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DOCUMENT RESUME ED 330 567 SE 052 077 TITLE The State of Mathematics Achievement in Montana: The Trial State Assessment at Grade Eight. INSTITUTION Educational Testing Service, Princeton, N.J.; National Assessment of Educational Progress, Princeton, NJ. SPONS AGENCY National Center for Education Statistics (ED), Washington, DC. REPORT NO ETS-21-ST-02; I53N-0-88685-14-9 PUB DATE Jun 91 NOTE 146p.; The entire Report consists of a cc.:mposite report, an executive summary, and 40 separate reports for 37 states, DC, Guam, and the Virgin Islands, respectively; see SE 052 055-096. AVAILABLE FROM Individual state reports are available directly from the assessment division of the appropriate State Department of Education. PUB TYPE Statistical Data (110) -- Reports - Research/Technical (143) EDRS PRICE MF01/PC06 Plus Postage. DESCRIPTORS Academic Achievement; Calculators; *Educational Assessment; Family Environment; *Grade 8; Homework; Junior High Schools; *Mathematics Achievement; Mathematics Instruction; Mathematics Skills; Mathematics Tests; National Proc.:rams; Problem E lying; Public Schools; *State Prylrams; Student Attitudes; Teacher Attitudes; 2glachel: Qualifications; Television Viewing IDENTIFIERS *Montana; National Assessment of EducAttonal Progress; *Numeracy; State Mathematics Assessments; Trial State Assessment (NAEP) ABSTRACT In 1990, the National Assessment of Educational Progress (NAEP) included a Trial State Assessment (TSA); for the first time in the NAEP's history, voluntary state-by-stite assessments (37 states, the District of Columbia, Guam, and the Virgin Islands) were made. The sample was designed to represent the 8th grade public school population in a state or territory. The 1990 TSA covered five mathematics content areas (numbers and operations; measurement; geometry; data analysis, statistics, and probability; and algebra and functions). In Montana, 2,488 students in 100 public schools were assessed. This report describes the mathematics proficiency of Montana eighth-graders, compares their overall performance to students in the West region of the United States and the nation (using data from the NAEP national assessments), presents the average proficiency separately for the five content areas, and summarizes the performance of subpopulations (race/ethnicity, type of community, parents educational level, and gender). To provide a context for the assessment data, participating students, their mathematics teachers, and principals completed questionnaires which focused on: instructional content (curriculum coverage, amount of homework); delivery of math instruction (availability of resources, type); use of calculators; educational background of teachers; and conditions facilitating math learning (e.g., hours of televisioa watched, absenteeism). On the NAEP math scale, Montana students had an average proficiency of 280 compared to 261 nationwide. Many.fewer students (Montana-23%; U.S.-12%) appear to have acquired reasoning and problem solvIng skills. (JJVCRW) NATIONAL CENTER FOR EDUCATION STATISTICS The STATE of _kg! A le. in MONIUNA The Trial. State Aotsessinent at Grade Eight THE NATION'S REPORT CARD r17'iZT PAn AVMLJLE U S DEPARTMENT OF EDUCATION e of dut at.ong, Resew,' ann Imp,orpment 1,1t IC IONAL RE SOURCA S NORUATON T CENTER tEP1C1 t,A COL nAS tx.e, feVfOtloceil as fe<e.verf foo, the OP,S0,, 0,colf,q,afion NI, not ,flar,ijeS fsa,e t r"acle ic m'sproo,, revrOduction 0,4111, _ - - of t? SlAfed fhS do( f-f cic, ne. psSaf fet,feSer0 omoat OE RI p0s.1,, 0, Prepared by Educational Tesfing Service under Contract with the National Center for Education Statistics Office of Edu.-,ational Research and Improvement US. Department of Education 2 What is The Nation's Report Card? THE NATION'S REPORT CARD, the NatMnal Assessment of Educational Progress tNIAEP). is the only nationally representative and continuing assessment of what America's students know and can do in various subject areas. Since 19(19. assessments have been conducted periodically in reading. mathematics, science, writing, history/geography. and other fields. By making obje:tive information on student performance available to policymakers at the national, state, and local levels. NAEP is an integral part of our nation's evaluation of the condition and progress of education. Only information related to academic achievement is collected under this program. NAEP guarantet!s the privacy of individual students and their families. NAEP is a congressionally inundated project of the National Center for Edueation Statistics. the U.S. Department of Education. The Commissioner of Education Statistics is responsible. by law, for carrying out the NAEP project through competitive awards to qualified organizations. NAEP reports directly to the Commissioner, who is also responsible for providing continuing reviews, including validation studies and solicitation of public comment. on NAEP's conduct and usefulness. In 1988. Congress created the National Assessment Governing Board (NAGB) to formulate policy guidelines for NAEP. The board is responsible for selecting the subjixt areas to he assessed, which may include adding to those specified by Congress: identifying appropriate achievement goals for each age and grade: developing assessment objectives: developing test specifications: designing the assessment methodology: developing guidelines and standards for data analysis and for reporting and disseminating results; developing standards and procedures for interstate, regional. and national comparisons; improving the form and use of the National Assessment; and ensuring that all items selected for use in the National Assessment are free from racial, cultural, gender, or regional bias. The National Assessment Governing Board Richard A. Boyd, Chairman Exeeluive Director Martha Holden Jennings Foundation Cleveland, Ohio Phyllis Williamson Aldrich Curriculum Coordinator Saratoga-Warren B.O.C.E.S. Saratoga Springs. New York Franck Aksander Associate Superintendent California Department of Education Sacramento, California David P. Battini High School History Teacher Cairo-Durham High School Cairo, New York Parris C. Battle Teacher Horace Mann Elementary School Miami. Florida Mary R. Blanton Attorney Cromwell, Porter, Blanton & Blanton Salisbury. North Carolina Boyd W. Boehije Attorney Gaass, Klyn. & Boehlje Pella, Iowa Linda R. Bryant Teacher Cireenway Middle School Teacher Center Pittsburgh. Pennsylvania Honorable Michael N. Castk Governor of Delaware Cartel State Office Building Wilmington, Delaware Honorabk Naomi K. Cohen State of Connecticut House of Representatives Legislative Office Building Hartford, Connecticut Chesier E. Finn, Jr. Professor of Education and Public Policy Vanderbilt University Washington. D.C. Michael S. Glade Wyoming State Board of Education Saratoga, Wyoming Christine Johnson Principal Abraham Lincoln High School knver. Colorado John Lindley Principal South Colby Elementary School Port Or0 -ad. Washington Carl J. Maser Director of Schools The Lutheran Church -- Missouri Synod International Center St. Louis, Missouri Mark D. Musick President Southern Regional Education Board Atlanta. Georgia Honorabk Carolyn Pollan Arkansas House of Representatives Fort Smith, Arkansas Matthew W. Prophet, Jr. Superintendent Portland Oregon School District Portland, Oregon Honorabk William T. Randall Commis.sioner of Education State Department of Education Denver. Colorado Dorothy K. Rich President Home and School Institute Special Projects Office Washington. D.C. Honorable Richard W. Riky Attorney Nelson, Mullins. Riley and Scarborough Columbia. South Carolina Thomas Topuzes Attorney Law Offices of Frank Rogozienski Coronado, California Herbert J. Walberg Professor of Education University of Illinois Chicago. Illino;s Assistant Secretary for Educational Research and Improvement (Ex-Officio) U.S. Department of Education Washington. D.C. Roy Truhy Executive Directin. NAGB Washington. D.C. 3 NATIONAL CENTER FOR EDUCATION STATISTICS The STATE of Mathematics Achievement in MONTANA The Trial State Assessment at Grade Eight THE NATION'S REPORT CARD Report No: 21-ST-02 June 1991 Prepared by Educational Testing Service under Contract with the National Center for Education Statistics Office of Educational Research and Improvement U.S. Department of Education 4 U.S. Department of Education Lamar Alexander Secretary Office of Educational Research and Improvement Bruno V. Manno Acting Assistant Secretary National Center for Education Statistics Emerson J. Elliott Acting Commissioner FOR MORE INFORMATION: Copies of the 1990 NAEP Trial.State Assessment's individual State reports are availabledirectly from the participating States. For ordering information, please contact the assessment division of yourSuite Department of Education. For oraering information on the composite report of results for the Nation and all State participants, or for single copies of the Executive Summary while supplies last, write: Education Information Branch Office of Educational Research and Improvement U.S. Department of Education 555 New Jersey Avenue, NW Washington, D.C. 20208-5641 or call 1-800-424-1616 (in the Washington, D.C. metropolitan area call 202-219-1651). Libnuy of Ccargrcss, Catalog Card Number: 91-61478 ISBN: 048685-14-9 The work upon which this publication is based was performed for the National Center for Education Statistics, Office of Educational Research and Improvement, by Educational Testing Service. Educational Testing Saviee is an equal opporumity/affirmative action employer. Educalional Taiing Service, ETS, and arc tegistered trademarks of Educational Testing Service. Table of Contents EXECUTIVE SUMMARY INTRODUCTION 7 Overview of the 1990 Trial State Assessment This Report 9 Guidelines for Analysis 12 Profile of Montana 14 Eighth-Grade School and Student Characteristics 14 Schools and Students Assessed 15 PART ONE How Proficient in Mathematics Are Eighth-Grade Students in Montana Public Schools/ 17 Chapter 1. Students' Mathematics Performance 1 8 Levels of Mathematics Proficiency 19 Content Area Performance 19 Chapter 2. Mathematics Performance by Subpopulations 24 Race/Ethnicity 24 Type of Community '7 Parents' Education I. vel 29 Gender Content Area Performance 33 THE 1990 NAEP TRIAL STAI.E ASSESSMENT 0 In PART TWO Finding a Context for Understanding Students' Mathematics Proficiency 37 Chapter 3. What Are Students Taught in Mathematics" 39 Curriculum Coverage 41 Mathematics Homework 42 Instructional Emphasis 45 Summary 48 Chapter 4. How Is Mathematics Instruction Delivered' 49 Availability of Resources 49 Patterns in Classroom Instruction 51 Collaborating in Small Groups 54 Using Mathematical Objects 55 Materials for Mathematics Instruction 56 Summary 59 Chapter 5. How Are Calculators Used? 60 The Availability of Calculators 62 The Use of Calculators 63 When To Use a Calculator 64 Summary 66 Chapter 6. Who Is Teaching Eighth-Grade Mathematics' 67 Educational Background 68 Summary 71 Chapter 7. The Conditions Beyond School that Facilitate Mathematics Learnmg and Teaching 73 Amount of Reading Materials in the Home 74 Hours of Television Watched per Day 75 Student Absenteeism 76 Students' Perceptions of Mathematics 78 Summary 79 PROCEDURAL APPENDIX 81 DATA APPENDIX ...97 THE 1990 NAEP TRIAL STATE ASSESSMENT NE NATION'S REPORT CARD EXECUTIVE SUMMARY In 1988, Congress passed new legislation for the National Assessment of FAlucational Progress (NAEP), which included -- for the first time in the projeet'slUstory -- a provision authorizing voluntary state-by-state assessments on a trial hasis, in addition to continuing its primary mission, the nation41 .issessments that NAEP has conducted since its inception. As a result of the legislation the 1990 NAEP program included a Trial State Assessrmnt Program in eighth-grade mithematics. National assessments in mathematics, readmg, writing, and science were conducted simultaneously in 1990 at grades four, eight, and twelve. For the Trial State Assessment, eighth-grade public-school students were assessed in each of 37 states, the District of Columbia, and two territories in February 1990. The sample was carefuliy designed to represent the eighth-grade public-school population in a state or territory. Within each selected school, students were randomly chosen to participate in the program. Local school district personnel administered all assessment sessions, and the contractjes staff monitored 50 percent of the sessions as part of the quality assurance progam designed to ensure that the sessions were being conducted uniformly. The results of the monitoring indicated a high degree of quality and uniformity across sessions. THE 1990 NAEP TRIAL STATE ASSESSMENT 1 Montana In Montana, 100 public schools participated in the assessment. The weighted school participation inte was 90 percent, which means that all of the eighth-grade students in this sample of schools were representative of 90 percent of the eighth-grade public .school students in Montana. In each school, a random sample of students was selected to participate in the assessment. As estimated by the sample, 0 percent of the eighth-grade public-school population was claskfied as limited English Proficient (LEP), while 7 percent had an Individualized Edueation Plan (IEP). An IEP is a plan, written for a student who has been determined to be eligible for special education, that typically sets forth goals and objectives for the student and describes a program of activities and/or related services necessary to achieve the goals and objectives. Schools were permitted to exclude certain students from the assessment. To be excluded from the assessment, a student had to be categorized as Limited English Proficient or had to have an Individualiied Education Plan and (in either case) be judged incapable of participating in the assessment. The students who were excluded from the assessment because they were categorized as IIP or had an IEP represented 0 percent and 2 percent of the population, respectively. In total, 2,486 eighth-grade Montana public-school students were assessed. The weighted student participation rate was 96 percent. This means that the sample of students who took part in the assessment was representative of 96 percent of the eligibk eighth-grade public-school student population in Montana. Students' Mathematics Performance The average proficiency of eighth-gra-1c public-school students from Montana on the NAFP mathematics scale is 280. This proficiency is higher than that of students across the nation (261). Average proficiency on the NAFP scale provides a global view of eighth graders' mathematics achievement; however, it does not reveal specifically what the students know and can do in the subject. To describe the nature of students' proficiency in greater detail, NAEP used the results from the 1990 national assessments of fourth-, eighth-, and twelfth-grade students to define the skills, knowledge, and understandings that characterize four levels of mathematics performance -- levels 200, 250, 300, and 350 -- on the NAEP scale. 9 2 THE 1993 NAEP TRIAL STATE ASSESSMENT In Montana, 100 percent of the eighth graders, compared to 97 percent in the nation, appear to have acquired skills in.volving simple additive reasoning and problem solving with whole numbers (level 200). However, many fewer students in Montana (23 percent) and 12 percent in the nation appear to have acquired reasoning and problem-solving skills involving fractions, decimals, percents, elementary geometric properties, and simple algebraic manipulations (level 300). The Trial State Assessment included five content areas -- Numbers and Operations; Measurement; Geometry; Data Analysis, Statistics, and Probability; and Algebra and Functions. Students in Montana performed higher than students in the nation in all of these five content areas. Subpopulation Performance In addition to the overall results, the 1990 Trial State Assessment permits reporting on the performance of various subpopulations of the Montana eighth-grade student population defined by race/ethnicity, type of community, parents' education level, and gender. In Montana: White students had higher average mathematics proficiency than did Hispanic or American Indian students. Further, a greater percentage of White students than Hispanic or American Indian students attained level 300. The results by type of community indicate that the average mathematics performance of the Montana students attending schools in areas classified as "other" was higher than that of students attending schools in extreme rural areas. In Montana, the average mathematics proficiency of eighth-grade public-school students having at least one parent who graduated from college was approximately 19 points higher than that of students whose parents did not graduate from high school. The results by gender show that eighth-grade males in Montana had a higher average mathematics proficiency than did eighth-grade females in Montana. In addition, a greater percentage of males than females in Montana attained level 300. Compared to the national results, females in Montana performed higher than females across the country; males in Montana performed higher than males across the country. , / THE 1990 NAEP TRIAL STATE ASSESSMENT 3 Montana A Context for Understanding Students' Mathematics Proficiency Information on students' mathematics proficiency is valuable in and of itself, but it becomes more useful for improving instruction and setting policy when supplemented with contextual information about schools, teachers, and students. To gather such information, the students participating in the 1990 Trial State Assessment, their mathematics teachers, and the principals or other administrators in their schools were asked to complete questionnaires on policies, instruction, and programs. Taken together, the student, teacher, and school data help to describe some of the current practices and emphases in mathematics education, illuminate some of the factors that appear to be related to eighth-grade public-school students' proficiency in the subject, and provide an educational context for understanding information about student achievement. Some of the salient results for the public-school students in Montana are as follows: About half of the students in Montana (50 percent) were in schools where mathematics was identified as a special priority. This is a smaller percentage than that for the nation (63 percent). In Montana, 46 percent of the students could take an algebra course in eighth grade for high-school course placement or credit. A greater percentage of students in Montana were taking eighth-grade mathematics (59 percent) than were taking a course in pre-algebra or algebra (37 percent). Across the nation, 62 percent were taking eighth-grade mathematics and 34 percent were taking a course in pre-algebra or algebra. According to their teachers, the greatest percentage of eighth-grade students in public schools in Montana spent 30 minutes doing mathematics homework each day; according to the students, most of them spent either 15 or 30 minutes doing mathematics homework each day. Across the nation, teachers reported that the largest percentage of students spent either 15 or 30 minutes doing mathematics homework each day, while students reported either 15 or 30 minutes daily. Students whose teachers placed heavy instructional emphasis on Algebra and Functions had higher proficiency in this content area than students whose teachers placed little or no emphasis on Algebra and Functions. Students whose teachers placed heavy instructional emphaEis on Numbers and Operations had lower proficiency in this content area than students whose teachers placed little or no emphasis on Numbers and Operations. 4 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana In Montana, 17 percent of the eighth-grade students had mathematics teachers who reported getting all of the resources they meded, while 21 percen* of the students were taught by teachers who got only some or none of the resources they needed. Across the nation, these figures were 13 percent and 31 percent, respectively. In Montana, 16 percent of the students never used a calculator to work problems in class, while 46 percent almost always did. In Montana, 38 percent of the students were being taught by mathematics teachers who reported having at kast a master's or education specialist's degree. This compares to 44 percent for students across the nation. About half of the students (47 percent) had teachers who had the highest level of teaching certification available. This is different from the figure for the nation, where 66 percent of students were taught by teachers who were certified at the highest level available in their states. Students in Montana who had four types of reading materials (an encyclopedia, newspapers, magazines, and more than 25 books) at home showed higher mathematics proficiency than did students with zero to two types of these materials. This is similar to the results for the nation, where students who had all four types of materials showed higher mathematics proficiency than did students who had zero to two types. About one-quarter of the eighth-grade public-school students in Montana (21 percent) watched one hour or less of television each day; 6 percent watched six hours or more. Average mathematics proficiency was lowest for students who spent sx hours or more watching television each day. ThE 1990 NAEP TRIAL STATE ASSESSMENT 5 THE NATION'S REPORT CARD INTRODUCTION As a result of legislation enacted in 1988, the 1990 National Assessment of Educational Progress (NAEP) included a Trial State Assessment Program in eighth-grade mathematics. The Trial State Assessment was conducted in February 1990 with the following participants: Alabama Arizona Arkansas California Colorado Connecticut Delaware District of Columbia Florida Georgia Hawaii Idaho Illinois Indiana Iowa Ohio Kentucky Oklahoma Louisiana Oregon Maryland Pennsylvania Michigan Rhode Island Minnesota Texas Montana Virginia Nebraska West Virginia New Hampshire Wisconsin New Jersey Wyoming New Mexico New York North Camlina North Dakota Guarn Virgin Islands ..7 3 THE 1990 NAEP TRIAL STATE ASSESSMENT 7 Montana This report describes the performance of the eighth-grade public-school students in Montana and consists of three sections: This Introduction provides background information about the Trial State Assessment a xl this report. It also provides a profile of the eighth-grade public-school students in Montana. Part One describes the mathematics performance of the eighth-gade public-school students in Montana, the West region, and the nation. Part Two relates students' mathematics performance to contextual information about the mathematics policies and instruction in schools in Montana, the West region, and the nation. Overview of the 1990 Trial State Assessment In 1988, Congress passed new legislation for the National Assessment of Educational Progress (NAEP), which included -- for the first time in the project's history -- a provision authorizing voluntary state-by-state assessments on a trial basis, in addition to continuing its primary mission, the national assessments that NAEP has conducted since its inception: The National Assessment shall develop a trial mathematics assessment survey instrument for the eighth grade and shall conduct a demonstration of the instrument in 1990 in States which wish to participate, with the purpose of determining whether such an assessment yields valid, reliable State representative data. (Section 406 (i)(2)(C)(i) of the General Education Provisions Act, as amended by Pub. L. 100-297 (20 LI.S.C. 1221e-1(i)(2)(C)(i))) As a result of the legislation, the 1990 NAEP program included a Trial State Assessment Program in eighth-grade mathematics. National assessments in mathematics, reading, writing, and science were conducted simultaneously in 1990 at grades four, eight, and twelve. For the Trial State Assessment, eighth-grade public-school students were assessed in each state or territory. The sample was carefully designed to represent the eighth-grade public-school population in the state or territory. Within each selected school, students were randomly chosen to participate in the program. Local school district personnel administered all assessment sessions, and the contractor's staff monitored 50 percent of the sessions as part of the quality assurance program designed to ensure that the sessions were being conducted uniformly. The results of the monitoring indicated a high degree of quality and uniformity across sessions. 8 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana The Trial State Assessnv-nt was based on a set of mathematics objectives newly developed for the program and patterned after the consensus process described in Public Law 98-511, Section 405 (E), which authorized NAEP through June 30, 1988. Anticipating the 1988 legislation that authorized the Trial State Assessment, the federal government arranged for the National Sc nce Foundation and the U.S. Department of Education to issue a special gxant to the Council of Chief State School Officers in mid-1987 to develop the objectives. The development process included cartful attention to the standards developed by the National Council of Teachers of Mathematics,' the formal mathematics objectives of states and of a sampling of local districts, and the opinions of practitioners at the state and local levels as to what content should be assessed. There was an extensive review by mathematics educators, scholars, states' mathematics supervisors, the National Center for Education Statistics (NCES), and the Assessment Policy Committee (APC), a panel that advised on NAEP policy at that time. The objectives were further refined by NAEP's Item Development Panel, reviewed by the Task Force on State Comparisons, and resubmitted to NCES for peer review. Because the objectives needed to be coordinated across all the grades for the national program, the final objectives provided specifications for the 1990 mathematics assessment at the fourth, eighth, and twelfth grades rather than solely for the Trial State Assessment in grade eight. An overview of the mathematics objectives is provided in the Procedural Appendix. This Report This is a computer-generated report that describes the performance of eighth-grade public-school students in Montana, in the West region, and for the nation. Results also are provided for groups of students defined by shared characteristics -- race/ethnicity, type of community, parents' education level, and gender. Defmitions of the subpopulations referred to in this report are presented below. The results for Montana are based only on the students included in the Trial State Assessment Program. However, the results for the nation and thc region of the country are baseu on the nationally and regionally representative samples of public-school students who were assessed in January or February as part of the 1990 national NAEP program. Use of the regional and national results from the 1990 national NAEP program was necessary because the voluntary nature of the Trial State Assessment Program did not guarantee representative national or regional results, since not every state participated in the program. National Council of Teachers of Mathematics, Curriculum and Evaluation Standards for School Mathematics (Reston, VA: National Council of Teachers of Mathematics, 1989). 4 7- THE 1990 NAEP TRIAL STATE ASSESSMENT 9 Montana RACE/ETHNICITY Results are preserted for students of different racial/ethnic groups based on the students' self-identification of their race/ethnicity according to the following mutually exclusive categories: White, Black, Hispanic, Asian (includins Pacific Islander), and American Indian (including Alaskan Native). Based on critelia described in the Procedural Appendix, there must be at least 62 students in a particular subpopulation in order for the results for that subpopulation to be considered reliable. Thus, results for racial/ethnic groups with fewer than 62 students are not reported. However, the data for all students, regardless of whether their racial/ethnic group was reported separately, were included in computing overall results for Montana. TYPE OF COMMUNITY Results are provided for four mutually exclusive community types -- advantaged urban, disadvantaged urban, extreme rural, and other -- as defined below: Advantaged Urban: Students in this group live in metropolitan statistical areas and attend schools where a high proportion of the students' parents are in professional or managerial positions. Disadvantaged Urban: Students in this group live in metropolitan statistical areas and attend schools where a high proportion of the students' parents are on welfare or are not regularly employed. Extreme Rural: Students in this group live outside metropolitan statistical areas, live in areas with a population below 10,000, and attend schools where many of the students' parents arc farmers or farm workers. Other: Students in this category attend schools in areas other than those defined as advantaged urban, disadvantaged urban, or extreme rural. The reporting of results by each type of community was also subject to a minimum student sample size of 62. PARENTS' EDUCATION LEVEL Students were asked to indicate the extent of schooling for each of their parents -- did not fmish high school, graduated high school, some education after high school, or graduated college. The response indicating the higher level of education was selected for reporting. 