"I forgot the formula:" How students can use coherence to reconstruct a (partially) forgotten equation

Introductory physics instruction emphasizes fluency with routine problem-solving procedures. However, even when applying these procedures, students frequently encounter challenges. This paper investigates how students navigate such moments when answering qualitative E&M problems in interviews. Students frequently noted they had partially forgotten a key equation on a problem involving RC circuits. We present focal cases that show how coherence-seeking approaches were employed to overcome this problem-solving challenge. In attempts to reconstruct these equations, participants were guided by identifying and chaining qualitative dependencies and seeking coherence between qualitative and mathematical understanding of the physical system. These moments of forgetting and reconstructing equations are a useful site for studying broader physics learning goals. While prior work investigates the use of mathematical sensemaking by examining how students respond to explicit prompts, our cases illustrate how students can spontaneously use mathematical sensemaking strategies. We reflect on these cases to consider how such adaptive reasoning can be a target for instruction and assessment.

💡 Research Summary

The paper investigates how introductory physics students cope when they cannot recall a key equation while solving a qualitative electromagnetism (E&M) problem, focusing on a four‑circuit RC‑circuit scenario. Drawing on the literature of mathematical sensemaking and coherence‑seeking, the authors argue that forgetting an equation provides a natural window into students’ adaptive expertise – the ability to modify and reconstruct knowledge rather than merely apply memorized procedures.

Data were collected through semi‑structured, one‑hour clinical interviews with 18 participants (nine from a community college and nine from an elite private university). All participants had completed an introductory E&M course. During the interviews, each student was asked to think aloud while answering a set of qualitative E&M questions, most notably the “Four Circuits Problem,” which requires comparing discharge time constants (τ = RC) across four circuits with differing resistances and capacitances. The researchers identified 12 instances in which students explicitly reported that they had “forgotten” the relevant expression (11 students, one student experienced two separate forget‑events).

The analysis proceeded in two stages. First, the researchers coded transcripts for moments where physical reasoning, mathematical reasoning, and coherence‑building were evident. Second, they examined the specific strategies students employed to reconstruct or compensate for the missing equation. The cases fell into two broad categories.

1. Partial Recall / Symbolic Form Filling – Students remembered fragments of the equation (e.g., the product form τ = RC, or that resistance appears in the denominator of a time constant) and used conceptual dependencies to fill in the missing pieces. Four participants (Gabi, Tom, a second Gabi, and Rene) exemplify this pattern. They identified qualitative relationships such as “if resistance doubles, the time constant doubles” or “larger capacitance means slower discharge,” then mapped these relationships onto the symbolic template □ · □. By checking dimensions and limiting cases (e.g., τ → 0 when R → 0), they verified the reconstructed form.

2. Complete Forgetting / Re‑derivation – Students could not retrieve any part of the formula and therefore resorted to first‑principles reasoning. Blake, Sam, and Riley illustrate this approach. Blake tried several lines of reasoning—conceptual arguments about current flow, analogical reasoning with mechanical damping, and attempts to write down an equation—before settling on the qualitative claim that a larger τ implies slower discharge. Sam and Riley abandoned symbolic manipulation altogether, instead reasoning mechanistically about charge flow, the role of resistance, and the definition Q = CV to decide which circuit retains the most charge after a given time.

Across all cases, the authors observed a “chaining” process: (a) identify a physical dependency, (b) articulate it qualitatively, (c) translate it into a symbolic form, (d) validate via dimensional analysis, limiting‑case checks, or known definitions. This chain mirrors the symbolic‑form framework described by Sherin (2001) but is shown here to arise spontaneously in a natural interview setting, without explicit prompts.

The findings have several pedagogical implications. First, instruction should shift from rote memorization of formulas toward activities that foreground the coherence between concepts and mathematical structures. Prompting students to articulate “what changes when X changes?” can cultivate the same sensemaking moves observed in the study. Second, integrating explicit training in dimensional analysis, limiting‑case reasoning, and symbolic‑form identification into regular problem‑solving practice may equip students to handle “forgetting” moments productively. Third, educators might treat instances of forgotten equations as diagnostic opportunities, designing assessments that reward the reconstruction process rather than merely the final correct answer.

In sum, the study demonstrates that when confronted with a lapse in recall, students can and do employ sophisticated, self‑generated sensemaking strategies that link qualitative physics reasoning with mathematical structure. By documenting these spontaneous coherence‑seeking moves, the paper extends the literature on mathematical sensemaking beyond prompted tasks and highlights a promising avenue for fostering adaptive expertise in introductory physics curricula.