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The Physics Inventory of Quantitative Literacy (PIQL) aims to assess students’ physics quantitative literacy at the introductory level. PIQL’ s design presents the challenge of isolating types of mathematical reasoning that are independent of each other in physics questions. In its current form, PIQL spans three principle reasoning subdomains previously identified in the research literature: ratios and proportions, covariation, and signed (negative) quantities. An important psychometric objective is to test the orthogonality of these three reasoning subdomains. We present results that suggest that students’ responses to PIQL questions do not fit this structure. Groupings of correct responses identified in the data provide insight into the ways in which students’ knowledge may be structured. Moreover, questions with multiple correct responses may have different responses in different data-driven groups, suggesting that the both the answer choice and the context of the question may impact how students (implicitly) relate various ideas. 

Mathematical reasoning flexibility across physics contexts is a desirable learning outcome of introductory physics, where the “math world” and “physical world” meet. Physics Quantitative Literacy (PQL) is a set of interconnected skills and habits of mind that support quantitative reasoning about the physical world. The Physics Inventory of Quantitative Literacy (PIQL), which we are currently refining and validating, assesses students’ proportional reasoning, co-variational reasoning, and reasoning with signed quantities in physics contexts. In this paper, we apply a Conceptual Blending Theory analysis of two exemplar PIQL items to demonstrate how we are using this theory to help develop an instrument that represents the kind of blended reasoning that characterizes expertise in physics. A Conceptual Blending Theory analysis allows for assessment of hierarchical partially correct reasoning patterns, and thereby holds potential to map the emergence of mathematical reasoning flexibility throughout the introductory physics sequence.