10 THE 1990 NAEP TRIAL STATE ASSFSSMENT Montana GENDER Results are reported separately for males and females. REGION The United States has been divided into four regions: Northeast, Southeast, Central, and West. States included in each region are shown in Fkgure 1. All 50 states and the District of Columbia are listed, with the participants in the Trial State Assessment highlighted in boldface type. Tenitories were not assigned to a region. Further, the part of Virginia that is included in the Washington, DC, metropolitan statistical arra is included in the Northeast region; the remainder of the state is included in the Southeast region. Because most of the students are in the Southeast region, regional comparisons for Virginia will be to the Southeast. FIGURE 1 I Regions of the Country NORTHEAST SOUTHEAST CENTRAL WEST Connecticut Alabama Illinois Alaska Dslaware Arkansas Indiana Arizona District of Columbia Florida Iowa California Maine ftorgla Kansas Colorado Maryland Kentucky Michigan HawaII Massachusetts Louisiana Minnesota Idaho New Hampshire Mississippi Missouri Montana New Jena), North Carolina Nebraska Nevada New York South Carolina North Dakota New Mexico Pinnsybrania Tennessee Ohio Oklahoma Rhode island Virginia South Dakota Oregon Vermont mot Virginia Wisconsin Texas Virginia Utah Washington Wyoming THE 1990 NAEP TRIAL STATE ASSESSMENT 11 Montana Guido lines for Analysis This report describes and compares the mathematics proficiency of various subpopulations of students -- for example, those who have certain demographic characteristics or who responded to a specific background question in a particular way. The report examines the results for individual subpopulations and individual background questions. It does not include an analysis of the relationships among combinations of these subpopulations or background questions. Because the proportions of students in these subpopulations and their average proficiency are based on samples -- rather than the entire population of eighth graders in public schools in the state or territory -- the numbers reported are necessarily estimates. As such, they are subject to a measure of uncertainty, reflected in the standard error of the estimate. When the proportions or average proficiency of certain subpopulations are compared, it is essential that the standard error be taken into account, rather than relying solely on observed similarities or differences. Therefore, the comparisons discussed in this report are based on statistical tests that consider both the magnitude of the difference between the means or proportions and the standard errors of those statistics. The statistical tests determine whether the evidence -- based on the data from the groups in the sample -- is strong enough to conclude that the means or proportions are really different for those groups in the population. If the evidence is strong (i.e., the difference is statistically significant), the report describes the group means or proportions as being different (e.g., one group performed higher than or lower than another group) -- regardless of whether the sample means or sample proportions appear to be about the same or not. If the evidence is not sufficiently strong (i.e., the difference is not statistically significant), the means or proportions are described as being about the same -- again, regardless of whether the sample means or sample proportions appear to be about the same or widely discrepant. The reader is cautioned to rely on the results of the statistical tests -- rather than on the apparent magnitude of the difference betwecn sample means or proportions -- to determine whether those sample differences are likely to represent actual differences between the groups in the population. If a statement appears in the report indicating that a particular group had higher (or lower) average proficiency than a second group, the 95 percent confidence interval for the difference between groups did not contain the value zero. When a statement indicates that the average proficiency or proportion of some attribute was about the same for two groups, the confidence interval included zero, and thus no difference could be assumed between the groups. When three or more groups are being compared, a Bonferroni procedure is also used. The statistical tests and Bonferroni procedure are discussed in greater detail in the Procedural Appendix. 12 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana ..11=11 It is also important to note that the confidence intervals pictured in the figures in Part One of this report are approximate 95 percent confidence intervals about the mean of a particular population of interest. Comparing such confidence intervals for two populations is not equivalent to examining the 95 percent confidence interval for the difference between the means of the populations. If the individual confidence intervals for two populations do not overlap, it is true that there is a statistically significant difference between the populations. However, if the confidence intervals overlap, it is not always true that there is not a statistically significant difference between the populations. Finally, in several places in this report, results (mean proficiencies and proportions) are reported in the text for combined groups of students. For example, in the text, the percentage of students in the combined group taking either algebra or pre-algebra is given and compared to the percentage of students enrolled in eighth-grade mathematics. However, the tables that accompany that text report percentages and proficiencies separately for the three groups (algebra, pre-algebra, and eighth-grade mathematics). The combined-group percentages reported in the text and used in all statistical tests are based on unrounded estimates (i.e., estimates calculated to several decimal places) of the percentages in each group. The percentages shown in the tables are rounded to integers. Hence, the percentage for a combined group (reported in the text) may differ slightly from the sum of the separate percentages (presented in the tables) for each of the groups that were combined. Similarly, if statistical tests were to be conducted based on the rounded numbers in the tables, the results might not be consonant with the results of the statistical tests that are reported in the text (based on unrounded numbers). 7 ;) THE 1990 NAEP TRIAL STATE ASSESSMENT 13 Montana Profile of Montana EIGHTH-GRADE SCHOOL AND STUDENT CHARACTERIS11CS Table 1 provides a profde of the demographic characteristics of the eighth-grade public-school students in Montana, the West region, and the nation. This profile is based on data cc _A,xted from the students and schools participating in the Trial State Assessment. TABLE 1 I Profile of Montana Eighth-Grade Public-School I Students PERCENTAGE OF STUDENTS - WM) MEP TRIAL STATE ASSESSMENT Montana West Nation DEMOGRAPHIC SUBGROUPS Race/Ettinicity White 87 ( 1.1) 53 ( 1.9) 70 ( 0.5) Black ( 01) T ( 2.0) 18 ( 0.3) Hispanic 3 ( 0.4) 21 ( 1.5) 10 ( 0.4) Asian 1 ( 0.3) 4 ( 1.3) 2 ( 0.5) American Indian 8 ( 1.1) 4 ( 2.3) 2 ( 0.7) Type of Conenunity Advantaged urban 2 ( 0.1) 14 ( 85) 10 ( 3.3) Disadvantaged urban 0 ( 0.0) 19 ( 7.5) 10 ( 2.8) Extreme rural 31 ( 2.8) 10 ( 3.8) 10 3.0) Other 56 ( 2.8) 58 (10.1) 70 ( 4.4) Parents' Eclucation Did not finish high sch001 5 ( 0.7) 10 ( 1.3) 10 ( 0.8) Graduated high school 23( 1.3) 19 ( 2.5) 25 ( 1.2) Some education after high school 23 ( 0.9) 18 ( 1.2) 17 ( 0.9) Graduated college 45 ( 1.3) 42 ( 4.0) 39( 1.9) Oendiw Male 51 ( 1.4) 55 ( 2.1) 51 ( 1.1) Female 49 ( 1.4) 45 ( 2.1) 49 ( 1.1) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within .± 2 standard errors of the estimate for the sample. The percentages for Race/Ethnicity may not add to 100 percent because some students categorized themselves as "Other." This may also be true of Parents' Education, for which some students responded "1 don't know." Throughout this report, percentages less than 0.5 percent are reported as 0 percent. 20 14 THE 1990 NAEP TRIAL STATE ASSESSMENT Monuna SCHOOLS AND STUDENTS ASSESSED Table 2 provides a profile summarizing participation data for Montana schools and students sampled for the 1990 Trial State Assessment. In Montana, 100 public schools participated in the assessment. The weighted school participation rate was 90 percent, which means that all of the eighth-gradc students in this sample of schools were representative of 90 percent of the eighth-grade public-school students in Montana. TABLE 2 J Profile of the Population Assessed in Montana EIGHTH-GRADE PUSUC SCHOOL PARTICIPATION WeIghted school participation rate before substitution Weighted school participation rate after substitution Number of schools originally sampled Number of schools not eligible Number of schools in original sample participating Number ot substitute schools provided Number of substitute schools participating Total number of participating schools 90% 124 8 1GO 4 100 EIGHTH-GRADE Pt/WC-SCHOOL STUDENT PARTICIPATION Weighted student participation rate after make-ups Number of students Selected to participate in the assessment Number of students withdrawn from the assessment Percentage of students who were of Limited English Proficiency Percentage of students excluded from the assessment due to Limited English Proficiency Percentage of students who had an Individualized Education Plan Percentage of students excluded from the assessment due to Individualized Education Plan status Number of students to be assessed Number of students assessed 0% 7% 2% 2,588 2,48$ THE 1990 NAEP TRIAL STATE ASSESSMENT 15 Montana In each school, a random sample of students was selected to participate in the assessment. As estimated by the sample, 0 percent of the eighth-grade public-school population was classified as Limited English Proficient (LEP), while 7 percent hae an Individualized Education Plan (IEP). An IEP is a plan, written for a student who has been determined to be eligible for special education, that typically sets forth goals and objectives fcr the student and describes a program of activities and/or related services neces;ary to achieve the goals and objectives. Schools were permitted to exclude certain students from the assessment. To be excluded from the assessment, a student had to be categorized as Limited English Proficient or had to have an Individualized Education Plan and (in either case) be judged incapable of participating in the assessment. The students who were excluded from the assessment because they were categorized as LEP or had an IEP represented 0 percent and 2 percent of the population, respectively. In total, 2,486 eighth-grade Montana public-school students were assessed. The weighted student participation rate was 96 percent. This means that the sample of students who took part in the assessment was representative of 96 percent of the eligible eighth-grade public-school student population in Montana. 16 THE 199U NAEP TRIAL STATE ASSESSMENT Montana PART ONE ME NATION'S REPORT CARD How Proficient in Mathematics Are Eighth-Grade Students in Montana Public Schools? The 1990 Trial State Assessment covered five mathematics content areas -- Numbers and Operations; Measurement; Geometry; Data Analysis, Statistics, and Probability; and Algebra and Functions. Students' overall performance in these content areas was summarized on the NAEP mathematics scale, which ranges from 0 to 500. This part of the report contains two chapters that describe the mathematics proficiency of eighth-grade public-school students in Montana. Chapter 1 compares the overall mathematics performance of the students in Montana to students in the West region and the nation. It also presents the students average proficiency separately for the five mathematics content areas. Chapter 2 summarizes the students' oveiall mathematics performance for subpopulations defined by race/ethnicity, type of community, parents' education level, and gender, as well as their mathematics performance in the five content areas. THE 1990 NAEP TRIAL STATE ASSESSMENT 37 Montana CHAPTER 1 Students' Mathematics Performance As shown in Figure 2, the average proficiency of eighth-grad: public-school students from Montana on the NAEP mathematics scale is 280. This proficiency is higher than that of students across the nation (261).2 FIGURE 2 I Average Eighth-Grade Public-School I Mathematics Proficiency NAEP Uathematics Scale 200 225 250 275 300 500 Average Proficiency Montana 1118 0.8) 1-1414 West 2S1 ( 2.8) Nation 281 ( 1.4) The standard eirors are presented in parentheses. With about 95 percent certainty, the average mathematics proficiency for each population of interest is within ±2 standard errors of the estimated mean (95 percent confidence interval, denoted by F4.4). If the confidence intervals for the populations do not overlap, there is a statistically significant difference between the populations. 2 Differenms reported are staustically different at about thc 95 percent certamty level. This means that with about 95 percent certainty there is a real difference in the average mathematics proficiency between the two populations of interest. 18 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana LEVELS OF MATHEMATICS PROFICIENCY Average proficiency on the NAliP scale provides a global view of eighth graders' mathematics achievement; however, it does not reveal the specifics of what the students know and can do in the subject. To describe the nature of students' proficiency in greater detail, NAEP used the results from the 1990 national assessments of fourth-, eighth-, and twelfth-grade students to defme the skills, knowledge, and understandings that characterize four levels of mathematics performance -- levels 200, 250, 300, and 350 -- on the NAFP scale. To define the skills, knowledge, and understandings that characterize each proficiency level, mathematics specialists studied the questions that were typically answered correctly by most students at a particular level but answered incorrectly by a majority of students at the next lower level. They then summarized the kinds of abilities needed to answer each set of questions. While defining proficiency levels below 200 and above 350 is theoretically possible, so few students performed at the extreme ends of the scale that it was impractical to defme meaningful levels of mathematics proficiency beyond the four presented here. Definitions of the four levels of mathematics proficiency are given in Figure 3. It is important to note that the definitions of these levels are based solely on student performance on the 1990 mathematics assessment. The levels are not judgmental standards of what ought to be achieved at a particular grade. Figure 4 provides the percentages of students at or above each of these proficiency levels. In Montana, 100 percent of the eighth graders, compared to 97 percent in the nation, appear to have acquired skills involving simple additive reasoning and problem solving with whole numbers (level 200), However, many fewer students in Montana (23 percent) and 12 percent in the nation appear to have acquired reasoning and problem-solving skills involving fractions, decimals, percents, elementary geometric properties, and simple algebraic manipulations (level 300). CONTENT AREA PERFORMANCE As previously indicated, the questions comprising the Trial State Assessment covered five content areas -- Numbers and Operations; Measurement; Geometry; Data Analysis, Statistics, and Probability; and Algebra and Functions. Figure 5 provides the Montana, West region, and national results for each content area. Students in Montana performed higher than students in the nation in all of these five content areas. THE 1990 NAEP TRIAL STATE ASSESSMENT 19 Montana FIGURF 3 J Levels of Mathematics Proficiency LEVEL 200 THE NAIRM'S REPORT CARD Simple Additive Reasoning and Problem Solving with Whole Numbers Students at this level have some degree of understanding of simple quantitative relationships involving whole numbers. They can solve simple addition and subtraction problems with and without regrouping. Using a calculator, they can extend these abilities to multiplication and division problems. These students can identity solutions to one-step word problems and select the greatest four-digit number in a list, In measurement, these students can read a ruler as well as common weight and graduated scales. They also can make volume comparisons based on visualization and determine the value of coins. In geometry, these students can recognize simple figures. In data analysts, they are able to read simple bar graphs. In the algebra dimension, these students can recognize translations of word problems to numerical sentences and extend simple pattern sequences. LEVEL 250 Simple Multiplicative Reasoning and Two-Step Problem Sok:ing I Students at this level have extended their understanding of quantitative reasoning with whole numbers from additive to multiplicative settings. They can solve routine one-step multiplication and division problems involving remainders and two-step addition and subtraction problems involving money. Using a calculator, they can identify solutions to other elementary two-step word problems. In these basic problem-solving situations, they can identify missing or extraneous information and have some knowledge of when to use computational estimation. They have a rudimentary understanding of such concepts as whole number place value, "even," "factor," and "multiple." in measurement, these Students can use a ruler to measure objects, convert units within a system when the conversions require multiplication, and recognize a numerical expression solving a measurement word problem. In geometry, they demonstrate an initial understanding of basic terms and properties, such as parallelism and symmetry. In data analysis, they can complete a bar graph, sketCh a circle graph, and use information from graphs to solve simple problems. They are beginning to understand the relationship between proportion and probtibility. In algebra, they are beginning to deal informally with a variable through numerical substitution in the evaluation of simple expressions. 20 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana FIGURE 3 I Levels of Mathematics Proficiency (continued) I LEVEL 300 Reasoning and Problem Solving Involving Fractions, Decimals, Percents, Elementary Geometric Properties, and Simple Algebraic Manipulations Students at this level are able to represent, interpret, and perform simple operations With fractions and decimal numbers. They are able to locate fractions and decimals on number lineS, simplify fractions, and recognize the equivalence between common tractions and decimals, including pictorial representations. They can interpret the meaning of percents lesS than and greater than 100 and apply the concepts of percentages to solve simple problemS. These students demonstrate some evidence Of using mathematical notation to interpret expressions, Including those with exponents and negative integers. in measurement, these students can find the perimeters and areas of rectangles, recognize relationships among common units of measure, and use proportional relationships to solve routine problems involving similar triangles and scale drawings. In geometry, they have some mastery of the definitions and properties of geometric figures and solids. in data analysis, these Students can calculate averages, select and interpret data from tabular displays, pictographs, and line graphs, compute relative frequency distributions, and have a beginning understanding of sample bias. In algebra, they can graph points in the Cartesian plane and perform simple algebraic manipulations such as simplifying an expression by collecting like terms, identifying the solution to open linear sentences and inequalities by substitution, and checking and graphing an interval representing a compound inequality when it is described in words. They can determine and apply a rule for simple functional relatiOns and extend a numerical pattern. rLEVEL 350 Reasoning and Problem Solving Involving Geometric Relationships, Algebraic Equations, and Beginning Statistics and Probability Students at this level have extended their knowledge of number and algebraic understanding to include some properties of exponents. They can recognize scientific notation on a calculator and make the transition between scientific notation and decimal notation. In measurement, they can apply their knowledge of area ar,d perimeter of rectangles and triangles to solve problems. They can find the circumferences of circles and the surface areas of solid figures. In geometry, they can apply the Pythagorean theorem to solve problems involving indirect measurement. These students also can apply then knowledge of the properties of geometric figures to solve problems, such as determining the slope of a line. In data analysis, these students can compute mear,_ from frequency tables and determine the probability of a simple event. In algebra, they can identify an equation describing a linear relation provided in a table and solve literal equations and a system of two linear equations. They are developing an understanding of linear functions and their graphs, as well as functional notation, including the cimposition of functions. They can determine the nth term of a sequence and give counterexamples to disprove an algebraic generalization. THE 1990 NAEP TRIAL STATE ASSESSMENT 21 Montana FIGURE 4 I Levels of Eighth-Grade Public-School I Mathematics Proficiency LEVEL 350 State Region Nation LEVEL 300 State Region Nation LEVEL 250 State Region Nation LEVEL 200 State Region Nation iemmwail 1-44'44 11.1.1.1101W4 P-1.04 , 1-114 Flool 144 0 20 40 so 80 0 ( 0.1) ( 0.4) 0 ( 0.2) 23 ( 1.4) 12 ( 2.4) 12 ( 1.2) 88 ( 0.9) 63 ( 2.8) 64 ( 1.6) 100 ( 0.1) 97 ( 1.0) 97 ( 0.7) 100 Percentage at or Above Proficiency Levels The standard errors are presented in parentheses. With about 95 percent certainty, the value fo i. each population of interest is within ± 2 standard errors of the estimated percentage (95 percent confidence interval, denoted by 1-1-4). If the confidence intervals for the populations do not overlap, there is a statistically significant difference between the populations. 22 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana FIGURE 5 I Eighth-Grade Public-School Mathematics I Content Ares Performance State Rejlon Nation State Region Nation State Region Nation State Region Nation State Region Nation 200 225 250 275 300 Avorage Prenciency 282 ( 1.0) 264 ( 2.6) 268 ( 1.4) 279 ( 1.4) 258 ( 3.0) 258 ( 1.7) 280 ( 0.8) 280 ( 2.6) 259 ( 1.4) 282 ( 0.8) 282 ( 3.6) 252 ( 1.8) 278 ( 0.9) 259 ( 2.4) 280 ( 1.3) SOO Mathematics Submits Proficiency The standard errors are presented in parentheses. With about 95 percent certainty, the average mathematics proficiency for each population of interest is within ± 2 standard errors of the estimated mcan (95 percent confidence interval, denoted by i-H). If the confidence tntervals for the populations do not overlap, there is a statistically significant difference between the populations. THE 1990 NAEP TRIAL STATE ASSESSMENT 23 Montana CHAPTER 2 Mathematics Performance by Subpopulations In addition to the overall state results, the 1990 Trial State Assessment included reporting on the performance of various subgroups of the student population defined by race/ethnicity, type of community, parents' education level, and gender. RACE/ETHNICITY The Trial State Assessment results can be compared according to the different racial/ethnic groups when the number of students in a racial/ethnic group is sufficient in size to be reliably reported (at least 62 students). Average mathematics performance results for White, Hispanic, and American Indian students from Montana are presented in Figure 6. As shown in Figure 6, White students demonstrated higher average mathematics proficiency than did Hispanic or American Indian students. Figure 7 presents mathematics performance by proficiency levels. The figure shows that a greater percentage of White students than Hispanic or American Indian students attained level 300. 3 (i 24 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana FIGURE 6 I Average Eighth-Grade Public-School Mathematics Proficiency by Race/Ethnicity The standard errors are presented in parentheses. With about 95 percent certainty, the average mathematics proficiency for each population of interest is within ± 2 standard errors of the estimated mean (95 percent confidence interval, denoted by I-1-4). If the confidence intervals for the populations do not overlap, there is a statistically significant difference between the populations. ! Interpret with caution the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). THE 1990 NAEP TRIAL STATE ASSESSMENT 25 Montana ME MN'S REPORT FIGURE 7 I Levels of Eighth-Grade Public-School CARD I Mathematics Proficiency by Race/Ethnicity LEVEL 300 State White Hispanic Amer. Indian Region White Hispanic Amer. Indian Nation White Hispanic Amer. Indian LEVEL 250 State White Hispanic Amer. Indian R9Ion White Hispanic Amer. Indian Nation White Hispanic Amer. Indian LEVEL 200 State White Hispanic Amer. Indian Ragion White Hispanic Amer. Indian Nation White Hispanic Amer. Indian 20 40 60 80 Percentage at or Above Proficiency Levels The standard errors are presented in parentheses. With about 95 percent certainty, the value for each population of interest is within ± 2 standard errors of the estimated percentage (95 percent confidence interval, denoted by 1-1-1). If the confidence intervals for the populations do not overlap, there is a statistically significant difference between the populations. Proficiency level 350 is not presented in this figure because so few students attained that level. ! Interpret with caution the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 100 26 THE 1990 NAEP TRIAL STATE ASSESSMENT 26 ( 1.6) 4 ( 2.9) 6 ( 2.2) 16 ( 3.2) 3 ( 1.6) P001 I *al 15 ( 1.5) 3 ( 1.1) 1 ( 2.3)1 91 ( 0.7) 70 ( 7.0) 80 ( 8.8) 74 ( 3.3) 41 ( 5.4) laist 74 ( 1.8) 41 ( 4.5) 45 (16.0)1 100 ( 0.1) 99 ( 0.0) 911 ( 0.9) 99 ( 0.8) 03 ( 2.0) *. OS ( 0.4) 93 ( 1.6) 97 ( 6.7)I Montana TYPE OF COMMUNITY Figure 8 and Figure 9 present the mathematics proficiency results for eighth-grade students attending public schools in areas classified as "other" and extreme rural areas. (These are the "type of community" groups in Montana with student samples large enough to be reliably reported.) The results indicate that the avexage mathematics performance of the Montana students attending schools in areas classifted as "other" was higher than that of students attending schools in extreme rural areas. FIGURE 8 Average Eighth-Grade Public-School Mathematics Proficiency by Type of Community MEP liathematics $eale 200 225 250 275 300 500 Average Proficiency Montana Extreme rural Other weSt Extreme rural Other Nation Extreme rural Other The standard errors are presented in parentheses. With about 95 percent certainty, the average mathematics proficiency for each population of interest is within ± 2 standard errors of the estimated mean (95 percent confidence interval, denoted by 1.41). If the confidence intervals for the populations do not overlap, there is a statistically significant difference between the populations. ! Interpret with caution -- the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. THE 1990 NAEP TRIAL STATE ASSESSMENT 27 FIGURE 9 LEVEL 300 sm. Ext. rural Other Ext. rural Other Nation Ext. rural Other LEVEL 250 State Ext. rural Other Rai lion Ext. rural Other Nation Ext. rural Other LEVEL 200 slaw Ext. rural Other 9ion Ext. rural Other Nation Ext. rural Other Levels of Eighth-Grade Public-School Mathematics Proficiency by Type of Community 20 40 60 80 Percentage at or Above Proficiency Levels The standard errors are presented in parentheses. With about 95 percent certainty, the value for each population of interest is within ± 2 standard errors of the estimated percentage (95 percent confidence interval, denoted by 54-4). tf the confidence intervals for the populations do not overlap, there is a statistically significant difference between the populations. Proficiency level 350 is not presented in this figure because so few students attained that level. ! Interpret with caution the nature of the sample does not allow aocurate determination of the variability of this estimated mean proficiency. 18 ( 1.7) 28 ( 1.7) ( 4.8)1 10 ( 1.8) S ( 2.3)I 12 ( 1.2) 118 ( 2.4) IS ( 0.9) 52 (12.8)1 112 ( 5.0) SS ( 8.2)1 64 ( 2.3) 180 ( 0.3) 100 ( 0,1) 00 ( 1.3)1 Se ( 1.7) 97 ( 2.8)1 97 ( 1.0) 100 28 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana PARENTS' EDUCATION LEVEL Previous NAEP findings have shown that students whose parents are better educated tend to have higher mathematics proficiency (see Figures 10 and 11). In Montana, the average mathematics proficiency of eighth-grade public-school students having at least one parent who graduated from college was approximately 19 points higher than that of students who reported that neither parent graduated from high school. As shown in Table 1 in the Introduction, a larger percentage of students in Montana (45 percent) than in the nation (39 percent) had at least one parent who graduated from college. In comparison, the percentage of students who reported that neither parent graduated from high school was 5 percent for Montana and 10 percent for the nation. FIGURE 10 I Average Eighth-Grade Public-School 1 Mathematics Proficiency by Parents' Education NAEP Mathematics Scat. 0 200 225 250 275 300 500 118101? CMS Average Pro &fancy Montana Hui HS non-graduate in( 2.3) 144 HS graduate 271 ( tS) HI Some college in( 1.1) College graduate ( 1.0) West HS non-graduate 31S ( 4.4) HS graduate in( 2.2) Some college ( 340) College graduate fin( 2.4) Nation 1404 hiS non-graduate 242 ( 2.0) 14 HS graduate 244 ( 1.5) P44 Some college 2120 ( 1.7) College graduate 274 ( 1.6) The standard errors are presented in parentheses. With about 95 percent certainty, the average mathematics proficiency for each population of interest is within ± 2 standard errors of the estimated mean (95 percent confidence interval, denoted by P4-I). If the confidence intervals for the populations do not overlap, there is a statistically significant difference between the populations. THE 1990 NAEP TRIAL STATE ASSESSMENT 2.9 Montana ,11. 