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A review is presented of the role of algorithms in learning and instruction. The paper first describes an algorithm as a procedure guaranteed to produce a correct result and lists its features as: 1) having a reasonably finite number of unambiguously defined operations; 2) having zero or more inputs from a specified domain; 3) having outputs with a relationship to the inputs; and 4) having operations sufficiently simple to be completed in a finite period. The second section reviews the use of algorithms to describe cognitive processes in learning and problem-solving; and the third, their use in developing instructional strategies. The fourth part of the paper examines the use of algorithms in task analysis; and the following section, their application to instructional materials. The concluding segment discusses research issues related to the use of algorithms in learning and instruction, including the order in which the operations of and paths through algorithms are best taught, the demonstration of the interrelationships among an algorithm's components, and methods of teaching students to synthesize the steps of an algorithm, including the retrogression approach. (Author/PB)

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)

This paper describes a teaching method, presently in the experimental stage, that advocates the use of a programed instruction model in the language laboratory, with students working in homogeneous groups of three. The author describes the technical construction and operation of the language laboratory which allows three students to work together and to use a programed instructional format. The students study new grammar points in microunits. Diagrams illustrate the working of the program, and the author provides an explanation of the steps in its operation. An example of how the program works is presented. (VM)

This paper distinguishes several ways in which phonemic information might be employed in spelling; examines some empirical data on the consequences of using these information sources; formulates hypotheses about the underlying processes involved; and suggests possible implications for the teaching (and learning) of spelling. Included are tables of the phoneme code with graphemic options and examples, Comparisons of Spelling by Elementary School Pupils with the Hanna-Rudorf Algorithm, and figures depicting the spelling and reading errors by means of a Generator-Test Process. The simulation technique discussed may be used to identify the spelling processes used by individual children and the information they hold in memory, and to diagnose and prescribe treatment for individual spelling problems. (Author/LS)

This monograph focuses on the influence of meaning theory on elementary school mathematics programs and on mathematics instruction. Theories of arithmetic instruction up through 1935 are described, and the philosophy of meaning theory, contributions and discussions by mathematics educators, and applications to actual instruction are delineated for the period from 1935 through 1960. A discussion of the implicit concern for meaning in the modern mathematics movement from 1960 to the present concludes the first section. The second section summarizes studies that establish and affirm the importance of and need for meaning and studies that explore the effect of teaching various procedures with meaning. The final section of the paper briefly discusses the implementation of mathematically meaningful instruction. An extensive bibliography is included. (DT)

This guide continues instruction of the three guides in SE 012 723 and of the two guides included in SE 012 724. Counting skills, appreciation of pattern and order, and understanding of place value are extended to numbers up to one million, and the associative, commutative and distributive laws are made explicit. Manipulation of equations is stressed throughout. This section contains a systematic approach to the development of automatic response in the basic number facts, and many activities are suggested for practicing these skills. Formal algorithms are discussed at length, but the teacher is warned not to develop these until the pupil is thoroughly familiar with place value, the laws of operation, the basic number facts, and manipulation of equations. There is also further work with fractions. The booklet concludes with a glossary and a summary of the contents of Sections A to F. (MM)

Words and word fragments from the computer-readable data bases "Chemical Abstracts Condensates" and "Biological Abstracts Previews" were analyzed in terms of length, number, and frequency of appearance to determine some parameters upon which inversion of these data bases could be predicated. Types (unique words or fragments) and tokens (all appearances of types) were counted and type:token ratios calculated. A KLIC (Key-letter-in-Context) Index was also generated from each of the data bases. The paper discusses the impact of the various counts, ratios and projections on the problem of inverting the data bases for retrospective search purposes. (Author)

This study has investigated some pattern recognition capabilities of devices consisting of arrays of cooperating elements acting in parallel. The problem of recognizing straight lines in general position on the quadratic lattice has been completely solved by applying parallel acting algorithms to a special code for lines on the lattice. The relation of the code to Farey series and continued fractions and the effects on the code of a line when the line is subjected to affine transformations were studied in detail. Algorithms for reducing straight line codes to a standard form were developed and made the basis of a line recognition process. Cellular automata were designed to carry out line recognition. Other cellular automata were designed to recognize topological connectedness, detect boundaries and approximate curves by straight line segments. (Author)

