Grades measure a student's demonstrated achievement of the measurable learning objectives of a course. Educational success, however, encompasses the broader and often less easily measurable growth a student makes toward a broad range of goals. These goals include the general goals of mathematical education, as well as the individual goals that students bring with them into a course.
What constitutes meaningful growth will necessarily differ from one student to another. Students enter our classrooms from vastly different starting points. They have different prior experiences with mathematics, different strengths and weaknesses, different reasons for taking a course, and different circumstances in their lives. As a result, the growth that is most meaningful for one student may look very different from the growth that is most meaningful for another.
For one student, meaningful growth might mean developing enough mathematical confidence to participate in class and attempt problems independently. For another, it might mean developing the mathematical skills necessary to pursue a particular career or further education. For another, it might mean learning to communicate mathematical ideas clearly or becoming more comfortable working through problems with other people.
Importantly, individualizing educational success does not mean lowering standards or allowing every student to define achievement entirely on their own terms. Rather, it means recognizing that students can make meaningful progress toward different educational goals from very different starting points.
If meaningful educational success depends on where a student begins and what they hope to accomplish, what role can a course grade play in measuring and encouraging that success?
Grades provide a concrete measure of demonstrated achievement and can communicate whether a student has met the measurable learning objectives of a course. For many students, working toward a high grade is both challenging and achievable, and along the way they make meaningful growth in their mathematical literacy. In this way, grades act as a proxy for achieving the goals of the course.
But a course has goals beyond those that can be measured by a grade. How do you measure the growth of mathematical confidence? How do you measure perseverance through hardship? How do you measure the ability to solve problems in a group? How do you take into account the vastly different starting positions of students as well as their vastly different current life circumstances? How do we take all of these factors into account without introducing bias?
Grades, therefore, necessarily capture only some of what education is trying to accomplish. They cannot definitively tell us whether a student has meaningfully grown in their mathematical literacy or made progress toward their educational goals.
This distinction matters, in large part, because what we measure inevitably influences what we value. Because grades are the most visible measure of success, students can understandably come to believe that getting a “high grade” is educational success. For some students, working toward a high grade supports meaningful learning throughout the semester. But for others, a focus on the grade can obstruct growth.
For this reason, part of an instructor's role is to help students understand that a grade is one measure of their educational experience, not the determination of their success.
There are many ways of helping students see and pursue their own version of meaningful growth. Instructors can make clear how what students are learning connects to meaningful growth [click here for more on “why are we learning this?”]. Instructors can construct assessments that ask students to demonstrate a more holistic understanding of mathematics [click here for more on effective problems]. An extremely effective strategy for helping students clarify their educational goals is also one of the simplest: talk early and often about what success means.
I ask every student I talk to about their goals for taking the course. I ask what they see as their weaknesses as a student. I talk through what I think are reasonable goals for the semester and how to move toward those goals. Some students engage seriously with these conversations and others just want to talk about the material.
If a student’s goal is to get through the class, I respect that and tell them that my goal is also to get them through the class, along with helping them feel more capable in mathematics than they did going in. I stress that these are very compatible goals.
Clearly, in a 500-person class, we will not be able to have deep one-on-one conversations with every student. We can, however, still talk explicitly about how success in a class is not simply getting a “high” grade. It is about learning and growing. Every time I talk about grades with the full class, I remind students that I do not consider them THE indicator of success.
Three stories of educational success
There are many ways of helping students see and pursue their own version of meaningful growth. Instructors can make clear how what students are learning connects to meaningful growth [click here for more on “why are we learning this?”]. Instructors can construct assessments that ask students to demonstrate a more holistic understanding of mathematics [click here for more on effective problems]. An extremely effective strategy for helping students clarify their educational goals is also one of the simplest: talk early and often about what success means.
I ask every student I talk to about their goals for taking the course. I ask what they see as their weaknesses as a student. I talk through what I think are reasonable goals for the semester and how to move toward those goals. Some students engage seriously with these conversations and others just want to talk about the material.
If a student’s goal is to get through the class, I respect that and tell them that my goal is also to get them through the class, along with helping them feel more capable in mathematics than they did going in. I stress that these are very compatible goals.
Clearly, in a 500-person class, we will not be able to have deep one-on-one conversations with every student. We can, however, still talk explicitly about how success in a class is not simply getting a “high” grade. It is about learning and growing. Every time I talk about grades with the full class, I remind students that I do not consider them THE indicator of success.
In the spring of 2026, I was adjunct faculty for a course at Cayuga Community College through the Cornell Prison Education Program (CPEP). We covered topics ranging from symbolic logic and set theory to probability and statistics. My students were incarcerated men between 30 and 70, all pursuing associate degrees and many hoping to continue their education after prison.
One student was particularly passionate about psychology and hoped to eventually earn a master's degree. He had learned to read only a year earlier and was still developing basic mathematical and reading intuition; for example, he did not yet have an intuitive sense that “one million” would begin with a “1.” The course was not designed with a student like him in mind. He struggled to read the textbook and homework problems, although when I read the problems aloud, we could often work through the mathematics together. He had no one to study with, and the constraints of education in prison meant that we could not provide any accommodations during exams. He earned an 8% on the first midterm.
But over the course of the semester, he continued to work hard. He made a friend to do homework with and began turning in every assignment, with each homework grade a little higher than the last. He spent time most days reading the textbook. He began speaking up in class, both asking and answering questions. His ability to pass the course ultimately came down to the final exam. When I asked him how he felt about it, he said, “I'm just going to work hard, and then we will see what happens.”
He passed with a D-. By almost any conventional measure, his performance in the course was poor. But he left the class a stronger reader, a more confident mathematics student, and better prepared to continue his degree. Most importantly, passing meant that he could continue funding his education with Pell grants while in prison, giving him more time and opportunity to keep learning.
When I taught a high school mathematics seminar, most of my students were seniors who had run out of mathematics courses at school and wanted another math credit. One student was a freshman who had already exhausted the high school mathematics curriculum and spent his free time reading Wikipedia pages about research-level mathematics.
He was exceptionally bright, and he knew it. He also seemed to be carrying some of the insecurities that come with being a teenager who feels different from his peers, and this manifested in a strong impulse to prove his intelligence to others in the room . When I introduced mathematical voting theory, he announced that he already knew how to prove Arrow's theorem, the proof I had planned to work toward with the class over the semester. His confidence often hindered his learning and made it difficult for him to recognize that other people in the room might know mathematics that he did not. He often assumed that if he disagreed with another student—or with me—then the other person must be wrong.
I wanted to give him an experience I had during my first semester of college. I took a course on special relativity and realized that mathematics and physics contained ideas far more complicated than anything I had imagined and that discovering how much I did not know was exciting rather than threatening.
So I began questioning his reasoning more deliberately than I normally would when working with high school students. When he made an argument, I would highlight missing steps, even when his conclusion was correct. Other times, when he insisted that I was wrong, I would ask him to give me a formal proof. Then I would continue the intended lesson with the other students, while he worked on trying to prove me wrong.
By the end of the semester, he still thought he was the smartest person in the room, but he had begun to recognize that being smart did not imply you can’t be wrong. He listened when I spoke, and a few times, instead of saying “you're wrong,” he said, “I don't understand that.” He also began genuinely collaborating on a research project with other students. He still thought his ideas were the best, but he had begun listening to theirs.