111E WON'S rPaRT FIGURE II I Levels of Eighth-Grade Public-School .JUID I Mathematics Proficiency by Parents' Education LEVEL 300 State HS non-grad. HS greduate Some college College grad. notion HS non-grad. HS graduate Some college College grad. Nation HS nOn-grad. HS graduate Some college College grad. LEVEL 250 State HS non-grad. HS graduate Some college College grad. RegIon HS non-grad. HS graduate Some college College grad. Nation HS non-grad. HS graduate Some college College grad. LEVEL 200 State HS non-grad. HS graduate Some college College grad. Region HS non-grad. HS graduate Some college College grad. NatIon HS non-grad. HS graduate Some college College grad. 20 40 80 80 Percentage at or Above ProfIcleocy Levels The standard errors are presented in parentheses. With about 95 percent certainty, the value for each population of interest is within ± 2 standard errors of the estimated percentage (95 percent confidence interval, denoted by 14-4). If the confidence intervals for the populations do not overlap, there is a statistically significant difference between the populations. Proficiency level 350 is not presented in this figure because so few students attained that level. 100 36 30 THE 1990 NAEP TRIAL STATE ASSESSMENT ( 2.8) 11 ( 1.7) 20 ( 3.1) 32 ( 1.8) 2 ( 2.3) 2 ( 1.3) 15 ( 2.8) ( 3.5) 1 ( 0.9) 5 ( 1.5) 12 ( 1.4) 21 ( 1.9) ( 3.7) ( 3.0) ( 1.9) ( 1.2) ( 8.8) ( 4.4) ( 4.1) ( 3.6) 37 ( 4.6) 56 ( 2.7) 71 ( 2.6) 78 ( 2.0) loo 00:30)) 100 ( 0.1) 100 ( 0.0) ( 2.2) ( 1.6) ( 0.7) ( 0.7) ( 1,9) ( 0.8) ( 0.7) ( 0.7) Montana GENDER As shown in Figure 12 eighth-grade males in Montana had a higher average mathematics pmficiency than did eighth-grade females in Montana. Compared to the national results, females in Montana performed higher than females across the country; males in Montana performed higher than males amass the country. FIGURE 12 I Average Eighth-Grade Public-School Mathematics Proficiency by Gender NAEP liathemalks Scale 200 225 250 275 300 500 Average Prosciana 1+4 eN Montana Male Fet1111116 West Male Female Nation Male ftil) Female The standard errors are presented in parentheses. With about 95 percent certainty, the average mathematics proficiency for each population of interest is within ± 2 standard errors of the estimated mean (95 percent confidence interval, denoted by 1-4-I). If the confidence intervals for the populations do not overlap, there is a vatistically significant difference between the populations. As shown in Figure 13, there was no difference between the percentages of males and females in Montana who attained level 200. The percentage of females in Montana who attained level 200 was greater than the percentage of females in the nation who attained level 200. Also, the percentage of males in Montana who attained level 200 was greater than the percentage of males in the nation who attained level 200. THE 1990 NAEP TRIAL STATE ASSESSMENT 31 Montana FIGURE 13 I Levels of Eighth-Grade Public-School Mathematics Proficiency by Gender LEVEL 300 State Male Female Region Male Female Nation Male Female LEVEL 250 State Male Female Region Male Female Nation Male Female LEVEL 200 Stat. Male Female Region Male Female Nation Male Female 0 20 40 SO 60 100 Percentage at or Above Proficiency Levels The standard errors are presented in parentheses. With about 95 percent certainty, the value for each population of interest is within ± 2 standard errors of the estimated percentage (95 percent confidence interval, denoted b; P+4). If the confidence intervals for the populations do not overlap, there is a statistically signthcant difierence between the populations. Proficiency level 350 is not presented in this figure because so few students attained that level. :3 32 THE 1990 NAEP TRIAL STATE ASSESSMENT 22 ( 1.9) 19 ( 2.4) 13 ( 3.1) 11 ( 2.2) 14 ( 1.7) 10 ( 1.3) ( 1.4) 117 ( 1$) 65 ( 4,1) 61 ( 3.2) 64 ( 2.0) 64 ( 1,8) 100 ( 0.1 ) 100 ( 0.1) 97 ( 1,2) ( 1.0) 97 ( 09) 97 ( 0.8) Montana In addition, a greater percentage of males than females in Montana attained level 300. The percentage of females in Montana who attained level 300 was greater than the percentage of females in the nation who attained level 300. Also, the percentage of males in Montana who attained level 300 was greater than the percentage of males in the nation who attained level 300. CONTENT AREA PERFORMANCE Table 3 provides a summary of content area performance by race/ethnicity, type of community, parents' education level, and gender. THE 1990 NAEP TRIAL STATE ASSESSMENT 33 Montana TABLE 3 I Eighth-Grade Public-School Mathematics Content Area Performance by Subpopulations AVERAGE MATHEMATICS PROFICIENCY OF STUDENTS 1100 NAEP TRIAL STATE ASSESSMENT timbers and Operations linalarallnana " " CM. Data Analysis, Inadades' and Probability AlgebraFunctionsand TOTAL State Region Nation RACUETHN1C1TY Mite State Region Nation lapanIc State Region Nation American Indian State Region Nation TYPE OF COMMUNITY Bennie rural State Region Nation Other State Region Nation Pralidsocw 2421 211 200 265 ( 1.0) 271 ( 3.2) 273 ( 1.8) 267 ( 3.1) 24$ ( 3.5) 248 ( 2.7) 25$ ( 3.4) 4.11 249 ( 7.8)1 280 ( 1.7) 254 ( 8.8)1 258 ( 4.3)1 253 ( 1.1) 262 ( 3,5) 206 ( 1.9) Pra0049nay Prolkienty 271 A1 2110 ( Prold0noV 0.esi Pracioncy 273 ( 0.9) ILO) 200 ( 2.0 20242 25S ( 2.4) 254 1.7) 254 ( 202 124) 200 ( 1.3) 243 ( 1.4) 263 ( 0.9) 215 ( 0.7) 201 ( OA) 207 ( 3.9) 267 ( 3.0) 272 ( 4.4) 267 ( 2.6) 207 ( 2.0) 267 ( 1.5) 272 ( 1.8) 266 ( 1.4) 257 ( 6.3) 261 ( 3.' 261 ( 3.7) 260 ( 3.0 239 ( 4.2) 245 ( 4.4) 240 ( 4.7) 243 ( 4.0) 231 ( 3.4) 243 ( 3.2) 230 ( 3.4) 243 ( 3.1) 256 ( 3.1) 256 ( 4.5) 255 ( 4.4) **ft) 1144 247 ( di)! 243 ( 8.8)1 242 ( 5.2)1 242 ( 4.0)1 275 ( 2.3) 27$ ( 1.7) 270 ( 2.2) 274 ( 1.6) 254 ( 4.6)! 252 $.4)1 253 ( 8.6)1 251 ( 8.5)1 254 ( 4.2)! 253 ( 4.5)t 267 ( 5.0)1 268 ( 4.8)1 282 ( 1.7) 251 ( 1.1) 284 ( 0.9) 280 ( 1.1) 255( 4.2) 258 ( 3,4) 250 ( 42) 258 ( 257 ( 2.4) 2541( 1.7) 281 ( 22) 261 ( 1.7) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. ! Interpret with caution -- the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 34 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE 3 I Eigbth- ti Public-School Mathematics (continued) I Content Area Performance by Subpopulations AVERAGE MATHEMATICS PROFICIENCY OF STUDENTS , 1990 NAEP TRIAL STATE ASSESSMENT . - timbers and Operations _ Measurement Geometry - Data Andysis, Mad Mkt 8116 Probability - Algebra and Functions . TOTAL Pralkiency !Wow *el Mem MI Mew Siolisiomv State 242 (1.0) 279 ( 1.41 200 ( 0.1 1 262 ( 0.1) 27111 Region 264 (2.6) Nation 2036 (1.4) 25$ } 221191 rot 1.si PARES rs' 200061106 NS non-graduate State 272 ( 3,4) 285 ( 3.9) 270 ( 264 ( 3.1) 264 ( Region 248 ( 4.2) 242 ( 82) 246 `, 4.P) 246 ( 6.2) 245 ( 6.1 Nation 247 ( 2.4) 237 ( 3.6) 242 ( 240 ( 3.1) 242 ( HS imitate State 274 ( 2.1) 266 ( 2.7) 271 ( 2.1) 272 ( 1.9) 269 ( 2.4) Region 254 ( 2.5) 245 ( 3.0) 251 ( 3.6) 249 ( 3.2) 250 ( 2.4) Nation 249 ( 1.1) 2411 ( 2.1) 252 ( 1.6) 233 ( 22) 253 ( 2.0) Some ciAlitge State 284 ( 2.0) 283 ( 2.3) 282 ( 2.7) 247 ( 1.9) 260 1.7) Region 272 ( 2.7) 268 ( 5.3) 264 ( 3.9) 271 ( 4.9) 264 3.2) Nation 270 ( 1.5) 264 ( 2.7) 262 ( 2.0) 209 ( 2.4) 263 2.2) Doi Mge graduate State 289 ( 1.4) 288 ( 1.6) 285 ( 1.2) 289 ( 1.1) 285 ( 1.1) Region 275 ( 2.7) 271 ( 3.0) 271 ( 2.3) 276 ( 4.3) 272 ( 2.6) Nation 278 ( 1.8) 272 ( 2.0) 270 ( 1.6) 276 ( 2.2) 273 ( 1.7) GENDER MMe State 285 ( 1.3) 285 ( 1.8) 253 ( 1.2) 265 ( 1.3) 279 ( 1.4) Region 264 ( 3.8) 263 ( 3.5) 261 ( 34) 264 ( 4.1) 200 ( 3.3) Notion 266 ( 2.0) 262 ( 2.3) 260 ( 1.7) 262 ( 2.1) 200(1.6) Femal State 279 ( 1.5) 274 ( 1.9) 276 ( 1.6) 279 ( 1.2) 277 ( 1.4) Region 263 ( 2.5) 252 ( 2.9) 259 ( 2.9) 200 ( 4.0) 259 ( 2.8) Nation 266 ( 14) 253 ( 1.6) 258 ( 1.5) 261 ( 1.9) 260 ( 14) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. THE 1990 NAEP TRIAL STATE ASSESSMENT 35 Montana THE NATION'S REPORT CARD PART TWO Finding a Context for Understanding Students' Mathematics Proficiency Information on students' mathematics proficiency is valuable in and of itself, but it becomes more useful for improving instruction and setting policy when supplemented with contextual information about schools, teachers, and students. To gather such information, the students r ating in the 1990 Trial State Assessment, their mathematics teachers, and the princip e.! other administrators in their schools were asked to complete questionnaires on policies, instruction, and programs. Taken together, the student, teacher, and school data help to describe some of the current practices and emphases in mathematics education, Illuminate some of the factors that appear to be related to eighth-grade public-school students' proficiency in the subject, and provide an educational context for understanding information on student achievement. It is important to note that the NAEP data cannot establish cause-and-effect links between various contextual factors and students' mathematics proficiency. However, the results do provide information about important relationships between the contextual factors and proficiency. The contextual information provided in Part Two of this report focuses on four major areas: instructional content, instructional practices, teacher qualifications, and conditions beyond school that facilitate learning and instruction -- fundamental aspects of the educational process in the country. THE 1990 NAEP TRIAL STATE ASSESSMENT 37 Montana Through the questionnaires administered to students, teachers, and principals, NAEP is able to provide a broad picture of educational practices prevalent in American schools and classrooms. In many instances, however, these findings contradict our perceptions of what school is like or educational researchers' suggestions about what strategies work best to help students learn. For example, research has indicated new and more successful ways of teaching and learning, incorporating more hands-on activities and student-centered learning techniques; however, as described in Chapter 4, NAEP data indicate that classroom work is still dominated by textbooks or worksheets. Also, it is widely recognized that home environment has an enormous impact on future academic achievement. Yet, as shown in Chapters 3 and 7, large proportions of students report having spent much more time each day watching television than doing mathematics homework. Part Two consists of five chapters. Chapter 3 discusses instructional content and its relationship to students' mathematics proficiency. Chapter 4 focuses on instructional practices -- how instruction is delivered. Chapter 5 is devoted to calculator use. Chapter 6 provides information about teachers, and Chapter 7 examines students' home support for learning. 40 tj 38 1 IE 1990 NAEP TRIAL STATE ASSESSMENT CHAPTER 3 What Are Students Taught in Mathematics? In response to the continuing swell of information about the poor mathematics achievement of American students, educators and policymakers have recommended widespread reforms that are changing the direction of mathematics education. Recent reports have called for fundamental revisions in curriculum, a reexamination of tracking practices, improved textbooks, better assessment, and an increase in the proportions of students in high-school mathematics programs.' This chapter focuses on curricular and instructional content issues in Montana public schools and their relationship to students' proficiency. Table 4 provides a profile of the eighth-grade public schools policies and staffing. Some of the salient results are as follows: About half of the eighth-gxade students in Montana (50 percent) were in public schools where mathematics was identified as a special priority. This compares to 63 percent for the nation. 3 Curtis McKnight, et al., The Underachieving Currkulum Assessing U.S. School Mathematics from an International Perspective, A National Report on the Second International Mathematics Study (Champaign, IL: Stipes Publishing Company, 1987). Lynn Steen, Ed. Everybody -aunts A Report to the Nation on the Future of Mathematics Education (Washington, DC: National Academy Press, 1989). . 4 THE 1990 NAEP TRIAL STATE ASSESSMENT 39 Montana In Montana, 46 percent of the students could take an algebra course in eighth grade for high school course placement or credit. About three-quarters of the students in Montana (77 percent) were taught mathematics by teachers who teach only one subject. About half (46 percent) of the students in Montana were typically taught mathematics in a class that was grouped by mathematics ability. Ability grouping was more prevalent across the nation (63 percent). TABLE 4 I Mathematics Policies and Practices in Montana 1 Eighth-Grade Public Schools PERCENTAGE OF STUDENTS 1900 NAEP TRIAL STATE ASSESSMENT Montana West Nation _ Percentage of eighth-grade students In public schools that Identified mathematics as receiving special emphasis in school-wide goals and objectives, Instruction, in-service training, etc. Percentage of elghth-grade public-school students who are offered a course in algebra for high school course placement or credit Percentage of eighth-grade students in public schools who are taught by teachers My° teach only mathematics Percentage of eighth-grade students in public schools who are usignad to a mathematics due by their abNIty in mathematics Percentage of eighth-grade students in public schools who receive fair or more hors of mathematics instruction per week Percentage Percentage Pmentailw 50 ( 3.0) 01 ( 8.8) 63 ( 5,9) 40 ( 2.0) 92 ( 4.7) 78 ( 4.6) 77 ( 3.0) 96 ( 1.6) 91 ( 3.3) 40 ( 2.9) 64 ( 8.3) 63 ( 4.0) 49 ( 2.9) 25 ( 5.9) 30 ( 4.4) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. THE 1990 NAEP TRIAL STATE ASSESSMENT Montana CURRICULUM COVERAGE To place students' mathematics proficiency in a curriculum-related context, it is necessary to =mine the extent to which eighth graders in Montana art taldng mathematics courses. Based on their revonses, shown in Table 5: A greater percentage of students in Montana were taking eighth-grade mathematics (59 percent) than were taldng a course in pre-algebra or algebra (37 percent). Across the nation, 62 percent were taking eigAth-grade mathematics and 34 percent were taking a course in pm-algebra or algebra. Students in Montana who were enrolled in pre-algebra or algebra courses exhibited higher average mathematics proficiency than did those who were in eighth-grade mathematics courses. This result is not unexpected since it is assumed that students enrolled in pre-algebra and algebra courses may be the more able students who have already mastered the general eighth-grade mathematics curriculum. TABLE 5 I Students' Reports on the Mathematics Class I They Are Taking PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 11190 NAP TRIAL STATE ASSESSMENT I Montana West Nation Percentage and Prilidenay Percenlage and Preidency Percentage and Preiciency What kind of mathematics class are you taking this year? Eighth-grade mathematics 59 ( 2.4) 63 ( 2.7) 62 ( 2.1) 27$ ( tO) 252 ( 2.4) 251 ( 1.4) Pro-algebra 26 ( 1.9) 15 ( 2.7) 19 ( 1.9) 251 ( 1.1) 200 ( 3.6) 272 ( 2.4) Affiebn 12 ( 1.5) 17 ( 1.5) 15 ( 1.2) 299 ( 3.8) 299 ( 4.5) 298 ( 2.4) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. The percentages may not total 100 percent because a small number of students reported taking other mathematics courses. THE 1990 NAEP TRIAL STATE ASSESSMENT 41 Montana Further, from Table A5 in the Data Appendix:* About the same percentage of females (36 percent) and males (38 percent) in Montana were enrolled in pre-algebra or algebra courses. In Montana, 36 percent of White students, 35 percent of Hispanic students, and 49 percent of American Indian students were enrolled in pre-algebra or algebra courses. Similarly, 38 percent of students attending schools in areas classified as "other" and 37 percent in schools in extreme rural areas were enrolled in pre-algebra or algebra courses. MATHEMATICS HOMEWORK To illuminate the relationship between homework and proficiency in mathematics, the assessed students and their teachers were asked to report the amount of time the students spent on mathematics homework each day. Tables 6 and 7 report the teachers' and students' responses, respectively. According to thfir teachers, the greatest percentage of eighth-grade students in public schools in Montana spent 30 minutes doing mathematics homework each day; according to the students, the greatest percentage spent either 15 or 30 minutes doing mathematics homework each day. Across the nation, according to their teachers, tilt:, largest percentage of students spent either 15 or 30 minutes doing mathematics homework each day, while students reported spending either 15 or 30 minutes daily. Further, as reported by their teachers (Table 6 and Table A6 in the Data Appendix): In Montana, 3 percent of the students spent no time each day on mathematics homework, compared to 1 percent for the nation. Momover, 3 percent of the students in Montana and 4 percent of the students in the nation spent an hour or more on mathematics homework each day. 4 For every table in the body of the report that includes estimates of average proficiency, the Data Appendix provides a corresponding table presenting the results for the four subpopulations -- race ethnicity, type of community, parents' education level, and gender. 42 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana The results by race/ethnicity show that 3 percent of White students, 3 percent of Hispanic students, and 6 percent of American Indian students spent an hour or more on mathematics homework each day. In comparison, 3 percent of White students, 3 percent of Hispanic students, and 5 percent of American Indian students spent no time doing mathematics homework. In addition, 4 percent of students attending schools in areas classified as "other" and 2 percent in schools in extreme rural areas spent an hour or more on mathematics homework daily. In comparison, 3 percent of students attending schools in areas classified as "other" and 3 percent in schools in extreme rural areas spent no time doing mathematics homework. TABLE 6 Teachers' Reports on the Amount of Time Students Spent on Mathematics Homework Each Day PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1900 NAEP TRIAL STATE ASSESSMENT Montana West Nation _ . Parasite", nd Proficiency Pwaentese Paraistase Sad Prialmay About how much time do students spend on mathematics homework each day? None 3 ( 44, ( 0.4) 44) 1 ( 0.3) 1 ( «14 ( 0.3) 16 minutes 38 ( 21) 42 ( 6.7) 43 ( 42) 260 ( 1.2) 255 ( 4.2) 250 ( 2.3) 30 minutes 50 ( 2.8) 43 ( 62) 43 ( 43) 263 ( 1.2) 284 ( 4.7) 203 ( 2.0) 46 minutes 0 ( 1.7) 9 ( 2.3) 1.10) 280 ( 2.6)1 270 ( 8.5)1 272 ( 51 )1 An hour or more 3 ( 0.5) ( 1.9) 4 ( 04) 252 (12.1) 276 ( 5.1)f The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. ! Interpret with caution the nature of the umple does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). THE 1990 NAEP TRIAL STATE ASSESSMENT 43 Montana TABLE 7 Students' Reports on the Amount of Time They Spent on Mathematics Homework Each Day PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 11100 NAEP TRIAL STATE ASSESSMENT Montana West Nation About how much time do you usually spend each day on mathematics homework? None 15 minutes 30 minutes 45 minutes An hour or more Panne lap Perverdege Percordefe end and end Proficiency kedelleiey Prefirgency ( OA) 279 ( 2.9) 29 ( 1.3) 254 ( 1.3) 33 ( 1.1) 280 ( 1.3) 18 ( 0.7) 280 ( 1.4) 13 ( 0.5) 275 ( 1.9) 12 ( 1.7) 254 ( 4.2) 31 ( 43) 283 ( 3.5) 2$ ( 1.7) 251 ( 2.9) 15 ( 1.8) 287 ( 4.2) 14 ( 1.1) 251 ( 43) The standard errors of the estimated statistics appear in parentheses. lt can be said witii about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. And, according to the students (Table 7 and Table A7 in the Data Appendix): In Montana, relatively few of the students (9 percent) reported that they spent no time each day on mathematics homework, compared to 9 percent for the nation. Moreover, 13 percent of the students in Montana and 12 percent of students in the nation spent an hour or more each day on mathematics homework. The results by race/ethnicity show that 11 percent of White students, 18 percent of Hispanic students, and 21 percent of American Indian students spent an hour or more on mathematics homework each day. In comparison, 9 percent of White students, 17 percent of Hispanic students, and 7 percent of American Indian students spent no time doing mathematics homework. 44 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana In addition, 12 percent of students attending schools in areas classified as "other" and 13 percent in schools in extreme rural areas spent an hour or more on mathematics homework daily. In comparison, 10 percent of students attending schools in areas classified as "other" and 8 percent in schools in extreme rural areas spent no time doing mathematics homework. INSTRUCTIONAL EMPHASIS According to the approach of the National Council of Teachers of Mathematics (NCTM), students should be taught a broad range of mathematics topics, including number concepts, computation, estimation, functions, algebra, statistics, probability, geometry, and measurement.' Because the Trial State Assessment questions were designed to measure students' knowledge, skills, and understandings in these various content areas -- regardless of the type of mathematics class in which they were enrolled -- the teachers of the assessed students were asked a series of questions about the emphasis they planned to give specific mathematics topics during the school year. Their responses provide an indication of the students' opportunity to learn the various topics covered in the assessment. For each of 10 topics, the teachers were asked whether they planned to place "heavy," "moderate," or "little or no" emphasis on the topic. Each of the topics corresponded to skills that were measured in one of the five mathematics content areas included in the Trial State Assessment: Numbers and Operations. Teachers were asked about emphasis placed on five topics: whole number operations, common fractions, decimal fractions, ratio or proportion, and percent. Measurement. Teachers were asked about emphasis placed on one topic: measurement. Geometry. Teachers were asked about emphasis placed on one topic: geometry. Data Analysis, Statistics, and Probability. Teachers were asked about emphasis placed on two topics: tables and graphs, and probability and statistics. Algebra and Functions. Teachers were asked about emphasis placed on one topic: algebra and functions. Nanona1 Council of Teachers of Mathematics. Currieultort and Evaluation Standards for School Alathonatk.t (Reston, VA: Natronal Council of Teachers of Mathematics, 1989). THE 1990 NAEP TRIAL STA] E ASSESSMENT 45 Montana The responses of the assessed students' teachers to the topic emphasis questions for each content area were combined to create a new variable. For each question in a particular content area, a value of 3 was given to "heavy emphasis" responses, 2 to "moderate emphasis" responses, and 1 to "little or no emphasis" responses. Each teacher's responses were then averaged over all questions related to the particular content area. Table 8 provides the results for the extreme categories -- "heavy emphasis" and "little or no emphasis" -- and the average student proficiency in each content area. For the emphasis questions about numbers and operations, for example, the proficiency reported is the average student performance in the Numbers and Operations content area. Students whose teachers placed heavy instructional emphasis on Algebra and Functions had higher proficiency in this content area than students whose teachers placed little or no emphasis on Algebra and Functions. Students whose teachers placed heavy instructional emphasis on Numbers and Operations had lower proficiency in this content area than students whose teachers placed little or no emphasis on Numbers and Operations. fir; i 46 THE 1990 NAEP TRIAL STATE ASSESSMENT .1111=1M. Montana TABLE 8 Teachers' Reports on the Emphasis Given to Specific Mathematics Content Areas .11I PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 11190 NAV TRIAL STATE ASSESSMENT 1- Montana West Won leacher °emphasis° categories by content areas Ihromia. Prellialsney Perusises lireiki10104, ParomMes Prolielony Numbers and Operations Heavy emphasis 40 ( 2.9) 42 ( 1.4) 49 290(2.0) 2117 IlLS) 290 1".ei tittle or no emphasis 14 ( 1.3) 13 ( 2.1) 15 ( 2.1) 293 ( 2.9) 291(6.9) 217 ( 3.4) Measunament Heavy emphasis 9 ( 1.0) 11 ( 2.3) 17 ( 3.0) 277 ( 5.7) 251 ( 7.7)4 250 ( 5.5) Little or no emphasis 23 ( 1.9) 36 ( 5.3) 33 ( 4.0) 2115 ( 3.3) 275 ( 8.3) 272 ( 4.0) Geometry Heavy emphasis 31 ( 2.5) 24 ( 8.3) 211 ( 3.5) 26e1 ( 1.5) 260 ( 25)4 2e0 ( 32) Little or no emphasis 13 ( 1.4) le ( 4.5) 21 ( 13) 279 ( 2.8) 277 (11.4)1 264 ( 54) Data Analysis,Statistics, and Probabillty Heavy emphasis 13 ( 2.3) 14 ( 3.7) 14 ( 22) 207 ( 3.0) 254 (10.8)1 259 ( 4,3) Little or no emphasis 59 ( 2.5) 54 ( 6.3) 53 ( 4.4) 281 ( 1.4) 262 ( 4.9) 261 ( 2,9) Algebra and Functions Heavy emphasis 58 ( 3.0) 43 ( 5.0) 48 ( 3.0) 281 ( 13) 277 ( 5.2) 275 ( 2.5) Little or no emphasis 10 ( 2.9) 23 ( 5.1) 20 ( 3.0) ( 3.2)1 243 ( 4.2)1 243 ( 10) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population or interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. The percentages may not total 100 percent because the "Moderate emphasis" category is not included. Interpret with caution the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. THE 1990 NAEP TRIAL STATE ASSESSMENT 47 Montana SUMMARY Although many types of mathematics learning can take place outside of the school environment, there are some topic areas that students are unlikely to study unless they are covered in school. Thus, what students are taught in school becomes an important determinant of their achievement. The information on curriculum coverage, mathematics homework, and instructional emphasis has revealed the following: About half of the eighth-grade students in Montana (50 percent) were in public schools where mathematics was identified as a special priority. This compares to 63 percent for the nation. In Montana, 46 percent of the students could take an algebra course in eighth grade for high-school course placement or credit. A greater percentage of students in Montana were taking eighth-grade mathematics (59 percent) than were taking a course in pre-algebra or algebra (37 percent). Across the nation, 62 percent were taking eighth-grade mathematics and 34 percent were taking a course in pre-algebra or algebra. According to their teachers, the greatest percentage of eighth-grade students in public schools in Montana spent 30 minutes doing mathematics homework each day; according to the students, most of them spent either 15 or 30 minutes doing mathematics homework each day. Across the nation, teachers reported that the largest percentage of students spent either 15 or 30 minutes doing mathematics homework each day, while students reported either 15 or 30 minutes daily. In Montana, relatively few of the students (9 percent) reported that they spent no time each day on mathematics homework, compared to 9 percent for the nation. Moreover, 13 percent of the students in Montana and 12 percent of students in the nation spent an hour or more each day on mathematics homework. Students whose teachers placed heavy instructional emphasis on Algebra and Functions had higher proficiency in this content area than students whose teachers placed little or no emphasis on Algebra and Functions. Students whose teachers placed heavy instructional emphasis on Numbers and Operations had lower proficiency in this content area than students whose teachers placed little or no emphasis on Numbers and Operations. 48 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana CHAPTER 4 rixs SIM 111/11111111111 all 111141111111111111111111111 SIM 111111111111111/ ii 1111111111111i111111111111/1111 1.111111111111111Willill RIM II NUM MEM :1111/.1111111 Nnossommuss. -ma swam MOM" ,If,gang AM. MIMI MINOS 11111111111111 How Is Mathematics Instruction Delivered? Teachers facilitate learning through a variety of instructional practices. Bocause a particular teaching method may not be equally effective with all types of students, selecting and tailoring methods for students with different styles of learning or for those who come from different cultural backgrounds is an important aspect of teaching.