STATEMENT OF PERMISSION TO COPY In presenting this paper in partial fulfillment of the requirements for an advanced degree at Montana State University, 1 agree that the Library shall make it freely available for inspec¬ tion. I further agree that permission for extensive copying of this paper for scholarly purposes may be granted by my major professor, or, in his absence, by the Director of Libraries. It is understood that any copying or publication of this paper for financial gain A COMPARISON OF TRADITIONAL AND MODERN APPROACHES TO BASIC MATHEMATICAL SKILLS GRADES THREE THROUGH EIGHT by CARL ROBERT SCHWERTFEGER A professional paper submitted in partial fulfillment of the requirements for the degree of MASTER OF EDUCATION with concentration in Secondary Education Graduate Dean MONTANA STATE UNIVERSITY Bozeman, Montana August, 1977 iii ACKNOWLEDGMENTS The researcher of this study wishes to express his sincere gratitude and appreciation to the following three groups of individuals: I. School District No. 1, Helena, Montana. (A) All members of the main administrative staff (and their secretaries), with special thanks to Director of Instruction Louis 0. Strand; (B) Special and Instructional Services Personnel—Signe Harlow, Roy E. Kallin, and Shirley Madsen; (C) All principals who gave advice or assistance, with special thanks to those who allowed the gathering of data at their schools; namely, (1) Henry Jorgensen—Central School; (2) Archie H. Lucht—Helena Junior High School; (3) Tom W. Miller—C. R. Anderson School; (D) All teachers who allowed, or assisted in, the gathering of data; namely, (1) At Central School—Grade 3—Debra Feller, Marilyn Giuliani; Grade 4—Lynn Huck, Charlotte Thomas, Eleanor Zimmerman; Grade 5—Margaret Holt, Stanton Howe, Patricia Park; Grade 6— Jacqueline Pace, Betty Sando; (2) At Helena Junior High School— Grade 7—Eric Feaver, Donna A. Morgan, Janet Olson; Grade 8—Ronald Dooley, Dallas 0. Miller, Berma Saxton; Counselor Margaret Reinart; (3) At C. R. Anderson School—Grade 7—Ed Canty, Patrick E. Rillahan, Tamara A. Ziegler; Grade 8—Maris Aldrich, Gary Barker, Merle Korizek; Librarian Louise Loucks. TABLE OF CONTENTS Page VITA . ii ACKNOWLEDGMENTS ..... iii TABLE OF CONTENTS ... . v LIST OF TABLES viii LIST OF FIGURES ix ABSTRACT x Chapter 1. INTRODUCTION . 1 INTRODUCTION 1 STATEMENT OF THE PROBLEM . . . . 6 CONTRIBUTION TO EDUCATION THEORY ... .... . . 6 GENERAL QUESTIONS .... 7 GENERAL PROCEDURE . 7 LIMITATIONS AND/OR DELIMITATIONS 9 DEFINITIONS 11 SUMMARY 13 2. REVIEW OF RELATED LITERATURE . 15 INTRODUCTION . 15 THE NEED FOR REVISION OF TRADITIONAL MATHEMATICS INSTRUCTION 15 PURPORTED ADVANTAGES OF MODERN MATHEMATICS INSTRUCTION .............. 17 vi Chapter Page MODERN MATHEMATICS CONTROVERSY AND RESEARCH FINDINGS 18 SUMMARY . . . . 18 3. PROCEDURES 23 INTRODUCTION 23 POPULATION DESCRIPTION AND SAMPLING PROCEDURE .... 24 CATEGORIES OF INVESTIGATION AND GRADE EQUIVALENT SCORE CONVERSIONS ........... 26 SPELLING TEST AS CONTROL 27 TWO METHODS OF COLLECTING DATA 29 TEST RELIABILITY 32 TEST VALIDITY 34 METHOD OF ORGANIZING DATA 35 GENERAL QUESTIONS 35 STATISTICAL HYPOTHESES ...... 36 PRECAUTIONS TAKEN FOR ACCURACY . 39 SUMMARY . 39 4. DESCRIPTION OF THE DATA * 41 INVESTIGATIVE PHASE OF THE STUDY 41 LONGITUDINAL PHASE OF THE STUDY . . . . . . . . . . . 50 SUMMARY ............. 60 5. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS 62 SUMMARY 62 vii Chapter Page CONCLUSIONS 63 RECOMMENDATIONS 65 REFERENCES 67 viii LIST OF TABLES Table Page 1. Mathematics Study Groups and Textbooks Produced for Various Grade Levels in 1961 . . . . . . ... . 3 2. Total Number of Classrooms and Pupils Involved in the Investigative Phase of the Study . 32 3. Correlation Coefficients for Three Subtests of the Iowa Tests of Basic Skills 33 4. Data Table of Sample Statistics—Grade 3 42 5. Data Table of Sample Statistics—Grade 4 44 6. Data Table of Sample Statistics—Grade 5 45 7. Data Table of Sample Statistics—Grade 6 . 46 8. Data Table of Sample Statistics—Grade 7 48 9. Data Table of Sample Statistics—Grade 8 . 49 ix LIST OF FIGURES Figure Page 1. City-Wide Total Arithmetic Test Scores on ITBS in Helena, Montana—Grade 3 51 2. City-Wide Total Arithmetic Test Scores on ITBS in Helena, Montana—Grade 4 52 3. City-Wide Total Arithmetic Test Scores on ITBS in Helena, Montana—Grade 5 54 4. City-Wide Total Arithmetic Test Scores on ITBS in Helena, Montana—Grade 6 55 5. City-Wide Total Arithmetic Test Scores on ITBS in Helena, Montana—Grade 7 57 6. City-Wide Total Arithmetic Test Scores on ITBS in Helena, Montana—Grade 8 59 X ABSTRACT The purpose of this study was to investigate whether or not today's students being taught modern mathematics in grades three through eight in Helena, Montana schools were as proficient in the basic skills of arithmetic as their predecessors who were taught with more traditional methods and materials fifteen years earlier. Grade level patterns, skill improvement trends, and district longitudinal tendencies were also examined and illustrated in connection with the research indicated above. The testing instrument employed was the Multi-Level Edition for Grades 3-9 of the Iowa Tests of Basic Skills (ITBS), forms 1 and 2. The three subtests that were administered and utilized in the study were (a) Arithmetic Concepts (Test A-l), (b) Arithmetic Problem Solving (Test A-2), and (c) Spelling (Test L-l) as a control. Pupils in grades seven and eight were tested at the beginning of the school year, while pupils in grades three, four, five, and six were tested at mid-year. (This same testing procedure was utilized with the traditional youngsters during the 1961-62 school year.) After Grade Equivalent scores from the 1961-62 school year were converted back into raw scores, all test results were programmed into a computer at Montana State University, Bozeman; and means, standard deviations, variances, F-Test and t-Test statistics were calculated. Results indicated the following: Third grade level: A significant difference was indicated, with modern math students scoring higher than traditional math students on both arithmetic subtests. Grades four through six: No significant differences were indicated in the results of the arithmetic subtests. Grades seven and eight: A significant difference was indicated, with traditional math students scoring higher than modern math students on both arithmetic subtests. No significant difference was indicated in the results of the spelling test in grades three through seven, but eighth grade traditionally- taught students scored significantly higher on this subtest than their counterparts taught with modern mathematics techniques. In the longitudinal phase of the study, which included district statistics from the 1959-1973 school years, it has been shown that total arithmetic test scores in grades three through eight have fallen by approximately forty percentile points. Chapter 1 INTRODUCTION During the first half of the twentieth century, man applied his knowledge of mathematics, science and technology in the production of many inventions. Among these were the airplane, medicine, artifi¬ cial rubber, radar, sonar, electronic computers, electronic sewing machines, automatic washers and dryers, television, air conditioning, the atomic bomb, jet propulsion, plastics, talking motion pictures, wireless telegraphy, etc. (The American Peoples Encyclopedia, 1962:195-7). When considering these accomplishments from the stand¬ point of research and applied mathematics, one would tend to agree with the reviewer who wrote, "... the amount of mathematics pro¬ duced in this century [prior to 1957] was substantially greater than that produced in all previous history" (Willoughby, 1969:768). In spite of these indicated advances in research and applied mathematics, it is interesting to note that, during the same time period, advances in the instruction of mathematics in the schools apparently had not kept pace. This fact was emphasized in the 1958 Rockefeller Report on Education which stressed the crisis in science and mathematics: 2 The fateful question is not whether we have done well, or whether we are doing better than we have done in the past, but whether we are meeting the stern demands and unparalleled opportunities of the times. And the answer is that we are not (Price, 1961:11). Howard Fehr, of Columbia University, commented that the only major changes in mathematics curriculum content during the first half of the twentieth century were the introduction of graphing into the algebra program and the pushing of some topics, such as trigonometry and part of analytic geometry and calculus, into lower grades (Willoughby, 1969:767). This, then, was the setting into which the "modern" mathe¬ matics was cast except for one additional major historical event. This event occurred on October 5, 1957, when the Soviet Union announced the launching of the first artificial earth satellite, "Sputnik." The net effect of this event in the United States was a renewed thrust, including much needed financial assistance, for various groups to produce a new and more effective curriculum in mathematics (Kline, 1974:20). Men with outstanding qualifications, such as former Yale Professor Edward G. Begle, began to assemble writing teams under acronyms such as SMSG (School Mathematics Study Group) and UICSM (University of Illinois Committee on School Mathematics). Dr. Brown, a mathematics specialist for the U.S. Office of Education, indicates the large amount of financial support that these new programs received from the Federal government: 3 The Federal government expects local school systems to plan and develop their own high school mathematics programs, but it considers improvement so important that through the National Science Foundation it has contributed more than $4 million to the School Mathematics Study Group for the develop¬ ment of sample textbooks. These books do not present a national curriculum; rather it is hoped that the material will serve as a guide to authors and to school systems attempting to improve their own programs (Brown, 1961:16). Dr. Brown includes a table showing the major groups involved in the production of textbook material in 1961. The table is reproduced below: Table 1. Mathematics Study Groups and Textbooks Produced for Various Grade Levels in 1961. 