In their recent article on teachers' proportional reasoning, Copur-Gencturk et al. (2022) draw attention to a type of strategy that they call "relative", lodged right between additive and multiplicative thinking. This strategy raised interest in our research team, as it aligned well and helped give stronger meaning to some strategies…

Mathematical writing is one way for primary students to communicate their mathematical thinking. Research in the field of writing has shown that to become an effective teacher of writing, preservice teachers must have experience engaging in the kinds of writing given to their students. The study reported in this paper explored how 27 preservice…

This article examines the mathematics learning experiences in proportional reasoning I created for pre-service teachers in my teacher education course in Canada. The approach aimed to enable pre-service teachers to work on and with their prior mathematics understandings, make connections that were particular to what emerged during group work, and…

Ginsburg (1977) observed that children typically develop surprisingly powerful informal (everyday) knowledge of mathematics and that mathematical learning difficulties often arise when formal instruction does not build on this existing knowledge. By using meaningful analogies teachers can help connect new formal instruction to students' existing…

The study reported on in this chapter describes the justifications that elementary and middle school prospective teachers (PTs) made as they examined the temperature story that a Grade 5 student posed for an integer subtraction number sentence. The ways that the PTs made sense of the student's story that used integer subtraction as distance are…

Whole number arithmetic is the foundation of higher mathematics and a core part of elementary mathematics. Awareness of pattern and underlying problem structure promote the learning of whole number arithmetic. A growing consensus has emerged on the necessity to provide students with the opportunity to engage in algebraic reasoning earlier in their…

An important decision that professional development (PD) facilitators must make when preparing for activities with teachers is to select an appropriate tool for the intended learning goals of the PD (Sztajn, Borko, & Smith, 2017). One important and prevalent tool is artifacts of student thinking (e.g. Jacobs & Philipp, 2004). In this paper…

Both additive and proportional reasoning are types of quantitative analogical (QA) reasoning. We investigated the development and nature of primary school children's QA reasoning by offering two missing-value word problems to 3rd to 6th graders. In one problem, ratios between given numbers were integer, in the other ratios were non-integer. These…

Before starting school, many children reason logically about concepts that are basic to their later mathematical learning. We describe a measure of quantitative reasoning that was administered to children at school entry (mean age 5.8 years) and accounted for more variance in a mathematical attainment test than general cognitive ability 16 months…

The study addresses the relational reasoning of different-aged children and how addition reasoning is related to problem-solving skills within addition and to reasoning skills outside addition. Ninety-two 5- to 8-year-olds were asked to solve a series of conceptually related and unrelated addition problems, and the speed and accuracy of all…

Reports on two studies that use new tasks to compare English children's use of strategies that reverse the order of addends in solving addition problems. Shows that knowledge of commutativity among young children is widespread, but does not establish a direct link between this knowledge and children's choice of addition strategies. (DSK)

On the basis of implications drawn from Piagetian theory, an approach to first-grade arithmetic which eliminated instruction and used only two kinds of activities (situations in daily living and group games) was implemented in an experimental context. Situations in daily living and group games, in contrast to "mechanical" worksheet…

This study investigated the relationships between Piagetian logical reasoning abilities with an information processing capacity, and first-grade children's performance on verbal addition and subtraction problems. An analysis of simple arithmetic problems indicated that several reasoning abilities identified by Piaget are needed to achieve…

Describes the use of spreadsheets in a third grade class to teach basic mathematical concepts by investigating the existence of vampires. Incorporates addition and multiplication skills, patterning, variables, formulas, exponential growth, and proof by contradiction. (LRW)

An editorial and abstracts for 12 research reports are contained in this issue. The editorial by Robert E. Reys focuses on the dangers of publishing poor research. The abstracts, each with a critique, concern research on a game for logical reasoning, wait-time and sex differences, vocabulary instruction on ratio and proportion, male-female…