M A T H E M A T I C S A C H I E V E M E N T IN T H E D O M I N I C A N R E P U B L I C : G R A D E 12 b y S A N D R A M . C R E S P O L U N A B . E d . (Hons) Pontificla Unlvers idad Catol ica M a d r e y Maestra , 1987. A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T F O R T H E D E G R E E O F M A S T E R O F A R T S i n the D E P A R T M E N T O F M A T H E M A T I C S A N D S C I E N C E E D U C A T I O N F a c u l t y of E d u c a t i o n W e accept this thesis as conforming to the required s tandard D a v i d F . Robkai l le D a v i d J . Ba teson T h o m a s L . Schroeder T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A September 1990 © S a n d r a M . Crespo L u n a , 1990. In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of MATHEMATICS AND SCIENCE EDUCATION The University of British Columbia Vancouver, Canada Date OCTOBER/9/1990. DE-6 (2/88) i i Abstract The general goal of the present study was to assess mathematics achievement at the end of Grade 12 i n the Dominican Republ ic , with part icular attention to school and regional differences, as well as gender differences. Also, gains in achievement were examined by comparing the achievement of students in Grade 12 to that of students f inishing Grade 11. In addition, the performance of Grade 12 students was compared to that of Grade 8 students as assessed in the Teaching and Learning of Mathematics in the Dominican Republic (TLMDR) study and to that of students from other countries in the Second International Mathematics Study (SIMS). The sample included 1271 students in Grade 12 and 1413 in Grade 11, distr ibuted over 49 schools. Three types of schools were sampled, public schools, and two kinds of private schools. They were urban schools located i n the twelve largest cities of the country. These cities were grouped into three regions of similar size. The mathemat ics test consisted of 70 mul t ip le - cho ice i tems distributed over two test forms. Students' scores were analyzed to assess how m u c h mathematics students in Grade 12 know. Grade 11 data were used as a surrogate for pre - test scores to est imate gains i n achievement. School means were used in an analysis of variance designed to examine the effect of school type and region on mathemat ics achievement. Males' and females' scores were used to analyze gender differences i n achievement at the item level, and within each of the school types and regions in the sample. Grade 12 students' responses to 14 items were compared to those of Grade 8 students. Finally, Grade 12 students' i i i responses to 10 items were compared to those of students from other countries in SIMS. Among the findings of this study were: 1. Students in Grade 12 scored poorly on the mathematics test. Grade 11 and Grade 12 students obtained similar achievement levels which indicated that the achievement gains between the two grades were very small. 2. School type and region were found to significantly affect mathematics achievement, but no interaction effect was found. 3. The comparison of school type means showed that only one type of private school significantly outperformed public schools. This type of school also outperformed the other type of private school. 4. The comparison of region means did not produce the predicted outcome. The pairwise comparisons showed that none of the regions was significantly different from the other, despite the fact that the region factor was significant. 5. The analysis of gender differences in mathematics achievement showed that males performed significantly better than females. At the item level, males outperformed females on only 19 items. Most of these items dealt with geometry, or were at the application level. 6. Gender differences favoring males were found to be independent of school type and region. 7. Comparison between Dominican Grade 12 and Grade 8 students revealed that mathematics achievement improved between the grades for most items. 8. Dominican performance was very poor on the SIMS items and it was far behind that of other countries. iv Several conclusions were highlighted by these results. First, the mathematics achievement of students finishing Grade 12 in the Dominican Republic was extremely low. However, achievement levels improved since Grade 8. Second, students attending one type of private school learned more mathematics than students attending public and the other type of private schools. Third, males outperformed females but differences were most marked in geometry and application items. Finally, the performance of Dominican students, even those in F-schools, was far below that of students from other countries. V Table of Contents Abstract i i List of Tables ix List of Figures xi Acknowledgements xv Chapter 1 Background l Education i n Developing Countries l The Dominican Republic 2 The Educational System of the Dominican Republic 4 Statement of the Problem 7 Research Questions 9 Sources of Data 10 Definition of Terms 12 Limitations 14 Organization of the Following Chapters 16 Chapter 2 Review of the Literature 17 Review of Related Studies 18 The Second International Mathematics Study 18 The Grade 8 Mathematics Study in the Dominican Republic 25 Educational Achievement i n Developing Countries 30 School and Region Variables i n Research of Academic Achievement 32 School Differences in Achievement 33 v i Gender differences i n Mathematics Achievement 34 Summary 39 Chapter 3 Methodology 41 Research Hypotheses 41 Development of the Instruments 43 The Table of Specifications 43 Selection of Test Items 45 Content Validity of the Test 47 Test Administration 50 Test Reliability 50 Sample Selection 52 Description of the Population 52 Selection Technique 53 Data Analysis 54 Test Scoring 54 Correction for Guessing 54 Statistical Analyses 55 Chapter 4 Results 59 Description of the Data 59 Research Question 1 60 Research Questions 2, 3, & 4 68 Research Question 5 73 Research Question 6 77 Research Question 7 77 v i i Research Question 8 82 Chapter 5 S u m m a r y and Discussion 85 S u m m a r y of Findings 86 Mathematics Content Students K n o w 86 Results of School a n d Regional Differences i n Mathematics Achievement 87 Results of Gender Differences i n Mathematics Achievement 88 C o m p a r i s o n of Dominican Grade 12 Performance with that of Grade 8 i n T L M D R 90 C o m p a r i s o n of D o m i n i c a n Grade 12 Performance with that of Other Countries i n SIMS (Population B) 91 Conclusions 91 Implications 93 Suggestions for Future Research 95 References 97 A p p e n d i x A : Mathematics Achievement Tests (Originals) 104 B : Mathematics Achievement Tests (Translated) 121 C : Test Items and M e a n Percent Responses of the Educat ional Systems Participating i n SIMS 137 D : T L M D R Test Items Identified and Grade 8 M e a n Percent Responses at the National and School Levels 148 E : Frequency Distribution of Teachers' O T L Responses 155 viii F: School Mean Scores and Students' Responses to Test Items 157 G: Summary ANOVA Tables and Tukey Comparisons for School Type and Region: Grade 11 164 H: Males' and Females' Test Results 167 I: Summary ANOVA Tables for Gender: Grade 11 169 J: Summary ANOVA Tables for Gender Differences within School Type and Region: Grade 11 and Grade 12. 173 K: Weighted Mean p-values of Grade 12 Responses to TLMDR Items and Comparison of Grade 12 and Grade 8 Responses 181 L: Weighted Mean p-values of Grade 12 Responses to SIMS Items and Comparison of Grade 12 with other Countries' Responses 189 i x List of Tables 1. Identification of SIMS Items 22 2. Distribution of Items by Content Area, Sub-area, and Cognitive Level 45 3. Sources of the Test Items 46 4. Distribution of Teachers' Responses and OTL percentages 49 5 . Total Test and Subtest Reliabilities, Means, Standard Deviations, and Standard Errors of Measurement 51 6. Easiest Items 61 7. Most Difficult Items 61 8. Test Results for Grade 11 and Grade 12 Students 65 9. Subtest Mean Percent Correct Scores for Grade 11 and Grade 12 66 10. Subtest Mean Percent Correct Scores by School Type i n Grades 11 and 12 68 11. A alysis of Variance Testing Main and Interaction Effects 7012. C o m p a r i s o n of School Type Means 71 13. Compar ison of Region Means 72 14. Analysis of Variance of Gender Differences 74 15. A N O V A of Differences between T o p 100 Ranked Males a n d Females 74 16. Test Items Showing Significant Gender Differences 75 17. S u m m a r y of Results by Gender and Content A r e a 75 18. S u m m a r y of Results by Gender and Cognitive Level 76 19. Identification of T L M D R Items 78 20. Grade 8 a n d Grade 12 Percent Correct o n C o m m o n Items 79 21. S I M S Items: Topics a n d Cognitive Levels Identified 84 F l . Grade 11 School M e a n Scores 158 F2. Grade 12 School M e a n Scores 159 F3. Grade 12 Students' Responses to Test A 160 F4. Grade 12 Students' Responses to Test B F5. Grade 11 Students' Responses to Test A F6. Grade 11 Students' Responses to Test B G l . Analysis of Variance Testing Main and Interaction Effects (Grade 11) G2. Comparison of School Means (Grade 11) G3. Comparison of Region Means (Grade 11) HI Test Results by Gender H2 Statistics of the Top Ranked 100 Males and Females II. One-way Analysis of Variance Testing Gender Differences on the Grade 11 data. 12. ANOVA of Differences between the Top Ranked 100 Males and Females (Grade 11) 13. Test Items with Significant Gender Differences (Grade 11) 14. Summary of Results by Gender and Content Area (Grade 11) x i i 15. Summary of Results by Gender and Cognitive Level (Grade 11) 172 J l . Results by Gender by School and by Region 174 J2. Analysis of Variance Testing Gender Differences in Grade 11 Public Schools 175 J3. Analysis of Variance Testing Gender Differences in Grade 11 Private F- Schools 175 J4. Analysis of Variance Testing Gender Differences in Grade 11 Private O- Schools 176 J5. Analysis of Variance Testing Gender Differences in Grade 11 i n the Region of Santo Domingo 176 J6. Analysis of Variance Testing Gender Differences in Grade 11 i n the Region of Santiago 177 J7. Analysis of Variance Testing Gender Differences i n Grade 11 in the Region of the next ten largest cities 177 J8. Analysis of Variance Testing Gender Differences in Grade 12 Public Schools 178 J9. Analysis of Variance Testing Gender Differences in Grade 12 Private F- Schools 178 J10. Analysis of Variance Testing Gender Differences in Grade 12 Private O- Schools J l l . Analysis of Variance Testing Gender Differences in Grade 12 in the Region of Santo Domingo J12. Analysis of Variance Testing Gender Differences in Grade 12 in the Region of Santiago J13. Analysis of Variance Testing Gender Differences in Grade 12 in the Region of the next ten largest cities KI. Weighted Mean of Percent Correct Responses for Grade 12 LI. Weighted Mean of Percent Correct Responses for Grade 12 xiv List of Figures 1. Organization of the Dominican Republic's educational systeme 5 2. Educational funnel i n the Dominican Republic 7 3. Framework for SIMS 19 4. Post-test achievement scores compared: Dominican schools and SIMS (Population A) International Mean 29 5 . Arithmetic achievement scores compared: TLMDR and SIMS (Population A) 30 6. Distribution of sampled schools 53 7. Number of students per strata 54 8. Estimated target Population 58 9. Comparison between Grades 11 and 12 65 X V Acknowledgements I wish to thank the members of my committee, Dr. David Robitaille, Dr. David Bateson, and Dr. Thomas Schroeder for their guidance, and encouragement throughout the process of planning, developing, and writing this thesis. I would also like to thank Dr. Robert Wilson for participating in my thesis defense. In addition, I would like to thank Dr. Eduardo Luna and Professor Sarah Gonzalez for providing the data used in this study, and for their encouragement, and support throughout my studies. I would also like to thank my friend and fellow student Thomas Garcia for his invaluable help with all my computer-related work. Finally, I would like to thank my family, and my friends back home, and new friends here in Canada for their encouragement, patience, and constant support. 1 CHAPTER 1: BACKGROUND Education in Developing Countries In the developing world, reducing illiteracy rates and expanding primary education to a larger proportion of the population are of first priority. However, the rapid population and enrollment growth set against a weak economy have left most of the third-world educational systems unable to provide education to the whole school-age population (Fuller & Heyneman, 1989). The economic crisis affecting most of the developing world has led to deep cuts in government expenditures for education (Fuller & Heyneman, 1989). This has caused most third-world educational systems to decline in quality while trying to expand formal education (Blat Gimeno, 1983). Such a lowering of academic standards contributes to differences in employment opportunities and levels of remuneration. Thus, the educational systems in these countries tend to perpetuate social inequalities rather than to eliminate them (Blat Gimeno, 1983). The efficiency of third-world school systems is affected by many different problems. For example, a shortage of qualified teachers is very common. This is due to the fact that generally in these countries teachers are not highly paid. This in turn makes teaching the least attractive professional choice for talented students (Schiefelbein, 1983). Other common problems are the high rates of grade repetition and dropouts (Blat Gimeno, 1983). It is not uncommon to find high degrees of grade repetition, even among the same pupils. Sooner or later, these students end up dropping out of school. The imbalance between rural and urban advancement is another 2 characteristic of developing countries (Adams & Bjork, 1969; Blat Gimeno, 1983). In these countries, there is a wide gap in wealth and range of opportunities between the few relatively large urban centres and the much more backward rural areas (Blat Gimeno, 1983). Equally unbalanced is access to education. Rural regions usually do not offer all school grades and have lower academic standards (Adams & Bjork, 1969; Blat Gimeno, 1983). In many third-world countries, female participation in education is very limited. The World Bank (1989) reported that females are the largest group underrepresented in post-primary education in Africa. However, in Latin American countries, male and female enrollment rates are almost uniform at all age levels. Even at tertiary ages (18-23), differential enrollment rates are moderate: males, 22 percent and females, 18 percent (Bowman & Anderson, 1982). The Dominican Republic The Dominican Republic is located in the Caribbean Sea. It shares the island of Hispaniola, second largest of the Caribbean islands, with Haiti. Hispaniola lies in the group of islands called the Greater Antilles, between Cuba and Jamaica on the west, and Puerto Rico on the east. Dominicans occupy the eastern two-thirds of the island, an area of 49,000 km 2 . The country is divided into 26 provinces including the Distrito Nacional where the capital, Santo Domingo, is located. The country has a population of approximately six million inhabitants, and a population growth rate of 3.6 percent per year [Censo Nacional de Poblacion, 1970), among the highest in Latin America. It also has a high populational density, about 100 inhabitants per km 2 . Another characteristic of the 3 country is that the population is relatively young: approximately 63 percent of the population is under 25 years of age. Population growth has been accompanied by a rapid process of urbanization resulting in a drastic change in the urban-rural population distribution. According to the 1970 census, the distribution was 40 percent rural and 60 percent urban. During the ten year interval from 1960 to 1970, a population growth rate of 6.4 percent was registered in urban areas compared to 1.5 percent in rural areas. With these data, it is estimated that in the 90's the current urban-rural distribution would be approximately 62 and 38 percent respectively. The country's mineral deposits of bauxite, nickel, gold, and silver are a source of economic wealth. However, the mainstay of the economy is agriculture, with sugar being the chief export crop. Decreases in the price of the main export products and inflation of the cost of imported goods have contributed to severe economic difficulties. In short, the Dominican Republic is a developing country with all the characteristics and problems common to developing countries. Adams and Bjork (1969) list the following among those characteristics and problems: high birth and death rates, poor nutrition, wealth concentrated in one or two large cities, and low per-capita income. In the Dominican Republic, the World Bank (1987) reported an annual per capita income of US$710. In addition, the Dominican Republic has an international debt of more than 3.2 billion dollars (Lora, 1984), a debt load which threatens the country's economy and its path to development. 4 The Educational System of the Dominican Republic In the Dominican Republic, administration of the educational system is centralized and is under the direction of the National Council of Education and the Minister of Education. The Secretaria de Estado de Education Bellas Artes y Cultos, SEEBAC (Ministry of Education, Fine Arts, and Religion) supervises and evaluates all schools. In addition to its central administration, SEEBAC has a regional structure which directs and administers educational services in different geographic zones of the country. There are nine regional directors, 76 district headquarters, and 403 centres for educational development called nucleos escolares (scholastic nuclei) which are the basic unit of the educational system in rural areas. Each nucleus has a central school which offers the first eight years of instruction, even if smaller schools under its jurisdiction cannot offer them. There are several types of schools in the country: public schools, totally financed by the government; semi-official schools, partially financed by the government; and private schools, which receive no financial aid from the government. Schools may also be classified by area into urban and rural. Urban schools are known to have better qualified teachers and higher standards of both teaching and accommodation than rural schools (Bell, 1981; Diaz Santana, 1987; Luna, Gonzalez, & Yunen, 1985a). State education through the secondary level is free. For the majority of children who receive any education at all, school begins at age six with admission to first grade (see Figure 1). In theory, but not in practice, pupils move up at the beginning of each scholastic year and reach Grade 6, the last year of primary education, by the age of 11. Figure 1 shows that 5 following primary are two intermediate grades, leading to the secondary level, which has four grades. Pupils who successfully complete the fourth year of secondary school qualify for entry into university. The secondary grades 1°, 2°, 3°, and 4° de bachillerato are referred to as Grades 9, 10, 11, and 12 throughout this study. Age Grades 22 Higher Education Not regulated by the government 21 20 19 18 17 Secondary Level 4° 16 3° 15 2° 14 1° 13 Intermediate 8° 12 7° 11 Primary Level 6° 10 5° 9 4° 8 3° 7 2° 6 1° 5 Pre-school Not regulated by State 4 3 Figure 1. Organization of the Dominican Republic's educational system. In many Latin American countries, several problems, such as low enrollment at primary level, affect the efficiency of the educational system (Brock, 1985). Blat Gimeno (1983) asserts that this problem is largely due to the shortage of schools to fulfill the demands. In the Dominican Republic, this problem is caused in part because many children register in school at a later age and many others repeat grades. This disrupts the regular flow of children through primary school and impedes other children from entering school (Diaz Santana, 1987). 6 The shortage of schools also results in a low rate of participation or retention. In 1974-75, the Ofictna Nacional de Estadistica (National Office of Statistics) reported that only 68 percent of the population at primary and intermediate levels; ages 5 to 14, was attending school. At the secondary level, only 38 percent of the population in the 15-19 age group was in school. Nevertheless, the number of classrooms has increased through the years; and, according to a SEEBAC report, in 1983 there were 33,933 classrooms of which 75 percent were in the public sector and 25 percent in the private sector. At the secondary level, SEEBAC reported that there were 7,435 classrooms (71 percent public, 29 percent private). Another problem affecting the educational systems of developing countries is the low retention rates (Adams & Bjork, 1969). A study carried out by Unesco in 18 Latin American countries showed that "Out of 1,000 children enrolled in first year primary education, fewer than 500 reached the fourth year" (Cited in Blat Gimeno, 1983, p.38). Similarly, in the Dominican Republic, Shiefelbein (1976) reported that for every 1,000 students enrolled in first grade, only 160 completed sixth grade. Of those 160 students who complete the elementary grades, only 120 registered in the intermediate grades (7th and 8th), and only 30 finished the secondary level (see Figure 2). Although it has been several years since the Shiefelbein report was completed, it is still the case that a large proportion of Dominican children receive little formal, school-based instruction (Luna & Gonzalez 1989). 7 No of students registered per year Grade (In thousands) Figure 2. Educational Funnel in the Dominican Republic. Statement of the Problem In recent years the quality of education in the Dominican Republic has been questioned by educators and concerned citizens. Diaz Santana (1987) asserts that, in the past two decades, the educational system has systematically declined in quality, especially public education. Three different studies (SEEBAC, 1984; SEEBAC-INTEC, 1984; Luna, Gonzalez, & Yunen, 1985) evaluating the quality of the outcomes of education in the country reached the same conclusion: education in the Dominican Republic is deficient at both the primary and the intermediate levels with the situation being worst in public schools which serve the poorest part of the population (Diaz Santana, 1987). At the high school level, it is assumed that the quality of education is equally low and deficient but no data has been available to verify this conjecture. Students graduating from high school and entering the university have been found to exhibit poor performance on the university entrance tests (Lora, 1984). They have been found to be inadequately prepared for the task of successfully completing their programs, and this 8 is reflected in the increasing numbers of drop-outs in the early years of study (Lora, 1984). Some universities have tried to upgrade students' competencies in the subjects taught in high school by establishing the first year of study as a make-up year (Lora, 1984). The Pontificia Universidad Catolica Madre y Maestra (PUCMM) and the Universidad Nacional Pedro Henriquez Ureha (UNPHU) are two of the universities in the country known to have installed a CU [Colegio Universitario, University College) year to review concepts taught in high school. In order to improve the quality of education in secondary schools in the Dominican Republic it is necessary to find out what students are taught, and what they have actually learned. Lora (1984) pointed out the need for research in the Dominican Republic to identify the causes of educational problems in pre-university years. During the 1982-83 school year, a study was conducted in the country at the Grade 8 level. That study, known as Teaching and Learning of Mathematics in the Dominican Republic (TLMDR) involved 116 schools, 160 teachers, and 5342 students. Among other findings, Luna, Gonzalez, and Yunen (1985a) reported that mathematics achievement in all areas was very low. They concluded that such low achievement levels among eighth graders, almost certainly implied that similarly low levels would be found among students completing secondary school. Although that study contributed to our understanding of the problems affecting mathematics education in the country, more research is needed to provide reliable data on which to base informed decisions regarding suggestions for improvements in the quality of education. As was stated in the General Report of the 1985 mathematics assessment in British Columbia, "Decisions about new directions or new emphases in educational 9 matters should be based upon a clear understanding of what students are learning and how they are being taught" (Robitaille, 1985, p. 2-3). This is true not only for the province of British Columbia, but for any educational system in need of change or improvement. In the Dominican Republic, changes are necessary to improve the quality of education, and these should not be made in a haphazard fashion. They should be based on sound research data and take into account the social and financial context in which learning is taking place. As noted by Beeby (1966), the first stage to improve the quality of any educational system is to assess its potential growth. The first step in that move is to determine the level of students' achievement reached by actual practice in the schools. The present study is a contribution to the research needed in the Dominican Republic. It is the first study of its kind, and is intended to provide information on what mathematics Grade 12 students have learned. In addition, it will also contribute to the growing literature of research on academic achievement in developing countries. The study arises from a concern that students finishing high school may not have learned the mathematics necessary for university entrance. This may be a grave problem because, although a great number of students drop out of the educational system prior to Grade 12, 85 percent of students finishing high school in the Dominican Republic enter university (Reported in Diagndstico del Sector Educativo, SEEBAC, 1984). Research Questions In the general context of assessing mathematics achievement at the Grade 12 level, the present study was designed to address the following 10 questions: 1. What mathematics content do Dominican students in Grade 12 know? 2. Do students from private schools score higher than their public school counterparts? 3. Do students in schools located in privileged regions score higher than students attending schools in less privileged regions? 4. Is there an interaction between school type and region which results in greater mathematics achievement for a particular combination? 