7th 8th 9th 10th 11th 12th SMSG X X X X X X UICSM X X X University of Southern Illinois X X Ball State Teachers College X X X Boston Series X University of Maryland X X As time passed from the I9601 s into the 1970 ’s and the newly- produced mathematics materials found their way into the majority of American schools, a curious thing began to happen. Sharp differences of opinion as to the merits of the innovations began to surface among 4 professional mathematicians, teachers, students, and the general public. Willoughby informs readers that a professor of mathematics named Morris Kline of New York University was one of the earliest critics of the "new mathematicsM (Willoughby, 1969:768). He indi¬ cates that Professor Kline "... has continued to suggested that an intuitive, constructive, applied approach would be more appropriate than the highly theoretical approach he believes underlies most of the [new] programs" (Willoughby, 1969:768). As standardized achieve¬ ment test scores began to decline in the public schools, Morris Kline wrote a book entitled Why Johnny Can*t Add. In his book. Dr. Kline is critical of the framers of the modern mathematics curriculum for their lack of experimentation with the new materials. He writes. One would think that the framers of the modern mathe¬ matics curriculum would have experimented with many groups of children and teachers and thus produced some evidence in favor of their programs before urging them upon the country. The sad fact is that most of the groups undertook almost no experimental work (Kline, 1974:124). Kline also produces quotations from several of the original framers of the new mathematics, i.e.. Professors Max Beberman and Edward G. Begle, in which both professors indicated doubts about the wisdom of their programs. Professor Beberman, originator of the UICSM group, went so far as to say that he feared a major national scandal was in the making (Kline, 1974:132-34). 5 Other noted scholars have also indicated similar concerns about the curriculum reform movement in mathematics. As early as March of 1962, the text of a petition, signed by Lars V. Ahlfors of Harvard University and sixty-four other mathematicians from various parts of the United States and Canada, appeared in The Mathematics Teacher. Commentary taken from the reprint of this petition appears below: Mathematicians may unconsciously assume that all young people should like what present-day mathematicians like or that the only students worth cultivating are those who might become professional mathematicians . . . the mathematics curriculum of the high school should provide for the needs of all students: it should contribute to the cultural back¬ ground of the general student and offer professional preparation to the future users of mathematics, that is, engineers and scientists, etc. To know mathematics means to be able to do mathematics . . . and, what may be the most important activity, to recognize a mathematical concept in, or to extract it from, a given concrete situation . . . mathematics, is linked to the other sciences and the other sciences are linked to mathematics, which is their language and their essential instrument. Mathematical thinking is not just deductive reasoning; it does not consist merely in formal proofs .... Extracting the appropriate concept from a concrete situation, generalizing from observed cases, inductive arguments, arguments by analogy, and intuitive grounds for an emerging conjecture are mathe¬ matical modes of thinking. Indeed, without some experience with such "informal" thought processes the students cannot understand the true role of formal, rigorous proof .... If pushed prematurely to a too formal level he may get discouraged and disgusted .... The best way to guide the mental develop¬ ment of the individual is to let him retrace the mental development of the race-retrace its great lines, of course, and not the thousand errors of detail. What is bad in the present high-school curriculum is not so much the subject matter presented as the isolation of 6 mathematics from other domains of knowledge . . . (Ahlfors and others, 1962:191-93). STATEMENT OF THE PROBLEM The problem of this study was to investigate and compare 1961-62 and 1976-77 student achievement in Helena, Montana schools (grades three through eight) on the basic skills of arithmetic as defined and measured by Forms 1 and 2 of the Iowa Tests of Basic Skills (ITBS). CONTRIBUTION TO EDUCATIONAL THEORY There are several ways in which this research has made a contribution to educational theory. These are listed below: 1. It has allowed school officials in Helena, Montana and in similar western localities to see whether or not declining scores in the basic skills of arithmetic are occurring here, as well as in the more populous areas of California, New Hampshire, and New York (Kendig, 1974:34). 2. It has allowed an objective examination of the concept that students studying modem mathematics will lag behind their traditional counterparts at lower grade levels but will improve their basic arithmetic skills by the time they reach the junior high level (Kendig, 1974:34). 7 3. It has allowed Helena, Montana school officials, through the longitudinal phase of the study, to objectively compare and contrast fifteen years of test results in the basic skills of arith¬ metic at grade levels three through eight. GENERAL QUESTIONS Was there a difference in the arithmetic test performance of modern-taught students vs. traditionally-taught students in grades three through eight as measured by Forms 1 and 2 of the ITBS? Did differences in performance indicate trends at grade levels three through eight? Did spelling test performance indicate a difference between groups when utilized as an indicator of general academic decline? Did longitudinal results of ITBS testing in Helena, Montana bear any resemblance to national trends as indicated in the literature? GENERAL PROCEDURE The researcher determined that an appropriate method for comparing effectiveness of modern vs. traditional mathematics instruc¬ tion was to find statistics within a school district that indicated performance levels of students taught under a traditional approach and then to test a comparable group of modern-taught students on 8 the same instrument and perform a statistical analysis of the results. It was fortunate that test statistics from the 1961-62 school year were available in the files of some Helena schools and that the researcher was able to obtain enough copies of the tests administered in those pre-modern math years to carry out testing in the investiga¬ tive phase of the study. Many hours were spent by the researcher in talking with building principals prior to the actual testing procedure in deter¬ mining availability of test statistics and staff involvement. Many hours were also spent by the researcher in gathering data from files at the DistrictTs Instructional Services Center in order to present the longitudinal portion of this study?s statistical analysis. In general, the following procedures were utilized: 1. Availability of 1961-62 test statistics was determined by personal interview with building principals. 2. Availability of test Forms 1 and 2 of the ITBS was deter¬ mined through the medium of telephone calls and letters to various headquarters of the Houghton Mifflin Company. Lynn Berntson, Director of the Measurement and Research Division of the Palo Alto Office was particularly helpful. 3. Availability and gathering of statistical information from the Helena District's Instructional Services Center was accomplished and utilized in the longitudinal phase of the study. 9 4. The actual testing procedure at two junior high schools (grades seven and eight) and one elementary school (all classrooms, grades three through six) was accomplished. 5. All 1976-77 data was gathered from corrected answer sheets and results were recorded on various data forms. All 1961-62 data that was gathered from the schools participating in the study was converted to useable form and recorded as above for 1976-77 data. 6. Final statistical analysis was accomplished through the use of a computer at Montana State University, Bozeman, Montana. Eric Strohmeyer, advisor to the researcher, receives credit for acquisition of this valued computer time. 7. Results were analyzed and reported by the researcher. LIMITATIONS AND/OR DELIMITATIONS The whole area of the affective domain was left virtually unexplored in this research. Only objective cognitive kinds of learning were measured. No item analysis showing percent of pupils answering the items correctly from the two groups was made available by the researcher for comparison and analysis. Statistics were not available from the 1961-62 school population to serve this purpose. 10 Since testing, which was done during the 1961-62 and 1976-77 school years, utilized a standardized test (i.e., ITBS), any special objectives developed locally may not be measured by it. Statistics, such as I.Q. Test scores, were not available for Helena, Montana schools, so it was necessary to control for signifi¬ cant changes in the general intellectual level* of the students with the spelling subtest of the ITBS. Significant population shifts* have been denied by district administrators, city officials, and chamber of commerce personnel. Henry Jorgensen, principal at Central School for twenty-two years, claims that effects of recent urban renewal on his school's attendance area have been minimal. At the junior high level the testing pro¬ cedure of this study should have compensated for any population shifting. Minority or ethnic influences* in the Helena schools have been estimated by counselor Robert Reiman of Helena High School to be less than 1 percent. Other various and sundry items such as recent innovations (e.g., open classroom concept), declining school and home discipline, effects of television, changes in priorities, and attitudes of students, *Due to the lack of "hard" relative evidence, this variable in the research is listed as a "possible" limitation. 