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This curriculum guide for grades K-6 is the first volume of a two-part series. It is meant to provide an ordered sequence of mathematical concepts from which teachers may organize an arithmetic program allowing for individual student progress with the greatest amount of individual attention. Each topic is arranged into levels based on the topic's content and not necessarily by grade level. Each level contains the following general categories: Concepts, Behavioral Objectives, References and Resources. The objectives are matched with textbooks referenced by pages and with specific resource materials to be used in the instruction. A list of activities that may be used for instruction is also provided at the end of each level. Topics covered include numeration, place value, addition, subtraction, multiplication, division and inequalities. Also provided is a list of 54 classroom games that are directly related to the topics included in this guide. For Volume II, see SE 017 305. (JP)

This investigation sought to determine the effect of emphasizing mathematical structure in the acquisition of computational skills by seven- and eight-year-olds. The meaningful development-of-structure approach emphasized closure, commutativity, associativity, and the identity element of addition; the inverse relationship between addition and subtraction; and place value. The control groups essentially used drill-type activities. The subjects were students from a middle class elementary school in Tampa, Florida who had the poorest performances on pretests. Analysis of covariance was used to analyze the data; results favored the experimental group. (Author/LS)

The slow learner is defined as a student whose IQ falls between 75 and 90 or who ranks below the thirtieth percentile of the student population in mathematical achievement. His characteristics and special needs are discussed in some detail. A presentation of selected methods and techniques for instructing slow learners to add and subtract non-decimal fractions follows. Behavioral objectives, specific topics, and an estimated schedule are given. Resources required for this instruction are described. The strategy employs concrete examples to the maximum extent possible. The laboratory approach and discovery methods are recommended. The nature and use of a particular teaching aid are discussed. The emphasis in evaluation is on each student's progress. (KM)

Two sequences of activities were developed to provide instruction on the algorithms for addition and subtraction of two-digit numbers. In the integrated sequence (I) the mechanics of "carrying" and "borrowing" were treated as a single process "regrouping." In the sequential treatment (S) the addition algorithm was developed before the subtraction algorithm. Students of two second-grade classes were randomly reassigned to either group S or group I. Profiles were generated by item sampling. Group means were estimated for addition, subtraction and total performance every three days. Also, on the eighteenth day all children were administered a 20-item achievement test. Overall differences in group performance were not significant. Some differences in performance on operations at specific times were significant and favored group S. (Author/MM)

This study explored the effects of visual (vertical and horizontal) and oral presentation modes upon simple mathematical computations (addition, subtraction, and multiplication). Seventy-two undergraduate education majors were employed as subjects. The placement of the process sign (left, middle, right) and whether a one or two digit number appeared first in the mathematical sentence was manipulated. The results demonstrated significant differences for modality, type of computation, sign and two-digit placement. Further, it appears that for the oral presentation, the process sign placed last is superior to the first position. (Author/MM)

Two sequences of activities were developed to provide instruction on the algorithms for addition and subtraction of two-digit numbers. In the integrated sequence (I) the mechanics of "carrying" and "borrowing" were treated as a single process "regrouping." In the sequential treatment (S) the addition algorithm was developed before the subtraction algorithm. Students of two second-grade classes were randomly reassigned to either group S or group I. Profiles were generated by item sampling. Group means were estimated for addition, subtraction and total performance every three days. Also, on the eighteenth day all children were administered a 20-item achievement test. Overall differences in group performance were not significant. Some differences in performance on operations at specific times were significant and favored group S. (A slightly different prepublication draft of this document was announced as ED 062 195.) (Author/MM)

Viewing the number system as a complex of interrelated words, three studies are described: (1) a study of the difficulty of simple addition and subtraction in different decimal positions; (2) a study of variations in the number of digits in the addend or minuend; (3) a study of the effect of irregular morphemes occurring in some number words. Findings proposed were (1) arithmetic tasks involve sequences of operations of reading and arithmetic called shifts; (2) different arithmetic performances require different shifts and thus some performances require more time to complete. (JG)

This teacher guide is part of the materials prepared for an individualized program for ninth-grade algebra and basic mathematics students. Materials written for the program are to be used with audiovisual lessons recorded on tape cassettes. For an evaluation of the program, see ED 086 545. In this guide, the teacher is provided with objectives for each topic area and guided to materials written for a given topic. Three short criterion tests are included for each topic covered. The work in this package covers addition and subtraction of simple and mixed fractions, multiplication and division of mixed fractions and work on determining the relative size of two fractions. This work was prepared under an ESEA Title III contract. (JP)

