I am fond of the student who asks, “Why are we learning this?” This question represents to me agency in their learning, as well the opportunity to convince them that what I am asking them to understand will be worth the effort.
There are, of course, straightforward answers to “Why are we learning this?” Sometimes students are learning something because they will use it in another course, in their careers, or in pursuing a particular goal. Sometimes it is simply a requirement for something else they want to accomplish. These are good reasons, and when they apply, I am happy to give them. In many classrooms, they are also the easiest answers to provide: if a student needs calculus to become an engineer, I can explain why calculus belongs in their engineering education.
But these answers do not address every version of the question. Why should the student jump through the particular hoops that the instructor has laid in front of them? Why should students learn math that they might not use in the future? Why should a student care about math at all? And, eventually, at the very top: why learn?
I don't think there is a single answer to these questions. Every student, ultimately, has to find their own reasons to learn. Here are some of the reasons I have found for learning mathematics.
Why learn?
During the summer before fifth grade, I was assigned the task of “representing one million”. I started with the goal of ripping up little pieces of stickers and sticking one million of them onto printer paper. I got to about 3000 stickers before I realized that this was not going to happen.
What was I supposed to learn from this experience? I already knew what one million was. I knew how to say it, write it, do arithmetic with it. But clearly I did not understand what one million meant. If I had, I would not have begun ripping up stickers.
The point of this assignment, I eventually realized, was not actually representing one million. It was to understand its magnitude. One million is a number that we encounter constantly: in textbooks, on television, in statistics, and in everyday expressions like “I've told you a million times.” As a fifth grader, I knew that one million was big. I did not yet know how big.
This assignment continues to impact the way I think about the world; how I understand money, mass casualties, history, science, how I go about setting goals for myself, how I think about knowledge, and how I think about failure. I remember walking into fifth grade on the first day of school, and being so embarrassed of my 3000 tiny pieces of stickers, and yet now I think of this as one of the best moments of learning in my life.
I think often about that experience because it captures what I value about learning: learning more allows you to see the world around you differently. We can know a fact without appreciating its implications, know that something works without understanding why, or be familiar with an idea without having considered it from another perspective. Learning gives us the opportunity to deepen and complicate our understanding of things we already encounter every day.
Why learn math?
Mathematics gives us useful tools. Many of my students see the goal of mathematics education as providing them with the technical mathematical skills they will use in their everyday lives and future careers. This is a legitimate answer to “why math?”. All chefs need proportional reasoning, all carpenters need geometry, all plasma physicists need general relativity, and all citizens need a baseline understanding of probability, statistics, and financial math.
More generally, mathematics teaches us how to arrive at understanding: how to take something that seems obvious, mysterious, or arbitrary and figure out why it has to be that way. It teaches us not only how to find a path to understanding, but how to explain it so that someone else can traverse the same path. A mathematical argument is, at its best, an invitation for another person to take the same journey from “I am unsure that if A, then B” to “I understand why it must be true that if A, then B.” Sometimes my friends and I joke that mathematicians are just kids who never stopped asking, “But why is that true?”.
I did not always understand mathematics this way. My answer to “Why math?” was, for many years, “Because I’m good at it”.
As I got older, it felt like my instructors and textbooks began pantomiming walking the path from A to B. They would take one step, then another, then another, and eventually arrive at B. I could imitate their steps, but I did not see how those steps formed a path.
Ultimately, I would be asked to use the formula at the end of the pantomime, and so I began to just tune the pantomime out. It was how I imagine babies listen to adults: “Bla bla bla food, bla bla bla no”. I stopped listening and used formulas to get praised.
But I began to feel uneasy: like at any moment my teacher could say, “Just kidding!! That is not how any of this works.” I knew how to get the right answers, but I did not understand what I was doing. I could follow the rules, but I didn't understand why the rules were the rules. Eventually, I wanted to understand mathematics in my bones. I wanted to know not just that there was a path from “if A” to “then B”, but why this path must exist and how I could convince others of this inevitability.