11 etc., are also listed as "possible" limitations on the results of this study. A delimitation of this study was the population groups under consideration. At the elementary level (grades three through six), only the student body of Central School was involved in the study. There are presently eight other elementary schools in the district where Forms 1 and 2 of the ITBS have not been recently administered. Two of these eight schools, Smith and Rossiter, were added to the district in 1966. This was written for the benefit of those who would arbitrarily make inference to the entire school district based upon Central School test results. Such inference at the junior high level, however, may not be eschewed as easily. Actually, overall district results are best indicated and compared with respect to time in the longitudinal phase of this study. DEFINITIONS Performance. This term refers to scores earned by pupils on three subtests of Forms 1 and 2 of the ITBS. Modem Mathematics Program. This terminology is generally thought of in connection with "up-to-date" course offerings in the study of mathematics. For purposes of this research, "up-to-dateness" included the following: 12 1. Textbooks and materials approved by administrative personnel in School District No. 1, Helena, Montana after the 1961-62 school year. Information provided by the Helena School District’s Instructional Services Center indicates that a ’’new arithmetic pro¬ gram" was begun during the 1962-63 school year in grades three through six, during the 1963-64 school year in grade seven, and during the 1964-65 school year in grade eight. Thus, by the time the 1964-65 school year began, the "new arithmetic program" was in effect in at least the elementary grades three through eight in Helena, Montana schools. 2. Teachers trained and experienced in teaching the concepts of the "new arithmetic program." It is assumed by the researcher that a minimum of thirteen to fifteen years of exposure to the "new arithmetic program" by teachers of the district, "up-to-date" teacher training in mathematics by the colleges for new staff members, and former exposure by experienced teachers coming into the district has enabled students to receive effective and efficient training in the skills of the "new arithmetic program." Traditional Mathematics Program. This terminology is generally thought of as consisting of programs, courses, textbooks, materials, and teacher training and experience prior to the advent of "Modem Mathematics Programs." For purposes of this research it has included all of the above in relation to the advent of the "new 13 arithmetic program" in the elementary schools of Helena, Montana. This "new arithmetic program" was introduced during the 1962-63 school year and the performance of students in the "traditional" program was based upon their test performance on the ITBS during the 1961-62 school year. Basic Skills of Arithmetic. These skills were operationally defined in this study to include items of a conceptual and problem solving nature as outlined in the Teacher’s Manual (Lindquist, 1964:47-51) for the Iowa Tests of Basic Skills. The eight major skill areas that were tested consisted of currency (money), Decimals, Equations, Fractions, Geometry, Measurement, Numerals, Per cents. Ratio and Proportion and Whole Numbers. SUMMARY School mathematics instruction during the first half of the twentieth century was accomplishing some of its objectives, but it was roundly criticized on occasion for being less than vibrant in its content and methodology. This was apparently not true of research and applied mathematics during the same time period. The launching of the Russian space satellite Sputnik in 1957 provided the thrust and financing to promote change both in the methods and mate¬ rials of school mathematics instruction. The "modern" changes were at first welcomed and quickly adopted by textbook publishers and 14 schools, but within the latter 1960,s and early 1970’s, more and more criticism was, and still is, being directed against the new materials for a variety of reasons. Declining scores on standardized tests have been used by opponents of the new mathematics programs to pressure for change. This study has investigated objectively whether or not declining test scores in basic arithmetic skills are a reality or fiction as measured by the Iowa Tests of Basic Skills in Helena, Montana schools (grades three through eight). Chapter 2 REVIEW OF RELATED LITERATURE INTRODUCTION In this chapter the reader will have an opportunity to examine strengths and weaknesses of both "traditional" and "modern" approaches to school mathematics instruction. This will be made possible by considering a variety of sources and viewpoints in the literature, with both philosophical and scientific rationale being utilized in a supporting or deprecating role. It is the goal of the researcher to present a balanced argument for both approaches inasmuch as this is possible. THE NEED FOR REVISION OF TRADITIONAL MATHEMATICS INSTRUCTION In May of 1974, the Conference Board of the Mathematical Sciences appointed a National Advisory Committee on Mathematical Education (NACOME) to prepare an overview and analysis of U.S. school- level mathematical education. Although this committee was primarily concerned with portraying current trends in curriculum and instruc¬ tion, its report does allude to some of the problems experienced with traditional mathematics instruction. Conceptual thought in mathematics must build on a base of factual knowledge and skills. But traditional school instruction far over-emphasized the facts and skills and 16 far too frequently tried to teach them by methods stressing rote memory and drill. These methods contribute nothing to a confused child's understanding, retention, or ability to apply specific mathematical knowledge. Furthermore, such instruction has a stultifying effect on student interest in mathematics, in school, and in learning itself (NACOME, 1975:24). These shortcomings of traditional mathematics instruction were known long before the 1940's, but after World War II public dissatisfaction became more apparent. Frank Kendig indicates. During the war, it seems, the military discovered that the nation's fighting men were sadly deficient in the mathe¬ matical skills necessary to wage modern war, and the armed services were forced to set up special training courses to remedy the situation. After the war, perhaps because of the need for mathematically skilled workers to run our expanding technology, numerous complaints were lodged against the country's system for educating its young in mathematics (Kendig, 1974:28,32). George L. Henderson, a consultant in mathematics education, reviewed those years and made comment in Today's Education on the process of rote learning: I vividly remember spending most of an elementary school year memorizing through sheer repetition the 144 multiplication facts .... I also remember having to memorize the rules for finding square root on four different occasions: once in elementary school, twice in high school, and once in college. The reason I had to memorize these rules four times was because I didn't really understand the process and so I couldn't recreate the procedure" (1975:44). 17 PURPORTED ADVANTAGES OF MODERN MATHEMATICS INSTRUCTION In order to correct for the inadequacies of learning mathe¬ matics by rote, textbook writers of modern curriculum materials began to require students to approach an understanding of the subject through the use of logical deductive proof, clear definitions, precise symbolism, set terminology, discovery, and unifying structure. Max Beberman, director of the UICSM study group, is generally credited with the earliest attempt at major curriculum revision. Esther C. Ortenzi writes, "By 1951, the University of Illinois, under a grant of $500,000 from the Carnegie Corporation, had already begun to study and recommend changes in teaching" (Ortenzi, 1964:7). The Illinois committee, in one of its High School Mathematics textbooks, gives the following "Operating Principles" behind the development of its materials: We believe that students should be given an opportunity to discover a great deal of the mathematics which they are expected to learn. Mathematical ideas, principles, and rules . . . you will not find rules displayed on pages immediately following discovery exercises ... it is part of the teacher’s job to determine when the students have discovered correct generalizations (UICSM, 1959:i-iii). It was believed by creators of the new materials that stu¬ dents learning mathematics in the manner outlined above would have such an advantage over their traditionally-taught predecessors that they would be able to learn more mathematics, and more profound 18 mathematics, in a shorter period of time. These sentiments are expressed in a 1965 administrator^ guide which was published by a team of representatives from The American Association of School Administrators (AASA), the National Council of Teachers of Mathe¬ matics (NCTM), the Association for Supervision and Curriculum Development (ASCD), and the National Association of Secondary School Principals (NASSP). The recommendations reported by Marshall H. Stone, professor of Mathematics at the University of Chicago, were the following: The grand goal proposed is to compress the mathematical program so that what is now taught over twelve years of school plus three of college can be completed by the end of high school; that is in twelve years. How do the authors plan to achieve this plan? The means proposed are essentially those which have been put forward by everyone else who has seen the need for this kind of compression: the introduction of a great deal more mathematics into the elementary school program; better use of the opportunities for moving ahead in grades seven and eight; a more or less drastic re-evaluation of topics to be included in the curriculum; a more tightly and skillfully organized presentation of the essential elements of school mathematics; and finally, more stimulating and efficacious pedagogical methods aimed at developing important insight into the structure of mathematics as well as basic