The Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) outlines ambitious goals for fraction learning, starting in third grade, that include the use of the number line model. Understanding and constructing fractions on a number line are particularly complex tasks. The current work of the authors centers on ways to successfully…

Tasks that ask students to label rational number points on a number line are common not only in curricula in the upper elementary school grades but also on state assessments. Such tasks target foundational rational number concepts: A fraction (or a decimal) is more than a shaded part of an area, a part of a pizza, or a representation using…

A steady stream of research has shown that many elementary school teachers have weak mathematical knowledge in some areas, including place value and fractions. Since a teacher's mathematical knowledge affects their students' performance, improving elementary school teachers' knowledge is critical. A better understanding of the mathematical…

Using Logo to work through a maze is described. Suggested programs to help students learn to debug programs are also included, along with the story of the original "computer bug." (MNS)

Assessment data and other research findings on fractions are briefly reviewed. These indicate that students are able to add and subtract fractions but have substantial difficulty finding and using common denominators. Selected results for nine-year-olds from the first and second national assessments of mathematics are used to portray what students…

This study involved 179 9-year-olds and 352 13-year-olds in New Zealand. Four addition and four subtraction tasks were set to ascertain students' ability to use the number line. Comparison of these data with results from the national assessment in the United States are given. (MNS)

Fourth-grade students in Title I mathematics took 40-problem tests to determine types of persistent errors made in addition and subtraction. Tests, administered to 117 students (addition) and 104 students (subtraction), were analyzed for basic fact, random, or procedural errors. Findings are reported and discussed, and recommendations for…

The informal subtraction strategies that children develop are discussed in detail, with the role of the counting-down strategy described. Problems and remedies for difficulties caused by counting backward and the double count are also presented. (MNS)

Diagnostic evaluation is a major way to help students overcome errors. How such evaluation can be done is discussed, with comments on what should be checked and recorded by the teacher. (MNS)

Some possible reasons why remediation is needed in mathematics instruction are presented. The need to diagnose the problem and then provide appropriate, effective remediation is discussed. Four guiding principles for diagnosis are presented, followed by six questions covering hypotheses about the source of the difficulty. Suggestions are then…

The need for teachers to be proficient in diagnosing pupil difficulty in mathematics is advocated. Some facets of student behavior to be observed by the teacher are noted, including reversals, lack of retention, and lack of differentiation. All are characteristics associated with dyslexia. Problems of dyslexic pupils are noted and some general…

Some research about students with learning disabilities that textbook publishers need to consider as they revise problem-solving materials is briefly cited. (MNS)

Examines a variety of mistakes found in published materials and practiced by classroom teachers. Provides examples of mistakes in reading time, place value, price notation, geometry, and language. (YP)

Students make errors in mathematics for a variety of reasons. It is the job of the educator to identify the source of these errors and to assist students in correcting them. The purpose of this monograph is to help identify, analyze, and evaluate students' computational proficiency. This monograph is divided into four sections: "Whole Numbers";…

Ten crucial mathematical concepts with which errors are made are listed, with methods used to teach them to high school students. The concepts concern order, place values, inverse operations, multiplication and division, remainders, identity elements, fractions, conversions, decimal points, and percentages. (MNS)