This report describes the initial evaluation of a text compression algorithm against computer assisted instruction (CAI) material. A review of some concepts related to statistical text compression is followed by a detailed description of a practical text compression algorithm. A simulation of the algorithm was programed and used to obtain compression ratios for a small sample of both traditional frame-structured CAI material and a new type of information-structured CAI material. The resulting compression ratios are to 1.5: 1 for both types of material. The simulation program was modified to apply the algorithm to the lesson files of a particular frame-structured CAI subsystem used in the Air Force Phase II Base Level System. The compression in this case was found to be 1.3: 1 because some uncompressible, frame-formating bytes were present in the lesson file. The modified simulation program was also used to take letter occurrence statistics on the text being compressed. From these, a theoretical compression was calculated using a probalistic model of the compression algorithm. Theoretical compression was within two percent of measured compression, thus verifying the model's applicability. (Author)

This document contains the reports summarizing the main discussion held during the March 1972 Computational Linguistics Conference. The first report, "Computational Linguistics and Linguistics," helps to establish definitions and an understanding of the scope of computational linguistics. "Integrated Computer Systems for Language" and "Computer-Oriented Grammars and Parsing" deal with technical issues and immediate concerns for research. "Machines and Speech" reports on a topic of emerging relevance in the field; lengthy consideration is given to reading machines for the blind and speech understanding systems, among other topics. "Language Performance (Psycholinguistics and Dialectology)" describes another fairly new application for computational linguistics. The last two reports, "Social Implications of Automatic Language Processing" and "Professional Ethics, Standards, and Education," involve issues related to the potential impact and implications of computational linguistics and its concerns. (VM)

Evaluation sheets and developmental checklists, prepared by three special education departments, are provided to facilitate continuous measurement of student progress in elementary school mathematics. One hundred forty-three objectives are given, and each is followed by a continuum-oriented set of worksheet-type pages on mathematics skills. The sheets were not intended to be used for seatwork or as a replacement for a school's present program in mathematics but to provide a fast means for evaluating each child's progress. No suggestions for diagnosis are included. The objectives cover content in number concepts; all operations with whole numbers, decimals, and fractions; adding and subtracting integers; commutative, associative, and distributive laws; problems in money, time, and units of measurement; and recognition of geometric figures. (DT)

One hundred seventy-six seventh grade students underwent a recorded interview where each was given a set of computational exercises and asked to say aloud his thinking as he worked them. The most frequently used strategies in computations with whole numbers and fractions are described in detail, an analysis of the nature of wrong answers is included, and characteristics of good and poor computers are listed and discussed. Thirteen conclusions are given, covering computational strategies, vertical vs. horizontal problem arrangement, mathematical vocabulary of students, estimating answers, and the technique of using recorded interviews in research. The computation problems given to the students are included in the report, and the appendices list all the wrong answers given with the accompanying verbal description by the student. (DT)

This series of four text-workbooks was designed for tenth grade mathematics students who have exhibited lack of problem-solving skills. Electric desk calculators are to be used with the text. In the first five chapters of the series, students learn how to use the machine while reviewing basic operations with whole numbers, decimals, fractions, and percents. The rest of the chapters present word problems in simple consumer mathematics, business activities, installment buying, banking, stocks and bonds, insurance, taxes, and utilities. A chapter on the use of formulas is included. (DT)

A project was undertaken to carry out experimental and methodological investigations on learning phenomena and psychometric methods relevant to instructional technology and computer-assisted instruction. The project's accomplishments are presented in this report, along with a listing of reports and products produced. The work of the project is reported in terms of five areas of effort: 1) adaptation of instructional environments to the learning characteristics of the individual trainee; 2) computer testing, computer algorithms, and language for experimental instructional systems; 3) exploration of techniques for learner manipulation of subject matter; 4) the development of concepts and techniques for the analysis of subject-matter structures; and 5) annual conferences on developments in learning relevant to instruction and development. (Author)

Gradient methods are employed in orthogonal oblique analytic rotation. Constraints are imposed on the elements of the transformation matrix by means of reparameterisations. (Author)