manipulative skills (AASA, ASCD, NASSP, AND NCTM, 1965:17,18). MODERN MATHEMATICS CONTROVERSY AND RESEARCH FINDINGS Earlier in this paper it was pointed out that opponents of modern mathematics programs have cited declining scores on standard¬ ized tests as an indication of the inefficaciousness of the "new” 19 math. Frank Kendig, a freelance writer who has a degree in mathe¬ matics, has written an article for the New York Times Magazine. In his article he produced the following facts related to this phenomenon: In California . . . sixth graders averaged 47 [compared with the test publisher’s "norm" of 50] on state-wide math tests given during the 1969-70 school year, the year California adopted the new math. The same tests were administered again in the 1971-72 school year, and this time California’s sixth graders averaged 38, a drop of nearly 10 points. According to New York’s 1970 Fleischmann Report, almost one-third of the state's sixth graders performed below standard in mathematics tests. In New Hampshire, Fernand Prevost, math consultant to the state's Department of Education, reports that "since the advent of modem math, the computation scores of students in this state have very definitely declined" (Kendig, 1974:34). Modernists usually counter facts and figures such as these in a number of ways. First of all, they will point out the fact that scores in mathematics are not declining in all schools, nor are math scores the only ones that are dropping on the various tests and sub¬ tests. NACOME indicates that, "Recent results from Rhode Island, Delaware, Mississippi, and Virginia suggest steadily improving mathe¬ matics performance in those states (NACOME, 1975:106). Also in the NACOME report, between 1963 and 1970 the ITBS has obtained mathematics scores which indicate general improvement in the lower grades but consistent and sizeable losses in the upper grades on both concepts and problem solving. Fairly consistent losses also occurred in 20 reading and some language skill areas during the same time period (NACOME, 1975:107). This phenomena of second and third graders scoring well on standardized tests and sixth graders on up doing poorly is subtan- tiated in several sources. The Oakland Tribune carried an article indicating that this was happening in California ("California Students Lag in Testing," 1974:1,8), and the NACOME report indicated that it was happening in the state of New York (NACOME, 1975:104). As one delves deeper into the literature, a plethora of unanswered questions appears: Is it true that elementary school teachers are holding back progress in "new" mathematics by their lack of prepara¬ tion in the new topics, and by their refusal to teach the recommended curricula in favor of time-honored computational skills (NACOME, 1975:11)? Is it true that all standardized test results are culturally biased, fail to test elements such as creativity and critical thinking and are administered by too many untrained teachers (Chicago Daily News Release, 1977:22)? Is it true that parents want more "basic skills" taught in the schools, not only in mathematics, but in other areas as well (Back to Basics in the Schools, 1974:87,88,91,92, 95)? Is it true that the "principal of gradualism" is the best approach to curriculum reform, with individual classroom teachers deciding on appropriate changes (Nygren, 1976:28-30)? Is it true that powerful national organizations are presently waging a campaign to prevent an over-reaction in the "Back to Basic Skills" movement (NCSM Position Paper, October, 1976)? 21 Is it true that conventional classrooms have many strengths relating to mathematics instruction which modern programs, such as IPI (Individually Prescribed Instruction), cannot approach (Lipson, 1976:11)? Is it true that the "new math" stereotypes the learning of youngsters more perniciously than the "old arithmetic" ever did (Phillipson, 1975:34)? Is it'true what French mathematician Jean Dieudonne indicates, . . . "it is quite clear that for 90% of school children of elementary and high school level, their need in adult life for any math above elementary arithmetic will be nill" (Kendig, 1974:33)? Is it true that at some point in time, such as the ninth grade level, giving a youngster a calculator is a good answer for his lack of basic skills (Kendig, 1974:34)? Is it true that no gaps exist between traditional and new math, but that the new math is simply an "enrichment" of the old (Ortenzi, 1964:11, Kline, 1974:109)? Is it true that when public preference is made known, as in the State of Maryland Public Poll, studying mathematics for mathematics’ sake receives low rankings, but mathematics studied for the solution of "real life problems" ranks very high (Hershkowitz, 1975:723-28)? SUMMARY This review of literature has emphasized the need for revision of traditional mathematics instruction, purported advantages of modern mathematics instruction, and controversy in the research findings. It was the purpose of the researcher to close with questions aimed at the latest controversial issues in order to stimulate thought regarding them within the mind of the reader and to serve as an 22 indicator of the many areas related to product vs. process education in mathematics which are very much unsettled and are in need of further research. Chapter 3 PROCEDURES INTRODUCTION This study was designed to investigate and compare 1961-62 and 1976-77 student achievement on two arithmetic subtests and one spelling subtest contained in Forms 1 and 2 of the Iowa Tests of Basic Skills. Testing was carried out in two junior high schools and one elementary school in Helena, Montana during the 1976-77 school year. Results of previous testing completed during the 1961-62 school year were gleaned from the files of the schools involved in the study. Detailed information is provided in this chapter regarding population, sampling procedure, categories of investigation, spelling test as control, two methods of collecting data, test reliability, test validity, method of organizing data, general questions, statis¬ tical hypotheses, analysis of data, and precautions taken for accuracy. Figures in Chapter 4 are utilized to illustrate the longitudinal trend of total arithmetic subtest scores on the ITBS, given in percentile ranks, as far back as the 1959-60 school year. Data collection for this phase of the research is described below. 24 POPULATION DESCRIPTION AND SAMPLING PROCEDURE The population of this study consists of the pupils in attendance at Helena, Montana schools during the time(s) of testing. Complete school populations (grades three through six) were tested at Central School during the 1961-62 school year and again during the 1976-77 school year. The entire Helena Junior High School (HJHS) population (grades seven and eight) was tested during the 1961-62 school year, but a stratified random sample was chosen as a comparison group from among the 1976-77 student body at Helena Junior High School and C. R. Anderson School.* When selecting the stratified random samples, care was exercised with regard to the following: both east and west sides of town contributed approximately equal numbers of pupils. Three classrooms of Social Studies-English students were randomly selected at the seventh and eighth grade levels. Discussion with building principals was beneficial in helping the researcher understand the flow patterns of students through their class schedules and helped to insure that no student had a chance to be involved in the testing procedure any more than one time, yet all students had the *C. R. Anderson School is a combination elementary-junior high school. 25 opportunity to be selected. It was decided that English-Social Studies classes would be utilized since students are placed in these classes by computer on a random basis. Mathematics classes were not utilized because students are placed in these classes on the basis of ability. (In the longitudinal phase of this study complete district statistics were available and utilized for grade levels three through eight.) From a testing standpoint, the researcher assumed comparability of the 1961-62 and 1976-77 populations based on the following: 1. Testing in both years was done in the same school system. 2. Testing in both years was done in the same schools and at the same grade levels. 3. Testing in both years utilized the same standardized testing instrument. 4. Testing in both years was completed during the same time of year. 5. Testing in both years at the lower grade levels (three through six) utilized the classroom teacher as test administrator. 6. Testing in both years involved the complete population, or a representative random sampling of the same, for all schools involved in the study. 26 7. Testing in both years included spelling test results for control of significant changes in general academic ability. CATEGORIES OF INVESTIGATION AND GRADE EQUIVALENT SCORE CONVERSIONS Populations and samples of students involved in the study were compared on a basis of achievement in relation to three subtests of the ITBS: Arithmetic Concepts (Test A-l), Arithmetic Problem Solving (Test A-2) and Spelling (Test L-l). Each of the arithmetic tests was timed at thirty minutes in length and the spelling test was timed at twelve minutes. All raw scores at a given grade level were treated statistically to determine central tendency, dispersion, t and F tests. Scores recorded on 1961-62 computer "list reports" were given in Grade Equivalent (G.E.) scores based upon a ten month school year. A student with a score of forty-six is considered to have a raw score corresponding to the fourth grade, sixth month. All of these G.E. scores had to be converted back to raw scores for comparison purposes. A conversion table was provided on the Scoring Mask for MRC (Measure¬ ment Research Center) Answer Sheets. When the exact G.E. score was not found in the table, the closest score to it was always utilized for conversion purposes. Whenever a G.E. score was found to exist directly in between two other G.E. scores, an alternating high-low 27 selection procedure was utilized. The highest or lowest score available was utilized for those few scores that extended beyond table limits. See Teacher^ Manual for description of extreme scores (Lindquist, Teacher’s Manual, 1964:21). SPELLING TEST AS CONTROL A. N. Hieronymous is a professor of Education and Psychology and Director of the Iowa Basic Skills Testing Program at the Univer¬ sity of Iowa. In a published letter which he wrote to Mr. Jerry Thiese, dated February 14, 1973, he included a table of median scores for each of the years 1965-72 inclusive showing a decrease in mathe¬ matics achievement. Included also are scores for Vocabulary, Reading, and Spelling. Hieronymous' reason for including these other scores along with the mathematics scores is given as follows: Because of the possibility of changes in the general level of achievement caused by factors as change in the nature of the school population, etc., data are also included for vocabulary, reading, and spelling to be used as "controls" (Hieronymous, Letter, 1973). In addition to Hieronymous * endorsement of spelling as a useful control, the researcher believes that there were other reasons to choose it: 1. Spelling words are generally chosen from vocabulary or reading word lists. These have been known to change very little with respect to time thereby giving spelling words a certain universality. 