Playing With Functions

For this assignment in my Creativity in Teaching and Learning class we were asked to develop a playful introduction to our topic area (functions). I came up with two introductions (because I couldn’t decide and both might be effective in different contexts).

The first introductions uses a technology that I’ve relied on several times throughout this course, desmos.com. What I’ve done is created a very basic smiley face in Desmos. The task for students is to create a “non-basic” smiley face using at least three different families of functions, at least ten functions, and at least one animation. They should strive to be as creative as possible. They should look to other pieces of art for inspiration. This is a good introduction to functions because students have a lot of flexibility to “play” with the mathematics. They’ll have to figure out how to shift functions vertically and horizontally as well as stretch and squeeze them. They’ll also have to figure out how to restrict domains so that the graphs are limited to just the face.  Having them do an animation will ensure that they will learn how to use sliders in an organic way (because they have to create an animation, not because I said “please create a slider). My hope is that this will help them see where sliders might be helpful in future problems. I also think, with reflection and future application, that this will provide a good foundation that I’ll be able to connect back to throughout the year.

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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?

How to Make a Human Drum kit

In my Masters of Educational Technology program at Michigan State we had the opportunity to host a Maker Faire. We broke up into groups and each group designed a maker “station”. Our group created a human drum kit and it turned out awesome! I want to share a “how-to” for building a human drum kit.

Purpose: The purpose of this activity is to leverage the power of a Makey Makey and Scratch programming to create a set up where one person in a group is a drummer, and each of the other people in the group are part of the drum kit (snare, cymbals, etc.). When the “drummer” touches the hand of the person connected to the snare wire it will complete the circuit causing the snare sound (in Scratch) to play. If you have a person for each part of the drum kit you will then have a fully operational drum set (made of people).

Materials Needed

  • Makey Makey
  • Several Alligator clips and connecting wires
  • Conductive thread (This has two uses, first it is sewn into the “drummers” head band, second it extends the connections between the Makey Makey and the parts of the drum kit.)
  • Pipe Cleaners (To create the bracelets that parts of the drum kit will wear.)
  • Copper Tape (To wrap around the pipe cleaners, so that the wristbands are conductive.)
  • A computer with working speakers that is running this Scratch Program

For the set up I will be referencing the diagram below. The blue dotted lines represent conductive thread connected to wristbands (pipe cleaners wrapped in copper tape) which must be touching human skin. The red dotted line is conductive thread connected from the “earth” part of the Makey Makey to a headband. The headband had conductive thread woven into it. The thread must be touching the forehead (skin). With the scratch program running on the computer a person only needs to touch the drummer for that instrument to sound. This completes the circuit which sends the signal to the computer.

A special note about the kick drum: You can use a wrist band and a person for the kick drum. We found that it worked better if we attached the blue kick drum wire to copper tape on the floor. Then when the drummer touched it with their bare foot they completed the circuit for the kick drum. This allowed for the kick drum to feel more natural. (You could also connect the blue wire to tin foil and wrap it around the person’s shoe if you didn’t want to go barefoot.)

Human Drum Kit Set up

 

General Suggestions

Here are a few suggestions after having been through the project. First, tape down the wires. This keeps them much more organized. Second, make sure there are many points of contact for the headband. Third, make sure no wires are touching the headband wire. This unintentionally completes the circuit. Last, make sure that the copper on the wristbands has a good contact with human skin. Without that you can’t complete the circuit.

Below are a few images and videos of the drum kit in action. If you have any questions at all please leave a comment or tweet me and I’d be happy to point you in the right direction.

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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.

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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.

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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.