Arithmetic Notation…now in 3D!

Arithmetic Notation…now in 3D! David Landy (dlandy@richmond.edu) Department of Psychology, University of Richmond Richmond, VA 23173 Sally Linkenauger (sal3g@virginia.edu) Department of Psychology, University of Virginia Charlottesville, VA 22904 Abstract partially described by the order of operations. In any equation, operations bind in the following order: parentheses, exponents, multiplications and divisions, additions and subtractions. This apparently arbitrary system appears to require explicit memorization, but in fact can be computed using basic perceptual-motor mechanisms such as grouping (Landy & Goldstone, 2007) and automatic attentional biases (Landy, Jones, & Goldstone, 2008). On this account, computing the answer to a math problem involves taking physical actions to transform abstract forms. If notations really are fundamentally abstract, then their implied physical structure is entirely given by their surface form: there is no sense in which any of the symbols are ‘close’ or ‘far away.’ However, if people indeed often reason by treating symbols as pictures of objects in space, then these objects must be laid out in some three- dimensional arrangement. Thus, it is at least possible that different symbols would be seen as closer or further away than others. In the next section, we outline some reasons to expect such differences. When people reason formally, they often make use of special notations—algebra and arithmetic are familiar examples. These notations are often treated as mere shorthand—a concise way of referring to meaningful mathematical concepts. Other authors have argued that people treat notations as pictures—literal diagrams of an imagined set of objects (Dorfler, 2003; Landy & Goldstone, 2009). If notations depict objects that exist in space, then it makes sense to wonder how they are arranged not just in the two visible dimensions, but in depth. In four experiments, we find a consistent pattern: properties that increase mathematical precedence also tend to make objects appear closer in space. This alignment of formal pressures and informal pressures suggests that perceived depth may play a role in supporting computational reasoning processes. Although our primary focus is documenting the existence of depth illusions in notations, we also evaluate several sources of information that might guide depth judgments: availability of an object for computational actions, formal syntactic structure, relative symbol salience and voluntary attention shifts. We consider relationships between these nonexclusive possible sources of information in guiding how people judge depth in mathematics. Reasons to Expect Illusions of Differential Depth Keywords: Mathematical cognition, embodied cognition, depth perception Introduction Special notations are ubiquitous markers of mathematical thinking. Often, these notations are treated as mere conventional patterns, which serve as the target of rule- learning systems such as generic production systems (e.g., (Koedinger, Alibali, & Nathan, 2008; Anderson, 2005; Koedinger & MacLaren, 2002). From this perspective, the exact format and layout of the expression doesn’t much affect how reasoning happens—what makes learning difficult is the rules, not the layout. However, growing evidence suggests a different account. It seems formal operations—from reasoning to logic to simple mathematics—are not always computed through the action of an abstract reasoning system, but instead make use of perceptual-motor systems typically involved in real-world perception and action. In the case of mathematics in particular, reasoning that on the surface appears to require formal operations can be simplified if reasoners treat notations as pictures of a physical scene (Dorfler, 2003; Landy & Goldstone, 2009). For instance, algebraic syntax has a hierarchical structure Actual equations and expressions are of course purely two-dimensional; thus actual depth experience should not directly inform perceived depth judgments with mathematical forms. Furthermore, accounts that treat notation as basically abstract predict no particular differences in apparent depth of different symbols. However, several factors might affect the perceived depth of symbols seen as objects that exist in space, which can be acted on in particular kinds of ways. One clear prediction is that symbols that afford action appear proximal relative to those that do not. Several studies have found that depth perception can be affected by the action capabilities of the observer (Linkenauger, Witt, Stefanucci, Bakdash, & Proffitt, 2009; Witt, Linkenauger, Bakdash, & Proffitt, 2008; Witt & Proffitt, 2005; Witt, Proffitt, & Epstein, 2005). Therefore, if solving a mathematical equation requires actions on the part of the solver, high precedence terms—those most available for actions—should generally seem most proximal in arithmetic expressions. Put simply, years of experience acting first on multiplications in expressions like will lead the multiplication to appear closer than the addition. We hypothesized, more generally, that terms and operation signs that were most immediately available for