It was not until I took a proofs course that I began to feel as though I understood. For the first time, I saw mathematics as something other than a collection of formulas and procedures. Proofs were a language for asking and answering THE question: But why is that true?
That feeling of knowing why something is true, of being able to stand on solid logical ground, is the reason I love mathematics. It took me years to develop that feeling, which I don't think is unusual. Learning to recognize what counts as a satisfying explanation, to distinguish a convincing argument from a merely plausible one, and to become comfortable asking “But why?” is itself something we learn.
I don't think everyone stops their “but why” journey at the same place (for instance, I'm pretty chill just accepting the axiom of choice). Different students will want, or be ready for, different kinds and levels of mathematical justification. Part of my job as an instructor is to help students to develop a sense of when they understand why something works to their own satisfaction, and when they need to interrogate their own understanding further.
Why learn this math?
As a student, I was unsurprisingly interested in mathematics for its own sake. A teacher never had to convince me that mathematics was worth my time. But I often found myself confused about why the topics in a course were arranged as they were, or why a particular topic was important. I remember feeling, in precalculus, that matrices seemed an unimportant detour. Imagine my surprise in linear algebra two years later.
A similar doubt was expressed from the other side of the classroom in my Spring 2026 Cornell Prison Education Program (CPEP) class. My CPEP students never needed much convincing that learning is valuable. They cared deeply about education: about expressing themselves, understanding the world, telling their stories, and ultimately getting out and building the lives they want to live.
When we got to the statistics unit, they were easily convinced that it was important to understand how statistics can be used to support a particular argument, and how different measures such as mean, median, mode, and range can tell different stories about a data set. One of my students even brought up my favorite example of how different a story the median and mode can tell, using average household incomes.
Then we moved on to standard deviation. Because the students had naturally understood the importance of all the other numerical descriptions of a data set, I thought I could jump directly into the weeds of standard deviation. This was a bad idea. I wrote that disgustingly unintuitive formula on the board, and I lost my class. Many students were rolling their eyes at each other, and Mr. H said, “This is stupid. We are never going to use this.”
I thanked Mr. H and did what I should have done at the start: I talked about why the spread of a data set might matter. I explained that, at the end of the day, I wanted them to be able to critically evaluate arguments that use standard deviation to support a claim. I did not expect them to compute standard deviation by hand more than a couple of times in their lives, all of which would probably be in that class. When I returned to the formula, I felt Mr. H was with me. The formula that had been so pointless a few minutes earlier suddenly demanded understanding.
When we teach mathematics, we are often too familiar with the connections between concepts. We know what skills students will need later, how different ideas fit together, and why a particular topic is worth the time it takes to learn. Students do not necessarily have access to that larger picture. A topic that feels to an instructor like an essential piece of a larger whole can feel to a student like an arbitrary detour. Part of my job as an instructor, then, is to make the why visible: to help students see why I have chosen to spend our limited time learning this particular thing.
Importantly, “Why?” does not always mean “Why will I need this in real life?” Sometimes the reason for learning a mathematical idea is more abstract. For example, in that same CPEP class, we spent a week working with number representations in bases other than ten. I did not expect my students to leave the class with a pressing need to do arithmetic in base seven, or even understand when other people use base seven. The value of the exercise was elsewhere: they had to take ideas about arithmetic that felt completely familiar and apply them in a context where some of their usual intuitions no longer worked. That experience of having to reconsider something familiar in an unfamiliar setting is a valuable learning opportunity to develop fluency, regardless of whether the new setting will come up again or not.
Why learn this math this way?
Instructors choose how a course is structured, what kinds of activities students will do, how they will work with one another, how often they will receive feedback, and how their learning will be assessed. And, just as students may not understand why they are learning particular material, they may not understand why they are being asked to learn it in a particular way.
I am currently teaching a mastery-based differential equations course. On the first day of class, I spent a lot of time explaining why the teaching team believes this structure promotes student learning. In previous years, the course has gone very badly because students lost confidence in the mastery-based system and came to believe that the teaching team had chosen it because it would be easier for the instructors or because it was a newfangled thing. This is a striking example of how failing to answer “Why learn this math this way?” can have negative consequences for student learning.
