Merlyn J. Behr

IDENTIFYING FRACTIONS ON NUMBER LINES

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Journal for Research in Mathematics Education.

Cognitive Restructuring Ability, Teacher Guidance, and Perceptual Distracter Tasks: An Aptitude-Treatment Interaction Study.

The aptitude-treatment interaction (ATI) study reported here explored the relationship between cognitive restructuring ability, as measured by the Group Embedded Figures Test (GEFT), and treatments varying in amounts of teacher guidance. It specifically investigated how these two variables affected performance on rational number tasks involving perceptual distracters. Perceptual distracter problems evolved from the work of the Rational Number Project (Behr & Post, 1981). Such problems contain visualstimulus information inconsistent with the demands of the task that must

Interaction Between Structure of Intellect Factors and Two Methods of Presenting Concepts of Logic.

The study reported is directly related to the work of Eastman (1975), which was a continuation of work begun by Carry (1968) and Webb (1971). Eastman obtained a significant interaction when the criterion measure of transfer was regressed on factors of general reasoning and spatial visualization. The Necessary Arithmetic Operations test and the Abstract Reasoning test of the Differential Aptitude Test Battery were used to measure these aptitudes, respectively. Two treatments, one graphical and the other analytical, were used to present methods for finding solution sets of quadratic inequalities. In the graphical treatment, concepts were presented in a symbolic-deductive mode. Eastman observed that factor analytic studies indicated that the measure for spatial visualization used by Carry and Webb (paper folding) had factor loadings on deduction. Conjecturing that the problem of deductive versus inductive presentation of the learning material might have been a confounding variable in the studies by Carry and by Webb, Eastman selected a different measure for spatial visualization, revised the instructional treatments to reflect a figural-inductive mode and a symbolic-deductive mode of presentation, and was successful in isolating an interaction. The present study was an attempt to generalize the results of Eastman to another mathematical content area. Two instructional treatments were written by the authors to present basic logical inference patterns and common logical fallacies. Three inference patterns were presented-modus ponens, modus tollens, and the law of hypothetical syllogism. The two treatments can be characterized as symbolic-deductive and figural-inductive. In the figural-inductive treatment the concept of Euler diagrams was introduced and used to determine whether a conclusion was possible according to one of the three inference patterns. The use of Euler diagrams reflects the figural nature of the treatment. The inductive nature of the treatment was reflected by the fact that the subjects learned to analyze premises by numerous examples without being given-either symbolically or verbally-a statement of the inference patterns. By contrast, the deductive nature of the other treatment was reflected by the fact that the inference patterns were given to subjects as rules prior to any exemplification of the rules, and were given in symbolic form. The symbolic nature of the treat

Development and Validation of Two Cognitive Preference Scales.

This paper reports the development of two scales to measure the cognitive preference of learners and a comparison of three populations based on these scale scores. For each scale, items were written which present mathematical concepts in two modes. The two modes were figural and symbolic for the FS scale, and inductive and deductive for the ID scale. The FS and ID scales were developed to determine whether learners’ cognitive preference was figural or symbolic, and inductive or deductive, respectively. The scales were administered to university freshmen and university senior preservice and inservice elementary school teachers. The scales exhibited high reliability, a good range of scores, and did discriminate between subjects. Significant differences were found among the three groups on each of the two scales. The discussion considers the possibility that a methods of teaching mathematics course contributed to the university seniors’ higher figural and inductive preference.

THE SYSTEM OF WHOLE NUMBERS

This chapter discusses the system of whole numbers. The concept of number develops from the repeated associations with collections of real objects, that is, collections of blocks, stones, and fingers. Every set that is an element of a given class is called a representative of that class or is said to represent the class. If equals are multiplied to equals, the results are equal. A whole number, x , is divisible by two only if the number named by the ones digit in the Hindu–Arabic numeral for x is divisible by two. A whole number, p , is prime only if it has exactly two factors. There is not a largest prime whole number; that is, the set of prime whole numbers is infinite. A pair of whole numbers can have many common multiples. A method for computing the greatest common factor of a pair of whole numbers that involves repeated application of the division algorithm is called the Euclidean Algorithm.

THE SYSTEM OF RATIONAL NUMBERS

This chapter discusses the system of rational numbers. Every equation of the form a + x = b , where a and b are elements of the system and x is a variable whose domain is that system, has a nonempty solution set. Every element of the system has an inverse in regard to addition. Construction of the rational-number system employs the system of integers exactly as the construction of the fractional numbers employed the system of whole numbers. Every rational number has an inverse in regard to addition and every nonzero rational number has an inverse in regard to multiplication; this is the property that distinguishes the system of rational numbers from the other number systems. The system of integers is isomorphic to a subsystem of the rationals, that is, the integers are embedded in the rational numbers as the whole numbers are embedded in the fractional numbers.

THE SYSTEM OF INTEGERS

This chapter discusses the system of integers and describes the need for a number system that is more extensive than the whole-number system and different from the fractional numbers. The system of whole numbers can be used in the construction of the system of integers. The construction of the integers parallels to some extent the construction of the whole numbers, and more so, the construction of the fractional numbers. A similar construction can be used to construct a more extensive number system than the integers. Every element of an equivalence class is called a representative of that class. If an integer and its inverse in regard to addition are added, the result is the identity in regard to addition. An important function on the set of integers, called the absolute value function, maps every integer onto a nonnegative integer.

THE SYSTEM OF FRACTIONAL NUMBERS

This chapter discusses the system of fractional numbers. A distinguished-partition is identified by, or associated with, an ordered pair of whole numbers. To determine the multiplicative inverse of a given nonzero fractional number, the fractional number is determined, such that the product of it and the given fractional number is the identity for multiplication. The sum or product of two fractional numbers can be determined by choosing any element or representative of each of the fractional numbers, and then determining the sum or product of these representatives. Fractions are used to determine sums and products of fractional numbers in a way that is analogous to the way finite sets are used to determine sums and products of whole numbers. The chapter describes an isomorphism between the system of whole numbers and a subsystem of the fractional numbers. The system of whole numbers is isomorphic to a subsystem of the system of fractional numbers.

SETS, RELATIONS, FUNCTIONS, AND OPERATIONS

This chapter discusses sets, relations, functions, and operations. The modern approach to the study of mathematics is to concentrate on mathematics as a system or structure, or as a collection of substructures. Four essential components are identified in a mathematical or logical system or structure; they are undefined terms, definitions, postulates, and theorems. The role of postulates is fundamental in a mathematical system. Postulates are assumptions that provide the foundation for the structure. The role and the use of postulates are not unique to mathematics. A one-to-one correspondence between two sets A and B is a pairing of elements of A and B, such that with every element of A exactly one element of B is paired, and with every element of B, exactly one element of A is paired. If two sets are equal, they are equivalent. Equivalent sets need not be equal. A relation is a set of ordered pairs. The set of first components is called the domain of the relation. The set of second components is called the range.

DECIMAL NUMERALS, INTRODUCTION: REAL NUMBERS

This chapter discusses decimal numerals and real numbers, and describes the construction of a number system that is more extensive than the system of rational numbers. The set of fractional numbers that can be named with terminating decimals is a subset of the set of fractional numbers. The number of decimal places in the decimal numeral of the product of two numbers is the sum of the number of decimal places in the decimal numerals for the two factors. The sum, difference, and product of two numbers, named by terminating decimals, is also a number that has a terminating decimal name. To every fractional number, there corresponds a unique repeating decimal. There is a one-to-one correspondence between the set of fractional numbers and a set of repeating decimal numerals. The set of irrational numbers can be characterized as the set of numbers that can be named by nonrepeating decimals.