5. Do males outperform females in mathematics? 6. Do males outperform females independently of school type and region in which the students are found? 7. How does the mathematics achievement of Grade 12 students compare with that of Grade 8 students in TLMDR? 8. How different is the performance of Grade 12 Dominican students from that of students from other countries which participated in the Second International Mathematics Study? Sources of Data The data for the present study were collected as part of an evaluation project of secondary mathematics in the Dominican Republic. Students from different types of schools and regions were sampled, and four major curriculum strands were tested: arithmetic, algebra, geometry, and trigonometry. Items were selected at three cognitive levels: computation, comprehension, and application. Seventy multiple-choice items, 11 distributed over two test forms, were administered to students in Grades 11 and 12. In addition, students' opportunity to learn was assessed by asking teachers whether or not they had taught the content measured by each item on the achievement test. To answer the first research question, data from Grade 11 and Grade 12 students were used. Grade 11 data were used as Grade 12 pre-test data to find out how much mathematics students learned during the school year. The comparison of public and private schools, as well as the comparison of regions were made using the means from participating schools. The research questions on gender differences were answered using the data from students' responses to the test items. To test whether gender differences were independent of the school and region in which the students were found, analyses were made using a sub-sample of students' scores made up of the students in each school and in each region, resulting in six data sets: one for each school and for each region. The comparison of the mathematics achievement obtained by Grade 12 students with that of Grade 8 students was made using data from 14 items used in the TLMDR study. The responses of Grade 8 students were obtained from the various summary reports of the TLMDR. Only the data from the urban schools reported in the TLMDR were used for comparison because all the schools in the present study were from urban areas. Finally, the performance of Dominican students in Grade 12 was compared to that of students from other countries that participated in the Second International Mathematics Study (SIMS: Population B). Comparisons were made on ten items drawn from the SIMS (Population B) achievement test (Robitaille & Garden, 1989, Appendix F). 12 Definition of Terms Several variables were investigated in this study. The dependent variable was mathematics achievement and the independent variables were school type, region, and gender. The tests were developed as part of an unpublished evaluation project carried out by Luna and Gonzalez of PUCMM, who also arranged the sampling procedures which are explained in detail in Chapter 3. This project is referred to hereafter as the Luna and Gonzalez (1988) secondary school project. Definitions of the terms used throughout this study are provided below: Mathematics achievement. Mathematics achievement refers to the achievement exhibited on the two mathematics tests used in the present study. Content areas. Content areas are the topics outlined in the curriculum guide as part of the program of study for a particular grade. Four content areas outlined in the secondary mathematics curriculum of the Dominican Republic were evaluated in the mathematics achievement test analyzed in this study. Sub-areas. Sub-areas are the specific topics included in each of the four content areas. The content areas with their respective sub-areas evaluated in the mathematics achievement test were the following: 1. A r i t h m e t i c 1.1 Real Numbers: properties and operations 1.2 Complex Numbers 2. Algebra 2.1 P o l y n o m i a l s 2.2 Roots and Radicals 2.3 Equations and Inequalities 3. Geometry 3.1 Eucl idean 13 3.2 Analytic 4. Trigonometry 4.1 Angles and triangles 4.2 Identities 4.3 Logarithms Cognitive Level. Cognitive level or behavior refers to the cognitive complexity of the item. Three cognitive levels were considered from the Wilson model for the evaluation of educational achievement: Computation, Comprehension and Application. These terms are defined by Wilson (1971) as follows: 1. Computation is the recall of specific facts and terminology or the ability to carry out algorithms according to learned rules. The emphasis is upon knowing and performing operations and not upon deciding which operation is appropriate. 2. Comprehension relates either to recall of concepts and generalizations or to transformation of problem elements from one mode to another. The emphasis is upon demonstrating understanding of concepts and their relationship. 3. Application requires recall of relevant knowledge, selection of appropriate operations, and performance of the operations. It requires the students to use concepts in a specific context and in a way they have practiced. School Type. Two main types of schools were included in this study: public and private. Public schools are totally financed by the government, whereas private schools receive no government funding. Within the private schools there are two different types: F-schools and O-schools. F-schools are autonomous to a certain degree as they are authorized by the Ministry of Education to set their own examinations. O- schools are not allowed to give such examinations and are supervised by the Ministry. The F- and O-schools were given these names originally in the 14 TLMDR as a remainder of their characteristics. F-schools are called with Faculty to give their own examinations, and O-schools are without the faculty to give such examinations. Region. Region refers to the geographical location of the schools. Schools in twelve regions of the Dominican Republic were included In this study. They were grouped into three categories: Santo Domingo, Santiago, and the next 10 largest cities in the country. Privileged region. Privileged region, in the context of the Dominican Republic, refers to a large city where the economy is prosperous and the standard of living is better than in most of the other regions in the country which are usually very poor. According to this definition, the regions sampled in the present study can be classified as privileged and less privileged. The capital city, Santo Domingo, is a privileged region because it is the largest city in the country and where most of the economic resources are produced and spent. The region of Santiago follows that of Santo Domingo in terms of size and wealth. The third region considered in this study, the next ten largest cities, can be characterized as the least privileged because these cities are smaller and have fewer economic resources than the previous two. Limitations In the planning and development of the goals for this study, several problems were identified, and these are discussed in this section. One limitation is that although the general goal is to assess mathematics achievement at the Grade 12 level, the sample used does not represent all students in Grade 12 attending public or private schools in the Dominican Republic. The schools in the sample are only urban schools from 12 cities 15 in the country. However, it has to be considered that all private schools in the Dominican Republic are located exclusively in urban areas (Jimenez, Lockheed, Luna, & Paqueo, 1989), and that most public schools in rural areas do not offer all the secondary grades (Bell, 1981). A second limitation of the study is that Grade 11 data have been used as Grade 12 pre-test data to estimate how much mathematics students learned during Grade 12. This could be invalid because Grade 11 students may not be comparable to Grade 12 students. The fact that they are not the same students is obvious. However, it can be argued that students in the sample are similar because Grade 11 and Grade 12 students were drawn from the same schools. Another related problem is the fact that 25 percent of the students drop out of school during the course of these two grades (see Figure 2) which affects the comparability of the samples. It is not clear why these students drop out or what their level of ability is. If they are low ability students at the bottom (or close to the bottom) 25 percent then the Grade 11 scores are an underestimate of the Grade 12 pre-test scores. Using these scores as a pre-test will inflate the estimates of what students in Grade 12 learn during the year. However, it can be argued that even longitudinal studies do not assure that the same students will still be in school at the end of the study. A third limitation is that mathematics achievement is evaluated by a set of only seventy test items. Even though all the precautions were taken to obtain a reliable and valid measure of mathematics achievement, the fact that mathematics knowledge is very broad cannot be forgotten. In addition, the mathematics achievement tests were set in a multiple-choice format. Multiple-choice tests are believed to give cues to students to 16 answer correctly and to give an advantage to some students who do better in this type of test than in other types of tests (Wood, 1977, cited in Nitko, 1983). Therefore, some may argue that the test used in this study was not a valid measure of mathematics achievement because it used a multiple-choice format. However, an advantage of multiple-choice tests is that they evaluate a greater variety of instructional objectives (Nitko, 1983). Another advantage of this type of tests is that students focus on reading and thinking, allowing the students to devote their time, not to the writing process, but to work over their ideas (Wood, 1977, cited in Nitko, 1983). Therefore, multiple-choice tests, opposed to essay tests, are less time consuming to the respondents. In addition, they are easily scored which allows the investigator to be able to manage large samples (Nitko, 1983). Organization of the Following Chapters A review of the literature pertaining to the research questions, a description of methodology, the results of statistical analyses, and a discussion of the findings of the study are found in subsequent chapters. In Chapter 2 the related literature is reviewed. Chapter 3 includes, the details of sample selection, development and administration of instruments, and data analysis. The results of the statistical analyses are discussed in Chapter 4. Finally, a summary of the results and also implications and recommendations are presented in Chapter 5. The original tests administered to the students with their respective translation are included in the appendices. 17 CHAPTER 2: REVIEW OF THE LITERATURE The present study is a follow-up, at the senior secondary level, to the TLMDR study conducted by Luna, Gonzalez, and Yunen at Grade 8 in the Dominican Republic. Among their findings, Luna, Gonzalez, and Yunen (1985a) reported that a small group of students attending private schools had higher achievement scores than students attending other types of schools. One of the goals of the present study was to investigate the significance of these differences, using a sample of Grade 12 students. An additional objective of the study was to investigate gender differences in mathematics achievement, at the senior secondary level, particularly between males and females attending the same type of school and from the same region. In addition, Grade 12 students' performance was compared to that obtained by Dominican students in Grade 8 and to the performance of students in the terminal year of high school in other countries. This chapter deals with the literature related to these areas of investigation. It is organized in four sections. First, there is a review of related studies: the Second International Mathematics Study (SIMS), and the "Teaching and Learning of Mathematics in the Dominican Republic" (TLMDR). Next, the literature with regard to educational achievement in third-world countries is discussed. This is followed by a review of studies on the effect and relationship of school and regional variables on academic achievement. Finally, the literature related to gender differences in mathematics achievement is reviewed. 18 Review of Related Studies The present study is a Grade 12 study of mathematics achievement parallel to the TLMDR at Grade 8 level, which in turn was a replication of the SIMS study. These two projects are reviewed because they have influenced the methodology and theoretical framework of the present study. For the purposes of this study, only those findings related to the questions of interest to the present investigation will be discussed. The Second International Mathematics Study In the early 1960's the International Association for the Evaluation of Educational Achievement (IEA) conducted an international survey of mathematics. This study is known as the First International Mathematics Study (FIMS). Twelve educational systems participated in FIMS, most of them highly developed European countries. Years later, in the early 1980's, IEA conducted a second study known as the Second International Mathematics Study (SIMS). SIMS included a wider variety of countries ranging from highly industrialized nations to third world countries, and thus provided a broader overview of mathematics achievement. The SIMS participants included the twenty educational systems listed below. Belgium (French and Flemish) Luxembourg Canada (British Columbia and Ontario) Netherlands England and Wales New Zealand Finland Nigeria France Scotland Hong Kong Swaziland Hungary Sweden Israel Thailand Japan United States 19 IEA was founded for the purpose of comparing the performance of schools and students in different countries. The primary aim of FIMS was to compare the outcomes of schooling of various educational systems, with mathematics achievement serving as the dependent variable (Robitaille & Travers, 1989). In that project, mathematics was chosen for convenience whereas in SIMS, mathematics was central to the entire investigation. SIMS represented an attempt to advance mathematics education through international comparisons of curricula, aims, attainments, and attitudes of teachers and students in a wide variety of countries (Travers, Garden, & Rosier, 1989). SIMS was designed to be an in-depth study of the mathematics curriculum at three levels (See Figure 3): the intended curriculum as transmitted by national or system level authorities; the implemented curriculum as interpreted and translated by teachers in a particular classroom; and the attained curriculum defined as what students have actually learned and what is manifested in their achievement and attitudes. COMPONENTS OF THE STUDY CURRICULUM DIMENSION FOCUS Curriculum Analysis II Classroom processes III Student Outcomes Intended Implemented Attained Educational System Schools and classrooms Students Figure 3. Framework for SIMS. 20 There were two target populations selected for the study, and these were designated as Populations A and B. Population A: All students in the grade (year, level) where the majority have attained the age 13.00 to 13.11 years by the middle of the school year. Population B: All students who are in the normally accepted terminal grade of the secondary education system and who are studying mathematics as a substantial part of their academic program. (Travers, Garden, & Rosier, 1989). For the purposes of this study, only Population B has been reviewed. At this level, only 15 educational systems participated and Thailand was the only developing country participating. The different configurations of Population B made comparisons among systems and interpretation of results difficult. For example, some systems were found to be more selective than others and to retain relatively fewer students in pre-university mathematics (Robitaille & Travers, 1989). In addition, in all of the participating countries, except for Hungary, mathematics is not compulsory for students in the terminal year of high school. Findings of SIMS A large amount of information was collected in SIMS and many findings were uncovered. However, only the findings related to the present investigation are discussed below. Curricular differences across systems Students' mathematics achievement was evaluated by means of a mathematics achievement test which was based on the results of the international curriculum analysis. The detailed topics and subtopics included in the Population B achievement test can be found in Robitaille 21 and Garden (1989). The major topics evaluated were: Sets, Relations and Functions Geometry Number Systems Elementary Functions and Calculus Algebra Probability and Statistics In all of the participating systems, the mathematics curriculum tended to include both algebra and calculus as important strands. Less emphasis was placed on geometry and trigonometry which were covered in earlier years (Robitaille & Travers, 1989). Curricular differences were found across systems in the area of probability and statistics, calculus and geometry. In Belgium (Flemish), Hungary, and Canada (British Columbia), little or no probability and statistics was included in the intended curriculum. Calculus was included in the terminal year mathematics curriculum in most of the participating countries, except in Canada (British Columbia), and was taught only in certain courses in Canada (Ontario) and the United States. All systems included geometry in their curriculum, but they differed in the kind of geometry taught and the approaches to teaching it (Garden, 1989). The topic of sets, relations, and functions received ratings of low importance in most systems, possibly because it was taught in earlier grades and may have been judged less important in the terminal grades (Garden, 1989). The topic of number systems consisted of two subsets of items: complex numbers, and common laws of number systems. Complex numbers was not included in the Population B curriculum in Canada (British Columbia), Hungary, or Israel, and was included in the curriculum of a relatively small proportion of students in the Scotland and Belgium (French and Flemish) samples. 22 In algebra, few items were rated as inappropriate. Within the topic of algebra, the subtopic of equations and inequations was rated as central to the Population B mathematics curriculum. Trigonometry was included within the topic of geometry and 7 out of 8 items were rated as appropriate in all systems. Students' performance Only ten SIMS items are included in this review because they are of interest to the present study. These items are identified in Table 1. The cognitive level of each item is identified as: (1) computation, (2) comprehension, or (3) application. In addition, the topics covered by the items are presented. The international mean percent correct is also reported in the table. The results for each of these items obtained by each of the participating systems are included in Appendix C. They show that in almost every item Japan or Hong Kong obtained the highest percent of correct responses. Table 1 Identification of SIMS Items SIMS (Pop B) Cognitive level Topic Int'l mean item number percent correct 012 3 Geometry 56 080 1 Algebra 60 038 3 Algebra 65 078 2 Number Systems 30 003 3 Number Systems 38 004 1 Algebra 41 081 3 Algebra 43 068 1 Geometry 49 097 3 Algebra 62 087 2 Algebra 71 23 On this set of ten items, the lowest item score was on item 078. This item was extremely difficult in most systems. Apart from Hong Kong and Japan which had p-values of 78 and 69 percent respectively, p-values ranged from 40 to 8. This is surprising because this item (see Appendix C) asks students to identify the Least Common Multiple (LCM) of two numbers expressed as powers. The easiest item, with the highest averaged SIMS p-value, was item 087. This item (see Appendix C) tests comprehension of logarithmic equations. The percentage of correct responses ranged from 96 to 40. It is worth noting that another difficult item on this set of ten items was item 003. This item tests the application of the commutative operation (see Appendix C). In this item, p-values ranged from 54 to 15 percent, excluding Hong Kong and Japan with scores of 76 and 74 percent respectively. Gender differences At the Population B level, many more males than females took mathematics courses in most systems, except in Hungary and Thailand. In Hungary, 62 percent of Population B was females, and in Thailand, the numbers of males and females were approximately equal. In the other systems, the proportion of females taking Population B mathematics ranged from 21 percent in Hong Kong to 44 percent in the Unites States. The overall pattern suggested by the data with the exception of Hungary and Thailand was that males are more likely to elect to study mathematics than are females. This means that a substantial degree of selection by gender has occurred by the end of secondary school in most systems (Wagemaker & Knight, 1989). Analyses of the data showed that boys outperformed girls on virtually 24 every subtest in almost every system (Robitaille & Travers, 1989). With the exception of the sets, relations, and functions subtest, overall group performance was greater for males than for females in all the major subtests. Probability and statistics was the foremost subtest in producing large mean differences favoring boys (Garden, 1989). In two specific subtests, algebra (computation) and analytical geometry, differences were smaller or favored girls in several systems. However, differences favoring females were of practical significance (Garden, 1989) only in Canada (British Columbia) and on only two subtests: sets, relations, and functions, and algebra (computation). Other findings The study of classroom processes uncovered many similarities and discrepancies across systems. The most common teachers' approach to teaching mathematics across systems was found to be the traditional "chalk-and-talk" style (Robitaille & Travers, 1989), using whole class instruction as the most common instructional technique. Teachers in all systems seemed to rely heavily on the prescribed textbook, and rarely gave differentiated instruction or assignments (Werry, 1989). There were large discrepancies among systems with regard to average class sizes (Robitaille & Travers, 1989): from 14 to 43 at Population B. Students' performance in some of the largest classes, especially those from Japan and Hong Kong, were among the highest obtained in the study. Students from Hong Kong were also among the youngest participants and their teachers were less than fully qualified to teach mathematics. The performance levels attained by Hong Kong students were exceptional given these limitations. 