Visual Salience of Algebraic Transformations

Abstract

for help with technical arguments in philosophy, connectionist theory, and mathematics, and to two editors and numerous anonymous reviewers of JRME who have contributed substantially to the shaping of this article.

Conditional inference and advanced mathematical study

“…In the example given in Figure 6, the notation '8' is an opaque representation for the concept not-q; in contrast, the notation 'not 1' is a transparent representation, as it shares salient features with the conditional 'if p then q': the presence of 1. In short, Evans and Handley's (1999) account of the affirmative premise effect suggests that participants struggle to see through the opaqueness of the notation used in the implicitly negated component of the task, in exactly the same way as the students studied by Kirshner and Awtry (2004), Lesh et al, and Zazkis and Liljedahl (2004) struggled to see through the opaqueness of the notations used in their tasks. However, the crucial difference between our study, and those of other researchers who have looked at opaque and transparent representations, is that unlike the arts students, the mathematics students in our sample (all undergraduates who had been highly successful at school level mathematics) were apparently not affected by the opaqueness of the representation of implicitly negated premises: they exhibited no affirmative negation effect on either negation-type.…”

“…In the domain of knot theory, for example, the Tait Conjectures remained unproved for much of the twentieth century, but upon the discovery of new and more transparent ways of representing knots -the Jones and HOMFLY polynomials -the conjectures were quickly solved. Kirshner and Awtry (2004) noted that representation systems for some mathematical concepts may be opaque in a slightly different way: rather than de-emphasising certain valid properties, they may emphasise other invalid properties. Kirshner and Awtry suggested that many of the widely documented 'mal-rules' applied by students when making algebraic manipulations are a consequence of such effects.…”

Formal notations are diagrams: Evidence from a production task

“…We have argued previously that a broadly similar interference of metric (non-order-related) spatial properties on syntactic judgments provides evidence that spatial processes and representations implement syntax in typical human judgments ; see also Kirshner, 1989;Kirshner & Awtry, 2004). To study the influence of perceptual grouping on mathematical reasoning, we tasked undergraduate participants with judging whether an algebraic equality was necessarily true.…”

Structural Reasoning

“…Research has also shown that some pedagogical approaches developed by Davydov et al (1994) advance PPG ways of thinking among elementary school students (Morris 2009); and that Ba careful mixture of free play, collaboration, questioning, revisiting the problem, and a call for justification^lead young students to not only observe structures but also to create structures (Maher 2009). On the other hand, Kirshner and Awtry's (2004) work mentioned earlier suggests that there might be cognitive factors involving Bvisual salience^that would unavoidably cause difficulties for students to construct and observe structures among algebraic expressions. More research is needed to better understand the cognitive process and epistemological and didactical obstacles (in the sense of Brousseau 2002) involved in cultivating structural reasoning among students.…”

“…They do not explicitly address the cognitive source of students' weaknesses, however. Kirshner and Awtry (2004) report an interesting finding related to structural reasoning among students as they are introduced to algebra, which might account for these weaknesses. They report that students' errors in algebra symbol manipulation stem from Bvisual salience of a rule, rather than just from its declarative complexity^(p. 231).…”