To understand misconceptions with rational numbers (i.e., fractions, decimals, and percentages), we administered an assessment of rational numbers to 331 undergraduate students from a 4-year university. The assessment included 41 items categorized as measuring foundational understanding, calculations, or word problems. We coded each student's…

This report describes the test performances of 26 students who displayed a variety of erroneous rules in completing a 48-item test of fraction addition problems, constructed to diagnose at least 88 projected erroneous rules. Cases were selected for their potential usefulness in designing and implementing an error-diagnostic testing system and in…

Four different conceptions of error-making from a pedagogical perspective are developed. Using errors as springboards for learning is advocated, so that better understanding of content and of the nature of mathematics results. (MNS)

This report illustrates procedures of item construction for addition and subtraction examples involving decimal fractions. Using a procedural network of skills required to solve such examples, an item characteristic matrix of skills analysis was developed to describe the characteristics of the content domain by projected student difficulties. Then…

The process of transition from a novice's state to that of an expert, in the domain of decimals, is described in terms of explicit, intermediate, and transitional rules which are consistent yet erroneous. Data from students in grades 6-9 are included. (MNS)

Results obtained from 11- and 15-year-olds on tests of understanding of decimal place value and its influence on addition and subtraction skills are reported and discussed. (MNS)

The Concepts in Secondary Mathematics and Science (CSMS) and Strategies and Errors in Secondary Mathematics (SESM) research projects based at Chelsa College, England, have shown the marked reluctance of secondary school students to use fractions when solving mathematical problems, even though they have been taught the topic for a number of years.…

This report illustrates a network of procedures which can be used to solve problems involving the addition and subtraction of fractions. This network, which is based on a skills hierarchy, is used to classify seven levels of student competency. The determination of student competency depends upon the careful construction of error-diagnostic tests.…

A theory of how people learn certain procedural skills is presented. It is based on the idea that the teaching and learning that goes on in a classroom is like an ordinary conversation. The speaker (teacher) compresses a non-linear knowledge structure (the target procedure) into a linear sequence of utterances (lessons). The listener (student)…