When presented with a problem in mathematics class, students often function as problem performers rather than problem solvers (Rigelman 2007). That is, rather than understanding the problem, students focus on using an operation to complete it. Students' tendencies to act as problem performers can prevent them from suggesting problem-solving…

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…

In a classroom teaching experiment, we explored fifth graders' ability to identify relationships in and between two patterns--what we may refer to as functional relationships. Conventional functions curricula are dominated by defining relationships between two patterns via a rule, describing how to find y or f(x) given a particular value for x…

Measurement is an important component of K-grade 12 mathematics curricula. The concepts of area and perimeter of polygons are first introduced in third grade and serve as the basis for teaching in the upper grades. Without a strong understanding of measurement, students will struggle to meaningfully grasp three-dimensional measurement concepts…

An operational understanding of the equal sign can hinder learning its relational meaning. After providing examples that illustrate the folly of expecting students who know the operational meaning of the equal sign to intuit the sign's relational meaning, this article provides teachers with strategies that will help them effectively introduce…

The Leading for Mathematical Proficiency (LMP) Framework (Bay-Williams et al.) has three components: (1) The Standards for Mathematical Practice; (2) Shifts in classroom practice; and (3) Teaching skills. This article briefly describes each component of the LMP framework and then focuses more in depth on the second component, the shifts in…

The advance of technology has caused many educators to question the time and energy expended for students to master the pencil-and-paper computation skills embodied in the long-division algorithm. In today's world, this mastery is truly a questionable goal. But understanding the conceptual infrastructure of the algorithm will add to students…

English Language Learners (ELLs) have an increasing presence in the nations' classrooms. From 1991 to 2004 the number of ELLs nearly doubled in grades K-12. Furthermore, new forms of accountability have school personnel concerned with the progress--or lack of progress--of the ELLs. "Principles and Standards" also reflects this concern: The Equity…

Effectively teaching mathematics to speakers of other languages requires teachers to recognize, validate, honor, and support the math that these students have already learned before entering a U.S. classroom. To do so, they must become aware of procedures, types of math instruction, and students' current performance levels. They should be…

Seeking more effective mathematics instruction, this author decided to incorporate Cognitively Guided Instruction (CGI) into first-grade classroom lessons. Students in CGI mathematics classrooms are prompted to use their prior knowledge to solve new problems, establish cognitive structures to which new learning can be connected, and be driven by…

Presents an approach for teaching the multiplication of fractions by helping students build a foundation on the basis of their informal knowledge. Suggests some action research ideas for teachers to try in their classrooms. (ASK)