# Math, Play, and a bit of Desmos Love

I’m sure a lot has been written about how Desmos is an incredibly valuable tool for the math classroom. I’d like to echo those sentiments and give a few observations from my classroom related to Desmos and play in mathematics.

My thoughts for this post stem from this activity. It’s pretty math heavy but the basic idea is that students spend time making conjectures and either verifying or invalidating them. When their conjecture is invalidated, they spend time tweaking their functions in Desmos to match the actual answer. These kinds of activities are becoming commonplace in my classroom and I’ve noticed a few positive shifts in my classroom environment. First, engagement has increased. I had a couple more high fives during this activity (and a bunch more in a previous activity) and 100% engagement in class. To be fair, this is AP calculus so it is the “better” students, but I am hard pressed to find 100% engagement when I give p. 102 #1-15 from the textbook. Second, they learn better! This shouldn’t be a big surprise since we know that allowing students to explore/play with a concept before direct instruction often yields better results (See page 58 of this research and this learning model from Ramsey Musallam if you don’t believe me). My students came up with these definitions of the chain rule without me ever saying the term “chain rule”, giving any direct instruction on the chain rule, and with minimal guidance from me. Don’t get me wrong, we followed it with a formal lesson on the chain rule with several examples and they will have an assignment from the book tomorrow, but I think laying the groundwork with the activity will pay dividends later. There’s also another pay off here…

Students are more motivated to come to class. They enjoy activities in which they are allowed to, if you can believe this, play with math. I don’t think this is possible without Desmos. The power in it is that it breaks down barriers for students. It’s like giving a student a canvas, paint, and brushes and saying “here, work with this for a while, make some conjectures, tinker with it, and tell me what you come up with at the end. ” Maybe I’ve drank a bit too much of the Lockhart cool-aid, but to me this is the essence of mathematics. I don’t agree with all of Lockhart’s points, but I think the link between curiosity, play, and learning is powerful and we can leverage that linkage in ways that create powerful learning experiences.

And if I haven’t convinced you yet, check out this excerpt from one of my student’s blog posts below.

“Writing this statement in question #4 required that I fully understood what was being shown in the lab. It made me think deeper about how I came to that conclusion. I believe that looking at the graphs, analyzing relationships between them, and forming a conjecture in my own words helped me to truly learn and understand the concept rather than just being given the relationships and rules and being asked to memorize them.  By completing this lab, I have improved in my ability to  identify what the graph of a function’s derivative should look like. I feel that I have a much better understanding of the relationship between functions and their derivatives and I know that I can look back at the examples in Lab 6 if I ever need help.”

I’d love to hear your thoughts on this. What are your experiences with learning and play? How does it fit into your content area or your classroom?

# 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.

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.

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 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.

References

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)

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.

# “Develop and Sell it” – A (slightly) More Creative Approach to Related Rates

As we moved through the calculus unit on related rates I searched for a way to bring more meaning to the topic and to make it more exciting. Some students were struggling and I wanted a solid review day before the quiz. I came up with this “Develop and Sell it” Activity. Essentially I gave students 4 related rates problems that were as “real world” as I could find. I had students work in groups of two to four and told them that they could only work with their group. They were told that they worked for a firm that worked out related rates problems for companies. Their task was to “develop” the solution and “sell” their reasoning and solution to the company.

Once they had found solutions that each group member agreed to, I assigned each student in the room a number 1-4. Students who received a number 1, for example, then congregated and compared solutions and methods to that problem. This then allowed students to see different perspectives on the same problem and “argue” about whose answer was right. It was a tremendously engaging activity that the students enjoyed and it relied on one important piece: I couldn’t help at all. I didn’t tell them if their answer was right, if their method was correct, or “what I would do in their situation.” They had to rely on their reasoning and justification. Taking myself out of the equation forced students to analyze each other’s work and critique each other’s reasoning. That was the best part of the activity!

In the future I would love to get actual questions from companies and industry. I’ve also considered video questions and other ways to make the problems more authentic.

As usual, any feedback you can give me would greatly appreciated!

(Here is the Pages Version of the Activity)

# Finding Areas in AP Calculus (without talking about calculus)

I’ve probably mentioned this elsewhere in my blog but one of my goals this year is to introduce each major topic using an exploration or by allowing students to “play” with the math. In that theme I considered different ways to introduce the topic of finding areas under curves in calculus.

