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This paper examines the gap between procedural fluency and conceptual understanding in mathematics learning through the lens of "hidden variables." In school mathematics, problem solving often prioritizes procedural execution under idealized assumptions with key variables suppressed; yet this practice can impede substantive understanding, especially in real-world and geometric contexts. Using a set of illustrative tasks-including origami puzzles, traffic-flow calculations, and counting images in mirrors-we investigate how implicitly embedded variables reshape learners' interpretations of a problem and the depth of understanding they attain. We argue that, rather than eliminating variables, introducing additional variables or increasing dimensionality can make the problem structure explicit and facilitate concept formation. Extending beyond notions of maxima and minima, we further analyze geometric ideas naturally characterized by saddle points, thereby probing mathematical mechanisms underlying concept development. These findings suggest pedagogical directions for fostering conceptual understanding beyond procedural competence and highlight the growing importance of creative problem comprehension in AI-rich learning contexts.