A common obstacle for students in the transition from arithmetic to algebra is developing a conceptual understanding of equations representing functions. Two experiments manipulated isomorphic problems in terms of their solution requirements (computation vs. interpretation) and format to test for understanding of linear functions. Experiment 1 provided problems in a story context, and found that performance on slope comparison problems was low, especially when problems were presented with equations. Experiment 2 tested whether performance on slope comparison problems improves when problem prompts include explicit mathematical terminology rather than just natural language consistent with the problem story. Results suggest that many undergraduate students fail to access the mathematical concept of slope when problem prompts are presented with natural language. Overall, the results suggest that even undergraduate students lack understanding of the slope concept and equations of linear functions, both which are foundational for advanced algebraic thinking.

Many people experience illusions of understanding for explanations of scientific phenomena (Rozenbleit & Keil,2002) and readers tend to be poor at gauging how well they have understood what they have read in expository science texts(Dunlosky & Lipko, 2007; Maki, 1998; Thiede, Griffin, Wiley, & Redford, 2009). The present line of research includesstudies demonstrating that metacomprehension accuracy may be especially poor when students are presented with texts thatinclude features such as diagrams, graphs, animations, and analogical examples. Although these adjuncts are meant to improvecomprehension, they can often lead to illusions of understanding. An important theme of this research is articulating the kinds ofinstruction and skills that students may need before they can learn effectively from expository science texts including graphicsor analogies.

Children display an early sensitivity to implicit proportions (e.g., 1 of 5 apples vs 3 of 4 apples) but have considerabledifficulty in learning the explicit, symbolic proportions denoted by fractions (e.g., 1/5 vs 3/4). Theoretically, reducingthe gap between representations of implicit vs explicit proportions would improve understanding of fractions, but littleis known about how the representations develop and interact with one another. To address this, we asked 163 third tofifth graders to estimate the position of proportionally-equivalent integers and fractions on number lines (e.g., 3 on 0-8number line vs 3/8 on 0-1 number line). We found that, with increasing age, children were more accurate and linear inrepresenting both integers and fractions. More importantly, childrens estimates of implicit and explicit proportions becamemore coherent, such that a childs estimates of fractions on a 0-1 number-line was a linear function of the same childsestimates of equivalent integers. This representational coherence independently predicted childrens fraction proficiency inother tasks, suggest- ing that building a coherent understanding of proportions is an educationally-important goal.