This case study investigated the relationship between five undergraduate interior design students' reasoning with numerical units and reasoning about length and area. The multiplicative concepts frame participants' coordination of numerical units. Differences were found between participants' reasoning about length and area, based on their…

Structural reasoning is a combined ability to "look for structures, recognize structures, probe into structures, act upon structures, reason in terms of general structures, and see how a piece of knowledge acquired resolves a perturbation experienced" (Harel and Soto, 2017). The purpose of this study was to explore the cognitive…

Whole number computations are a critical skill that serves as a foundation upon which higher-order concepts in mathematics are taught to children. To facilitate their instruction, educators often use multiple representations to support a child's cognition. Representations with physical manipulatives are widely studied through a graduated…

Three iterative, 18-episode design experiments were conducted after school with groups of 6-9 middle school students to understand how to differentiate mathematics instruction. Prior research on differentiating instruction (DI) and hypothetical learning trajectories guided the instruction. As the experiments proceeded, this definition of DI…

Multiplicative thinking is widely accepted as a critically important 'big idea' of mathematics that underpins much mathematical understanding beyond the primary years. It is therefore important to ensure not only that children can think multiplicatively, but that they can do so at a conceptual rather than procedural level. This paper reports on a…

Developing multiplicative reasoning is an important milestone for elementary school students, which influences their learning of later mathematical concepts (Hackenberg and Tillema 2009). For children to conceptually understand multiplication, one should move beyond merely counting by ones to dealing with composites (twos, fives, etc.) and other…

The number sequences describe a hierarchy of students' concepts of number. This research uses two defining cognitive structures of the number sequences--units coordination and the splitting operation--to model middle-grades students' abilities to write linear equations representing the multiplicative relationship between two unknowns. Results…

This small study sought to determine students' knowledge of multiplication and division and whether they are able to use sets of bundling sticks to demonstrate their knowledge. Manipulatives are widely used in primary and some middle school classrooms, and can assist children to connect multiplicative concepts to physical representations.…

We address the question: How can a student's conceptual transition, from attending only to singleton units (1s) given in multiplicative situations to distinguishing composite units made of such 1s, be explained? We analyze a case study of one fourth grader (Adam, a pseudonym) during the course of a video recorded cognitive interview. Adam's case…

Over the course of her teaching career, this author learned to create environments in which both the teacher and learners embrace understanding. She introduced new concepts with a general question or word problem and encouraged students to find solutions with a strategy of their choice. By using this instructional method, she allowed her students…

The author tells the story of an exploration he undertook, what he learned, and the questions he was able to answer as a result. Thinking, on the part of the learner, is complex, far from explicit, and some might say intangible. But, by recognising certain "clues" it might be possible to begin to understand how different types of thinking…

Many children in the U.S. initially come to understand the equal sign operationally, as a symbol meaning "add up the numbers" rather than relationally, as an indication that the two sides of an equation share a common value. According to a change-resistance account (McNeil & Alibali, 2005b), children's operational ways of thinking…

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…

In an 8-month teaching experiment, I investigated how 4 sixth-grade students reasoned with reversible multiplicative relationships. One type of problem involved a known quantity that was a whole number multiple of an unknown quantity, and students were asked to determine the value of the unknown quantity. To solve these problems, students needed…

This study of fourth-grade students and teachers explores mathematics teaching and learning that focuses on discovering and modeling algebraic relationships. The study has two parts: an investigation of how students learn to construct algebraic statements and models for comparisons and measurement situations in the multiplicative domain, and an…