Last week at NCTM 2018, I presented some thoughts from a radical Algebra II experiment I ran some years ago. While Algebra II was the center piece, the topic is broader. What story are you trying to tell with your curriculum? How do you integrate pieces so that students can see a theme in the skills and concepts they learn throughout the year? Why should what you teach in September matter in April?
This is a summary of those ideas from my talk.
The textbook models of Algebra II are pretty consistent. A whole parade of function types, done in isolation, over the course of a school year. Your experience probably looks something like this:
I dutifully followed a script similar to this for four years of Algebra II. But there were lingering problems. In the Fall of 2012, I included this as part of my final exam:
I made what I thought was a fair assumption: we had covered linear, linear systems, absolute value, and radical functions throughout the fall system and students should be able to work with them when mixed to together.
I was completely wrong. Students had forgotten fundamental algebra mechanics and were thrown trying to remember not only how the non-linear ones were dealt with differently, but even what they were in the first place. The topics we discussed had been so spaced out that students had made no valid connection that equation solving is a universal thought, different functions merely introduce new mechanics.
Second, we had this section in the book on parent functions. There was an activity that accompanied it:
Students spent a class period matching equations, words, and graphs together to form a parent function library of sorts. Then we never talked about this activity again. It served no purpose other than to say "ooh, activity!"
I spent the summer of 2013 frustrated with Algebra II and determined to find a better way. There was good material in this course, but it was being handled in such an awful way. The messaging was all wrong. If you take the Texas State Standards for Algebra II and make a word cloud, you get something like this:
Clearly, the individual functions themselves are not the most important players here. Equations and Inequalities are the most frequently used terms. Incidentally, you see how small "real world" is there? Probably could stand to fix that too.
I revisited the parent function activity and thought: what if this was the entire course? What if we talked about parent functions at the very beginning of school and referenced it throughout the year? What if we focused on not just solving equations with one function at a time, but with many functions at time?
Pivot Algebra Two was born. Let's have students see how a set of skills play out with many different kinds of functions all at once. I divided the curriculum into two parts. Part 1: linear, radical, quadratic, and absolute value. Part 2: log, exponential, rational, polynomial. In each part, we would do overviews of the skills in play and slowly spiral up the difficulty.
In addition to taking tons of times to play with equation mechanics (roughly 6 weeks on linear, quadratic, absolute value, and root functions with increasingly higher difficulty), students got to see connections between their work and graphs. I don't know about you, but prior to this experiment, my kids hated graphing. It was a tedious process that seemed unrelated to anything. In this vision of Algebra II, I was going to give graphs a purpose.
When it was time to start a month of graphing, they groaned. As the ones that came before, they hated graphing and didn't understand the point. But then I took some classwork we had done previously. At this point they all agreed that solving equations was super easy. I graphed one side of the equation, and then graphed the other. I clicked the intersection points. MINDS BLOWN. The graph and the equations are related! I can check my answer easily! I don't need your help anymore!
It was a beautiful moment. I used it as a jumping off point to more complex algebra. In both our look at advanced quadratics and logarithms, we started with the graphs. We know that the intersection point of two graphs is equivalent to the solution of the equation. What algebra do we need to prove that?
Then we summarized what we learned about our first four functions:
The efficiencies in this system were apparent pretty quickly. I was no longer reteaching simple algebra moves. Kids had them down. We could tackle mechanics previously unheard of in Algebra II. We even put logarithm properties to good use with problems like this:
There is a lot of overhead in this problem. You have to understand how logarithms combine, how they are undone, that the resulting quadratic has two solutions, and that it's possible not all of those solutions are valid. And, to my sheer astonishment, the kids were telling me this was easy. Absolutely bonkers to think that this problem, which take a good 5-6 minutes of kid time to work, was easy! I'd ask if they need any help and they'd say, nope, I can just graph it.
These connections also help improve their ideas of equivalence. We could take any line in this process, graph it, and see that the intersection points had the same x-value. We were manipulating a log into an equivalent parabola.
The Longterm Lesson
This line of thinking, finding the story in my curriculum, continues to pay off. Because of this exercise, I have improved my ability to weave concepts together in other courses. I no longer teach Algebra II, now I'm primarily focused on Calculus, but the fundamental thought behind this experiment remains. Calculus has a story, and students should be able to see it.
Calculus is even better suited to these ideas. Limits, Derivatives, and Integrals interplay with each other in all kinds of ways. What if students saw that relationship from the very beginning of the course? What if they didn't have to wait 4 months before they even heard the word "integral?" How much better could they be at seeing those relationships?
I encourage you to take a hard look at what you teach. What are the big ideas? What should students be able to do when it's all over? How can you give them an overview of their year in the first month of school?
For more detailed information, I chronicled everything about my radical journey through Algebra II. Please contact me if you need anything.