° An inspection of the availability arid use of resources for mathematics education can provide insight into how and what students are learning in mathematics. To provide information about how instruction is delivered, students and teachers participating in the Trial State Assessment were asked to report on the use of various teaching and learning activities in their mathematics classrooms. AVAILABILITY OF RESOURCES Teachers' use of resources is obviously constrained by the availability of those resources. Thus, the assessed students' teachers were asked to what extent they were able to obtain all of the instructional materials and other resources they needed. National Council of Teachers of Mathematics, Professional Standards.for the Teaching of Mathematics (Reston, VA: National Council of Teachers of Mathematics, 1991). THE 1990 NAEP TRIAL STATE ASSESSMENT 49 Montana From Table 9 and Table A9 in the Data Appendix: In Montana, 17 percent of the eighth-grade students had mathematics teachers who reported getting all of the resources they needed, while 21 percent of the students were taught by teachers who got only sane or none of the resources they needed. Across the natior, , these figures were 13 percent and 31 percent, respectively. In Montana, 19 percent of students attending schools in areas classified as "othet" and 14 percevt in schools in extreme rural areas had mathematics teachers who got all the resources they needed. By comparison, in Montana, 22 percent of students attending schools in areas classified as "other" and 20 percent in schools in extreme rural areas were in classrooms where only some or no ICSOUTCCS were available. Students whose teachers got all the resources they needed had mathematics achievement levels similar to those whose teachers got only some or none of the resources they needed. TABLE 9 I Teachers' Reports on the Availability of I Resources PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY - , 111100 NAEP TRIAL STATE ASSESSMENT Montana Iftst Mahon - Which of the following statements is true about how well supplied you are by your school system with the Instructional materials and other resources you need to teach your class? I get all the resources I need. I get most of Ow resourcos I med. I get some of none of the resources I need Peee.--ereago No:1May Ponentego and and and Proloiancy Prelidancy Prolitiona 17 ( 22) 15 ( 5.2) 13 ( 2.4) 265 ( 1.9) 261 ( La) 205 ( 4.2) 02 ( 3.0) 52(3.8) Se ( 4.0) 250 ( 1.1) 266 ( 4.1) ( 2.0) 21 ( 2.7) 23 ( 6.4) 31 ( 4.2) 2110 ( 1.3) 257 ( 3.7)1 261 ( 2.0) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sar-tple. Interpret with caution -- the nature of the sample does not allow accurate determination of the vari lity of this estimated mean proficiency. ta 50 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana PATTERNS IN CLASSROOM INSTRUCTION Research in education and cognitive psychology has yielded many insights into the types of instructional activities that facilitate students' mathematics learning. Increasing the use of "hands-on" examples with concrete materials and placing problems in real-world contexts to help children construct useful meanings for mathematical concepts are among the recommended approaches.' Students' responses to a series of questions on their mathematics instruction provide an indication of the extent to which teachers are making use of the types of student-centered activities suggested by researchers. Table 10 presents data on patterns of classroom practice and Table 11 provides information on materials used for classroom instmction by the mathematics teachers of the assessed students. According to their teachers: More than half of the students in Montana (60 percent) worked mathematics problems in small groups at least once a week; relatively few never worked mathematics problems in small gxoups (8 percent). The largest percentage of the students (62 percent) used objects like rulers, counting blocks, or geometric shapes less than once a week; relatively few never used such objects (1 percent). In Montana, 62 percent of the students were assigned problems from a mathematics textbook almost every day; 6 percent worked textbook problems about once a week or less. About half of the students (47 percent) did problems from worksheets at least several times a week; about one-quarter did worksheet problems less than weekly (30 percent). Thomas Romberg, "A Common Curriculum for Mathematics," individual Differences and the Common Curriculum Eighty-second Yearbook of the National Society for the Study of Education (Chicago, University of Chicago Press, 1983). THE 1990 NAEP TRIAL STATE ASSESSMENT 51 Montana TABLE 10 I Teachers' Reports on Patterns of Mathematics Instruction PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1900 MEP TRIAL STATE ASSESSMENT Montana West Nation _ or penandamo 'Ad Peramtags and Peranulais and About how often do students work problems in small groups? Pirsialinay Pralialanay Praliokosy At least ones a week 130 ( 3.0) ST ( 8.9) 50 ( 4.4) 282 ( 1.2) 262 ( 4.2)! 260 ( 2.2) Less than once a week 32 ( 2.9) 39 ( 7.6) 43 ( 4.1) 290 ( 1.9) 266 ( 4.5) 264 ( 2.3) Now ( 1.8) 8 ( 2.0) 290 ( 2.3)! 277 ( 5.4)1 About how often do students use objects Percentage Pengentage Percentage like rulers, counting blocks, or geometric awl end and solids? Pnalidiancy Pralaiency Proildancy At least once a week 37 ( 2.6) 262 ( 1.7) 34 ( 8.2) 256 ( 4.9)! , 254 ( 3.2) LOU Man once a weak 82 ( 2.6) 57 ( 6.4) 69 ( 3.9) 281 ( 1.0) 265 ( 4.0) 263 ( 1.9) New ( 0.4) .4* ( 8 ( 3.0) .4") 9 ( 2.6) 282 ( 5.9)1 The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. ! Interpret with caution the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 52 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE 11 I Teaches' Reports on Materials for I Mathematics Instruction PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY - , 1800 MAEP TRIAL STATE ASSESSMENT Montana West Nation About how often do students do problems Proontage mid Peramtage Penteitags and from textbooks? ProMow fralloissiv Pradany Almost ovary day 92 ( 3A) 55 ( to) 62(3.4) 234 ( 1.0) 270 ( LS) 1.1) Swami Onos a week 32 ( 2-5) ( SA) ( 3.1) 276 ( 1.4) 251 ( 5.2) 254 ( 2.9) About once a woak or loss 6 ( 27$ ( 1.6) ( 4A) ( *41 ( 1.6) 200 ( 5.1)4 About how often do students do problems on worksheets? Perantsge amd Peranbie and Permits. ant Itelidancy MO:1mm At toast swami limas a weak 47 ( 3.3) 25 ( 5.2) 34 ( 3.6) 280 ( 1.1) 258 ( 4.3)I 256 ( 2.3) Mout ones a wok 22 ( 2.1) 34 ( 4.6) 33 ( 3.4) 262 ( 2.7) 258 ( 4.1) 260 ( 2.3) Lass than moldy 30 ( 2.2) 41 ( 5.6) 32 ( 3.6) 281 ( 0.8) 274 ( 4.2) 274 ( 2.7) 5 The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population if within ± 2 standard errors of the estimate for the sample. ! Interpret with caution - the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). The next section presents the students' responses to a corresponding set of questions, as well as the relationship of their responses to their mathematics proficiency. It also compares the responses of the students to those of their teachers. tr) THE 1990 NAEP TRIAL STATE ASSESSMENT 53 Montana COLLABORAUNG IN SMALL GROUPS In Montana, 39 percent of the students reported neves working mathematics problems in small groups (see Table 12); 31 percent of the students worked mathematics problems in small groups at least once a week. TABLE 12 I Students' Reports on the Frequency of Small Group Work PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 16$0 NAEP MAL STATE ASSESSMENT Montana Most Nation - - Pagodas and Pr* !Mom Paresnlop and Prot/Macy Parvedie End Proadmv How often do you work in small groups in your mathematics class? Al toast one* a win* 31 ( 13) 96 ( 4.6) 26 ( 2.5) 260 ( 1.4) 254 ( 42) 2941 ( 2.7) Loss than once a week 30 ( 1.3) 2.5) 2$ ( 14) 232 ( 1.3) 271 ( 3.1) 2117 ( 2.0) Now 30 ( 1.6) 36 ( 4.8) 44 ( 9.2) 2$0 ( 1.5) 256 ( 2.0) 261 ( 1.6) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. Examining the subpopulations (Table A 12 in the Data Appendix): In Montana, 31 percent of students attending schools in areas classified as "other" and 33 percent in schools in extreme rural areas worked in small groups at least once a week. Further, 31 percent of White students, 27 percent of Hispanic students, and 36 percent of American Indian studmts worked mathematics problems in small groups at least once a week. Females were as likely as males to work mathematics problems in small groups at least once a week (31 percent and 32 percent, respectively). 54 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana USING MATHEMATICAL OBJECTS Students were asked to report on the frequency with which they used mathematical objects such as rulers, counting blocks, or geometric solids. Table 13 below and Table A 13 in the Data Appendix summarize these data: About one-quarter of the students in Montana (27 percent) never used mathematical objects; 36 percent used these objects at least once a week. Mathematical objects were used at least once a week by 37 percent of students attending schools in areas classified as "other" and 35 percent in schools in extreme rural areas. Males were as likely as females to use mathematical objects in their mathematics classes at least once a week (37 percent and 35 percent, respectively). In addition, 37 percent of White students, 30 percent of Hispanic students, and 29 percent of American Indian students used mathematical objects at least once a week. TABLE 13 I Students' Reports on the Use of Mathematics I Objects PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS F 1FICIENCY _ 1800 NAEP TRIAL STATE ASSESSMENT Montana West Nation How often do you work with objects hke rulers, counting blocks, or geometric solids in your mathematics class? Percentage and Proficiency Percentage and Prolidency Percentage and Prolidancy At least once a week 38 ( 1.1) 30 ( 3.5) 28 ( 1.8) 282 ( 1.2) 200 ( 4.0) 258 ( 2.6) Less than once a week 38 ( 1.2) 28 ( 1.8) 31 ( 12) 282 ( 1.3) 209 ( 2.7) 21I9 ( 1,5) Never 27 ( 1.2) 36(3.3) 41 ( 22) 278 ( 1.5) 258 ( 2.8) 259 ( 11) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within i 2 standard errors of the estimate for the sample. G THE 1990 NAEP TRIAL STATE ASSESSMENT 55 Montana MATERIALS FOR MATHEMATICS INSTRUCTION The percentages of eighth-grade public-school students in Montana who frequently worked mathematics problems from textbooks (Table 14) or worksheets (Table 15) indicate that these materials play a major rvle in mathematics teaching and learning. Regarding the frequency of textbook usage (Table 14 and Table A 14 in the Data Appendix): About three-quarters of the students in Montana (79 percent) worked mathematics problems from textbooks almost every day, compared to 74 percent of the students in the nation. Textbooks were used almost every day by 78 percent of students attending schools in areas classified as "other" and 79 percent in schools in extreme rural areas. TABLE 14 I Students' Reports on the Frequency of I Mathematics Textbook Use PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY _ 1900 NAP TRIAL STATE ASSESSMENT Montane West Nation _ How often do you do mathematics problems from textbooks in your mathematics class? nareantaga and Proloimay Parcantap and Peaddinay Permits,* end Proltaioncy Almost every day Ta ( 1.5) 71 ( 3.5) 74 ( 1.9) 262 ( 0.8) 267 ( 2.4) 267 ( 12) Several times a week 13 ( 1.0) 15 ( 1.5) 14 ( 0.8) 2/9 ( 1.1) 251 ( 2.4) 252 ( 1.7) Abota awe a week or less ( 1.0) 14 ( 3.1) 12 ( 1.8) 206 ( 3.2) 242 (11.2)1 242 ( 4.5) The standard errors of the estimated StatistieS appear in parentheses. it can he said with about 95 percent certainty that, for each population of interest., the value for the entire population is within ± 2 standard errors of the estimate for the sample. I. Interpret with caution the nature of the sample does not allow accurate determination of the variability of this estimated mean proriciency. 61 56 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana And, for the frequency of worksheet usage (Table 15 and Table A 15 in the Data Appendix): Less than half of the students in Montana (32 percent) used worksheets at least several times a week, compared to 38 percent in the nation. Worksheets were used at least several times a week by 30 percent of students attending schools in areas classified as "other" and 36 percent in schools in extreme rural areas. TABLE 15 I Students' Reports on the Frequency of I Mathematics Worksheet Use PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY MO MEP TRIAL STATE ASSESSMENT Montane West Nation - How often do you do mathematics problems on worksheets In your mathematics class? Powesedaw and Proliciency Pseverdies sad Prannalage end At least several times a weak 32 ( 2.1) 35 ( 4.0) 3$ ( 2.4) 278 ( 1.3) 250 ( 4.2) 293 ( 2.2) About once a week 29 ( 1.2) 23 ( 2.6) 25 ( 1.2) 280 ( 1.4) 262 ( 2.1) 261 ( 1.4) Less than weekly 39 ( 2.1) 41 ( 4.1) 37 ( 2.5) 284 ( 0.9) 270 ( 3.4) 272 ( 1.9) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. Table 16 compares students' and teachers' responses to questions about the patterns of classroom instruction and materials for mathematics instruction. G THE 1990 NAEP TRIAL STATE ASSESSMENT 57 Montana TABLE 16 Comparison of Students' and Teachers' Reports on Patterns of and Materials for Mathematics Instruction PERCENTAGE OF STUDENTS 1890 NAEP TRIAL STATE ASSESSMENT Montana West Nation Patterns of classroom Instruction Perasaaage eindlits teectIers Pereentap Samiteals Tendons Percentap Nugent. Tetbdiere Percentage of students who work mathematics problems In small groups At least once a week 31 ( 1.7) 60 ( 3.0) 35 ( 4.1) 57 ( 6.9) 20 ( 2.5) 50 ( 4.4) Less than once a week 30 ( 1.3) 32 ( 2.9) 29 ( 2.8) 39 ( 7.8) 28 ( 1.4) 43 ( 4.1) Never 39 ( 1.6) ( 1.9) 36 ( 4.8) 3 ( 2.2) 44 ( 29) 3 ( 2.0) Percentage of students who use objects like Merv, counting blocks, or geometric solids At least once a week 36 ( 1.1) 37 ( 2.6) 36 ( 3.5) 34 ( 6.2) 28 ( 1.8) 22 ( 3.7) Less than once a week 36 ( 1.2) 62 ( 2.6) 25 ( 1.8) 57 ( 6.4) 31 ( 12) 89 ( 3.9) Never 27 ( 12) 1 ( 0.4) 36 ( 3.3) 8 ( 3.0) 41 ( 2.2) 9 ( 2.6) Materials for mathematics Pasmeniage Percentage Percentage Instruction Stueents Teachers Inman.* Teachers Students Teacture Percentage of students rho use a mathematics todbook Almost every day 79 ( 1.5) 82 ( 3.1) 71 ( 33) 55 ( 6.0) 74 ( 1.9) 02 ( 3.4) Several times a week 13 ( 1.0) 32 ( 2.5) 15 ( 1.5) 38 ( 5.1) 14 ( 0.8) 31 ( 3.1) About once a week or less ( 1.0) 6 ( 1.8) 14 ( 3.1) 9 ( 4.9) 12 ( 1.8) 7 ( 1.8) Percentage of students who use a mathematics worksheet At least several times a week 32 ( 2.1) 47 ( 3.3) 35 ( 4.0) 25 ( 5.2) 38 ( 2.4) 34 ( 3.8) About once a week 29 ( 12) 22 ( 2.1) 23 ( 2.6) 34 ( 4.0) 25 ( 1.2) 33 ( 3.4) Less than weekly 39 ( 2.1) 30 ( 2.2) 41 ( 4.1) 41 ( 5.6) 37 ( 24) 32 ( 3.6) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within 2 standard errors of the estimate for the sample. 58 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana SUMMARY Because classroom instructional time is typically limited, teachers need to make the best possible use of what is known about effective instructional delivery practices and resources. It appears that mathematics textbooks and worksheets continue to play a major role in mathematics teaching. Although there is some evidence that other instructional resources ani practices are emerging, they are not yet commonplace. According to the students' mathematics teachers: More than half of the studer .s in Montana (60 percent) worked mathematics problems in small groups at least once a week; relatively few never worked in small groups (8 percent). The largest percentage of the students (62 percent) used objects like rulers, counting blocks, or geometric shapes less than once a week, and relatively few never used such objects (1 percent). In Montana, 62 percent of the students were assigned problems from a mathematics textbook almost every day; 6 percent worked textbook problems about once a week or less. About half of the students (47 percent) did problems from worksheets at least several times a week; about one-quarter did worksheet problems less than weekly (30 percent). And, according to the students: In Montana, 39 percent of the students never worked mathematics problems in small groups; 31 percent of the students worked mathematics problems in small gioups at least once a week. About one-quarter of the students in Montana (27 percent) never used mathematical objects; 36 percent used these objects at least once a week. About three-quarters of the students in Montana (79 percent) worked mathematics problems from textbooks almost every day, compared to 74 percent of students in the nation. Less than half of the students in Montana (32 percent) used worksheets at least several times a week, compared to 38 percent in the nation. THE 1990 NAEP TRIAL STATE ASSESSMENT 59 Montana CHAPTER 5 How Are Calculators Used? Although computation skills are vital, calculators -- and, to a lesser extent, computers -- have drastically changed the methods that can be used to perform calculations. Calculators are important tools for mathematics and students need to be able to use them wisely. The National Council of Teachers of Mathematics and many other educators believe that mathematics teachers should help students become proficient in the use of calculators to free them from time-consuming computations and to permit them to focus on more challenging tasks. The increasing availability of affordable calculators should make it more likely and attractive for students and schools to acquire and use these devices. Given the prevalence and potential importance of calculators, part of the Trial State Assessment focused on attitudes toward and uses of calculators. Teachers were asked to report the extent to which they encouraged or permitted calculator use for various activities in mathematics class and students were asked about the availability and use of calculators. s National Assessment of Educational Progress, Mathematics Objectives 1990 Assessment (Princeton, NJ: Educational "resting Servia, 1988). National Council of Teachers of Mathematics, Curriculum and evaluation Standards for School Mathematics (Reston, VA: National Council of Teachers of Mathematics, 1989). 60 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana Table 17 provides a profile of Montana eighth-grade public schools' policies with regani to calculator use: In comparison to 33 percent across the nation, 57 percent of the students in Montana had teachers who allowed calculators to be used for tests. A greater percentage of students in Montana than in the nation had teachers who permitted unrestricted We of calculators (32,percent and 18 percent, respectively). TABLE 17 I Teachers' Reports of Montana Policies on I Calculator Use PERCENTAGE OF STUDENTS 1000 NAEP TRIAL STATE ASSESSMENT Pagans W1 Nation Percentage of elghth-grade students In public schoolS whoae teachers permit the unrestricted use of calculators Percentage of eighth-grade students In public schools whose teachers permit the use of calculators tor tests Percentage of eighth-grade students In public schools whose teachers report that students have access to calculators awned by the school Percentage Percentage Ponseetais 32 ( 2.7) 20 ( 4.2) 15 ( 3.4) 57 ( 3.1) 48 ( 8.8) 33 ( 4.5) 62 ( 2.9) 72 ( 7.4) 58 ( 4.6) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. i''-` ) 0 THE 1990 NAEP TRIAL STATE ASSESSMENT 61 Montana THE AVAILABILITY OF CALCULATORS In Montana, most students or their families (98 percent) owned calculators (Table 18); however, fewer students (56 percent) had teachers who explained the use of calculators to them. From Table A 18 in the Data Appendix: In Montana, 55 percent of White students, 51 percent of Hispanic students, and 58 percent of Amesican Indian students had teachers who explained how to use them. Females were as likely as males to have the use of calculators explained to them (53 percent and 58 percent, respectively). TABLE 18 Students' Reports on Whether They Own a Calculator and Whether Their Teacher Explains How To Use One PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY MO NAEP TRIAL STATE ASSESSMENT Montana West Nation Do you or your family own a calculator? Perondaga and 98 ( 0.3) 281 ( 0.8) 2 ( 0.3) Pan:anises and Proficiency 56 ( 2.0) 27e ( 1.0) 44 ( 2.0) 283 ( 1.5) Poreantaga and Pndkilimay 98 ( 0.8) 263 ( 2.6) Paraentaaa. and Prolidoncy SO ( 3.4) 260 ( 2.7) 41 ( 3.4) 2es ( 3.0) Paresatsge and Prelkdanki 9? ( OA) 263 ( 1.3) 3 ( 0.4) 234 ( 3.8) Paresalage and Prellidevey 40 ( 2.3) 25$ ( 1.7) 51 ( 2.3) 266( 1.5) Yes No Does your mathematics teacher explain how to use a calculator for mathematics problems? Yes No The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 62 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana THE USE OF CALCULATORS As previously noted, calculators can free students from tedious computations and allow them to concentrate instead on problem solving and other important skills and content. As part of the Trial State Assessment, st its were asked how frequently (neva, sometimes, almost always) they used calcuuktors for working problems in class, doing problems at home, and taking quizzes or tests. As reported in Table 19: In Montana, 16 percent of the students never used a calculator to work problems in class, while 46 percent almost always did, Some of the students (11 percent) never used a calculator to work problems at home, compared to 35 percent who almost always used one. About one-quarter of the students (30 percent) never used a calculator to take quizzes or tests, while 21 percent almost always did. TABLE 19 I Students' Reports on the Use of a Calculator I for Problem Solving or Tests PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY MO NAEP TRIAL STATE ASSESSMENT Montana West Nation Percentage and Proficiency Percentage and ProfidancY Percentage and andkiencY How often do you use a calculator for the following tasks? Working problems In class Almost always 48 ( 1.7) 53 ( 2.1) 48 ( 1.6) 277 ( 0.9) 255 ( 2.6) 254 ( 1.5) Never 18 ( 1.7) 14 ( 2.4) 23 ( 1.9) 285 ( 2.1) 265 ( 3.0) 272 ( 1.4) Doing problems at home Almost always 3$ ( 1.2) 29 ( 1.7) 30 ( 1.3) 283 ( 1.2) 263 ( 3.3) 261 ( 1.8) Never 11 ( 0.9) 19 ( 1.6) 19 ( 0.9) 279 ( 2.5) 258 ( 3.7) 263 ( 1.8) Taking quizzes or tests Almost always 21 ( 1.4) 25 ( 1.6) 27 ( 1.4) 278 ( 1.3) 25a ( 3.9) 253 ( 2.4) Never 30 ( 1.7) 22 ( 9.0) 30 ( 2.0) 287 ( 1.3) 270 ( 33) 274 ( 1.3) The standard errors of the estimated statistics appear M parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. The percentages may not total 100 percent because the "Sometimes" category is not included. THE 1990 NAEP TRIAL STATE ASSESSMENT 63 Montana WHEN TO USE A CALCULATOR Part of the Trial State Assessment was designed to investigate whether students know when the use of a calculator is helpful and when it is not. There were seven sections of mathematics questions in the assessment; however, each student took only three of those sections. For two of the seven sections, students were given calculators to use. The test administrator provided the students with instructions and practice on how to use a calculator prior to the asses&ment. During the assessment, students were allowed to choose whether or not to use a calculator for each item in the calculator sections, and they were asked to indicate in their test booklets whether they did or did not use a calculator for each item. Certain items in the Lalculator sections were defined as "calculator-active" items -- that is, items that req iiied the student to use the calculator to determine the correct response. Certain other items were defined as "calculator-inactive" items -- items whose solution neither required nor suggested the use of a calculator. The remainder of the items were "calculator-neutral" items, for which the solution to the question did not require the use of a calculator. In total, there were eight calculator-active items, 13 calculrtor-neutral items, and 17 calculator-inactive items across the two sections. However, because of the sampling methodology used as part of the Trial State Assessment, not every student took both sections. Some took both sections, some took only one section, and some took neither. To examine the characteristics of students who generally knew when the use of the calculator was helpful and those who did not, the students who responded to one or both of the calculato i. sections were categorized into two groups: High -- students who used the calculator appropriately (i.e., used it for tile calculator-active itenr and did not use it for the calculator-inactive items) at least 85 percent of the time and indicated that they had used the calculator for at least half of the calculator-active items they were presented. Other -- students who did not use the calculator appropriately at least 85 percent of the time or indicated that they had used the calculator for less than half of the calculator-active items they were presented. ti 64 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana The data presented in Table 20 and Table A20 in the Data Appendix are highlighted below: About the same percentage of students in Montana were in the High group as were in the Other group. About the same percentage of males and females were in the High group. In addition, 54 percent of White students, 51 percent of Hispanic students, and 43 percent of American Indian students were in the High group. TABLE 20 1 Students' Knowledge of Using Calculators PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1080 NAEP TRIAL STATE ASSESSMENT Montana West Nation "Calculator-use" group iterlsaifts and Pro &slew Pereembile Paramisis and and Pee Waxy Preacioncy 53 ( 2.1) SI ( 2.5) 42 ( 1.3) 218 ( 12) 273 ( 2.7) 272 ( 1.5) 47 ( 2.1) 55 ( 1.3) 275 ( 1.5) 253 ( 2A) 255 ( 1.5) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. 7 () THE 1990 NAEP TRIAL STATE ASSESSMENT 65 Montana SUMMARY Given the prevalence of inexpensive calculators, it may no longer be necessary or useful to devote large portions of instructional time to teaching students how to perform routine calculations by hand. Using calculators to replace this time-consuming process would create more instructional time for other mathematical skill topics, such as problem solving, to be emphasized. The data related to calculators and their use show that: In comparison to 33 percent across the nation, 57 percent of the students in Montana had teachers who allowed calculators to be used for tests. A greater percentage of students in Montana than in the nation had teachers who permitted unrestricted use of calculators (32 percent and 18 percent, respectively). In Montana, most students or their families (98 percent) owned calculators; however, fewer students (56 percent) had teachers who explained the use of calculators to them. In Montana, 16 percent of the students never used a calculator to work problems in class, while 46 percent almost always did, Some of the students (11 percent) never used a calculator to work problems at home, compared to 35 percent who almost always used onc. About one-quarter of the students (30 percent) never used a calculator to take quizzes or tests, while 21 percent almost always did. "/ 66 TIIE 1990 NAEP TRIAL STATE ASSESSMENT Montana CHAPTER 6 Who Is Teaching Eighth-Grade Mathematics? In recent years, accountability for educational outcomes has become an issue of increasing importance to federal, state, and local governments. As part of their effort to improve the educational process, policymakers have reexamined existing methods of educating and certifying teachers.9 Many states have begun to raise teacher certification standards and strengthen teacher training programs. As shown in Table 21: In Montana, 38 percent of the students were being taught by mathematics teachers who reported having at least a master's or education specialist's degree. This compares to 44 percent for students across the naf.on. About half of the students (47 percent) had mathematics teachers who had the highest level of teachhig certification available. This is different from the figure for the nation, where 66 percent of the students were taught by mathematics teachers who were certified at the highest level available in their states. About three-quarters of the students (77 percent) had mathematics teachers who had a mathematics (middle school or secondary) teaching certificate. This compares to 84 percent for the nation. 9 National Council of Teachers of Mathematics, Pr *vsional Standards for the n'aching of Mathernatir.s ;Reston, VA: National Council of Teachers of Mathematics, 1991). P 0-1 THE 3990 NAEP TRIAL STATE ASSESSMENT 67 Montana TABLE 21 I Profile of Eighth-Grade Public-School Mathematics Teachers PERCENTAGE OF STUDENTS 1660 NAEP MAI. STATE ASSESSMENT Montana Wed Nation Percentage of students whose mathematics teachers repelled having the following degrees Sachet lor's degree Master's or specialist's degree Doctorate or professional degree Percentage of students whom mathematics teachers have the faillowing typos of teaching cortincates that ars rectignktod by Montana No regular certification 1 0.1) Regular certification but less than the highest available 52 2.6) Highest certification available (permanent or long-term) 47 2.6) Percentage of students *two mathematics teachers have the following types of teaching certificates that aro recognized by Montana Mathematics (middle school or secondary) 17 ( 3.0) 61 ( 3.0) Education (elementary or middle School) 23 ( 3.0) 9 ( 2.6) Other 0 ( 0.1) 2 ( 1.3) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. EDUCATIONAL BACKGROUND Although mathematics teachers are held responsible for providing high-quality instruction to their students, there is a concern that many teachers have had limited exposure to content and concepts in the subject area. Accordingly, the Trial State Assessment gathered details on the teachei-s educational backgrounds -- more specifically, their undergraduate and graduate majors and their in-service training. 68 THE 1990 NAEP 'TRIAL STATE ASSESSMENT Montana Teachers' responses to questions concerning their undergraduate and graduate fields of study (Table 22) show that: In Montana, 50 percent of the eighth-grade public-school students were being taught mathematics by teachers who had an undergraduate major in mathematics. In comparison, 43 percent of the students across the nation had mathematics teachers with the same major. Some of the eighth-grade public-school students in Montana (19 percent) were taught mathematics by teachers who had a graduate major in mathematics. Across the nation, 22 percent of the students were taught by teachers who majored in mathematics in graduate school. TABLE 22 I Teachers' Reports on Their Undergraduate and Graduate Fields of Study PERCENTAGE OF STUDENTS _ 11190 NAEP TRIAL STATE ASSESSMENT Montana West Nation What was your undergraduate major? Paramlasa Peroante. Percentage Mathematics 50 ( 2.8) 31 ( 5.9) 43 ( 3.9) Education 38 ( 2.9) 34 ( 35 ( 3.8) Other 12 ( 2.3) 35 (5.5) 22 ( 3.3) What was your graduate major? Parcantaga Porcontage Peraintage MaMematics 19 ( 1.7) 19 ( 4.7) 22 ( 3.4) Education 43 ( 2.8) 36 (44) 34 ( 34) Other or no graduate Wel study 38 ( 3.0) 45 ( 54) 40 ( 34) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. 