Different presentations of a mathematical concept can support learning in complementary ways.

Abstract

Previous research has found that different presentations of the same concept can result in different patterns of transfer to isomorphic instances of that concept. Much of this work has framed these effects in terms of advantages and disadvantages of concreteness or abstractness. We note that mathematics is a richly structured field, with deeply interconnected concepts and many distinct aspects of understanding of each concept, and we discuss difficulties with the idea that differences among presentations can be ordered on a concrete-abstract dimension. To move beyond this, we explore how different presentations of a concept can affect learning of subsequent concepts and assess several distinct aspects of understanding. Using the domain of elementary group theory, we teach adult participants a group operation using a visuospatial or an arithmetic presentation. We then teach them concepts that build upon this operation. We demonstrate that our presentations differentially support learning complementary aspects of the system presented. We argue that these differences arise from the fact that each presentation supports learning by connecting to different systems of reasoning learners are already familiar with, and that it is these connections to extant knowledge systems, rather than differences in concreteness versus abstractness, that determine whether a presentation will be helpful. Furthermore, we show that presenting both presentations and encouraging participants to recognize the relationship between them improves performance without requiring additional time, at least for some participants. Educational Impact and Implications StatementThe details of how a concept is taught can have far-reaching effects on students' learning. Using abstract algebra with adult subjects, we show that 2 presentations of a concept that connect to different types of students' prior knowledge can have advantages and disadvantages for later learning that builds on the target concept. We show that one possible solution to the dilemma of choosing which presentation to use is giving students both concepts and explaining how they are related to each other. In summary, when designing pedagogical materials, we should consider not only how they affect learning of the present concept, but also how they support learning of future concepts, and use multiple complementary presentations rather than searching for a single ideal one.

How prior knowledge, learning, teaching and assessment affect students’ achievements in Mathematics

“…The result was consistent with some previously reported works, who argued that higher prior knowledge, comprehensive learning approach, problem-based teaching, and higher assessment impact scores predict higher basic-learning skill scores (Baumert et al, 2017a;Fries et al, 2019;Gearing & Hart, 2019;Gersten et al, 2009;Hakyolu & Ogan-Bekiroglu, 2016;Han et al, 2015;C. H. Hill et al, 2015;Kilion, 2016;Lampinen & McClelland, 2018;Linsell et al, 2012;Nguyen, 2016;Rimbey, 2013;Santagata & Yeh, 2014). In conclusion, H # 5: The higher prior knowledge, comprehensive learning approach, problem-based teaching, and higher assessment impact scores predict higher basic-learning skill scores, is been supported.…”

“…Hence, it is evidenced that approaches to instruction and curriculum design, formative assessment data, feedback to students, prior knowledge, pedagogical knowledge, individual learning support, teachers' mathematical knowledge, problem-based lessons, on-going professional development for teachers impact student's basic-learning skill in mathematics. Garet et al (2016) revealed that math content knowledge and instructional practice were generally not correlated with student math achievement, but Lampinen and McClelland (2018) pointed out that pedagogical materials affect learning. Project-based learning instruction, as well as diagnostic assessment information, influenced student achievement in mathematics Han, Capraro, and Capraro (2015); Linsell et al (2012).…”

Transforming task representations to perform novel tasks

“…If the different levels of abstraction share structural features, sharing computation should improve generalization. Homoiconicity could also support the ability to build abstractions recursively on top of prior abstractions, as humans do in mathematical cognition (15)(16)(17). Although homoiconicity is not a necessary part of metamapping, we suggest that homoiconic approaches will be beneficial and verify this empirically, see Polynomials section).…”

One Instructional Sequence Fits all? A Conceptual Analysis of the Applicability of Concreteness Fading in Mathematics, Physics, Chemistry, and Biology Education

“…External representations can support and constrain but may also hinder learning and reasoning in characteristic ways. That is, different external representations or combinations thereof affect what is processed and encoded (Knuuttila 2011;Lampinen and McClelland 2018;Louca and Zacharia 2012). For example, the use of graphs instead of tables leads to more flexible performance when interpreting data, but graphs can also induce interpretive bias (Braithwaite and Goldstone 2013).…”

“…Second, MERs can constrain interpretation by virtue of familiarity or their inherent properties. For example, Lampinen and McClelland (2018) argue that different representations Fig. 1 The different functions of MER as conceptualized in the DeFT framework by Ainsworth (2006) support the learning of complementary aspects of a target system (e.g., a scientific principle or a theory) by triggering different reasoning systems.…”