Beginning with the results of Girschick on the asymptotic distribution of principal component loadings and those of Lawley on the distribution of unrotated maximum likelihood factor loadings, the asymptotic distributions of the corresponding analytically rotated loadings is obtained. The principal difficulty is the fact that the transformation matrix which produces the rotation is usually itself a function of the data. The approach is to use implicit differentiation to find the partial derivatives of an arbitrary orthogonal rotation algorithm. Specific details are given for the orthomax algorithms and an example involving maximum likelihood estimation and varimax rotation is presented. (Author)

Using string-manipulation algorithms, FORTRAN computer programs were designed for analysis of written material. The programs measure length of a text and its complexity in terms of the average length of words and sentences, map the occurrences of keywords or phrases, calculate word frequency distribution and certain indicators of style. Trials of the programs, in studies of readability and reading rate, in aiding editors, in grading essays, and in identifying sources of response bias in multiple choice tests, demonstrate the potential applications of these algorithms in educational research and the usefulness of augmenting FORTRAN's computational facilities with character-processing capability. (Author)

The purpose of this project was to determine if the method of subtraction of integers taught to seventh grade students affected their mathematics achievement or retention. Computation, concept, and problem-solving sections of the California Achievement Test were given as pretests and posttests. An investigator-constructed test of the addition and subtraction of integers was also used as a pretest, posttest, and as a retention test one month after completion of the treatments. The project was divided into two studies. The first study involved 140 students and three teachers, and compared the Complement Method of subtraction to the Related Facts Method. Results showed a statistical difference in the area of concepts, favoring the group taught the Related Facts Method. No statistical differences were found in the areas of computation, problem-solving, or addition and subtraction of integers. The second study involved 90 students and two teachers, and compared the Complement Method of subtraction to the Systems Method. Statistical difference on the retention of the subtraction of integers favored the Systems Method. Statistical difference on the retention of the addition of integers favored the Complement Method. No statistical differences were found in any areas of immediate achievement in the second study. (Author/DT)

This paper describes the notation and underlying organization of an augmented, recursive-transition network grammar and illustrates how such a grammar is a natural medium for expressing and explaining a wide variety of facts about the psychological processes of sentence comprehension. A general discussion of transformational grammar and psycholinguistics serves as background information. Details on the structure, characteristics, and operation of the augmented, recursive-transition network grammar are provided. There is a discussion concerning the formalization of perceptual strategies and how these strategies can be naturally represented in transition network grammars. A concluding section explains the justification of transition network models. A list of references is included, and diagrams illustrate the working of the grammar in several problem areas. (VM)

This is the first workbook-text in a two-part series written for deaf students. It is remedial in nature, aimed at the secondary level, and covers addition, subtraction, multiplication, and division of whole numbers. The use of the number 10 in explaining the concepts presented is stressed throughout. For the second workbook, see SE 015 828, and for the teacher's guide, see SE 015 829. (DT)

The Maryland Refutation Proof Procedure System (MRPPS) is an interactive experimental system intended for studying deductive search methods. Although the work is oriented towards question-answering, MRPPS provides a general problem solving capability. There are three major components within MRPPS. These are: (1) an inference system, (2) a search strategy and (3) a base clause selection strategy. The "inference system" is based on the resolution principle and performs the logical deductions specified. The user may select from a wide variety of refinements of resolution. The "search strategy" directs the deductions to be made by selecting from clauses already generated those that have a best merit. The "base clause selection strategy" determines which facts and general axioms to select from the data base. Such a clause may be selected regardless of whether or not it has the best merit. Heuristic techniques are applied within each of the three major components. This technical report describes the current implementation of MRPPS. It describes each of the components and how they are integrated into what has been termed the "Q" algorithm. (Author/NH)

A new basic algorithm is discussed that may be used to do factor analysis by any of these three methods: (1) unweighted least squares, (2) generalized least squares, or (3) maximum likelihood. (CK)

In this paper a reliable method is found for approximating the value of the Bivariate Normal Correlation Coefficient, rho, given values of the joint probability and the normal deviates, h and k, or the related areas. This technique finds useful application in the computation of the tetrachoric correlation coefficient, r, when the underlying distributions may be assumed to be normal. (Author)