25 The Grade 8 Mathematics Study In the Dominican Republic TLMDR was a replication of SIMS at the Grade 8 level (Population A). It was designed to study the mathematics curriculum at the intended, implemented, and attained levels. The intended curriculum was analyzed using the Grade 8 mathematics program and approved textbooks. To study the implemented curriculum, a questionnaire was administered to teachers to gather information about their mathematical background, methods of teaching, years of experience, and the opportunity their students had to learn the topics considered in the study. The attained curriculum was studied through the analysis of students' answers to a mathematics achievement test on the topics outlined in the curriculum guide: arithmetic, algebra, geometry, measurement, and descriptive statistics. Students were tested at the beginning and the end of the school year to obtain a measure of growth. The TLMDR study was conducted during the 1982-83 school year, and it was the first national survey of its kind to be conducted in the country. The purpose of the study was to explore the influence of factors such as school and regional resources on learning. Schools were classified into different types according to their status with the Ministry of Education, and regions were classified according to their degree of urbanization using number of inhabitants as the categorizing criterion. Researchers were also interested in studying the influence of several non-school factors, such as social, demographic, and other environmental characteristics, on achievement (Luna, Gonzalez, & Wolfe, 1990). The SIMS definition of Population A which called for "students in the grade (year level) with modal age of 13 years" was not used in the TLMDR, since the modal age of Dominican students attending Grade 8 public 2 6 schools is 15 years. The target population of the study was redefined as "the students registered in Grade 8 of the Traditional Program, or in the second year at the middle level of the Reformed Program" (Luna, Gonzalez, & Yunen, 1985a). Findings A large amount of information was collected in TLMDR. Many findings were uncovered about teachers' characteristics, teaching practice, and students' background and achievement. For the purposes of the present investigation, only the related findings are discussed. Mathematics Teaching Luna, Gonzalez, and Yunen (1985a) reported that a teacher of a typical Grade 8 classroom is approximately 33 years of age and has 11 years of teaching experience. Only one in every four teachers has the academic preparation required by the Ministry to teach mathematics at that level. The typical teacher has a heavy teaching load: 34 periods per week (each period is 45 or 50 minutes long). Usually, teachers work in different sessions: mornings, afternoons, and sometimes even in the evening sessions. It was not surprising to find that teachers rely basically on the textbook, since they have little time to prepare thoughtful, well-designed lessons. Most of the mathematics teachers indicated that their students had not mastered the prerequisite mathematics, and that they were going to review most of the topics taught in previous years. In addition, it was found that the total number of hours dedicated to teaching mathematics was less than the number specified in the curriculum guide. For these reasons, the content of the program could not be completely developed in the classes, nor taught to the desired depth (Luna, Gonzalez, & Yunen, 27 1985a). Given such conditions, one would expect mathematics achievement to not attain the desired standards. Not surprisingly, this was found to be the case. The low mathematics performance of Dominican children could be partially explained by the fact that students' opportunities to learn mathematics and the conditions in which learning occurs are very poor. Students' achievement Luna, Gonzalez, and Yunen (1985a) reported that the overall achievement in each of the mathematics topics tested was very low. They also reported that growth in mathematics achievement, or the difference between the post- and pre-test, was very poor. In addition, the national averages on the post-test in the areas of ratio, proportion, and percentage, and of measurement were reported to be lower than chance. School type and regional differences in achievement In the Grade 8 study, different types of schools and regions were sampled. The comparison of the scores obtained by students attending the different schools showed that students attending private F-schools had higher achievement scores than students attending other types of schools. Moreover, F-school students scored higher on the pre-test than students from other schools did on the post-test. The researchers also found that schools located in larger towns and cities (urban schools) had more resources and better qualified teachers than rural schools. Another finding was that only one of every four Dominican children had his or her own mathematics textbook, and this situation varied considerably for students in the different types of school. In private schools the ratio is 3:5. In public urban schools the ratio is 1:5, and it is 1:6 in public rural schools. This is a worrying matter because availability of 28 textbooks is one of the school-resource variables associated with higher achievement levels (Heyneman, Farrell, & Sepulveda; 1981). Students attending private and public schools were found to be of different socioeconomic status (SES). Luna, Gonzalez, and Wolfe (1990) refer to the students in Grade 8 in the Dominican Republic as an "elite with schooling", because only a small number of children who enter primary first grade complete eight years of schooling. Only 7 percent of the population entering first grade completes the eighth grade in the expected time. Approximately 85 percent of the "elite with schooling" attend public schools and have low socioeconomic status, albeit higher than the majority of poor people in the country (Luna, Gonzalez, & Wolfe, 1990) . The remaining 15 percent attend private schools. This small population attends mainly private O-schools (10 percent) and these students are a mixture of low and middle SES. Five percent of the "elite with schooling" attend F-schools and these students are of high and middle SES. As mentioned before, children in these F-schools had better achievement levels than children in other schools. Therefore it seems that the already economically advantaged children are receiving the best education in the country. Gender differences Gonzalez and Luna (1984) found that boys outperformed girls on many items. The greatest differences were found in the topics of natural numbers and ratio, proportion, and percentage. Differences in attitudes towards mathematics were also found. Boys indicated that mathematics is a male domain, while girls indicated that mathematics is an appropriate domain for both genders. More boys than girls perceived mathematics as 29 being important to society, and more boys than girls thought they could learn mathematics. Luna, Gonzalez, and Yunen (1985a) suggested that boys' and girls' differences in achievement could, in part, be a consequence of the social and cultural environment of the Dominican Republic. International comparisons The mathematics achievement test administered in the TLMDR contained 91 items common to the SIMS final set of items (Luna & Gonzalez, 1984). These items were used to compare the performance of Dominican children with that of students from other countries. It was found that the national mean score of the Dominican Republic was lower than the average international score in each of the evaluated areas. This is shown in Figure 4. In the figure, the achievement levels of Dominican children in F-, O-, and public schools are compared to the international mean on each mathematics subtest. The bar graph shows that even private F-schools were far behind the average international scores. 100 n • Public S O-schools 0 F-schools • Int'l Mean Arithmetic Algebra Geometry Descriptive Measurement Statistics Figure 4. Post-test achievement scores compared: Dominican schools and SIMS (Population A) International Mean. 30 The achievement levels of the Dominican Republic were lower than other developing countries in SIMS in all evaluated subtests. This is portrayed in Figure 5 for the arithmetic subtest. The national mean for arithmetic in the Dominican Republic was lower than that of Nigeria, Swaziland, and Thailand. Japan Canada (B. C) Netherlands Belgium (Flemish) j France Hungary Canada (Ontario) Belgium (French) • ^ Hong Kong 7 Scotland United States England/Wales Israel • Luxembourg ' 7 Finland New Zealand Thailand Nigeria Sweden Dominican F-schools Swaziland Dominican Republic ~ 7 62 ^ 61 2^ 60 z 5 9 c 59 2^58 Z57 57 55 54 52 52 ^ 4 8 Z47 20 40 60 80 100 Figure 5. Arithmetic achievement scores compared: TLMDR and SIMS (Population A). Educational Achievement in Developing Countries Education in developing countries confronts many problems which affect the efficiency of their school systems to the detriment of educational quality. In the World Bank report (1989) on education in Africa, it was pointed out that stagnation of enrollments and erosion of quality are the main educational issues in Africa nowadays. Fuller and Heyneman (1989) pointed out that the baseline level of quality in developing nations is very low relative to the United States or western European countries. In this context, educational quality refers to students' achievement 31 and material resources available to schools. In third world countries, research in this area has used indirect evidence, such as supplies of inputs like books and other learning materials, to measure educational quality. It has been found that these learning inputs are critically low in Africa (World Bank, 1989). Fuller and Heyneman (1989) reported that many third world schools lack basic textbooks, desks, and simple writing materials. Another indirect measure of educational quality used in research is to look at the levels of education spending by different systems. The erosion of quality is apparent when industrialized and third world countries are compared. Fuller and Heyneman (1989) reported that, from 1970 to 1980, spending levels in industrialized countries rose from $1,229 to $2,257 per student. In middle income countries the level rose moderately, from $135 to $180. In the poorest countries, it declined from $122 to $81 per student. Levels of spending allocated to instructional materials is another indicator of educational quality. Again, differences are apparent when developed and developing countries are compared. For example, Fuller and Heyneman (1989) reported that in Bolivian primary schools, the annual spending on instructional materials was 80 * per student, and $4 in Brazil. These contrast with industrialized countries, such as Sweden, where annual expenditures on learning materials exceed $300 per student. These differences In resource levels are disturbing, especially because basic instructional materials, such as textbooks, appear to have a significant impact on academic achievement (Fuller & Heyneman, 1989; Heyneman, Farrell, & Sepulveda, 1981) Very few studies have been conducted on students' academic performance in the developing world. In Africa, the level of achievement 32 has been found sufficiently poor to be a cause of serious concern (World Bank, 1989). Thorndike (1974, cited in Fuller & Heyneman, 1989) reported findings from the first international evaluation of achievement and pointed that only "1 out of 10 Third World students at age 14 was as literate in the language of instruction as the average pupil from an industrialized country". In Chile and India, average reading scores were half the level found in industrialized nations. School and Region Variables in Research on Academic Achievement Some researchers have pointed out the importance of studying an educational system within its own social context to better understand the determinants of academic achievement (Luna & Gonzalez, 1989; Luna, Gonzalez, & Wolfe 1990; Murnane, 1984; Theisen, Achola, & Boakari, 1983). In the literature dealing with social factors influencing achievement, four general explanatory variables have been identified: background characteristics of the learner, school resources, school and classroom environment, and the general social and cultural context in which instruction takes place. Theisen, Achola, and Boakari (1983) reported that although school effect variables such as teacher quality and school resources on the one hand, and socioeconomic status and home environment variables on the other, explain substantial variance in achievement, their impact varies according to the context in which achievement is measured. For example, it has generally been found that school resources tend to account for more variance in less-industrialized countries than they do in more industrialized nations (Heyneman, 1976; Farrell, 1977; Heyneman & Loxley, 1982). 33 Regional resources are also suggested to have an effect on academic achievement in developing countries (Theisen, Achola, & Boakari, 1983). In these countries, the richer regions are more likely to be able to support more and perhaps better educational resources for teachers and students. In addition, regional and school resources are important variables which may have a combined effect on academic achievement. Studies in several African countries (cited in Theisen, Achola, & Boakari, 1983) reported a strong association between region and school quality. School Differences in Achievement Studies of academic achievement comparing public and private schools have been conducted in some educational systems. In the United States, the most widely known study is that of Coleman, Hoffer, and Kilgore (1982) in which they concluded that private (Catholic) schools were more effective than public schools in helping students to acquire cognitive skills. From this study many arguments arose about the research methodology and the interpretations which led to their conclusion. Nonetheless, it was recognized that there are important differences in the quality of education offered in different kinds of schools in the United States (Murnane, 1984). In developing countries there have been several studies which have uncovered differences between private and public schools. In Chile, Schiefelbein and Farrell (1973) found that textbook availability has a stronger relationship with academic achievement among Grade 8 children attending private schools than among children attending public schools. In Sri Lanka, Niles (1981) found that schools tended to reinforce the differences between socioeconomic groups, because children of higher 34 status were more likely to attend schools with better resources and more highly qualified teachers. Jimenez, Lockheed, and Wattanahua (1987), using the SIMS data from Thailand, reported that, after holding selection and background factors constant, students in private schools had an unconditional advantage over public school students in mathematics performance. In the Dominican Republic, Luna, Gonzalez, and Yunen (1985a) found that a group of private school students had pretest scores surpassing post-test scores of students in other types of schools. In addition, the achievement gain during the academic year was found to be greater for that small group of students in private schools than for students in other schools. Gender Differences in Mathematics Achievement Gender differences in academic achievement have been extensively studied in the fields of education and psychology. Most of these studies have taken place in the United States, and it has generally been found that girls do better in verbal and linguistic studies than boys. Boys generally show stronger numerical and spatial aptitudes, and perform better in tests of mathematical reasoning (Armstrong, 1981; Benbow & Stanley, 1982, 1983; Maccoby, 1966; Maccoby & Jacklin, 1974). In mathematics achievement, the National Assessment of Educational Progress (NAEP, 1975) reported that from age 13 onwards, boys generally do better than girls. Lockheed, Thorpe, Brooks-Gunn, Casserly, and McAloon (1985) reviewed 31 studies of mathematics performance of children in Grades 4 to 8 and found that results were mixed and drew no consistent picture. Some studies reported differences favoring females; others reported differences favoring males. However, a large body of 35 research reported no significant differences between males and females in these age groups. Male superiority in mathematics achievement cannot be generalized to all areas of mathematics. Marshall (1984) found among sixth graders that girls performed better than boys on computation items and that boys outperformed girls on story problem items. Hanna (1986) found no gender differences in the subtests of arithmetic, algebra, and probability and statistics. She reported differences favoring boys in geometry and measurement among eighth graders in Ontario. The major discrepancies in mathematics performance arise in two general areas. One involves spatial ability and judgment, and the other involves problem solving behavior. Linked to spatial abilities, there are specific topics such as geometry, measurement, and proportionality which appear to present consistent difficulty for girls (Badger, 1981). In problem-solving tasks, it has been found that girls consistently fail to follow a problem through its conclusion (Armstrong, 1981), and they are less alert to implausible solutions (Wood, 1976). Many theories have been advanced to explain why such differences exist, but none of them is wholly satisfactory (Wood, 1976). Perhaps it is not possible to single out one factor as the prime cause. Instead there may be a combination of factors which influence performance to varying degrees. These theories will be discussed briefly to provide an overview of the literature on gender differences in mathematics achievement. Gender differences in mathematics achievement have been linked to spatial abilities. This linkage was established because, as in the case of mathematics achievement, girls and boys show little difference in performance on spatial tasks during childhood, but boys start 36 outperforming girls at age 13-14 and tend to increase this advantage throughout adolescence (Maccoby & Jacklin, 1974). Fennema (1974) explains that most concrete and pictorial representations of arithmetical, geometrical and algebraic ideas rely heavily on spatial abilities. Research studying gender differences in spatial ability has usually found a male advantage. Boys with high spatial abilities tend to perform better than girls in mathematics achievement tests (Tracy, 1987). Nevertheless, women appear to be more able than men to convert their spatial ability into mathematics achievement (Ethington & Wolfe, 1984). Since spatial abilities are considered an innate trait, it has been suggested that gender differences in spatial skills are linked to a genetic sex factor. According to this view, gender differences in spatial abilities would be apparent from childhood. However, they are usually manifested during adolescence. In trying to account for this developmental aspect, some researchers have associated spatial ability differences to changes in the levels of sex hormones during adolescence. Others have suggested that spatial ability differences are due to differential rates of brain lateralization between males and females. No systematically consistent evidence has been found in support of these views, and no recent research has addressed this problem (Badger, 1981). Fennema and Sherman (1977) have argued that, although gender differences in spatial ability may be due to biological factors, they are strongly influenced by environmental influences. Studies have proven that girls' performance in spatial tasks improves with training and practice (Goldstein & Chance, 1965; Sherman, 1980). Tracy (1987) has argued that children's toy-playing habits are gender-typed and may have an effect on the development of spatial abilities and 37 mathematics and science achievement. She argues that girl-typed toys are oriented toward domestic pursuits and that they do not encourage manipulation, construction, or movement through space. These feminine playing habits seem to place females in a disadvantaged position. Nash (1975, cited in Tracy, 1987) found that children with a masculine sex role orientation had higher spatial ability than those with a feminine sex role orientation. Attitude toward mathematics is another sex-related variable found to be related to students' mathematics achievement. It is generally found that females' attitudes towards mathematics are favorable in the lower grades, and by the end of secondary school their attitudes have become less favorable. Another common pattern of female attitudes is reflected in their withdrawal from mathematical activities (Badger, 1981). This have been explained through the stereotyping of sex roles in society. Social theorists have argued that, since the time spent on any task is relative to the importance and value assigned to it, tasks which are considered not useful or sexually appropriate will be ignored in favor of more appropriate or useful tasks (Nash, 1978). One of the most significant factors which distinguishes girls' and boys' attitudes towards mathematics is girls' lower estimation of their own ability. Fennema and Sherman (1978) found that boys have more self-confidence in their abilities to learn mathematics than girls. In addition, there is some indication that there is a correlation between girls' success in mathematics and their refusal to regard mathematics as a male domain (Sherman, 1980; Preece, 1979). In summary, spatial skills seems to be related to mathematics achievement. Boys show higher spatial abilities than girls. It may be the 38 case that girls do not have the opportunity to develop their spatial skills during childhood due to the type of toys they play with, yet these skills improve with training. Consequently, the learning environment, not innate differences, seems to be a strong factor influencing gender differences in mathematics achievement. In addition, it seems that children's socialization with stereotyped sex roles may affect girls' motivation to study and to do well in mathematics. Gender differences in mathematics achievement seem to be strongly related to social and cultural sex role expectations. These expectations are mediated through the type of school organization and the type of community in which the school is located (Keeves, 1973). Some researchers have found that the occurrence of gender differences in mathematics achievement varies from school to school (Badger, 1981; Fennema and Sherman, 1977; Wood, 1976). However, not many investigations have focused on finding the reasons why such variations occur. It might be that there are certain types of schools where males and females are motivated to do well in school subjects regardless of their gender. In the same manner, there are certain schools where school subjects are sex-typed, as noted by Dale (1974, cited in Badger, 1981). He found that students in coeducational schools tended to identify schools subjects according to their sexual appropriateness. Therefore, research studying gender differences within schools might prove fruitful. Accordingly, Bowman and Anderson (1982) have pointed out the need to study differences in boys' and girls' schooling between regions within countries, and among ethnic and socioeconomic sub-populations. Most gender-differences research has focused on white students (Hart & Stanic, 1988), neglecting the possibility of different patterns of 39 mathematics achievement among boys and girls within other racial or socioeconomic groups. However, Yando, Seitz, and Zigler (1979, cited in Hart & Stanic, 1988, p.27) studied the performance among black males and females, and they found that black female students obtained higher achievement levels than black male students. This finding leads one to believe that gender differences in mathematics achievement among children within other sub-populations have been overlooked and that more research is needed in this area. Summary SIMS findings on students' performance are sensitive to differences in the content and emphasis in the curricula of the participating systems. Mathematics achievement has to be interpreted within the educational context in which it occurs. It is reasonable to expect low levels of achievement in places where the content of the test items had not been taught. Similarly, high achievement levels can be expected when the content tested has been emphasized in the curricula. One might anticipate levels of achievement to be high when educational conditions, such as availability of textbooks and teachers' qualifications, are positive. When they are less positive, such as in the developing countries participating in SIMS, levels of achievement are usually low. In the TLMDR, Grade 8 students' mathematics achievement and growth was very poor. When compared to the achievement levels of other systems participating in SIMS, Dominican children were left behind, even by children in other developing countries. These low achievement levels found in Dominican children and the developing countries which participated in SIMS are not rare in third world countries. Academic 40 achievement in developing countries is generally very low when compared to academic standards of developed nations. Taking all this into consideration, it would be no surprise to obtain low levels of mathematics achievement among Dominican Grade 12 students. In the literature dealing with comparisons of private and public school students' academic achievement, it has generally been found that private school students obtain higher achievement levels than their public school counterparts. Private and public schools differences are even more likely to occur in third world countries. Private schools usually have better physical resources and better qualified teachers than public schools. These school resources are known to have greater impact on academic achievement in less industrialized countries. With respect to regional differences in achievement, the literature suggests that richer regions provide teachers and students with better educational resources. Moreover, it is suggested that regional and school resources may jointly exert an effect on students' academic achievement (Theisen, Achola, & Boakari, 1983). The review of the literature on gender differences in mathematics achievement provides evidence of differences in favor of males at the secondary level. However, differences are usually found in specific content areas, especially in those closely related to spatial abilities. The literature also suggests that these differences are possibly related to social and cultural factors. In addition, it is also suggested that gender differences would be smaller, maybe imperceptible, in schools, regions or communities where excellence is promoted equally among males and females. 41 CHAPTER 3: METHODOLOGY The general purpose of this study was to assess mathematics achievement at the end of Grade 12 in the Dominican Republic. In particular, the goals were to explore school type and regional differences in mathematics achievement, as well as gender differences in achievement. The data were obtained from Luna and Gonzalez (1988) secondary school project in the Dominican Republic conducted during the 1988-89 school year. Details of the instrumentation and sampling procedures are available from Luna and Gonzalez (1988). The selection of the sample and the construction of the test were arranged by Dr. Eduardo Luna and Professor Sarah Gonzalez of PUCMM. The mathematics achievement test was administered to students in a sample of intact Grade 11 and Grade 12 classes. Since eleventh and twelfth graders were chosen from the same schools, results from Grade 11 were treated as a pretest. In this chapter, the research hypotheses are stated first, followed by a brief description of the development of the instruments. Then, details of the selection of the sample, test scoring, and data analyses are presented. Research Hypotheses Several research hypotheses guided this investigation. The literature on academic achievement in developing countries indicated that usually achievement in the least industrialized countries is very low. No hypothesis is formally stated for the first research question because it is of an exploratory nature. Only four hypotheses have been formally stated. The review of the literature provides convincing evidence that private schools obtain higher achievement levels than public schools. It also 42 suggests that richer regions of poor countries have an educational advantage over smaller and poorer regions. In addition, it was suggested that these two variables may have a combined effect on achievement. Based on the literature on gender differences in mathematics achievement, it is anticipated that males will perform better than females. These hypotheses can be stated as follows: Hj: Both types of private schools will obtain significantly higher achievement levels than the public schools. H 2 : The region of Santo Domingo will obtain a significantly higher achievement score than any of the other two regions. H 3 : There will be a school type by region interaction effect on mathematics achievement. H 4 : Males will significantly outperform females in the mathematics achievement test. In the literature, there is no evidence that gender differences in achievement occur only in certain types of schools and regions. It is suggested, however, that gender differences are related to social and cultural sex role expectations which are mediated through the type of school organization. In this study it was thought that perhaps gender differences might be negligible in private schools. Private schools are designed to serve students of high socioeconomic status. These students are more likely to have highly educated parents with mothers working outside their homes. Therefore, parents may be less rigid in their sex role differentiation which is then transmitted through the schools. 