Following the standard form: Effects of equation format on algebraic modeling

Abstract

When people construct algebraic equations to represent quantitative relations, they often reverse the roles of the variables (6S = P instead of 6P = S). Results from three experiments show that a major reason for such reversal errors is people's adherence to, and interpretation of, the Standard Form of algebraic models. College students constructed, selected, and interpreted algebraic models that either had the standard multiplication format (MF: ax = y), or the mathematically equivalent division format (DF: y/a = x). A large minority of participants constructed reversed MF models, whereas most participants who were instructed to use DF equations constructed correct models (Experiment 1). Most participants understood that MF and DF models of the same relation should be mathematically equivalent (Experiment 2), but they drew a conceptual distinction between these models (Experiment 3). The authors discuss the impact of notational conventions on people's interpretation and use of representational tools.

Abstract numeric relations and the visual structure of algebra.

“…These may include images, conventionalized structures such as number lines, or candidate expression fragments adapted or selected from those previously encountered (Sherin, 2001;Fisher, et al, 2011). These simulations are made specific by a process of alignment: we suggest that structure mapping (Gentner, 1983) is used to identify entities in text or images with those in equations.…”

“…Recently, Fisher, et al (2011) demonstrated that the reversal rate of typical students-and-professors style problems sharply decreases when students are forced to use an equation structure that creates a high-quality alignment between text and equation. In their experiments, students were encouraged to construct division models such as for sentences such as "There are six students for every professor.…”

“…In the alignment view espoused here and in Fisher et al (2011), the successful procedure involves two (non-sequential) tasks: selecting an appropriate equation frame, and aligning selected frames with the source situation. If a reasoner succeeds in selecting an appropriate frame for a multiplicative relational equation (regardless of whether the operation chosen is multiplication or division), in the absence of alignable information the reasoner will have an equal chance of producing correct and inverted equations.…”

“…This paper focuses on one essential step: the creation of a notational expression-in this case, an equation-that models a situation. One possible account suggests that much of the time this equation-construction process involves selecting an appropriate template-say, a memorized equation pattern such as F=ma, y=mx+b, ax 2 +bx+c=0, or base+change (Carlson-Radvansky & Logan, 1997;Fisher Borchert, & Bassok, 2011;Sherin, 2001)-and applying structure-mapping (Gentner, 1983;Hummel & Holyoak, 1997) to align that structure with other representations of the scene, such as written descriptions, pictures, or mental simulations (Bassok, Chase, and Martin, 1998). For example, Abstract numeric relations 5 asked to find the total weight of a case with 6 apples, each apple weighing 10 ounces, one might retrieve an a times b frame, and fill in the mapping a=6, b=10 via alignment to yield 6*10.…”

POETIC: Interactive solutions to alleviate the reversal error in student–professor type problems

“…Many researchers have attributed student difficulty to a tendency for a "directtranslation" approach (Clement, 1982;Fisher et al, 2010;Hegarty et al, 1995;Wollman, 1983). This strategy is used when students attempt to make the sequence of algebraic symbols match the sequence of objects in a word problem.…”