28 Albert J. Harris, Ph.D., calls this to the reader’s attention as he writes. Nevertheless, there is a common core which makes up perhaps as much as 90% of the running words in ordinary reading matter, and which can be identified by means of consulting available word lists (Harris, 1974:354). 2. Lindquist and Hieronymous in the ITBS Administrator * s Manual indicate that their spelling words are taken from word lists supplied by scholars such as Betts, Gates, Thorndike, Rinsland, and Horn (Lindquist, 1964:32). 3. Harris indicates that, Reading and spelling are closely associated because many of the abilities required for one are also required for the other. The correlation between scores on reading tests and scores on spelling tests usually falls in the range of .80 to .85 (Harris, 1947:5). 4. There is a close relationship between errors in word analysis in reading and errors in spelling. Not only are poor readers usually poor spellers, but they also tend to make the same kind of errors in spelling that they make in recognizing words (Harris, 1947:190). 5. Harris and Dechant indicate that there is a link between reading ability and general intelligence. There is unquestionably a substantial relationship between mental age and learning to read . . . (Harris, 1947:25). I.Q. . . . is significant in that it puts a ceiling upon individual achievement . . . [and it] ... is an important long-range predictor of the child's performance . . . [But] . . . intelligence is a more important determinant of reading success in the later grades than in the earlier grades. In the later grades, reading scores are an expression of profi¬ ciency in content-area reading. Content-area reading 29 generally requires greater use of those skills that we associate with intellectual activity (Dechant, 1964:40). The last reason for choosing the spelling test as a control is its economy in time and administration. In conclusion, a link has been shown to exist between spelling and reading and between reading and general intelligence. This in turn may lead one to surmise that if one is interested in establishing a single control for decline in the general academic level of a given population, spelling may be at least as good a variable as any other for this purpose. TWO METHODS OF COLLECTING DATA Two methods were utilized in the collection of data because there were essentially two areas of concern in the research; an investigative phase which involved testing and comparing data, and a longitudinal phase which was concerned with utilization of school district records and representing data already collected. In the investigative phase of the study, testing was done with Forms 1 and 2 of the ITBS, at.different grade levels, during different times of the year. This was in keeping with the previous research so that a valid comparison could be made between performances. Form 1 test booklets were utilized in collecting data in grades three, five, and seven, while Form 2 test booklets were utilized in collecting 30 data in grades four, six, and eight. Note: Forms 1 and 2 are equivalent forms of the same test (Lindquist, Administrator's Manual, 1964:46). After a stratified random sampling of the population was selected, with students at the seventh grade level and students at the eighth grade level for both Helena Junior High School and C. R. Anderson School, the testing was begun. This took place at the beginning of the school year in October. Grades three, four, five, and six at Central School were tested at midyear, in early February. At the junior high school level, the researcher conducted some of the testing personally in the school library. All elementary school testing at Central School was done with the aid of the classroom teacher. In order to insure uniformity of test administration, an instruction sheet was presented to each teacher which was based on the instructions in the Teacher's Manual of the ITBS (Lindquist, Teacherfs Manual, 1964:8-17). Teacher's Manuals were also included with the test booklets and MRC answer sheets for additional reference purposes if needed. Since most of the teachers had administered ITBS tests in the past, this part of the research actually proceeded quite uneventfully. After the tests were administered, the answer sheets were corrected by hand with MRC correcting masks by the researcher. Grade Equivalent scores from the 1961-62 school year were converted to raw scores and then all data was recorded on standard IBM coding sheets; 31 cards were punched; and the computer at Montana State University, Bozeman, accomplished the calculations necessary for analysis. Record files at School District No. 1 Instructional Services Center contained percentile rank scores for entire district arith¬ metic test totals, grades three through eight, as far back as the 1959-60 school year. These statistics appear diagrammatically in various figures found in Chapter 4. The table below gives an indication of the number of class¬ rooms (CLRMS) and pupils involved in the investigative phase of this study during the 1961-62 and 1976-77 school years at the various grade levels. 32 Table 2. Total Number of Classrooms and Pupils Involved in the Investigative Phase of the Study. 1961-62 School Year School Grade Level 3 4 5 6 7 8 Central //CLRMS 3 4 5 3 School //Pupils 90 100 132 96 Helena //CLRMS 8 8 Jr. H.S. //Pupils 252 242 1976-77 School Year School Grade Level 3 4 5 6 7 8 Central //CLRMS 2 3 3 2 School //Pupils 52 62 57 49 Helena //CLRMS 3 3 Jr. H.S. //Pupils 67 61 C. R. //CLRMS 3 3 Anderson School //Pupils 70 71 TEST RELIABILITY The Administrator’s Manual of the ITBS indicates that two methods of estimating reliability were used to obtain the data provided. The first was the split-halves (odds-evens) method, and the second was 33 the test-retest method involving equivalent test forms. Reliability coefficients were estimated by using the Spearman-Brown formula in the first situation and Pearson's product-moment correlation in the second situation. Although reliability coefficients obtained by correlating scores from equivalent forms are considered superior to those derived by the split-halves procedure (Lindquist, Administrator's Manual, 1964:38), both are presented below in table form for all sub¬ tests administered at grade levels three through eight. Table 3. Correlation Coefficients for Three Subtests of the Iowa Tests of Basic Skills. Grade Level Type of r Determined Test A-l Test A-2 Test L-l 3 Split-Halves .84 .80 .90 Test-Retest .79 .72 .81 4 Split-Halves .86 .80 .90 Test-Retest .80 .74 .82 5 Split-Halves .86 .82 .90 Test-Retest .83 .73 .86 6 Split-Halves .86 .81 .91 Test-Retest .83 .71 .90 7 Split-Halves .88 .79 .91 Test-Retest .86 .69 .88 8 Split-Halves .88 .79 .92 Test-Retest .84 .69 .86 34. Upon examining the reliability coefficients and the procedures described above, Virgil E. Herrick, Professor of Education, University of Wisconsin, Madison, Wisconsin, reports in Burros* Fifth Mental Measurements Yearbook: Because of the length of the tests, one would expect the reliability coefficients to be high and they are. They range from .84 to .96 for the major tests and from .70 to .93 for the subtests .... These data were derived from scores obtained at the beginning of the school year. One would expect that coefficients obtained from midyear and end of year testings would be even higher (Herrick, 1959:31,32). TEST VALIDITY Validity is concerned with the question of whether the ITBS really measures what it purports to measure. Virgil E. Herrick evaluates this phase of the test by stating: A major strength of this battery is its curricular validation. Besides the usual widespread administration of sample test items and the establishment of discrimination and difficulty indexes, extremely careful identification and definition of the skill processes being tested was done before test items were devised. This aspect of test develop¬ ment is not usually undertaken with such care, and the authors are to be commended for the way the curricular validation of their test items was done. School staffs attempting to improve their curriculum in the skill areas could use with profit the definitions of the skill objectives developed by the Iowa Staff. These curricular analyses are found in the Teacher*s Manual . . . Here each basic skill is analyzed, the test items related to it identified, and corresponding teaching suggestions made (Herrick, 1959:32). 