I felt like each time I’ve either learned it or taught it, this idea is just dropped on the student. It’s actually a profound idea and technique that we use to find these areas. I wanted to solidify the idea that, by using areas of “normal” shapes, we can get decent estimations of areas of abnormal. In addition, I wanted students to see that the more shapes you use and the smaller the shapes the more accurate your measurement of the area. To do this, I gave students four shapes that had varying degrees of “squigglyness”. They had to use a ruler and formulas they already new to get measurements for the areas as accurate as possible. They also had to explain their method for finding the areas.

The activity went really well. I found a lot of value in not helping at all. Students asked “what’s the best way to find the area of this?” and I said “I don’t know.” I made sure to point out that there was no correct method for finding the area and many students used different methods. We finished by comparing all the areas in this google doc and discussing who had the most accurate method. What I loved about the activity is that students engaged in problems with no obvious answers that required them to think critically. It was then a natural segue into this activity, where we look at finding areas under curves.

Below are some samples of the students’ work. I definitely enjoyed the different methods and thought processes that students demonstrated. As usual, any feedback you can give would be much appreciated!

# I want to assess my students on more than just skills… I think

This is my second year of teaching AP calculus. Last year I felt like my students weren’t getting a full understanding of the conceptual underpinnings of calculus. This year, I’ve been taking a little bit more time with concepts, Implementing more activities that aren’t skills practice, but ask students to dig deeper into the math. I’m trying very hard to get students to talk about the math more. (See my last post.) Also, my students are doing metacognitive journaling every week via a blog. This is another technique I’m using to try to get kids to think deeper about the concepts. In the journaling and in the conversations it seems like I’m seeing good conceptual understanding. However, when I gave the most recent quiz I saw that my students seem to be lacking in applying those concepts to new situations. Let me explain more.

The problems that students practiced over and over, the skills problems, seemed to go pretty well. The problems that ask students to explain concepts directly, also seemed to go pretty well. However, there were some skills type problems that were really asking students to take the concepts and apply them in a slightly different way. My thoughts in writing those problems were that students would have the tools they needed to solve them, they just needed to pull the right tools out of the box and apply them. Either my students didn’t know which tools to use, or they weren’t entirely sure of what the tools they had were used for.

There seems to be a disconnect. My students can practice something over and over and over again and replicate those processes on the test. (Maybe this isn’t surprising.) My students seem to be able to grab on to conceptual underpinnings and explain them. However they struggle to apply the concepts in new situations. I’m still not entirely sure of how to bridge that gap. How do I put students in a position to be successful on those types of problems?

# If You Can’t Do it by Yourself… Crowdsource It! (How I handle Questions over assignments in AP Calculus)

Last year my AP calculus class ran like many math classes. There was a lecture and maybe an exploration or lab, then some independent practice time, and whatever students didn’t get done they took home as homework. Then the following day I would take questions from the class and answer as many as I could before the lesson had to begin.

Every year (or trimester for that matter) I agonize over how I should structure daily work and the grading of that work in my class. This summer was no different, especially in calculus. I wanted to strike a balance between gaining formative knowledge for me and allowing students enough independent practice, all while trying to incorporate more collaboration.

One of my professors in college took an interesting approach to assessing homework. Each day we would arrive to class and fill out our part of a spreadsheet. For each homework problem we would answer the question “Do you feel confident enough in your answer to present it to the class?” He would then select a couple students to present their solutions to certain problems to the class. This method was certainly different, and had a number of problems associated with it (for instance, I have no idea how he established a grade for us using this method) but I certainly thought the question that he was asking us was interesting.

So, in the middle of July as I am writing idea webs on a whiteboard in the bedroom my wife comes in and gives me the following idea. She said “Why don’t you ask the students that same question but, ask them before they come to class. Then when they get the class, have them teach each other in small groups.” From there, I looked at how technology could help me aggregate this “confidence data” And I worked out the details of how this would look in class. (You can see the flowchart that I made for my students here.)

It basically works like this: students have time at the end of each hour to work on their assignment. Once they finish their assignment they fill out a Google form that looks like this: This gives me a big picture of how the assignment went as well as which specific problems were most difficult for students. When students get to class each pod is assigned a problem (Click for diagram). For instance, Pod 1 might be assigned problem #52. Then, for 10 to 15 minutes students are to go to whichever pod is assigned the problem(s) they struggled with, and work on that problem. If students fully understood the assignment they are to go around and help other students with the assignment. Everyone has something to do. (I should also note that I have several “mega” whiteboards that are laid out on each pod when students come to class.)