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


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.


Conceptual Change Essay (Abstract)

In this essay I look at the differences between how experts and novices approach problems and what we, as educators, can learn from these differences. I then look at a couple teaching methods that can support learning processes, namely metacognition and appropriate use of time to help students transfer knowledge.

My Sentence

For our first assignment in the MAET here at Michigan State we were asked to come up with our “sentence”. For an explanation on this, see the video below.

It took me a bit of brainstorming and several sentences that I eventually trashed, but I think I’ve settled on the sentence below. My Sentence


In addition to coming up with a sentence we were also asked to find a fitting picture to overlay the sentence onto. Let me explain my picture and my sentence a bit. The sentence activity made me think about looking back at my life and how I’d want to sum it up. I hope that I look back and can say that I was leader and helped foster positive change wherever I worked. I think that a major part of my job as an educator, regardless of discipline, is to help students learn to think. We need a society that can really think critically and problem solve. We face a tremendous number of problems as a society and many of them are getting bigger and will impact future generations in a significant way. A society that can think, problem solve, and innovate is incredibly important. As a math teacher, I try hard to show students the importance of perseverance and divergent thinking. I think these are incredibly valuable and if I can look back and say that I helped students develop these characteristics consistently then I will be happy with the legacy I’ve left.

The picture is a graph of a chaotic function. If you zoom into the middle of it you see infinitely many oscillations at an increasing frequency. This looks, well… chaotic. I think this can be representative of the education world. It’s easy, if we don’t step back, to feel like we are in a maze of acronyms, new theories, mandates, and people that have the silver bullet to all things education. When we step back and look at this function, we start to see order. The height of the oscillations is governed by the black dotted lines. As educators I think it’s important to be able to step back and look at the broad patterns and trends. It’s important to see the big picture. If we zoom in too far for too long, we can lose focus of our primary goal. The key, I believe, is to be able to cut through the chaos, to find what is most valuable to student learning.

“Opening up” Math Class

In an effort to write more I’m going to be posting shorter posts on things that are on mind regarding education and mathematics. Writing helps me process and refine my ideas and I believe it will make me a better educator.

I often think about “opening up” my math class. By “opening up” I mean developing my class in such a way that students have time to explore ideas (preferably ideas that are of interest to them, but also concepts that are in the standards).  In this setting students would be encouraged to do a number of things on a regular basis.

First, they’d be encouraged to explore wrong answers. If a student got an answer wrong they would take time to figure out why, and represent the correct solution in multiple ways (graphing, algebraically, numerically, verbally, etc.). We so often don’t have time for this and don’t value this type of exploration. I think that should change.

Second, they’d be encouraged to take ideas further on their own, in class. A good example is synthetic division vs. long division of polynomials. We always tell students that synthetic division only works in certain situations, but what about that student that wants to know why? How do we support that student? Because if that student is allowed to explore that idea he/she will likely come away with an understanding of polynomials that is far deeper than if I just told him/her the reason. (God forbid the student came up with a reason I hadn’t thought of!)

Third, students would be encouraged to work on meaningful tasks involving mathematics in small groups. These might be “real world” projects or, equally valuable, deep explorations in mathematics. The objective for the group would be not only to solve the problem(s) but to be able to communicate the solution in a meaningful (dare I say visually meaningful and appealing) way.

I do some of this on a small scale in my various classes, but I am quite often up against two major adversaries: the curriculum and time. Although I am up against this, I think that if I “opened up” my class my students would become better thinkers, communicators, and self-motivated learners. In general I think they’d become more mathematically minded and I think it is incredibly valuable to have a society of mathematically minded individuals (more on this in a future post!). I think this is why educators have to be creative, take risks, and embrace technology. That combination, for me, has been powerful in helping me to take what steps I have toward the “open” math class.

If I think of more ways in which math class could be opened up I will be sure to update. Please give me your feedback and ways in which you “open up” your class (math or otherwise)!


Multiple Methods for a Simple Problem

For each video I have students watch I ask them, among other things, to submit one question they have after watching the video. After a student watched the “Solving Logarithmic and Exponential Equations” video, he submitted the following question.

How would you solve 8^x = 16^x ?

The following day I used this question in our WSQ chat over the video. What appeared to be a simple problem revealed some interesting solutions. I’ve provided the main types of solutions that I found.

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I anticipated the second method from most students but only two of the five groups approached it that way. All methods are valid and what I really liked is that not only were they different methods but also different thought processes that led to the method/solution.

I would love to hear your feedback or observations that you have seen in your class!