Investigated was the development of a diagnostic and individualized remedial program for prospective elementary teachers in the area of mathematical skills. Subjects were 120 students in the pre-methods course on number systems during the Winter and Spring 1971 terms at the Ogontz Campus, Pennsylvania State University. All subjects were given a series of tests prior to beginning the course, and were retested with the same tests at the end of the course. Students scoring below criterion on the pre-test diagnostic test were randomly assigned to an experimental group for remedial treatment or to a control group. Post-testing revealed an increase in diagnostic scores, some improvement in self-evaluation, and an improvement in attitude scores. However, no significant differences in test score gains were found between the experimental and the control groups. Because students in the experimental group attended the remedial clinic in very irregular patterns, the investigator concludes that test gains were not related to the experimental treatment. It was found possible to diagnose arithmetic deficiencies and to prescribe appropriate remedial treatment, although carrying through the remediation program posed several problems. Appended are facsimiles of test materials utilized. (JG)

It is argued that many investigators utilize the Tucker and Messick (1963) Model with no intention of looking for individual differences or, after utilizing the model, draw improper inferences. An example is given illustrating the difficulties which result from improper use of the model. Several proper methods are outlined. (Author)

A model and a computer program for performing conjoint measurement is developed. (AG)

Improvements of the Gaussian quadrature in conjunction with the Newton-Raphson iteration technique (TM 000 789) are discussed as effective methods of calculating the bivariate normal correlation coefficient. (CK)

The second part of the four-part report of research on the development of a computerized, phrase-structure grammar of modern Hebrew describes the application of a generalized complex-constituent-phrase structure grammar to modern Hebrew. This volume discusses the details of the grammar: limitations, input and output, symbols, rules, tree diagrams, word classification, syntax, and orthography. An appendix provides a Hebrew-English dictionary. The grammar presented here provides new material for teacher training in the form of a transformational-type, theoretical model of modern Hebrew that views the language as an integrated whole. By studying the language this way, teachers of Hebrew can acquire a better understanding of the deep structure of the language. The formal presentation of the grammar, however, may not be the best form for training teachers. For related reports see FL 002 627, FL 002 629, and FL 002630. (Author/VM)

This first part of a four-part report of research on the development of a computerized, phrase-structure grammar of modern Hebrew presents evidence to demonstrate the need for material to train teachers of Semitic languagues in the theory of grammar. It then provides a discussion of the research already done on the application of computational grammars to artificial and natural languages. Research procedures are discussed. Following a section on computational grammars, there is discussion of grammar theories and of several grammars which might be suitable for generating and analyzing Hebrew sentences. The general requirements of complex-constituent-phrase structure grammar are outlined and methods for applying it to Semitic languages are discussed. A list of references is provided. For related reports see FL 002 628, FL 002629, and FL 002 630. (VM)

The final part of a four-part report of research on the development of a computerized, phrase-structure grammar of modern Hebrew describes the computerized algorithm for analyzing the sentences generated based on a complex-constituent-phrase structure grammar. The first section here discusses a structural model for modern Hebrew; the second provides a detailed description of the procedure for analyzing sentences; the third gives a detailed description of the computer program for the procedure; the last section describes the tests and the verifications of the algorithm. Appendix A lists the grammar rules of Hebrew syntax for the analysis of sentences. Appendix B gives the source-language listing of the computer program and subprograms used. Appendix C provides a sample output. Appendix D gives examples of the exhaustive snytactic analysis of the computer. For related reports see FL 002 627, FL 002 628, and FL 002 629. (VM)

Procrustes rotation involves fitting a factor pattern matrix to a specified target matrix in factor analysis. These rotations are useful for the investigator who wishes to see how well his data can be made to fit a hypothesized factor pattern matrix. The mathematical problems involved in these transformations are outlined and computer algorithms for both oblique and orthogonal solutions are presented. (CK)

Stochastic approximation algorithms for least square error approximation to density and distribution functions are considered. The main results are necessary and sufficient parameter conditions for the convergence of the approximation processes and a generalization to some time-dependent density and distribution functions. (Author)

