I'm starting to love teaching trig. In the past few terms, we've ended the unit with a GeoGebra project where we model monthly temperatures in Wisconsin. The project is a great way to see if students have connected the ideas to the real world and the students actually end up thinking it's a little cool. However, I don't like having to do a week or two of abstract, context-free trig before really getting to students to see why it matters. So, to try something a bit different, I put together an intro task where they look at some more real data - in this case, a fellow teacher's monthly gas bill. You can find that task here. I give them this graph and a few questions:
How many years of data is represented on the graph? How do you know?
Circle the point associated with the most expensive bill.
How much did he pay?
Which month of the year do you think that was? Why?
Put a box around a point associated with an “average” bill.
How much did he pay?
Which month of the year do you think that was? Why?
On the graph, sketch a curve that fits the data.
Let’s convert that graph into a different form. Create a table for months 6 - 29.
How would you describe the pattern of the gas bills?
Predict what Mr. Bruns’ gas bill will be in the 40th month. Show or explain your reasoning.
Predict what Mr. Bruns’ gas bill will be in the 100th month. Show or explain your reasoning.
Hmm, it would be awfully nice if we had a function that could do some of the work for us...
From here, I'm thinking that we would go on to develop a "cost" function, connect it to a circle, and then get to a point where there is a motivation to develop the unit circle, sine, and cosine:
I'm going to try this out this week - we'll see how it goes!
We've been working on our algebra 2 curriculum over the past year, both to align it with the Common Core and make it seems less like a random collection of function things. Last year we had the idea to make it very data-based; the idea was to have students gather a ton of data, and then base our study of functions on those data sets.
There are a few problems with this approach. For one, our district decided that prep periods are a luxury that cannot be afforded, so we don't have the time to prepare such a radical departure from our current curriculum.
Second, and more problematically, I think this would put way too much emphasis on the applied part of mathematics, and would make it difficult to meaningfully teach any aspects that aren't as practical. If we're motivating the study of functions based on answering questions about real data, I'm not sure how I'm going to help students understand removable discontinuities. Or motivate them to solve difficult equations algebraically when a graphical method yields a solution quickly. And so on.
Mathematics is not only about answering practical questions about the real world and data. It is certainly useful for that, but it's also much more. Thus a solely data-based approach to teaching algebra 2 would be a mistake, in my opinion.
So we need a more flexible approach if we're going to be true to all of mathematics. Thinking about the essence of our algebra 2 course, and really algebra in general, what we're dealing with is relationships. Linear, polynomial, rational, radical, exponential, logarithmic, and trigonometric, to be specific. We learn about these relationships by looking them in different lights - algebraically, graphically, and numerically.
Currently, a major problem with algebra 2 is that, to students, it feels like a ridiculously long laundry list of things to learn. I get the feeling that they don't understand that the equations, graphs, and tables that we work with represent the same relationships. I try to teach the connections, but they're mostly lost on students. And it's not surprising - the focus of the course isn't on understanding the relationships, it's on the set of techniques and facts about them. Without understanding the relationships themselves.
So, my idea: refocus the course on the relationships. One way to do that might be to start numerically. For example, we might start with the following:
I think this would really put the focus on the relationship itself. We're dealing with two quantities, and there is some way to predict one from the other. However, it is difficult to figure that out simply with numbers, so translating the problem into different forms - equations and graphs - takes on more meaning.
Our new semester starts in about a month; I'll let you know how it goes. Any wisdom you have to offer would be appreciated.
I haven't blogged for a long time, and I'll explain that in relatively short time. But in the mean time, I have a brief thought to share.
I've really, really been struggling lately to find my passion for teaching pure mathematics. I'm very passionate about teaching computational thinking skills and statistics - ideas and methods that clearly are relevant to developing an informed citizenry - but I've been questioning the point of teaching apathetic teenagers rather abstract mathematics.
Well, I've found the beginning of a cure in, of all places, a graphic novel. This weekend I've been reading Logicomix, a story about the life and ideas of Bertrand Russell.
For those of who don't know, or have forgotten, Russell's main role in the story of mathematics was his attempt to create a solid foundation of all of mathematics in his "Principia Mathematica", co-authored with Alfred Whitehead. This graphic novel does a beautiful job of connecting the ideas in mathematics to the questions that make our existence so wonderfully interesting.
I need to remember, I'm teaching ideas. And these ideas have a context; they are part of the story of real people attempting to more fully understand our place in the universe.
Every teacher has a handful of characteristics that make him or her unique. While your teaching style may share characteristics with other teachers, it's the unique combination that makes you different. It's interesting to figure out where those come from.
Right before I started my first year of teaching, we did this activity:
Put a post-it note in the middle of a blank piece of paper.
Think of your four favorite teachers you've had.
Write their names on the paper around the edges of the post-it note.
Now think about each teacher individually. Choose one word that describes why you wrote that teacher's name.
Remove the post-it note.
Ta da! How similar does that look to your teaching style?
Fun, faith, inquiry, & reach
Fun (high school Spanish teacher)
I wasn't convinced I wanted to be fluent in Spanish, but we had so much fun in this class that we didn't realize how much we were learning. I recognize that now with how much Spanish I still remember, even though its chief use is when we go out to eat.
Faith (high school computer science teacher)
He challenged us, a lot. But it was always obvious that he truly believed and knew that we could do what he threw at us. And you know what? We always did. After a while, I just started to figure that if he was assigning it, I could do it.