35 METHOD OF ORGANIZING DATA All raw scores from the investigative phase of the study were recorded on data sheets which were utilized to punch data cards for computer analysis. Tables provided in Chapter 4 indicate the results of calculations and printouts supplied by the computer. N-values, means, standard deviations, variances, F-test and t-test statistics were reported in the data tables. In these tables, statistically significant values are indicated with two asterisks. Various figures in Chapter 4 were utilized by the researcher in the presentation of data for the longitudinal phase of the study. It will be noted by the reader that norms provided for school averages in the Administrator’s Manual were reported at each grade level in percentile ranks for comparison purposes. These were related to beginning of the year averages (grades seven, eight) and mid-year averages (grades three through six) (Lindquist, 1964:59-69). GENERAL QUESTIONS Did differences in performance indicate trends at grade levels three through eight? Did longitudinal results of ITBS testing in Helena, Montana bear any resemblance to national trends as indicated in the literature? 36 STATISTICAL HYPOTHESES Questions regarding the investigative phase of the research were translated into the following suitable hypotheses: There was no significant difference between mean V yT = H l: PT ^ yM H 2 2 aT =aM H 1* T a_2 ^ a. 2 M achievement levels of traditionally-taught (y^) and modern-taught (y^) students as measured by the ITBS, Forms 1 and 2. There was a significant difference between mean achievement levels of traditionally-taught and modem-taught students as measured by the ITBS, Forms 1 and 2. There was no significant difference between score dispersions (or score variances) of traditionally- 2 2 taught (aT ) and modern-taught (a^ ) students as measured by the ITBS, Forms 1 and 2. There was a significant difference between score dispersions of traditionally-taught and modern- taught students as measured by the ITBS, Forms 1 and 2. 37 ANALYSIS OF DATA The means of the two distributions of scores (i.e., traditional vs. modern) were compared with a t-test for two independent samples, and differences in variances were compared with an F-ratio. All results of statistical applications are presented in tabular form in Chapter 4 of this study. Equations illustrating the computational requirements are presented below. I. Descriptive A. Arithmetic Mean ZX EfX x = Tor N 7 Variance (S ) and Standard Deviation (S) (Unbiased estimate for sample statistics) .2 _ E ( X-t) 2 ' ”N-l II. Testing Hypotheses A. t-Ratio Test statistic: D = 2 . Sum of squares: SS = Degrees of Freedom: df = ^+^-2 Level of significance: a = .05 38 Error Term : _ -'ll SSJ+SS2 /! A Xl‘^2 f NI+N2-2 'Nl V Calculated t-Ratio: t = X1 ' X2 xrx2 B. F-Ratio Test Statistic: F = 1 (numerator) (denominator) Degrees of Freedom: df = n^-1, n2“^ Level of Significance: a = .05 Calculated F-Ratio: F . n n--l,n2-l 551 (Greater n^-1 Variance) 552 (Lesser n2-l Variance) C. Error Possibilities in Testing Hypotheses H is true H is not true o o Reject Type I Error Correct H 0 P (Type I Error) = a Decision Retain Correct Type II Error H 0 Decision P (Type II Error) = 3 a and 3 are inversely related in that an increase in the level of a produces a corresponding decrease in the level of 3 and vice versa. All testing in this study was at the a = .05 level of significance. Note: 39 PRECAUTIONS TAKEN FOR ACCURACY Before administering the three subtests of the ITBS, teachers received a sheet of detailed instructions as to proper procedure. These instructions were taken directly from the ITBS Teacher’s Manual. (The Teacher*s Manual was also available if questions arose.) All tests were administered according to testmaker's specifications. All raw data was programmed into a computer at Montana State University, Bozeman, Montana in order to insure proper and accurate calculations. SUMMARY This chapter has outlined procedures which were followed in order to complete the research involved in this study. Populations and samples from the 1961-62 and 1976-77 school years in Helena, Montana were described. The ITBS was indicated as the testing instru¬ ment, and subtests A-l, A-2, and L-l were indicated as the criterion for performance. Data from the investigative phase was analyzed with appropriate statistical techniques to determine means, variation, t-test, and F-test results. Statistical hypotheses were indicated for null and alternative situations and precautions taken to insure accuracy were included. 40 Statistics from the longitudinal phase of the study were referred to in appropriate figures in Chapter 4. Chapter 4 DESCRIPTION OF THE DATA In this chapter data has been described for both investigative and longitudinal phases of the study through utilization of tables and figures. These will allow the reader to readily compare, contrast, and interpret results of the research and hypothesis testing. INVESTIGATIVE PHASE OF THE STUDY Table 4 depicts data at the third grade level for ITBS sub¬ tests A-l, A-2, and L-l. 42 Table 4. Data Table of Sample Statistics—Grade 3 Iowa Tests of Basic Skills (Forms 1 and 2) School Year Subtest A-l* Subtest A-2* Subtest L-l* N 1961-62 90 90 90 1976-77 52 52 52 X 1961-62 21.5 13.1 20.9 1976-77 23.5 18.1 21.0 Q 1961-62 5.9 4.7 6.8 1976-77 4.0 5.0 5.7 s2 1961-62 35.4 22.2 46.5 1976-77 16.7 25.0 33.0 Calculated F 2.11** 1.12 1.41 Calculated t 2.13** 5.99** .04 *Note: Subtest A-l refers to Arithmetic Concepts Subtest A-2 refers to Arithmetic Problem Solving Subtest L-l refers to Spelling **These values in the table are significant at the a = .05 level of significance for two-tail hypothesis testing. An interpretation of the data in Table 4 indicates the following: 1. Mean achievement levels on both Arithmetic Concepts and Arithmetic Problem Solving subtests were significantly higher for 43 grade three modern-taught students than for those taught with tradi¬ tional methods. 2. The variability of traditionally-taught students at this grade level was significantly higher than it was for the modern-taught students on the Arithmetic Concepts subtest. 3. No statistically significant difference was indicated for the spelling test, although modern-taught students achieved a higher mean score and were less variable than traditionally-taught students at the third grade level. Table 5 depicts data at the fourth grade level for ITBS sub¬ tests A-l, A-2, and L-l. 44 Table 5. Data Table of Sample Statistics—Grade 4 Iowa Tests of Basic Skills (Forms 1 and 2) School Year Subtest A-l Subtest A-2 Subtest L-l N 1961-62 100 100 100 1976-77 62 62 62 X 1961-62 22.3 17.5 27.3 1976-77 21.4 17.0 27.0 1961-62 5.6 4.6 7.5 D 1976-77 5.5 5.1 7.1 q2 1961-62 32.3 21.6 56.4 b 1976-77 30.7 26.2 50.2 Calculated F 1.04 1.21 1.12 Calculated t 1.09 . 66 .16 An interpretation of the data in Table 5 indicates the following: 1. There is no statistically significant difference in mean achievement for modern or traditionally-^taught students in ITBS sub¬ test A-l, A-2, or L-l at the fourth grade level. Traditionally-taught students had a slightly higher mean score on each of three subtests. 2. The variability of traditionally and modern-taught stu¬ dents was not statistically significant on the three subtests at this grade level. 45 Table 6 depicts data at the fifth grade level for ITBS sub¬ tests A-l, A-2, and L-l. Table 6. Data Table of Sample Statistics—Grade 5 Iowa Tests of Basic Skills (Forms 1 and 2) School Subtest Subtest Subtest Year A-l A-2 L-l N 1961-62 132 132 132 1976-77 57 57 57 X 1961-62 25.8 15.5 23.7 1976-77 25.4 16.7 24.7 c 1961-62 6.4 4.9 6.9 1976-77 6.7 5.3 9.8 s2 1961-62 40.7 24.7 47.7 1976-77 45.0 28.5 97.4 Calculated F 1.10 1.15 2.04** Calculated t .38 1.58 .79 **These values in the table are significant at the a = .05 level of significance for two-tail hypothesis testing. An interpretation of the data in Table 6 indicates the following: 1. There is no statistically significant difference in mean achievement for modern or traditionally-taught students in ITBS subtest 46 A-l, A-2, or L-l at the fifth grade level. Modem-taught students at this level have shown slightly higher mean scores in the Arithmetic Problem Solving and Spelling Subtests. 2. Modern-taught students display more variability in all three subtests of the ITBS at the fifth grade level. This difference is one of significance only on the spelling test. Table 7 depicts data at the sixth grade level for ITBS sub¬ tests A-l, A-2, and L-l. Table 7. Data Table of Sample Statistics—Grade 6 Iowa Tests of Basic Skills (Forms 1 and 2) School Subtest Subtest Subtest Year A-l A-2 L-l N 1961-62 96 96 96 1976-77 49 46 46 X 1961-62 1976-77 28.2 27.8 18.9 18.9+ 30.2 30.4 1961-62 7.6 5.7 9.9 b 1976-77 7.2 6.4 9.2 s2 1961-62 57.7 32.7 98.1 b 1976-77 52.2 40.8 85.9 Calculated F 1.11 1.24 1.14 Calculated t .25 .00 .19 47 An interpretation of the data in Table 7 indicates the following: 1. There is no statistically significant difference in mean achievement for modern or traditionally-taught students in ITBS sub¬ test A-l, A-2, or L-l at the sixth grade level. Modern-taught students at this level are reflective of modern-taught fifth graders in that both achieved a slightly higher mean score in the Arithmetic Problem Solving and Spelling subtests. 2. The variability of traditionally and modern-taught stu¬ dents was not statistically significant on the three subtests at this grade level. Table 8 depicts data at the seventh grade level for ITBS subtests A-l, A-2, and L-l. 48 Table 8. Data Table of Sample Statistics—Grade 7. School Year Iowa Tests of Basic (Forms 1 and 2 Skills ) Subtest A-l Subtest A-2 Subtest L-l 1961-62 252 252 252 N 1976-77 137 137 137 Y 1961-62 24.9 14.8 26.9 A 1976-77 20.2 12.8 25.7 q 1961-62 6.1 3.5 8.3 1976-77 7.3 4.5 9.5 S2 1961-62 37.2 12.2 68.4 1976-77 53.7 20.9 89.8 Calculated F 1.44** 1.71** 1.31 Calculated 6.70** 4.88* 1.39 t **These values in the table are significant at the a = .05 level of significance for two-tail hypothesis testing. An interpretation of the data in Table 8 indicates the following: 1. Mean achievement levels on both Arithmetic Concepts and Arithmetic Problem Solving subtests were significantly higher for grade seven traditionally-taught students than for those taught with modern methods. 49 2. The variability of modern-taught students at this grade level was significantly higher than it was for the traditionally- taught students on both arithmetic subtests. 