This post is available as a podcast here: https://podcasters.spotify.com/pod/show/s-p01/episodes/Math-e2b6mks

You can do it. Yes, you can. This post is a summary of conversations that I have with students, strangers, and friends about math. See, there are really only two responses when I tell someone that I teach math. 1) “Oh, I loved math, there’s always a right answer.” or 2) “I liked math, but I also hated it because I was bad at it.” Naturally, there are variations on these themes – but I think the assumptions of the speaker are built on a broken foundation. Math is not about rote calculations, but that’s the way it’s been taught through much of our recent history. Math, like learning anything, is about resilience (growth mindset is key!)

Let’s address the first response first: the “right answer.” Perhaps, on some level, they’re correct. There is a “right answer,” but the way that I teach math is about the reasoning not the answer. The “old” ways of teaching math placed heavy emphasis on arriving at the correct answer. In fact, some teachers would only grade the answers, without ever checking the work. We have moved beyond those days, thank goodness. There is more emphasis on the work, but standardized testing is still designed around the answer.

There is movement toward emphasis on the work rather than the answer, and this is a movement I’m solidly within. The work shows the reasoning, and I think that previous teaching and grading techniques lost many students whose reasoning was good. Those people in response 2 were often lost in the mathematics classes emphasizing computation and answers. The conversations usually go something like this:
Person: “I liked math, but I was bad at it.”
Me: “Why do you say you were bad at it?”
P: “I couldn’t get the steps; it never made sense.”
Me: *Queue long speech about how they were probably capable of the mathematics, but it used to be taught in a way that lost a lot of people…

Why do people struggle with the steps? The way in which the math is taught. I usually assure them that math isn’t about the steps, it’s about the reasoning. The way that math was taught emphasized the steps because the operations provide a method to follow, but the reason to learn mathematics is for the reasoning. Reasoning is found in the thinking, finding logical methods and connections to solve the problem. This is why active learning has become a popular teaching technique – it engages the students in reasoning to learn the mathematics.

What is incredible to me is that I find the students who struggle with steps excel at reasoning. Why? The reason they struggle with steps is because they need the reasoning to make sense out of the steps before they stick. If they do not understand the reasoning behind the steps, they struggle to remember which steps to use and when. It’s all about connections and reasoning. The two groups are not mutually exclusive, but I do find a significant number of students who excel at calculations tend to struggle with the reasoning – because they’ve learned that math is all about the steps, and they don’t dig any deeper.

When it comes to the skill that is going to make someone excel – it’s always the reasoning. The calculations come from the reasoning, so if a student figures out their way to reason through a problem – the calculations are easier to apply. Employers and graduate schools want students who can reason because the reasoning skills give the students an ownership and empowerment to solve a new problem, not just rote problems from their classes.

Algebra is a classic example of this issue. There are jobs that require the applicant to solve some basic algebra problems from an algebra course, but the employers find that the completion of basic algebra problems doesn’t translate to the application of algebra in the job. An applicant can ace the test, but then struggle to solve a similar problem on the job that isn’t pre-defined as an algebra problem. The real-world application of that knowledge requires reasoning. This is why word problems cause so many issues. First, they are an attempt at modeling reality, but rarely fit it. Second, students struggle with word problems, yet they are not tested (nor taught) to the same level as basic computations.

Thus the reason for the mathematical modeling component of mathematics teaching. Mathematical modeling is the study of real-world application problems utilizing the associated mathematics. It’s a bridge between the computations and the application in the real world. When students solve realistic, open-ended problems it aligns with the application of mathematics in jobs. This prepares them to be successful when they go into the world and have to solve new problems for their employer. Isn’t that what we should be preparing them for? How math actually applies in the world. I think so.

My story this week is about my greatest struggle in math.

My favorite math analogy came from one of my high school teachers. When I was taking Calculus my senior year of high school, my teacher talked about “walls” in math. He said that we all hit walls in math, it’s just a matter of where. We hit walls, and it takes extra work to climb over them. He said, maybe someday you’ll hit a wall so high you don’t make it over, and that’s okay. He hit his big wall sometime in college, but I’ve forgotten his story – here’s mine.

Real analysis was a wall, but I made it over that wall. Complex Variables was my biggest wall in math, and considering the fact that I recently taught that class and loved every bit of the content – I eventually made it over that wall. When I took Complex Variables in undergrad, it never fully made sense. I passed the class, but I felt like I was drawn along and never grasped it. This is a feeling that I am guessing is like those that struggle earlier in math – it’s the experience of a “wall.” The class may pass by, but the knowledge is missing.

I needed to pass an exam in Complex Variables (with an A) for my PhD, so that meant I had to learn it. I sat in on a couple days of the undergraduate course at my graduate school, but I really worked through the book on my own. I think Laurent series and residues were the last key pieces that were not taught in my undergraduate course. When I looked back at my textbook from undergrad, that’s where the bookmark stopped. So, those were the classes I attended. After teaching Calculus, Complex Variables all came together. See, Complex Variables is basically like doing calculus with complex numbers. Sure, it’s more complicated, but the foundation is calculus.

I think all our walls in math stem from an underlying fraction in the foundation of our knowledge. If arithmetic doesn’t make sense, algebra is going to be a wall. If geometry doesn’t make sense, trigonometry is going to be a wall. If algebra doesn’t make sense, then calculus is going to be a wall, and so on. My recommendation, if you find yourself struggling mathematically, is to look back at your foundation – without a solid foundation, any structure will not be sound. When I spent all that time studying for my graduate exam in calculus, and taught calculus, I solidified my foundation. Then, when I tried to teach myself complex variables – it made sense, and it stuck. I rocked that exam, and I think it’s entirely because I secured my foundation before building on it.

I still love using that metaphor: learning new things in math is a lot like one of those obstacle courses with the walls that you have to climb over. Some walls, in the beginning, you might get over quickly. As you continue, you get stronger, but the walls also get higher. The balance is ensuring that you strengthen the foundational knowledge as much as possible so that your strength and endurance get you over the next wall. Otherwise, you might find yourself stuck, or falling. You can still put in the work to get over the wall, it’s up to you if it’s worth that work. It’s about resilience, not gifts (growth mindset.) You can do it!

One of my students once asked me what drugs I was on to enjoy calculus so much; I told them “my drug of choice is math.” Learning is a high of its own, if you love learning, any topic can give you joy.

I believe in my students, and I believe in you. Love, -S.