Inquiry (high school physics teacher)
This was my introduction to a truly inquiry-based modeling class. He drove a lot of my peers mad because he wouldn't just give us the answers, but he was one of the first teachers to introduce the idea that we're fully capable of creating our own knowledge.
Reach (high school orchestra teacher)
I remember sitting in orchestra after a recent concert and getting the new music. We would look at it, try to play some of it, and sometimes barely make it through a measure. Follow this up with one of my most vivid memories from high school - the director tearing up on stage after finishing the last note. That's what I mean by reach - she showed us what we could do.
The really powerful part
I had a student come in during lunch last year, and she saw the post-it note hanging above my desk. "What's that?", she asked. I briefly explained it to her.
Her response? "Yeah, that really does fit you."
Just think about that. The qualities of my favorite teachers not only influence how I like to teach, but my students recognize them in my teaching. These teachers were so amazing that they're impacting my students.
You know how there's that saying, "be careful what you ask for - you might just get it?"
It just happened to me.
School starts on Tuesday, and my block algebra 2 class just got moved to the Mac lab. Every day. Instead of teaching in a normal, boring math classroom with 28 deskchairs, four off-white walls, and some corny math posters, I am going to be in a room with 30 gigantic iMacs, along with with a side room that has a half dozen circular tables with chair. I couldn't ask for a much better setup.
I have five days to figure out some kind of rough plan for how I'm going to approach and teach this class differently. Feel free to skip the next section if you don't care about what led to this.
I was frustrated last year with the limited access to technology that my classes I had, and it didn't look like it was going to get any better. So I decided to take matters into my own hands, and I started collecting old computers from friends and family. I cleaned them up, installed Ubuntu or lubuntu, along with GeoGebra, Scratch, and matplotlib for Python.
That project is coming along. It would be great to have one or two more, but I should enough functional computers in my classroom for groups of four to be able to share a computer with GeoGebra, Scratch, and Python.
Then our tech person came into my classroom today. She asked a polite question about how my mini-lab was coming, and then said, "I don't want you to have to do this. I mean it's great that you are, but you shouldn't have to. The Mac lab is open block D - what class do you have then? Algebra 2? Ok, you're teaching that class in the Mac lab."
Umm, what? Really? This is going to take a day or two to sink in.
I want to underscore, I am ecstatic about this. I'm not complaining. It's just that school starts in five days, and I don't want to squander this. There is so much that I could do - I just don't know what that is yet, or how I'm going to do it.
I don't want to just list the specific tools that I'm going to use. I want to develop a picture of what I want my students doing that is different than the standard paper-and-pencil based algebra 2. Then I'll figure out what programs, services, whatever that will help us accomplish that.
So, my list:
Work with large data sets
Researching real problems
Presenting with digital resources
Learning to become self-directed learners
Doing all of the above collaboratively
I could probably add more, but I think that's a decent set of ideas. I'll certainly try to give computational thinking a central role in the class.
Where I need help is making this happen in algebra 2. How can I adapt our unit on polynomials to a classroom with this kind of access to technology? What can I do with solving radical equations?
If you have any ideas, please comment, tweet, Google+, or email me. I promise to share what I do and give credit where it's due. The six units we teach are polynomials, rational functions, radical functions, exponential & logarithmic functions, trigonometric functions, and statistics.
Seriously. Whether it's just an idea you have or a lesson you've taught ten times, if you're reading this, please share. I've been given the blessing/curse of what I wanted, and I want to make sure I actually take advantage of it.
I don't mean problems. I mean building. Creating. Writing. Designing. I think it can be amazing [read: tragic] how little students actually produce in a typical math class - no wonder math is often perceived as abstract and disconnected from the real world.
Having students do more is good for about a million reasons. They provide a firm foundation to build on their prior knowledge and experience, formulate questions, climb the ladder of abstraction, and experience how math is connected to the world they live in. And the best part? The result of their growth is concrete. It's not a completed worksheet, or a good grade on a test. Maybe it's a video, or a presentation, or a programmed simulation, or whatever. The point is that it is something much more real.
That all being said, there are reasons why projects aren't the norm in math. I know some schools have taken bold steps and are letting going of pre-determined curricula in place of project-based learning and student-driven inquiry, but they're the minority. And not my school.
In a typical public school, I think these are the main challenges for using projects in math:
Finding projects that fit the unit.
Student expectations of what math class is.
Presenting the projects to students.
Assessing the projects.
My plan is not to solve all of these challenges this year. My hope is to create some space and a framework that allows me to insert projects more frequently.
First project: Transformations with Scratch
I participated in a CS4HS workshop this summer and was introduced to Scratch. The best way to learn what it is is to play with it. Go ahead, it's a free download, cross-platform, and fun.
The first project of the year for my geometry students is going to be creating a 30 second animation with Scratch. I like this it introduces programming as well as gives students a deeper experience with transformations.
Here's the handout for students that I will use. The front side presents the task and details to the student; the back contains the rubric by which the projects will be assessed.
I could greatly use some critical feedback before I give this a go. Design, language, rubric, whatever. The "product" part of the rubric definitely needs improvement, but I'm having trouble describing what it is that I want students to produce.
After this project, I plan on tweaking the setup and then using this as my template for the rest of the projects. Hopefully it will making creating new ones quicker and easier!