7 THE 1990 NAEP TRIAL STATE ASSESSMENT 69 Montana Teachers' responses to questions concerning their in-service training for the year up to the Trial State Assessment (Tabk 23) show that: ln Montana, 38 percent of the eighth-grade public-school students had teachers who spent at least 16 hours on in-service education dedicated to mathematics or the teaching of mathematics. Across the nation, 39 percent of the students had teachers who spent at least that much time on similar types of in-service training. Relatively few of the students in Montana (5 percent) had mathematics teachers who spent no time on in-service education devoted to mathematics or the teaching of mathematics. Nationally, 11 percent of the students had mathematics teachers who spent no time on similar in-service training, TABLE 23 I Teachers' Reports on Their In-Service Training PERCENTAGE OF STUDENTS 111110 NAEP TRIAL STATE ASSESSMENT I Montana Wast Nation During the last year, how much time in total have you spent on in-service education in mathematics or the teaching of mathematics? None One to 16 hours 15 hours or mom Pimento. Pirtientage Prventess 5 ( 1.0) 11 ( 3.0) 11 ( 2.1) 58 ( 3.0) 45 ( 7.0) 51 ( 4.1) 38 ( 2.8) 44 ( 8.9) 39 ( 3.8) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is withm ± 2 standard errors of the estimate for the sample. 70 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana SUMMARY Recent results from international studies have shown that students from the United States do not compare favorably with students from other nations in mathematics and science achievement." Further, results from NAEP assessments have indicated that students' achievement in mathematics and science is much lower than educators and the public would like it to be." In curriculum areas requiring special attention and improvement, such as mathematics, it is particularly important to have well-qualified teachers. When performance differences across states and territories are described, variations in teacher cpinlifications and practices may point to areas worth further exploration. There is no guarantee that individuals with a specific set of credentials will be effective teachers; however, it is likely that relevant training and experience do contribute to better teaching. The information about teachers' educational backgrounds and experience reveals that: In Montana, 38 percent of the assessed students were being taught by mathematics teachers who reported having at least a master's or education specialist's degree. This compares to 44 percent for students across the nation. About half of the students (47 percent) had mathematics teachers who had the highest level of teaching certification available. This is different from the figure for the nation, where 66 percent of students were taught hy mathematics teachers who were certified at the highest level available in their states. In Montana, 50 percent of the eighth-grade public-school students were being taught mathematics by teachers who had an undergraduate major in mathematics. In comparison, 43 percent of the students across the nation had mathematics teachers with the same major. Some of the eighth-grade public-school students in Montana (19 percent) were taught mathematics by teachers who had a graduate major in mathematics. Across the nation, 22 percent of the students were taught by teachers who majored in mathematics in graduate school. `° Archie E. Lapointe, Nancy A. Mead, and Gary W. Phillips, A World of Differences An international Assessment of Mathematics and Science (Princeton, NJ: Center for the Assessment of Educational Progress, Educational Tesung service, 1988). " Ina V.S. Mulhs, John A. llossey, Eugene U. Owen, and Gary W. Phillips, The State of Mathematics Achievement NA Ers 1990 Assessment of the Nation and the Thal Assessment of the States (Princeton, NJ: National Assessment of Educational Progress, Educational Testing Service, 1991). THE 1990 NAEP TRIAL STATE ASSESSMENT 71 Montana In Montana, 38 percent of the eighth-grade public-school students had teachers who spent at least 16 hours on in-service education dedicated to mathematics or the teaching of mathematics. Across the nation, 39 percent of the students had teachess who spent at least that much time on similar types of in-service training. Relatively few of the students in Montana (5 percent) had mathematics teachers who spent no time on in-service education devoted to mathematics or the teaching of mathematics. Nationally, 11 percent of the students had mathematics teachers who spent no time on similar in-service training. 72 THE 1990 NAEP TRIAL STATE ASSESSMENT CHAPTER 7 The Conditions Beyond School that Facilitate Mathematics Learning and Teaching Because students spend much more time out of school each day than they do in school, it is reasonable to expect that out-of-school factors greatly influence students' attitudes and behaviors in school. Parents and guardians can therefore play an important role in the education of their children. Family expectations, encouragemert, and participation in student learning experiences are powerful influences. Together, teachers and parents can help build students' motivation to learn and can broaden their interest in mathematics and other subjects. To examine the relationship between home environment and mathematics proficiency, students participating in the Trial State Assessment were asked a series of questions about themselves, their parents or guardians, and home factors related to education. THE 1990 NAEP TRIAL STATE ASSESSMENT 73 Montana AMOUNT OF READING MATERIALS IN THE HOME The number and types of reading and refesence materials in the home may be an indicator of the value placed by parents on learning and schooling. Students participating in the Trial State Assessment west asked about the availability of newspapers, magazines, books, and an encyclopedia at home. Average mathematics proficiency associated with having zero to two, three, or four of these types of materials in the home is shown in Table 24 and Table A24 in the Data Appendix. TABLE 24 I Students' Reports on Types of Reading Materials in the Home PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY _ MO NAEP TRIAL STATE ASSESSMENT Montana West Nation , _ Does your family have, or receive on a regular basis, any of the following items: more than 25 books, an encyclopedia, newspapers, magazines? Zero to two types Tires types Four typos Poesintage Proicknoy Poroonlado and Prat:Now Paraminia sad Praliekear 12 ( 0.7) 24 ( 15) 21 ( 1.0) 2eS ( 2.1) 245 ( 4.1) 244 ( 2.0) 32 ( 12) 31 ( 1.4) 30 ( 1.0) 277 ( 1.8) 251 ( 2.4) 251 ( 1.7) 55 ( 1.0) 45 ( 111) 44 ( 1.3) 215 ( 0.5) 273 ( 3.2) 272 ( 1.5) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population Is within 2 standard errors of the estimate for the sample. The data for Montana reveal that: Students in Montana who had all four of these types of materials in the home showed higher mathematics proficiency than did students with zero to two types of materials. This is similar to the results for the nation, where students who had all four types of materials showed higher mathematics proficiency than did students who had zero to two types. 74 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana A smaller percentage of Hispanic and American Indian students had all four types of these reading materials in their homes than did White students, About the same percentage of students attending schools in areas classified as "other" as in extreme rural areas had all four types of these reading matesiab in their homes. HOURS OF TELEVISION WATCHED PER DAY Excessive television watching is generally seen as detracting from time spent on educational pursuits. Students participating in the Tr State Assessment were asked to report on the amount of television they watched each day (Table 25). TABLE 25 I Students' Reports on the Amount of Time Spent i Watching Television Each Day PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1990 MAEP TRIAL STATE ASSESSMENT Montana West Nation Posueniago and Proficiency Passentago and Prelkiettoy petvantago and Prakdoncli How much television do you usually watch each day? One hour or less 21 ( 0.9) 14 ( 1.5) 12 ( 0.8) 290 ( 1.5) 260 ( 3.6) 209 ( 2.2) Two hours 30 ( 1.2) 20 ( 1.6) 21 ( 0.9) 255 ( 1.2) 265 ( 3.6) 205 ( 1.5) Throw hours 24 ( 1.0) 20 ( 1.2) 22 ( 0.5) 2711 ( 1.2) 282 ( 3.2) 205 ( 1.7) Pot, to !Ivo hours 20 ( 1.1) 29 ( 1.7) 2$ ( 1.1) 24'5 ( 1.6) 263 ( 2.9) 2eo ( 1.7) 613/ hours or mars ( 0.6) 16 ( 2.0) 16 ( 1.0) 261 ( 2.9) 246 ( 2.6) 245 ( 1.7) The standard errors of the estimated statistiCs appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for thr entire population is within ± 2 standard errors of the estimate for the sample. THE 1990 NAEP TRIAL STATE ASSESSMENT 75 Montana From Table 25 and Table A25 in the Data Appendix: In Montana, average mathematics proficiency was lowest for students who spent six hours or more watching television each day. About one-quarter of the eighth-grade public-school students in Montana (21 percent) watched one hour or less of television each day; 6 percent watched six hours or more. About the same percentage of males and females tended to watch six or more hours of television daily. However, a smaller percentage of males than females watched one hour or less per day. In addition, 5 percent of White students, 10 percent of Hispanic students, and 13 percent of American Indian students watched six hours or more of television each day. In comparison, 22 percent of White students, 14 percent of Hispanic students, and 12 percent of American Indian students tended to watch only an hour or less. STUDENT ABSENTEEISM Excessive absenteeism may also be an obstacle to students' success in school. To examine the relationship of student absenteeism to mathematics proficiency, the students participating in the Trial State Assessment were asked to report on the number of days of school they missed during the one-month period preceding the assessment. From Table 26 and Table A26 in the Data Appendix: In Montana, average mathematics proficiency was lowest for students who missed three or more days of school. Less than half of the students in Montana (40 percent) did not miss any school days in the month prior to the assessment, while 21 percent missed three days or more. In addition, 20 percent of White students, 17 percent of Hispanic students, and 37 percent of American Indian students missed three or more days of school. 76 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana Similarly, 22 percent of students attending schools in areas classified as "other" and 19 pesvent in schools in extreme rural areas missed three or more days of school. TABLE 26 I Students' Reports on the Number of Days of I School Missed PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1090 NAEP TRIAL r :11E ASSESSMENT Madam Wad Nation 1 How many days of school did you miss last month? Ons or two days Three days or more Pialkiemy fralicismiv aid holk.may 40 ( 1.3) 43 ( 2.7) 45 ( 1.1) 244 ( 1.4) ( 245 ( 1.9) 11(12 ) 282 ( 1.0) $0 ( 295 ( 14) 34) 295 041.4 21 ( 0.8) 27 ( 1.1) 23 ( 1.1) 272 ( 2.0) 250 ( 3.1) 250 ( 1.9) The standard errors of the estimated statistics appeur in parentheses. It can be mid with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. c 2 THE 1990 NAEP TRIAL STATE ASSESSMENT 77 Montana STUDENTS' PERCEPTIONS OF MATHEMATICS According to the National Council of Teachers of Mathew,c.it - learning mathematics should require students not only to master essential skills and concepts but also to develop confidence in their mathematical abilities and to value mathematics as a discipline.'2 Students were asked if they agreed or disagreed with five statements designed to elicit their perceptions of mathematics. These included statements about: Personal experience with mathematics, including students' enjoyment of mathematics and level of confidence in their mathematics abilities: I like mathematics; I am good in mathematics. Value of mathematics, including students' perceptions of its present utility and its expected relevance to future work and life requirements: Almost all people use mathematics in their jobs; mathematics is not more for boys than for girls. The nature of mathematics, including students' ability to identify tht salient features of the discipline: Mathematics is useful for solving everyday problems. A student "perception index" was developed to examine students' perceptions of and attitudes toward mathematics. For each of the five statements, students who responded "strongly agree" were given a value of 1 (indicating very positive attitudes about the subject), those who responded "agree" were given a value of 2, and those who responded "undecided," "disagree," or "strongly disagree" were given a value of 3. Each student's responses were averaged over the five statements. The students were then assigned a perception index according to whether they tended to strongly agree with the statements (an index of 1), tended to agree with the statements (an index of 2), or tended to be undecided, to disagree, or to strongly disagree with the statements (an index of 3). Table 27 provides the data for the students' attitudes toward math.:matics as defmed by their pereption index. Thr following results were observed for Montana: Average mathematics proficiency was highest for students who were in the "strongly agree" category and lowest for students wly- were in the "undecided, disagree, strongly disag...,!e" category. About one-quarter of the students (30 percent) were in the "strongly agree" category (perception index of 1). This compares to 27 percent across the nation. Some of th., students in Montana (20 percent), compared to 24 percent across the nation, were in the "undeciied, disagree, or strongly disagyee" category (perception index of 3). " National Council of Teachers of Mathematics, Curriculum and Evaluation Standards for School Mathematics (Reston. VA: National Council of Teachers of Mathematics, 1989). 78 THE 1990 NAEP TRIAL STATE ASSESSMENT TABLE 27 1 Students' Perceptions of Mathematics PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY IWOO NAEP TIM STATE ASSESSMENT Student °perception index" groups Strongly agree (*perception Index" of 1) AS iv ("percep(ion index" of 2) Undecided, (Meagre*, dranaly disagree (.perception Index" ot 3) 271 1 44 2011 2.4) 2419(2M 25(24) 2111.1113 The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. SUMMARY Some out-of-school factors cannot be changed, but others can be altered in a positive way to influence a student's learning and motivation. Partnerships among students, parents, teachers, and the larger community can affect the educational environment in the home, resulting in more out-of-school reading and an increased value placed on educational achievement, among other desirable outcomes. The data related to out-of-school factors show that: Students in Montana who had four types of reading materials (an encyclopedia, newspapers, magazines, and more than 25 books) at home showed higher mathematics proficiency than did students with zero to two types of materials. This is similar to the results for the nation, where students who had all four types of materials showed higher mathematics proficiency than did students who had zero to two types. C., THE 1990 NAEP TRIAL STATE ASSESSMENT 79 Montana About one-quarter of the eighth-grade public-school students in Montana (21 percent) watched one hour or less of television each day; 6 percent watched six hours or more. Average mathematics proficiency was lowest for students who spent six hours or more watching television eitch day. Less than half of the students in Montana (40 percent) did not miss any school days in the month prior to the assessment, while 21 percent missed three days or more. Average mathematics proficiency was lowest for students who missed three or more days of school. About one-quarter of the students (30 percent) were in the "strongly agree" category relating to students' perceptions of mathematics. Average mathematics proficiency was highest for students who were in the "strongly agree" category and lowest for students who were in the "undecided, disagree, strongly disagree" category. 80 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana THE NATION'S REPORT CARD PROCEDURAL APPENDIX This appendix provides an overview of the technical details of the 1990 Trial State Assessment Program. It includes a discussion of the assessment design, the mathematics framework and objectives upon which the assessment was based, and the procedures used to analyze the results. The objectives for the assessment were developed through a consensus process managed by the Council of Chief State School Officers, and the items were developed through a similar process managed by Educational Testing Service. The development of the Trial State Assessment Program benefitted from the involvement of hundreds of representatives from State Education Agencic5 who attended numerous NETWORK meetings, served on committees, reviewed the framework, objectives, and questions, and, in general, provided important suggestions on all aspects of the program. Assessment Design The 1990 Trial State Assessment was based on a focused balanced incomplete block (BIB) spiral matrix design -- a design that enables broad coverage of mathematics content while minimizing the burden for any one student. In total, 137 cognitive mathematics items were developed for the assessment, including 35 open-ended items. The first step in implementing the BIB design required dividing the entire set of mathematics items into seven units called blocks. Each block was designed to be completed in 15 minutes. C; THE 1990 NAEP TRIAL STATE ASSESSMENT 81 Montana The blocks were then assembled into assessment booklets so that each booklet contained two background questionnaires -- the first consistir; of geveral background questions and the second consisting of mathematics background questions -- and three blocks of cognitive mathematics items. Students were given five minutes to complete each of the background questionnaires and 45 minutes to complete the three I5-minute blocks of mathematics items. Thus, the entire assessment required approximately 55 minutes of student time. In accordance with the BIB design, the blocks were assigned to the assessment booklets so that each block appeared in exactly three booklets and each block appeared with every other block in one booklet Seven assessment booklets were used in the Trial State Assessment Program. The booklets were spiraled or interleaved in a systematic sequence so that each booklet appeared an appropriate number of times in the sample. The students within an assessment session were assigned booklets in the order in which the booklets were spiraled. Thus, students in any given session received a variety of different booklets and only a small number of students in the session received the same booklet. Assessment Content The framework and objectives for the Trial State Assessment Program were developed using a broad-based consensus process, as described in the introduction to this report.' The assessment framework consisted of two dimensions: mathematical content areas and abilities. The five conte,4 areas assessed were Numbers and Operations; Measurement; Geometry; Data Analysis, Statistics, and Probability; and Algebra and Functions (see Figure A l). The three mathematical ability areas assessed were Conceptual Understanding, Procedural Knowledge, and Problem Solving (see Figure A2). Data Analysis and Scales Once the assessments had been conducted and information from the assessment booklets had been compiled in a database, the assessment data were weighted to match known population proportions and adjusted for nonresponse. Analyses were then conducted to determine the percentages of students who gave various responses to each copitive and background question. Item response theory (IRT) was used to estimate average mathematics proficiency for each jurisdiction and for various subpopulations, based on students' performance on the set of mathematics items they received. IRT provides a common scale on which performance can be reported for the nation, each jurisdiction, and subpopulations, even when all students do not answer the same set of questions. This common scale makes it possible to report on relationships between students' characteristic- (based on their responses to the background questions) and their overall performance in the assessment. I ational Assessment of Educational Progress, Mathematics Objectives 1990 Assessment (Pnnceton, Educational Testing Service, 1988). 82 THE 1990 NAEP TRIAL STATE ASSESSMENT 411 Montana FIGURE AI I Content Areas Assessed THE NATION'S RD REPORT Numbers and Operations This content area focuses on students' understanding of numbers (whole numbers, fractions, decimals, integers) and their application to real-world situations, as well as computational and estimation situations. Understanding numerical relationships as expressed in ratios, proportions, and percents is emphasized. Students' abilities in estimation, mental computation, use of calculators, generalization of numerical patterns, and verification of results are also included. IMeasurement This content area focuses on students' ability to describe real-world objects using numbers. Students are asked to identify attributes, select appropriate units, apply meaSurement conogits, and communicate measurement-related ideas to others. Questions are included that require an ability to read instruments using metric, customary, or nonstandard units, with emphasis on precision and acuracy. Questions requiring estimation, measurements, and applications of measurements of length, time, money, temperature, mass/weight, area, volume, capacity, and angles are also included in this content area. Geometry This content area focuses on students' knowledge of geometric figures and relationships and on their skills in wo,.king with this knowledge. These skills are important at all levels of schooling as well as in practical applications. Students need to be able to model and visualize geometric figures in one, two, and three dimensions and to communicate geomet: ic ideas. In addition, students should be able to use informal reasoning to establish geometric relationships. Data Analysis, Statistics, and Probability This content area focuses on data representation and analysis across all disciplines and reflects the importance and prevalence of these activities in our society. Statistical knowledge and the ability to interpret data are necessary skills in the contemporary world. Questions emphasize appropriate methods tor gathering data, the visual exploration of data, and the development and evaluation of arguments based on data analysis. Algebra and Functions This content area is broad in scope, covering algebraic and functional concepts in more informal, exploratory ways for the eighth-grade Trial State Assessment. Proficiency in this concept area requires both manipulative facility and conceptual understanding: it involves the ability to use algebra as a means of representation and algebraic processing as a problem-solving tool. Functions are viewed not only in terms of algebraic formulas, but also in terms of verbal descriptions, tables of values, and graphs. Lb THE 1990 NAEP TRIAL STATE ASSESSMENT 83 Montana FIGURE A2 Mathematical Abilities The following three categories of mathematical abilities are not to be cOnstrued as hierarchical. For example, problem solving involves interactions between conceptual knowledge :'d procedural skills, but what Is considered complex problem solving at one grade level may considered conceptual understanding or procedural knowledge at another. I.Cotceptual Understanding Students demonstrate conceptual understanding in mathematics when they provide evidence that they can recognize, label, and generate examples and counterexamples ot concepts: can use and interrelate models, diagrams, and varied representations of concepts: can identify and apply principles; know and can apply facts and definitions; can compare, contrast, and integrate related concepts and principles; can recognize, interpret, and apply the signs, symbols, and termS used to represent concepts: and can interpret the assumptions and relations involving concepts in mathematical settings. Such understandings are essential to performing proCedures in a meaningful way and applying them in problem-solving situations. Procedural Knowledge Students demonstrate procedural knowledge in mathematics when they provide evidence of their abdity to select and apply appropriate procedures correctly, verify and justify the correctness of a procedure using concrete models or symbolic methods, and extend or modify procedures to deal with factors inherent in problem settings. Procedural knowledge includes the various numerical algorithms in mathematics that have been created as tools to meet specific needs in an efficient manner. It also encompasses the abilities to read and produce graphs and tables, execute geometric constructions, and perform noncomputational skills such as rounding and ordering. Problem Solving In problem solving, students are required to use their reasoning and analytic abilities when they encounter new situations. Problem solving includes the ability to recognize and formulate pr7blems: determine the sufficiency and consistency of data: use strategies, data, modals, and relevant mathematics: generate. extend, and modify procedures: use reasoning (i.e,, spatial, inductive, deductive, statistical, and proportional): and judge the reasonableness and correctness of solutions. E 5 84 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana A scale ranging from 0 to 500 was created to report performance for each content area. Each content-area scale was based on the distribution of student performance across all three grades assessed in the 1990 national assessment (grades 4, 8, and 12) and had a mean of 250 and a standard deviation of 50. A composite scale was created as an overall measure of students mathematics proficiency. The composite scale was a weighted average of the five content area scales, where the weight for each content area was proportional to the relative importance assigned to the content area in the specifications developed by the Mathematics Objectives Panel. Scale Anchoring Scale anchoring is a method for defining performance along a scale. Traditionally, performance on educational scales has been defmed by norm-referencing -- that is, by comparing students at a particular scale level to other students. In contrast, the NAEP scale anchoring is accomplished by describing what students at selected levels know and can do. The scale anchoring process for the 199(1 Trial State Assessment began with the selection of four levels -- 200, 250, 300, and 350 -- on the 0-to-500 scale. Although proficiency levels below 200 and above 350 could theoretically have been defined, they were not because so few students performed at the extreme ends of the scale. Any attempts to define levels at the extremes would therefore have been highly speculative. To defme performance at each of the four levels on the scale, NATI' analyzed sets of mathematics items from the 1990 assessment that discriminated well between adjacent levels. The crise, ia for selecting these "benchmark" items were as follows: To define performance at level 200, items were chosen that were answered correctly by at least 65 percent of the students whose proficiency was at or near 200 on the scale. To define performance at each of the higher levels on the scale, items were chosen that were: a) answered correctly by at least 65 percent of students whose proficiency was at or near that level; and b) answered incorrectly by a majority (at least 50 percent) of the students performing at or near the next lower level. The percentage of students at a level who answered the item correctly had to be at least 30 points higher than the percentage of students at the next lower level who answered it correctly. THE 1990 NAEP TRIAL STATE ASSFSSMENT 85 Montana Once these empirically selected sets of questions had been identified, mathematics educators analyzed the questions and used their expert judgment to characterize the knowledge, skills, and understandings of students performing at each level. Each of the four proficiency levels was defined by describing the types of mathematics questions that most students attaining that proficiency level would be able to perform successfully. Figure 3 in Chapter I provides a summary of the levels and their characteristic skills. Example questions for each level are provided in Figure A3, together with data on the estimated proportion of students at or above each of the four proficiency levels who correctly answered each question.' Questionnaires for Teachers and Schools As part of the Trial State Assessment, questionnaires were given to the mathematics teachers of assessed students and to the principal or other administrator in each participating school. A Policy Analysis and Use Panel drafted a set of policy issues and guidelines and made recommendations concerning the design of these questionnaires. For the 1990 assessment, the teacher and school questionnaires focused on six educational areas: curriculum, instructional practices, teacher qualifications, educational standards and reform, school conditions, and conditions outside of the school that facilitate learning and instruction. Similar to the development of the materials given to students, the policy guidelines and the teacher and school questionnaires were prepared through an iterative process that involved extensive development, field testing, and review by external advisory gxoups. MATHEMATICS TEACHER QUESTIONNAIRE The questionnaire for eighth-grade mathematics teachers consisted of two parts. The first requested information about the teacher, such as race/ethnicity and gender, as well as academic degrees held, teaching certification, training in mathematics, and ability to get instructional resources. In the second part, teachers were asked to provide information on each class they taught that included one or more students who participated in the Trial State Assessment Program. The information included, among other things, the amount of time spent on mathematics instruction and homework, the extent to which textbooks or worksheets were used, the instructional emphasis placed on different mathematical topics, and the use of various instructional approaches. Because of the nature of the sampling for the Trial State Assessment, the responses to the mathematics teacher questionnaire do not necessarily represent all eighth-grade mathematics teachers in a state or territory. Rather. they represent the teachers of the particular students being assessed. 