43 For the last two research questions, there are non-directional hypotheses. Similar to the first question, these last two were designed to explore, first, how much change in achievement has occurred since the TLMDR Grade 8 study; and second, how Dominican performance compares to that of other countries. Development of the Instruments In the process of developing the mathematics achievement test Luna and Gonzalez (1988) first constructed a table of specifications. The table consisted of a two-dimensional matrix, content by cognitive levels, designed to assist in weighting the topics and cognitive objectives. Once the table was established, test items were selected to fill the cells. The Table of Specifications Luna and Gonzalez (1988) reported that two documents were used as references to identify the objectives to be evaluated in the achievement test. The first was the National Program of the Mathematics Curriculum (the Curriculum Guide), and the SIMS working paper Second IEA Mathematics Study: Suggested Tables of Specification for the IEA Math Tests (Wilson & Weinzweig, 1977). In addition, since these documents referred to the prescribed or intended curriculum, they were cross-referenced with the content that teachers identified as having been taught in the secondary grades. Fifty-five school teachers, from Grades 9 to 12 and from different types of schools, were asked to list the topics they had taught during the current year by Luna and Gonzalez (1988). The table of specifications was established using the results from this exercise. It provided a rationale for weighting the topics and cognitive 44 levels within the table, and for selecting the number of items in each cell. Luna and Gonzalez (1988) described these criteria as follows: 1. Since arithmetic objectives are taught and reviewed in the first two years of secondary school, and they are used in the last two years when applying more advanced concepts, it was decided to allocate 20 percent of the items to this area. 2. Allocation of the remaining 80 percent of the test was based on the proportion of time spent on each major topic in Grades 9 to 12. For example, 1.5 years of the four-year secondary program is spent on geometry. Hence, the weighting for geometry was: (1.5/4) x 0.8 = 0.3, or 30 percent of the items. Similarly, a weighting of 30 percent of the items was assigned to algebra, and of 20 percent to trigonometry. 3. The National curriculum guidelines stressed the computation level more than the other cognitive levels. Therefore, the test was heavily weighted toward computation-level items (50 percent). Of the remaining 50 percent, 27 percent was assigned to comprehension-level items, and 23 percent to application. The final test forms are reproduced in Appendix A, and the test translations are provided in Appendix B. The distribution of items by content and cognitive level is presented in Table 2. 45 Table 2 Distribution of Items by Content Area, Sub-area, and Cognitive Level Computation Comprehension Application Total 1. Arithmetic 7 3 5 15 (20%) 1.1 Real Numbers: prop, and operations. 3 2 5 1.2 Complex Numbers 4 1 2. Algebra 10 6 4 20 (30%) 2.1 Polynomials 2 4 1 2.2 Roots and radicals 4 2 3 2.3 Equations and Inequalities 4 3. Geometry 10 6 4 20 (30%) 3.1 Euclidean 6 5 4 3.2 Analytic 4 1 4. Trigonometry 8 4 3 15 (20%) 4.1 Angles and triangles 4 2 1 4.2 Identities 1 2 2 4.3 Logarithms 3 Total 35 (50%) 19 (27%) 16 (23%) 70 (100%) Selection of Test Items Items were selected to meet the criteria identified in the table of specifications. They were drawn from existing sources such as the SIMS, the TLMDR, and the British Columbia Grade 12 Provincial Examinations. In cases where appropriate items were not available from those sources, new ones were developed. Table 3 shows the number of items selected from each source. Two criteria were considered in the selection of items: an acceptable range of p-values (0.4 to 0.7), and a discrimination index greater than 0.25, based on the performance of students in other jurisdictions on those 46 items. This included countries which participated in SIMS and the province of British Columbia in Canada. Table 3 Sources of the Test Items Source Number of Items SIMS (Population B) 10 TLMDR 14 B. C. Grade 12 Provincial Examinations 15 New 16 Other sources 15 Total 70 The items were administered to 94 students from three schools in the Santiago area in a timing pilot. The pilot involved administration of two test forms, each consisting of 35 multiple-choice items. It was found that the average time taken to write Form A was 51.7 minutes, and 39.3 minutes to write Form B. Using the pilot data, the two final test forms were then developed, each consisting of 35 multiple-choice items. These forms were arranged to be as parallel as possible by assigning each a similar weighting of items as identified in the table of specifications, and assigning items with similar ranges of known p-values to each form. Once the items were separated into two groups, they were randomly ordered on each form, except for the first item which was chosen to be relatively easy. 47 Content Validity of the Test Content validity is defined as the process of establishing the adequacy of content sampling (Wiersma, 1986). The mathematics achievement test used in this study is taken to be a valid measure of knowledge of secondary school mathematics in the Dominican Republic because the questions it contains were drawn from an analysis of the prescribed curriculum, cross-referenced with the content teachers identified as having been taught in those grades. Another procedure for determining content validity is for experts to examine the appropriateness of the items to measure the intended trait (Ghiselli, Campbell, & Zedeck, 1981). In the present case, the mathematics teachers of the students who answered the test could be considered as experts who could validly judge the adequacy of the test items, since they should know what mathematics their students have been taught. In the Luna and Gonzalez (1988) secondary school project, the mathematics teachers of the sampled schools/classrooms responded to an opportunity-to-learn (OTL) questionnaire. The OTL questionnaire asked the teachers whether they had taught the objectives measured by each test item included in the students' achievement tests. For each and every item, teachers were asked whether the content and objective evaluated: - (A) was considered pre-requisite and therefore had not been taught or reviewed during the school year {Prerequisite); - (B) was reviewed during the school year [Reviewed); - (C) was taught as new content [Taught as New); - (D) would be taught in subsequent courses [Next courses); or - (E) was not taught [Not taught). 48 The frequency distribution of responses of the Grade 12 mathematics teachers to the OTL questionnaire is provided in Appendix E. Teachers' responses were classified into "Taught" and "Not Taught" categories to obtain a simplified pattern of responses reported in Table 4. The Prerequisite, Reviewed, and Taught as New responses were grouped under the "Taught" category, and the Next Courses and Not taught responses were clustered into the "Not taught" category. The percent of teachers answering that the content evaluated had been taught (OTL%) is reported for each item in Table 4. The OTL percentage was calculated based on the number of teachers who answered the question (omitting the number of teachers who did not respond). Table 4 shows that the range of item OTL percentage was 65 to 100. Only 7 of the 70 items received low OTL ratings (below 80 percent). It is important to note that 3 out of the 5 items in the analytic geometry sub-area, A27, B25, and B31, received low OTL ratings (76, 70, and 65 percent respectively). Therefore, this is to be kept in mind when looking at students' responses to those items. The other 4 low-rated items (A24, A34, B14, and B30) did not form an obvious pattern, except that they were at the application level on various sub-areas. However, the mean OTL percentage for all items was 92 percent. This and the fact that the majority of items received an OTL rating above 80 percent leads to the conclusion that the content areas evaluated in the mathematics achievement tests were appropriate measures of mathematics achievement at the Grade 12 level. Table 4 Distribution of Teachers' Responses and OTL percentages I T E M Taught Not taught O T L % A l 44 0 100 A2 42 1 98 A3 38 4 90 A4 41 1 98 A5 41 2 95 A6 39 4 91 A7 43 1 98 AS 42 2 95 A9 42 1 98 A10 41 3 93 A l l 43 0 100 A12 39 4 91 A13 42 1 98 A14 38 5 88 A15 36 6 86 A16 37 5 88 A17 39 4 91 A18 42 1 98 A19 35 7 83 A20 42 1 98 A21 39 4 91 A22 36 6 86 A23 42 1 98 A24 31 12 72 A25 43 0 100 A26 41 2 95 A27 31 10 76 A28 42 0 100 A29 41 2 95 A30 41 2 95 A31 42 1 98 A32 40 2 95 A33 38 5 88 A34 31 11 74 A35 37 6 86 BI 43 0 100 B2 42 1 98 B3 36 7 84 B4 42 1 98 B5 37 4 90 B6 42 1 98 B7 40 2 95 B8 41 1 98 B9 41 1 98 BIO 40 2 95 B l l 40 2 95 B12 43 0 100 B13 42 0 100 B14 31 11 74 B15 38 4 90 B16 42 0 100 B17 36 3 92 B18 42 0 100 B19 38 3 93 B20 40 1 98 B21 40 0 100 B22 36 5 88 B23 41 0 100 B24 36 5 88 B25 26 14 65 B26 34 6 85 B27 41 0 100 B28 39 2 95 B29 40 0 100 B30 28 10 74 B31 28 12 70 B32 39 1 98 B33 38 2 95 B34 40 1 98 B35 40 1 98 50 Test Administration Each student completed both test forms, answering a total of 70 items. The students were given a recess between the two tests, and they had a maximum of 1 hour and 15 minutes to complete each test. The allotted time permitted the majority of the students to complete the tests. This is supported by the fact that 88 percent of the students responded to all the items in Test A, and 80 percent answered all the items in Test B. The remaining 12 and 20 percent did not respond to the last item in tests A and B respectively. The tests had written instructions on how to respond to the items as well as an example showing how to mark the answers. The test administrator read these instructions aloud after distributing the test to the students. Students were not allowed to use calculators during the tests and they were instructed to work quickly and carefully. They were also encouraged not to waste time on items they considered too difficult. Test Reliability The reliability coefficient of the achievement test was computed using Hoyt's ANOVA procedure which was performed using the computer program LERTAP (Nelson, 1974). Hoyt's procedure uses the internal-consistency method of estimating reliability (Nelson, 1974). It is algebraically equivalent to KR-20 and Cronbach's alpha. The reliability of the mathematics achievement test was calculated using the responses of Grade 12 students to the two test forms. The data files of both tests were combined in a single file. The reliability coefficient for the total set of items, as well as subtest reliabilities, were calculated. 51 Results are presented in Table 5. Table 5 Total Test and Subtest Reliabilities, Means, Standard Deviations, and Standard Errors of Measurement Number of Items M e a n Standard Devia t ion Hoyt Re l iab i l i ty Standard E r r o r Total Test 70 17.9 8.1 0.84 3.3 Subtest 1: A r i t h m e t i c 15 4.3 2.2 0.54 1.5 Subtest 2: J Algebra 20 5.2 2.7 0.59 1.7 Subtest 3: Geometry 20 5.8 3.4 0.71 1.8 Subtest 4: Trigonometry 15 2.5 1.8 0.36 1.4 The total test had a reliability coefficient of 0.84. The reliabilities of the subtests ranged from a low 0.36 to a high 0.71. It is not clear why the subtests obtained such low reliabilities. However, it is known that test reliabilities are influenced by a number of factors including length of the test and item difficulty. In relation to the length of the subtests of the mathematics test, each of the subtests included only a small number of items. This might have affected the subtest reliabilities. In the relation between reliability and item difficulty, only items with middle difficulty levels can have high, positive discriminations, and the more items with high, positive discrimination items in a test, the higher the reliability of the test (Nitko, 1983). In the mathematics achievement test, there were several items, 52 particularly in the subtest of trigonometry, which had high difficulty levels, producing low discrimination indices, which, in turn, affected the reliability of the subtests. Sample Selection Description of the Population The sample used in this study was selected from the population of intact Grade 12 classrooms of urban schools located in the regions of Santo Domingo, Santiago, and the next ten largest cities. The schools were of three types: public, private F-, and private O-schools. All the schools considered in the target population were urban schools because of two reasons. First, all private schools are urban. Second, rural schools usually do not offer all the secondary school grades. In the Dorninican Republic schools operate on three shifts: mornings, afternoons, and evenings. Since the evening classes are usually designated for adults, only students attending morning and afternoon classes were considered as part of the target population. Students attending private and public schools are known to be of different socioeconomic status (SES). Students of low SES mainly attend public schools, whereas students of middle and some low SES attend private O-schools and students of high and middle SES attend private F-schools (Luna, Gonzalez, & Wolfe, 1990). Selection Technique In selecting the sample, Luna and Gonzalez used a stratified random sampling technique. The target population was stratified according to 53 school type and region, resulting in nine cells. It was designed to represent three types of schools (public, private F, and private O), and 3 regions of similar populational size (Santo Domingo, Santiago, and the next 10 largest cities). Both Grade 11 and Grade 12 classes were chosen from the same school, one of each per school. When the schools had more than one class per grade, one classroom was randomly selected. The sampling procedure resulted in the selection of a sample of 49 schools: 1413 students in Grade 11, and 1271 in Grade 12. The strata could not be equally represented since there were errors in the information provided by the Ministry of Education concerning the number of students in Grades 11 and 12 attending the schools of interest. Figures 6 and 7 present the distribution of schools and students in each stratum in the obtained sample. 8 F <8 o Region <$> c& 0 & 15 16 18 15 17 17 49 Grade 12 Region o F J & o 15 16 18 15 17 17 49 Grade 11 SD Santo Domingo Sgo Santiago Other Next 10 largest cities P Public schools F Private Type F O Private Type O Figure 6. Distribution of sampled schools. 54 Region 8 & 64° p 182 161 117 460 F 145 171 139 455 0 73 131 152 356 400 463 408 1271 SD Santo Domingo Sgo Santiago Other Next ten largest Region 194 149 152 495 £ o 154 185 174 513 116 146 143 405 464 480 469 1413 P Public schools F Private Type F O Private Type O Figure 7. Distribution of students. Data Analysis Test Scoring The student responses were entered into a computer file at the Centro de Investigaciones (Research Centre) at PUCMM. The data were then sent to the investigator on diskette with the corresponding code book. The test answer keys were reviewed by the investigator and no errors were found. Correction for Guessing The test scores used for the analyses were based on number right responses. No correction for guessing was applied to the scores because of two reasons. First, according to Garden and Robitaille (1989), there are two testing circumstances in which random guessing is likely to occur on some test items. One is speeded tests in which some students randomly choose an answer to items which they did not have time to complete. The other is when students are instructed to guess the answer to the items 55 they do not know. In the present case, students were not invited to guess their answers, and they had sufficient time to complete both test forms. Second, an examination of the pattern of students' responses to the test items indicated that students did not seem to have blindly guessed their answers. It was noticed that many students left questions unanswered which means that students elected not to answer the questions they did not know instead of guessing them. It was also noticed that many students chose the same wrong answer to many items. This indicated that students were not guessing, they were confident when choosing the wrong answers. Statistical Analyses To examine the question of what mathematical content students know, descriptive statistics were used. Mean percent correct responses were calculated for Grade 11 and Grade 12 students for each of the content areas and sub-areas. Means were also calculated for students in each type of school. For testing differences in achievement among schools and regions, a 3x3 (region-by-school type) analysis of variance was performed with the school mean score being the unit of analysis. Post-hoc comparisons were made using the Tukey B 2 variance approach. A level of significance of 0.05 was chosen for both ANOVA and Tukey B 2 procedures. This a level was used because 0.05 is the most commonly chosen value (Glass & Hopkins, 1984), especially in behavioral sciences. Gender differences in mathematics achievement were examined by first conducting a one-way analysis of variance, using gender as the independent variable. Differences favouring males were found at the 0.05 56 significance level. It was observed that females outnumbered males in the total sample as well as in each cell. This was thought possibly to act as a confounding variable since it might be the case that low ability males drop out of school earlier than females. In order to compensate for this possibility, the ANOVA procedure was then performed on a sample formed by the top 100 ranked males and the top 100 ranked females. The result showed that gender differences still existed between these two groups in favor of males. These differences were investigated further at the school type and regional levels. Analysis of variance was performed for each school and region to examine gender differences within the same sub-populations. Six one-way ANOVAs were performed at the 0.05 level of significance. Gender differences were also investigated at the item level. Chi-square tests were performed on each test item. In this case, the level of significance chosen was 0.01 to prevent making a type II error due to the repetition of the same test 70 times. In addition, Yates "correction for continuity" was not applied to the chi-square values, since Camilli and Hopkins (1977, cited in Glass & Hopkins, 1984) found that Yates correction "is not only unnecessary, but causes the already conservative values for a to be even more conservative" (p. 288). Comparative analyses were made for the items in the mathematics achievement test drawn from the SIMS (Population B) and the TLMDR. The SIMS participating countries' mean percent responses were obtained from Robitaille and Garden (1989), they are included in Appendix C. The TLMDR items are identified in Appendix D. Also included are bar graphs showing the mean percent of correct responses of students in public, F-, and O- schools, as well as the national mean obtained in the TLMDR study. 57 Mean percent correct of Grade 8 students were obtained from Luna, Gonzalez, and Yunen (1984a; 1984b; 1984c; 1985b; 1985c; 1985d). The national mean percent correct was calculated for TLMDR urban schools, since the present study included only this type of schools. For each item in the comparison, the national mean percent correct of students in Grade 12 was estimated from the percent of correct responses obtained by students in the sample. In this manner a better estimate of the mathematics achievement score of the total population of urban Dominican twelfth graders was obtained. The percent of correct responses (p) of students in the sample was weighted by the number of students in the target population. The estimated target population is presented in Figure 8. This information was provided by the Ministry of Education of the Dominican Republic for the 1988-89 school year. The weighted mean percent of correct responses for the Grade 12 population was obtained by using the formula below. 3 3 n i j P i j IX ^Population j^-_ i=l J=l n j In the formula, the proportion of correct responses for each cell is represented by p j j t and the number of students in each cell is nowhere i represents school type cells; j. represents region cells. N is the total number of students in the target population. 58 Region Region « a. u OT 6 0 p 2689 762 4985 8436 F 2429 661 1024 4114 O 4597 572 1769 6938 9715 1995 7778 19488 2. S 6 0 o* 6 p 3385 917 6260 10562 F 2662 744 1121 4527 O 5070 572 2053 7695 11117 2233 9434 22784 Grade 12 Grade 11 Figure 8. Estimated Target Population. The formula can be worked in a comprehensive manner. First the weighted p-values for each type of school were calculated. Then, these values were used to calculate the weighted p-value for the total population. This was done using the general formula and the distribution of students per region and school. The weighted p-value for each school type was calculated using the formulas presented below. PP = PF = P o 2698p„ + 762p12 + 4985p13 8436 2429p n + 661p12 + 1024p13 4144 4597p n + 572p12 + 1769p13 6938 In the formulas, Pp , p F , and p Q represent the weighted mean percent correct for public, F- and O-schools respectively. These values were calculated by weighting the mean percent correct of each type of school in the sample by the number of students in the target population attending that type of school. Using these values, the p-value for the population was then calculated using the formula below. PPopulatlon 8436pP + 4114pF + 6938pQ 19488 59 CHAPTER 4: RESULTS In this chapter, the description of the data, the results of the mathematics achievement tests, and of the statistical analyses are reported. They are presented following the order of the research questions. First, pseudo-growth in mathematics achievement is examined by comparing the scores obtained by students in Grade 11 and Grade 12. Second, the analysis of school type and regional differences in mathematics achievement is reported, followed by a third section containing the analysis of gender differences. Finally, the performance of Grade 12 students is compared, initially, to the performance of Dominican Grade 8 students from the TLMDR; and then, to that of other countries which participated in the SIMS (Population B). Description of the Data The sample used in this study consisted of 1271 students in Grade 12 and 1413 in Grade 11, distributed over 49 urban schools in the Dominican Republic. Schools were of three types: private F-, private O-, and public schools. They were located in the largest twelve cities of the country. Students' scores were used in the descriptive analyses of the data to answer the first research question. Scores from Grade 11 students were used as Grade 12 pre-test scores to estimate change in achievement. School means were used in the analysis of school type and regional differences in achievement for research questions 2, 3, and 4. Males' and females' scores were used in the analysis of gender differences in mathematics achievement for question 5. In addition, males' and females' responses to each of the test items were used in an 60 item-by-gender analysis. Males' and females' scores were then compared within each type of school and each region in the sample for question 6. Finally, students' responses to 24 test items were used for the last two research questions, 7 and 8. Grade 12 students' responses were weighted by the number of students in the population attending each type of school. Responses to 14 of these items were compared to Dominican Grade 8 students' responses obtained in the TLMDR study. Students' performance on 10 items was compared to that of other countries that participated in SIMS. Research Question 1 What mathematics content do Dominican students in Grade 12 know? The test results indicated that Grade 12 students have not learned most of the mathematics content prescribed in the secondary school curriculum. The mean score obtained by Grade 12 students was very low: 17.8 (25 percent). The mean scores obtained by each of the three types of schools and the three regions were also computed and they were similarly low. The highest achievement levels were found in the F-schools (30 percent), and in the region of Santiago (29 percent). These low scores were unexpected because teachers' OTL responses (reported in Chapter 3) indicate that students in Grade 12 had the opportunity to learn the evaluated content. Students' responses to the test items showed that only 4 test items were found to be "easy" with over 60 percent of the students answering correctly. Most of the items had low p-values. Seven items were extremely difficult for the majority of the students with 10 percent or less 61 of the students answering correctly. Tables 6 and 7 present the results for these items. Table 6 Easiest Items Item Topic Cognitive Level p-value O T L % A 2 A r i t h m e t i c A p p l i c a t i o n 78 98 A 2 4 Algebra A p p l i c a t i o n 65 72 B7 Algebra Computation 71 95 B9 A r i t h m e t i c A p p l i c a t i o n 67 98 Table 7 Most Difficult Items Item Topic Cognitive Level p-value O T L % A 1 7 Trigonometry Comprehension 6 91 A 3 2 Algebra Computation 9 95 B5 A r i t h m e t i c Comprehension 2 90 B6 Algebra Computation 6 98 B14 Algebra A p p l i c a t i o n 8 74 B24 A r i t h m e t i c Computation 4 88 B26 Algebra A p p l i c a t i o n 8 85 Table 6 shows that the percent of correct responses and teachers' rating of the four items are both as might be expected. However, Table 7 shows that a high percentage of teachers rated these items as being taught, yet the students found them extremely difficult. 