Language-responsive support for multiplicative thinking as unitizing: results of an intervention study in the second grade

Abstract

Multiplicative thinking involves the ability to coordinate bundled units on a more abstract level than additive thinking and implies the identification of the different meanings of the multiplier and the multiplicand. The transition from additive to multiplicative thinking, however, constitutes an obstacle for many children. Specific formulations that are typically used in classroom discourse for talking about multiplicative tasks and situations (e.g., ‘3 times 4’ or ‘3 lots of 4’) might inhibit meaning-making processes because they do not address the idea of unitizing. A language-responsive introduction to multiplication that addresses the core idea of unitizing and that uses phrases such as ‘3 times 4 means you have 3 fours’ may help to overcome these problems. In the study presented in this paper, three second grade primary school teachers joined a teacher program to introduce multiplication in their classes (n = 66) by addressing meaning-making phrases. Another 58 second graders taught by teachers without this teacher program served as the control group. A specially developed multiplication test gave insight into the children’s understanding of multiplication as unitizing immediately after the intervention (posttest) and nearly three months later (follow-up test). We found significant differences between the intervention and control groups in the multiplication posttest. These differences could be underlined in the follow-up test. Our results indicate that a language-responsive teaching intervention that focuses on meaning-making processes can lead to long-term insights and help to develop multiplicative thinking as unitizing.

Untersuchung der Effekte spezifischer Designelemente in Online-Fortbildungen: Am Beispiel von Systematisierungsvideos für die Entwicklung fachdidaktischen Wissens

“…The key meaning of multiplication is counting in units, in other words, 3 × 5 is to be interpreted as three units of five each (Götze and Baiker 2021) in graphical representations such as dot arrays or as jumps on the number line (Barmby and Milinkovic 2011). The PD session aimed at enabling teachers to unpack and notice the lack of deep understanding in students' incomplete explanations such as "the dot array matches 3 × 5 because here is 3 and here is 5."…”

“…The PD session aimed at enabling teachers to unpack and notice the lack of deep understanding in students' incomplete explanations such as "the dot array matches 3 × 5 because here is 3 and here is 5." Classroom interventions in which the unit structure is repeatedly articulated by teachers and students (in phrases such as "the dot array consists of three fives" or "three rows of five each") have been shown to lead to significantly higher learning gains than interventions with K the same tasks and graphical representations, but without meaning-related phrases articulated (Götze and Baiker 2021). This means that explicating the unit structure is a crucial part of effective instructional strategies for enhancing students understanding of multiplication.…”

Conjecturing is not all

“…In the PD sessions preparing the teaching experiment for enhancing students' understanding of the meaning of multiplication, Estelle was attentive to the unpacking of this overall learning goal into its constituent content elements: the counting in units and the language for explaining how the dot array is connected to the multiplication by "three sets of fives" (Götze & Baiker, 2021). When reporting from her classroom trials later on, Estelle reported that she could also enhance students' learning progress in these detailed content elements, so that learning progress in these unpacked content elements served her as fine-grained success categories.…”

What is mathematics teaching talk for? A response based on three sites of practice in mathematics education

“…The studies covered by these surveys (e.g., Ernst-Slavit & Mason, 2011;Götze & Baiker, 2021) have made considerable advances in our learning about language-responsive content teaching. We would add, however, that in both these surveys, the inherent tensions in the work around 'modelling the mathematics register' or 'acting as a language model' in mathematics teaching talk are underscored, and we suggest that this is a function of what is prioritised and focused on across the articles reviewed.…”