2 Since there were insufficient numbers of eighth-grade questions at levels 200 and 350, one of the questions exemplifying level 200 is from the fourth-grade national assessment and one exemplifying level 350 is from the twelfth-grade national assessment. 86 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana FIGURE A3 I Example Items for Mathematics Proficiency Levels Level 200: Sknple Additive Reasoning and Problem Solving with Whole Numbers EXAMPLE 1 A / Tries Ger WM real o loose Gig Osib 0 Sobest 7. La& bad dose Uses wan ashhe mo we sod doom Mono lusds ha so shore gem 1 tios flak sok bee with dr boot si logio aeons We& 'smoke brow 1010 is St Ths kw look the ousio bah Cis The hos Irak doe pit bails V no les s4ch dos saber bats 0I7 *a cadre sdl. EXAMPLE 2 DOXIS OF MIT PICKED AT FARAWAY FARM Tam Med 11.. Oars OM' wak Lasso Owiwevii 0. HOW allay WIMP oi mein woos picked as Thurodayi OD SS 60 70 40 90 I don't tzar. Grade 4 OveraN Percentage Percentage Correct S22 65 91 Grade 4 Overall Percent/go Percentage Correct MI MI 75 91 Grade 8 Overall Percentage Percentage COffict 78 87 fl2 Correct 73% for Anchor Levels: S2 114 ico 0% for A :f4'4 Levels: 122 204 100 Correct 89% tor Anchor Levels: MI nil 98 100 Level 250: Simple Multiplicative Reasoning and Two-Step Problem Solving I Montana FIGURE A3 Example Items for Mathematics Proficiency Levels (continued) EXAMPLE 1 7. What is the value of n 4- 5 when a 3 Answer. EXAMPLE 2 KAM COLOR ARM ILSULTS Ca. si oh 17 SO 11 The uWe thine shows the mks s Nom of hair Woe. Co At cipcle below, ails $ code 'soh to iDisIxtSISthe &as la she table. Lthei oath pen *I the dick mop %rub olbe thethet Mu Wee. Ltd you two Os, eskailitor ea this isseitlos? Yes EXAMPLE 3 h. Utilise is pschio4 balchalle Imo boxes. Each box holds 6 baultellt. She Us 24 bells WWI& number mew* will kelp hot linel oue how luny beim she will othel CD 24 6 0 cl) 24 4 , s, 24 + 6 0 op r a* 0 CD I &Wang:1w. Grade Cverall Percentage Correct: 75% Paroantaos Comsat for Anchor Laval:: MI IN MO MI 28 89 95 98 Grad* 8 Overall Pencentage Correct: 73% Percentage Correct for Ancbor ardi 202 es 92 92 Grad* 8 Overall Percentage Correct: 77% Percentage Correct for Anchor live's: /29 MO 3119 37 71 95 100 88 THE 19 N NAM' TRIAL STATE ASSESSMENT Montana FIGURE A3 I Example Items for Mathematics Proficiency Levels (continued) Level 300: Reasoning and Problem Solving involving Fractions, Decimals, Percents, Elementary Geometric Properties, and Simple Algebraic Manipulations EXAMPLE 41 Ia. WIbith As Saw* sham ohs mole si Noolvo dig Inaftie OM' die U.* t QD EXAMPLE 2 lo the said woo Au doss Is s oar is ism leo/ lopossocsi by milt Isio&I S hullos load. U ea mat sash is loot at loos 3S low wool* k rsorosoico.; Li 3 Wad !soda Law way Mau WO 0 II 0 0 I 0 7 eV DO roe use die osicuistor so ibi gosationt 0 Ws 0 No c 4 Grade 8 Choral! Percentage Correct: 80% Peroentage Correct for Anchor Levels: 2012 Mt 242 HQ 33 49 77 90 Grade 12 Overall Percentage Correct: 75% Percentage Correct for Mellor Lai/leis: MD 222 222 214 48 79 95 Grade 8 Overaii Percentage Percentage Correct MI 17 48 Correct: 59% for Anchor WW2: 222 252 88 99 BEST COPY AVAILABLE Montana FIGURE A3 I Example Items for Mathematics Proficiency Levels (continued) Level 350: Reasoning and Problem Solving Involving Geometric Relationships, Algebraic Equations, and Beginning Statistics and Probability EXAMPLE 1 01014.10so 14-17 joko io the hilIoo sus omen. of dot 4 woos 0 I 2 2 16. II Ala putts of doufigmes j cxyasout. bow fosoy dos ll bc ia ghe )30th fout1 CD 100 CP 10 CID 199 2D0 4) 201 EXAMPLE 2 IT. typluo how you foonyl you, UMW./ suestion 1 6. Ammer Grad. 8 Overall Percentage Correct 34% Percentage Correct for Maher Levels: Z:?2 2/11 2Ciit 201/ 13 19 53 88 Grade 12 Overall Percentage Correct: 40% Percentage Correct for Mchor Levels: B2Q 2110 202 22 48 so Grad* 8 Overall Percentage Correct: 15% Percentage Correct for Anchor Levels: 22Q 221/ 112 1 4 28 74 Grads 12 Overall Percentage Correct: 27% Percentage Correct for Anchor Levels: 222 Zig A22 122 3 22 74 90 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana SCHOOL CHARACTERISTICS AND POLICIES QUESTIONNAIRE An extensive school questionnaire was completed by principals or other administrators in the schools participating in the Trial State Assessment. In addition to questions about the individuals who completed the questionnaires, there were questions about school policies, course offerings, and special priority areas, among other topics. It is important to note that in this report, as in all NAEP reports, the student is always the unit of analysis, even when information from the teacher or school questionnaire is being reported. Having the student as the unit of analysis makes it possible to describe the instruction received by representative samplet of eighth-grade students in public schools. Although this approach may provide a differeat perspective from that which would be obtained by simply collecting information from a sample of eighth-grade mathematics teachers or from a sample of schools, it is consistent with NAEP's goal of providing information about the educational context and performance of students. Estimating Variability The statistics reported by NAFP (average proficiencies, percentages of students at or above particular scale-score levels, and percentages of students responding in certain ways to background questions) are estimates of the corresponding information for the population of eighth-grade students in public schools in a state. These estimates are based on the performance of a carefully selected, representative sample of eighth-grade public-school students from the state or territory. If a different representative sample of students were selected and the assessment repeeted, it is likely that the estimates might vary somewhat, and both of these sample estimates might differ somewhat from the value of the mean or percentage that would be obtained if every eighth-grade public-school student in the state or territory were assessed. Virtually all statistics that are based on samples (including those in NAEP) are subject to a certain degTee of uncertainty. The uncertainty attributable to using samples of students is referred to as sampling error. like almost all estimates based on assessment measures, NAEP's total group and subgroup proficiency estimates are subject to a second source of uncertainty, in addition to sampling error. As previously noted, each student who panicipated in the Trial State Assessment was administered a subset of questions from the totil set of questions. If each student had been admiMstered a different, but equally appropriate, set of the assessment questions -- or the entire set of questions -- sommhat different estimates of total group and subgroup proficiency might have been obtained. Thus, a second source of uncertainty arises because each student was administered a subset of the total pool of questions. n THE 1990 NAEP TRIAL STATE ASSESSMENT 91 Montana In addition to reporting estimates of average proficiencies, proportions of students at or above particular scale-score levels, and proportions of students giving various responses to background questions, this report also provides estimates of the magnitude of the uncertainty associated with these statistics. These measures of the uncertainty are called standard errors and are given in parentheses in each of the tables in the report. The standard errors of the estimates of mathematics proficiency statistics reflect both sources of uncertainty discussed above. 'The standard errors of the other statistics (such as the proportion of students answering a background question in a certain way or the proportion of students in certain racial/ethnic groups) reflect only sampling error. NAEP uses a methodology called the jackknife procedure to estimate these standard errors. Drawing Inferences from the Results One of the goals of the Trial State Assessment Program is to make inference about the overall population of eighth-grade students in public schools in each participating state and territory based on the particular sample of students assessed. One uses the results from the sample -- taking into account the uncertainty associated with all samples -- to make inferences about the population. The use of confidence intervals, based n the standard errors, provides a way to make inferences about the population means and proportions in d manner that reflects the uncertainty associated with the sample estimates. An estimated sample mean proficiency ± 2 standard errors represents a 95 percent confidence interval for the corresponding population quantity. This means that with approximately 95 percent certainty, the average performance of the entire population of interest (e.g., all eighth-grade students in public schools in a state or territory) is within ± 2 standard errors of the sample mean. As an example, suppose that the average mathematics proficiency of the students in a particular state's sample were 256 with a standard error of 1.2. A 95 percent confidence interval for the population quantity would be as follows: Mean ± 2 standard errors = 256 ± 2 (1.2) = 256 ± 2.4 = 256 - 2.4 and 256 + 2.4 = 253.6, 258.4 Thus, one can conclude with 95 percent certainty that the average proficiency for the entire population of eighth-gade students in public schools in that state is between 253.6 and 258.4. Similar confidence intervals can be constructed for percentages, providea that the percentages are not extremely large (greater than 90 percent) or extremely small (less than 10 percent ). l'or extreme percentages, confidence intervals constructed in the above manner may not be appropriate and procedures for obtaining accurate confidence intervals are quite complicated, 92 '1 ile. 1990 NAEP I !UAL STATE ASSESSMENT Analyzing Subgroup Differences in Proficiencies and Proportions In addition to the overall results, this report presents outcomes separately for a variety of important subgroups. Many of these subgroups are defined by shared characteristics of students, such as their gender, race/ethnicity, and the type of community in which their school is located. Other subgroups are defined by students' responses to background questions such as About how much time do you usually spend each day on mathematics homework? Still other subgroups are defmed by the responses of the assessed students' mathematics teachers to questions in the mathematics teacher questionnaire. As an example, one might be interested in answering the question: Do students who reported spending 45 minutes or more doing mathematics homework each day exhibit higher average mathematics proficiency than students who reported spendk IS minutes or less? To answer the question posed above, one begins by comparing the average mathematics proficiency for the two groups being analyzed. If the mean for the group who reported spending 45 minutes or more on mathematics homework is higher, one may be tempted to conclude that that group does have higher achievement than the group who reported spending 15 minutes or less on homework. However, even though the means differ, there may be no real difference in performance between the two groups in the population because of the uncertainty associated with the estimated average proficiency of the groups in the sample. Remember that the intent is to make a statement about the entire population, not about the particular sample that was assessed. The data from the sample are used to make inferences about the population as a whole. As discussed in the previous section, each estimated sample mean proficiency (or proportion) has a degree of uncertainty associated with it. It is therefore possible that if all students in the population had been assessed, rather than a sample of students, or if the assessment had been repeated with a different sample of students or a different, but equivalent, set of questions, the performances of various groups would have been different. Thus, to determine whether there is a real difference between the mean proficiency (or proportion of a certain attribute) for two groups in the population, one must obtain an estimate of the degree of uncertainty associated with the difference between the proficiency means or proportions of those groups for the sample. Ti estimate of the degree of uncertainty -- called the standard error of the difference between the groups -- is obtained by taking the square of each group's standard error, summing these squared standard errors, and then taking the square root of this sum. Similar to the manner in which the standard error for an individual group mean or proportion is used, the standard error of the difference can be used to help determile whether differences between groups in the population are real. The difference between the mean proficiency or proportion of the two groups ± 2 standard errors of the difference represents an approximate 95 percent confidence interval. If the resulting interval includes zero, one should conclude that there is insufficient evidence to claim a real difference between groups in the population. If the interval does not contain zero, the difference between groups is statistically significant (different) at the .05 level. s THE 1990 NAEP TRIAL STATE ASSESSMENT 93 Montana As an example, suppose that one were interested in determining whether the average mathematics proficiency of eighth-grade females is higher than that of eighth-grade male; in a particular state's public schools. Suppose that the sample estimates of the mean proficiencies and standard crews for females and males were as follows: Group Average Proficiency Standard Error -. Female 259 2.0 Male 255 2.1 The difference between the estimates of the mean proficiencies of females and males is four points (259 - 255). The standard error of this difference is 2.02 + 232 = 2.9 Thus, an approximate 95 percent confidence interval for this difference is Mean difference ± 2 standard errors of the difference = 4 2 . (2.9) = 4 ± 5.8 = 4 - 5.8 and 4 + 5.8 = -1.8, 9.8 The value zero is within this confidence interval, which extends from -1.8 to 9.8 (i.e., zero is between -1.8 and 9.8). Thus, one should conclude that there is insufficient evidence to claim a difference in average mathematics proficiency between the population of eighth-grade females and males in public schools in the state.3 Throughout this report, when the mean proficiency or proportions for two groups were compared, procedures like the one described above were used to draw the conclusions that are presented. If a statement appears in the report indicating that a particular group had higher (or lower) average proficiency than a second group, the 95 percent confidence interval for the difference between groups did not contain zero. When a statement indicates that the average proficiency or proportion of some attribute was about the same for two groups, the confidence interval included zero, and thus no difference could be assumed between the groups. 'Me reader is cautioned to avoid drawing conclusions solely on the basis of the magnitude of the differences. A difference between two groups in the sample that appears to be slight may represent a statistically significant difference in the population because of the magnitude of the standard errors. Conversely, a difference that appears to be large may not be statistically significant. The procedure described above (especially the estimation of the standard error of the difference) is, in a strict sense, only appropriate when the statistics being compared come from independent samples. For certain comparisons in the report, the groups were not independent. In those cases, a different (and more appropriate) estimate of the siandcrd error of the difference was used. Ct, 94 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana The procedures described in this section, and the certainty ascribed to intervals (e.g., a 95 percent confidence interval), are based on statistical theory that assumes that only one confidence interval or test of statistical significance is being performed. However, in each chapter of this report, many different groups are being compared (i.e., multiple sets of confidence intervals are being analyzed). When one considers sets of confidence intervals, statistical theory indicates that the certainty associated with the entire set of intervals is less than that attributable to each individual comparison from the set. If one wants to hold the ce.-tainty level for the set of comparisons at a particular level (e.g., .95), adjustments (called multiple comparison procedures) must be made to the methods described in the previous section. One such procedure -- the Bonferroni method -- was used in the analyses described in this report to form confidence intervals for the differences between groups whenever sets of comparisons were considered. Thus, the confidence intervals in the text that are based on sets of comparisons are more conservative than those described on the previous pages. A more detailed description of' the use of the Bonferroni procedure appears in the Trial State Assessment technical report. Statistics with Poorly Determined Standard Errors The standard errors for means and proportions reported by NAEP are statistics and therefore are subject to a certain degree of uncertainty. In certain cases, typically when the standard error is based on a small number of students, or when the group of students is enrolled in a small number of schools, the amount of uncertainty associated with the standard errors may be quite large. Throughout this report, estimates of standard errors subject to a large degrec of uncertainty are followed by the symbol "!". In such cases, the standard errors -- and any confidence intervals or significance tests involving these standard errors -- should be interpreted cautiously. Further details concerning procedures for identifying such standard errors are discussed in the Trial State Assessment technical report. Minimum Subgroup Sample Sizes Results for mathematics proficiency and background variables were tabulated and reported for groups defmed by race/ethnicity and type of school community, as well as by gender and parents' education level. NAEP collects data for five racial/ethnic subgroups (White, Black, Hispanic, Asian/Pacific Islander, and American Indian/Alaskan Native) and four types of communities (Advantaged Urban, Disadvantaged Urban, Extreme Rural, and Other Communities). However, in many states or territories, and for some regions of the country, the number of students in some of these groups was not sufficiently high to permit accurate estimation of proficiency and/or background variable results. As a result, data are not provided for the subgyoups with very small sample sizes. For results to be reported for any subgroup, a minimum sample size of 62 students was required. This number was determined by computing the sample size required to detect an effect size of .2 with a probability of .8 or greater. 100 THE 1990 NAEP TRIAL STATE ASSESSMENT 95 Montana The effect size of .2 pertains to the true difference between the average proficiency of the subgroup in question and the average proficiency for the total eighth-grade public-school population in the state or territory, divided by the standard deviation of the proficiency in the total population. If the true difference between subgroup and total group mean is .2 total-group standard deviation units, then a sample size of at least 62 is required to detect such a difference with a probability of .8. Further details about the procedure for determining minimum sample size appear in the Trial State Assessment technical report. Describing the Size of Percentages Some of the percentages reported in the text of the report are given quantitative descriptions. For example, the number of students being taught by teachers with master's degrees in mathematics might be described as "relatively few" or "almost all," depending on the size of the percentage in question. Any convention for choosing descriptive terms for the magnitude of percentages is to some degree arbitrary. The descriptive phrases used in the report and the rules used to select them are shown below. Percentage Description of Text in Report I p = 0 None 0 < p s 10 Relatively few 10 < p s 20 Some 20 < p S 30 About one-quarter 30 < p s 44 Less than half 44 < p 5_ 55 About half 55 < p 5_ 69 More than half 69 < p 5_ 79 About three-quarters 79 < p S. 89 Many 89 < p < 100 Almost all p = 100 All _ .. 96 THE 1990 NAEP TRIAL STATE. ASSESSMENT Montana THE NATION'S REPORT CARD DATA APPENDIX For each of the tables in the main body of the report that presents mathematics proficiency results, this appendix contains corresponding data for each level of the four reporting subpopulations race/ethnicity, type of community, parents' education level, and gender. 102 THE 1990 NAEP TRIAL STATE ASSESSMENT 97 Montero TABLE A5 I Students' Reports on the Mathematics Class They Are Taking PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY MO NAEP TRIAL Eigh In-grade STATE ASSESSMENT Maihematics Pre-algabra TOTAt State Nation 36SEEM_CITY Mita State Nation Mapank State Nation American Indian State Nation TYPE CIF COMMUNITY &tram nral State Nation Other State Nation 10 is 4;1;1 $72 2.4 ( 24) 61 ( 2.4) 200 50 ( 2.5) 25 ( 1.0) 50 ( 9.0) 75 ( 4,41 240 ( 24) 43 (5.8) 252 ( 4.6) $4 ( 5.7) aim ( e«.) 00 ( 6.3) 2.4) 74 ( 4.5) 241) ( 3.1)1 59 ( 2.3) 279 1.0) 01 2.2) 251 2.0) 24 ( 2.0) 225 ( 41.2 ( 2.4 2r7 ( 27 ( 9.5) 3.9) (144 041 311( 5.0) 261 ( 4.0) $ ( 7.2) .4*) 25 ( 4.7) 203 ( 2.8)1 14 ( 5.0) 26 ( 2.0) 291 OA) 20 2.1) 272 23) 12 ( 1.5) 302 ( 34) ( 1.5) 300 ( 2.3) 32) *I* *el ( 1.5) Imp ( «in 13 ( 5.0) 0.11 .44) ( 2.7) .44 ( ihwt) 12 ( 4.1) 263 ( 72)1 ( 22) ( *se) 12 ( 1.1) 300 ( 24) 10 ( 1.4) 294 ( 2.7) ',11MmIlIMMIr The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 pacent cenainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the ample. The percentages may not total 100 percent because a small number of students reported taking other mathematics courses. ! Interpret with caution - the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size it insufficient to permit a reliable estimate (fewer than 62 students). 1 C 3 98 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana ...=111 ..1m..1 TABLE AS 1 Students' Reports on the Mathematics ChisS (continued) They Are Taking PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 11110 NAEP TRIAL STATE ASSESSMENT Mgha-grads ilalhenance - Prwa Ware Algebra _ TOTAL State Nation Of 2.1 2$1 1.4 PAREIUNCENS KS non-graduen State Nation KS 'vain. State Nation Sense collage State NaVon College graduate State Nation IMO U. State Nation Female State Nation 3.5) 200 211 n SS .. ( on et* oira 13 1 ) 3.4) 3 1.11 241 2.1 on 03 ( t 26 ( 5.1) (17 24$ ( la NO 1 213 70 2 276 ( 2011 22 ( 1$ ( 24 277 ( 5.2) do. «Nri 0 1.5) 1 1.1) 21 ( 1.9 257 ( 2.1) 61 ( 203 2.6 60 1 3.1 21 2.9 25 290 $ty 41 12 23) 15 13) 270 ( 2.4) 295 3.2) 55 ( 3.2) 211 ( 2.6) 15 ( 1.6) 264 ( 1.3) 206 ( 13) 53 ( 2.7) 21 ( 2.3) 304 ( 3.7 24 ( 1.1 250 ( 1.5) 271 ( 24) 303 ( 2.3 58 ( 2.9) 281 ( 1.3) 13 ( 2.1) 252 ( 1.11) ( 2.6) 275 ( 1.4) 11 ( 2.6) 251 ( 1.5) 251 2.5) 285 ( 1.8) 1$ ( 14) 275 ( 24) 20 ( 2.3) 278 ( 2.1) 20 ( 2.3) 260 ( 3.0) 13 ( 1.4) 301 ( 4.0) 15 ( 1.2) 220 ( 2.5) 10 ( 1.9) 290 ( 4.2) 15 ( 1.7) 293 ( 2.8) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within 2 standard errors of the estimate for the sample. The percentages may t t total 100 percent because a small number of students reported taking other mathematics courses. ! Interpret with caution - the 112-are of the sample does not allow accurate determination of the variability of this estimated mean proficiency. ' Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 1 4 THE 1990 NAEP TRIAL STATE ASSESSMENT 99 Montana TABLE A6 Teachers' Reports on the Amount of Time Students Spent on Mathematics Homework Each Day PERCENTAGE Of STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1111* NAEP TRIAL STATE ASSESSMENT - Nom 16 Minutes 30 Minuiss 46 Minis. I An Naar ar Mars TOTAL State 10 2 020( 1.2 411( 4.2 300 ( Nation RAOVETNNICITY White State Nation 14111Panic State Nation American Indian State Nation TYPE OF COMMUNITY Extreme rurat State 3 ( OA) 1 ( 222 ( 12 SS 13) 220 1 *** ( 4") 2112 ( 22 2112 2.7) 1 ( 03) VT 70 ii 3A 3 ( 21 34 ( ILO) IS ( 71 ) "de cm* 11 1 1 1 01) 48 1 7.II) 34 ( RS) 13 1 245 ( 3.444 251 ( 4.2)4 ser i see *Mr ( .rev. 5 ( 2.1) 35 ( 0.5) 45 ( 7.0) 9 ( SA) "P" ( 5") 291 ( 49)4 25$ ( SAP "11 ( "*) 0 , 0.0) 74 (31S) 22 (2112) 0 ( 0.0) rt. ( in 540 ( ..45) at* c on ( ***) Nation Other State Nation 43 ( 7.4) 277 ( 1.II)I 0$ (141) 253 ( 5.4)4 34 ( 2.2) 211 ( 1.4) 37 ( 4,3) 258 ( 3.1) 47 ( 8.1) 200 ( 2.3) 14 (101) 14,) 204 1.8) 40 Si) 53 21) 205 ( 2.5) 5 ( 2.0) .44 $ ( 5.8) ( 2,2) 54* 10 2.4) 278 ( 1.0 s 2 ( 1.6) *45(5.5) 10 ( 73) 4 ( 0.3) 4 ( 1.1) 2$2 MAY The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. Interpret with caution - the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. ' Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 1 r t.) 100 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE Ab Teachers' Reports on the Amount of Time (imntinued) Students Spent on Mathematics Homework Each Day PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1990 NAEP TRULL STATE ASSESSINENT , Nolte 15 Minutes 30 Minutes 45 115kades . An Hour or Mont . TOTAL. .91,00911, rialzdancy 3 04) 4. 1 03) gel 2.1) 1 ( 04) «.41 5 ( 1.4) «e «44) ( 0.5) ( «in 4 ( 0.5) HP* ( ( 0.9) 44. ( 2 ( 0.4) ( 0.3) ( ".) 3 ( 0.5 ) ( 1 ( 0.3) Mr* ( trirl 3 ( 0.4) *I* ( *41 I ( 0.4) Poisems99 an0 Prellielemy $I ( 2,0) 200 ( 12) 43 ( 4.2) 268 ( 2.3) 39 ( OA) elk.) 49 ( 83) 240 ( 2.5) 41 ( 3.9) 272 ( 13) 43 ( 5.2) 249 ( 3-1) 37 ( 4.1) 262 ( 1.7) 44 ( 5.4) 266 ( 2.6) 38 ( 3.3) 265 ( 1E) 40 ( 4.7) 265 ( 25) 37 ( 2.8) 252 ( 1.4) 44 ( 4.4) 257 ( 2.9) 39 ( 33) 277 ( 1.5) 41 ( 44) 255 ( 23) Porotatigs and frolialow W ( 2115 ( 12 43 ( 200 ( 2.0) 49 ( 7.5) ( 141 40 ( 8.1) 248 ( 3.7) 44 ( 3.7) 271 ( 2.4) 44 ( 5.5) 258 ( 23) 50 ( 4.1) 2118 ( 3.2) 43 ( 5.8) 270 ( 3.8) 52 ( 3.3) 290 ( 1.5) 44 ( 41) 277 ( 3.0) 52 ( 2 8) 287 ( 1.4) 43 ( 4.3) 260(2.0) 4$ ( 33) 279 ( 2.3) 43 ( 4.7) 264 ( 2.8) Pmedlage and trelialow II) 290 mop 10 1.9) 272 ( 53)1 ( 42) *el 8 ( 1.7) t 7 ( 2.8) .4,41 9 ( 3.1) 44* ( ( 1.5) 7 ( 2.1) Orli ( 5 (1.8) *.it 441 11 I, 2.3) 287 ( 6.1)1 5 ( 1.8) ( 1.9) 273 ( 7.3)1 ( 2.0) 278 ( 3.8)1 ( 2.0) 272 ( 5.7)1 Pwoolios and PralkiWICY ( 04) 282 (12.1) 4 ( 0.9) 270 ( 5.1)1 2 1.41 4 ( 1.3)) 3 ( 0.9) 14* ( 41 3 ( 1.0) 2 ( 1.1) ( 4 ( 1.0) 3 ( 0.5) 5 ( 1.3) 41 3 ( 0.6) 5 ( 1.3) 279 71/4 3 ( 0.8) ( *41 4 ( 0.9) ( ***) State Nation PARENTS' EDUCATION 143 noriraduats State Nation It5 graduate State Nation Sento college State Nation College graduate State Natioi GENDER M. State Nation Female State Nation The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is rtithin ± 2 standard errors of the estimate for the sample. ! Interpret with caution -- the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). THE 1990 NAEP TRIAL STATE ASSESSMENT 101 Montana TABLE A7 I Students' Reports OD the Amount of Time They I Spent on Mathematics Homework Each Day PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 11100 MEP TMAL STATE AlISESSMENT None 15 Mknomo 30 Mimi's M Minsios An Hour or Mono Nation Beigiangant Nle State Nation Moo* State Nation Arassican Indian State Natim ECEIZSZLIMM bongo nra1 State Nation Other State Nation el so 12 11101 1$1 V* Li tal $.1t It IS 15 2741 ` 7 $9 9011) 37 5.4) 14 4.1 11 SA 37 30 .12;1 17 Li 14 71.; 240 344 241 ( 43) ( 10 I 44 12 tit) ipLi 30 (i0.0) 27 ( 111.7) 24 114.2) 9 ( 41.1 «e« ( 1.5) 91 ( 270 ( 1.7) 203 2.2 200 3.5 ( 2.3) 30 4.6 10 ( 09) 20 ( 14) 271 3.1) 2E4 ( LS) ( 1.0) $0 ( 12) 3$ ( 270 31 255 23 251 32 1.7 2.0 2.9 5.1 1.7 1.3 ) 15 ( 271 ( 1$ ( 10 AS 15 1.3) 2.4) 3.9) 0.8) t7) 1.1) 13 267 7 ( 270 ( 1$ ( ( 24) 2413 ( LS) 214 ( 2.3 207 2.1) 250 ( 4.2) 1.5) 2.7) 2.2) 1.1) 0.9) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within 2 standard errors of the estimate for the sample. ! Interpret with caution - the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. "* Sample size is insufficient to permit a reliable estimate (fewer than 62 students). r 102 THE 1990 NAEP TRIAL STATE ASSESSMENT Maxima TABLE A7 I Students' Reports on the Amount of Time They (wiltinued) I Spent on Mathematics Homework Each Day PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1SS0 NAB MAL STATE ASSESSMENT 15 INNAss 30 Mutes 41 Wide' An Near w Mom 121ti State Nation DiRnatiffiriAngii I99 non-gradate State Nation HS graduate State Nation letne college State Nation College graduate State Nation GENDER, maw State Nation Female State Nation teItt.o.. *kW - No* 249 ow 24 30 4.1 17 241 ( 4.2) 250 .6170 i 1.1 32 2 i .8*. 