This article describes a process called "Backing Up" which is a way to preassess student understanding of a topic and gauge student readiness to move forward in the learning process. This process of backing up begins with using responses to a word problem to identify categories of students' understandings in relation to the expectations…

Understanding the decimal system is challenging, requiring coordination of place-value concepts with features of whole-number and fraction knowledge (Moloney and Stacey 1997). Moreover, the learner must discern if and how previously learned concepts and procedures apply. The process is complex, and misconceptions will naturally arise. In a…

In this article, the authors describe how their first-grade class used the book "Inch by Inch" (Lionni 1960)--the story of an inchworm that escapes from a hungry robin by proving he is useful for measuring--as inspiration for a lesson focused on exploring length measurement through examining the meaning of an inch. The authors designed a…

Based on a study with six elementary school teachers and 115 students in grades 3, 4, and 5, this article discusses ways that teachers might support an understanding of variable as students explore pattern-finding activities. The practices emerged from interviews with teachers and a yearlong observation of teaching algebraic thinking during an…

As teachers seek activities to assist students in understanding division as more than just the algorithm, they find many examples of division as fair sharing. However, teachers have few activities to engage students in a quotative (measurement) model of division. Efraim Fischbein and his colleagues (1985) defined two types of whole-number…

For students to develop meaningful conceptions of fractions and fraction operations, they need to think of fractions in terms other than as just whole-number combinations. In this article, we suggest two powerful images for thinking about fractions that move beyond whole-number reasoning. (Contains 5 figures.)

Discusses misconceptions about constructivism by identifying related myths such as students should always be actively and reflectively constructing, manipulatives make learners active, and cooperative learning is constructivist. One central goal of constructivism should be that students become autonomous and self-motivated in their learning.…