Scholarship of Teaching and Learning in Mathematics
Developing symbol sense, conceptual understanding, and productive comfort with ambiguity in mathematics.
Overview
Students often learn to manipulate mathematical notation before they become comfortable interpreting what that notation means. A correct calculation can therefore hide a fragile conceptual understanding, especially when a familiar idea is written in an unexpected form or when the role of a symbol changes from one expression to another.
My Scholarship of Teaching and Learning project explores how designed instruction can help students develop stronger symbol sense: the ability to recognize structure, interpret notation, connect different representations, and explain the meaning behind a procedure. The goal is not simply to reduce errors, but to help students become more flexible and independent readers of mathematics.
“Ironically for a discipline touted as precise, the student of mathematics has to develop a tolerance for ambiguity. Pedantry can be the enemy of insight.”
— Gila Hanna
This perspective treats ambiguity as something that can be used productively. Students should learn standard notation and conventions, but they should also be prepared to reason through expressions that are unfamiliar, incomplete, or open to more than one initial interpretation.
Instructional approach
The project uses a post-instructional refinement approach. Students first encounter a mathematical idea through lecture, worked examples, and ordinary practice. Once the basic procedure is familiar, a closely related question changes one feature of the notation, representation, or underlying assumption. For example, consider
\[f(x)=x+\int_0^1 2x\,dx.\]A student may calculate the integral mechanically without noticing that (x) appears both as the free variable in (f(x)) and as the variable of integration. Rewriting the expression as
\[f(x)=x+\int_0^1 2t\,dt\]does not change its value, but it makes the different roles of the symbols visible. The purpose of the example is not merely to correct notation. It asks students to distinguish between a variable that remains free and a dummy variable whose role is confined to an integral. A second example compares a continuous function with a discrete sequence:
\[f(x)=\sin(\pi x), \qquad a_n=\sin(\pi n).\]Although the formulas look almost identical, they define very different mathematical objects. The function (f(x)) varies continuously, while the sequence stays $0$ for every positive integer (n). Students can also differentiate (f(x)), whereas an expression such as (\frac{d}{dn}a_n) is not defined in the same ordinary sense because (n) is a discrete index. This comparison encourages students to examine the domain and meaning of a symbol instead of relying on the visual appearance of a formula. Similar variations can be used with summations. For instance, comparing
\[\sum_{n=1}^{k}n \qquad\text{and}\qquad \sum_{n=1}^{k}k\]helps students see that the index (n) changes from term to term, while (k) remains fixed within the sum. Expanding the expressions reveals that they are structurally different even though they use the same letters.
Evaluation and broader impact
The effectiveness of this approach can be studied through student explanations, recurring error patterns, short diagnostic questions, and entry-and-exit surveys that compare confidence with actual performance. This makes it possible to examine not only whether students feel more comfortable with notation, but also whether their interpretations become more accurate and their explanations become more precise.
The expected outcomes include clearer mathematical communication, fewer notational errors, stronger connections between symbolic and conceptual representations, and greater awareness of one’s own understanding. The project also aims to produce reusable examples, diagnostic questions, and short instructional activities that can be adapted across undergraduate mathematics courses. If you are interested in designing a version for your own courses, please contact me.