Effective mathematics teaching practices include not only probing student thinking and selecting student strategies to share but also facilitating discourse and encouraging reflection about important mathematical ideas (National Council of Teachers of Mathematics [NCTM], 2014, pp. 36-37). Encouraging students to use precise academic vocabulary to…

There has been a consensus that students' conceptual understanding of mathematical operations (such as multiplication) can be developed through communication about multiple representations. However, learning opportunities have often appeared to be limited to surface translations (in which only obvious similarities such as numbers have been…

Students encounter mathematics word problems as early as kindergarten and continue to see them throughout their schooling experience. Schema instruction with an attack strategy can support students to successfully navigate word-problem solving. Schemas help students categorize word problems by similar characteristics. To better support students…

This article explores the role that gestures play in the development of mathematical understanding. Using Pirie Kieren's (1994) notion of image making and Lunney Borden's (2011) idea of verbing mathematics, we share two examples of how students respond to teacher requests to demonstrate what they know about arrays.

The importance of fostering in students the requisite language to understand what is being communicated and how to communicate their understanding requires educators to conceptualize themselves as teachers of language and content. It is possible to engage in activities of the mathematics classroom and through that participation engage in language…

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…

In the last decades, many research projects have indicated that, especially for children with mathematical difficulties, procedural training programs seem to be very helpful in promoting learning of multiplicative routines such as the times table. However, there is less knowledge about whether and how a conceptual understanding of multiplication…

Teaching simulations such as scriptwriting tasks offer a novel way to support pre-service teachers in gaining diagnostic competencies and in guiding students to work through difficulties. We first report on the theory-driven development of a scriptwriting task, specifically the theoretically-founded language- and accompanying content-related…

This study explores components of ambitious teaching practices for teaching multiplicative properties that can give teachers opportunities to learn through a quick images activity in a cycle of enactment and investigation. Throughout the cycle, it is found that the participants have opportunities to learn the following components of ambitious…

The debate on how to understand such expressions of multiplication entailing "times" as "n times more than" and "increase (by) n times" has been on and off in China since the 1980s. A review of literature seems to suggest that despite early-stage divergence in understanding, there is a general consensus among the…

As today's classrooms become more and more diverse, there is a growing need to explore the intersection between English Learners (ELs) and students with learning disabilities (LD) in the content-specific instruction of mathematics problem solving. The aim of this study was to determine which types of instructional scaffolds may be used by math…

The relationships between three critical elements, and the associated mathematical language, to assist students to make the critical transition from additive to multiplicative thinking are examined in this article by Chris Hurst.

In this case study, we examine the usage of language -- how teachers used and regulated their language when teaching English language learners (ELLs) with learning disabilities (LD) how to solve mathematics multiplication problems. We focus on types of scaffolds used by teachers to identify how scaffolding helps ELLs with LD build better…

In this study, we considered how middle school teachers understood the relationship between fractions and ratios. We used two instruments to collect data from 11 teachers and relied on Knowledge in Pieces as a lens for considering understandings teachers have and how coherent those understandings are. From our analysis, we developed three main…

This study examined discussions centered on precise mathematical language use in two fifth grade classrooms. Drawing on episodes from lessons in which teachers focused on encouraging mathematics reasoning, our analysis examines the relationship between precise language use and mathematical justifying. We present three classroom episodes that…