Helping Students to Mathematical Literacy
Every student enters our classroom with a unique mathematical background and their own reasons for being there. For some, our course is their most exciting class this semester; for others, it is a general education requirement standing between them and their degree. Outside the classroom, students are balancing jobs, family responsibilities, illnesses, grief, and the countless other realities that shape how much time and energy they can devote to learning.
At the same time, they have unprecedented access to mathematical resources: worked solutions, video lectures, and increasingly sophisticated AI tools. Given the competing demands on their time and energy, many students ask, "If technology can do mathematics for me, why should I learn it?"
This is a new version of an old question: why should students invest the effort to understand mathematics when there are easier ways to get answers? If we understand the purpose of mathematics education to be the production of correct answers, then this question is difficult to answer.
However, the mathematic educators that I admire have shown me that the lasting value of mathematics education lies in developing mathematically literate people: people who can confidently solve problems, evaluate their own reasoning and the reasoning of others, and communicate mathematical ideas clearly. Under this interpretation of mathematics education, every student, regardless of their background, goals, or current circumstances, can gain lifelong benefits from engaging with mathematics. I strive to align every aspect of my teaching with this view of mathematics education, from how I discuss math, assign problems, and assess student work.
Making Purpose Explicit
Misconceptions about the goals of a course can undermine students' motivation to learn, so as I plan each lesson, I keep in mind the student who asks, "Why are we learning this?"
When working with high school students on an ungraded research project, I came to realize that they misunderstood demonstrating progress as the goal. They would tell me that they had read materials but could not answer follow-up questions about them and they created nonsensical presentation slides with AI to show me that they were working. They were approaching the project as many students have learned to approach school: their job was to produce evidence for their teacher, rather than to take ownership of their own learning. We had a frank conversation about how the project was intended to support their intellectual growth, not to impress me. Although this conversation did not suddenly make the project intrinsically interesting to them, for the rest of the semester we were able to have interesting and fruitful conversations about math.
"Why are we learning this?" is not a question students can always answer for themselves, especially early in their educational development. As instructors, we should regularly discuss why we spend time learning particular topics, justifying our reasoning, working collaboratively, revising our thinking, and anything else we do in class.
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Recognizing Students’ Capacity for Growth
Understanding the purpose of mathematics education is important, but students must also believe that they are capable of learning it. One of the greatest barriers to developing mathematical literacy is the belief that mathematical ability is an innate talent rather than a skill that can be developed. I work to challenge the idea that there are "people who are good at math" and "people who are bad at math." My goal is for students to see mathematical confidence as something people build, rather than something they possess or lack.
One way I do this is by talking about mathematics as a language: the language of formal reasoning. Like any language, mathematical fluency develops through sustained practice: reading, writing, speaking, and thinking mathematically. This metaphor has worked especially well for many of my incarcerated students, many of whom have recently become passionate poets and creative writers and thus already understand that fluency develops through practice. [Click here for more on mathematical confidence.]
Another large barrier to student growth is the belief that progress only counts as success if it would also count as success for someone else. Because every student enters the classroom with different preparation, experiences, and goals, I encourage students to consider what success means for them individually within the shared expectations of the course. For some students, success may mean earning an A; for others, it may mean passing the course, rebuilding confidence after negative experiences with mathematics, or consistently completing their assignments. Especially in one-on-one conversations with students, I assist them in discovering what success looks like for them and help identify meaningful next steps toward that goal.
I’ve seen the importance of this approach every time I've taught. For instance, one of my Calculus II students initially defined success as earning an A. However, he had attended a high school where students were not given traditional exams. After his first midterm did not reflect the understanding he had demonstrated in class, we discussed how developing test-taking skills could be a meaningful goal for the remainder of the course. By the end of the semester, although he felt his grade did not fully reflect his understanding of the material, he told me that he felt proud of his progress and more prepared for future mathematics courses. [Click here for more examples of educational success.]
Beyond how I talk about math with the students, I shape the design of assignments and assessments around the ideology that mathematical education is to help students become mathematically literate.
Designing Assignments and Assessments
I strive to design assignments that provide a point of entry for every student while also encouraging continued growth. One approach I have found particularly successful is organizing homework into three levels. Level 1 problems help students develop fluency with essential concepts and techniques. Level 2 problems ask students to connect ideas or adapt familiar techniques to new situations. Level 3 problems are optional extra credit and extend beyond the expected course learning objectives, providing opportunities for students who have the time and interest to pursue deeper challenges.
Although students may find different levels of problems challenging, I want every student to engage with the broader goals of mathematical literacy. Therefore, I write problems at all levels with both the course material and these broader goals in mind. One type of question I particularly value asks students to identify errors or gaps in mathematical arguments, reinforcing the idea that mathematics is fundamentally about logical reasoning rather than obtaining correct answers. [Click here for more on effective problems.]
If I want my assignments and assessments to help students develop and demonstrate these habits of mathematical thinking, effective feedback and grading must focus on the mathematical understanding that students communicate, not simply whether they arrive at a correct final answer. I believe a student's conceptual understanding cannot be assessed if they do not communicate their reasoning; a correct answer without explanation provides little evidence of how a student approached a problem, while an incorrect answer accompanied by thoughtful reasoning often reveals meaningful understanding. Accordingly, my grading places greater weight on what students' work demonstrates about their mathematical thinking, including careful reasoning, clear communication, and thoughtful reflection.
Continuing Beyond My Classroom
Ultimately my class is a small part of students’ mathematical education. They will leave my classroom to become scientists, teachers, carpenters, artists, business owners, and engaged citizens, continuing to learn and use mathematics in ways that I cannot predict. My goal is that they feel better equipped to continue learning: more confident in their reasoning, more comfortable communicating mathematical ideas, and better prepared to think critically about unfamiliar problems.