Algebra 2 scope and sequence

At long last, we conclude our tour of course outlines. The biggest problem I had with the way I taught algebra 2 last year was that each topic was taught in isolation. Once we finished quadratics, we never really went back and did anything with them. And then we learned about polynomials, and forgot about them. And then we learned about rational functions, and forgot about them.

So my primary goal for this upcoming year was to come up with a curriculum that—surprise!—flowed and referred back to previous topics. (It turns out that all of my courses last year suffered from the same problems….)

With that in mind, I finally settled on aligning the course around what I’m calling the Five Expectations. They are:

1. Create and solve equations.
2. Graph functions.
3. Transform and combine functions.
4. Describe and interpret functions and their graphs.
5. Find and use inverses.

In many ways, the course outline that I came up with looks a lot like the traditional one I used last year. In the end, I decided that the problem wasn’t that the sequencing of units was wrong, but that there weren’t any threads flowing throughout the course to bring these various function families together. My hope is that the Five Expectations will focus my teaching and the students’ work in such a way that each class of functions is seen as a new tool for doing the mathematics that the Expectations describe.

Of note about the algebra 2 standards list is that there is a lot of material listed that is review. I ultimately decided to leave it all in, because (1) the rest of the department teaches those topics in algebra 2 and (2) I noticed last year that my students still really struggled with the review material. Your mileage may vary.

As with the algebra 1 and geometry outlines and standards documents: feel free to use in any way you find useful. If you have suggestions for making the documents better, please share!

Algebra 1 scope and sequence

Many of the same problems that plagued geometry last year also affected algebra 1. The largest of which (to me) was a lack of flow. Just like in geometry, it is easy to teach units in algebra and then only come back to them on a surface level, if at all. So with the new curriculum outline, I wanted to have some sort of logic to the unit order: one unit should hopefully flow to the next somewhat naturally, and later units should build on and use earlier units.

As always, feedback welcomed, and feel free to use and abuse to your heart’s content. If you do make improvements, I’d love to see them!

Video games in the classroom

I’ve been thinking a lot lately about developing soft skills in my students. Although I think it is tremendously important to teach my students a lot of math, I think it is equally important to help them develop into well-rounded human beings. As an added bonus, these softer skills are in most cases essential to further academic development.

I’ve often struggled with the disconnect between how students confront their lives outside of school versus inside. Most of my students seem to treat school as a mystical place where learning happens completely differently from outside of school. I think it’s as a direct result of this that many students struggle in school, even as they are successful learners in the “real world”. I want to help my students see that good learning frequently looks the same, no matter where it happens.

Enter the video game.

Most of my students play video games. Even if all they play is casual iOS or Facebook games, my students understand games and what it means to be successful in them. And the thing is, the same skills that allow players to be successful in video games allow people to be successful in life. Games encourage perseverance, learning from failure, critical thinking and problem solving, cooperation, goal setting and follow-through, exploration, experimentation, intrinsic motivation, and playfulness—all of which are associated with real-world success. If you have any interest at all in the intersection of video games and learning, I can’t recommend enough Jane McGonigal’s book, Reality is Broken.

I really want to spend a whole period or two in the beginning of the year playing a game and explicitly linking it to the process of learning and developing a growth mindset about math. I’m still looking for the perfect iOS game, because in a perfect world, none of my students will actually know how to play the game when we start. Which means no Angry Birds, Candy Crush, etc.

If I was still in a computer lab next year, I’d use Perspective. This game is especially nice because it’s short—about an hour and a half, depending on your puzzle solving skills—and radically different from most other games. The point of the game is to lead the blue man to the goal by manipulating the camera perspective. Blue areas are safe for the blue man, and orange areas kill him immediately on impact. If the camera is closer to the man, he appears smaller; if it is further away, he grows in size. The concepts of foreground and background, safety and danger, are always in flux, as they depends entirely on how the camera is oriented. It is a very difficult game to describe, but is entirely self-consistent and makes more sense as you play and make sense of the game world.

Here’s how I envision class going. Students open the game and have no idea what to do. I invite them to play and try to beat the game, and then set them free. After a short time, I ask students to stop and reflect for a moment, in writing, on their thoughts as they’ve been playing. We might talk about our thoughts so far, and then I’d let them go again. I am 95% sure that this would eventually involve students asking each other for help and guidance. I would make sure that this looked more like guiding and teaching than taking-the-controller. We would stop again as the average student was about 50%–75% done with the game to write and share more reflections. As students finished, I would ask them to reflect one last time and then write a little on what they think this has to do with math. Then we’d talk as a whole group about learning in the game, learning outside of school, and learning in our math classroom. As the year progressed, we’d have a shared, successful learning experience in the game that we could go back to whenever learning math became difficult.

Enter the Common Core.

Here’s what students are doing as they play the game: making sense of problems (and the game world) and persevering in solving them; reasoning abstractly; constructing viable arguments, critiquing the reasoning of others, and attending to precision as they work together; and using appropriate tools strategically as they learn to make use of the tools the game provides to solve each puzzle. If two days of class time can impress upon students the importance of these skills the way I think they can, then I can’t think of a better use of that time in the beginning of the year.

Geometry Scope and Sequence

Last year was my first year teaching at the high school, and it was definitely an interesting experience. One of the most frustrating things for me was that I didn’t make the move until August, which did not leave much time for planning. Whereas in the past I had come into the year with a list of standards and an idea of how my courses would flow, I went into last year with, essentially, the textbook.

Geometry was the course that suffered the most for my lack of preparation. While our textbook (Holt) isn’t horrible, I never really felt like the year had any sort of flow. And if I felt that way as the teacher, I don’t want to think about how the students felt. In particular, my biggest regret is that I don’t believe my students ever really developed a strong understanding of proof.

So the goal for this year was to write a curriculum that was logical, had a flow to it, and actually developed students’ ability to write and understand proofs. In writing the curriculum, I went back and reviewed the Van Hiele model. In light of this, the course attempts to start at Level 0–1 before slowly moving to Level 3–4 as the year progresses. My hope is that each individual section will do the same.

My second largest failing last year was that too many topics were taught and then dropped. In a course like geometry, where I am asking students to refer back to previous theorems, concepts, and definitions on a daily basis, constant review is critical. To this end, the beginning of the year is filled with some introductory material which will be called upon and extended throughout the year. It is my hope that each section will have many natural opportunities to go back and review, use, and extend earlier sections.

I wanted to focus on the overall flow of the course, so there are no unit lengths, assignments, “I can…” statements, or assessments. Instead, each section has a brief description of the overall goal and then a few points on the specific theorems, constructions, and understandings that students should learn. Of course, that means it’s probably not very useful to anyone, but I’m sharing it anyway. I would love any feedback that anyone wants to share.