62 The most difficult item was B5 with only 2 percent of the students selecting the correct answer. This item asked: If the inverse of a is \ , then the inverse of 10+5i is: a Percent selecting choice (2) (4) (76) (10) This item evaluated knowledge of number operation in the system of complex numbers. The percent of students selecting each choice is presented in parenthesis. It is shown that 76 percent of the students selected choice C which shows that students were misled by the irrelevant information given in the item. Item B24 was also very difficult for the students: only 4 percent answered it correctly. This item asked: Simplify (2+i)(3-i) Percent selecting choice A 5 (17) B 5+1 (19) C 6 (12) D 6 - i 2 (39) * E 7+1 (4) Item B24 also dealt with operations with complex numbers. The most frequently selected answer was choice D which indicates that * A B c D _2_ J_ 25 " 25 1 15 1 5 1 1 1 1 0 + 5 1 10 5 1 63 students mistakenly recognized a factorization exercise from the case of factorization (a+b)(a-b)= a 2 - b 2 . These two examples indicate that students did not learn how to perform operations with complex numbers even if they were taught to do so. The other three items in the subtest of complex numbers were examined. In Appendix F (Table F3), it is shown that in Items A3, A15, and A19 students p-values are also very low: 26, 11, and 17 percent respectively. These results show that students did not learn the content even though the majority of teachers agreed that the content had been taught. When examining students' responses to the items it was clear that they did not know most of the mathematical content evaluated in the achievement tests. It is possible that the items were rated incorrectly by teachers. Perhaps teachers rated the items by glancing at the topic and not taking into account the cognitive complexity of the items. Results from this study show that teachers' OTL ratings are not a reliable measure to predict students' achievement in the Dominican Republic. Scores from Grade 11 students were used as twelfth graders' pre-test scores to estimate how much mathematics students have learned during a year in Grade 12. This was done keeping in mind that inferences and conclusions have to be made with caution because there are many variables (e.g. family background, and students' abilities) acting differently on each group, and these variables were not controlled in this study. Table 8 presents the mean scores, standard deviation, minimum score (min), and maximum score (max) obtained by students in Grade 11 and Grade 12. Results are also reported for each type of school and region. Students in Grade 11 obtained a mean score of 15.4 (22 percent). 64 In Grade 12, students' mean score was 17.8 (25 percent). Both scores are disappointingly low with a difference of 3 percentage points. This difference indicates that students in Grade 11 know almost as much mathematics as students in Grade 12. In the Grade 11 sample, standard deviations are smaller than in the Grade 12 sample. It seems that Grade 11 students are more homogeneous than twelfth graders. This is also noticeable in the range of the scores: from 0 to 47 in Grade 11 and from 0 to 64 in Grade 12. However, it is worth noting that the pattern of the results (means and standard deviations) is similar for both grades. For example, F-schools obtained the highest mean score in both grades. Similarly, O-schools had the lowest mean scores. Within the regions, Santiago obtained the highest mean scores; the other two regions scored similarly low in both grades. This similarity suggests that Grade 11 and Grade 12 samples might be comparable. A comparison between Grade 11 and Grade 12 schools and regions is presented in Figure 9. The graph shows that differences in achievement, between Grade 11 and Grade 12 students, are minimal even between the same types of schools and regions. However, it can be seen that Grade 11 students attending private F-schools obtained higher achievement scores (M = 18.1) than Grade 12 students attending public (M = 16.3) and private O-schools (M = 15.8). Table 8 Test Results for Grade 11 and Grade 12 Students n Mean S.D. a min max Grade 11 1413 15.4 6.35 0 47 Public Sch 495 14.4 5.47 2 34 F-Schools 513 18.1 6.54 2 42 O-Schools 405 13.2 5.89 0 47 Santo Dgo. 464 14.4 6.68 0 42 Santiago 480 17.7 6.16 0 47 Othersb 469 14.1 5.55 2 33 Grade 12 1271 17.8 8.12 0 64 Public Sch. 460 16.3 6.72 0 43 F-Schools 455 21.0 9.25 0 64 O-Schools 356 15.8 6.93 0 44 Santo Dgo. 400 17.3 9.43 0 61 Santiago 463 20.3 7.31 4 64 Othersb 419 15.7 6.81 0 41 Note. Maximum possible score= 70. a S.D. = standard deviation. Others = Next ten largest cities. Figure 9. Comparison between Grades 11 and 12. 66 Mean scores for each of the sub-areas were calculated and are reported in Table 9. It was decided to further examine these scores to find out whether the levels of achievement were as low in the subtests as they were in the total test. Table 9 shows that there are three sub-areas with achievement levels over 30 percent in Grade 12: real numbers, polynomials, and Euclidean geometry. The table also presents the mean percent obtained by Grade 11 and Grade 12 students in each sub-area. Differences in achievement between the two grades is very small in all subtests. The lowest difference in achievement is found in trigonometry. The greatest difference is found in algebra, specifically in the sub-area of polynomials. Table 9 Subtest Mean Percent Correct Scores for Grade 11 and Grade 12 Subtest # items Grade 11 Grade 12 1. Arithmetic: 15 25 29 Real Numbers 10 34 37 Complex Numbers 5 7 12 2. Algebra: 20 21 26 Polynomials 6 28 35 Roots & Radicals 7 13 18 Equations and 7 23 26 Inequalities 3. Geometry: 20 26 29 Euclidean 15 29 33 Analytic 5 17 19 4. Trigonometry: 15 16 17 Angles & triangles 5 20 21 Identities 5 13 13 Logarithms 5 15 15 6 7 Students' subtest scores were calculated for public, private F-, and private O-schools, and they are reported in Table 10. The goal was to avoid an over-generalization of the results, and find out whether a particular type of school would obtain higher achievement levels on any particular subtest. This was the case in the F-schools which obtained achievement levels over 40 percent in three subtests: real numbers, polynomials, and Euclidean geometry. It is important to notice that O- and public schools had their highest subtest scores on those same subtests, but they were lower than those of the F-schools. Considering that the difference in the total test score between Grade 11 and Grade 12 students was only 3 percentage points, a difference of 5 percent or more will be considered of educational importance. The 5 percent or more difference in achievement between the two grades in private F-schools is found in four sub-areas: polynomials (10 points), complex numbers (7 points), euclidean geometry (6 points), and roots and radicals (5 points). In private O-schools, there are differences in three sub-areas: polynomials (7 points), angles and triangles (6 points), and roots and radicals (5 points). In public schools, there are differences in only two subtests: polynomials (7 points), and roots and radicals (5 points). These findings suggest that, although students may have been taught the same mathematics content, students attending F-schools appear to have learned more than students in other schools. 68 Table 10 Subtest Mean Percent Correct Scores by School Type in Grades 11 and 12 Public Schools F-Schools O-Schools Subtest G n Gi2 G i l Gl2 G i l Gi2 Real Numbers 31 34 39 42 30 34 Complex Numbers 8 11 7 14 7 11 Polynomials 28 33 31 41 25 32 Roots & Radicals 12 17 16 21 11 16 Equations and Ineq. 21 23 24 28 19 23 Euclidean Geometry 26 28 35 41 23 27 Analytic Geometry 16 17 20 23 14 16 Angles & Triangles 19 21 22 23 13 19 Identities 11 13 15 14 11 11 Logarithms 12 12 21 20 11 14 Research Questions 2, 3, & 4 Do students from private schools score higher than their public school counterparts? Do students in schools located in privileged regions score higher than students attending schools in less privileged regions? Is there an interaction between school type and region which results in greater mathematics achievement for a particular combination? To answer these questions, a two-way analysis of variance (ANOVA) was performed with school type and region as the independent variables. The dependent variable was mathematics achievement as exhibited on the mathematics achievement tests. It is important to note that the school mean, not student's score, was the unit of analysis used in the ANOVA computations. The school was the unit of analysis because intact 69 classrooms, not students, were randomly selected for inclusion in the sample. In this manner, the ANOVA assumptions (normality, independence, and homogeneity of variance of the error term) which are observed in random samples, are better ensured. It has to be mentioned that using the school means as the unit of analysis has its drawbacks. The fact that less subjects are included in the sample reduces the power of the statistical tests, that is the probability of rejecting a false hypothesis (type II error). However, this in turn reassures the significance of whatever findings, if any, are found. It can be ensured that any significant finding uncovered by the statistical tests really exists. The ANOVA test was used to determine whether significant differences in mathematics achievement existed among the three types of schools, and among the three regions in the sample. It was also designed to find out whether or not there was a school-by-region interaction effect on mathematics achievement. The omnibus F for the interaction effect (see Table 11) was not significant, contradicting the hypothesized school-by-region effect on mathematics achievement. It was concluded that there is not a statistically significant interaction effect between school type and region on mathematics achievement. The omnibus F obtained in the ANOVA statistics (see Table 11) indicated that there was a significant region effect, and a significant school type effect on mathematics achievement. A multiple comparison approach was then carried out to learn which means differed significantly from which other means. Tukey tests were then performed on each independent variable for the comparison of means. 70 Table 11 Analysis of Variance Testing Main and Interaction Effects Source of Sum of Degrees of Mean variation squares freedom square F Total Explained 612.467 8 76.558 3.74 * Main Effects 532.215 4 133.054 6.51 * A (Region) 208.687 2 104.344 5.10* B (School type) 353.898 2 176.949 8.65 * AB 80.252 4 20.063 0.98 Error terms (Res) 818.091 40 20.452 Totals withln-group 1430.558 48 29.803 Note. Significant F's (a = 0.05) needed for significance are: F(2,40) = 3.23, and F(4,40) = 2.61. Mean scores for each of the 49 schools are reported in Appendix F. *p<0.05. The Tukey B 2 test for multiple comparison was used to test the first hypothesis posed in Chapter 3: Hi: Both types of private schools will obtain significantly higher achievement levels than the public schools. The mean scores of the public, private F-schools, and private O-schools were tested against each other to find out which pairs of means were significantly different. It was found that the F-schools obtained a significantly higher achievement level than the public schools. However, an unexpected finding arose; the mean of the private O-schools was not significantly different from that of the public schools. More surprising was 71 that the F-schools obtained a significantly higher achievement level than the O-schools. In Table 12, the schools' means, and the F statistics for the Tukey tests are reported. Table 12 Comparison of School Type Means School Type Mean Comparison F F 21.1 FvsP 8.77* Public (P) 16.3 FvsO 7.06* O 15.2 PvsO 0.46 Note. Tukey B 2 needed for significance at a = 0.05 is TB 2 (3,40) = 5.9. *p<0.05. The ANOVA statistics reported on Table 11 suggested that the region variable was a significant factor acting on mathematics achievement. The Tukey B 2 test was used to test the second research hypothesis: H 2 : The region of Santo Domingo will obtain a significantly higher achievement score than any of the other two regions. Results of the Tukey test, reported in Table 13, contradicted this research hypothesis. None of the comparisons were significant for the Grade 12 sample. It was therefore concluded that there were no significant differences among the sampled regions on their levels of mathematics achievement. However, this has to be taken with caution because, as noted earlier, the power of the test was reduced by using 72 school means as the unit of analysis. Table 13 Comparison of Region Means Region M e a n Compar ison F Santiago (Sgo) 20.0 Sgo vs SD 4.83 Santo Domingo (SD) 16.5 Sgo vs Other 3.89 Other cities (Other) 15.7 SD vs Other 0.25 Note. Tukey B 2 needed for significance at a = 0.05 is 7 B 2 (3.40) = 5.9. The statistical analyses were also performed on the Grade 11 sample to find out whether similar results would be obtained, and these are reported in Appendix G. They are similar to the results reported for Grade 12, with one exception: the comparison of region means. It was found that schools located in the region of Santiago (M = 17.0) performed significantly higher than the schools from Santo Domingo (M = 14.1), TB2 (3,40) = 6.93, p < 0.05. This finding also contradicts the research hypothesis, but it leads to a different conclusion: students in the region of Santiago performed significantly better than those in the region of Santo Domingo. The hypothesis that the region of Santo Domingo would outperform the other two regions in the mathematics achievement test was rejected. In Grade 11 the region of Santiago significantly outperformed the region of Santo Domingo. In Grade 12 that particular comparison yielded high but not significant F values for the Tukey test at the 0.05 level of significance. It is possible that these differences exist, but the test was not powerful 73 enough to uncover them. Added to that is the fact that the Tukey test is very conservative which means that very high F-values are needed for significance. Research Question 5 Do males outperform females in mathematics? This question was examined using several approaches. First, a one-way analysis of variance (ANOVA) was performed, with gender as the independent variable and mathematics achievement as the dependent variable. Then, gender differences were investigated at the item level by performing chi-square tests on each item. The omnibus F obtained in the ANOVA, reported in Table 14, indicated that males (M = 19.8) significantly outperformed females (M = 16.7) in the mathematics achievement test. It was found that females outnumbered males, and this might have been a confounding variable. For example, it might be the case that low-ability males drop out of school sooner than low-ability females. To compensate for this possibility, the ANOVA statistics were then performed on a sub-sample of 200 students; the top 100 ranked males and females. Results show that gender differences still existed between these two groups in favor of males. Table 15 shows that males (M = 33.2) had significantly better mathematics achievement scores than females (M = 29.8), F (1, 198) = 12.43, p < 0.05. This finding supports the fourth research hypothesis and leads to the conclusion that males obtained significantly higher mathematics achievement scores than females. In addition, similar results were obtained when the ANOVAs were performed on the Grade 11 sample. Those results are reported in Appendix I. 74 Table 14 Analysis of Variance of Gender Differences Source of „ c Degrees of S u m of squares _ , r , Mean square F v a r i a t i o n freedom Between 2 877.474 1 2 877.474 45.53 * W i t h i n 79 942.564 1265 63.196 T o t a l 82 820.038 1266 65.419 Note. 1271 cases were processed, 4 cases were missing. Males' and females' mean scores are presented i n Appendix H . *p<0.05. Table 15 ANOVA of Differences between Top 100 Ranked Males and Females Source of o r Degrees of S u m of squares „ . , , , , Mean square F v a r i a t i o n freedom ^ Between 584.820 1 584.820 12.43 * W i t h i n 9 319.160 198 47.066 T o t a l 9 903.980 199 49.769 Note. Males' and females' mean scores are listed i n Appendix H . *p<0.05. Male and female differences in achievement were investigated further by comparing their performance on each test item. This was done to obtain more information about which specific topics these gender differences occurred. Chi-square tests ( x2) were performed on each test item at a 0.01 level of significance. Results show that all test items showing significant gender differences were in favor of males. They are 75 reported in Table 16, and summaries of these results are presented in Tables 17 and 18. Table 16 Test Items Showing Significant Gender Differences Cognitive M a l e Female Item C o n t e n t 3 l e v e l " (p-value) (p-value) D C x2 A4 Art 1 24 14 10 21.98 A8 Geo 2 36 25 11 18.8l] A l l Geo 1 47 36 11 15.29* A13 Geo 3 53 43 13 21.75* A18 Geo 1 49 40 9 10.10] A23 Geo 2 49 37 12 18.84* A29 i Tri 1 24 17 7 9.76 A33 Art 3 38 19 9 56.5o] A35 Geo 3 41 30 11 15.69* B4 Ari 3 42 27 15 31.56] B8 Geo 1 33 19 14 31.78] B9 Ari 3 73 64 9 12.59* B12 Geo 3 46 31 15 27.10* B15 Geo 3 38 26 12 20.61* B18 Alg 2 31 23 8 9.03* B19 Geo 2 34 20 14 33.08 B23 Alg 1 32 23 9 12.0l] B27 Geo 1 50 35 15 26.6l] B33 Geo 2 36 25 11 19.86* a Content Art = Arithmetic Alg = Algebra Geo = Geometry Tri= Trigonometry. D Cognitive level 1 = Computation 2 = Comprehension 3 = Application. c D= difference. *p<0.01. Table 17 Summary of Results by Gender and Content Area Content Area # of items # of non signif icant items # items i n favor of Males(%) # items i n favor of Females(%) A r i t h m e t i c 15 11 4(27) 0 Algebra 20 18 2(10) 0 Geometry 20 8 12(60) 0 Trigonometry 15 14 1(7) 0 Total Test 70 51 19 (27) 0 76 Table 18 Summary of Results by Gender and Cognitive Level # of non signif icant items Cognitive Level # of items # items i n favor of Males(%) # items i n favor of Females(%) Computat ion Comprehension A p p l i c a t i o n Total Test 35 19 16 70 18 14 9 51 7 (20) 5(26) 7(44) 19(27) 0 0 0 0 Results showed that males outperformed females on 19 of the 70 items (p < 0.01). However, no significant gender differences were found on 51 items, 73 percent of the test. Table 17 shows that males scored significantly higher on 12 out of the 20 geometry items. That is 60 percent of the geometry items and 17 percent of the total test. On the other subtests, there were only a few items showing significant gender differences. Table 18 shows that males outperformed females on 7 of the 16 application items. According to these findings the most marked gender differences appear to lie in the content area of geometry and in the cognitive level of application. It is important to note that this same pattern was found in the Grade 11 results (see Appendix I). 77 Research Question 6 Do males outperform females independently of school type and region in which the students are found? To answer this question, gender differences were investigated within each of the schools and regions in the sample. It was thought that there might be a type of school in which males and females did not differ significantly in their levels of mathematics achievement. The literature discussed in Chapter 2 suggested the possibility of finding schools in a particular region in which gender differences were not significant. To test this research hypothesis, a one-way analysis of variance was performed with gender as the dependent variable and mathematics achievement as the independent variable. The ANOVA was performed on each school and on each region in the sample. The F-statistics obtained in each of the ANOVAs were significant (p < 0.05). Results are reported in Appendix J . These results indicated that there are significant gender differences in mathematics achievement independent of the school type and of the region in which the students are found. Research Question 7 How does the mathematics achievement of Grade 12 students compare with that of Grade 8 students in TLMDR? The achievement test included 14 items drawn from the TLMDR. These items dealt with the content areas of arithmetic and geometry. The cognitive level and the topic evaluated by each of these items are identified in Table 19. 78 Table 19 Identification of TLMDR Items Grade 12 item T L M D R item Cognitive level Topic A l 0-30 1 A r i : decimals A 2 2-23 3 A r i : Natural A 4 4-27 1 A r i : power of 10 A 3 3 1-11 3 A r i : percentage B I 0-31 1 A r i : fractions B4 3-31 3 A r i : fractions B9 0-35 3 A r i : proportions A 1 3 0-20 3 Geo: similar triangles A 3 0 0-1 1 Geo: volume A 3 5 0-6 3 Geo: area B12 2-4 3 Geo: angles B15 1-32 3 Geo: similar triangles B19 2-24 2 Geo: measurement B35 4-14 1 Geo: volume Note. A r i = Arithmetic and Geo= Geometry. The percent of correct responses obtained by students in Grade 8 and Grade 12 on these items are presented in Table 20. The responses of Grade 8 and Grade 12 students attending F-, O-, and public schools are also included. Grade 12 students' responses were weighted by the number of students in the population attending each type of school in the sample, and these were derived using the formulas listed in Chapter 3. They are reported in Appendix K. Bar graphs comparing Grade 8 and Grade 12 performances are also reported in Appendix K. 79 Table 20 Grade 8 and Grade 12 Percent Correct on Common Items Public Schools F-Schools O-Schools National Item G8 G12 D a G8 G12 D a G8 G12 D a G8 G 12 D a Arithmetic A l 22 42 20 46 50 4 29 35 6 25 41 16 A2 37 68 31 66 84 18 37 74 37 39 73 34 A4 12 13 1 14 30 16 13 12 -1 13 16 3 A33 47 18 -29 59 34 •25 46 14 •32 48 20 -28 Bl 12 30 18 42 60 18 28 34 6 16 38 22 B4 24 24 0 33 38 5 24 26 2 24 28 4 B9 41 61 20 56 79 23 42 54 12 42 62 20 Geometry A13 31 41 10 48 59 11 32 37 5 33 44 11 A30 19 36 17 27 52 25 27 35 8 21 39 18 A3 5 11 19 8 33 46 13 8 27 19 12 28 16 B12 13 36 23 18 47 29 11 21 10 13 33 20 B15 24 23 -1 32 46 14 21 18 3 24 26 2 B19 8 19 11 15 39 24 7 11 4 8 21 13 B35 13 17 4 15 19 4 10 14 4 13 16 3 a D= difference (G 12 - G 8) In the set of arithmetic items, it is striking to note that on item A33 Grade 8 students scored higher than Grade 12 students. However, there is a slight difference between the TLMDR item and item A33 given to Grade 12 students which may have caused this difference. The original TLMDR item read as follows: 80 Two stores have a television sale. One of them offers a 10% discount and the other one offers a 15% discount. What is the difference in price between the two stores for a regularly priced television of $100? A $5 B $10 C $15 D $85 E $90 Item A33 (see Appendix B) is identical to the above item except that it asked for the difference in price between the two stores for a television which costs $200. Although these two problems have to be worked out in exactly the same way, they derive different answers. The answer to the TLMDR item is choice A ($5) and the answer to item A33 is choice B ($10). Notice that to obtain the correct answer to the TLMDR item, students might have worked the problem incorrectly (subtracting 10% from 15%) and yet obtain the correct answer. When looking at twelfth graders' answers to this item, it was noticed that 44 percent of Grade 12 students chose distractor A (See Table F3 in Appendix F). Therefore, it seems that Grade 12 students tried to solve this problem using an incorrect procedure, and not by subtracting the 10% and 15% of $200 ($30-$20). It is difficult to assess how much change in achievement has occurred in this particular item. It is impossible to tell whether Grade 8 students used the correct procedure and obtained the correct answer or whether they used the incorrect procedure and still got the correct answer. Table 20 shows that, from the set of arithmetic items, A4 and B4 produced the smallest differences in achievement at the national level. At 81 the school level, Item A4 (power of ten) produced almost no difference in achievement between the two grades for public and O-schools. However, F-schools obtained a difference in achievement of 16 percent. On Item B4 (operations with common fractions), differences between the grades were very low: 0 to 5 percent for all schools. On Item A2 (operations with natural numbers), the difference in achievement between the two grades was high in all the schools. On Item B9 (proportion), differences between the two grades were 23 and 20 percent in F- and public schools respectively, and 12 percent in O-schools. On Item B l (common fractions), public and F-schools reached an 18 percent difference but O-schools had a difference of only 6 percent. On Item A l (operation with decimals), F- and O-schools obtained a low difference. Nonetheless, in the case of F-schools, 50 percent of Grade 12 students answered this item correctly, but only 35 percent of O-school students answered correctly. For the set of geometry, Items B15 (similar triangles) and B35 (volume of a prism) produced the smallest differences at the national level: 2 and 3 percent respectively. However, F-schools obtained a 14 percent difference on Item B15. On B35, differences were small for every school. Item B12 (angles) produced the largest difference between the two grades on the set of geometry items. Differences were large in all schools: 29 percent in F-schools, 23 percent in public schools, but only 10 percent in O-schools. Overall, differences in achievement between Grade 8 and Grade 12 students were over 10 percent for 9 out of 13 items (excluding item A33). Only two items showed little change in achievement from Grade 8 to Grade 12 in all schools. These items were B4 testing arithmetic, common fractions; and B35 testing geometry, volume of a prism. In the O-schools, 82 differences in achievement between the grades were small with only few exceptions. Public schools had small differences between Grade 12 and Grade 8 students on 5 items. F-schools had small differences on 3 items. The differences between F-, O-, and public schools found in TLMDR and among Grade 12 schools, previously reported in this study, are also evident when Grade 8 and Grade 12 schools are compared. Students in Grade 8 F-schools had similar or higher p-values than Grade 12 students in O-, and public schools on almost every item. This is similar to the earlier findings when Grade 11 and Grade 12 students' performances were compared. Students in F-schools seemed to have an advantage over the other schools even before classes started. However, some of the bar graphs in Appendix K show that the differences between the schools are smaller in Grade 12 than in Grade 8. The number of items used to compare Grade 8 and Grade 12 performances are too few to be conclusive. The fact that a lot of dropping out takes place between these two grades does not make the comparison any easier. With this in mind, the findings obtained in this analysis still can be useful in providing a general idea of what is happening in the Dominican Republic's educational system. Question 8 How different is the performance of Grade 12 Dominican students from that of students from other countries In SIMS (Population B)? Ten items from the Second International Mathematics Study (SIMS), Population B, were included in the achievement test administered to Dominican twelfth graders. These items were reviewed in Chapter 2 and are identified in Table 21. The percentage of correct responses obtained 83 by students from the SIMS participating systems are compared to the responses of Dominican students in Grade 12. The p-values obtained at the National level and at the school level (public, F- and O-schools) are compared to that of SIMS systems. These comparisons are presented in Appendix L. In addition, item p-values as well as their weighted p-values are provided for each school in Appendix L. On all of these items, Dominican students performed very poorly in comparison with those from other countries. Only for the two arithmetic items was the Dominican Republic's performance comparable to that of some other countries. However, the performance of all the educational systems in SIMS, with the exception of Japan and Hong Kong, was very poor on these two items. On the algebra items, all Dominican students, even those from private F-schools, were left far behind by students from other countries. The same pattern was found on the two trigonometry items. It is not surprising that Dominican students are far behind students from developed countries. However Thailand, another developing country, also obtained higher achievement levels than the Dominican Republic did on these items. These findings are similar to those reported by Luna and Gonzalez (1984) that achievement levels of Dominican students in Grade 8 were far behind SIMS (Population A) systems, even those from developing countries. 