272 ( 1 2 294 ( 2 27 ( 277 9 31 ( 1 9 ( 1 0 ( 1.1) $1 2.3 ) IS tA *do ( .4,6 223 ..204 *** ( 11,10 Ma 9 ( 12} SO 2.1 a 14 211t1 274 2115 ( SS) 275 ( 2.0) 271 ( 22 279 ( SA 271 24) 209 ( *2) 291 ( 1.4 227 2.1 2155 1 10 ( 1.1) 29 ( 14 1 7 ( OA) 31 ( 3.4 111 2 1$ 1.2 292 i 14 1 205 3.9) 204 2.1) ase 24 INS 222 3.9) 237 LS) 294 12 11 1.1) 11 1.1) 30 12) S4 2.4) 32 20 12 19 12 14 41 I 11 ( 2211( 4.1 273 ( 4.0) 2030 ( 14) 277 1.7 275 ( 1.0 273 3.5 240 ( 4.1) 203 ( 1.5) NO 2.0) 2IN 1 2.4 219 *3 7 ( OA) 29 ( 2.0) 34 ( $6 1.7 1$ ( 13) 17 ( 1.0 14 ( 13 1:3 ( OA) 211 ( 1.8) The standard errors of the estimated statistics appear in parenthesei. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). THE 1990 NAEP TRIAL STATE ASSESSMENT 103 Montana TABLE A8 I Teachen' Reports on the Emphasis Given To Specific Mathematics Content Areas PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY tit= NAEP TRW STATE ASSESSMENT timbers and Operations Maasurament Own fry Heavy Emphasis Little or No EMphasis Heavy Emphasis Uttie or No Em aslsJ Heavy Emphasis Little or No Emphasis State Nation 862fidifflrM %Mite Nation 217 2E1 13 206 .22 4.0 MO 14 ne 14 9 02 1.11 .2;3 410 44 '44' 14 ft 4A) ill 14; State 3411 240 111 07 4.3 215 13) 223 ,16) Niapanic *tr. g 41 ifez? 6 La.5.4 State Nation d .41) I 1 ( *) MS *** " Irb ** 4.4)1 ) 471.17) 23( 4.11 SA) 24 4 Amens= Indian State Nation 254 ( SAN ( ( NWT 45 ( 4.0) 11 ( 42) 11 ( 3.1) 29 7.3) 2$ ( 4,$) 44 (115) I .0.71} 7 17) 13 15.5) ! (0") 243 106)1 *** ( ..e.) es* ( ipih, *a/ 10 (10.7I TYPJL2F COMMUNITY Ulnae rural State 2111 2.6 12A 205 (101 44 7.4 $ 14 ( 111 22 3.7) 14 ( Nation 53 12.4 3.11 0 (41 32 11.7 9 11) II 7.11 ) 276 4A 257 7.1 INN! NI OM* f11/1 2.5 al ...) Itee Other Nation 130 2.3 MO 72 4.11 2.5 14 286 3.4) 247 10 22 10 3.41 34 6.3) 26 4.0 24 4.3 250 7.1 270 ( 4.111) 926 52 14 1.7 6 ( OA 41 2.4) 36 2.9) 11 ( 03) State AMMMIC The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. The pescentages may not total 100 percent because the "Moderate emphasis" category is not included. Interpret with caution - the n2ture of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 104 THE 1990 MARI' TRIAL STATE ASSESSMENT Montana TABLE A8 I Teachers' Reports on the Emphasis Given to (continued) I Specific Mathematics Content Areas PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1990 NAEP TRIAL STATE ASSESSMENT Nurtiben and Operations Measurement Geometry heavy Emphasis Little or No Emphasis , Heavy Emphasis Little or No Emphasis Heavy Emphasis Little or No Emphasis TOTAL Percentage and Pro Odom Parontege and Prvildency Porantae Poundage and and Prole lona P. necioncy Percentags and Praliciany Paranta. ind Prolkiency State 40 ( 2.6) 14 ( 1.3) 9 ( 1.0) 33 ( 31 ( 2.5) 13 ( 1A) 280 ( 2.0) 23 ( 2.9) 277 ( 5.7) 285 ( 3.3 ) 288 ( 1.5) 279 ( 2.8) Nation 49 ( 31) 15 ( 2.1) 17 ( 3.0) 33 ( 4.0 28 ( 3.8) 21 ( 3.3) 280 ( 1.8) 287 ( 3.4) 250 ( 5.8) 272 ( 4.0) 260 ( 3.2) 264 ( 5.4) PARENTS' EDUCATION NS non-graduate State 32 ***) ( *al Wit* ( *** *al 111 ( *el Nation 00 ( 0.9) 7 ( 2.3) 22 ( 5.3) 25 ( 5.3) 32 ( 6.3) 20 ( 0.7) 251 ( 3.4) ***) .41 ( 441 ( *el HS graduate State 44 ( 3.4) 9 ( 2.1) 10 ( 1.3) 32 ( 4.0) 26 ( 2.9) 14 ( 2.2) 273 ( 2.8) ( '1') 262 (10.8) 269 ( 4.7) 279 ( 3.9) 265 ( 5.1) Nation 55 ( 4.8) 11 ( 2.6) 17 ( 3.9) 27 ( 5.0) 27 ( 4.5) 24 ( 5.1) 259 ( 2.9) ( ") 251 ( 8.1)1 253 ( 4.7)4 255 ( 4.2) 246 ( 4.8y Som college State 30 ( 4.1) 16 ( 1.7) 6 ( 1.8) 30 ( 2.9) 29 ( 31) 11 ( 1.7) 280 ( 3.1) 301 ( 4.1) ( 444) 293 ( 51) 285 ( 42) 288 ( 4.8) Nation 47 ( 4.4) 17 ( 3.3) 12 ( 2.7) 39 ( 5.5) 27 ( 5.0) 23 ( 4.1) 265 ( 2.8) 264 ( 4.1)? ( ) 279 ( 4.5) 202 ( 4.8)? 270 ( 4.7) Cottage graduate State 40 ( 3.4) 16 ( 2.0) 8 ( 1.1) 36 ( 2.6) 35 ( 3.5) 14 ( 1.0) 288 ( 2.3) 297 ( 3.8) 289 ( 5.3) 293 ( 3.6) 290 ( 2.81 288 ( 3.5) Nation 44 ( 4.1) 19 ( 2.4) 16 ( 3.3) 37 ( 3.6) 26 ( 3.4) 21 ( 2.9) 269 ( 2.0) 298 ( 3.4) 264 ( 7.2)l 283 ( 3.8) 270 ( 3.3) 260 ( 8.4) GENDER Male State 37 ( 2.7) 14 ( 1.9) 9 ( 12) 33 ( 2.2) 31 ( 3.2) 14 ( 1.5) 282 ( 2.0) 299 ( 4.1) 279 ( 51) 290 ( 5.0) 288 ( 2.4) 281 ( 3.4) Nation 48 ( 261 ( 4.1) 2.5) 14 ( 237 ( 2.1) 4.4) 17 ( 258 ( 3.3) 6.7) 32 ( 275 ( 3.9) 4.8) 29 ( 263 ( 4.1) 3.8) 20 ( 2er ( 3.3) 8.8) Femal State 42 ( 3.4) 13 ( 1.8) 8 ( 1.8) 34 ( 2.4) 30 ( 3.1) 11 ( 1.8) 278 ( 2.9) 286 ( 4.3) 275 ( 7.4)1 280 ( 31) 283 ( 2.8) 277 ( 3.8) Nation 51 ( 3.9) 15 ( 2.4) 17 ( 3.2) 35 ( 4.3) 27 ( 3.9) 23 ( 3.5) 260 ( 2.0) 286 ( 3.3) 241 ( 54) 288 ( 4.1) 250 ( 3.3) 263 ( 5.0) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is withM ± 2 standard errors of the estimate for the sample. The percentages may not total 100 percent because the "Moderate emphasis" category is not included. Interpret with caution - the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 1 1 0 THE 1990 NAEP TRIAL STATE ASSESSMENT 105 Montana TABLE A8 I Teachers' Reports on the Emphasis Given To (cmitinued) I Specific Mathematics Content Areas PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1990 NAEP TRIAL STATE ASSESSMENT Data Analysis, Statistics, and Probability Algebra and Functions Heavy Emphasis Little or No Emphasis Heavy Emphasis , Little or No Emphasis TOTAL Percentage and Proficiency Percentage and Proficiency Permit:spa and Proficiency Penoentage and Proficiency State 13 ( 2.3) 59 ( 2.5) 58 ( 3.0) 10 ( 2.9) 287 ( 3.0) 281 ( 1.4) 281 ( 1.5) 268 ( 3.2)1 Nation 14 ( 2.2) 53 ( 4.4) 46 ( 3.6) 20 ( 3.0) 2$9 ( 4.3) 261 ( 2.9) 275 ( 2.5) 243 ( 3.0) RACESETH ICITY White State 13 ( 2.3) 58 ( 2.6) 58 ( 3.0) 9 ( 2.7) 288 ( 3.0) 284 ( 1.3) 285 ( 1.6) 271 ( 2.7)1 Nation 14 ( 2.4) 53 ( 5.0) 4a ( 4.2) 18 ( 2.8) 276 ( 4.1) 271 ( 3.1) 281 ( 3.0) 251 ( 3.3) Hispanic State 5 ( ( 3.1) "4) 62 58 ( 8.3) .41 8 ( 44-4 3.3) 4.0.) Nation 56 ( 246 ( 6.3) 4.4) 46 ( 257 ( 5.9) 4.0)1 18 ( 044. ( 4.2) American Indian State 7 ( 3.1) 66 ( 6.7) 53(7.5) 16 ( 6.6) 4" ( 4") 257 ( 5.8) 253 ( 5.8) Nation 3 ( 4.2) 82 (29.1) *** ( 16 (21.5) «a* ( .4*) 87 .41 TYPE OF COMMUNITY Extreme rurai State ( 5.8) 00 ( 279 ( 6.6) 3.0) 4$ ( 277 ( 6.7) 3.3) 15 ( 7.9) ..*) Nation 5 5.4) 85 (16.9) 33 ( 8.1) 42 (18.0) ( 254 ( 6.7)l 241 ( 5.9)t Other State 1$ ( 2.2) 58 ( 2.8) 65 ( 2.1) 8 ( 1.2) 289 ( 2.8) 282 ( 1.4) 283 ( 1.8) 263 ( 3.7) Nation 15 ( 2.9) 53 ( 5.2) 47 ( 4.3) 17 ( 3.3) 267 ( 4.7) 260 ( 3.4) 276 ( 2.8) 245 ( 4.4)1 The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. The percentages may not total 100 percent because the "Moderate emphasis" category is not included. Interpret with caution -- the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *" Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 106 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE A8 I Teachers' Reports on the Emphasis Given To (wntinued) i Specific Mathematics Content Areas PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY IWO NAEP TRIAL STATE ASSESSMENT Data Ma Von, Statistics, and Probability Algebra and Functions Heavy Emphasis Little or No Emphasis Hesavy Emphasis Little or No Emphasis TOTAL Percents,* and Pro Money PIWOOrage and Proadency Penmeall and Prolidancy peraanlage and Proficiency State 13 ( 2.3) 59 ( 2.5) 58 ( 3.0) 10 ( 2.9) 287 ( 3.0) 281 ( 1A) 281 ( 1.5) 266 ( 32)1 Nation 14 ( 2.2) 53 ( 4.4) 46 ( 3.6) 20 ( 3.0) 209 ( 4.3) 261 ( 2.9) 275 ( 2.5) 243 ( 3.0) PARENTS' EDUCATION HS non-graduat State 17 ( 4.9) 60 ( 4.8) 55 ( 5.5) 6 ( 2.0) 264 ( 4.6) 264 ( 4.2) Nation 9 ( *** 3.0) 53 ( 240 ( 7.7) 6.2) G4r1 29 ( ( 6.9) 44) HS graduate State 14 ( 3.6) 59 ( 271 ( 3.4) 2.5) 54 ( 273 ( 3.7) 2.2) 11 ( 3.0) *ate) Nation 17 ( 3.7) 54 ( 5.4) 44 ( 4.8) 23 ( 3.9) 261 ( 8,0)1 247 ( 2.9) 265 ( 3.5) 239 ( 3.4) Some college State 11 ( IP** ( 2.3) ***) 58 ( 288 ( 3.7) 3.3) 56 ( 282 ( 3.7) 2.9) 9 ( 3.4) ***) Nation 13 ( 2.5) 57 ( 5.8) 48 ( 4.8) 17 ( 3.1) 270 ( 3.7) 278 ( 3.0) College graduate State 12 ( 2.2) 58 ( 2.8) 61 ( 3.4) 10 ( 3.3) 293 ( 4.4) 287 ( 1.7) 289 ( 1.5) 272 ( 5.0)1 Nation 15 ( 2.4) 53 ( 4.4) 50 ( 3.9) 18 ( 2.4) 282 ( 4.5) 275 ( 3.8) 288 ( 0) 249 ( 4.0) GENDER Male State 14 ( 2.4) 58 ( 2.6) 58 ( 3,5) 10 ( 2.7) 292 ( 3.6) 284 ( 1.6) 283 ( 2.2) 267 ( 3.0)1 Nation 13 ( 2.2) 54 ( 4.7) 44 ( 4.1) 22 ( 3.6) 275 ( 5.8) 260 ( 3.5) 276 ( 3.2) 243 ( 3.0) Female State 12 ( 2.6) 59 ( 3.0) 58 ( 3.3) 9 ( 3.3) 281 ( 3.2)1 278 ( 1.9) 280 ( 2.0) 269 ( 4.5)1 Nation 16 ( 2.4) 53 ( 4.5) 48 ( 3.6) 18 ( 2.9) 263 ( 4,4) 262 ( 2,8) 274 ( 2.7) 244 ( 3.9) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. The percentages may not total 100 percent because the "Moderate emphasis" category is not included. ! Interpret with caution -- the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a rehable estimate (fewer than 62 students). 1 2 THE 1990 NAEP TRIAL STATE ASSESSMENT 107 TABLE A9 I Teaches' Reports on the Availability of Resources PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATiCS PROFICIENCY ISSO NAEP TMAL I OM MI So Rowan I I OM Mast of Ma I Clot Sawa ar Nana of STATE ASSESSMENT Need Nalwarcas I Need No Ramouroas I Need State Nation BASIMIESIU MOM State Nation iiispanic State Nation Amartain Indian State Nation nnieeatiMODI &Irmo rural State Nation Omer State Nation 14 ( U) ta 731 ;11 34 7.7) - 411 2441 1.7)1 214) 1244 244 ( 2441 13 ( 77 I ft .40 10 ( U) 0( 7.4) 72 (2r11) 22 120.71 oh* ( 2113 2.7)1 14 4.41) 2 2.6) eei 3.111 2.4 11 2.011 205 10 2.2 NI ( 6.5) 271 ( 2.2) 54 (104) 280( 84)1 50 ( 3.1) 2$1 ( 14) SII ( EA) 2414 ( 2.1) 20 ( 4.0) 277 ( 24)) 43 (10.3) 257 ( 5.0)1 22 ( 3.2) 202 1.1) 2213 4.2) 54) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population it within ± 2 standard errors of the estimate for the sample. I Interpret with caution the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). I 7 3 108 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE A9 J Teachers' Reports on the Availability of (continued) Resources PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1860 NAEP TRIAL I Get AN the ItillOWCAIS 1 1 (1111 Wet OS MS 1 Get law sr None of STATE ASSESSMENT Need Remands I Need the Reeserces I Need TOTAL State Nation PARENTS EDUCATtON HS ooniraduate State Nation HS graduate State Nation Some college State Nation Whigs graduate State Nation GENDER M. State Nation Female State Nation 111.1840.0 ,c80818mW 17 ( 1.8 13 ( 1.4 205 ( 42) "1"111:111.411. Ofivialsoir 280 t.111 al 4.0 02 15 ( SI) 441) 88 ( MIS ( 54 ( 244 ( 24 5.7 2.7) 1$ ( 2.7) 81 ( 3.4) 278 ( 2.1) 270( 2.4) 10 ( 2.5) 54 ( 42) 253 ( 4.8)4 25$ ( 12) 18 ( 33) $2 ( 44) 294 ( 4.8); 282 ( 1.8) 13 ( 3.3) 82 ( 4.3) 2e0 ( 2.5) 17 ( 2.2) 2es ( 2.0) 83 ( 28$ ( 3.1) 1.3) 15 ( 2.9) 56 ( 4.9) 276 ( 54)4 275 ( 2.2) le ( 2.1) 82 ( 33) 267 ( 2.4) 264 ( 12) 13 ( 2.6) 57 ( 4A) 284 ( 5.0)1 205 ( 2.8) 18 ( 2.6) 62 ( 3.2) 263 ( 2.7) 278 ( 1.8) 13 ( 2.4) 55 ( 44) 206 ( 3A) 264 ( 2.0) 21 .343414 38 13 943 23 22 25$ ( 22 22 ( 201 ( 2.8 2$ ( 4.1 on ( 3.8) 18 ( 5.1) 209 ( 24) 30 ( 5.1) 273 ( 3.7) 21 ( 3.1) 2613 ( 2.4) 30 ( 4.0) 254 ( 3.3) 20 ( 2.7) 277 ( 2.1) 32 ( 4.7) 257 ( 3.0) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within * 2 standard errors of the estimate for the sample. I Interpret with caution -- the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 1 4 THE 1990 NAEP TRIAL STATE ASSESSMENT 109 Montana TABLE AlOa I Teachers' Reports on the Frequency of Small I Group Work PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 111110 NAEP TRIAL STATE ASSESSMENT I I At Laast Owe a Wank Less Than Om* a Week Now TOTAL INWOMINNIM AINIMMIIRS GO ( $.0) 14401101410112, SI ant Prolkleacv State 242 ( 1.2) 20131 11:11 Nation 40 ( 4A) 43 4,1) 6 2.0) 200 ( 22) 294 ( 211) 277 ( 5.4)4 SMail_4NICITY %Ma State 00 ( 3.1) 32 ( 3.1) 9 ( 1.9) 245 ( 1.2) 244 1.4) 241 ( 2.6)4 Nation 49 ( 4.6) 43 4.5) ( 2.3) 245 ( 2.7) 271 2.2) 285 ( 4.9)4 HIspank State 54 ( 45) .+4 041 34 ( 8.5) iNhe 4.041) ( 00. ( Nation 64 ( 7.2) 246 ( 2.6) 32 ( 8.9) 247 ( 8.3)4 4 ( 1.4 d.,41 Anworlext Indian State 56 ( 8.4) 267 ( 46)4 34 ( 5.4; 254 ( 5.8)4 6 ( 2.9)) Nation 18 (24.3) ( 441 80 (27.2) 444 2 ( $1) .44.) TYPE Of COMMUNITY Extrante rural State 54 ( 7.4) 34 ( 7.0) 12 ( 5.6) 277 ( 2.1)1 277 ( 4,0)1 281 ( 52)! Nation 35 t14.6) 255 ( 5.5)1 56 (17.1) 258 ( 5.9)1 ( 6.6) 4144, Other State 85 ( 22) 30 ( 2.1) 5 ( 0.4) 244 ( 1.5) 252 ( 2.3) Nation 50 ( 4.4) 44 ( 4.5) ( 1.6) 244.4 (2.4) 264 ( 25) 277 ( 6.3)1 The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. ! Interpret with caution -- the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. I** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 110 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE A:Oa I Teachers' Reports on the Frequency of Small (continued) I Group Work PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY MO !IAEA =AL STAT SSE11111 ENT E A At Least Ones a Weak Less Thee Once a Week - Never 12104, State Nation 31,41420,4, pielaisiow 222 IS) 1:11 20 4A) 43 4.1 200 3.2) 204 23) 14111111102010 oil Pn$41444y '2110 23 277 ( SA EMETEIESAIEN N$ nonirsdisste State ( 7.1) 20 6.5) 4 ( 25) 271 ( 3.4) *** emt) Nation 10 ( 14) 6.531 1.41 le graduate 244 ( 3.2) 24430 State 5$ 270 1 34 ( 44) 273 ( 2.6) i0 2.9) Nation 41 4.1 41 ( 5.1) 0 C 2.5) some =foga 252 ( 2.1 257 ( 2.7) State 57 ( 265 ( 2.2 35 ( 3.2) 203 ( 3.0) 2.8) .*) Nation $1 ( 5.2 42 ( &I) ( 2.3) 266 ( SA) 286 ( 3.2) College graduate State 63 ( 3.2 29 ( 24) 8 ( 1.7) 219( 1.4) 287 ( 2.5) 286 ( 3.3p Nation 46 ( 52) 43 ( 4.4) 11 ( 2.7) 271 ( 2.6) 276 ( 3.0) 265 ( 4.1)/ 0040ER M. State CO ( 3.8) 33 ( 34) 7 ( 1.7) 248 ( 1.5) 293 ( 24) 279 ( 3.1)1 Nation 50 ( 4.5) 42 ( 4.0) ( 2.1) 281 ( 3.0) 285 ( 3.1) 278 ( 5.3)1 Female State 80 ( 3.2) 31 ( 3.2) 9 ( 2.1) 277 ( 1,3) 277 ( 3.7) 241 ( 3.7)1 Nation 50 ( 4.7) 43 ( 4.7) ( 2.1) 259 ( 2.2) 283 ( 2.1) 275 ( 6.6)1 The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. ! Interpret with caution - the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 1 6 THE 1990 NAEP TRIAL STATE ASSESSMENT 111 Montana TABLE A 10b I Teachers' Reports on the Use of Mathematical I Objects PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY MO NAEP TRIAL STATE ASSESSMENT Al Least Once a Week Lees Than Once a We* Never TOTAL Poressise and 11411444a4y $7 (2.0) 242 1.7) 22 3.7) for043144. 2:11 State Nation 254 ( 3.2) 253 1 RAMTNNICITY Mite State 31 ( 2.8) ( 2.5) 208 ( 1.5) 213 ( Nation 17 ( 4.0) 72 ( 4.2 201 ( SA)! ( 2.1 Hispanic State 43 ( 7.8) 511 7.7) Nation 30 ( 7.5) 55 ( 247 ( 3.8) 245 ( 3.8)1 American Indian State 22 ( 8.5) 73 ( 115) Mir ( 201 ( 3.4) Nation 71 (34.8) 22 (34.8) TYPI Of COMMUNITY ladreme rural State 32 ( 64) 67 1 Oh) 275 ( 3.7) 279 ( 1.6) Nation 27 (14.9) ( *on 05 (143) 262 ( 2.8)1 Other State 40 ( 2.7) 50 ( 23) 235 ( 2.0) 211 ( 1.4) Nation 19 ( 4.3) 72 ( 5.0) 253 ( 3.9)1 263 ( 2.2) 1 (OA) 441 1101 2.7) 21115 guy ( 2.8) ".) 5 ( 2.2) Mr* (011r9 01 0.0) *MP *41 1 ( O.?) ( ( 3.9) ( *41 1 ( 0$) 251 ( 7.41 The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within t 2 standard errors of the estimate for the sample. ! Interpret with caution - the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *" Sample size is insufficient to permit a reliable e-'imate (fewer than 62 students). 112 THE 1990 NAEP TRIAL STATE ASSESSMENT Montww TABLE AlOb I Teachers' Reports on the Use of Mathematical (c*ntinued) I Objects PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 11100 NAEP TRIAL STATE ASSESSMENT At Least Ones a Vittek LOIS Than Once a Meek Never TOTAL, Pementage and lorseskagl Pirosolsgs SRI Prolioisecy Percaulses and Widow State $7 ( 282 ( 2.5) 11) 82 ( 261 ( 1.0 *H. 0.4) 041 Nation 22 ( 37) Oa ( 2.5) 254 ( 3.2) 203 ( 1,9) 242 ( 5.9)1 PARENTS' EDUCATION NS non-graduate State 36 ( 82) 81 ( 6.1) 3 ( 12) 269 ( 3.1) Nation 25 (( 5.6) 06 ( 243 ( 7.2) 2,2) 9 ( &SI .40) HS graduate State 32 ( 3.3) 06 ( 3.2) 2 ( 0.8) 269 ( 2A) 273 ( 1.7) ( Nation 23 ( 248 ( 4.11) 4.0)I TO ( 255 ( 5.3) 22) 7 ( 2.8) (4 ) Sem cealge State 38 ( 3.4) 00 ( 3-3) 286 ( 2.5) 262 ( 2.1) Nation 18 ( 281 ( 4.0) 4.4)! 73 ( 269 ( 4.3) 2.3) 9 ( 2.4) 0..) College graduate State 39 ( 3.4) 00 ( 3.4) 1 ( 0.3) 219 ( 1.8) 267 ( 1.5) Nation 20 ( 3.9) 09 ( 3,7) 11 ( 2.5) 266 ( 3.5)1 274 ( 22) 297 ( 4.2)1 GENDER Mato State 36 ( 284 ( 3.2) 2,9) 03 ( 265 ( 32) 1.1) 1 ( 0.3) 4.1 Nation 22 ( 41) 09 ( 4.1) 8 ( 2.0) 255 ( 4.1) 265 ( 2.1) 287 ( 72)! Renal' State 38 ( 3.0) 00 ( 3.1) 2 ( 0.6) 280 ( 22) 270 ( 1.5) Nation 21 ( 3.6) 89 ( 4.2) 10 ( 3.3) 254 ( 3.3) 2t? ( 1.9) 278 ( 8.0)1 The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. 1. Interpret with caution - the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 1 s THE 1990 NAEP TRIAL STATE ASSESSMENT 113 Montww TABLE AI la I Teachers' Reports on the Frequency of Mathematics Textbook Use PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 16190 NAEP TRIAL. STATE ASSESSMENT Ahmed Every Day Several Times a tffilat a About Once Week or LAM Niait State Nation awl Poldisionw :2$4 SA 267 t* Porsemesp Pnikflaaat 2 278 11 3.1 264 ( 24) BagalaNTY White State 63 ( 3.1) 31 ( 2.6) 286 ( 1.1) 261 ( 1.2) Nation 84 ( 3,7) 26 ( 3.2) 272 ( 1.9) 264 ( 2.4) Hispanic State 50 ( 9.0) *en 39 (( 9.1) Nation 61 ( el) 32 ( 53) 251 ( 3.1) 240 ( 4.3)I American Indian State 51 ( 7 .0) 45 ( 7.3) 283 ( 4.0) 250 ( 4.2)1 Nation 15 (25.11) *** ( 63 (263) TYPE Of COMMUNITY Extreme rural State 61 ( 0.7) 31 ( 4.6) 281 ( 1.6) 271 ( 3.8) Nation SO (10.0) 40 (10.0) 208 ( 4.0)1 247 ( 7.0)1 Other State 02 ( 24) 32 ( 3.0) 285 ( 1.4) 278 ( 1.3) Nation 63 ( XS) 31 ( 3.5) 287 ( 2.3) 255 ( 3.1) Peresaisips Noe Preildwity 14) 279 1.4 240 5.1 ( 14) 262 ( 3-1)1 ( 23) 264 ( 5,4)4 $ ( 2.6) ( mon 9 ( 2.3) 114.11 ( MO) 4 ( 23) 44,) 2 ( 3.0) 4* ( 7 ( 54) ( 441 8 ( 0.7) 279 ( 3.8) 0 ( 14) 257 ( 54)I The stardard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. Interpret with caution -- the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 114 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE Alla I Teaches' Reports on the Frequency of (continued) Mathematics Textbook Use PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1990 NAEP TRIAL STATE ASSESSMENT Aknoet Every Day Several Then a Wolk About Ottoo a Week or Lass TOTAI- State Nation PARENTS' EDUCATKIN KS non-graduate State Nation fitt graduate State Nation Sam coils,' State Nation College grutuat State Nation GENDER Male State Nation Female State Nation PerialMage 11Ri 01110101111113/ 05 ( 6.3) 269 ( 2.3) 07 ( 5.5) 245 ( 3.2) 01 ( 4.1) 274 ( 1.5) 61 ( 4.4) 257 ( 2.5) 62 ( 4.2) 2118 ( 2.3) 68 ( 42) 272 ( 2.7) 61 ( 3.1) 290 ( 1.4) Of ( 4.0) 261 ( 2.2) 61 ( 3.1) 287 ( 1.5) 80 ( 3.7) 2ea ( 2.1) 62 ( 3.4) 281 ( 1.3) 85 ( 3.6) 266 ( 1.6) 0101110161190 and trelklem. I 111 254 2.9 28 ( 5.5) ( don 27 ( 5.2) 32 ( 3.1) 266 ( 3.4) 34 ( 3.7) 250 ( 2.9) 33 ( 3.9) 277 ( 2.1) 26 ( 3.7) 258 ( 5.2) 33 ( 2.6) 263 ( 2.0) 31 ( 3.9) 265 ( 3.1) 33 ( 23) 280 ( 2.0) 33 ( 34) 256 ( 3.6) 31 ( 2.7) 272 ( 2.0) 28 ( 3.3) 253 ( 2.5) illanspiege iM6 241/1100181 II 'MS 9.0 7 1.0 280 9..1 ( 23) 7 ( 2.7) 15) *** 141 ( 20) e 1.9)) to* 1.6) 3.1) .hen *** 7 1.2) 261 ( 8.7)4 ( 2.1)) 7 ( 2.2) ( The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. Interpret with caution -- the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). THE 1990 NAEP TRIAL STATE ASSESSMENT 115 Montana TABLE Al lb I Teachers' Reports on the Frequency of Mathematics Worksheet Use PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1S110 NAEP TRIAL Al Laist Swami Tknas STATE ASSESSMENT a Ws* About Oncto a Weak Less than %%aft , TOTAL Paressdase and firsidistray feraudaga and Pralidelav ParaNdep and Predidsony State 47 ( 3.3) 22 ( 2.1) 00 ( 22) 2$0 ( 1.1) 2$2 ( 2.7) 241 ( OA) Nation 34 ( 3.8) 3$ ( 3.4) 32 ( SA) 250 ( 2.3) 200 ( 2.3) 274 ( 2.7) RACE/ETHNICITY Mite State 4$ ( 3.4) 23 ( 2.2) 30 ( 23) 282 ( 1.1) 288 ( 2.1) 254 ( 1.0) Nation 32(4.1) 33 ( 3-5) 35 ( 3.8) 264 4 21) 264 ( 2.7) 279 ( 2.9) Hispanic State 40 ( 8.7) ( *41 17 ( 8.5) 044 ( 43 ( 9.1) ( *41 Nation 41 ( 7.7) 2Sf 5.3) 33 ( 7.5) 242 ( 12)1 244 ( 5.1)1 257 ( 2.3)' American Indian State 49 ( 7.9) 200 ( 4.1)1 20 ( Si) 244 ( 9.9)1 31 ( 6.0) ..**) Nation 10 (113.6) 76 (362) 0114 ( 444 ) 13 (18.5) ( 0,1 TYPE OF COMMUNITY Extreme nwal State 59 ( 7.5) 18 ( 4.8) 23 ( 48) 280 ( 1.7)1 269 ( 7.1)1 277 ( 2.0) Nation 27 (14.3) 49 (12.7) 24 (WA) 258 ( 6.7)1 ( Other State 42 ( 2.8) 24 ( 1.9) 34 (2.1) 279 ( 1.5) 288 ( 2.5) 282 ( 1.0) Nation 30 ( 44) 35 ( 43) 36 ( 42) 256 ( 3.3) 259 ( 2.8) 272 ( 2.9) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population Is within ± 2 standard errors of the eptimate for the sample, 1. Interpret with caution - the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufacient to permit a reliable estimate (fewer than 62 students). 1 2 1 116 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE Al lb Teaches' Reports on the FIN (continual) I Mathematics Worksheet Use f PERCENTAGE OF 5 DENTS AND A'ffERAGE MATHEWTt5 PROFICIENCY MU NAEP TRIAL STATE ASSESSMENT At Lead Sew* naps Week About Once a Week - Lass Sun Weekly State Nation 1261MILMEN&USE NS nsn-graduate State Nation AS graduate State Nation Same college State Nation CoStge graduate State Nation OENWR M. State Nation FORUM Stine Nation SS 2,4 all 19 2410( Iwt) Nod Preltrimmy 2AI 3111 SA 274 2.7) *a 44 29 &Si 40 ( s. 38 I Ili 230 ( 3.2) *ft ( 2$0 ( 4.5)I :0 I 42.1i 35 53) 20$ ) 21 ( 2.2) 3$ 27304 I 234.7)) 30 4.8) 250 ( 2.8) 250 ( 2. ) 263 ( 3.4) 47 ( 11.8) 22 ( 2 31 ( 2.7) 242 ( 2.5) 2170 ( 210 ( 2.8) 33 ( 201 ( 32 ( 4 22033$71 422.8.11 4.73 50 ( 3.5) 22 ( 2. 29 ( 2.5) 1.3) 35 3.1) 290 ( 2 11 26 ( 1.0) 32 ( 264 2.9) 271 ( 24 299 ( 2.9) 3$ ( 3.5) 50 ( 3,8) 283 ( 1.2) $5 ( 4.1) 257 ( 3.2) 45 ( 3,8) 278 ( 1.8) 34 ( 4,1) 254 2.1) 20 2.7 287 4 35 25 ( 2 270 ( 3 32 ( 3. rAl ( 2 The standard errors of the estimated statistics appear in parentheses. certainty that, for each population of interest, the value for the entire po of the estimate for the sample. ! Interpret with caution - the nature determination of the variability of this estimated mean proficiency. *** reliable estimate (fewer than 62 students). THE 1990 NAEP TRIAL STATE ASSESSMENT 30 ( 27) 284 ( 1.8) 31 ( 3,5) 275 ( 3,2) 31 ( 2.1) 278 ( 1.3) 34 ( 4,1) 273 ( 2.8) t can be said with about 95 percent ulation is within ± 2 standard errors the sample does not allow accurate ample size is insufficient to permit a 117 Montana TABLE Al2 I Students' Reports on the Frequency of Small I Group Work PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1 1100 NAEP TRIAL STATE AS SESSMENT At Least Ones a WiaA Leas Than One* a Wm* New TOTAL State Nation 291 300 1.4 1 210 RACE/ETHNICITY *tits State Nation Hispanic State Nation Amarican Indian State Nation TYPE QF COMMU4ITY Extrema rural State Nation Odor State Nation OM 31) 2014 131 $i 1.0 27 2.9 1.7 11,*1) 242 ( SA 37 ( 5,21 1 Si 1 29 272 ( 14) 22 24 250 ( SA) SO ( 5.5) 27 ( 3.7) 252 ( 4.0)I VIM 31 ( 5.1) S$ ( ***) 111441) 33 ( 3.9) 29 ( 2.1) 276 ( 2.7) 200 ( 2.1) 34 (10.6) 27 ( 3.5) 249 ( 5.3)I 264 ( 3.5)i 31 ( 1.7) Si ( 14) 252 ( 1.5 263 ( 1.7) 27 ( .02} 26 ( 1.7) 200 ( 3.3 264 ( 2.1) 240 2A 121 El 41 $11 217 5.1) 5.0) 335.0) ow SS ( S.5) 277 ( 23) SS (11A) 250 ( 6.2)I IS ( 1.5) 200 ( 14) 41 ( 3.3) 262 ( 22) 4 The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the vidue for the entire population is within t 2 standard errors of the estimate for the sample. ! Interpret with caution -- the nature of the sample does not allow accurate detertni.nation of the variability of this estimated mean proficiency. *" Sample size is Msufficient to permit a reliable estimate (fewer than 62 students). 118 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE A 12 I Students' Reports on the Frequency of Small (mitinued) I Group Work PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY STATE ASSESSMENT MAD TRIAL AI Least Ono a Week Lass Than Once a Week New /Ea State Nation MagatliffAlt_ON HS nan-graduata State Nation Hi graduate State Nation S ense college State Nation College graduate State Nation OENDER M. State Nation Fem.* Stire Nation foregalk. Problems/ $1 ( 13) NO 1A) 2$ 24) 258 2.7) 40 ( GI) 444 ( .4) 29 ( 4.5) 242 ( 3.4) 30 ( 2.9) 242 ( 2.4) 26 ( 3.0) 251 ( 3.7) 29 ( 2.5) 225 ( 2.1) 27 ( 3.9) 285 ( 31) 32 ( 22 2116 ( 2.0) 211 ( 3.0) 270 ( 2.7) 32 ( 2.1) 202 ( 1.7) 31 ( 21) 259 ( 3.3) 31 ( 1.9) 277 ( 2.0) 20 ( 2.4) 257 ( 2.8) rod Pralidemqt $0 (tS) 222 (1,$) Perameass and Orideimlif ell 241( 14 ) 44 ( 2.9 267 ( 2.0) 261 ( 25 ( 04) 35 ( 5.9) ( 4.) 0.11r41, ( Oil 29 ( 3.0) 42 ( 4.5) 244 ( 3.0) 242 ( 2.7) 30 ( 2.3) 40 ( 2.8) 271 ( 3.3) 274 ( 2.4) 28 ( 1.11) 43 ( 3.4) 261 ( 21) 252 ( 1.7) 31 ( 3.1) 40 ( 2.8) 287 ( 3.0) 279 ( 2.4) 27 ( 2.4 46 ( 3.8) 268 ( 3.3) 268 ( 2.1) 30 ( 1.9) 38 ( 2.2) 290 ( 14) 266 ( 1.7) 2$ ( 1.9) 44 ( 31) 278 ( 2.8) 275 ( 22) 32 ( 2.1) 37 ( 14) 200 ( 2.2) 263 ( 1.5) 20 ( 1-7) 41 ( 24) 2011 ( 21) 262 ( 1.8) 29 ( 1.7) 41 ( 1.9) 279 ( 1.3) 278 ( 2.2) 27 ( 1.11) 47 ( 3.2) 200 ( 1.7) 260 ( 1.9) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within * 2 standard errors of the estimate for the sample. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). THE 1990 NAEP TRIAL STATE ASSESSMENT 119 Montana TABLE A 13 I Students' Reports on the Use of Mathematics Objects PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1690 AMP TRIAL TE ASSIIIIIMENT STA At Load Once a Week Loss Than Once a Week. Plover 12M12 State Nation MadatiaM WINN State Nation diaper* State Nation American Indian State Nation Typg OF COMMUNITY Extreme nrid State Nation Other State Nation Pansmitro torieriew ( 1.2) Oa 13) $1 1.2) 20. 15) Peromease Old $1911$01/49$ 27 ( 279 ( 41 ( 2,2 2311( 1.9) 37 ( 1.3) 37 ( 1.2) 26 1M5 ( 1.2) 205 ( 13) so Vt 13) 33 ( 1A) 40 2116 ( 23) 275 ( 1.9) ( SO ( 53) 31 ( ILO) Se ( 9.11) 6.11 ( 001 f ill IMb ( OM) 1111 ( 4.2) 23 i 2.0) 40 ( 4.0) 241 ( 4A) 253 ( 4.3) 240 ( 1A) 29 ( 4A) 23 ( 3.4) 257 ( 4.2) 36 ( 5.1) 257 ( 5.4) 36 ( 3.4) 04* ( 4.43 37 ( 12) .... ( .**) 2$ f LA) ... ( *An 35 ( 2.2) 4ff 2.0) 23 ( 2.5) 279 ( 2.5) 27$ ( 1.7) 274 ( 3.3) 21 ( 3.1) ( 44,41 37 ( 4.7) 262 f 4.7)1 43 ( 5.0) 251 ( 5.2)( 37 ( 1.4) $4 ( 1.6) 29 ( .6) 2413 ( 1.5) 265 ( 1.1) 277 ( 1.6) 27 ( 2.0) $1 ( 1.4) 41 ( 2.4) 259 ( 2A) 270 ( 1.8) 200 ( 2.2) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. ! Interpret with caution -- the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *" Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 120 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE A13 I Students' Reports on the Use of Mathematics (c4mtinued) I Objects PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1910 MEP TRIAL STATE ASSESSMENT At Least Once a Weak Lass Than Onoa a Weak Mow TOTAL lIt 1.1) ( 1.2) 20 ( 1.11) State Nation minnizsarm 21$ ( 2A) 141 mergracktate State 31 ( 4.1) Nation 27 ( 4.2) 237 ( 3.0) 1111 graduate State 32 ( 2.8) 271 ( 2.3) Nation 27 ( 2.7) 250 ( 2,4) Soma cottage State 37 ( 2.5) 2161 2.7) Nation 29 ( 29) 281 ( 3.5) Collage graduate State 40 ( 1.5) 217 ( 1.3) Nation 30 ( 23) 269 ( 3.0) GENDER Maio State 37 ( 1.7) 264 ( 1.3) Nation 32 ( 2.0) 256 ( 2.9) Female State 35 ( 1.9) 279 ( 2.1) Nation 25 ( 2.0) 257 ( 3.0) ( .40 ( SIM 34 $.1) 24 2.7 ;4:012.3 253 ( 3.5 24047 31 ( 2.8) ( 2/2 ( 2.7) 270 ( 2.11 111 ( 2.4) 43 ( 3.3 216(2.7) 263 ( 2.1) 37 ( 2.1) 29 ( 1.9) 213 ( 1.8) 2/9 ( SM 38 ( 2.3) 36 ( 2.8) 274 ( 2,2) 203 ( 2.1) 38 ( 1.9) 25 ( 2.0) 290 ( 1.5) 264 ( 2.2) 32 ( 2.0) 38 ( 2.8) 271 ( 2.0) 275 ( 2.0) 35 ( 1.8) 27 ( 1.9) 206(2.1) 241 ( 2.1) 30 ( 13) 3111( 2.2) 271 ( 2.1) 200 ( 1.6) 37 ( 1.4) 27 ( 1.5) 270 ( 1,3) 272 ( 1.0) 31 ( 1.9) 414 ( 2.6) 266 ( 1.5) 257 ( 1.9) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students), THE 1990 NAEP TRIAL STATE ASSESSMENT 121 Montana TABLE A14 I Students' Reports on the Frequency of Mathematics Textbook Use PERCENTAGE Of STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY - 11190 NAEP 'TRIAL STATE ASSESSMENT Newt Every Day Several Times a Week About Onc Week or e a LOU a TOTAL earameaga ame Prwealang 71(1.5) 242 ( OA) 74 ( 1.1) 2ti ( 1,2) PeeReelees PrweWataaff 13( 1.0) 14) 14 OA) 252 1.7) flet,***11. 000110111gat See (3.2) 12 ( 14) 242 ( 4.5) State Nation RACEIETHNICITY Witte State 79 ( 1.6) 13 ( CO) 1 ( 1.0) 265 ( Me) 242 ( 1.7) 270 ( 3.4) Nation 711( 2.5) 13 ( 0.8) 11 ( 2.2) 274 ( 14) 258 ( 2.2) Hispanic State OA) 23 ( 5.2) 9 ( 4.0) '41 Nation 61(3.7) 21 i 2.9) 17 ( 2.7) 249 ( 2.3) 242 ( 5.1) 224 ( 3.4) American Indian State 78 ( 4.3) 14 ( 3.7) ( 2.2) 251 ( 3.7) *** 44P*) *4114P) Nation 61 ( 4.4) .. ( 22 ( 30) 17 ( 4.0) TYPE OF COMMUNITY Darlene rural State 79 ( 2.0) 13 ( 1.7) 7 ( 1.8) 278 ( 1.9) 279 ( 3.0) 237 ( 8.1)1 Nation ee (113) 203 ( 4.2)1 15 ( 3.6) 444 ( 4 17 ( 5.2) aaa) Other State 78 ( 1.7) 13 ( 1.1) a ( 1.2) 204 ( 0.9) 279 ( 2.2) 205 ( 3A) Nation 75 ( 2.2) 14 ( 1.0) 10 ( 1.9) 26? ( 1.6) 252 ( 2.0) 239 ( 4.3)1 The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. I Interpret with caution - the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 1 r: 122 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE A 14 I Students' Reports on the Frequency of (continued) Mathematics Textbook Use PERCENTAGE OF STUDENTS AND, AVERAGE MATHEMATICS PROFICIENCY _ IWO NAIND TRIAL TAU ASSEINUENT Almost Dray Day Several Timm a Weak . About Moe a Week er Lass Air TOTAL Se Nation muntsasass NI nen-gradusts State Nation NS graduate State Nation enta college State Nation College graduate State Nation oeNDER U. State Nation Female State Nation 71 ( 20 ( $45 ( 24) 23) 10 1$ tee $.0) *On 2 .0) ( al 11 (43) 411 11 ( «le) 77 ( 272 ( 3.4) 1.7) 14 272 ( 1.6) ( 111 71 ( SS) 111 ( 1A 13 ( 22) 254 ( 1D) 248 ( 3.2 239 ( 3.4)1 40 ( 205 ( 2.3) 1$ 200 ( 2.3 ( 3.2) 5 ( 1.4) oedi ( «ft 00 ( 2.0 11 ( 1.2 9 ( 1.7) 270 ( 1.9 79 ( 1.7) 13 ( 1.3) 1 ( 1.0) 209 ( 1.1) 2$5 ( 3.0) 276 ( 4.2) 77 ( 2.7) 13 ( 0.9) 10 ( 2.3) 279 ( 1.6) 200 ( 2.3) 237 ( 6.4)1 77 ( 1.7) 14 ( 1.2) ( 1.1) 2116 ( 1.1) 242 ( 2.7) 209 ( 4.3) 72 ( 2.4) 13 ( 1.2) 12 ( 2.1) 20. ( 14) 242 ( 24) 242 ( 61) 00 ( 2.0) 13 ( 1.3) 7 ( 1.2) 279 ( 1.3) 276 ( 2.2) 261 ( 54) 78 ( 1.1) 13 ( 1.0) 11 ( 1,6) 205 ( 13) 250 ( 24) 242 ( 3.8) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. ! Interpret with caution -- the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insuffic4ent to permit a reliable estimate (fewer than 62 students). 