Today's my last Friday off for summer, and I was thinking about the start of the school year and I thought it'd be good to get a quick post about teaching Algebra 2 again.
I love the subject; it's just the right combination of easy and hard, boring and exciting, useless and useful, tediousness and shortcuts. But a lot of students don't like it because it's "hard". Not just the procedures, but the concepts start to become harder too. And most are coming from a relatively easy subject in Algebra 1 and are not prepared for the big jump in difficulty of content that Algebra 2 is.
We try to prepare for this by starting the year off with a review of important Linear Equations topics, something most students know and (somewhat) remember from Algebra 1. Linear equations are easy because linear thinking is easy -- it's predictable, it's consistent, it's just a straight line. We often think linearly, in that you do this, then that, then this happens, then you do that.
But unfortunately, life doesn't happen linearly. We might like to think that it does, but it doesn't. As the movie Prometheus says, "Nature doesn't build in straight lines" (except crystals apparently). Nonlinear thinking is much, much harder than linear thinking, mostly because nonlinearity is harder than linearity -- it's unpredictable sometimes, it's deceiving, it sneaks up on you. And, in my opinion, the biggest reason why students think Algebra 2 is "harder" than Algebra 1, is because of the jump from living in Linear World, to being thrust into the Land of the Nonlinear.
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Anyways, back to my post. So I did my recent usual process of transcribing my rambling thoughts via Google Docs, and pasted them into GPT-4 asking it to create a blog post, and this is what it came up with:
Title: The Nonlinear Path of Teaching: A Journey into the Complexities of Algebra 2
Being an Algebra 2 teacher is an exciting challenge. It is my eighth year diving into this subject, and each academic year brings its own set of learnings, surprises, and revisions. Teaching this subject has helped me evolve and learn how to navigate through the often complex and intricate world of mathematics.
In the past, students traditionally went from Algebra 1 to Geometry and finally to Algebra 2. This sequence offered a logical progression of concepts, with each year's lessons building upon the last. However, a few years ago, the school district made a change – Algebra 2 was moved ahead of Geometry. This shift had its advantages and its fair share of hurdles.
One of the reasons I favor this change is because the beginning of Algebra 2 mirrors much of what is taught in Algebra 1, serving as a comprehensive review. We revisit concepts like change, slope, and linear equations. We explore the y equals x parent function, different forms of linear equations, and delve into systems of equations. It's a revisit of the fundamentals before we progress to more complex subjects.
As we dive deeper into systems of equations, we talk about real-life applications, like business. For instance, the idea of 'revenue minus cost' helps students understand concepts like the break-even point. I think it would be beneficial to encourage students to think about setting up a business, considering their desired monthly earnings, and working out the equations accordingly. This kind of project-based learning could be an interactive and engaging way to comprehend systems of equations.
However, there's a flip side to this rearrangement. Algebra 2 is perceived to be more challenging than Geometry. So, when students transition from the relatively easy Algebra 1 to the more complex Algebra 2, and then revert to a somewhat easier subject like Geometry, it changes the learning dynamic.
The shift also brought a change in the student demographic. Previously, I mainly taught sophomores, but with the changes, I see more freshmen in Algebra 2. It may not seem significant, but the difference in maturity between 14 and 15-year-olds can be substantial.
When students enter high school, they often carry the energy of middle schoolers – goofy, unsure, and trying to fit in. However, over time, they begin to emulate their senior peers, evolving in the way they dress, speak, interact with friends, and even in their attitude towards learning. It's important to lean on examples of sophomore maturity over freshman immaturity in class, encouraging positive behavior whenever possible.
In the aftermath of the global pandemic, I have noticed that students' preparation in Algebra 1 and pre-Algebra has been somewhat compromised. This change has led to a reduction in the rigor of my classes, making it difficult to push the curriculum as much as before. The challenge, then, is to gradually increase the rigor, so that the students who may be less prepared can catch up.
High school becomes a serious affair as students transition to junior year. The realization dawns that grades matter, that test scores matter, and suddenly, their attitude towards academics undergoes a transformation. The focus sharpens, and their dedication to homework and test preparation intensifies. Though it is rare to see Juniors in Algebra 2, their serious demeanor and focused attitude permeate the classroom.
Now, let's delve into the crux of Algebra 2. The subject becomes genuinely challenging when we transition from linear subjects to non-linear ones. This is a monumental shift, and one of the reasons why Algebra 2 seems so much harder than Algebra 1. It's difficult for students to make this transition, from linear thinking to nonlinear thinking, because our minds are programmed to project linearly, which is simpler and more intuitive.
We start with quadratics, where we multiply factors to create a quadratic equation. The quadratic formula may initially seem daunting, but with a systematic approach – starting with the discriminant – it becomes less intimidating and easier to grasp.
As we progress to more complex equations like cubics, rational functions, and composite functions, we introduce synthetic division, a more efficient alternative to long division. We also navigate the complicated process of simplifying radicals, a task that students often find hard, given its open-ended nature.
As we move deeper into Algebra 2, we explore exponential equations and introduce logarithms, the inverse of an exponential equation and the key to solving exponential equations. Logarithms are typically a difficult concept for students, but understanding them as the only way to solve for the exponent helps demystify them.
