This article evaluates the plausibility of synthesizing theory of knowledge objectification (Radford, 2003) with equity research on mathematics education. I suggest the cognitive phenomenon of mathematical inference as a promising locus for investigating the types of agency that equity-driven scholars often care for. In particular, I conceptualize students' appropriation of semiotic-cultural artifacts (e.g., algebraic symbols and forms) to objectify their pre-symbolic inferences as conditional on their agency to carefully and incrementally construct personal meaning for these artifacts. To empirically ground this emerging approach, this study focuses on algebraic generalization (as a type of mathematical inference) and applies Radford's framework to video data of two iterations of an instructional intervention conducted in a high school program for academically at-risk youth. I analyze and compare students' acts of appropriation/objectification during whole-class conversations centered on pattern-finding tasks, in relation to the instructional mode adopted for each of the iterations--"direct instruction" vs. "inquiry-based." The analysis shows that the implementation involving inquiry-based instruction enabled more equitable access to opportunities for agency-as-mathematical inference, whereas the implementation involving direct-instruction was ostensibly more productive. Implications for future equity research involving cognition-and-instruction analyses are discussed.