This teacher guide is part of the materials prepared for an individualized program for ninth-grade algebra and basic mathematics students. Materials written for the program are to be used with audiovisual lessons recorded on tape cassettes. For an evaluation of the program, see ED 086 545. In this guide, the teacher is provided with objectives for each topic area and guided to materials written for a given topic. Three short criterion tests are included for each topic covered. The work in this package provides practice with addition and subtraction of whole numbers, reviews the commutative and associative properties for addition and provides work on estimation of solutions to problems. This work was prepared under an ESEA Title III contract. (JP)

This teacher guide is part of the materials prepared for an individualized program for ninth-grade algebra and basic mathematics students. Materials written for the program are to be used with audiovisual lessons recorded on tape cassettes. For an evaluation of the program, see ED 086 545. In this guide, the teacher is provided with objectives for each topic area and guided to materials written for a given topic. Three short criterion tests are included for each topic covered. The work in this package reviews writing fractions as decimal numbers and presents problems on addition and subtraction of decimal numbers. This work was prepared under an ESEA Title III contract. (JP)

Material included in this bulletin provides an overview of research studies which pertain to teaching addition and subtraction of whole numbers in the primary grades. The studies mentioned relate directly to the following questions: (1) What foundation for addition and subtraction do children have upon entering school? (2) What is the relative difficulty of addition and subtraction facts? (3) Should addition and subtraction be introduced at the same time? (4) What type of problem situation should be used for introductory work with subtraction? (5) How can number facts be taught effectively? (6) How should subtraction with renaming be taught? (7) What is the role of drill in teaching addition and subtraction? A list of selected references is supplied at the end of the paper. (FL)

In the winter of 1965, an experimental course in Elementary Number Theory was presented to a 6th grade class in the Hosmer School, Watertown, Massachusetts. Prior to the introduction of the present material, students had been exposed in class to such topics from the University of Illinois Arithmetic Project as lattices, number lines, frame equations, and linear affine transformations. The present materials are concerned with such mathematical concepts as (1) fundamental operations involving integers, (2) division of integers which included remainders, factorization, and the Sieve of Eratosthenes, and (3) number systems in bases 2, 7, 10, and 12. Teacher and student materials that were used for a period of 14 weeks are included. [Not available in hardcopy due to marginal legibility of original document.] (RP)

This programed mathematics textbook is for student use in vocational education courses. It was developed as part of a programed series covering 21 mathematical competencies which were identified by university researchers through task analysis of several occupational clusters. The development of a sequential content structure was also based on these mathematics competencies. After completion of this program the student should be able to: (1) know that "difference" indicates the operation of subtraction, (2) order any set of fractions, (3) subtract a small fraction of the form a/b where 0 is less than (a,b) when these are less than 100 from a larger fraction with the same denominator, (4) subtract a small fraction of the form a/b, where 0 is less than (a,b) and these are less than 100, from a larger fraction of the same form with unlike denominators, (5) subtract two literal fractions with common denominators, (6) subtract two literal fractions with unlike denominators, and (7) subtract a small mixed number from a larger one of the form Xa/b where 0 is less than (X,a,b) when these are less than 100. The material is to be used by individual students under teacher supervision. Twenty-six other programed texts and an introductory volume are available as VT 006 882-VT 006 909, and VT 006 975. (EM)

This booklet, one of a series, has been developed for the project, A Program for Mathematically Underdeveloped Pupils. A project team, including inservice teachers, is being used to write and develop the materials for this program. The materials developed in this booklet include (1) addition, subtraction, and multiplication in modulo seven (related to the days of the week), (2) congruency and equivalence classes, and (3) some basic properties of whole numbers for the indicated operations in modulo seven. (RP)

This booklet, one of a series, has been developed for the project, A Program for Mathematically Underdeveloped Pupils. A project team, including inservice teachers, is being used to write and develop the materials for this program. The materials developed in this booklet include (1) the addition and subtraction of whole numbers on the number line, (2) the addition and subtraction of integers on the number line, and (3) the idea of inequality. Accompanying these booklets will be a "Teaching Strategy Booklet" which will include a description of teacher techniques, methods, suggested sequences, academic games, and suggested visual materials. (RP)

