Hopefully, through several iterations of this lesson, I can actually accomplish the objective above.