Exploring the way rational expressions trigger the use of “mental” brackets by primary school students

Connecting characterizations of equivalence of expressions: design research in Grade 5 by bridging graphical and symbolic representations

“…Although the empirical scope was still limited by a small sample size, we believe that the identified content trajectory (in Fig. 8) of successively extending and connecting the characterizations is of major relevance for algebra education research as the discovered bridging characterization seems to have a high epistemic potential to support students' development of conceptual understanding of transformation of expressions by perceiving deeper structures in a highly problematic field (Linchevski & Livneh, 1999;Papadopoulos & Gunnarsson, 2020).…”

“…The focus is solely on surface structures to determine which operation to complete first, yet not on systemic structures. • In a transformational perspective, students view expressions mainly as sets of symbols that are to be manipulated according to transformation rules, which can be arbitrary or welljustified by drawing upon robust understandings of structure (Kieran, 2004;Linchevski & Livneh, 1999;Papadopoulos & Gunnarsson, 2020). For correctly enacting transformations, students need to recognize the surface structures of the expression.…”

“…Students' tendency to adopt operational rather than relational perspectives is also expressed in their focus on procedures and surface structures rather than systemic structures while transforming expressions (overview in Kieran, 2007). Their ability to correctly identify symbolic surface structures within the expressions depends on form and order of the elements of the represented expression (Papadopoulos & Gunnarsson, 2020), which applies for expressions with and without variables (Linchevski & Livneh, 1999). The former difficulty can partially be traced back to students' difficulties in connecting symbolic substructures with parts of structured figures (E A -S A and E A -S B ; Malle, 1993;Wilkie & Clarke, 2016).…”

Fraction Symbols and their Relation to Conceptual Knowledge

“…Thus, students who are more attentive to the perceptual cues associated with fraction conventions (i.e., the orthography) may also have a better understanding of fraction symbols. This view is supported by research showing that attention to perceptual cues in equations is related to equation solving (Hoch & Dreyfus, 2004;Landy et al, 2014;Landy & Goldstone, 2007;Papadopoulos & Gunnarsson, 2020). For example, the equation 3 + 5 × 4 can also be written 3 + (5 × 4).…”

“…For example, the equation 3 + 5 × 4 can also be written 3 + (5 × 4). The brackets are unnecessary, but they helped second and third graders attend to the order of operations and successfully solve the equations (Marchini & Papadopoulos, 2011, as cited in Papadopoulos & Gunnarsson, 2020). Similarly, narrowly spacing precedent operations within equations, for example, 3 + 5×4 versus 3 + 5 × 4, helped students attend to the order of operations and thus better discriminate between the correct (i.e., 23) and incorrect (i.e., 32) solutions (Landy & Goldstone, 2007).…”