Conceptual design requires conceptual understanding, what we call design insight. Novel designs require novel insights. Such understanding is usually hard-won through difficult analysis. We propose algorithmic approaches to developing insight, utilizing dimension reduction techniques. To provide such tools, we generalize the active subspace to describe field quantities of interest, drawing novel connections with the proper orthogonal decomposition. We present applications of the proposed insight-generating strategy to problems from aircraft design. Using the proposed methods, we can 'discover' the lift dependence on the camber line through an automated procedure, and recover similar observations in the problems of airfoil and structural design.