TO DETERMINE THE EFFECT OF INSTRUCTIONAL APPROACHES FOR SUBTRACTION AND/OR TIME OF SYMBOLIZATION UPON (1) ACHIEVEMENT IN ARITHMETIC, (2) ATTAINMENT OF NUMBER CONCEPTS, (3) ATTITUDES TOWARD ARITHMETIC IN GRADES ONE AND TWO, THIS REPORT ASSESSES FOUR EXPERIMENTAL CLASSES, THERE WERE TWO FIRST GRADE CLASSES OF 20 PUPILS EACH AND TWO SECOND GRADE CLASSES OF 17 PUPILS EACH. THE INSTRUCTIONAL APPROACH USED FOUR TREATMENTS (1) PPW-E (PART-PART-WHOLE WITH EARLY SYMBOLIZATION), (2) PPW-L (LATE SYMBOLIZATION), (3) TA-E (TAKE-AWAY WITH EARLY SYMBOLIZATION), (4) TA-L (LATE SYMBOLIZATION). A CONCEPT ATTAINMENT TEST WAS DESIGNED FOR USE AT THE BEGINNING AND END OF GRADE ONE. OTHER TESTS WERE ALSO ADMINISTERED BEFORE, DURING, AND AFTER GRADES ONE AND TWO. THE TESTS USED WERE THE CONCEPT ATTAINMENT TEST (BEGINNING AND END OF GRADE ONE), THE STANFORD ACHIEVEMENT TEST (END OF GRADE TWO), A SUBTRACTION APPLICATIONS AND TRANSFER TEST (END OF GRADES ONE AND TWO), LORGE-THORNDIKE INTELLIGENCE TEST (BEGINNING OF GRADE ONE), AN ARITHMETIC ATTITUDE SCALE (THREE TIMES IN EACH GRADE), SUBTRACTION FACTS TEST (END OF GRADE TWO), AND A LOGIC TEST (END OF THE STUDY). THERE WERE NINE CONCLUSIONS. (1) GRADES ONE AND TWO CAN LEARN USING PPW. (2) TA WAS MORE EFFECTIVE FOR TEACHING SUBTRACTION SKILLS AFTER GRADE ONE BUT, AFTER GRADE TWO, THE EFFECT OF TA AND PPW EQUALIZED. (3) AFTER GRADE TWO, PPW WAS SUPERIOR IN TEACHING APPLICATIONS OF SUBTRACTION. (4) AFTER GRADE TWO, PPW WAS SUPERIOR IN PROBLEMS REQUIRING TRANSFER. (5) PPW WAS SUPERIOR IN RELATING ADDITION AND SUBTRACTION. PPW FOSTERED PARTITIONING. (6) TA WAS INITIALLY EASIER TO TEACH. (7) HIGHER ACHIEVEMENT FOLLOWED EARLY SYMBOLIZATION. (8) CHANGES IN ATTITUDE WERE NOT PRODUCED BY ANY OF THE FOUR TREATMENTS. (9) NO DIFFERENCES WERE ATTRIBUTABLE TO SEX. FUTURE RESEARCH USING HIGHER GRADES, LARGER NUMBERS OF STUDENTS, AND OTHER MATHEMATICAL TOPICS SHOULD BE DONE. (LG)

THIS BOOKLET, ONE OF A SERIES, HAS BEEN DEVELOPED FOR THE PROJECT, A PROGRAM FOR MATHEMATICALLY UNDERDEVELOPED PUPILS. A PROJECT TEAM, INCLUDING INSERVICE TEACHERS, IS BEING USED TO WRITE AND DEVELOP THE MATERIALS FOR THIS PROGRAM. THE MATERIALS DEVELOPED IN THIS BOOKLET INCLUDE (1) ADDITION AND SUBTRACTION WITH WHOLE NUMBERS, (2) PATTERNS AND PROCEDURES IN MULTIPLICATION, AND (3) DIVISION AS A PROCESS OF FINDING A MISSING FACTOR. ACCOMPANYING THESE BOOKLETS WILL BE A "TEACHING STRATEGY BOOKLET" WHICH WILL INCLUDE A DESCRIPTION OF TEACHER TECHNIQUES, METHODS, SUGGESTED SEQUENCES, ACADEMIC GAMES, AND SUGGESTED VISUAL MATERIALS. (RP)

