Algebra 2 & relationships

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:

{ (1, -3), (10, 1019), (-1, -4.5), (?, 59), (2, -1), (3, 3), (0, -4), (0.5, ?) }

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.

Rekindling why I teach mathematics

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.

I haven't been teaching that at all.

[NBI] My teaching story

This post is part of the new blogger initiative.

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:

1. Put a post-it note in the middle of a blank piece of paper.
2. Think of your four favorite teachers you've had.
3. Write their names on the paper around the edges of the post-it note.
4. Now think about each teacher individually. Choose one word that describes why you wrote that teacher's name.
5. Remove the post-it note.

Ta da! How similar does that look to your teaching style?

Here's mine:

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.

That's the kind of teacher I want to be.

HELP! (or, be careful what you wish for)

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.

The backstory

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.

Ideas

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
• Programming
• Researching real problems
• Presenting with digital resources
• Writing
• Learning to become self-directed learners
• Live backchanneling
• 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.

Please.

Transformations with Scratch

Students need to do more

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.

Challenges

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: 

1. Finding projects that fit the unit.
2. Student expectations of what math class is.
3. Presenting the projects to students.
4. Assessing the projects.
5. Instructional time.
6. Planning time.

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.

Scratch project

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!

[NBI] Modeling Temperatures with GeoGebra

This post is part of the new blogger initiative.

The more that I grow as a teacher, the more I believe in teaching modeling with mathematics.

That might sound somewhat silly. I've always known, in some sense, that modeling is a good idea. They certainly encouraged it in my prep program and it's a significant part of the Common Core. However, when you're actually making instructional decisions for the day or week, modeling can be tough to choose. It takes more class time, and in some ways, can seem "softer".

But every time I have my students complete a good modeling activity, those thoughts just seem ridiculous. I see real struggle. I see connections get made and broken. I see students conjecture. I see them contextualize and decontextualize. Beyond the truly deep learning, I haven't encountered a better assessment tool.

Monthly temperatures in Wisconsin

One of the better lessons that I taught during my first year of teaching was a modeling lesson. We were finishing up a unit on trig functions and I *thought* that my students had learned the transformations. I wanted them to see that what we learned really does show up in the world, so I grabbed the last five years of monthly temperatures in Wisconsin from the Online Climate Data Directory. I then entered that data in the spreadsheet of GeoGebra, created a list of points, and distributed the GeoGebra file to my student. (I would love for my students to be able to handle the data, but I made an executive decision based on time-constraints).

I know that it's supposed to look like this, but it still kind of amazes me. 

For the activity itself, I distributed the following halfsheets to guide students through the techy aspects of the activity:
Temperature Directions

Basically, I wanted students to find an equation that fits the data as accurately as possible, interpret the different parts of their equation in the context of monthly temperatures, and present their work in a professional looking document. As we had done very little work of this type, I created a template with Google Docs to use.
Temperature in Wisconsin

Results

It wasn't perfect, but it was good. The insights into students' thinking were amazing. For example, it turned out to be very difficult for my students to interpret the period as one year, or that the midline represents the average temperature. I had a number of very good conversations trying to help students make these connections.

Additionally, I really enjoyed watching students while they were fitting equations to their data. I had considered setting up the equation for them with sliders, but I'm glad that I didn't. There was value in having them manually type in and change constants in the appropriate places in their equation. Students were checking their notebooks to figure out how to change the amplitude, or just tinkering to see if it worked. Really, either was fine with me. 

In the end, I hope trig was a bit more concrete, and I had a much better idea of what my student understood.

Next time

Instead of ending the unit with this, I will start with it. Trig would then come from the real world and students would be playing with all of the transformations before I teach the vocabulary. We could then discuss the ideas of trig transformations with a solid context to refer to and draw comparisons to other places in the world - "How do you think the amplitude of the monthly temperatures in Wisconsin compares to Cuba's? Why? What about the period?"

Next time.

Geometry summer curriculum work

This summer our department was given some curriculum hours to develop/reshape our geometry courses in light of the Common Core State Standards (CCSS). I'll assume that you know something about this - I just want to share the process we went through and where we are currently.

At the beginning of the summer, we had a consultant give a short workshop for math teachers in our district. To be perfectly honest, I didn't find it incredibly helpful - possibly because I'm not that long removed from my certification program, where the CCSS were the only standards that we worked with.

But, I did take one quite useful idea from the workshop - a process for breaking down the standards into something that we can get a better grasp on. I turned that process into a sensibly designed Google Docs template to facilitate our summer work.

So how did this work? Well, as a starting point, we decided to base all of our work of the suggested "pathway" from appendix A of the CCSS for mathematics. Within each unit, we looked at the different clusters. We decided that, for the most part, we're going to teach mini-units based on clusters.

For example, here is the first cluster:

Reading through this it makes sense, but this is where is becomes more real. How we turn this into what we're teaching? We considered just making these our direct standards, but there was some messiness there. Concepts overlap, others would be difficult to assess, and so on.

This is where breaking them down became quite helpful. For each cluster, we created a document that broke down each standard into the what and how. Looking at these all together, then, we would create our standards. We used some standards pretty much directly. Others we would combine, tweak, and interpret. Some standards we decided were things we would do/teach in class, like G-CO.4, but we wouldn't make them standards for assessment.

For the above cluster, this is how we broke the standards down, and these are the standards we developed:

1. Geometric definitions
   Identify & precisely define angle, circle, perpendicular, parallel, & line segment based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
2. Transforming figures Reflect, rotate, and translate figures.
3. Identifying transformations Identify sequences of transformations that carry a given figure onto another.
4. Transformations as functions Describe a transformation as a function in the coordinate plane.

Now these are standards that I'm more confident that I can teach and assess.

What we're currently working on & where we're going

We're done with everything above, and we're finishing writing small formative assessments for every standard. We're then going to pull them together, tweak them, and create the summative assessments. Our school implements a 80% summative, 20% formative grading policy, so we're somewhat forced to break things down like this. I probably wouldn't otherwise.

This was all rather mundane work, and I have a hard time calling it "important", especially considering what we might have done with our time otherwise. However, whatever gripes you may have with national standards, and I have several, we are legally obligated to teach them, are students are going to be judged on how well they know them, and we're likely going to be judged on how well our students do. So, in some manner of speaking, this was important work to complete.

There's clearly something missing from this model. It treats math as the sum of the standards, when it should be so much more than that. At the beginning of the summer, we were hoping to include a "cluster question" within every cluster of standards that would be an interesting problem, application, project, or whatever that uses the main idea of the cluster. We didn't have time to develop them, but those could help.

Incorporating the practice standards will help. I'm going to try to use them daily to guide my teaching. But what I really want to do is use this work as the base for turning geometry into a project-based course. We'll have a solid grasp of what our students need to learn, along with some of the how. With this foundation laid, I think we could do something really special while respecting our commitment to teach the standards.