84 Table 21 SIMS Items: Topics and Cognitive Levels Identified Grade 12 Item number SIMS (Pop B) Item number Cognitive level Topic B2 078 2 A r i : multiples B3 003 3 A r i : properties of operations A 3 2 080 1 A l g : rationalization A 3 4 038 3 Alg : equations B6 004 1 Alg: exponents B14 081 3 Alg : radicals B26 097 3 Alg : equations B28 087 2 Alg : logarithms A 1 6 012 3 Trig: identities B22 068 1 Trig: angles and triangles Note. A r i = Arithmetic; Alg = Algebra; and Trig = Trigonometry. 85 CHAPTER 5: SUMMARY AND DISCUSSION The general goal of this study was to assess mathematics achievement at the Grade 12 level in the Dominican Republic. Five specific areas were examined, all concerning content which students had been taught in Grade 12; school type and regional differences in mathematics achievement; gender differences in mathematics achievement; comparison of Grade 12 performance with that of Grade 8 students in TLMDR; and finally a comparison of Grade 12 performance with that of students from other countries which participated in SIMS (Population B). First, test results were used to assess how much mathematics Grade 12 students know. Grade 11 data were used as a surrogate for pre-test scores to estimate how much mathematics students had learned in Grade 12. Second, the effect of school type and region on mathematics achievement were examined. This was followed by an analysis of gender differences in achievement. In addition, an item-by-gender analysis was performed to localize the domains in which these differences were found. Gender differences were also examined within each type of school and each region to investigate whether differences between males and females were independent of the school and region in which students were found. Finally, twelfth graders' performance was compared with that of Grade 8 students in TLMDR, and with that of other countries which participated in SIMS. In the following paragraphs, a summary of the findings and the conclusions based on those findings are presented. Implications of the results are then discussed and finally, suggestions for future research are provided. 86 Summary of Findings Mathematics Content Students Know Students in Grade 12 scored very poorly on the achievement test. This was surprising because the content tested was selected from the prescribed curriculum, and the items were rated as having been taught by the teachers. The reasons why this happened are not clear, but it is possible that teachers judged the test items only by looking at the content and did not take into consideration their complexity. Another finding was that Grade 11 students obtained mathematics achievement levels similar to those of Grade 12 students. Also, students in Grade 11 attending F-schools scored higher than students in Grade 12 attending O- and public schools. This suggests that students entering Grade 12 in F-schools know as much, or even more, mathematics than students finishing Grade 12 in a public or a private O-school. Students obtained the highest achievement scores in three subtests: real numbers, polynomials and Euclidean geometry. At the school level, these were the sub-areas in which students had their highest achievement scores, with F-schools outperforming O- and public schools. This finding suggests that although students seem to have been taught the same mathematics content, there are some other variables positively influencing the mathematics achievement of students attending F-schools. Differences in achievement between Grade 11 and Grade 12 students were very small in all subtests; the smallest differences were in the trigonometry subtests, and the largest in the polynomials subtest. At the school level, the sub-area of polynomials produced the highest achievement scores in every school. This may be due to the emphasis that 87 is placed on Algebra topics in the mathematics curriculum of Grade 12. F-schools also obtained a greater difference in achievement between the two grades in this subtest than the other two types of schools. School Type and Regional Differences in Mathematics Achievement In the analysis of school type and regional differences in achievement, it was found that school type and region were significant variables. However, the two variables seem to act independently of each other since no interaction effect was found. The comparison of the schools' means provided interesting insights. Only one type of private school scored significantly higher than public schools; the private F-schools. The F-schools also obtained significantly higher achievement levels than the other type of private school; O-schools. Unexpectedly, public schools and private O-schools were not significantly different in their levels of mathematics achievement. Perhaps the similarity of private O- and public schools is caused by the selection effect of the educational system on the public schools. The fact that many students drop out of the educational system prior to Grade 12, and that the highest drop-out rates are found in public schools, may have helped in selecting the most able students in the public schools. These students might also be of a higher socioeconomic status than the ones who dropped out, and this variable is known to positively affect academic achievement. These two factors may have caused public and O-school differences in mathematics achievement to disappear at the Grade 12 level. The analysis of differences among regions using the Grade 12 data showed that the means were not significantly different from each other. 88 This contradicted the hypothesis that the region of Santo Domingo would outperform the other two regions in the mathematics achievement test. When the regions of Santo Domingo and Santiago were compared using the Grade 11 sample, Santiago significantly outperformed Santo Domingo (p < 0.05). In Grade 12 that comparison yielded high but not significant F values for the Tukey test when a = 0.05. It is not clear why the Tukey test for the comparison of regions was significant for the Grade 11 sample and not for the Grade 12 sample. It is possible that there are real differences between Santiago and the other regions but they failed to show in the Grade 12 sample. It was mentioned before that the power of the test was hampered by choosing school means as the unit of analysis. In addition, Tukey tests are known to be very conservative and require a large F-value to be significant. These two factors might have masked real differences. Gender Differences in Mathematics Achievement The analysis of gender differences in mathematics achievement showed that males performed significantly better than females (p < 0.05). This was also supported by the finding that the 100 top ranked males significantly outperformed the 100 top ranked females in the sample (p < 0.05). When comparing males' and females' responses to the test items, no significant differences were found for 51 out of the 70 test items (73 percent of the test). There were 19 items which showed significant gender differences, all favoring males (p < 0.01). Twelve of those items dealt with geometry, and seven were at the application level. This led to the conclusion that the most marked gender differences occur in 89 geometry and at the cognitive level of application. Differences found in geometry may be linked to differences in males' and females' spatial visualization abilities. Differences at the cognitive level of application may be related to males' and females' differences in problem-solving abilities (Armstrong, 1981; Wood, 1976). Gender differences were investigated at the region and school level to find out whether these differences existed independently of the school and the region in which the students were found. In the review of the literature, it was suggested that there might be certain sub-populations in which gender differences would be negligible. In the case of the present study, significant gender differences in favor of males were found in each of the regions in the sample and in each of the three types of schools. It was concluded that males outperformed females independent of the school type and of the region. However, it is worth noting (see Appendix J , Table Jl ) that females in F-schools (M=19.2) scored 2 points higher than males in public (M=17.3) and O-schools (M=17.5). The literature review suggested that gender differences may be affected by social and cultural factors. Therefore, differences in favor of males were not a big surprise because the socialization of boys and girls with rigid sex-role differentiations is wide spread in the Dominican culture. Theisen, Achola, and Boakari (1983) stated that, in the least industrialized countries, the fact that women are tacitly relegated to subordinate roles in society might be expected to influence girls' socialization as intellectually inferior of boys. This view was supported in the present study. The finding that females in certain schools scored higher than males 90 in other types of schools indicates that females have the ability to perform as well as males in mathematics. But when they were compared to males in the same schools they had lower scores. This also indicate that there might be a socialization effect influencing females' mathematics achievement. Comparison of Dominican Grade 12 Performance with that of Grade 8 in TLMDR Comparison between Dominican Grade 12 and Grade 8 students' achievement on 14 items revealed that students performance improved. The proportion of correct responses changed by at least 10 percentage points between the two grades on 9 out of 13 items. However, four items showed little change in achievement. This is an indication that students still have not learned some of the mathematical content taught in Grade 8. Another finding was that the gain in achievement over four years seems to be greater in private F- and public schools than in private O-schools. On almost every item, students in Grade 8 F-schools had similar or even higher p-values than that of Grade 12 students in O-, or public schools. This is an indication that differences among the schools build up over the years and not in one or two years. The F-school advantage might be caused by the type of educational experiences students encounter in that type of school organization. However, the private school advantage over the other schools seems to decrease with the grades. It was found that the differences between the schools are smaller in Grade 12 than in Grade 8. 91 Comparison of Dominican Grade 12 Performance with that of Other Countries in SIMS (Population B) Dominican performance on the 10 SIMS items was very poor. Dominican students were far behind when compared to other SIMS participating systems. Even the performance of F-schools, although higher than the performance of the other two types of schools, did not approach the level of the other countries. The performance of Dominican students were higher than some of the other systems on only two items. These items evaluated arithmetic content: least common multiple (B2), and properties of operations (B3). However, the performance of most of the SIMS systems, with the exception of Hong Kong and Japan, was poor on those items. Conclusions Finishing Grade 12 in the Dominican Republic is a privilege of very few students. In fact, only 3 percent of the students who enter primary first grade finish high school. These students constitute a privileged minority with schooling (Luna, Gonzalez, & Wolfe, 1990). On the other hand, 12 years of formal education does not appear to ensure an adequate level of mathematics learning. Results of this study have shown that mathematics achievement of students in Grade 12 is extremely low. These findings are similar to the findings at the Grade 8 level in TLMDR. There are several conclusions highlighted by the results of this study. First, the gain in mathematics achievement from Grade 11 to Grade 12 is small. This is particularly noticeable in the complex numbers subtest which is taught only in Grade 12. The results in this subtest showed that 92 students did not learn this content although teachers agreed that it had been taught. However, there are some students who learn more mathematics than others. This is noticed in students attending three types of schools in the country. Students in private F-schools appear to learn more mathematics than students in public or O-schools. This conclusion is supported by the finding that Grade 11 students in F-schools obtained higher achievement levels than students in Grade 12 attending O- and public schools. This was also highlighted when Grade 12 students were compared to Grade 8 students (TLMDR) on 14 items. The achievement gains from Grade 8 to Grade 12 were considerably higher in F-schools and public schools than in O-schools in the set of 14 items. On the other hand, the differences in achievement between the schools seem to decrease throughout the grades. More evidence was found when the means of the 3 school types were compared. Students in private F-schools were found to know more mathematics than students attending public or O-schools. Students attending private O-schools seem to learn as much mathematics as students in public schools. However, this does not mean that public and private O-schools provide the students with the same type of educational experiences. It is very likely that the similarity between O- and public schools is due to the selection effect in the public schools. However, the achievement levels of F-school students proved to be very poor when compared to students from other countries. The mathematics performance of Dominican students was far behind that of students from the SIMS participating systems. Even though the comparison of performance was made on only 10 items, it shows that 93 mathematics achievement in the Dominican Republic is very low. Another finding of this study was that the regional location of the schools does not affect mathematics achievement. Schools, whether in Santo Domingo, Santiago, or other large cities, are homogeneous in their mathematics achievement levels. However, in the Grade 11 sample, the region of Santiago produced higher mathematics achievement levels than Santo Domingo or the next ten largest cities. It is speculated that there might be real differences among the regions because the test used was very conservative and was not very powerful. Differences between males' and females' mathematics achievement were also found. These differences were most marked in the area of geometry and the cognitive level of application. It was concluded that this might be related to males' and females' differences in spatial visualization and problem solving abilities. Differences were found in all types of schools and in all regions. They may be related to the rigid social and cultural sex-role expectations which are so widely spread in the Dominican culture. Implications The low levels of mathematics achievement found in the present study among twelfth graders have several implications. Students have a poor mathematics background when they enter a university. This, combined with the fact that mathematics is an important tool in the technological development of modern societies, implies that the Dominican Republic must continue to import foreign technology. The demand for skilled professionals that are needed to continue along the path of technological advancement will probably not be met by students 94 within the Dominican Republic. Unless the appropriate steps are taken to improve the quality of education in the Dominican Republic, there is little chance for development. The finding that Grade 12 students in F-schools obtain higher mathematics achievement levels than students in public and O-schools suggests that F-school students constitute an elite within the "elite with schooling". The fact that students attending F-schools are economically and educationally in an advantageous position suggests that the educational system is widening the gap between the social classes. The finding that private O-schools and public schools have equally low achievement levels cannot be overlooked. It suggests that students paying tuition for their education are not getting the type of instruction they expect. It implies that students attending O-schools are paying for the same education that public school students obtain for free. In the present study, no regional differences can be claimed. The region variable was a significant factor acting upon mathematics achievement. However, the multiple comparisons were mixed. These were hot significant for the Grade 12 sample, and significant for the Grade 11 sample. Therefore, this is an area that needs further investigation. Gender differences in mathematics achievement were also found in this study. The finding that these differences are most marked in geometry and on application items indicates that females do not have the opportunity to practice their spatial visualization and problem solving abilities in the Dominican schools. Unless the proper steps are taken, gender differences in mathematics will continue to deepen, and most of the mathematically-oriented professions will continue to be dominated by males in Dominican society. 95 Suggestions for Future Research The results of this study open many doors for future investigations of academic achievement in the Dominican Republic. First, the fact that teachers' OTL ratings did not match with the achievement levels obtained by their students suggests that more research is needed to find out what is happening in the classrooms. A suggestion is that, in addition to asking the teachers whether the content evaluated had been taught or not, teachers should also be asked what percentage of their students they expect would answer the item correctly. The present investigation showed that students learn little mathematics content in Grade 12. It is suggested that the secondary mathematics curriculum should be revised and updated. Another question which needs to be addressed in this area is the extent to which teachers follow, use, and obtain help from the official curriculum. In addition, research focusing on classroom observations might also help in clarifying what goes on in the classroom which prevents the learning of mathematics from happening. The low mathematics achievement levels found among Dominican twelfth graders calls for more studies investigating which factors promote mathematics achievement in the Dominican setting. The availability of textbooks, teachers' use of instructional time, amount and type of curriculum covered, and teachers' training have been identified to positively affect academic achievement (Fuller & Heyneman, 1989). Some of these variables might also affect mathematics achievement in the Dominican Republic. Research on this area might prove to be fruitful in helping to improve mathematics achievement in the country. 96 The conclusion that F-school students perform better than students in other types of schools opens the question of why these differences exist. More research is needed to investigate which variables contribute to the higher mathematics achievement levels In private F-schools. In addition, which of these variables can be implemented in the other types of schools to improve their achievement levels. In addition, the fact that only the largest twelve cities in the country were included opens the question of whether this finding is generalizable to smaller urban areas. Gender differences in mathematics achievement is another interesting area for research in the Dominican Republic. The question of why these differences exist needs to be addressed. This could be accomplished by surveying or interviewing students about their perception of mathematics in relation to gender and society. Moreover, research on gender differences should focus on ways of reducing these differences in the Dominican setting. In addition, females' and males' differences in spatial visualization and problem solving abilities is another area open for future research. Comparison of achievement between developing countries, like the Dominican Republic, and developed countries is not fair. 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No se oermite el uso de libros, ni de materiales de referenda, ni de calculadoras. 3. Veamos ahora algunos ejemplos de como deben contestarse las preguntas en esta prueba. En cada pregunta se te presentan cinco (A, B, C, D, E) o cuatro (A, B, C, D) alternativas como posibles respuestas. de las cuales UNA SOLA es la respuesta correcta. Encierra en un circulo la alternativa correcta en cada pregunta. Ejemplo 1 27-19 es igual a: ® 8 B 12 C 16 D 18 E Ninguna de las anteriores Como 27 -19 = 8, entonces encerramos en un circulo la alternativa A. Ejemplo g . 6 X 3 es igual a: A 3 B 9 C 15 D 18 4. Trabaja tan rapido y cuidadosamente como puedas. No se pierden puntos por escoger una respuesta que parece razonable. No pierdas tiempo en preguntas que consideres muy diffciles. 5. El tiempo maximo que tienes para resolver las preguntas en esta prueba es 1 hora y 15 minutos. Cuando termines de realizar esta parte de la prueba tendras un receso de 15 minutos y luego empezaras a trabajar con la segunda parte de la misma. 6. Cuando tu profesor te diga que comiences, pasa a la pagina siguiente y empieza. 106 1. 0.40 X 6.38 es igual a: A 0.2552 B 2.452 C 2.552 D 24.52 E 25.52 2. Miguel es 4 anos mayor que Rita. Rita es 11 anos menor que Daniel. Daniei tiene 1 anos. iCuantos anos tiene Miguel? A 3 anos B 5 anos C 14 aiios D 19 anos E 27 anos 3. (2 + 3i)2 es igual a: A -5 B 13 C -5 + 12i D 13 + 12i 4. 3.23 X 105 es igual a: A 0.0000323 B 3.23000 C 32,300 O 323.000 E 32,300.000 5. Dado que y600 es aproximadamente 24.4949, una aproximacion de V60,000 es: A 2.44949 B 244.949 C 2,449.49 D 24,494.9 E 244.949 6. c A V2 B iCual es la longitud exacta del lado "b" en el triangulo de arriba? A 1 B 2 _ 2 V T ^ 3 - D £ -1A 107 7. Si x3 - 8 = (x - 2) (x2 + n + 4), el valor de n es: A x B 2x C -2x D -x E 4X 8. En el grafico siguiente m1 es paralela a m2. La medida del Z x es igual a: 9. A 3 0 8 B 4 0 s C 20° 0 8 0 s Cuando x3 - 8x + 3 se divide entre x + 3, el cociente es: A x 2 - 1 1 x + 36 B x 2 - 5 x - i 2 C x 2 - 3x + 1 D x 2 + 3x + 6 2A 10. I Cual expresidn exponencial simplificada es igual a: • —_ ? V: s X3 B xs 1 C X2 11. 4 - -3 2 " -1 - . -I I I l •4 -3 -2 - 1 -I I !— h -1 2 3 4 ' P i,Cuaies son las coorcenadas del punto P? A (-3, 4) B (-4, -3) C (3. 4) D (4, -3) E (-4,3) 108 12. I Cual de las siguientes ecuaciones es equivalente a: x log 3 + 7 log 3 = 2 log 5? A (x + 7) 3=2 S B 3 ?=5 2 C 3 =5 14 D 3x + 21 = 10 13. Los triarrgulos PQR y STU son semeiantes. iCual es la longitud del segmento SU? A 5 B 10 C 12.5 D 15 E 25 3A • 4 x 2 8 A n B 2n C 2 3 0 2 2 n - 3 E 2 n - 1 es igual a: 15. En los numeros complejos. el conjunto de todas las raices o soluciones de la ecuacidn (x2 + 4) (2x -1) = 0 esta form ado por: i_ 2 , 2. -2 2i.' -2i 2 2i, -2i • - . 2 i , -2i 2 16. iCual o cuales de las siguientes iguaidades es (son) verdadera (s) para todos los valores de 9 para los que las funciones estan definidas? I sen (-6) = - sen 9 II cos (-9) = - cos 9 III tan (-9) = - tan 9 A Solo \ B Solo il C Solo III D Solo l y III E Solo II y III 109 17. Si 1809 <X< 3609 y tan x = 2, entonces cos x es igual a: A - V T C D E 18. 2 V T 1 V T 1 V T 2 V T Los catetos de un triangulo rectangulo miden 6 centimetros y 8 centimetros. ^Cuai es la longitud de la hipotenusa en centimetros? A V48 B 7 C 100 0 10 19. Si a + bi y c + di son numeros complejos (donde a, b, c y d son numeros reales), y a + bi = c + di, entonces: A a+c=b+d B a = c y b = d C a = b y c = d D i = a-c d-b E Ninguna de las anteriores 20. Si un angulo de la base de un triangulo isosceles es el doble del angulo que no esta en la base, la medida en grados del angulo que no esta en la base es: A 45s B 36s C 60a D 30s 21. Una expresion simplificada igual a 1 1 gg . cos 9 + 1 cos 9-1 A 2 2 B 2 csc 9 C 0 D - 2 csc2e 22. Si y es directamente proporcional a x, e inversamente proporcional al cuadrado de w, entonces para alguna constante k se cumple que: A B C D y = kx kw x = k x V w y = -zr E y = J L T k w 4A 23. _ La recta MN es paralela a las rectas OPy QR; y la recta KL es una secante. <j,Cual de los siguientes angulos es diferente del angulo 2? » > R A 9 B 4 C 6 D 3 24. Cuando se abren al mismo tiempo dos Haves de agua P y Q, se llena un tanque en 2 minutos. Cuando se abre solo P el tanque se llena en 4 minutos. Cuando se abre solo Q el tanque se llena en: A menos de dos minutos B 2 minutos C 3 minutos 0 4 minutos E mas de 4 minutos 25. i,Cuai es el valor de la expresidn x3 + x2 - 2x -1 cuando x = -2? A 15 B 7 C -1 D -9 E -17 26. La distancia entre los puntos (-2, 5) y (3, - l)es: A vTf B V37 C Y4? D veT 27. La pendiente de la recta 3x = -2y - 7 A f-B 4 C -I D I 5A I l l 28. La medida de Z A es el doble de la de su complemento. ^Cual es la medida de Z A? A 120a B 60 J C 30 a D 15s 29. I Cual es el valor de x si log x 8 = 3 ? A I 8 B 512 r log 3 log 3 D 2 30. El volumen de una caja rectangular cuyas dimensiones son: 10 cm de largo, 10 cm de ancho y 7 cm de alto, es igual a: A 27 cm3 B 70 cm3 C 140 cm3 D 280 cm3 E 700 cm3 32. Si x > 0, y > 0, y x * y, entonces ._ 1 — = v x - V / es igual a: 6A 31. Dado que tan 9 = j y que el lado final dei angulo 9 esta en el III cuadrante, ^cual es valor de sen 9 ? B C D 3 " 5 3 5 4 5 A B C D E 33. V T - / V T - V 7 X - y 1 V * - y 1 1 V T V 7 V T + V 7 2 X 2 Dos tiendas tienen baratillos de televisores. Una de ellas ofrece un 10% de descuento y la otra un 15% de descuento. ^Cual es la diferencia de precio entre ambas tiendas para un televisor cuyo precio regular es $200? A S5 B Sio C $15 D $85 E $90 34. Un cierto numero de estudiantes debe acomodarse en una pension. Si se colocan dos estudiantes por habitacion, entonces se quedan'an dos estudiantes sin habitacion. Si se colocan tres estudiantes por habitacion, entonces se quedan'an dos habitaciones desocupadas. ^Cuantas habitaciones hay en la pension? A 6 B 8 C 10 D 12 E 14 35. • 1 unidad de superficie Cada uno de los cuadrados pequenos de la figura mide una unidad de area. ^Cual es el area aproximada de la region sombreada? A 23 unidades de superficie B 20 unidades de superficie C 18 unidades de superficie D 15 unidades de superficie E 12 unidades de surjerficie 7A 113 P0NTIF1CIA UNIVERSIDAD CATOLICA MADRE Y MAESTRA Ctntro La.tmoamtnc.ano <£& Investigation y 'DesarroUo en 'Education Matematica Centra de Investigaciones Poblacidn B - Prueba B INSTRUCCIONES A LOS ESTUDIANTES 1. El objetivo de esta prueba es determinar los conocimientos matematicos que tienen los estudiantes de bachiilerato. Esta prueba tiene dos partes. Esta es la segunda de ellas. 