THIS BOOKLET, ONE OF A SERIES, HAS BEEN DEVELOPED FOR THE PROJECT, A PROGRAM FOR MATHEMATICALLY UNDERDEVELOPED PUPILS. A PROJECT TEAM, INCLUDING INSERVICE TEACHERS, IS BEING USED TO WRITE AND DEVELOP THE MATERIALS FOR THIS PROGRAM. THE MATERIALS DEVELOPED IN THIS BOOKLET INCLUDE (1) NUMBER RELATIONSHIPS, (2) EQUIVALENT FRACTIONS, (3) ADDITION AND SUBTRACTION OF RATIONAL NUMBERS, (4) LEAST COMMON DENOMINATORS, AND (5) SUPPLEMENTARY ACTIVITIES WITH RATIONAL NUMBERS. ACCOMPANYING THESE BOOKLETS WILL BE A "TEACHING STRATEGY BOOKLET" WHICH WILL INCLUDE A DESCRIPTION OF TEACHER TECHNIQUES, METHODS, SUGGESTED SEQUENCES, ACADEMIC GAMES, AND SUGGESTED VISUAL MATERIALS. (RP)

Solution processes for simple arithmetic word problems are examined in three dimensions: the order of mention, the identity of the unknown, and the type of verb. The influence of mathematical and linguistic factors in these processes are discussed. (BJG)

Examination of models for representing integers demonstrates that formal operational thought is required for establishing the operations on integers. Advocated is the use of many models for introducing negative numbers but, apart from addition, it is recommended that operations on integers be delayed until the formal operations stage. (JP)

Reported is a study on solution methods used by third-grade students as they solve open sentences involving addition and subtraction. A list of methods that children might use is compared to a frequency tally of those methods used by a group of 32 students who solved 16 problems. (JP)

Discussed is a way of playing with numbers that leads to interesting results and provides well-disguised drill. (JP)

Three game activities are detailed. Informal work with addition and subtraction on a number line and graphing concepts are achieved with two simple dice games, and the distribution of monetary tokens is used to illustrate the division algorithm. (JP)

Four ideas are presented, each a variation of the tic-tac-toe game. Recognizing three addends is the goal of the primary level game; experiences with basic facts and fractions are objectives for upper levels. Each worksheet includes objectives, directions, and comments. (LS)

This form rates childrens' grasp of basic mathematical concepts and procedures. See TM 001 111 for details of the program in which it is used. (DLG)

The relationship between children's performance in solving open sentences and the factors of sentence type, number size, and context was investigated. The study also identified and classified the methods of solution used. Thirty-two third graders were individually given an orally administered test on open sentences. Four types of open sentences were used: N + a = b, a + N = b, a - N = b, and N - a = b, where N was the placeholder. Half of the test items included a verbal problem presented with the open sentence, while the remainder of the items gave only an open sentence. Multivariate analysis of variance was used. Results showed that open sentences of the type N - a = b were significantly more difficult than those with smaller constants. Children used an average of five different solution methods on the items. (Author/DT)

The study concerns the performances of first grade children in solving problems involving subtraction situations and the relationship of these performances to levels of conservation of numerousness and to I.Q. groups. The subjects were grouped into levels of conservation by a pretest administered prior to the main test of problem solving in subtraction. The effects of aids and transformations on the performances of the children in the problem solving test were also examined. An analysis of the results yielded the following: (1) children in the highest level of conservation of numerousness performed significantly better than those in the third and fourth levels; (2) there were significant differences among the performances of children in the three I.Q. groups; (3) there were significant differences among the twelve groups of children formed by levels of conservation and I.Q. groups; (4) the pretest of conservation of numerousness was found to be a better predictor of success in problem solving than the group I.Q. test; and (5) subtraction problems without aids or without transformations were significantly more difficult than problems with aids or with transformations. (Author/JG)

The Mathematics Evaluation Materials Package (MEMP) is a set of objectives and companion test items for mathematics education in Grades 4 to 6. MEMP was developed for classroom use and emphasizes increased teacher and principal decision-making and flexible curricular guidelines. The most significant feature in MEMP, which distinguishes it from other evaluation devices and strategies, is the variation in the patterns of objectives that it accomodates. Educational objectives that are stated in terms of student performance, and that are coupled with companion test items, provide the basic elements for measuring student achievement in a manner that is different from standardized testing and, in most cases, different from usual programs of pupil testing. MEMP can be used for designing tests covering short units of work or for constructing longer tests. It can be especially useful during changes in curricula because the various curriculum options that are open to teachers may be compared, as they are tried, in their effects on student performance. The instructional objectives with companion items, answers, and inventories that are models of the types of papers that can be prepared by teachers for evaluation purposes are provided. (Author/DB)