Of course, explaining the reasoning behind a course policy does not mean that the reason will be satisfying to students. One question a student had about the course structure was why their optional, ungraded homework would not receive any feedback. This meant that students would not receive feedback on their work until the first assessment. The answer was that the teaching team did not have the person-hours to provide that feedback. This was a frustrating answer, but it was also an honest one. Sometimes the answer to “Why are we learning this way?” is simply that we are working within constraints. In my experience, students respect that honesty, even when they are not happy with the constraint itself.
Sometimes, even after hearing our justifications, students have good reason to believe that a particular course structure will not promote their learning. My favorite example of this is group work, because as an instructor I use group work a lot, but as a student I always dreaded group work. I often processed thoughts more slowly than my peers, and so working in a group felt like turning math into a spectator sport. I learned to disengage during group work because I had already decided that there was no way I would be able to contribute. It felt like group work was getting in the way of learning mathematics, and whenever possible I would work on my own, even if everyone else was working in groups.
I sometimes still struggle with having mathematical conversations in real time when I haven't had a chance to prepare for them. Every year that I teach, I get a little better at it, but I am not sure I will ever become someone who can instantly produce a polished mathematical response to every unexpected question. But what I have realized is that this is okay. I was not wrong when I felt that group work disrupted my ability to have that “Eureka!” moment, but I now realize that the point of group work, and of mathematics education more broadly, is not to be the person who solves the problem first.
If I want students to develop trust in their own mathematical reasoning, they can’t do mathematics in a vacuum. If I want them to learn to communicate mathematics, they need opportunities to communicate. If I want them to recognize that mathematical ideas can be challenged, revised, and strengthened, they need opportunities to encounter half-baked ideas and challenge them. Some of this can happen through long-form writing, critique, and revision. But learning to have a mathematical conversation, to explain an idea in real time, respond to someone else's reasoning, and change your own thinking when the conversation demands it, are vital to learning mathematics.
Note that recognizing, and communicating, the value of a particular way of learning does not mean that every student will find that way equally accessible. Even though I have a good reason for asking students to work in groups, that does not absolve me of responsibility for the students for whom group work is particularly difficult. In fact, it makes that responsibility more important. If I believe there is something valuable for students to gain from a mathematical conversation, I work to help students find a way to access that value.
I had a student in my Spring 2026 CPEP class who consistently turned in A work, but spent much of class either in wide-eyed panic or entirely focused on his own paper. I would watch him hunched over his paper, trying to tune out everything happening around him, and I saw myself. When I talked to him about it, he explained that he has ADHD and is very sensitive to noise: thinking while other people talk was extraordinarily difficult.
We talked about how he could still get something out of conversations without having to process everything in real time. During group work, I suggested that he try simply taking notes on the conversation. He didn't need to follow every point as it happened or immediately contribute his own ideas. He could listen, write down what seemed important, and then go back and process it after class. What mattered was not whether he participated in group work in the same way as his classmates. What mattered was that he had a way to engage with the critical listening and mathematical thinking that the group work facilitates [Click here for more on defining success individually].
This student's experience with group work, as well as my own, is emblematic of how ‘Why this way?’ has two parts. First, how can an instructor realistically help students generally improve their mathematical literacy and understanding of the course material? And, secondly, how can an instructor adjust that plan to better help students who need something different? Importantly, as my experiences with group work have taught me, supporting a student who struggles with a particular way of learning does not mean abandoning that approach altogether. It means identifying what I want the student to gain from the experience and finding a way to make that learning accessible to them.
There is a temptation, when a student asks “Why are we learning this?”, to treat it as a question that needs to be answered quickly so that we can get back to the learning. But the more time I spend trying to answer this question for myself and with my students, the more I have grown to realize that seriously engaging with the question is a vital part of learning. The question asks us to examine what we think is worth knowing, why we think it is worth knowing given our limited time and energy, and what it means to learn it. These are questions that students and instructors can, and should, work through together as a natural and important part of the learning process.