Strategies in Cross-Notation Comparison

This preregistration describes Study 2 of a two-study research project investigating how adults understand and compare rational number magnitudes across symbolic notations—fractions, decimals, and percentages—and why many individuals continue to struggle with these comparisons despite extensive formal mathematics education. Cross-notation magnitude understanding is a foundational component of mathematical proficiency, supporting the evaluation of numerical reasonableness. Prior research demonstrates that cross-notation magnitude knowledge predicts fraction and decimal arithmetic performance, pre-algebra and algebra achievement, and overall mathematical success beyond what is explained by within-notation fraction knowledge alone (e.g., Braithwaite & Sprague, 2021; Hurst & Cordes, 2016; Park & Esposito, 2022; Schiller & Siegler, 2023; Schiller et al., 2025). Moreover, it mediates the relation between within-notation fraction understanding and fraction arithmetic, suggesting that it plays a critical integrative role in rational number reasoning rather than simply reflecting symbolic familiarity. Despite its importance, performance on cross-notation magnitude comparison tasks reveals persistent and systematic difficulties, even among educated adults. Accuracy is reliably lower when individuals compare magnitudes across notations than within a single notation, and responses frequently exhibit directional biases, such as overestimating percentages relative to equivalent fractions or decimals (“percentages-are-larger” bias; Schiller et al., 2025). The present project addresses a critical gap in the literature by focusing on the strategies individuals use during cross-notation magnitude comparison and by testing whether strategy use plays a causal role in successful performance. Study 1, which has already been completed, examined adults’ spontaneous strategy use during a cross-notation magnitude comparison task. Participants reported the strategy they used on each trial, allowing for trial-level analyses of strategy effectiveness. Strategy use emerged as a strong predictor of success: benchmarking and percentage-translation strategies were associated with higher accuracy and reduced susceptibility to the percentages-are-larger bias relative to less effective strategies. Individual differences analyses further revealed distinct strategic profiles, with participants who predominantly used benchmarking or percentage translation outperforming those who relied heavily on other strategies. Building on these completed findings, the present preregistration concerns Study 2, which experimentally tests whether strategy use causally influences cross-notation magnitude performance. Participants will be randomly assigned to receive brief instruction in either a benchmarking strategy or a percentage-translation strategy. Performance and strategy use will be assessed before and after instruction using the same cross-notation magnitude comparison task. This design allows us to evaluate whether targeted strategy instruction improves accuracy, whether instructional gains depend on problem features such as notation and numerical distance, and whether individuals with less effective baseline strategy profiles benefit disproportionately from instruction. In addition, the study examines whether instruction leads to changes in participants’ spontaneous strategy use, providing evidence that performance improvements reflect strategic shifts rather than general practice effects. Across both studies, trial-level accuracy and strategy use are analyzed using mixed-effects models that account for variability across participants and problem types. Measures of approximate magnitude sense, exact calculation ability, and within-notation comparison skill are included to clarify which underlying competencies support learning from instruction. The expected outcomes of this research are threefold. First, it will provide a strategy-based account of cross-notation magnitude understanding that explains persistent individual differences. Second, it will offer causal evidence that brief, targeted instruction can improve rational number magnitude understanding and reduce systematic biases, particularly for individuals who initially rely on less effective strategies. Third, the findings will inform instructional approaches aimed at fostering flexible and transferable numerical reasoning across symbolic representations, contributing to broader efforts to improve rational number understanding in adolescence and adulthood.