# “Feeling” Functions

Edit: In my original post there was a small mistake in the proof. I have now corrected it. Thanks to Carleton for pointing it out!

For this assignment in my Creativity in Teaching  and Learning class I had to come up with a way to “feel” my concept (functions). “Embodied thinking”, or the idea that “feeling” a concept can help us understand it in a useful and deeper way is at the heart of this assignment.

Getting a feel for mathematics can happen in a number of ways. I want to discuss a couple of the methods, starting with how real world situations are translated into mathematics. This assignment actually spurred an idea for a project in class in which I give a group of students a position versus time graph and have them walk out the graph as if they were the particle being described by the graph. I recorded it, put the videos on Youtube, and then had the other groups sketch graphs based on the videos. We then held a competition to see who could get the most accurate graph and also which group did the best at walking out their function. Position v. time graphs, which come up frequently in calculus, became much more real. It truly gave the functions a feel. My hope is that when students look at these graphs in the future they might imagine how a particle would feel while tracing the graph and that might help them get glimpses into the velocity and acceleration of the particle. This would be similar to how Robert and Michele Bernstein, authors of Sparks of Genius, point out that Stanislaw Ulam, a mathematician who worked on the atomic bomb, apparently “imagined the movements of atomic particles visually and proprioceptively” (1999). Below are a couple of the videos along with the graphs that went with them.

# 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?

# The GIF(t) of Curiosity

Recently I played around with Snagit on one of the class Chromebooks and discovered how easy it was to create a GIF. I then tried to figure out how to leverage this in the classroom. What I came up with was a prompt based on the GIF below.

The prompt essentially asked students to recreate the GIF. To accomplish this they had to “get under the hood” of the mathematics. This required them to generalize (they’re used to finding secant lines at concrete points) and that was very difficult for them. We rarely ask students to generalize and when we do, it’s usually is in context of a “critical thinking” book problem that gets skipped. Worse than that, often the teacher ends up doing the problem for them at the beginning of the next class. And even if a student does try it, unless it’s an odd problem, they usually can’t see if their generalization was correct until the next day.

# Observe and Re-imagine

My current project in CEP 818 – “Creativity in Teaching and Learning” was to observe an object tied to my topic area (functions in mathematics) and to re-imagine it, appealing to a different sense than originally intended.

First, a note on the importance of observation:  Robert and Michele Bernstein, authors of Sparks of Genius, point out that “All knowledge begins in observation. We must be able perceive our world accurately to be able to discern patterns of action, abstract their principles, make analogies between properties of things, create models of behavior, and innovate fruitfully .”

And now, the re-imagining of the graphing calculator.

Any math student that graduated from high school in the last two decades probably used one and I can guess (although I might be wrong in some cases) that it was used to get answers quickly and easily. In fact, as I was observing this device from every possibly angle and sense (okay, I didn’t lick it…) I noticed that it screamed simplicity. It is a no frills, unexciting, almost heartless object. Take 30 seconds and look at the image below. There is not much about that calculator that screams “Hey you, let’s do math and it’ll be awesome!” To be clear, it is incredibly powerful, but it is often underutilized as an exploration device and over used as a “shortcut machine”.

This may seem irrelevant to the learning of mathematics but it doesn’t help the plight of the math teacher that is trying to motivate students. I hadn’t considered this before, but I started thinking “Is this a device that I want to use?” I love mathematics but this device, although it is simple, is not elegant and not a device that I look forward to using. It is fraught with ugliness. From the black and green display to keys that seem randomly colored, there is much to be desired in terms of design. I believe that if we redesigned this device to be aesthetically pleasing and still powerful, functions and mathematics in general would be more accessible.

If you haven’t bought into this yet, consider your favorite technological device. My guess is it’s your phone (and if it isn’t, humor me for a minute). First, I bet the ratio of the sides is close to the Golden Ratio, which creates the most visually appealing rectangle. I would also guess that it fits nicely in your hand, that the interface is intuitive yet attractive, and that it balances power with beauty. The feel and design of this device make you want to use it. What if a calculator had that same appeal for math students? (The GIF below represents the potential of moving mathematics to the devices we use most often.)

In addition, this is not how I see mathematics.  I see math as a beautiful framework within which we can describe the natural world (and even if we couldn’t, it is still beautiful). Functions are like an engine that can be constantly modified to describe an infinite number of situations. If we can find ways to relay the beauty in mathematics and functions to students then I think (and admittedly I’m aiming high) we could ultimately shift the culture to one that views math as elegant and powerful, not just an obnoxious requirement to get a diploma.

A note on the above GIF: Each one of these graphs (with the exception of the smiley face) is a function or two functions graphed together. I used desmos.com to create the graphs. I think that tools like Desmos, which runs on all smartphones (the devices we actually like to use) can help students to visualize functions as well as other mathematics. (Here are some of the functions I used.)

References

Bernstein, R., & Bernstein, M. (1999). Sparks of genius: The thirteen thinking tools of the world’s most creative people (p. 30). Boston, Mass.: Houghton Mifflin.

