Let’s get one thing out of the way right from the start. I'm a big fan of math, and pretty much always have been. I find it interesting, useful, beautiful, and challenging, and if you spend any amount of time with me, I will probably dedicate a portion of our conversation to convincing you that this is true. And yet: mathematics as a discipline is not always a one-to-one match with the pieces of math content required by our state standards. So yes, there are topics that I don’t really like teaching or that I think should be removed from the curriculum (I hope I’m not alone in this!). But let's consider why this happens.
When I find myself with a strong dislike toward a particular topic, it’s usually for one of four reasons.
The content doesn’t really make sense to me (quotient remainder theorem and modular arithmetic in college – the only time I seriously reconsidered my mathematics major!)
The content feels super abstract or needlessly complicated (rational functions and polynomial division)
The content feels like a one-off skill that’s disconnected from previous things we’ve learned or to the big ideas of mathematics (simplifying radicals, roots and rational exponents)
The content feels arbitrary (quotient rule, definitions)
The reality of teaching and writing lessons is that we can’t just ignore the topics we find monotonous or dull. Instead, we have to dig in deeper and figure out what we’ve been missing about these topics. And let me tell you what I’ve learned from doing that: it’s me, hi, I’m the problem, it's me.
When I come across topics I don’t really like, it’s easy for me to say that the math itself is tedious or dry or overly complicated. But what is most often the case is that the approach to teaching that topic is tedious or dry or overly complicated. Perhaps we only focus on one aspect of the topic (e.g. the Algebra 1 standard related to the topic, and not what came before it or will come after it) and thus it seems random or out-of-place. Or we assume this is one of those topics that we just have to “cover” but that we can’t expect students to actually understand. Perhaps we never really understood it ourselves, or we’re missing some of the ways it connects to other topics. We might have a flat, one-dimensional view of the topic because we’re so focused on the skill we need students to master, or the SAT question we’re trying to prepare students for.
When I sense myself getting resentful about having to teach a certain topic, that's a signal to me that I probably don’t understand the content as well as I should and that it’s time to do a deep dive into the topic.
At no time has this been more true than when I had to make a lesson on roots, radicals, and rational exponents.
Roots and Rational Exponents: The Lesson I Did Not Want to Write
I’ll be honest. I have never liked this topic.
Who cares that √32 can be “simplified”? And Is 4√2 really that much simpler? For whom? Certainly the technology we have today doesn’t care!
And students are also expected to know that
Neat.
This standard hit many of my “reasons why I don’t like a topic” hot buttons. An abstract skill only made worse by the kinds of complicated expressions our old textbook suggested students should be able to simplify for “practice”. And like most definitions, the choice about how to define an nth root seemed…arbitrary?
I made a compelling argument for why this topic shouldn’t be included in the Math Medic curriculum (luckily, my team talked me down). But this topic showed up in math standards across the board, so we couldn’t just ignore it out of personal preference (or conviction). What was the solution? I had to dig in deeper. I knew I had to help students see the logic behind the decision for defining nth roots in the way they are defined. And I knew that I had to show that simplifying radicals was really just another piece of the big (and crucially important!) mathematical idea of equivalence. So what was the turning point in my journey writing this lesson I did not want to write? Logic puzzles.
Specifically, the logic puzzles that I adore, sometimes called “Einstein puzzles” where you are given a series of categories, and an equal number of options within each category, and your goal is to figure out which options are linked together based on a series of given clues. They are solved using simple logical processes, which is exactly the approach I took to the lesson.
This, by the way, is the antidote to the “arbitrary” argument. A definition or rule may feel arbitrary, but only until you consider that any other definition would not work. Something may feel like a random choice, but it’s a meaningful choice in that it is the only candidate left standing after considering all the other options.
Next we had to work on rewriting expressions like √32. Traditionally, we have only ever wanted students to know that this simplifies to 4√2. But thinking with the equivalence-as-a-big-idea hat, isn’t it more important that students understand that something can be rewritten in ways that look different, but are mathematically equivalent? The focus then becomes less whether 4√2 is “simpler” than √32, but whether we can prove that they are in fact equal. And when we speak of proof, we have to think of a logical sequence of statements, and nothing feels less logical than saying √32 magically becomes 4√2. What are the principles at play here? What is legal math we can do to rewrite √32? How else could we depict this same quantity?
My next question became how we could get students to think through these alternate ways of writing √32 without heavily guiding them through each step. They probably wouldn’t naturally start thinking about all the factors of √32 or translate back and forth between roots and rational exponents. But they should be able to identify whether one expression represents a valid (legal) rewrite of another.
Enter: Birds of a Feather.
(This lesson is part of our Algebra 1 course, in our unit on working with nonlinear functions. Check it out here. In a previous lesson, students learned about simplifying expressions with integer exponents.)
I began brainstorming ways students could logically work out equivalence. What if each student got an expression that looked different and had to find other students in the room that belonged to the same “flock” (i.e. were mathematically equivalent)? I didn’t say how many flocks there would be (it ended up being only 3) and how many “birds” would be in one flock (up to 12!). Students would be up and out of their seats and they would be using some legitimate logic skills to come up with the main idea of the lesson on their own.
Why This Lesson Works
The debrief of this lesson is also crucial, because it helps students name and notice the
mathematical ideas at play and the interconnection between these ideas. A big aspect of the lesson is asking students to justify why they belong to a particular flock (i.e. why they are equivalent to the other expressions in the set) and the debrief formalizes some of these justifications. Additionally it helps students see the nuance between statements that are “true” and those that are “helpful”. We purposefully included other equivalent expressions that were not strictly on the path to “simplifying” the radical. For example, rewriting √32 as √(8*4) is not strictly necessary in order to simplify it to 4√2. But if the big idea is equivalence, and not just following a procedure to write an expression in the teacher’s preferred way, we had to posit the idea that there are (infinitely) many ways to rewrite an expression. Which of those rewrites are useful is a whole other story, and one worth having our students think about! Check out the full lesson below.
So, I officially recant my statement of disliking teaching roots and rational exponents. It used to feel like a “one-off” and unnecessary in the grand scheme of things, but that’s because I hadn’t appreciated the logic behind the topic, and I hadn’t situated the lesson in the broader theme of equivalence.
This brings me to our lesson design principles that we draw upon whenever we teach or create a lesson on a topic we don’t love (or any topic, really):
Make it simple.
Make it logical.
Make it obvious.
Make it new.
Make it connect.
You probably see at least a couple of these at play in the “Birds of a Feather” lesson. The idea is to make what is inherently interesting about the topic more accessible to students, thus allowing them to experience how intrinsically rewarding it is to figure something out for themselves, and cultivating those a-ha moments. We’ll dig into these principles and how they apply to other content more in future blog posts!
So which topics are your least favorite? Now could be their redemption era!