2. No se permite el uso de libros, ni de materiales de referencia, ni de calculadoras. 3. Veamos ahora algunos ejemplos de como deben contestarse las preguntas en esta prueba. En cada pregunta se te presentan cinco (A, B, C, D, E) o cuatro (A, B, C, D) alternativas como posibles respuestas, de las cuales UNA SOLA es la respuesta correcta. Encierra en un ci'rculo la alternativa correcta en cada pregunta. Ejemolo 1 27-19 es igual a: ® 8 B 12 C 16 D 18 E Ninguna de las anteriores Como 27 -19 = 8, entonces encerramos en un circulo la alternativa A. 4. Trabaja tan rapido y cuidadosamente como puedas. No se pierden puntos por escoger una respuesta que parece razonable. No pierdas tiempo en preguntas que consideres muy dificiles. 5. El tiempo maximo que tienes para resolver las preguntas en esta prueba es 1 hora y 15 minutos. Ejemplo 2 6 X 3 es igual a: A 3 B 9 C 15 D 18 6. Cuando tu profesor te diga que comiences, pasa a la pagina siguiente y empieza. 114 1. 2 3 - + 3- es igual a: 5 o A B C D E _5_ 13 J5_ 40 _6_ 40 10 15 11 40 2. Si a = 2 3 - 5 2 - 7 y b = 3 2 • 5 3 • 7 2 • 11, entonces el minimo comun multiplo (m.cm. ) de a y b es: A 5 2 - 7 B 2 3 3 2 • 5 2 • 7 • 11 C 2 3 3 2 • 5 3 • 7 2 • 11 D 2 3 3 2 • 5 s • 7 3 • 11 E 2 3 - 3 2 • 5 6 • 7 2 • 11 3. Una operacidn * en el conjunto de los numeros reales es conmutativa si, para dos numeros reales cualesquiera x e y, x * y = y * x. ^Cual de las operaciones definidas a continuacidn es conmutativa? A x • y = x + xy B x « y = x -y C x • y = x (x+y) D x • y = xy (x + y) E x*y = x2 + xy2 + y4 4. En una clase hay 35 estudiantes. Se sabe que j de ellos va a la escuela en guagua y 2 2 en bicicleta. ^Cuantos van a la escuela utilizando otros medios de transports? A 7 B 14 C 21 D 28 E 35 5. Si el reciproco de a es ^ entonces el recfproco de 10 + 5i es: 2_ _ _ i _ 25 ' 25 _2_ J_ 15 15 A B C D 6. Si a * 0, entonces 3a2 x 2a 2 es igual a: 10 5 1 1 . 10 " 5 ' A 6a "I B 6a C 0 D 1 E 6 1B 115 7. El dueno de un supermercado tiene x libras de cafe en un almacen. Vende 15 libras y luego recibe un nuevo paquete que pesa 2y libras. ^Cuantas libras de cafe tiene ahora? A x - i 5 - 2 y B x + i5 + 2y C x - 1 5 + 2y D x + l 5 - 2 y E Ninguna de las anteriores. 8. En el triangulo dado. ST es paralela a RQ si: R 5 S B Z 2 + = 180s C Z3 = / 4 D Z 1 + Z 2 + Z 3 = 180" 9. Si hay 300 calorias en 100 gramos de un alimento, ^cuantas calorias hay en 30 gramos del mismo alimento? A 90 B 100 C 900 D 1,000 E 9,000 2B 10. Las soluciones de la ecuacion (2x- 1) (x + 2) = 0son: A 1. -2 B -1, -2 C -2 ,2 2.-1 E - 2 . 1 11, Si x = — , entonces log x es igual a: A log2 a - log b B 2 (log a - log b) C 2 log a - log b D 2 log -b E log a 2 - b 12. En la figura, las rectas PQ y RS se cortan en O. La suma de las medidas de los angulos 1 y 2 es: A 15° B 30° C 60" D 180 g E 3 0 0 5 116 13. cos (90° + 0) es igual a: A - cos 3 B sen 9 C - sen 9 D cos 9 14. Si x, y son numeros reales, ipara cuales x puede definirse y por: V 9 V ' A Cualquier x, excepto x = 3. B Cualquier x, excepto x = 3 y x = -3. C x < -3 6 x > 3. D -3 < x < 3. E x<3. 15. Un joven de 5 pies de estatura esta parado cerca de un poste telefdnico en un terreno llano. Cuando la sombra del joven es de 3 pies de largo, la sombra dei poste es de 45 pies de largo. i,Cual es la altura en oies del poste telefonico? A 24 B 27 C 30 D 60 E 75 1 16. Simplifica >= 2 - V3 denominador: A -VT B VT C 2 + VT D 7 e 2 + YT racionalizando el 17. i Cual es el valor de x si log :x = - 2? A H . B 1 C 4 D No tiene solucidn 18. Si a = 20, b=0, c = 10, x = 3, y = 12, entonces el valor de 2aby + 2cx es: A 100 B 160 C 400 O 640 E Ninguna de las anieriores 3B 19. En la regla de la figura, la lectura indicada por la flecha esta entre: A 51 y 52 B 57 y'58 , C 60 y 62 D 62 y 64 E 64 y 66 20. Las soluciones de de 4x2 - 7x -2 = 0 son: A 4 ,2 B --,2 4 C 2,4 D 1,-2 4 E - i a 21. 9K ^Cuai es el valor en grados de - y radianes? A 36 2 B 729 C 324« D 648« 118 22. O S Q En la figura de arriba, PQJ.OQ yRSlOQ. Si lamedidadeOQ y OR es 1, y 9 es la medidade ZPOQ, entonces la medida del segmento PQ es igual a: A sen 8 B cos9 C tan 9 D 2 sen e E 1 - cos 9 2 3 . En el sistema de ecuaciones, x + 2y =3 3x - y = -5 <j,cual es el valor de y? A -2 B -1 O -1 E 2 24. Simplifica (2 + i) (3 - i) A 5 B 5 + i C 6 D 6 - i2 E 7 + i 25. Si las rectas y, = rr^ x + b, y y2 = m2x + b2 son paralelas, entonces: A b 1 = b 2 B y = x C m, = - — D m 1 = IT>2 E m , a l rri2 26. Un tren que viaja a 50 km/h salid de la estacidn 3 horas antes que un tren ex-preso que viaja en la misma direccion a 90 km/h. ^Cuantas horas le tomara al tren expreso sobrepasar al otro tren? A §. 9 8 9 5 C 11 5 D V5 4 E H 58 119 2 7 . En el triangulo rectangulo siguiente la longitud de PQ es: Q A 1 cm B 3cm C 5cm D 7cm 13 c m 12 c m 28. Si log A = n, entonces log A 2 es igual a: A n + 2 B n2 C -^ 2 D 2n E n-2 2 9 . ; En cual cuadrante se encuentra el lado final de un angulo en posicidn estandar de -185s ? A I B n C ni D rv 3 0 . Los lados de un triangulo estan en la razdn de 2:3:4. <,Cual es el coseno del angulo mayor? A T B r C D 1_ 4 7 8 31 ^En que punto intersecta la recta 2y + 7x -17 = 0 el eje y? A (0, 17) B (o.'.£) C (..fl D |-f,0) IZ-.o 3 2 . Si a > 0, entonces es igual a: 5 A a « 5 B as C a 5 " 5 a 5 " 5 D E a 6S 33. En la figura siguiente, PQRS es un paralelogra-mo. iCual angulo debe tener la misma medida que el angulo k? T S R A t B m C n D o 34. i,Cual es el maximo comun divisor de las siguientes expresiones? x3 - x x3 + 2x2 + x A x B x + 1 C X(X + 1) D X(X + 1)2(X-1) E x3 El volumen del prisma de la figura es A 90 centimetros cubicos B 300 centimetros cubicos C 325 centimetros cubicos D 600 centimetros cubicos E 650 centimetros cubicos 7B A P P E N D I X B Mathematics Achievement Tests (Translated) Test A Test B 122 Identification of the Mathematics Test Items C o m p u t a t i o n C o m p r e h e n s i o n A p p l i c a t i o n F o r m A F o r m B F o r m A F o r m B F o r m A F o r m B A r i t h m e t i c 1 3 4 15 19 1 24 2 5 2 33 3 4 9 A l g e b r a 7 22 25 32 6 7 10 16 23 32 9 10 14 18 20 34 24 34 14 26 G e o m e t r y 11 18 27 28 30 8 25 27 31 35 8 20 23 26 19 33 13 35 12 15 T r i g o n o m e t r y 12 29 31 13 17 21 22 29 17 21 11 28 6 16 30 123 0.40 X 6.38 is equal to: A 0.2552 B 2.452 C 2.552 D 24.52 E 25.52 Miguel is 4 years older than Rita. Rita is 11 years younger than Daniel. Daniel is 12 years old. How old is Miguel? A 3 years old B 5 years old C 14 years old D 19 years old E 27 years old (2 + 3i)2 is equal to: A -5 B 13 C -5 + 12i D 13 + 12i 3.23 X 10 is equal to A 0.0000323 B 3.23000 C 32,300 D 323,000 E 32,300,000 5. Given that VbOO is approximately 24.4949, an approximation for -^60,000 is: A 2.44949 B 244.949 C 2,449.49 D 24,494.9 E 244,949 A ^ B What is the exact length of side "b" in the triangle above? A 1 B 2 If x 3 - 8 = (x - 2)(x2 + n + 4 the value of n is A x B 2x C -2x D -x E 4x In the following graph m, is parallel to m 2 . The measure of Z x is equal to: B 40° C 20° D 80° When x3 - 8x + 3 is divided by x + 3, the quotient is: A x -11 x + 36 B x2 -5x -12 C x 2 -3x + 1 2 D x +3x+6 10. Which simplified exponential expression is equal to: A r x £ 1 ^ 7 B 11 4 34-— l — I — -4 -3 -2 -I 1 2 3 4 -4 What are the coordinates of point P? A (-3,4) B (-4,-3) C (3,4) D (4,-3) E (-4,3) 12. Which equation below is equivalent to: x log 3 + 7 log 3 = 2 log 5 ? . 3 5 A (x + 7) =2 D 7 2 B 3 =5 C x+7 2 3 =5 D 3x + 21 = 10 13. Q 1 / V 2 2.5 Triangles PQR and STU are similar. How long is segment SU? B C D E 10 12.5 15 25 125 14. 4 x 2 " is equal to: 8 A n n-1 B 8 C 2 T D 2 2n-3 E 2' n-1 15. In the complex numbers, the set of a!! the roots or solutions of the equation (x2 + 4)(2x-1) = 0 is formed by: A 1 B - 2 - 2 C l , 2 l - 2 i 2 D 2,-21 E --,2\,-2 2 16. Which of the following is (are) true for all values of 0 for which the functions are defined? I sin (-0) = - sin 0 II cos (-0) = - cos 0 III tan (-0) = - tan 0 A I only B II only C III only D I and III only E II and III only 126 17. If 180 < x <360° and tan x = 2, then cos x is equal to: A - A/5 2 V5~ 18. The sides of a right triangle measure 6 and 8 cm respectively. What is the length (in cm) of the hypotenuse? A yfw B 7 C D 100 10 19. If a+bi and c+di are complex numbers (where a,b,c and d are real numbers), and a+bi=c+di, then: A a+c = b+d B a = c and b = d C a = b and c = d _ a - c E None of the above 20. If an angle in the base of an isosceles triangle is the double of the angle which is not in its base, the size in degrees of the angle not in the base is: A 45° B 36° C 60° D 30° 21. A simplified expression for — J 1— is: COS0 + 1 COS0-1 B 2csc e C 0 2 •2csc 8 22. If y is directly proportional to x, and inversely proportional to the square of w, then for some constant k it is true that: kx w B v = M_ 1 x C y = kxVw" kx w • + k w 127 2 3 . The line MN is parallel to OP and QR;an< the line K L ' s a secant. Which of the following angles is different from angle 2? A K A 15 IvL- 5j/6 B 7 9/10 0 m 3/4 C -1 7/8 1/2 D -9 E -17 A 9 B 4 C 6 D 3 24. When water is running out of two taps P and Q at the same time, a tank is filled in 2 min-utes. When only P is open, the tank is filled in 4 minutes. When only Q is open, the tank is filled in: A less than two minutes B 2 minutes C 3 minutes D 4 minutes E more than 4 minutes 2 5 . What is the value of the expression x3 + x2-2x-1,whenx = -2? 26. The distance between the points (-2,5) and (3,-1) is: A Jvf B C 74? D M 27. The slope of the line 3x = -2y - 7 is: A i 3 B - 1 2 C _1 3 D 3 2 E 3 7 128 28. The size of ZA is the double of its complement. What is the size of Z A ? A 120° B 60° C 30° D 15° 29. What is the value of x if log x 8 = 3 ? A 3 8 B 512 C ^ 8 log 3 D 2 30. The volume of a rectangular box with dimensions: 10 cm long, 10 cm wide and 7 cm high, is equal to: A 27 cm3 B 70 cm C 140 cm3 D 280 cm E 700 cm 31. Given that tan0 = and that the termina side of angle q is in quadrant III, what is the value of sin q? B 4 "5 3 5 3 5 4 5 32. If x > 0, y > 0, and x * y, then is equal to: A ^ + Vy~ x-v B Vx-Vy" Vx-Vy c D x-v 1 Vx-y 1 1 Vx" Vy Vx + Vy~ 2 2 x -v 3 3 . Two stores have a television sale. One of them offers a 10% discount and the other one offers a 15% discount. What is the difference in price between the two stores for a regularly priced television of $200?. A $5 B $10 C $15 D $85 E $90 34. A certain number of students are to be accommodat-ed in a hostel. If two students share each room, then two students will be left without any room. If three students share each room, then two rooms will be left unoccupied. How many rooms are there in the hostel? A 6 B 8 C 10 D 12 E 14 3 5 . j j 1 square unit Each of the small squares in the figure measure one square unit. What is the area of the shaded figure to the nearest square unit? A 23 square units B 20 square units C 18 square units D 15 square units E 12 square units 130 1. 2 3 g + g is equal to: A — 13 B ^ 40 c — 40 10 15 E 31 40 2. If a=23 5 2 - 7 and b = 32 V-72- 1 then the least common multiple (LCM) of a and b is: A B C D E 2 5 1 3 2 2 2 -3 -5 -7-11 3 2 3 2 2 - 3 - 5 7 - 11 3 2 5 3 2 3 -5 7 -11 3 2 6 2 2 - 3 - 5 - 7 -11 3 . An operation * on the set of real numbers is commutative if, for every two real numbers x and y, x * y = y * x. Which of the operation defined below is commutative? A x*y = x + xy B x*y=x-y C x*y = x (x + y) D x * y = xy (x + y) 2 ' x txy^ + y E x*y= 2 2 • 4 4. There are 35 students in a class. 1 of 5 them come to school by bus, another f-5 come by bicycle. How many come to school by other means? . A 7 B 14 C 21 D 28 E 35 5. If the inverse of a is then the inverse a of 10 + 5 i is A 25 25 B C D 15 15 10 5 JL_ i i 10 5 6. If a * 0, then 3a 2 x 2a 2 is equal to: A 6a" 7 B 6a C 0 D 1 E 6 7. The owner of a supermarket has x pounds of coffee stored. He sells 15 pounds and then receives a new package of coffee which weighs 2y pounds. How many pounds of coffee does he have now? A x-15-2y B x + 15 + 2y C x -15 + 2y D x +15 - 2y E None of the above 8. In the following triangle, ST is parallel to RQ if R 9. B Z2 + Z5 = 180° C Z3 = Z4 D Zl + Z2 + Z3=18( If there are 300 calories in 100 grams of certain food, how many calories are there in 30 grams of that food? A B C 90 100 900 D 1,000 E 9,000 10. 131 The solution set for the equation (2x-1)(x + 2) = 0is: A 1,-2 B -1,-2 C -2,-2 D 2,-E 11. 1 2 4 If x = y , then log x is equal to: A B C D E log a-logb 2 (log a-log b) 2 log a - log b 2log 1 loga2 - b In the figure, lines PQ and RS are intersected in 0. The sjum of the measures of angles 1 and 2 is: A 15° B 30° C 60° D 180° E 300° 13. cos (90° + 0) is equal to: A -cose B sine C - sin e D cose 14. If x and y are real numbers, for which x can you define y by y= A All x, except x = 3 B All x, except x = 3 and x = -3 C x < -3 or x > 3 D -3<x<3 E x<3 15. On the level ground, a boy 5 feet tall casts a shadow 3 feet long. At the same time a nearby telephone pole 45 feet high casts a shadow the length of which, in feet, is B 24 27 C 30 D 60 E 75 132 16. 1 Simplify denominator - V I VI 2 +VI 2 +VI rationalizing the 7 2 +Vi 5 17. What is the value of x if iog1 x = - 2? 2 B -C 4 D No solution 18. Ifa = 20,b = 0,c=10,x = 8,y = 12, then the value of 2aby + 2cx is: A 100 B 160 C 400 D 640 E None of the above 19. On the above ruler, the reading indicated by the arrow is between: A 51 and 52 B 57 and 58 C 60 and 62 D 62 and 64 E 64 and 66 20. The solutions of 4x 2 -7 x - 2 = 0 are: A 4,2 B C 2,4 D i - 2 2 1 . What is the value of 9JC radians in degree measure? 5 A 36° B 72° C 324° D 648° 22. In the figure above, PQ 1OQ and R S I O Q . If the measures of O Q and of O R equal 1, and 0 is the measure of Z P O Q , then the measure of segment P Q is equal to: B C D E sine cose tane 2 sine 1 - cose 2 3 . In the equation system, x + 2y = 3 3x - y = -5 what is the value of y? A -2 B C D E -1 24. Simplify (2 + i) (3 - i) A 5 B 5 + i C 6 D 6-iZ E 7 + i 2 5 . If the lines y^n^x+b, and y2 = m2x+b2 are parallel, then: A B C b1 = b 2 y = x m =-1 J_ m i = m 2 E m = - L 1 m_ 26. A freight train traveling at 50 km/h leaves a station 3 hours before an express train which travels in the same direction at 90 km/h. How many hours after leaving will it take the express train to overtake the freight train? A B C D E 5 9 9 5 12. 5 15 4 l i 4 135 27. In the following rectangle triangle, the length of PQ is: B 3 cm C 5 cm D 7 cm 28. If log A = n, then log A is equal to: A B C D E n + 2 2 n n_ 2 2n n-2 29. In what quadrant does the terminal side of a standard position angle of -185° lie? A I B II C in D rv 30. The sides of a triangle are in the ratio of 2:3:4. What is the cosine of the largest angle? A B C D 31. In what point does the line 2y + 7x -17 = 0 intercept the y-axis? A (0,17) B (..-* C ('•?) D (-*.. E 12 ) 32. If a>0,then v a is equal to A B C D E 5-6 6-5 l a 5 6 3 3 . In the following figure, PQRS is a parallelogram. What angle should have the same measure as angle k? A B C D m 34. What is the greatest common divisor of the following expressions? xNx x3+2x2+x A x B x + 1 C D E x(x + 1) 2 x(x + 1) (X - 1) 35. 10 cm The volume of the prism in the figure is: A 90 cubic centimeters B 300 cubic centimeters C 325 cubic centimeters D 600 cubic centimeters E 650 cubic centimeters A P P E N D I X C Test Items and Mean Percent Responses of the Educational Systems Participating in SIMS 138 Item A16or SIMS 012 Which of the following is (are) true for all values of 6 for which the functions are defined? I sin (-9) = - sin 8 II cos (-0) = - cos 6 III tan (-0) = - tan 9 A I only B II only C III only * D I and III only E II and III only Hong Kong England/Wales Japan Belgium (Flemish) Canada (Ontario) Sweden Belgium (French) Finland Scotland Israel New Zealand United States Hungary Thailand Canada (B. C) - * 6 9 L . - * 6 3 ^ 63 / „ 6 1 ^ 5 9 ^ 5 9 / ^ ou _ 4 6 * 42 s ^ 3 9 „ 3 9 / 2P 36 e — - — 1 — • — 1 — • — 1 — • — 1 — • — f 20 40 60 80 100 Mean Scores 139 Item A32 or SIMS 080 If x > 0, y > 0, and x * y, then —~=-—j=-is equal to: V x - V y A B C D V x " + V 7 X - V V x " - V 7 X - V 1 Vx -y 1 1 V x ~ V>T + Vy~ 2 2 X - V Japan Hong Kong Finland England/Wales Belgium (Flemish) Canada (Ontario) Canada (B. C.) Belgium (French) Israel Sweden New Zealand Thailand Scotland Hungary United States s s ' jTST 64 63 63 ; 7 4 5 ^ 52 ^48 48 ^ 42 ~* 42 44 20 — I -40 -7 92 ^ 9 2 60 80 =7 100 Mean Scores Item A34 or SIMS 038 A certain number of students are to be accommo-dated in a hostel. If two students share each room, then two students will be left without any room. If three students share each room, then two rooms will be left unoccupied. How many rooms are there in the hostel? A 6 * B 8 C 10 D 12 E 14 Hong Kong Japan Sweden England/Wales Belgium (French) Israel Finland Scotland Belgium (Flemish) Hungary Canada (Ontario) New Zealand Thailand United States Canada (B. C.) •T 85 85 •> •s 64 •s * 64 •s ^63 •s •f •s _ ^ 5 5 •s -j 53 •s •/• " 7 46 7 70 69 20 40 60 80 100 Mean Scores Item B2 or SIMS 078 If a = 2 3 • 5 2 - 7 and b = 3 2 • 5 3 • 72 • 11, then the least common multiple (LCM) of a and b is: A 5 7 B 3 2 2 •3 • 2 5 7 • 1 1 C 3 2 2 •3 3 5 2 7 • 1 1 D 3 2 2 •3 5 5 7 3 • 1 1 c 3 2 6 2 C 2 •3 5 7 • 1 1 Hong Kong Japan Thailand Hungary Canada (B. C.) Belgium (Flemish) England/Wales United States Israel Belgium (French) Finland Canada (Ontario) Sweden Scotland New Zealand 0 20 40 60 80 100 Mean Scores Item B3 or SIMS 003 An operation * on the set of real numbers is commutative if, for every two real numbers x and y, x * y = y * x. Which of the operations defined below is commutative? A x * y = x + xy B x * y = x - y C x * y = x (x + y) * D x * y = xy (x + y) E 2 2 4 x * y = x +xy +y Japan Hong Kong England/Wales Finland Sweden New Zealand Scotland Belgium (Flemish) Thailand Canada (Ontario) United States Israel Belgium (French) Canada (B. C.) Hungary Mean Scores Item B6* or SIMS 004 1 -1 lfa*0, then 3a2 x 3a 2 is equal to: A 9a 4 B 9a C 0 D 1 "Item B6 This item has a slight variation from the original SIMS. L -1 If a * 0, then 3a 2 x2 a 2 is equal to: 1 A 6a ' B 6a C o D 1 * E 6 Hong Kong Japan Sweden England/Wales Belgium (French) Finland Belgium (Flemish) Canada (B. C.) Scotland Canada (Ontario) New Zealand Israel United States Hungary Thailand Mean Scores Item B14 or SIMS 081 If x and y are real numbers, for which x can you define y by A All x, except x = 3 B All x, except x = 3 and x = -3 C x < -3 or x > 3 * D -3< x<3 E x<3 Finland Belgium (Flemish) Hong Kong Belgium (French) Japan Hungary England/Wales Canada (Ontario) Israel Sweden Thailand New Zealand United States Scotland Canada (B. C.) Mean Scores Item B22 or SIMS 068 / In the figure above, P Q 1 0 Q and / R S i OQ • If the measures of OQ z § 5 and of OR equal 1, and 0 is the measure of ZPOQ , then the measure of segment PQis equal to: A sin 6 B cos 6 C tan 9 D 2 sine E 1 - cos e ^ jfi 89 Hong Kong H H • H ^ ! ^ ^ ^ H H ^ H B H • H ^ H H S ^ ^ H B S ^ ^ ^ ^ 3 ^ I , I , , ^ Israel J * r M M ^ ^ ^ M l ^ ^ M * l ' ^ ^ ^ ^ M ' ^ ^ ^ w w * ^ ^ 7 2 Japan ] " P^H^H^H!^HBSHHi^HI^Hi^^"^^ England/Wales B H H ^ ^ ^ H H H » f Finland J * y g B B i ^ — ^ ^ ^ = 1 55 Sweden ]^HHHH^MHHHHHHHi^^| Belgium (Flemish) B H H S ! ^ S H B H ^ r f 4 Hungary ^ Belgium (French) B B H ^ S H H H ^ 3 8 Canada (Ontario) R S ^ ^ B B ^ 3 8 New Zealand B H S H S ^ ^ Thailand ^ ^ ^ 3 2 United States ]^HHSSHHB"H^^29 Canada (B. C.) J ^ S B B 1 ^ ^ B ^ 2 7 Scotland " " ^ T < 1 • 1 >——1 1 1 1 r 0 20 40 60 80 100 Mean Scores Item B26 or SIMS 097 A freight train traveling at 50 km/h leaves a station 3 hours before an express train which travels in the same direction at 90 km/h. How many hours after leaving will it take the express train to overtake the freight train? 5 A 9 B ! ° 'i E 18 4 Hong Kong Japan Sweden Israel England/Wales Finland Belgium (Flemish) New Zealand Canada (B. C.) Hungary Scotland Canada (Ontario) Belgium (French) United States Thailand Z 93 6b - 7 7 3 71 61 7 5 7 56 l 20 40 T 55 ^ 54 7 5 4 P 50 60 I 80 =7 100 Mean Scores Item B28 or SIMS 087 If log A = n, then log A is equal to: A n + 2 2 * B n j i C 2 D 2n E n -2 Hong Kong England/Wales Japan Sweden Israel Finland Belgium (Flemish) Belgium (French) New Zealand Canada (Ontario) Thailand Hungary Canada (B. C.) Scotland United States 7" y 61 4CT 20 40 T r 2 5 50 - I — 60 ~ £ 96 87 ^ 8 6 ~81 80 7 7 8 74 71 -"I -80 =7 100 Mean Scores A P P E N D I X D TLMDR Test Items Identified, and Grade 8 Mean Percent Responses at the National and School levels Identification of TLMDR Items Grade 12 item T L M D R item cognitive level topic A l 0-30 1 A r i t h m e t i c A 2 2-23 3 A r i t h m e t i c A 4 4-27 1 A r i t h m e t i c A 1 3 0-20 3 Geometry A 3 0 0-1 1 Geometry A 3 3 * 1-11 3 A r i t h m e t i c A 3 5 0-6 3 Geometry B I 0-31 1 A r i t h m e t i c B4 3-31 3 A r i t h m e t i c B9 0-35 3 A r i t h m e t i c B12 2-4 3 Geometry B15 1-32 3 Geometry B19 2-24 2 Geometry B35 4-14 1 Geometry 150 1. 0.40 X 6.38 is equal to: A 0.2552 B 2.452 C 2.552 D 24.52 E 25.52 o CD o o I 100 n 80 60 40 20 VN 1 Al m Ii Publ ic Schools Private F Private O U r b a n T L M D R VD CM O T L M D R 2. Miguel is 4 years older than Rita. Rita is 11 years younger than Daniel. Daniel is 12 years old. How old is Mi-guel? A2 A B C D 3 years old 5 years old 14 years old 19 years old o a> o u 100 -i 80 60 40 20 to 1 m VO I to CO CO to O T L M D R 4. 3.23X10 is equal to A 0.0000323 B 3.23000 C 32,300 D 323,000 E 32,300,000 o <D V_ 1_ o u 100 -l 80 60 40 20 to A4 to to in oi O T L M D R 151 30. The volume of a rectangular box with dimensions: 10 cm long, 10 cm wide and 7 cm high, is equal to: A 27 cm 3 B 70 cm C 140 cm 3 3 D 280 cm * E 700 c m 3 A 3 0 o l _ 1_ O I o . 100 n 80 60 40 20 oo cb 1 in o CM1 1 O TLMDR 152 33. Two stores have a television sale. One of them offers a 10% discount and the other one offers a 15% discount. What is the dif-ference in price between the two stores for a regular priced television of $100. A $5 B $10 C $15 D $85 E $90 o <D 1-l _ O U I 100 80 60 40 20 vO A33 oo cd in I m 9 -O TLMDR 35. '4 | I 1 iquOT unit Each of the small squares in the figure measure one square unit. What is the area of the shaded figure to the nearest square unit? A B C D E 23 square units 20 square units 18 square units 15 square units 12 square units u a> »_ i _ o u I 100 -i 80 60 40 20 A35 ID to i I oo O TLMDR 153 4. There are 35 students in a class. J - of 5 p them come to school by bus, another — 5 come by bicycle. How many come to school by other means? A 7 B 14 C 21 D 28 E 35 o u 100 n 80 60 40 20 B4 CM VO IO to I I CM CM to •vj CM O T L M D R 9. If there are 300 calories in 100 grams of certain food, how many calories are there in 30 grams of that food? A 90 B 100 C 900 D 1,000 E 9,000 12. In the figure, lines PQ and RS are intersected in O. The sum of the measures of angles 1 and 2 is: A B * C D E 15° 30° 60° 180° 300° o u u CD 1_ O u 100 n 80 60 40 20 0 100 80 60 40 20 B9 CM VO VO IO CM CM O T L M D R B12 to to — P F to O T L M D R 154 15. On the level ground, a boy 5 feet tall casts a shadow 3 feet long. At the same time a nearby telephone pole 45 feet high casts a shadow the length of which, in feet, is A 24 B 27 C 30 D 60 E 75 o <D O u 19. V 50 | 1WJ l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l On the above ruler, the reading indicated by the arrow is between: A B C D * E 35. 51 and 52 57 and 58 60 and 62 62 and 64 64 and 66 o <D 1_ 1_ o u I * B C D E The volume of the prism in the figure is: A 90 cubic centimeters 300 cubic centimeters 325 cubic centimeters 600 cubic centimeters 650 cubic centimeters o o u 100 -i 80 60 40 20 100 n 80 60 40 20 0 100 n 80 60 40 20 1 in B15 K) 1 O <N CM O TLMDR B19 <N iri 1 O TLMDR B35 CO U3 o i O TLMDR A P P E N D I X E Frequency Distribution of Teachers' OTL Responses Frequency Distribution of Teachers' OTL Responses ITEM Prerequisite Reviewed Taught as New Next courses Not taught No response A l 3 5 9 0 0 0 5 A2 2 6 14 2 0 1 6 A 3 3 4 31 4 0 7 A4 14 13 14 0 1 7 A 5 2 6 12 3 0 2 6 A 6 15 8 16 0 4 6 A 7 13 15 15 0 1 5 A8 32 6 4 0 2 5 A 9 3 0 7 5 0 1 6 A 1 0 8 12 21 1 2 5 A l l 12 22 9 0 0 6 A 1 2 13 9 17 2 2 6 A 1 3 2 9 8 5 0 1 6 A 1 4 8 14 16 4 1 6 A 1 5 1 1 34 3 3 7 A 1 6 14 15 8 2 3 7 A 1 7 2 0 8 11 2 2 6 A 1 8 21 18 3 1 0 6 A 1 9 3 1 31 6 1 7 A 2 0 3 3 7 2 1 0 6 A21 14 10 15 2 2 6 A 2 2 9 7 20 2 4 7 A 2 3 3 6 4 2 1 0 6 A 2 4 21 7 3 3 9 6 A 2 5 2 0 12 11 0 0 6 A 2 6 13 14 14 1 1 6 A 2 7 7 2 22 6 4 8 A 2 8 28 11 3 0 0 7 A 2 9 16 10 15 1 1 6 A 3 0 28 7 6 2 0 6 A31 15 15 12 1 0 6 A 3 2 8 15 17 1 1 7 A 3 3 3 7 1 0 1 4 6 A 3 4 21 3 7 4 7 7 A 3 5 30 4 3 3 3 6 B l 2 5 18 0 0 0 6 B 2 2 3 18 1 1 0 6 B 3 12 7 17 3 4 6 B 4 3 3 8 1 1 0 6 B 5 5 4 28 3 1 8 B 6 11 11 20 0 1 6 B 7 24 11 5 0 2 7 B 8 3 0 6 5 1 0 7 B 9 3 3 4 4 0 1 7 BIO - 7 11 22 1 1 7 B l l 18 8 14 1 1 7 B12 3 5 5 3 0 0 6 B13 19 11 12 0 0 7 B14 2 5 24 7 4 7 B15 2 5 6 7 1 3 7 B16 7 18 17 0 0 7 B17 14 9 13 1 2 10 B18 32 8 2 0 0 7 B19 2 8 5 5 0 3 8 B20 6 13 21 1 0 8 B21 2 6 9 5 0 0 9 B22 18 9 9 1 4 8 B23 ,5 18 18 0 0 8 B24 0 1 35 3 2 8 B25 6 2 18 6 8 9 B26 16 5 13 2 4 9 B27 24 12 5 0 0 8 B28 17 8 14 0 2 8 B29 18 16 6 0 0 9 B30 16 6 6 3 7 11 B31 2 2 24 3 9 9 B32 5 10 24 1 0 9 B33 34 3 1 0 2 9 B34 22 13 5 0 1 8 B35 2 6 8 6 1 0 8 A P P E N D I X F School Mean Scores and Students' Responses to Test Items Table F l Grade 11 School Mean Scores Region School School Code n Mean ! 1 43 15.09 1 1 2 32 14.88 1 1 3 40 12.55 1 1 4 48 10.42 1 1 5 31 13.61 1 2 1 30 15.00 1 2 2 33 18.27 1 2 3 37 25.00 1 2 4 29 20.97 1 2 5 25 12.08 1 3 1 23 9.61 1 3 2 24 13.46 1 3 3 23 12.96 1 3 4 25 11.72 1 3 5 21 6.38 2 1 1 31 14.94 2 1 2 34 20.06 2 1 3 37 19.73 2 1 4 33 16.27 2 1 5 14 14.71 2 2 1 40 24.47 2 2 2 26 18.92 2 2 3 37 15.19 2 2 4 44 19.39 2 2 5 38 16.39 2 3 1 21 21.57 2 3 2 14 13.64 2 3 3 32 15.16 2 3 4 7 14.86 2 3 5 21 12.62 2 3 6 17 15.71 2 3 7 34 16.68 3 1 1 35 13.94 3 1 2 24 16.96 3 1 3 41 10.98 3 1 4 20 10.50 3 1 5 32 12.72 3 2 1 39 17.33 3 2 2 24 14.25 3 2 3 40 14.20 3 2 4 23 23.43 3 2 5 24 16.33 3 2 6 24 15.96 3 3 1 30 12.47 3 3 2 10 11.80 3 3 3 24 12.63 3 3 4 17 6.71 3 3 5 22 11.73 3 3 6 40 14.47 Table F2 Grade 12 School Mean Scores Region School School Code n Mean 1 ! 