Interaction between pupils' cognitive styles and instructional method in effecting achievement was studied. The instructional variables were sequencing and use of manipulative materials. Ninety sixth-grade students were ranked on Sigel's Cognitive Style Test. The upper and lower 36 were defined as high and low analytic pupils, respectively. Nine high and nine low analytic pupils were randomly assigned to one of the following treatments: inductive manipulative, inductive nonmanipulative, deductive manipulative and deductive nonmanipulative. On four consecutive days, addition and subtraction of positive and negative integers were taught in 25-minute sessions. Immediate posttests of retention and transfer were administered on the fifth day. Four weeks later, delayed posttests of retention and transfer were administered. Analysis of variance provided the following results: (1) pupils taught using manipulative materials scored significantly higher (p less than .05) on the immediate retention measure than did pupils taught without manipulative materials; (2) inductive sequencing with the use of manipulative materials resulted in significantly higher (p less than .05) scores on immediate retention than did deductive sequencing with such materials; and (3) pupils identified as having low analytic cognitive styles scored significantly higher (p less than .05) on transfer measures when taught inductively rather than deductively. (Author/JP)

The purpose of this study was to investigate the effects of three instructional strategies on arithmetic achievement. The instructional strategies used in the study were drill, drill with feedback, and mixed drill. The materials were remedial in nature and were presented via computer-assisted instruction. The subjects for the study were junior high school students whose performance on diagnostic tests indicated that they lacked computational skill in addition and subtraction. An experimental and a control group were utilized, and data were obtained from the administration of pre- and post-tests. Among the findings were (1) all treatments produced significant increases in scores on the immediate post-test at the .05 level, (2) all treatment groups had significantly higher scores on both the prerequisite and terminal objectives for both addition and subtraction while the control group had no significant changes in scores, (3) the drill with feedback and control groups did not have significantly higher scores on the delayed post-tests on the prerequisite objectives, and (4) the fact that all groups, including the control group, had significantly higher scores on the delayed post-tests on the terminal objectives indicates that some effects other than those of the treatment may have been operating. (RP)

Reported are the results of a study to determine if specific errors in subtraction occur when students demonstrate ability to apply selected decimal numeration system principles. A secondary purpose was to examine and compare errors made by various subsets of the sample population characterized by grade level, arithmetic achievement, mental maturity, and mean achievement level of the school attended. Knowledge of place value and subtraction skills were measured by another-designed instruments. A positive correlation (.68) existed between performance on place value and subtraction tests. In general, high achievers and high mental maturity groups made significantly fewer (.05) errors than the corresponding low groups. Low achievers generally were unable to apply decimal numeration principles. (Author/RS)

The purpose of this study was to analyze the relationship of a child's performance in solving addition and subtraction problems using concrete embodiments compared to his performance using the symbolic mode. A random sample of 144 second grade children were tested. It was found that children proficient in regrouping addition and subtraction examples on three or four embodiments scored significantly higher on the written tests than children not proficient in using concrete materials, and significant correlations were found between the number of embodiments children were able to regroup and their performances. (Author/DT)

A BOOKLET DESIGNED FOR ELEMENTARY SCHOOL STUDENTS TO BE USED INDEPENDENTLY FROM AND IN ADDITION TO THE REGULAR CLASSROOM CURRICULUM IN MATHEMATICS IS PRESENTED. THE FIFTH- OR SIXTH-GRADE STUDENT IS GIVEN PRACTICE IN WORKING PROBLEMS IN SUBTRACTION AND DIVISION. SHORT CUTS AND HINTS ARE PRESENTED THROUGH EXAMPLES, SAMPLE PROBLEMS, AND ANSWERS. THE TEACHER IDENTIFIES POTENTIAL USERS, ORIENTS THEM TO THE AREA TO BE EXPLORED, AND STIMULATES AND MAINTAINS THEIR INTEREST IN SUSTAINED INDEPENDENT EFFORT.