The Effects of Idealized and Grounded Materials on Learning, Transfer, and Interest: An Organizing Framework for Categorizing External Knowledge Representations

Effects of Instructional Guidance and Sequencing of Manipulatives and Written Symbols on Second Graders’ Numeration Knowledge

“…With instructional guidance, on the other hand, we predicted that students in all sequence conditions would benefit relative to those in the no-guidance conditions, but that the symbols-first condition would benefit the most because there are few cues in the context that would scaffold the meaning they attribute to the symbols used in the standard algorithm. We predicted that the symbols-first condition would benefit the most from instruction because of the research indicating that idealized external representations are not as effective for learning and immediate performance as those that are "grounded" in representational features that are relevant to the task at hand (Belenky & Schalk, 2014) [14]. In this study, written symbols are less grounded than base-ten blocks because they do not possess perceptual characteristics that are as manifestly connected to concepts of numeration.…”

“…In contrast, the term "abstract" has typically referred to representations that do not embody elements of the context and can be considered "idealized" because they do not resemble their referents (Belenky & Schalk, 2014;Schnotz, Baadte, Müller & Rasch, 2010) [14,17]. According to a number of pedagogical theories, concrete representations are best presented before abstract ones for optimal learning outcomes (Freudenthal, 1991;Hoong, Kin & Pien, 2015) [18,19].…”

“…A review by Fyfe et al (2014) [8] demonstrates that the preponderance of evidence supports a concrete-to-abstract sequence in mathematics instruction. Concrete or otherwise "grounded" representations (see Belenky & Schalk, 2014) [14] can activate prior knowledge, which in turn can support the interpretation of subsequently presented abstract symbols, which by themselves are less transparent and as such more difficult to interpret (Fyfe et al, 2014;Koedinger & Nathan, 2004) [8,24].…”

“…A closer examination of the research literature suggests, however, that the issue of representational sequencing remains an open question (Belenky & Schalk, 2014) [14]. Some scholars have argued that under certain conditions, concrete representations are best presented after abstract representations.…”

“…A considerable body of research has focused on students' use of representations-e.g., diagrams, pictures, abstract symbols-in mathematics and science (Belenky & [14][15][16]. Because manipulatives can also be seen as representations (McNeil & Jarvin, 2007;Uttal et al, 1997 [2,3]), we hypothesized that the instructional principles that have emerged from the research on representations more broadly would also apply to children's mathematics learning with manipulatives.…”