# This Quote From a Student Made Me Smile

Additionally, Mr. Cresswell didn’t hold us in a right-wrong mindset while working on the problem – we were focused more on discussion.

# Considering Standards Based Learning

For over a year I’ve considered switching to standards based grading in AP Calculus. I’ve read, listened to conversations, and thought about how it could change my classroom in a positive way. My problem always seems to come with implementation in the traditional grading world. To try to pull out the advantages of SBG in this context I read Frank Noschese’s post on Keep It Simple Standards Based Grading. That post informed much of the system I’m about to lay out.

I want to write about how I’m considering implementing SBG this year in calculus (you might call it SBG Lite). I’d love feedback from people that have implemented SBG and can help me troubleshoot this before I dive in.

The Plan

First, a student’s grade will be broken into two pieces. The first piece I call the standards portion and is worth 90% of a student’s grade. The second piece is a blog, which is 10% of a student’s grade. I believe the blog should be a component by itself and has value beyond my standards. You can see how I evaluate my reflective learning blogs here.

I’ll lay out the standards portion first. This is basically the SBG portion of the grade. Last year, as I went through each unit and wrote out all the individual standards, so now I have all the standards typed up. I am going to follow Frank’s Yes/No method. Either “yes” you mastered the standard or “no” you haven’t mastered the standard. My test/quizzes are spiraled anyway, so topics from the first unit show up on tests in future units. This allows students to master topics even if they don’t master it the first time. If a student starts out with a yes (on the first assessment for example) and then misses the standard on the next assessment he/she will be moved to a “no”. In other words, students can slide back and forth from yes to no and no to yes, throughout the year. The idea is that I want students to be making sure they understand even the oldest concepts.

I don’t think there’s anything crazy about the above structure (but please, if there is then definitely let me know in the comments). However, I’ve added a bit more to each standard. In order to get a “yes” on a standard you have to master the objective portion of the standard AND the communication portion of the standard. The idea is that there is more to understanding the standard then just being able to do a problem on a test. You also need to be able to communicate the concepts. Check out the image below to give you an idea of how it’s laid out. (The CCC refers to how I assess homework on a daily basis. Check out this post for more on that.)

All of this information will be kept in a spreadsheet that is shared with each individual student. This way students will always know where their grade stands. I’ve provided a sample of that spreadsheet and posted it below. (Clicking the image takes you to the actual spreadsheet).

To summarize: Each student’s final grade is broken into two marking periods and a final exam (the weighting is 3/7 + 3/7 + 1/7 = final trimester grade). Each marking period grade will be broken down as outlined above (90% standards grade, 10% blog grade). Each standard is broken into two parts (objective portion and communication portion). If a student hasn’t mastered one part, then they don’t get a “yes” for that standard. Students must continue to demonstrate mastery throughout the year as any standard can slide back to a “no”.

As I mentioned above, the main purpose of this post is to get feedback on this system and help troubleshooting before I implement it. Any thoughts or ideas you have would be greatly appreciated!

# 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.

# It’s the Soft Skills!

This trimester I don’t teach in a flipped classroom. Precalc is done for the year, AP calculus is not flipped, and Algebra II is taught by myself and another teacher and is typically done very traditionally. This being the case, my goal this trimester is to “inject” solid higher level thinking activities into my lessons. Last year when I taught it (the first time) was very traditional (Lecture, assignment, repeat). The catch 22 of this that I haven’t had these students all year. They are coming to me from traditional classrooms and are not familiar with the different format of my classes (mega whiteboards, relaxed deadlines, higher level thinking, collaboration, etc.). I wanted to reflect on a few things that I noticed in the last couple days that I may have taken for granted.

Collaboration

For some reason, in the back of my mind, I just assumed that juniors in high school knew how to collaborate. This is not true. At least it isn’t true for my students. As you can see in the image my class is set up in pods and each day they come to class with a mega whiteboard and a few markers at their pod. (This is to encourage collaboration on “normal” days, not just days in which we have special activities.) Today I gave them an activity to help them discover the connection between combinations, binomial expansion, and Pascal’s triangle. I prefaced the activity with an emphasis on the need to collaborate and to share ideas with each other. I noticed that even the “best” groups struggled with this.

There were several specific problems I noticed. First, students didn’t use the mega whiteboards very often. Even when I explicitly said, “hey, this would be a good problem to do on the board.” Second, there was minimal communication between group members that “got it” and those that were still struggling. Third, there was minimal critique of each others work. for instance if one person had the correct answer, in many case everyone else just copied down the answer.

Tomorrow I will be making a point to talk about the best ways to collaborate. I failed to recognize their lack of skills in this area and I need to do a better job of setting them up for success in the area of collaboration. I also need to continue to try to build a community in which wrong answers are not shunned but are view as just a step in the learning process.

Communication

This is intimately connected to collaboration, but my students ability to communicate mathematics needs improvement. I think the more I help my students with this the better collaboration will be. How can I expect students to collaborate if they can’t communicate the math to each other?

I hope that I’ll continue to see improvement in these areas as the trimester continues. Please give me any ideas that you have to increase students’ ability to communicate and collaborate. I’d love to hear them!

# “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)!