Causal Models Frame 1 Running head: CAUSAL INTERPRETATION OF EQUATIONS Causal Models Frame Interpretation of Mathematical Equations

We offer evidence that people construe mathematical relations as causal. The studies show that people can select the causal versions of equations, and that their selections predict both what they consider most understandable and their expectations of change. When asked to write down equations, people have a strong preference for the version that matches their causal model. Causal Models Frame 3 Causal Models Frame Interpretation of Mathematical Equations Empirical phenomena afford multiple representations that each highlight a different aspect of empirical structure. Even though mathematical equations can provide a concise and exact representation of how variables relate, they need not represent causal structure. Mathematical relations are symmetric, X = Y is identical to Y = X. Causal relations are not, X causes Y is not the same as Y causes X. Moreover, the representation of causal structure is essential to intervene on the world and influence its dynamics. In these senses, causal structure is more demanding than purely mathematical structure. Nevertheless, or perhaps as a result, people impose causal structure to understand (diSessa, 1993; Hunt & Minstrell, 1994; Reif & Allen, 1992). We propose they impose causal structure even when they need not: in the interpretation of mathematical representations. We hypothesize that people prefer equations whose variables are in an order that matches the causal relations that generate the phenomena represented. People impose a causal frame even when it distorts a representation. For example, people assert a unidirectional causal relation between variables that interact in a more dynamic fashion. The circuit between a battery and a light is often incorrectly understood as unidirectional, with energy flowing from the battery to the light (Andersson, 1986; Driver, Guesne & Tiberghien, 1993; Reiner, Slotta, Chi & Resnick, 2000). Teaching correct causal interpretations often helps form a deeper understanding (White, 1993). Pearl (2000) suggested that not all versions of equations are cognitively equivalent. People will learn and use equations more effectively to the extent that these match their causal model. Supporting evidence is that people consider the left side of an Causal Models Frame 4 equation the outcome of something changing on the right (Sherin, 2001). The hypothesis predicts that, in the case of a three-variable equation matches a single causal model, the variable that is an effect of the other two variables should be written on the left and the two causes on the right. For example, the equation that relates pressure (P), weight (W) and area (A), can be written three different ways: P = W/A, A = W/P and W = AP. However, P = W/A should seem more intuitive, because the most natural naïve causal model represents pressure as the effect of weight and area, not as a cause of them. This is suggested by the intuition that if weight were increased, pressure would increase. The equation that solves for A is not as understandable; the intuition that increasing weight would increase area is weaker. Through a series of studies, we tested whether people have a preference for the version of equations that matches causal understanding. For these studies, we used 16 equations from various fields, such as physics, biology and economics (see Table 1 for full list of equations). They were grouped into 3 classes: The first type (Causal) were the equations with a clear causal model (e.g., P = W/A). These are the equations that represent causally related phenomena that are familiar, so most people understand what causes what. We assumed that the causal models of these simple situations are consistent across our population of Brown University undergraduates. The second type (Unclear) were equations with unclear causal models. These are equations that represent phenomena that are not very familiar, but for which causal models are assumed to exist. Therefore, we expected different people to think that they are related in different ways. One example is the equation that relates thermal efficiency (η), work done during one cycle (W), and heat added during one cycle (Q): η = W/Q. Some people might think that Causal Models Frame 5 increasing the thermal efficiency would increase the work done during one cycle. Conversely, others might think that increasing the work done during one cycle would increase the thermal efficiency. The third type (Noncausal) are equations that have no causal model. These are equations that represent geometric relations or scale transformations with no causal interpretation. An example of this type would be the equation that gives the relation between temperature in Kelvin and Celsius: K = oC + 273.