3. No statistically significant difference was indicated for the spelling test, although traditionally-taught students had scores with a higher arithmetic mean and lower variability statistic. Table 9 depicts data at the eighth grade level for ITBS subtests A-l, A-2, and L-l. Table 9. Data Table of Sample Statistics—Grade 8 Iowa Tests of Basic Skills (Forms 1 and 2) School Year Subtest A-l Subtest A-2 Subtest L-l N 1961-62 243 243 243 1976-77 132 132 132 X 1961-62 25.3 16.8 26.2 1976-77 20.3 13.2 21.6 1961-62 6.7 3.7 8.3 S 1976-77 7.5 4.1 9.8 s2 1961-62 45.6 13.9 69.9 1976-77 57.0 17.4 95.9 Calculated 1.25 1.25 1.37** F Calculated t 6.61** 8.57** 4.79** **These values in the table are significant at the a = .05 level of significance for two-tail hypothesis testing. 50 An interpretation of the data in Table 9 indicates the following: 1. Mean achievement levels on Arithmetic Concepts, Arith¬ metic Problem Solving and Spelling subtests were significantly higher for grade eight traditionally-taught students than for those taught with modern methods. 2. The variability of modern-taught students at this grade level was significantly higher than traditionally~taught students on the Spelling subtest. On the Arithmetic subtests, the scores of modern-taught eighth graders indicated more variability, but not by a statistically significant amount. 3. The statistically significant lower mean score earned by modern-taught students on the Spelling subtest may indicate that modern-taught students had less general academic ability than their traditional counterparts. LONGITUDINAL PHASE OF THE STUDY Figure 1 illustrates city-wide total Arithmetic test scores on the ITBS in Helena, Montana. Median percentile ranks are marked x for grade three achievement levels during 1959-72 school years. Unless otherwise indicated, testing at this grade level was completed during the mid-year. 51 Percentile Ranking 100-- 90- SO¬ TO ■ x x x # X X X X eo- so- 40- Test 2/71 x XX X Median National Norm Level 30 ’ 20- 10- o- —i > > i 1 1 1 1 1 1 1 l I I I 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 School Year Figure 1. City-Wide Total Arithmetic Test Scores on ITBS In Helena, Montana—Grade 3. //New Mathematics program began at this grade level during year indicated +Restandardization of ITBS Norms An interpretation of the data illustrated in Figure 1 indicates the following: 1. "Modern" mathematics was first taught to Helena grade three students during the 1962-63 school year. 2. Restandardization of ITBS norms were first applied to Helena students during the 1964-65 school year. 3. Total ITBS Arithmetic Test scores have fallen approximately 40 percentile points during the time span indicated. 52 Figure 2 illustrates city-wide total Arithmetic Test scores on the ITBS in Helena, Montana. Median percentile ranks are marked x for grade four achievement levels during 1959-73 school years. Unless otherwise indicated, testing at this grade level was completed during the mid-year. 100 - 90 - 80 - 70 - x x X Percentile Ranking 60 - 50 • 40 ■■ 30 ■■ x + X X Test 2/71 x x Test x X Fall x X Median National Norm Level 20 ■■ 10 0 ■■ i 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 School Year Figure 2. City-Wide Total Arithmetic Test Scores on ITBS in Helena, Montana—Grade 4. //New Mathematics program began at this grade level during year indicated +Restandardization of the ITBS Norms An interpretation of the data illustrated in Figure 2 indicates the following: 53 1. "Modern" mathematics was first taught to Helena grade four students during the 1962-63 school year. 2. Restandardization of ITBS norms first applied to Helena students during the 1964-65 school year. 3. ITBS Testing was completed in the fall of 1973. 4. Total ITBS Arithmetic Test Scores have fallen approximately 40 percentile points during the time span indicated. Figures 3 illustrates city-wide total Arithmetic test scores on the ITBS in Helena, Montana. Median percentile ranks are marked x for grade five achievement levels during 1959-73 school years. Unless otherwise indicated, testing at this grade level was completed during the mid-year. 54 100- X 90- h X X // X XXX 80- + Test 2/71 70- X X X x X Percentile 60- x Median Ranking 50- National Norm Level 40- Test Fallx 30- 20- 10- 0- -4 " i 1 i i y- 1 i i 1 1 ♦ t «- >■ 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 School Year Figure 3. City-Wide Total Arithmetic Test Scores on ITBS in Helena, Montana—Grade 5. //New Mathematics program began at this grade level during year indicated +Restandardization of the ITBS Norms An interpretation of the data illustrated in Figure 3 indicates the following: 1. "Modern" mathematics was first taught to Helena grade five students during the 1962-63 school year. 2. Restandardization of ITBS norms first applied to Helena students during the 1964-65 school year. 3. ITBS testing was completed in the fall of 1973. 55 4. Total ITBS Arithmetic test scores have fallen approximately 40 percentile points during the time span indicated. Figure 4 illustrates city-wide total Arithmetic Test scores on the ITBS in Helena, Montana. Median percentile ranks are marked x for grade six achievement levels during 1959-73 school years. Unless otherwise indicated testing at this grade level was completed during the mid-year. Percentile 100 y 90 -- 80 70 -- 60 -- Ranking 50 -. 40 -■ x x X + X X X X Te®t 2^ Median Nationalx Norm Level Test Fall x 30 -- 20 -• 10 -- 1 1 1 1 1 1 1 1 1 1— 3 64 65 66 67 68 69 70 71 72 73 School Year Figure 4. City-Wide Total Arithmetic Test Scores on ITBS in Helena, Montana—Grade 6. U -- 59 60 61 62 //New Mathematics program began at this grade level during year indicated +Restandardization of the ITBS Norms 56 An interpretation of the data illustrated in Figure 4 indicates the following: 1. "Modern" mathematics was first taught to Helena grade six students during the 1962-63 school year. 2. Restandardization of ITBS norms first applied to Helena students during the 1964-65 school year. 3. ITBS testing was completed in the fall of 1973. 4. Total ITBS Arithmetic Test scores have fallen approxi¬ mately 50 percentile points during the time span indicated. Figure 5 illustrates city-wide total Arithmetic Test scores on the ITBS in Helena, Montana. Median percentile ranks are marked x for grade seven achievement levels during 1959-1971 school years. Unless otherwise indicated, testing at this grade level was completed at the beginning of the school year. 57 100 ■■ 90 ■■ 80 70 -■ 60 -- Percentile Ranking 50 - - 40 -- 30 -- Test 2/71 X Median National Norm Level 20 -- 10 -- 0 4- -4- 4- -t- I' ■ -I » 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 School Year Figure 5. City-Wide Total Arithmetic Test Scores on ITBS in Helena, Montana—Grade 7. //New Mathematics program began at this grade level during year indicated +Restandardization of the ITBS Norms An interpretation of the data illustrated in Figure 5 indicates the following: 1. "Modern" mathematics was first taught to Helena grade seven students during the 1963-64 school year. 2. Restandardization of ITBS norms first applied to Helena students during the 1964-65 school year. 58 3. ITBS testing was completed in Feburayr of the 1970-71 school year. 4. Total ITBS Arithmetic Test scores have fallen approximately 50 percentile points during the time span indicated. Figure 6 illustrates city-wide total Arithmetic Test scores on the ITBS in Helena, Montana. Median percentile ranks are marked x for grade eight achievement levels during 1959-73 school years. Unless otherwise indicated, testing at this grade level was completed at the beginning of the school year. 59 100 90 80 70 60 Percentile Ranking 40 30 20 10 0 x X X X X #+ X X X Test 2/71 x Median National Norm Level 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 School Year Figure 6. City-Wide Total Arithmetic Test Scores on ITBS in Helena, Montana—Grade 8. //New Mathematics program began at this grade level during year indicated +Restandardization of the ITBS Norms An interpretation of the data illustrated in Figure 6 indicates the following: 1. "Modern" mathematics was first taught to Helena grade eight students during the 1964-65 school year. 2. Restandardization of ITBS norms first applied to Helena students during the 1964-65 school year. 60 3. ITBS testing was completed in February of the 1970-71 school year. 4. Total ITBS Arithmetic Test scores have fallen between 30 and 40 percentile points during the time span indicated. SUMMARY This chapter contains statistical data relative to both investigative and longitudinal phases of the study. Tables have been provided which indicate calculated t and F values for purposes of hypothesis testing at the various grade levels three through eight. In the investigative phase of the study, the following tables have indicated statistically significant results: Table 4—Grade 3: Mean achievement level of modern-taught students was significantly greater than that of traditionally-taught students on both arithmetic subtests. Traditionally-taught students had more variable scores than modern-taught students on subtest A-l. Tables 5, 6, and 7—Grades 4, 5, and 6: No statistically significant difference was indicated on any subtest except in Table 6—Grade 5 where spelling test scores of modern-taught students were more variable than those of traditionally-taught students. Table 8—Grade 7: Mean achievement level of traditionally- taught students was significantly greater than that of modern-taught 61 students on both arithmetic subtests. Scores of modern-taught stu¬ dents on these same subtests were significantly more variable. Table 9—Grade 8: Mean achievement level of traditionally- taught students was significantly greater than that of modern-taught students on ITBS subtests A-l, A-2, and L-l. Modern-taught students had more significantly variable scores on the spelling subtest. In the longitudinal phase of the study, total (i.e., combined) scores for ITBS Arithmetic subtests A-l and A-2 were pre¬ sented for grades three through eight. Figures were used to illustrate city-wide achievement levels during 1959-73 school years. In all instances, median scores show declines of approximately 40 percentile points. Chapter 5 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS SUMMARY This study was begun in an attempt to answer some of the controversial questions appearing in popular and professional litera¬ ture. It was not the intent of the researcher to add to the controversy but to try to objectively determine where the truth lay. The "General Questions," framed and presented in Chapter 1, reflect a genuine desire to discover whether students being taught modem mathematics have as much ability as their traditional counter¬ parts in solving arithmetic problems requiring the use of basic skills that so many parents of today1s pupils came to know and value. In addition to this question, it became obvious that another question had to be asked: If modern-taught youngsters do not have these basic arithmetic skills at the lower grade levels, possibly because they have not had an opportunity for much exposure to them, will they gain enough knowledge of them as they progress in school, so that by the time they are in the upper grades, their performance will not vary significantly from that of their traditional counterparts in the application of these skills to problem solving? The investigative phase of this study deals with these questions, and objective results of data gathering and hypothesis testing are presented in Chapter 4. 