A general method for estimating the unknown coefficients in a set of linear structural equations is described. In its most general form the method allows for both errors in equations (residuals, disturbances) and errors in variables (errors of measurement, observational errors) and yields estimates of the residual variance-covariance matrix and the measurement error variances as well as estimates of the unknown coefficients in the structural equations, provided all these parameters are identified. Two special cases of this general method are discussed separately. One is when there are errors in equations but no errors in variables. The other is when there are errors in variables but no errors in equations. The methods are applied and illustrated using artificial, economic and psychological data. (Author)

Procedures for grouping students into homogeneous subsets have long interested educational researchers. The research reported in this paper is an investigation of a set of objective grouping procedures based on multivariate analysis considerations. Four multivariate functions that might serve as criteria for adequate grouping are given and discussed; a method for optimizing these functions is also described. The set of procedures is illustrated through application to data from two samples of students, each student with scores on either ten or eleven subtests of a criterion referenced mathematics inventory. The results indicate that the procedures discussed provide a promising means for grouping students to minimize classroom heterogeneity. (Author)

The system described in this report is designed for use as a computational tool that allows a linguist to develop and study methods for generating surface strings from an underlying semantic structure. Initial findings with regard to form-determiners (such as voice, form, tense, and mood), some rules for embedding sentences, and some attention to pronominational substitution are reported. The report discusses how a particular semantic network can be used in conjunction with a grammar to generate sequences of coherent English sentences. The paper first outlines the form of the grammar and lexicon and describes a random generation algorithm that produces syntactically well-formed nonsense. It then shows how the semantic nets are used to control the ordering of grammar rules and lexical selections to generate meaningful discourse. There is also a section on some of the methods and rules for generating multi-sentence discourse. Special emphasis is given to the questions and problems that arise. A list of references is provided.For additional information, see FL 002 344. (Author/VM)

This report documents results of a two-year effort toward the study and investigation of the design of a prototype system for Chinese-English machine translation in the general area of physics. Previous work in Chinese-English machine translation is reviewed. Grammatical considerations in machine translation are discussed and detailed aspects of the Berkeley grammar II and lexicography are considered. Procedures for interlingual transfer are described in detail. The input and output systems, structure and logic of the syntax analysis system, and auxiliary diagnostic processes are covered as are the means for analyzing processed text. A bibliography is provided along with further details on flow charts, documentation, trees, coding, and sentence generation in the appendixes. (VM)

Two papers are included as Part Four of this report on Salton's Magical Automatic Retriever of Texts (SMART) project report. The first paper: "A Controlled Single Pass Classification Algorithm with Application to Multilevel Clustering" by D. B. Johnson and J. M. Laferente presents a single pass clustering method which compares favorably with more expensive clustering algorithms. The method is tested using the ADI collection of 82 documents and the Cranfield 424 Collection. The results are compared to full search and to results obtained by searching clusters produced by Dattola's algorithm. The second paper: "A Systematic Study of Query-Clustering Techniques: A Progress Report" by S. Worona describes an experiment using various techniques of query clustering on the Cranfield 424 document collection and gives some preliminary results. Several methods of evaluating the performance of clustered searches in the context of query-clustering are discussed. Some observations are also made concerning use of the SMART system as implemented at Cornell University. (For the entire SMART project report see LI 002 719, for parts 1-3 see LI 002 720 through LI 002 722, for part 5 see LI 002 724.) (NH)

Designed for the student who has acquired basic computational skills with non-negative rational numbers, this guidebook on minimum course content seeks further development of computational skills with fractions. General goals and performance objectives, a course outline, teaching strategies, sample test items, and a list of six references are included. The quin is based on chapters from the text, "Essentials of Mathematics 2", by Sobel, Maletsky and Hill. (DT)

Designed for the student who has acquired basic computational skills with non-negative rational numbers, this guidebook delineates minimum course content to further develop students' computational skills with whole numbers. Place value and estimation are also covered. General goals, performance objectives, a course outline, suggested teaching strategies, sample test items, and a list of six references are provided. The quin is based on chapters from the text, "Essentials of Mathematics 2", by Sobel, Maletsky and Hill. (DT)