Building and Understanding Multiplicative Relationships: A Study of Prospective Elementary Teachers

Abstract

This study focused on understanding the evaluation of the area of a rectangular region as a multiplicative relationship between the lengths of the sides. The analysis of data from a whole-class teaching experiment involving prospective elementary teachers resulted in the development of a description of the quantitative reasoning involved. Important aspects of this reasoning include the anticipation of a rectangular array of units as the structure of the area quantity and the subsequent constituting of the units based on the linear measures. The analysis of the prospective teachers' understandings proved useful in generating additional problems. The teachers' engagement with these problems resulted in their developing more complete understandings of the constitution of area units.

Preservice teachers’ knowledge mobilized in solving area tasks

“…In Task 4 PSTs have difficulties in accepting area conservation which is mainly due to the tendency towards the use of calculations and formulas, and to the unfamiliarity with non-prototypical triangles (Baturo & Nason, 1996;Caviedes et al, 2022b;Simon & Blume, 1994). However, in Tasks 1, 2, 3 and 5 the use of such property is implicit in the justifications and procedures used and PSTs do not present major difficulties.…”

“…The diversity of situations in which area measurement can appear and the different representations used in area measurement processes allow us to infer an underlying complexity (Caviedes et al, 2021). This complexity is associated with specific PSTs difficulties, which are related to a scarce variety of procedures for solving area tasks and a tendency to the use of formulas (Baturo & Nason, 1996;Caviedes et al, 2022a;Simon & Blume, 1994); to the lack of acquisition of geometric properties and principles ; to the lack of understanding of units of measurement (Chamberlin & Candelaria, 2018); and to the relationship that exists between perimeter and area (Livy et al, 2012). For example, Hong and Runnalls (2020) report that PSTs present difficulties in accepting area conservation in non-prototypical figures, since lacking numerical values to compare areas of triangles equivalent in area, but different in shape, they justify their answers based on visual estimation.…”

Investigation of the Specialized Content Knowledge of the Pre-service Primary School Teachers about Multiplication

“…The rectangular model supports pre-service teachers' understanding of the multiplication algorithm when conceptualizing multipliers as 'partial' products (Whitacre & Rumsey, 2020). Simon and Blume (1994) investigated prospective teachers' interpretations of multiplication on the rectangular model. They found that the interpretations of the prospective teachers were limited in presenting justification for the relationship between the equal groups meaning of multiplication and length × width.…”

The Analysis of a Novice Primary Teacher's Mathematical Knowledge in Teaching: Area Measurement

“…In the same way, the National Mathematics Advisory Panel (2008) states that teachers must know both the mathematical content that they are teaching and how this content is related to previous and future topics. Although the mathematical knowledge in teaching on different topics has been the subject of an increasing amount of research (Ball, 1990;Enochs & Gabel, 1984;Even, 1993;Even & Tirosh, 1995;Huang & Kulm, 2012;Isiksal, 2006;Roche & Clarke, 2012;Tekin-Sitrava, 2014), little research on area measurement has been conducted in Turkey and other countries (Baturo & Nason, 1996;Menon, 1998;Murphy, 2010;Simon & Blume, 1994;Tierney, Boyd, & Davis, 1990). In addition, when these studies were examined in detail, it was seen that they were conducted with pre-service teachers.…”