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)

Technology

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.

Pedagogy 

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.

References

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.

 

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.

 

Maker Lesson (Revision): Combining Like Terms

Our task this afternoon was to create a lesson plan in our content area that involved the maker kits we played with this morning. (See the video below for the fun we had this morning.) My partner and I bounced a lot of ideas around and definitely felt the pressure of frustration as we were coming up with the lesson. Ultimately though, we came up with an inquiry based lesson utilizing circuits, which provides students with immediate feedback, forces them to think before answering questions, uses gaming as a motivator, and forces students to think metacognitively throughout the activity. You can see the lesson plan here and our objective below.

Screen Shot 2014-06-19 at 11.52.26 PM

The inquiry piece of the lesson is probably the most important. We are asking students to look at several possible solutions in each station. As is most often the case in mathematics, there is structure behind every correct answer. It is on the student to create hypotheses, test them, and then explain the structure that yielded the correct answer. This phase of the lesson is supported by Bransford, Brown, and Cocking (1999) as they mention that “it can 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 et al, 1999, p. 58). This is precisely our aim in the lesson. We want students to experiment with different possibilities and begin to, after numerous opportunities, draw out the underlying structure in the mathematics.

Beyond the inquiry focus of the lesson, a couple other aspects are worth mentioning as I think they are incredibly valuable to learning. First, students get immediate feedback on their reasoning. We would stress early on in the activity that students should justify a choice prior to selecting that choice. They should explain that reasoning. Then they test the reasoning and benefit from immediately knowing if they need to rethink their reasoning or if it was correct. This feedback, coupled with our continuous feedback from monitoring the students Google Doc reflections and conversations, provides an incredibly valuable, diverse feedback loop that supports students learning throughout the activity (Bradsford et al, 1999, p. 59).

 

This lesson assumes that students are coming to the activity understanding the concept of a variable with coefficients. They should also have a surface level understanding of exponents. I think when we first designed the lesson we didn’t fully consider the prior knowledge students would need to get the full benefit of the activity. As Bransford points out, constructing new knowledge from existing knowledge means teachers need to consider “incomplete understandings” and “false beliefs” about a concept (p. 10). As a revision to the lesson I’m not sure that I would do any direct instruction over the needed concepts, but I would pay close attention to their reflections during each station. I can then help individual students to identify their misconceptions and hopefully eliminate the early misconceptions in the context of combining like terms. This is akin to when Bransford discusses a misconception about the world being flat. The danger is that the student, given new information (the world is actually round) constructs new knowledge that is incorrect (the world is like a pancake on a sphere) (Bransford et al, 1999, p. 10). In the context of the activity I would be monitoring for prior misconceptions and helping to effectively shape them into new, correct knowledge.

In addition to a modification in the way we approach their prior knowledge, I think I would extend this activity to another day. On the second day students would create their own circuit boards and then test each others. Since the creation of the circuit boards is fairly straightforward, I don’t think the math would get lost in the technology. Asking students to create their own problems will force them to do a number of things that are valuable to their learning. First, students would be encouraged to use several of the structures they discovered the previous day. This would force them to go back and evaluate the information they recorded in their reflections. In addition, in trying to make their board “tricky”, they will likely reflect on their misconceptions (that have hopefully been cleared up) and build those into the circuit board as possible choices. This act of metacognition and reflection allows students to “recognize the limits of one’s current content knowledge, and then take steps to remedy the situation” (Bradsford et al, 1999, p. 47). This is how experts approach problems and is often not how novices approach problems. The “day 2” piece of this activity helps students to move in the direction of thinking like experts and helps them construct a deeper understand of combining like terms.

Here are some images of our circuit board that would be utilized in the lesson.

IMG_0038 IMG_0039

References

Bransford, J. D., Brown, A. L., & Cocking, R. R. (1999). How people learn: Brain, mind, experience,     

and school. Washington, D.C.: National Academy Press.

Making Review Less boring

We are closing out our probability unit and instead of giving my students 30 problems of review to complete in class I designed a “station” activity. The station format idea actually came from a couple colleagues of mine, and it really helps to get kids up and moving. Also, as the title implies, makes review a little less boring.

I wanted to put a creative twist on this so I came up with the following station activity. There entire activity is self contained, meaning that you don’t need our textbook to use it as I designed the problems. You should be able to run it as is, or modify/improve it as you like.

Also, Ted-Ed deserves it’s own paragraph for it’s awesomeness. Now that I’ve actually gone through and used it to flip a lesson (or part of an activity really), I’m really excited about using it next year in my flipped classroom (or even my non-flipped classes)!

In addition, my good friend and colleague Eric Beckman, recorded the activity for me. Here is my reflection, and the activity resources:

Station Reflection

Stations 1 & 2: In these stations students were asked to watch two Ted-Ed lessons the night before (Station 1, station 2). I then used some of the provided questions, and created my own, for them to answer after watching the lessons. I loved that I get great data on their responses and that students can participate in discussions. I can review all student responses, both open ended and multiple choice, as you can see in the screen shot below. I can also give feedback to the open ended questions, and students will be notified when I give that feedback. You can also download all the responses as a CSV file. The discussions centered around the problems seemed really thoughtful, so I was happy with that. The videos also provide a different perspective on the concepts for the students, which I think was helpful for some.

ted ed layout Ted ed feedback

This did however take them more time than I anticipated. I had one student from each group create an account, which took time, and some of the questions were tougher for them then I anticipated. Next time I will have them set up with Ted-Ed accounts when they come to class, and will also likely reduce the number of problems they have to answer. Because some groups took a while, some groups didn’t complete all the stations.

Stations 3-5: These were the basic probability problems. Students did well on these to varying degrees. I could’ve given these problems all as one station, but breaking it into multiple stations broke it up for that students. Sometimes simple things like that make math more approachable for students. It’s also important to have the key available for students so they can get instant feedback if I’m not available.

Station 6: This station asked students to solve two problems and then create two short video lessons using my iPad to explain their solution. Every time I do this I get mixed results, but the good results outweigh the bad. The downside is that students really don’t like doing it so some push back a bit. However, it forces students to take their understanding to the next level. They will learn it better if they are forced to teach it. It makes them take an extra step in understanding, as they don’t want to explain it incorrectly on record, or mess up and have to re record it. I need to do more of this, as the students that really try get a lot out of creating short lessons explaining concepts.

Stations 7 & 8: More practice problems, similar to stations 3-5.

Activity Resources

Station Packet (PDF)                    Station Packet (Pages)                Station Packet (Word) – I make no promises about formatting….

Station Packet Answer Key

Ted-Ed Lesson for Station 1        Ted-Ed Lesson for Station 2        (These are editable, so feel free tow tweak to fit your needs)

 

“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)!

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