Functions are out, variables are in. Multiplying variables v. adding variables. I could not get a satisfactory model for functions. My ideas all ended up being lighted multiple choice questions where students determine if an equation is a function or not. I decided to switch to a simpler idea that was more specific, and my students have consistently had trouble with the difference between the exponent rules when adding variables v. multiplying them. After I switched ideas, the prototype was done in an hour. A good sign.
The day the refreshers came to give feedback I was nervous, though I know from experience that I always benefit from allowing others to see my work. Also, I wasn’t sure that exponents could be demonstrated using circuits or that it was even a decent idea, but I was sure that my students have trouble with them, and this seemed like a simple idea to start with.
My feedback group liked my clearly defined goal: to have students understand the difference the product rule of exponents by contrasting it with the principle of combining like terms. In a nutshell:
x+x+x=3x and xxx=x3
The idea is that students create one circuit that represents 3x and a second circuit that represents x3. The prototypes follow.
For x3(notice 3 lights in one circuit, one battery required):
For 3x (3 lights in 3 circuits, 3 batteries required):
I am so happy with my idea, and my group asked me some excellent questions, focused mostly around implementation. How am I going to work the lesson logistically? How do I tell the students what I want them to do? How am I going to introduce circuits? If I model my version, won’t they all just mimik mine?
They also presented ideas: What if each student made one circuit and had to find their “partner”, so 4x would have to find x4? What if the battery has an x on it to represent the base? What if they do one circuit individually, then do a second one in groups, or maybe just do this in groups to begin with?
This feedback I recieved got me thinking about the logistics of how to implement this into my classroom, like the knitty gritty, what-do-I-do-next, logistics. I teach in a very traditional method with lots of direct instruction, so introducing a project like this is new to me and my students. Furthermore, my students are mostly poor. Last year I read the book A Framework for Understanding Poverty by Ruby Payne. One of her ideas is that students with a poverty mindset need clearly laid out instructions because they have never been taught how to learn. A project like this could overwhelm someone who doesn’t know how to go about figuring something new out.
With this in mind, I considered how I do something new:
- I get an idea that inspires me.
- I get excited about it.
- I go the the internet to figure out what tools I need to do the project.
- I stop reading instructions about ⅓ of the way through (due to boredom or something more interesting walking by or a new post on Facebook)
- I start trying to make my project.
- I get frustrated.
- I try harder.
- I get angry and frustrated.
- I give up for a time.
- If I am really dedicated, I will then go back and read the instructions or finish watching the video.
- I return to the task and successfully complete my project.
Making my prototypes was like this. I dumped the contents on my desk, I started cutting copper wire and sticking it on the paper, pulling it up and trying to make it stick in a new location. I went through about 3 pieces of paper and even tried drawing the circuits on the paper at one point. Long story short, there was a big mess, little strips of paper all over my desk and evidence of where adhesive had been removed from my paper.
I would really like to be able to set out a bunch of supplies and a computer for my students, tell them what I’d like their product to be, say “go”, and watch them eagerly figure out how to make paper circuits that represent monomials.
These are not the students I teach. My students give up, don’t try, and are not interested in learning new things. One reason is that they don’t know how to learn. These types of students need clear cut, accurate instructions with correct vocabulary used. I think I will make my “how to” about how to teach paper circuits to students who have no idea how to go about learning something so far outside of their comfort zone.
Here is the draft of the steps I have so far as a play-by-play guide for students to learn how to create a paper circuit:
- Read the Paper Circuit instructions (link to PDF) all the way through before beginning.
- Watch a video (here’s one) to see someone in action and get an idea of what it will look like.
- Have 3 pieces of string, an LED bulb and a battery out. Arrange them in a basic series circuit, using the PDF above as a guide.
- When it look like it’s supposed to, cut copper wire the length of the string, but DO NOT TAKE OFF THE ADHESIVE!! The circuit will work without being attached to the paper. It’s a bit awkward, but it will work. If it just won’t stay still, you can use a small piece of tape over a small section of the copper wire to hold it in place.
- When the circuit lights up (it is VERY important to keep checking to make sure your bulb lights up- this is the point, after all!), and the circuit is arranged the way you’d like it, NOW, peel off the adhesive and attach your circuit to the paper.
- Test again, and if it’s still working, decorate all you want!
- Follow the same process for a parallel circuit, including watching the video again with an eye to these specifically.
- Once the basic system is mastered, it will get easier to make variations, more complicated decorations, etc.
Right now my lesson plan will go something like this:
- warm up- review exponent rules, 6 problems
- introduce paper circuits- make a fun one for practice
- Warm Up – 4 problems on exponent rules
- Introduce the task of representing 2 monomials with paper circuits
- Make the 2 paper circuits in partners (everyone makes 2 circuits)
- Clean up
- Formative assessment: give warm up from Day 1 again to see any improvement
- Present circuits to the class
- Write explanation of both circuits to be published on class web page
Payne, R. K. (2001). A framework for understanding poverty. Highlands, Tex: Aha! Process.