1 f's THE 1990 NAEP TRIAL STATE ASSESSMENT 123 Montana TABLE A1S I Students' Reports on the Frequency of Mathematics Worksheet Use PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY MO NAEP TRIAL sun mussuar At Least Several Timm a Weak ._ About Once a Weak Leas Thin Weakly TOTAL State Nation BAMETEELY White State Nation Hlipank State Nation American Indian State Nation TYPE OF COIMMUNITY Extreme rural State Nation Other State Nation ihrbkiolat I se 1,221 2$ 12) 1.4) 25 12) 2491 14) $22 32 ( 23) 20 ( 13) sitlis 210 ( 1.1) 25 ( 2.2) 144 1 1 41 202 ( 2.5) WM 1.5 271 2$ ( 2$( 4.5) 45 1 ... ( .4* I *In 44 ( 4.1 2$ 230 ( 33) 247 31 ( 4.0) 31 24$ ( 5.0) 250 41 ( 4.2) 30 ae ( 5.2) 275 ( 3.1) 42 (10.1) 249 ( 4.0)1 30 ( 2.4) 277 ( 12) 36 ( 2.9) 252 ( 3.0) ( 3.4) $2 ( 2.3) 241 ( 4.7) 33 ( 3.2)4 205 (11.3) 23 24 ( 2.4) 277 ( 3.0) 30 ( 4.4) 250 ( 3.4)1 31 ( 111) 251 ( 14) 25 ( 12) 261 ( 2.1) ( 43 ( 13 ( 5$) ( 4.0S (124) 40 ( 544) 200 ( 2$ ( 7.5) 207 ( 7.3)1 30 1.0) 2111 1.0) 30 29) 272 14) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. ! Interpret with caution -- the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a rehable estinnte (fewer than 62 students). 124 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE A15 1 Students' Reports on the Frequency of (continued) I Mathematics Worksheet Use PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY MO NAEP TRIAL STATE ASSESSMENT At Least Several Times a Week About Onc a Week Leas Than Wieldy TOTAL Permodepo add Pralitioacy Verodoliso and Podidiesay Persold. and Pregkdietv State $2 ( Li) 2Sf 1,2) 39 ( 2.1) 27$ ( 1$ ) 220 (1.4) 214 ( 0.9) Nation 3$ ( 2.4) 25 (12) 37 ( 24) 253 ( 22) 201 ( 1.4) 272 ( 1.9) PARENTS' EDUCATION HS non-graduete State 2S ( 5.3) 30 ( 5.0) 45 (( 14) Nation 41 ( 4.5) 30 ( 2.7) 22 ( 4.0) 235 ( 3.1) 243 ( 2.7) 253 (2.8) HS graduate State 31 ( 34) 30 ( 2.4) 39 ( 2.9) 206 ( 2.6) 270 ( 3.3) 276 ( 2.1) Nation 40 ( 3.2) 29 ( 22) 32 ( 3.6) 247 ( 2.7) 256 ( 2.5) 262 ( 22) Som college State 3$ ( 3.4) 2$ ( 3.1) 39 ( 3.8) 271 ( 2.2) 268 ( 35) 26$ ( to) Nation 34 ( 3.4) 26 ( 22) 40 ( 33) 259 ( 2.3) 269 ( 2.8) 271 211) College gracluat State 33 ( 2.7) 29 ( 1.8) 30 ( 2.5) 284 ( 1.4) 286 ( 1.7) 291 ( 1.5) Nation 38 ( 2.8) 22 ( 1.8) 41 ( 2.6) 264 ( 2.6) 273 ( 2.5) 285 ( 23) GENDER M. State 33 ( 2.3) 28 ( 1.8) 39 ( 2.5) 279 ( 1 A) 264 ( 2.0) 287 ( 13) Nation 39 ( 2.7) 2$ ( 1.6) 35 ( 2.7) 253 ( 2.7) 263 ( 2.3) 274 ( 2.4) Female State 31 ( 2.6) 29 ( 1.6) 40 ( 24) 273 ( 2.0) 277 ( 2.4) 2S1 ( 13) Nation 37 25) 25 ( 1.5) 38(23) 253 ( 2.1) 250 ( 1 A) ( 2.2) The standard errors of the esfunated statistics appear in parentheses. It ean be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample, *** Sample size is insufficient to permit a reliable ettimate (fewer than 62 students). 1 3o THE 1990 NAEP TRIAL STATE ASSESSMENT 125 Montana TABLE AIS Students' Reports on Whether They Own a Calculator and Whether Their Teacher Explains How to Use One PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1110 NMP TMAL STATE ASSESSMENT . , Own a Calculator Teacher Opiates Calculator Use Yes I No Yes , I No Mai State Nation Emaniffma woe state Nation Hispanic State Nation ARIENiCal Indian State Nation TYPE OF COMMUNITY Extrema niral State Nation Mee State Nation gi2it :311 17 ( 0.4) 2113 ( 1.3) 91 ( 03) 214 ( OA) 10 70 031.5i2 90 ( el) 264 (3.1) 12 (1.2) 245 2.7) 65 ( 13) 257 ( 3.4) N ( 3.1) eee frollakmar 1 ( 0.3) ( eel 2 03) 10 ( 0.7) grop. ,44) ( 1.2) 44 5 1) ( 1".) 6 ( 3.1) 44e ( eee) ad !Isom, 214 49 ( 2.p 2511 1.7) SS ( 2.2) 212 ( 1.0) 46 ( 2.6) 211 ( 13) Si ( 6.5) **IP 44/ 3.41 54 ( 5.6) 253 ( 3.9) 71 (16.7) vv. ( 161 141610111,06 44 ( 213 ( 1 111 ( 2.3 2116 ( 1.5) 45 ( 2.2) 215 ( 1.6) 54 ( 2.0) 273 ( 1$) 49 ( 6.5) *MI 37 ( 245 ( 2.9) 42 ( 53) 261 ( 4.6)1 29(197) *44, ) a6 ( OA) 278 ( 1.$) 00 ( 1.3) 257 ( 34)1 96 ( 0.3) 202 ( 1.0) 01(0.5) 213 ( 1.7) 1 4 to, 2 *14(444) 5 2$3 ( OA) ***) ( 1.3) ( 1M1 ( 0.3) ( 0.5) ( 54) 57 ( 270 ( 42 ( 251 ( 55 ( 210( 50 ( 251 ( 5,0) 2.0) 8.7) 41)1 11) 1.1) 2.7) 2,1) 43 ( 279 ( 56 ( 261 ( 45 ( 244 ( 50 ( 206 ( 5.0) 2.2) e.1) 4.4)1 13) 2.0) 2.7) 20) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. ! Interpret with caution - the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *" Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 126 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE AlS Students' Reports on Whether They Own a (wiltinued) Calculator and Whether Their Teacher Explains How To Use One PERCENTAGE Of STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1460 NAEP TRIAL STATE ASSEUMUT _ Own a Cakaatar ,. Teacher Scabs Calm Mar Use Yes No I Yes . No _ TOTAL State Nation PAR$TS' EDUC HS te State Nation NS graduate State Nation Some college State Nation College graduate State Nation OENKR M. State Nation Female State Nation Nes Stalkissio OS ( 241 ( 97 ( 243 ( 99 ( 240 ( 92 ( 243 ( 0.3) 0.4) 04) 1.3) 2.3) 2.3) 1.5) 2.0) 2 ( 0.3) imp ( ( 0.4) 234 ( 3.4) ( do* ( iC 1.0 2:1.1242 11(0.8) 271 ( 1.8) 97 ( OA) 255 ( 1.5) 11(0.5) 2 ( 3 ( ( 0.4) 0.4) 0.5) $4 ( 2. 270 ( 54 ( 3.0 252 ( 1A) se ( ti 263 ( 1.7) 242 ( 1.7 96 ( 0.9) 4 ( 0.9) a ( 3.2 23$ ( 14) ( 205 ( 2.4) 99 ( 0.4) 1 ( 0.4) 55 ( 2.2 287 ( 1.1) ( 245 ( 1.3) 99 ( 275 ( 0.2) 1.8) 1 ( 0.2) ( mkt ) 46 266 ( 2.6 ( 2 2 98 ( 0.4) 2 ( OA) 56 2.1) 254 ( ) 2$1 1.7) 97 ( 0.5) 3 ( 0.5) 31 ( 2.5. 234 ( 1.7) .441 2$1 ( 2.1) 00 ( 04) ( 0.4) 53 ( 2.5) 27$ ( 13) 27$ ( 1.4) 97 ( 0.5) 3 ( 04) 47 ( 24) 282 ( 1.3) ( 2511 ( 1.7) 51 2.3) 14) 243 42.14 2? R4) 47 4i ( 2.7) 273 ( 1.7) 40 ( s.a) 256 ( 2.0) 44 ( 3.1) 265 ( 2.7) 52 ( 3.2) 264 ( 2.2) 45 ( 2.2) 290 ( 1.7) 54 ( 2.6) ( 26014) 42 ( 2.1) 287 ( 1.7) 49 2.9) 206(2.1) 47 2.5) 279 ( 1.8) 53 ( 2.5) 263 ( 1.6) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). THE 1990 NAEP TRIAL STATE ASSESSMENT 127 Montalto TABLE A19 I Students' Reports on the Use of a Calculator I for Problem Solving or Tests PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY INO NAEP TRIAL STATE ASSESSMENT Wad Problems in Class Doing Problems at Homo Taking Quizzes or Tests Almost Always Neve r Almost Always Never Almost Always 4 Never TOTAL. State Nation RACEfETHNICITY White State Nation Hispanic State Nation American Indian State Nation TYPE OF COMMUNITY Extra'''. rural State Nation Other State Nation Perom44. persim4904 Poivoilme P4ralmbps Powlso Poombee 4414 and sed 4444 494 eml Posioinev Polielem 4/41941.444 114.4114*NW Pro04192149 PrskieNW 44( 19 171 277 ( 0.9 44 ( 1.5 254 ( 14) 2123 14 46 ( 1.7) 1C ( 1.7) 281 ( 1.0) 287 ( 2.2) 40 ( 1.7) 24 ( 2.2) 262 ( 1.7) 278 ( 1.3) 49 ( 6.3) 15 ( 5.2) .N ( .4.41 9.4.444 ( 404.) 51 ( 23) 16 ( 34) an ( 2.3) 252 ( 3.3)I 48 ( 5.9) 15 ( 44.2) 251 ( 4.0) *** ( ***) 33 ( Se) 23 ( 4.9) 44 ., ( 444444) 4.4.4 ( ***) 45 ( 3.1) 17 ( 4.5) 274 ( 2.1) 279 ( 2.1$ 46 ( 7.4) 29 ( 6.5) 246 ( 4.3)1 208 ( 47 ( 2.0) 14 ( 1.4) 279 ( 1.0) 297 ( 23) 48 ( 13) 22 ( 2.0) 254 ( 2.1) 272 ( Ca) 351 1.2 11 0.9) 21 1.4) 30( 1.7 1.2 279 2.5) 270 (1.3) 217( 1.2 :I 19 09) 27 (1.4) 30( 2.0 (11.1 292 1.2) 2511( 2.4) 274( 13 SS ( 1.4) 10 ( 21 ( 1.4) 31 ( 13) 236 ( 1.1) 285 ( 2.0 279 ( 1A) 290 ( 1.4) $1 ( 1,5) 15 ( 1.2 25 (13) 82 ( 28) 270 ( 1.7) 20 ( 28) 283 ( 2.0) 278 ( 1-2) 26 ( 73) op. i 4.4,4) 22 ( 63) 4.4,4 ( 494) 23 ( 5.8) 444 ( In 29 ( 6.3) i 441 26 i 3.2) 21 ( 2.1) 26 ( 22) 22 i 3.1) 296 ( 4.8) 244 ( 3.1) 237 ( 3.2) 256 ( 4.2) 32 ( 4.1) 258 ( 5.0)4 14 ( 3.5) - ( -) 26 ( 4.3) 250 ( Lep 22 ( 5.0) .-- ( -) 15 ( 4.9) 32 (10.1) 20 ( 62) 21 ( 7.3) ( .....) 444 ( *41 *44. ( in 444 ( 441 31 ( 2.3) 10 ( 1.7) 24 ( 3.5) 215 ( 4.6) 278 ( 2.8) 280 ( 18) 275 ( 2.4) 284 ( 1.7 20 ( 2.5) 23 ( 33) 24 ( 6.6) 37 ( 8.3 ( -) 263( 4.4$ ( ) 270 ( 4.0 37 ( 1,4) 11 ( 1.1) 20 ( 1.3) 30 ( 1.2 295 ( 1.5) 279 ( 3.3) 276 ( 1.5) 239 ( 1.7 32 ( 1.7) 18 ( 1.1) 27 ( 1.8) 29 ( 2.1 263 ( 2.3) 263 ( 2.8) 253 ( 2.7) 275 ( 13 The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. The percentages may not total 100 percent becauee the "Sometimes" category is not included. ! Interpret with caution - the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). r.. - 128 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana =1. TABLE A19 1 Students' Reports on the Use of a Calculator (continued) I for Problem Solving or Tests PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1920 NAEP TEAL STATE ASSES$MENT , Westing Problems in Class .. Doing Problems at Name - Taking Quizzes or Tests Almost I _ Always Never Almost I Always .. Never Almost 1 Atways , NeVef 121M State Nation feanniasano. KS non-graduate State Nation 149 eradiate State Nation Soma college State Nation College graduate State Nation Mate State Nation Female State Nation 0,66411006600 14111"1".. 8100 600 Prailisting tfinsisketv Prolimipow alnilhOmpf 1,00014,1W ,. 277 AO M 42 11 _If g sol 11 21.4.1 20.1) Ai .1144 or 1.3 4. 1.5 , 22 1.2 30 1.3 12 ( OA} 27 1.4 IQ 2.0 264 1.2 271i 1.4 18111 IA 21121( 13 2811 2A 274 1.3 50 ( IA ) 12 ( 2.9) 3.1 44* .4.) 84 $A 21 3.11) 240 2.2 n 4$ ( Si) 17 ( 249( 2.0) 275 ( 2.11 52 ( 2.5) 20 ( 2.4 249 ( 1.4) 285 ( 2.7) 44 ( 2.7) 13 ( 2.3) 273 ( 2.3) 218 ( 5.4) 48 ( 2.4) 28 ( 2.2) Nil ( 2.1) 272 ( 2.5) 47 ( 2.4) 17 ( 1.9) 364 ( 1.4) 292 ( 2.4) 45 ( 1.9) 25 ( 2.4) 255 ( 1.7) 284 ( 13) 47 2.1) 14 ( 280 1.4) NO ( 3.4 50 1.7) 20 ( 2.0 255 1.9) 275 ( 2.2) 45 2.0 17 ( 2.4) 274 ( 1.4 281 ( 2.5) 46 2.0 26 ( 2.1) 252 1.7 269 ( 11) 29 ( 5.11) 4.,.... .9711 13 ( aildt) 23 ( 4.4) 281 02 1 2.21 $8t 1 Zill 244 SA 244 4.2 237 ( 2.3 101 ( 2.5) 15 ( 2.3) 24 ( 2.5) 273 ( 2.1) 21$ ( 5.9) 297 i 29 ( 1.9) 11 ( 1.5) 2$ ( 1.11 250 ( 2.4) 25$ ( 2.4) 248 ( 2A 32 ( 3.0) $ ( 19) 21 ( 24) 255 ( 2.9) """ ( ") 279 ( 3.8) 24 ( 2.0) 20 ( 13) 2e( 2.4) 297 ( 3.0) 28$ ( 12) 255 ( le) 39 ( 13) 10 ( 1.3) 20 ( 1.7) 289 ( 14) 289 ( 3.2) 283 ( 1.9) 33 ( 2.0) 18 ( 1.4) 26 ( 1.8) 274 ( 2.2) 278 ( 2.8) 26$ ( 2.6) 33 ( 1.8) 12 ( 1.0) 20 ( 1.5) 287 ( 2.1 280 ( 2.9) 278 ( 2.1) 29 ( 1.8 19 ( 1.3) 27 ( 1.5) 264 ( 2.8) 263 ( 2.5) 250 ( 10) 37 ( 13) 10 ( 1.4) 23 ( 1.1) 279 ( 2.1) 278 ( 3.9) 273 ( 1.5) 32 ( 1.6) 18 ( 1.2) 27 ( 1.5) 250 ( 1.7) 261( 24) 251 ( 24 ) 22 ( 3.3) .... i Hi 24 ( 3.2) 251 ( 4,15) 20 ( 3.2) 279 ( 29) 27 ( 2.2) 216 ( 2.0) 29 ( 3.0) 247 ( 4.2) 35 ( 2.5) 275 ( 2.0) 31 ( 2.0) 295 ( 1.4) 33 ( 2.7) 285 ( 2.0) 27 ( 1.7) 294 ( 1.9) 26 ( 2,1) 277 ( 1.a) 33 ( 2.3) 262 ( 1.8) 33 ( 2.1) 271 ( 1.5) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. The percentages may not total 100 percent because the "Sometimes" category is not included. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). THE 1990 NAB? TRIAL STATE ASSESSMENT 129 Montana TABLE A20 I Students' Knowledge of Using Calculators PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY MO NAP TRIAL ITATE ASSEISMENT - Itigh "Caloutater4iss" Group _ Muir "Calaisier-tiar Ora. MIK State Nation sectanmsm White State Nation Nispani, State Nation American Indian State Nation TYPE Of COMMUNITY Extreme nevi State Nation Other State Nation 14 ( al) 2211 ( 1.4 VS 44 ( 141 21 277 ( 1.1 310 !: 2$4 4.2) 311 1 Loi 42) ite 1 1 iii ( 43 OS ( 43 ( 4.5) 57 ( 44) 285 ( 4.3) 11 ifti .g 20 (12.0) 44. I ***) sob ( ) 51 ( 2.1) 41) ( 2.1) 243 ( 11) 272 2.1) $0 ( 5.2) 61 5.4) 222 ( 44)1 241 42)1 ( 2,9) 47 ( 2.5) 2e ( 4.7) 275 ( 1.11) 42 ( 1.4) 511 ( 1.4) 271 ( 19) 255 ( 2.0) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. Interpret with caution -- the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). " ; 130 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE A20 I Students' Knowledge of Using Calculators (continued) I PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1300 NAEP TRIAL I E ASES STAT S SMENT IRO "Ca twister-We" Group "Calculator-Uses Grow TOTAL State Nation PARERTS' gpucAnc* HS non-graduate State 47 ( 8.4) 53 ( 8.4) Nation 34 ( 3.31 40 ( 3.3) 24$ ( 4.4) 242 ( 2.4) HS graduate State 47 ( 2.4) 53 ( 2S) Nation 40 ( 2.2) 40 ( 22) Some college ,04Witkfte. 47.1 $ 15.11 1.3) 255 ( 14) ColMge graduate State 50 ( 2.7) 44 ( 2.7) 291 ( 1.8) , 293 ( 2.2) Nation 48 ( 2.0) 54 ( 2.0) 282 ( 2.1) 28$ ( 1.9) GENDER Male State 51 ( 3.1) 49 ( 3.1) 291 ( 1.5) 277 ( 1.7) Nation 39 ( 20) 61 ( 2,0) 274 ( 2.0) 258 ( 2,3) Female State 58 ( 24) 44 ( 24) 281 ( 14) 273 ( 2.0) Nation 45 ( 1.8) 65 ( 14) 289 ( 1.7) 254 ( 1.3) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. *** Sample siu is insufficient to permit a reliable estimate (fewer than 62 studenu). THE 1990 NAEP TRIAL STATE ASSESSMENT 131 Montana TABLE A24 I Students' Reports on Types of Reading I Materials in the Home PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY MO NAEP TRIAL TE ASSEISMENT STA Zero to Two TYPits _ Throw TYPIK Fair TYiws TQTAL ParteillP Priediewi 12 ( 0.7) 259 ( 2.1) 21 ( 1,0) 244 ( 2.0) 10 ( OA) 274 ( 2.2) 10 ( 1.1) 251 ( 22) 22 ( 7.0) am. ( 641 44 ( 3.0) 237 ( 3.4) 20 ( 4.0) 249 ( 4.2) 29 (11.1) em ( *#1 13 ( 1.0) 205 ( 3.2) 17 ( 4.9) ( 12 ( 1.0) 272 ( 2.8) 22 ( 1.5) 244 ( 2.6) State Nation Bagainingan State Nation Hispanic State Nation Anworican State Nation TYPE OF COMMUNITY Extreme rural State Nation Other State Nation Pommies* PrOakacv 32 ( 1.2) 277 30 1.0) 255 1.7) 31 ( 13) 251 ( 14) 29 ( 1.3) ( 1,5) 38 ( 8.3) ( +41 30 ( 2.4) 244 ( 4.3) 40 ( 5.0) 2S5 ( 4.3) 40 ( 4.9) ( 41 33 ( 14) 275 ( 2.2) 33 ( 3.2) 253 ( 43)t 33 ( 1.6) 278 ( 22) 30 ( 1.3) 259 ( 22) Peromills Proisiesty 52 ( 1.0) 266 ( 0.4) 48 ( 1.3) 272 ( 14! 511 ( 1.1 227 ( OA) 36 ( 1.3) 276 ( 1.7) 40 ( 5.4) edne .D40) 26 ( 2.3) 233 ( 2.4) 34 ( 5.6) 263 ( 4.4)4 31 ( 0.2) gin 54 ( 1.6) 252 ( 14) 50 ( 5.1) 263 ( 5.6)1 55 ( 1.2) 258 ( 0.9) 48 ( 1.5) 272 ( 1.7) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample, ! Interpret with cautioii - the nature of the sample does not allow accurate determination of the variability of this estir...ated mean proficiency. * Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 13k THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE A24 I Students' Reports on Types of Reading (continued) Materials in the Home PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 10110 NAEP TRIAL STATE ASSESSMENT Zero to Two Typos Three Types Four npos TOTAL Peramags ftellolenay 0101280011 aid Prailkamay State 12 ( 0,7) 32 ( 12) ( 2.1) 277 ( 1,6) Nation 21 ( 1.0) 30 ( 1.0) 244 ( 2,0) 258 ( 1.1) PARENTS' FOUCATKIN Hil nan-graduate State 26 ( *FR ( 4.8) 111**) 43 ( 6.3) Nation 47 ( 4.0) 28 ( 3.0) 240 ( 3.4) 243 ( 3.3) NI graduate State 17 ( 1.5) 37 ( 2.6) 262 ( 3.8) 272 ( 3.1) Nation 26 ( 22) 33 ( 1.9) 248 ( 22) 253 ( 2.7) 8ortw college State 11 ( 1.3) 38 ( 2.() 271 ( 4.8) 282 ( 2.5) Nation 17 ( 1.5) 32 ( 1.7) 251 1 4.0) 282 ( 2.8) Coif's. graduate State 8 ( 0.9) 25 ( 1.8) 281 ( 3.7) 281 ( 2.5) Nation 10 ( 254 ( 0.8) 2.8) 28 ( 1.8) 2es ( 2 5) GENDER Male State 12 ( 0.9) 33 ( 1.5) 273 ( 24) 281 ( 2.3) Nation 21 ( 1.5) 31 ( 1.5) 244 ( 2.3) 259 ( 2.1) State 12 ( 1.1) 32 ( 1.9) 286 ( 3.5) 273 ( 2.0) Nation 22 ( 1.2) 29 ( 1.4) 244 ( 2.2) 251 ( 1.9) PINVINIMOS Pnii0411019 55 (1 ( 0.11 411 ( 1.3 '272 ( 15) !it 4.3) 25 I 24 246 ( 33) 45 ( 2.6) 274 ( 1.9) 40 ( 1.7) 260 ( 2.1) SO ( 2.3) 266 ( 2.0) 51 ( 2.0) 274 ( 1.9) 87 ( 1.5) 290 ( 1.2) 82 ( 2.0) 280 ( 1.8) 55 ( 1,3) 288 ( 1.1) 48 ( 1.4) 273 ( 2.0) 58 ( 1.7) 262 ( 1.4) 49 ( 1.9) 270 ( 1.7) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. *** Sample size is insuricient to permit a reliable estimate (fewer than 62 students). 1 S THE 1990 NAEP TRIAL STATE ASSESSMENT 133 Montana TABLE A25 I Students' Reports on the Amount of Time Spent I Watching Television Each Day PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1890 NAEP TRIAL STATE ASSESSMENT _ One Hour or Less Two Hours I Three Koos I Far t o Rye Hass Six Heirs or More TOTAL Poosalage sad ProOdeng 21 ( 0.9) 22012 0141 209 ( 2.2) 22 ( 0.9) 292 ( 1.4) 13 ( 1.0) 276 ( 2.5) 14 ( 4.1) eke ( 441 14 ( 2.4) fro *on 12 ( 3.0) *al 13 ( 5.0) glio) 19 ( 1.8) 285 ( 3.3) tio* *al 21 ( tO) 292 ( 1.6) 12 ( 1.0) 268 ( 2.6) Pereadage Pidialency $0 ( 1.2) 285 ( 12) 21 ( 0.9) 26$ ( 14) 30 ( 1.3) 267 ( 1.1) 23 ( 1.2) 275 ( 2.2) 22 ( 6.4) tnielt 20 ( 2.5) 245 ( 3.2) 25 ( 3.1) 17 ( $A) 4,to ,po) 28 ( 1.4) 282 ( 1.8) 30 ( 1.8) 266 ( 1.7) 21 ( 1.0) 209 ( 2.3) Veresolage ad Paddatcy 24 ( 1.0) 27$ ( 1.2) 22 ( OA) 265 ( 1.7) 24 ( 1.1) 281 ( 1.2) 24 ( 1.1) 272 ( 1.9) 27 ( 5.6) oo) 19 ( 2.4) 242 ( 5.6) 25 ( 3.7) .6.0* doo) 21 (10.5) 27 ( 1,5) 276 ( 1.7) 22 ( 1.3) 278 ( 1.9) 23 ( 12) 265 ( 2.1) Paromaile and Prolkisacy 20 ( 1.1) 275 ( 1.6) 26 ( 1.1) 260( 1.7) 19 ( 1.1) 276 ( 1.6) 27 ( 1.4) 267 ( 1.7) 31 ( 3.1) 247 ( 3.5) 25 ( 3.7) 254 ( 5.4)1 thio ( too) ( 1.7) 271 ( 3.0) 26 ( 2.7) 256 ( 3.6)1 20 ( 1.5) 277 ( 1.8) 27 ( 1.2) 250 ( 2.2) liersemlaga aM Prollaisocy ( 0.6) 261 ( 24) 16 ( 1.0) 245 ( 1.7) 5 ( 0.6) 264 ( 3.4) 12 ( 1.2) 253 ( 2.6) 10 ( 3-5) do. ( ..00) 17 ( 1.7) 236 ( 3.8) 13 ( 3.3) .441 .4o) 5 ( 1.0) 254 ( 4.6) 7 ( 0.7) 263 ( 3.7) 17 ( 1.4) 246 ( 2.5) State Notion RAa HNIC1TY White State Nation Hispanic State Nation American Indian State Nation TYPE OF COMMUNITY Extreme rural State Nation Other State Nation The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the enure population is within ± 2 standard errors of the estimate for the sample. ! Interpret with caution -- the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *I'. Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 1 134 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE A25 I Students' Reports on the Amount of Time Spent (continued) I Watching Television Each Day PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROPCIENCY 1910 NAV TRIAL STATE ANIESSUEXT , Ono Noir sr Lass Illso Hours Theo Hours Four 10 Ms Hours lb( Hours or Moro 191A State Nation 26BIBILENSAM?N HS nor-grodusto State Nation HS aroduste State Nation S ense caego State Nation College grockato State Nation GENIX4 M. State Nation FonsaIo State Nation 24,011fory 21 12 ( 111.11 1$ ( 210 ( 1 ( 240 ( 17 ( 295 ( 10 ( 11** 24 ( 294 ( 17 ( 262 ( 18 ( 293 ( 11 ( 269 ( 23 ( 267 ( 14 ( 269 ( Parlislonsr ad PrOsisnew Prelbelsw aNd Prallelsowi 021 f OA) 2.2) 51) 2.2) OH) 2.9 1.0 4.7) 2.0) 3.1) 1.4) INk11) 1.5) 2.0) 1.3) 2.6) 1.3) 2.3) 0.9) 3.3) 1.6) 1.9) 1.1) 23) la) 316 12) 21 0.9) 2111 1.1) 24 ( 4.11 It* ( 20 ( 3,1) ,m *es) 23 ( 2.0) 275 ( 3.0) 17 ( 1.4) 257 ( 23) 36 ( 2.0) 2111 ( 3.4) 25 ( 2.4) 275 ( 2.7) 31 ( 2.3) 269 ( 1.7) 22 ( 1.6) 260 ( 2.5) 28 ( 1.8) 268 ( 1.7) 22 ( 1.2) 207 ( 2.6) 31 ( 2.2) 251 ( 2.4) 20 ( 1.3) 269 ( 2.2) ( 12 11 22 ( 01 201 (1. 20 ( 3.9) 21 ( 21) 29 ( 2.4) 371 ( 2.0) 23 ( 2.0) 220 ( 32) 22 ( 22) 240 ( 2.?) 23 ( 2.6) 2.0 ( 3.5) 23 ( 1.5) 263 ( 2.1) 23 ( 1.1) 277 ( 2.2) 25 ( 2.0) 279 ( 1.7) 22 ( 1.0) 267 ( 22) 23 ( 1.6) 276 ( 1.7) 23 ( 1.4) 264 ( 1.8) IA) 276 1.6) 21 1.1) 2410 1.7) 21 ( 4.7) ( *el 2$ ( 2.9) 244 ( 32) 22 ( 291 2.9 321 2.3 253 ( 23) 22 ( 2.0) 275 ( 2.0) 26 ( 2.2) 261(23) 17 ( 1.8) 282 ( 2.8) 25 ( 1.5) 270 ( 2.4) 22 ( 13) 261 ( 2.1) 2$ ( 1,3) 262 ( 2.1) 18 ( 1.6) 267 ( 23) 26 ( 1.6) 258 ( 1.9) 6( 0.41) 201 ( 22) 16 ( 1.0) 245 ( 11) 14 ( 3.0) ( .41 20 ( 3.4) ok ( 1.6) .1*. 0441 19 ( 1.6) ( 3.0) 5 ( 1.4) 44. ( 14 ( 1.5) 242 ( 3.4) et* ( 12 ( 1.1) 255 ( 32) ( 1.0) 246 ( 4.6) 17 ( 1.5) 248 ( 2.5) 5 ( 0.8) 15 ( 12) 241 ( 22) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ÷ 2 standard errors of the estimate for the sample. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). THE 1990 NAEP TRIAL STATE ASSESSMENT 135 Montana TABLE A26 I Students' Reports on the Number of Days of School Missed PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1800 NAEP 'TRIAL STATE ASSESSMENT None Ono or Two Days Throe DIP or More TOTAL istionlaips ard Prato" 40( 13) State 284 ( 1.4) Nation 4$ ( 1.1) ( 13) WE/ETHNICITY Mite State 41 ( 1.5) 210 ( 14) Nation 43(1.2) 273 ( 19) Hispanic State 29 ( 7.1)) Nation 41 ( 3.3) 245 ( 4.6) American Indian State 20 ( 4.4) 263 ( 5.3)1 Nation 23 ( 0.6) TYPE OF COMMUNITY Extreme rural State 41 ( 2.3) 281 ( 1.9) Nation 43 ( 4.4) 257 ( 4.1)I Other State 40 ( 1.7) 288 ( 1.9) Nation 45 ( 1.3) 285 ( 2.2) fleumires and Prallolow 214 2:1111( 1,5) Parsiolsia awl 4Prallakmay 30 ( 1.3) 20 ( OM 2115 ( 02) ( la) 34 ( 1.2) 23 ( 12) 272 ( 1.7) 251 ( 2.1) 54 ( 0.5) 17 ( 4.7) cm. .14./ ( 44) 32 ( 22) 27 i 2.6) 250 ( 3.3) 295 ( 3.1) 31 ( 5.7) 37 ( 4.0) 240 ( 4.4) 250 ( 3.9) 39 ( 5.1) M 5.2) 4,41 40 ( 1.9) 278 ( 1.8) 32 ( 4.2) 244 ( 6.8)l 38 ( 1.7) 283 ( 1.3) 32 ( 1.1) 2418 ( 1.8) 19 ( 2.0) 209 ( 3.4) 25 ( 3.9) *41 22 09) 273 2.5) 2? 1.1) 251 ( 24) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. ! Interpret with caution . - the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 136 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE A26 I Students' Reports on the Number of Days of (continued) School Missed PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY 1NO NAEP TRIAL STATE ASSESSMENT None One or Two Days Three Days or Mora TOTAL Portents. awl Proficiency State 40 < 1.3) 244 ( 1.4) Nation 45 ( 1.1) 266( 1.8) PARENTS EDUCATION NS non-gracklate State 33 ( 5.0) Nation 36 ( 3.2) 245 ( 3.0) HS graduate State 43 ( 2.8) 277 ( 1.3) Nation 43 ( 2.1) 255 ( 2.0) Some college State 36 ( 2.3) 284 ( 3.7) Nation 40 ( 1.8) 270 ( 3.0) College graduate State 42 ( 2.1) 290 ( 1.9) Nation 51 ( 1.8) 275 ( 2.1) GENDER Male State 44 ( 1.6) 287 ( 1.5) Nation 47 ( 1,8) ( 2.0) Female State 36 ( 2.2) 281 ( 1.7) Nation 43 ( 1.4) 264 ( 23) iterundsle *WI Proliciemy SO 262 1.0 32 00 200 1.5) 34 ( 5.4) 26 ( 3.1) 249 ( 3.3) 36 ( 2.6) 271( 2.2) 31 ( 1.9) 257 ( 2.6) 43 ( 2.4) 267 ( 1.4) 37 ( 1.6) 271 ( 2.5) 30 ( 2.5) 286 ( 1.6) 33 ( 1.2) 277 ( 1.7) 36 ( 1.4) 283 ( 1,8) 31 ( 1.4) 287 ( 2.1) 42 ( 1.7) 280 ( 1.5) 32 ( 1.1) < 1.7) Porasula. and Prolk4oky 21 ( 0.11) 272 ( 2.0) 23 < 1.1) 250 ( 1.0) 33 ( 4.5) 38 ( 63) 237 ( 3.1) 22 ( 2.8) 250( 4.1) 27 ( 1.0) 249 ( 2.4) 21 ( 2.1) 273 ( 2.6) 23 ( 1.6) 253 ( 3.1) 10 ( 1.5) 264 ( 2.3) 16 ( 1.3) 2155 ( 3.1) 21 ( 1.4) 278 ( 3.5) 22 ( 14) 250 ( 2.6) 22 ( 1.3) 208 ( 2.4) 25 ( 1.3) 250 ( 1.8) The standard errors of the estimated statistics appear in parentheses. lt can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). THE 1990 NAEP TRIAL STATE ASSESSMENT 137 Montana TABLE A27 I Students' Perceptions of Mathematics PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY IWO NAEP TRIAL STATE ASSESSMENT Womb/ Agree AV Undecided, Disagree, Siren* Disagree Jr TOTAL Pers1966.1 and Pre lolsogy Peramines adi Prelicidecy liertediage and Prdidefacy State 30 ( 1A) 50 ( 1.4) 20 ( 1.1) 202 ( 1.2) 278 (.1.0) 267 ( 1.6) Nation 27 ( 1.3) 49 ( 1.0) 24 ( 1.2) 271 ( 13) 262 ( 1.7) 251 ( 1.8) RACEIETHNiCITY %Mite State 30 ( 1.4) 51 ( 1.5) 111 ( 1.1) 204, ( 1.2) 262 ( 1.0) 271 ( 1.4) Nation 28 ( 1.8) 43 ( 1.3) 26 ( 13) 279 ( 2.0) 272 ( 1.8) 257 ( 2.0) NisPank State 38 41 ( 4.5) ..64) 45 ( 7.0)) 17 ( 4.9) rtm,) Nation 24 ( 23) ( 2.6) 28 ( 2.1) 257 ( 5.5) 244 ( 2.2) 230 ( 3.8) American Imilan State 28 ( 5.2) 40 ( 4.2) 33 ( 43) ( **I ) 256 ( 4.3) 247 ( 4.4)1 Nation 1 23 *OA ( 7.4) ( 48 .pw (14.9) .44) 29 (mtip ( 9.5) 04*) TYPE Of COMMUNITY EN:trent* mai State 32 ( 3.5) 47 ( 2.8) 21 ( 23) 287 ( 1.4) 278 ( 2.1) 265 ( 3.0) Nation 34 ( 2.8) 49 ( 2.2) 17 ( ;.4) 270 ( 3.9)1 262 ( 4.1)1 ...... ( ...**) ONW State 29 ( 03) 52 ( 1.8) 19 ( 1.2) 204 ( 1.7) 260 ( 1.2) 268 ( 13) Nation 27 ( 1.4) 48 ( 1.2) 25 ( 1.4) 271 ( 2.4) 263 ( 2.2) 250 ( 1.9) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 stanrlard errors of the estimate for the sample. ! interpret with caution -- the nature of the sample does not allow accurate determination of the variability of this estimated mean proficiency. *** Sample size is insufficient to permit a reliable estimate (fewer than 62 students). / 138 THE 1990 NAEP TRIAL STATE ASSESSMENT Montana TABLE A27 I Students' Perceptions of Mathematics (continued) I PERCENTAGE OF STUDENTS AND AVERAGE MATHEMATICS PROFICIENCY , 1990 NAEP TRIAL STATE ASSESSMENT StroisidY *Rim AV.* Undecided, Disagree, Stanch Ditumiree TOTAL Mi Pm Galaxy PgrailleaP and PireNdage sod 941414811Cy State 30 ( 1.4) 50 ( 14) 20 ( 1.1) 292 ( 12) 279 ( 1.0) 267 ( 1A) Nation 27 ( 1.3) 49 ( 1.0) 24 ( 1.2) 271 ( 1.9) 262 ( 1.7) 251 ( 1.8) PARENTS' EDUCATION HS non-graduate State 19 ( *** 3.7) *** 43 ( .44 ( 4.3) 441 37 ( 4.3) 41 Nation 20 ( 2,6) 50 ( 3.3) 30 ( 3.6) *44 ( *AM) 243 ( 2.6) 238 ( 4.3) HS graduate State 23( 2.3) 54 ( 2.5) 23 ( 1.9) 285 ( 2.2) 270 ( 2.0) 261 ( 3.7) Nation 27 ( 2.1) 47 ( 2.3) 26 ( 2.0) 262 ( 2.7) 255 ( 2.3) 245 ( 2.4) Some college State 34 ( 2.3) 49 ( 2.6) 17 ( 2.3) 294 ( 2.7) 261 ( 2.1) 268 ( 2.9) Nation 26 ( 2.5) 47 ( 2.4) 25 ( 1.8) 274 ( 3.1) 267 ( 1.9) 258 ( 3.2) College graduate State 34 ( 2.1) 51 ( 2.0) 15 ( 1.4) 295 ( 1.5) 265 ( 1.4) 276 ( 2.5) Nation 30 ( 2.3) 51 ( 18) 19 ( 1.6) 280 ( 2.4) 274 ( 2.2) 296 ( 2.5) GENDER Mal. State 30 ( 1.6) 50 ( 2.0) 20 ( 1.7) 295 ( 1.4) 282 ( 1.7) 271 ( 1.9) Nation 26 ( 1-5) 48 ( 1.2) 24 ( 1.4) 273 ( 2.3) 263 ( 2.0) 251 ( 2.4) Female State 30 ( 2.2) 51 ( 2.1) 19 ( 1.7) 289 ( 1.7) 27$ ( 1.4) 2133 ( 2.5) Nation 26 ( 1.7) 50 ( 1.7) 25 ( 1.9) 269 ( 2.1) 262 1.8) 252 ( 1.9) The standard errors of the estimated statistics appear in parentheses. It can be said with about 95 percent certainty that, for each population of interest, the value for the entire population is within ± 2 standard errors of the estimate for the sample. *" Sample size is insufficient to permit a reliable estimate (fewer than 62 students). 4 4 THE 1990 NAEP TRIAL STATE ASSESSMENT 139 Acknowledgments The design, development, analysis, and reporting of the first Trial State Assessment was truly a collaborative effort among staff from State Education Agencies, the National Center for Education Statistics (NCES), Educational Testing Service (ETS), Westat, and National Computer Systems (NCS). The program benefitted from the contributions of hundreds of individuals at the state and local levels Governors, Chief State School Officers, State and District Test Directors, State Coordinators, and district administrators who tirelessly provided their wisdom, experience, and hard work. Finally, and most importantly, NAEP is grateful to the students and school staff who participated in the Trial State Assessment. Special recognition is due the Council of Chief State School Officers (CCSSO) for its considerable contributions to the program, especially its management of the National Assessment Planning Project. That project resulted in the mathematics framework and objectives for the assessment and recommendations about reporting the results of the program. In particular, we note the significant contributions of Ramsay Selden, Director of the State Education Assessment Center for the CCSSO and the members of the Steering, Mathematics Objectives, and Analysis and Reports Committees of the National Assessment Planning Project. The Trial State Assessment was funded through NCES, in the Office of Educational Research and Improvement of the U.S. Department of Education Emerson Elliott, NCES Acting Commissioner, provided consistent support and guidance. The staff -- particularly Gary Phillips, Eugene Owen, Stephen Gorman, Maureen Treacy, and Raul Garza worked closely and collegially with ETS, Westat, and NCS staff and played a crucial role in all aspects of the program. The members of the National Assessment Governing Board (NAGB) and NAGB staff also deseive credit for their advice and guidance. We owe a great deal to the Mathematics Item Development and Mathematics Scale Anchoring Panels. These people -- from school districts, colleges and universities, and State Education Agencies -- worked tirelessly to help ETS staff develop the assessment and a framework for interpreting the results. Under the NAEP contract to ETS, Archie Lapointe served as the project director and Ina Mullis as the deputy director. Statistical and psychometric activities were led by John Mazzeo, with consultation from Eugene Johnson and Donald Rock. John Barone managed the data analysis activities; Jules Goodison, the operational aspects; Walter MacDonald and Chancey Jones, test development; David Hobson, the fiscal aspects; and Stephen Koffler, state services. Sampling and data collection activities were carried out by Westat under the supervision of Renee Slobasky, Keith Rust, Nancy Caldwell, and the late Morris Hansen. The printing, distribution, and processing of the materials were the responsibility of NCS, under the direction of John O'Neill and Lynn Zaback. The large number of states and territories participating in the first Trial State Assessment introduced many unique challenges, including the need to develop 40 different reports, customized for each jurisdiction based on its characteristics and the results of its assessed students. To meet this challenge, a computerized report generation system was built, combining the speed and accuracy of computer-generated data with high resolution text and graphics normally found only in typesetting environments. Jennifer Nelson created the system and led the computer-based development of the report. John Mazzeo oversaw the analyses for this report. John Ferris, David Freund, Bruce Kaplan, Edward Ku lick, and Phillip Leung collaborated to generate the data and perform analyses. They were assisted by Drew Bowker, Laura McCamley, and Craig Pizzuti. Debra Kline coordinated the efforts of the data analysis staff. Stephen Koffler wrote the text for the report. Kent Ashworth was responsible for coordinating the cover design and fmal printing of this report. Special thanks are also due to many individuals for their invaluable assistance in reviewing the reports, especially the editors who improved the text and the analysts who checked the data. U.S. GOVERNMENT PRINTING OFFICE 1491 0 - 293-276 QL 8 WK. 22) 1 6