63 Much of the literature indicates that arithmetic scores are falling on standardized achievement tests throughout the nation. To find out whether this was true in Helena, Montana was a goal of the researcher. The longitudinal phase of the study dealt with this question, and Figures 1 through 6 of Chapter 4 present an objective answer. At all grade levels the trend indicated is one of declining ITBS total Arithmetic Test scores. During the 1959-73 school years, median percentile rank scores have dropped approximately forty points CONCLUSIONS In Table 9 of Chapter 4 the results of hypothesis testing indicate that modern students at the third grade level in Central School are out-performing their traditional counterparts in the basic skills of arithmetic. In grades four, five, and six no significant difference was indicated as a result of hypothesis testing, but if one considers only mean achievement, in grades three through six, it appears that traditional students surpass the modern students in arithmetic concepts but are overshadowed by their modern counterparts in arithmetic problem solving. In grades seven and eight traditional students have shown significantly higher levels of achievement on both the arithmetic concepts and the arithmetic problem solving subtests of the ITBS, Forms 1 and 2. This researcher made no attempt to compare students 64 of C. R. Anderson School and Helena Junior High School separately to their traditional counterparts since students from each of these schools were collectively grouped in order to make a representative 1976-77 junior high school sample. There is a mild tendency indicated which shows that students with significantly lower mean achievement scores also tend to be more varied or diverse in their abilities as well. The spelling test, utilized as a control for significant gains or declines in general academic ability levels, was only significant at the eighth grade level. This suggests that some of the decline in the arithmetic scores at that level may possibly be due to students possessing less general academic ability. At any rate, the notion of a "catch-up" effect occurring in the schools where modern math is being taught appears to be mere wishful thinking. The results of this testing seem to indicate that just the opposite occurs, i.e., if modern youngsters are ahead or holding their own on the basic skills of traditional arithmetic at lower grade levels, when they continue to study modern mathematics, without special supplementary work on these skills, they will fall further behind as they progress through the grades. This contention seems to be supported with other test results as well. For example, on November 15, 1974, the Oakland Tribune carried an article which indicated that "California's sixth grade students are also trailing far behind the national averages on 65 their tests ("California Students Lag in Testing," 1974:1,8). A similar result is reported in Overview and Analysis of School Mathe¬ matics by NACOME. This source indicates that since 1966 the New York State Department of Education has discovered in their fall achievement testing that, Only 18% of the state’s third graders fell below the previous 23% cutoff point. However, at grade 6, 32% of the students and at grade 9, 34% of the students performed below the 1966 reference point. Thus in some sense mathe¬ matical performance improved in the third grade and declined in the sixth and ninth grades from 1966 to 1973 (NACOME, 1975:104). The longitudinal phase of this study, presented in Figures 1 through 6 of Chapter 4, indicates that since the early 1960’s arith¬ metic test scores have declined by approximately 40 percentile points at all grade levels. This may be an indication that Helena, Montana schools are following national trends. Some of this decline can undoubtedly be attributed to a restandardization of the ITBS norms, and some of it may be attributed to other reasons. RECOMMENDATIONS 1. Results of the investigative phase of this research indicate that, at upper grade levels, students being taught modern mathematics achieve significantly lower standardized test scores on basic arithmetic skills than do traditionally-taught students. For this reason the researcher recommends that teachers utilize diagnostic 66 and supplementary materials at this level in an effort to improve abilities of students in their demonstrated areas of "weakness." 2. Results of the longitudinal phase of this research indicate that there has been a general trend toward lower standardized arithmetic achievement test scores in grades three through eight during the past fifteen or twenty years. Although the majority of the scores are presently located near the "comfortable" median national norm level and the variables affecting the decline are difficult to define and treat, this researcher recommends that plans be implemented for reversing the present downward trend for the benefit of Helena students and community. 3. Throughout this professional project, the researcher has continually been impressed by the divergency of professional and lay¬ men's opinions, as well as conflicting scientific reports, regarding the state of "modem" mathematics in the schools during the last fifteen or twenty years. Perhaps this divergency is a reflection of the American society itself, but then it may simply be an indica¬ tion that much more objective and meaningful research must be conducted. This is stated as a recommendation. 4. For those desiring to utilize this study as a basis for further research, it is recommended that the more worthy questions posed at the end of the "Review of Literature" section be considered. REFERENCES Ahlfors, Lars V., and Others. "On the Mathematics Curriculum of the High School," The Mathematics Teacher, LV (March, 1962), 191-193. American Association of School Administrators, Association for Supervision and Curriculum Development, National Association of Secondary School Principals, and National Council of Teachers of Mathematics. Administrative Responsibility for Improving Mathe¬ matics Programs. Washington, D.C.: October, 1965. "Back to Basics in the Schools," Newsweek, October 21, 1974, pp. 87, 88, 91, 92, 95. Brown, Kenneth E. "The Drive to Improve School Mathematics," The Revolution in School Mathematics. Washington, D.C.: National Council of Teachers of Mathematics, 1961. "California Students Lag in Testing," Oakland [California] Tribune, November 15, 1974, pp. 1, 8. Chicago Daily News Release. "SAT, CAT, PSAT: Batteries of Aptitude Tests Raise Questions Outside the Classroom," The Independent Record, Helena, Montana, May 8, 1977, p. 21. Conference Board of the Mathematical Sciences. National Advisory Committee on Mathematical Education (NACOME). Overview and Analysis of School Mathematics Grades K-12. Washington, D.C.: Conference Board of the Mathematical Sciences, 1975. Dechant, Emerald V. Improving the Teaching of Reading. Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1964. Harris, Albert J. How to Increase Reading Ability. 2nd ed. New York: Longmans, Green and Co., Inc., 1947. Henderson, George L. "Mathematics Yesterday, Today, and Tomorrow," TodayTs Education. Washington, D.C.: National Education Association, January-February, 1975. Herrick, Virgil E. "Review of Iowa Tests of Basic Skills," The Fifth Mental Measurements Yearbook. Highland Park, N.J.: The Gryphon Press, 1959. 68 Hershkowitz, Martin, Mohammad A. A. Shami, and Thomas E. Rowan. "Mathematics Goals: What Does the Public Want?" School Science and Mathematics, LXXV, No. 8 (December, 1975), pp. 723-728. Hieronymous, A. N. Letter to Mr. Jerry Thiese. February 14, 1973. Kendig, Frank. "Does New Math Add Up?," The New York Times Magazine. January 6, 1974, pp. 14-35. Kline, Morris. Why Johnny Can’t Add: The Failure of the New Math. First Vintage Books Edition, New York: February, 1974. Lindquist, E. F., and A. N. Hieronymus. Iowa Tests of Basic Skill, Manual for Administrators, Supervisors, and Counselors. Boston: Houghton Mifflin Company, 1964. . TeacherTs Manual (ITBS). Iowa City: State University of Iowa, 1964. Lipson, Joseph I. "Hidden Strengths of Conventional Instruction," The Arithmetic Teacher. Vol. 23, No. 1 (January, 1976), pp. 11-15. National Council of Supervisors of Mathematics. "Which Way to the Basics?" Minneapolis, Minnesota: October, 1976. Nygren, Burton M. "How to Survive Those innovations' that Nearly Ruined the Schools in the Sixties," The American School Board Journal, Vol. 163, No. 1 (January, 1976), pp. 28-30. Ortenzi, Esther C. What is Modern Math. Portland, Maine: J. Weston Walch, 1964. Phillipson, Prudence. "Inflexible Maths," The Times Educational Supplement. No. 3126, London: April 25, 1975, p. 34. Price, Baley G. "Progress in Mathematics and its Implications for the Schools," The Revolution in School Mathematics. Washington, D.C.: National Council of Teachers of Mathematics, 1961. The American People's Encylcopedia. XI, New York: Grolier Inc., 1962. Willoughby, Stephen S. Encyclopedia of Educational Research. 4th ed., New York: The MacMillan Co., 1969.