1 35 17.03 1 1 . 2 18 17.33 1 1 3 47 13.21 1 1 4 42 15.00 1 1 5 40 9.97 1 2 1 41 20.54 1 2 2 23 19.87 1 2 3 39 33.28 1 2 4 25 25.28 1 2 5 17 15.18 1 3 1 16 10.44 1 3 2 17 9.76 1 3 3 13 18.92 1 3 4 11 11.82 1 3 5 16 10.44 2 1 1 27 15.89 2 1 2 37 21.46 2 1 3 39 23.44 2 1 4 37 20.08 2 1 5 21 15.19 2 2 1 20 33.10 2 2 2 45 21.69 2 2 3 33 17.85 2 2 4 37 17.70 2 2 5 36 21.64 2 3 1 12 23.42 2 3 2 5 16.60 2 3 3 27 15.07 2 3 4 11 16.82 2 3 5 21 17.29 2 3 6 25 20.60 2 3 7 30 23.27 3 1 1 36 14.69 3 1 2 16 20.94 3 1 3 13 14.08 3 1 4 21 10.95 3 1 5 31 15.48 3 2 1 24 13.38 3 2 2 14 20.21 3 2 3 34 16.62 3 2 4 12 29.08 3 2 5 22 16.18 3 2 6 33 16.42 3 3 1 46 14.26 3 3 2 10 13.40 3 3 3 12 9.50 3 3 4 24 13.79 3 3 5 15 10.20 3 3 6 45 18.33 160 Table F3 Grade 12 Students' Responses to Test A Item A B C D E No Respc A l 23 4 40 5 24 5 A 2 4 78 4 3 8 3 A 3 16 24 26 15 0 19 A 4 25 22 6 17 23 7 A 5 9 28 17 14 17 16 A 6 15 16 19 23 0 28 A 7 15 26 17 6 21 16 A 8 20 29 28 14 0 9 A 9 8 14 48 12 0 19 A 1 0 10 18 23 24 0 25 A l l 34 5 9 40 5 7 A 1 2 27 9 22 26 0 16 A 1 3 11 5 48 18 9 9 A 1 4 60 17 5 2 7 10 A 1 5 23 19 11 13 5 29 A 1 6 18 12 15 20 13 21 A 1 7 12 15 6 12 28 26 A 1 8 29 10 9 43 0 8 A 1 9 19 17 9 15 27 13 A 2 0 20 15 32 19 0 14 A 2 1 7 16 49 17 0 12 A 2 2 18 10 7 13 32 20 A 2 3 22 7 17 41 0 12 A 2 4 6 17 1 65 7 4 A 2 5 9 10 44 16 6 15 A 2 6 37 14 9 8 0 32 A 2 7 6 17 14 9 25 29 A 2 8 45 31 9 2 0 13 A 2 9 23 6 36 20 0 14 A 3 0 39 5 5 1 42 7 A 3 1 11 29 22 12 0 25 A 3 2 9 7 36 20 7 22 A 3 3 44 26 11 6 3 10 A 3 4 35 28 7 9 7 13 A 3 5 32 7 5 34 10 12 Note. Numbers are i n percentage. Correct answers are i n bold. 161 Table F4 Grade 12 Students' Responses to Test B Item A B C D E No Resp Bl 41 7 7 1 43 2 B2 28 16 19 18 8 12 B3 14 28 11 19 8 20 B4 12 32 24 14 4 14 B5 2 4 76 10 0 9 B6 54 26 3 2 6 9 B7 3 11 71 2 9 4 B8 14 17 24 31 0 13 B9 67 11 6 3 7 7 BIO 24 12 15 20 12 17 B l l 18 12 14 25 17 14 B12 6 22 37 14 13 8 B13 16 21 11 42 0 11 B14 14 24 10 8 16 29 B15 6 15 12 23 30 14 B16 22 10 42 9 5 13 B17 9 27 10 38 0 16 B18 14 26 3 15 32 10 B19 4 41 7 19 25 4 B20 21 17 13 15 7 27 B21 21 21 24 8 0 27 B22 20 13 16 13 20 19 B23 32 15 8 5 27 13 B24 17 19 12 39 4 9 B25 27 14 9 17 9 24 B26 27 22 10 8 6 27 B27 27 10 41 12 0 11 B28 11 57 5 12 3 13 B29 4 19 49 13 0 15 B30 28 19 14 13 0 26 B31 22 17 13 14 7 28 B32 33 27 8 7 6 19 B33 27 11 29 21 0 11 B34 23 8 7 13 33 16 B35 32 18 13 9 8 20 Note. Numbers are in percentage. Correct answers are in bold. 162 Table F5 Grade 11 Students' Responses to Test A Item A B C D E No Respt A l 21 4 40 5 23 8 A2 3 77 4 2 12 2 A3 17 31 18 12 0 21 A4 14 18 7 20 29 11 A5 8 20 18 18 15 21 A6 12 12 23 23 0 30 A7 14 23 16 6 23 18 A8 19 28 25 19 0 10 A9 12 12 33 15 0 27 A10 9 15 20 23 0 32 A l l 32 7 12 35 5 10 A12 23 10 20 28 0 19 A13 9 8 42 20 11 11 A14 58 22 4 1 5 10 A15 27 23 5 5 3 37 A16 21 11 11 18 12 27 A17 8 14 5 7 28 37 A18 27 12 10 41 0 10 A19 18 11 11 13 27 21 A20 22 11 31 19 0 17 A21 8 16 47 14 0 15 A22 13 9 6 14 28 29 A23 24 7 18 37 0 14 A24 7 17 1 61 10 4 A25 8 14 36 18 5 19 A26 34 12 8 8 0 38 A27 5 14 11 7 28 35 A28 48 28 8 2 0 14 A29 21 6 35 18 0 20 A30 47 7 5 2 30 10 A31 7 24 26 11 0 32 A32 6 7 34 16 5 32 A33 44 22 11 7 5 12 A34 37 27 8 8 9 11 A3 5 33 6 7 29 11 14 Note. Numbers are in percentage. Correct answers are in bold. 163 Table F6 Grade 11 Students' Responses to Test B Item A B C D E No Respc B l 53 8 7 1 30 2 B2 29 15 15 19 7 14 B3 15 28 10 17 7 23 B4 14 30 27 12 5 13 B5 3 8 66 11 0 12 B6 62 18 4 2 3 11 B7 5 12 66 3 9 5 B8 15 17 23 30 0 14 B9 66 10 7 4 6 7 BIO 23 15 18 19 7 20 B l l 18 11 14 27 16 15 B12 8 21 31 21 11 8 B13 15 23 11 39 0 11 B14 17 24 8 6 11 35 B15 8 12 15 24 25 15 B16 25 12 32 7 7 16 B17 8 23 11 43 0 16 B18 20 13 4 12 40 11 B19 4 51 7 17 17 4 B20 23 16 12 13 7 30 B21 21 23 21 8 0 28 B22 20 14 11 15 17 23 B23 41 14 6 4 20 15 B24 19 32 9 28 0 12 B25 28 12 11 14 7 29 B26 30 19 12 9 6 25 B27 24 9 41 17 0 9 B28 12 58 4 11 2 13 B29 5 19 44 15 0 17 B30 22 20 13 16 0 29 B31 17 15 13 16 6 34 B32 19 29 12 8 9 23 B33 28 13 27 21 0 11 B34 21 8 6 11 37 18 B35 37 21 10 7 7 19 Note. Numbers are i n percentage. Correct answers are In bold. A P P E N D I X G Summary ANOVA Tables and Tukey Comparisons School Type and Region: Grade 11 165 Table G l Analysis of Variance Testing Main and Interaction Effects (Grade 11) Source of Sums of Degrees of M e a n v a r i a t i o n squares freedom square F Total Explained 363.952 8 45.494 4.60 * M a i n Effects 336.688 4 84.172 8.51 * A (Region) 123.533 2 61.766 6 .24* B (School type) 233.101 2 116.551 11.78 * A B 27.264 4 6.816 .69 Error terms (Res) 395.775 40 20.452 Totals within-group 759.727 48 29.803 Note. Significant F ' s (a = 0.05) needed for significance are: F(2,40) = 3.23, and F(4,40) = 2.61. * p<0.05 166 Table G2 Comparison of School Means (Grade 11) School Type M e a n Compar ison F F 17.9 F v s P 9.31 * Public (P) 14.4 P vs O 1.81 O 13.0 F v s O 12.22 * Note, Tukey B 2 needed for significance at a = 0.05 is TB2 (3,40) = 5.9. *p< 0.05 Table G3 Comparison of Region Means (Grade 11) Region M e a n Compar ison F Santiago (Sgo) 17.0 Sgo vs SD 6.93* Santo Domingo (SD) 14.1 Sgo vs Other 3.53 Next... (Other) 13.9 SD vs Other .04 Note. Tukey B 2 needed for significance at a = 0.05 Is TB2 (3,40) = 5.9. *p<0.05. A P P E N D I X H Males' and Females' Test Results 168 Table HI Test Results by Gender G i l G12 Gender n Mean n Mean Male 533 16.9 486 19.8 Female 880 14.5 781 16.7 Note. Maximum score=70 Table H2 Statistics for the Top Ranked 100 Males and Females G i l G12 Gender n Mean n Mean Male 100 26.9 100 33.2 Female 100 25.6 100 29.8 Note. Maximum score=70 A P P E N D I X I Summary ANOVA Tables for Gender: Grade 11 170 Table II One-way Analysis of Variance Testing Gender Differences on the Grade 11 data. Source of v a r i a t i o n S u m of squares Degrees of freedom Mean square Between W i t h i n T o t a l 1 809.272 54 516.292 56 325.564 1 1411 1412 1 809.272 38.637 39.891 46.83 * p<0.05 Table 12 ANOVA of Differences between the Top Ranked 100 Males and Females (Grade 11) Source of v a r i a t i o n S u m of squares Degrees of freedom Mean square Between W i t h i n T o t a l 83.205 3 706.790 3 789.995 1 198 199 83.205 18.721 19.045 4.44 *p<0.05 171 Table 13 Test Items with Significant Gender Differences (Grade 11) Cognitive Male Female Item Content a level 5 (p-value) (p-value) D c A5 Ar i 1 24 18 6 * 7.11 A l l Geo 1 40 31 9 11.31 A18 Geo 1 47 37 10 12.74* A19 Ar l 1 14 9 5 9.00* A23 Geo 2 44 33 11 18.77* A24 Alg 3 67 57 10 14.68* A30 Geo 1 34 27 7 8.99* A33 A r l 3 30 17 13 32.40* B4 Ar i 3 36 27 9 12.13* B8 Geo 1 32 18 14 35.96* B12 Geo 3 37 28 9 12.18 B15 Geo 3 30 23 7 8.99* B19 Geo 2 24 13 11 25.39 B21 Trl 1 25 18 7 9.44 B25 Geo 1 17 12 5 7.41 B33 Geo 2 32 24 8 9.02* B35 Geo 3 27 18 9 14.98 a Content Arl = Arithmetic Alg = Algebra Geo = Geometry Tri= Trigonometry. Cognitive level 1 = Computation 2 = Comprehension 3 = Application. 0 D = difference *p< 0.01 172 Table 14 Sunwiary of Results by Gender and Content Area (Grade 11) # of non # significant # significant Content Area # items significant items favoring items favoring items Males (%) Females (%) A r i t h m e t i c 15 11 4(27) 0 Algebra 20 19 1(5) 0 Geometry 20 9 11 (55) 0 Trigonometry 15 14 1(7) 0 Total Test 70 51 17 (24) 0 Table 15 Summary of Results by Gender and Cognitive Level (Grade 11) # of non # significant # significant Cognitive Level # items signif icant items favoring items favoring items Males (%) Females (%) Computat ion 35 27 8(23) 0 Comprehension 19 16 3(16) 0 A p p l i c a t i o n 16 10 6(38) 0 Total Test 70 51 17 (24) 0 A P P E N D I X J Summary ANOVA Tables for Gender Differences for School Type and Region: Grade 11 and Grade 12 Table J l Results by Gender by School and by Region Males Females n Mean n Mean Grade 11 Public schools 188 F-schools 202 O-schools 143 Santo Domingo 173 Santiago 199 Others3 161 Grade 12 Public schools 165 F-schools 186 O-schools 135 Santo Domingo 174 Santiago 169 Othersa 143 15.4 307 13.8 19.5 311 17.3 15.1 262 12.2 15.8 291 13.7 18.5 281 17.0 16.1 308 13.1 17.3 293 15.8 23.7 269 19.2 17.5 219 14.8 19.6 223 15.5 22.1 294 19.3 17.3 264 14.8 a Others= next ten largest cities 175 Table J2 Analysis of Variance Testing Gender Differences in Grade 11 Public Schools Source of variation Sum of squares Degrees of freedom Mean square Between Within Total 314.257 14 216.741 14 530.998 1 493 494 314.257 28.837 29.415 10.90 * *p<0.05 Table J3 Analysis of Variance Testing Gender Differences in Grade 11 Private F-Schools Source of variation Sum of squares Degrees of freedom Mean square Between Within Total 596.148 21 159.116 21 755.263 1 511 512 596.148 41.407 42.491 14.40 * p<0.05 Table J4 Analysis of Variance Testing Gender Differences in Grade 11 Private O-Schools Source of S u m of squares Degrees of Mean square F v a r i a t i o n freedom Between 788.308 1 788.308 24 .44* W i t h i n 12 999.089 403 32.256 T o t a l 13 787.398 404 34.127 p<0.05 Table J5 Analysis of Variance Testing Gender Differences in Grade 11 in the Region of Santo Domingo Source of S u m o f s q u a r e s Degrees of Mean square F v a r i a t i o n freedom Between 465.366 1 465.366 10.70 * W i t h i n 20 093.590 462 43.493 T o t a l 20 558.957 463 44.404 *p<0.05 177 Table J6 Analysis of Variance Testing Gender Differences in Grade 11 in the Region of Santiago Source of „ f Degrees of A / r„ o r, _„,,_„, ^ Sum of squares 6 Mean square F variation freedom Between 254.061 1 254.061 6.93 * Within 17 534.687 478 36.683 Total 17 788.748 479 37.137 *p<0.05 Table J7 Analysis of Variance Testing Gender Differences in Grade 11 in the Region of the next ten largest cities Source of o r Degrees of A / . „ „ „ „ . „ . « , c Sum of squares 6 Mean square F variation freedom Between 941.935 1 941.935 32.73* Within 13 442.304 467 28.784 Total 14 384.239 468 30.736 *p<0.05 178 Table J8 Analysis of Variance Testing Gender Differences in Grade 12 Public Schools Source of v a r i a t i o n S u m of squares Degrees of freedom Mean square Between W i t h i n T o t a l 222.572 20 310.418 20 532.989 1 456 457 222.572 44.540 44.930 5.00* *p<0.05 Table J9 Analysis of Variance Testing Gender Differences in Grade 12 F- Schools Source of v a r i a t i o n S u m of squares Degrees of freedom Mean square Between W i t h i n T o t a l 2 177.377 36 279.445 38 456.822 1 453 454 2 177.377 80.087 84.707 27.19 * *p<0.05 179 Table J10 Analysis of Variance Testing Gender Differences in Grade 12 O- Schools Source of S u m o f s q u a r e s Degrees of Mean square F variation freedom Between 644.091 1 644.091 14.07 * Within 16 116.846 352 45.786 Total 16 760.938 353 47.481 *p<0.05 Table J l l Analysis of Variance Testing Gender Differences in Grade 12 in the region of Santo Domingo Source of 0 , Degrees of wO O T , «,„,,„,._ F Sum of squares 6 Mean square F variation freedom Between 1643.840 1 1 643.840 19.66 * Within 33 031.253 395 83.623 Total 34 675.093 396 87.563 *p<0.05 180 Table J12 Analysis of Variance Testing Gender Differences in Grade 12 in the region of Santiago Source of Sum of squares Degrees of Mean square F variation freedom Between 854.500 1 854.500 16.70 * Within 23 587.867 461 51.167 Total 24 442.367 462 52.906 *p<0.05 Table J13 Analysis of Variance Testing Gender Differences in Grade 12 in the region of the next ten largest cities Source of S u m o f s q u a r e s Degrees of Mean square F variation freedom Between 561.098 1 561.098 12.39 * Within 18 347.786 405 45.303 Total 18 908.885 406 46.574 *p<0.05 A P P E N D I X K Weighted Mean p-values of Grade 12 Responses to TLMDR Items and Comparison of Grade 12 and Grade 8 Responses Table KI Weighted Mean of Percent Correct Responses for Grade 12 Santo Next ten weighted Domingo Santiago largest cities mean A l Public 28.57 51.55 35.04 42 F-Schools 54.48 35.09 38.85 50 O-Schools 31.51 44.27 38.16 35 National 41 A2 Public 70.88 90.06 63.25 68 F-Schools 86.90 88.89 72.66 84 O-Schools 73.97 79.39 71.05 74 National 73 A4 Public 12.64 21.22 11.11 13 F-Schools 40.00 17.54 13.67 30 O-Schools 12.33 15.27 10.53 12 National 16 A13 Public 36.81 46.58 42.74 41 F-Schools 64.14 50.29 51.80 59 O-Schools 32.88 54.96 43.42 37 National 44 A30 Public 34.07 46.58 35.04 36 F-Schools 55.17 46.78 46.76 52 O-Schools 36.99 46.56 27.63 35 National 39 A33 Public 15.93 31.06 17.09 18 F-Schools 35.86 43.27 22.30 34 O-Schools 8.22 28.24 22.37 13 National 20 A3 5 Public 16.48 27.95 18.80 19 F-Schools 49.66 54.39 32.25 46 O-Schools 23.29 50.38 28.95 27 National 28 Bl Public 28.57 54.66 26.50 30 F-Schools 70.34 50.29 39.57 59 O-Schools 32.88 41.22 34.87 34 National 38 Table continues on next page. Santo Next ten weighted Domingo Santiago largest cities mean B4 Public 26.92 40.37 20.51 24 F-Schools 41.38 39.77 29.50 38 O-Schools 26.03 41.22 21.71 26 National 28 B9 Public 57.14 72.67 61.54 61 F-Schools 82.76 79.53 69.78 79 O-Schools 52.05 67.94 53.95 54 National 62 B12 Public 30.22 41.61 37.61 36 F-Schools 52.41 50.88 33.09 47 O-Schools 17.81 32.82 24.34 21 National 33 B15 Public 21.43 32.30 23.08 23 F-Schools 51.03 43.86 33.81 46 O-Schools 17.81 26.72 14.47 18 National 26 B19 Public 9.89 34.16 22.22 19 F-Schools 42.76 32.16 33.81 39 O-Schools 9.59 24.43 11.84 11 National 21 B35 Public 12.64 17.39 18.80 17 F-Schools 18.62 23.39 16.55 19 O-Schools 10.96 22.90 17.11 14 National 16 184 1. 0.40 X 6.38 is equal to: P F 0 National 2. Miguel is 4 years older than Rita. Rita is 11 years younger than Daniel. Daniel is 12 years old. How old is Mi-guel? A2 A B C D 3 years old 5 years old 14 years old 19 years old 4. 3.23X10 is equal to A 0.0000323 B 3.23000 C 32,300 D 323,000 E 32,300,000 u o o cx o CD o o 100 "1 100 i 80 60 40 20 • Grade 8 Grade 12 • Grade 8 H Grade 12 rJ P F 0 National 185 13. P F 0 National * C 12.5 D 15 E 25 30. The volume of a rectangular box with dimensions: 10 cm long, 10 cm wide and 7 cm high, is equal to: A 27 cm3 3 B 70 cm C 140 cm3 3 D 280 cm A30 E 700 cm o <D 1_ L . O O I 1 0 0 -i • Grade 8 Grade 12 P F 0 National 186 33. Two stores have a television sale. One of them offers a 10% discount and the other one offers a 15% discount. What is the dif-ference in price between the two stores for a regular priced television of $200. A $5 B $10 C $15 D $85 E $90 35. o CD I_ L. O O I 'A '/>* z 4 Li 4 • 1 square unit Each of the small squares in the figure measure one square unit. What is the area of the shaded figure to the nearest square unit? A B C D E 23 square units 20 square units 18 square units 15 square units 12 square units o CD 1_ 1_ O o I o. 100 n 80 60 40 20 100 n 80 60 A33 • Grade 8 M Grade 12 0 National A35 • Grade 8 M Grade 12 F 0 National BI \ + is equal to: 5 8 A B D 5_ 13 _5_ 40 _6_ 40 1P_ 15 i i 40 100 -i 80 60 o CD L. S 40 i o. 20 -• Grade 8 H Grade 12 P F 0 National 187 There are 35 students in a class. J- of 5 p them come to school by bus, another — 5 come by bicycle. How many come to school by other means? B4 A 7 B 14 C 21 D 28 E 35 100 -i o o I a. • Grade 8 Grade 12 P F 0 National If there are 300 calories in 100 grams of certain food, how many calories are there in 30 grams of that food? B9 A B C D E 90 100 900 1,000 9,000 In the figure, lines PQ and FS are intersected in O. The sum of the measures of angles 1 and 2 is: A 15° B 30° C 60° D 180" o 100 -i 80 -40 20 • Grade 8 Grade 12 P F 0 National B12 • Grade 8 M Grade 12 P F 0 National E 300° 188 15. On the level ground, a boy 5 feet tall casts a shadow 3 feet long. At the same time a nearby telephone pole 45 feet high casts a shadow the length of which, in feet, is 100 -i 80 -A 24 60 -B 27 rect • C 30 •cor 40 -D 60 20 -E 75 • 19. ° 50 j 100 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l On the above ruler, the reading indicated by the arrow is between: A B C D E 51 and 52 57 and 58 60 and 62 62 and 64 64 and 66 o CD 35. * B C D E The volume of the prism in the figure is: A 90 cubic centimeters 300 cubic centimeters 325 cubic centimeters 600 cubic centimeters 650 cubic centimeters <_> CD o o 100 -i 80 60 S 40 i o. 20 -100 n 80 60 40 20 B15 • Grade 8 M Grade 12 0 National B19 • Grade 8 M Grade 12 0 National B35 • Grade 8 M Grade 12 0 National A P P E N D I X L Weighted Mean p-values of Grade 12 Responses to SIMS Items and Comparison of Grade 12 with other Countries Table LI Weighted Mean of Percent Correct Responses for Grade 12 weighted m e a n Santo Domingo Santiago Next ten largest cities A16 Publ ic F-Schools O-Schools Dom. Rep. Int'l Mean A32 Publ ic F-Schools O-Schools Dom. Rep. Int'l Mean A34 Publ ic F-Schools O-Schools Dom. Rep. Int'l Mean Publ ic F-Schools O-Schools Dom. Rep. Int'l Mean Publ i c F-Schools O-Schools Dom. Rep. Int'l Mean Publ ic F-Schools O-Schools Dom. Rep. Int'l Mean Publ ic F-Schools O-Schools Dom. Rep. Int'l Mean B2 B3 B6 B14 19.23 20.69 9.59 7.14 19.31 4.11 18.68 28.97 15.07 14.29 19.31 10.96 15.38 17.24 9.59 2.2 11.03 2.74 3.85 9.66 4.11 24.84 26.9 18.32 8.07 11.7 8.4 36.65 38.01 39.69 17.39 21.05 24.43 21.74 24.56 23.66 7.45 7.02 4.58 9.32 11.11 8.4 15.38 19.42 15.79 6.84 9.35 6.58 14.53 20.86 30.92 17.95 15.11 27.63 23.08 14.39 14.47 3.42 6.47 4.61 8.55 3.6 7.89 17 21 12 16 56 7 16 5 8 60 18 28 21 21 65 17 19 16 17 30 21 18 12 17 38 3 9 3 5 41 7 8 5 7 43 Table continues on next page. Santo Next ten weighted Domingo Santiago largest cities mean B22 Public 9.89 9.94 14.53 13 F-Schools 26.21 18.71 15.83 22 O-Schools 5.48 18.32 17.11 10 Dom. Rep. 14 Int'l Mean 49 B26 Public 5.49 10.56 6.84 7 F-Schools 8.97 12.87 10.07 10 O-Schools 4.11 6.11 5.92 5 Dom. Rep. 7 Int'l Mean 62 B28 Public 6.04 8.7 13.68 11 F-Schools 17.93 14.04 15.11 17 O-Schools 8.22 16.03 7.89 9 Dom. Rep. 11 Int'l Mean 71 ItemA16 or SIMS 012 Which of the following is (are) true for all values of 8 for which the functions are defined? I sin (-0) = - sin e II cos (-9) = - cos 0 III tan (-0) = - tan 0 A I only B II only C III only * D I and III only E II and III only Hong Kong England/Wales Japan Belgium (Flemish) Canada (Ontario) Sweden Belgium (French) Finland Scotland Int'l Mean Israel New Zealand United States Hungary Thailand Canada (B. C.) F-schools Public Schools Dominican Republic O-schools 100 Mean Scores Item A32 or SIMS 080 If x > 0, y > 0, and xny, then is equal to: Vx + y/T x -y Vx - Vy x-y 1 V x - y i i vx" v7 Vx + yfj 2 2 x -y V x - V7 Japan Hong Kong Finland England/Wales Belgium (Flemish) Canada (Ontario) Canada (B. C.) Int'l Mean Belgium (French) Israel Sweden New Zealand Thailand Scotland Hungary United States F-schools Dominican Republic Public School O-schools 1: z 66 64 63 63 9 52 z I: 48 48 45 44 42 42 I 20 40 —T~ 60 z 7* 92 92 I— 80 100 Mean Scores Item A34 or SIMS 038 A certain number of students are to be accommo-dated in a hostel. If two students share each room, then two students will be left without any room. If three students share each room, then two rooms will be left unoccupied. How many rooms are there in the hostel? A 6 * B 8 C 10 D 12 E 14 Hong Kong Japan Sweden England/Wales Belgium (French) Israel Int'l Mean Finland Scotland Belgium (Flemish) Hungary Canada (Ontario) New Zealand Thailand United States Canada (B. C) F-schools Dominican Republic O-schools Public Schools 7* 7* 85 85 7 70 69 XV\V\V\V\\\\VW\\\\-VsWy 64.9 64 J 63 20 40 60 - I — 80 100 Mean Scores Item B2 or SIMS 078 If a = 2 3 - 5 2 - 7 and b= 3 2 • 53 • 7* • 11, then the least common multiple (LCM) of a and b is: 2 •3 5 7 • 11 3 2 2 •3 3 5 2 •7 • 11 3 2 2 •3 S 5 3 •7 • 11 3 2 2 •3 6 5 2 •7 • 11 Hong Kong Japan Thailand Hungary Canada (B. C.) Int'l Mean Belgium (Flemish) England/Wales United States Israel Belgium (French) F-schools Finland Dominican Republic Public Schools O-schools Canada (Ontario) Sweden Scotland New Zealand 0 20 40 60 80 100 Mean Scores Item B3 or SIMS 003 An operation * on the set of real numbers is commutative if, for every two real numbers x and y, x * y = y * x. Which of the operations defined below is commutative? A x * y = x + xy B x*y=x-y C x*y = x (x + y) * D x * y = xy (x + y) E x * y = x2 + xy2 + y4 Japan Hong Kong England/Wales Finland Sweden Int'l Mean New Zealand Scotland Belgium (Flemish) Thailand Canada (Ontario) United States Israel Belgium (French) Public Schools Canada (B. C.) F-schools Dominican Republic Hungary O-schools A ^ V \ \ ^ \ ^ ^ 7 38.1 { - ^ 38 7 39^ y<9 20 -1— 40 7 7 4 60 i 80 100 Mean Scores 197 Item B6* or SIMS 004 If a * 0, then 3a2 x 3a 2 is equal to: A B C D * E 9a * 9a 0 1 9 "Item B6 T h i s Item h a s a slight variat ion f r o m the original S I M S . If a *0, then 3a Jx 2a equal to: A 6 a ^ B 6a C 0 D 1 * E 6 Hong Kong Japan Sweden England/Wales Belgium (French) Finland Belgium (Flemish) Int'l Mean Canada (B. C.) Scotland Canada (Ontario) New Zealand Israel United States Hungary Thailand F-schools Dominican Republic O-schools Public Schools 7 80 47 46 45 43 A \ ^ \ \ \ ^ ^ \ ^ ^ \ y 4 i . i 5* 40 19 ? 18 j iy " " "V 29 7 s - T -20 40 - T -60 80 100 Mean Scores Item B14 or SIMS 081 If x and y are real numbers, for which x can you define y by y = j , ? V 9 - X 2 A All x, except x = 3 B All x, except x = 3 and x = -3 C x<-3 or x > 3 * D -3< x<3 E x<3 Finland Belgium (Flemish) Hong Kong Belgium (French) Japan Hungary England/Wales Canada (Ontario) Israel Int'l Mean Sweden Thailand New Zealand United States Scotland Canada (B. C.) F-schools Public Schools Dominican Republic O-schools 100 Mean Scores Item B22 or SIMS 068 In the figure above, PQ J. OQ and RS J- OQ . If the measures of OQ and of OR equal 1, and 6 is the measure of z^ POQ , then the measure of segmen PQ is equal to: * A sin 6 B cose C tane D 2 sin 8 E 1 - cos e Hong Kong Israel Japan England/Wales Finland Sweden Int'l Mean Belgium (Flemish) Hungary Belgium (French) Canada (Ontario) New Zealand Thailand United States Canada (B. C.) Scotland F-schools Dominican Republic Public Schools O-schools z / 48 7 7 89 72 55 20 40 60 80 100 Mean Scores Item B26 or SIMS 097 A freight train traveling at 50 km/h leaves a station 3 hours before an express train which travels in the same direction at 90 km/h. How many hours after leaving will it take the express train to overtake the freight train? B C ^ 5 * D — 4 E 18 4 Hong Kong Japan Sweden Israel England/Wales Finland Int'l Mean Belgium (Flemish) New Zealand Canada (B. C.) Hungary Scotland Canada (Ontario) Belgium (French) United States Thailand F-schools Public Schools Dominican Republic O-schools 7 Z7 93 T?6 7 ^ 56 ^ 55 754 50 20 40 I— 60 80 100 Mean Scores Item B28 or SIMS 087 If log A = n, then logA is equal to: A n + 2 2 * B n n C 2 D 2n E n-2 Hong Kong England/Wales Japan Sweden Israel Finland Belgium (Flemish) Belgium (French) New Zealand Infl Mean Canada (Ontario) Thailand Hungary Canada (B. C.) Scotland United States F-schools Dominican Republic Public Schools O-schools 0 20 40 60 80 100 Mean Scores
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Mathematics achievement in the Dominican Republic : grade 12 Crespo Luna, Sandra M. 1990
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Title | Mathematics achievement in the Dominican Republic : grade 12 |
Creator |
Crespo Luna, Sandra M. |
Publisher | University of British Columbia |
Date Issued | 1990 |
Description | The general goal of the present study was to assess mathematics achievement at the end of Grade 12 in the Dominican Republic, with particular attention to school and regional differences, as well as gender differences. Also, gains in achievement were examined by comparing the achievement of students in Grade 12 to that of students finishing Grade 11. In addition, the performance of Grade 12 students was compared to that of Grade 8 students as assessed in the Teaching and Learning of Mathematics in the Dominican Republic (TLMDR) study and to that of students from other countries in the Second International Mathematics Study (SIMS). The sample included 1271 students in Grade 12 and 1413 in Grade 11, distributed over 49 schools. Three types of schools were sampled, public schools, and two kinds of private schools. They were urban schools located in the twelve largest cities of the country. These cities were grouped into three regions of similar size. The mathematics test consisted of 70 multiple-choice items distributed over two test forms. Students' scores were analyzed to assess how much mathematics students in Grade 12 know. Grade 11 data were used as a surrogate for pre-test scores to estimate gains in achievement. School means were used in an analysis of variance designed to examine the effect of school type and region on mathematics achievement. Males' and females' scores were used to analyze gender differences in achievement at the item level, and within each of the school types and regions in the sample. Grade 12 students' responses to 14 items were compared to those of Grade 8 students. Finally, Grade 12 students' responses to 10 items were compared to those of students from other countries in SIMS. Among the findings of this study were: 1. Students in Grade 12 scored poorly on the mathematics test. Grade 11 and Grade 12 students obtained similar achievement levels which indicated that the achievement gains between the two grades were very small. 2. School type and region were found to significantly affect mathematics achievement, but no interaction effect was found. 3. The comparison of school type means showed that only one type of private school significantly outperformed public schools. This type of school also outperformed the other type of private school. 4. The comparison of region means did not produce the predicted outcome. The pairwise comparisons showed that none of the regions was significantly different from the other, despite the fact that the region factor was significant. 5. The analysis of gender differences in mathematics achievement showed that males performed significantly better than females. At the item level, males outperformed females on only 19 items. Most of these items dealt with geometry, or were at the application level. 6. Gender differences favoring males were found to be independent of school type and region. 7. Comparison between Dominican Grade 12 and Grade 8 students revealed that mathematics achievement improved between the grades for most items. 8. Dominican performance was very poor on the SIMS items and it was far behind that of other countries. |
Subject |
Mathematics -- Study and teaching (Secondary) -- Dominican Republic |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-10-27 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0098389 |
URI | http://hdl.handle.net/2429/29585 |
Degree |
Master of Arts - MA |
Program |
Education |
Affiliation |
Education, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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