My favorite part of the Algebra 2 curriculum is teaching compound interest, a concept that finds practical application in everyday life. Exponential growth, which underpins compound interest, is a critical concept we encounter more often than linear or quadratic equations. It is a manifestation of the nonlinear world we inhabit.
The challenge with Algebra 2, as I mentioned before, is the shift from linear to nonlinear thinking. As we dive deeper into nonlinear subjects like quadratics, higher-order problems, radicals, and exponentials, it forces us to expand our thinking.
In conclusion, teaching Algebra 2 is both an exciting and complex
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That's decent, but I just wasn't feeling it for some reason. So I started to make some changes:
Revise the last answer. I didn't like the style. Create an essay in the style of Paul Graham (from Y combinator) and make sure the entire essay implicitly revolves around the central idea that Algebra 2 is harder than Algebra 1 because the students have to jump from linear thinking to nonlinear thinking. And humans aren't good at thinking nonlinearly. And make the essay 3000 words
then
better, but revise that to mention more nonlinear content, and remove the parts about their ages and them switching from algebra 1 to geometry to algebra 2 to now algebra 1 to algebra 2 to geometry. really drive home the focus of the essay: the difficulty of nonlinear thinking
then
better, but revise to remove the part about linear and nonlinear systems of equations, and include more discussion about the nonlinearity of the real world and why this makes Algebra 2 so important, how it sets the foundation for how they'll do in all higher math subjects and understanding compound interest could save you a lot of money in your life. so while Algebra 2 is harder because it's nonlinear, it's so much more important to truly understand and grasp the topics
and it turned into a math-word salad, conveying meaning while lacking any somehow. But it was presentable, and I thought about posting it here, as well as potentially having students read it during the first week of school and critique/discuss etc.
So I had this great thought: "You know what would be cool, if it wasn't so preachy and was more in the form of an older mentor giving you advice. Then students would care more" so I tried one last revision...
better, but revise to make it in the tone of a convincing older brother talking to his high school freshman or sophomore little sibling and trying to offer advice on why they should care about Algebra 2. but make sure to not sound too preachy, we want to be in the tone of an older mentor that cares about them, not a parent that's scolding them
This revision came out so bad and cringey that I knew I had to post it. I doubt that I'll show it to the students though:
Hey, kiddo,
I know math isn't necessarily the most exciting thing in your life right now - trust me, I've been there too. But I wanted to talk to you about something, specifically Algebra 2. I know, I know, sounds dull, right? But stick with me for a moment.
See, Algebra 2 is kind of like that challenging video game level you've been trying to beat for ages. It's a tough nut to crack, way more difficult than Algebra 1. And, yeah, I understand the frustration, especially when you're shifting from the neat world of Algebra 1 - all straightforward and linear - to the more chaotic realm of Algebra 2, where things aren't quite as straightforward. It’s kind of like learning to ride a bike and then suddenly being handed a unicycle. It's a big leap, no question about it.
In Algebra 2, the equations stop behaving like obedient soldiers marching in straight lines. Instead, they start twisting and turning, mimicking the unpredictability of a roller coaster ride. Parabolas, growth and decay models, logarithmic scales, and those peculiar things called asymptotes - they’re all part of this new, seemingly daunting world. I remember when I first faced these concepts, and I won't lie, it was tough. But, it’s essential, not just for your math education, but for how you view the world.
You know how life isn’t always predictable, right? Well, that's pretty much how things work outside of the classroom too. You see, the world isn't linear. It's complex and interconnected, much like the concepts in Algebra 2. Whether it’s the spread of a viral video, the growth of your favorite band's popularity, or even the twists and turns of the stock market, it's all nonlinear.
One example that might catch your interest is compound interest - you know, that financial stuff. Imagine you've saved up some money from your summer job. If you understand compound interest, you'd know that it doesn't just increase in a straightforward line. Instead, it grows exponentially, multiplying faster and faster the more you have. And guess what? That's a concept straight from Algebra 2. Understanding this could mean the difference between having just a bit extra cash in the future and a small fortune.
I get it, it's a lot to process. But Algebra 2 is like a key that opens doors to all sorts of cool stuff. It's the foundation for higher math subjects, like calculus and statistics, that are crucial in fields from engineering and physics to game design and computer science. So, if you've ever dreamt about designing your own video game or building a revolutionary app, Algebra 2 is your first step.
I know it seems challenging. It's a massive step up from what you're used to. But trust me, that's what makes it valuable. It teaches you to think in different ways, to embrace complexity, and to tackle ambiguity. It trains your brain to handle the real world, and honestly, that's a superpower.
Look, I'm not saying this to overwhelm you. I just want you to understand why Algebra 2 matters. It’s not some abstract subject designed to torture students, but a tool that helps you understand the world better, handle your finances smartly, and opens up a world of opportunities. So, when you find it challenging, remember it's not just about passing an exam. It's about growing, learning, and preparing for the road ahead.
Remember, every challenge you face is an opportunity to learn and grow. Algebra 2 may seem like a steep mountain now, but once you've conquered it, you'll be able to handle anything life throws at you. Trust me on this one.
You've got this, kid.