Creative Education 2013. Vol.4, No.10, 663672 Published Online October 2013 in SciRes (http://www.scirp.org/journal/ce) http://dx.doi.org/10.4236/ce.2013.410095 Copyright © 2013 SciRes. 663 Exploring Teaching Performance and Students’ Learning Effects by Two Elementary Indigenous Teachers Implementing CultureBased Mathematics Instruction WeiMin Hsu1, ChihLung Lin2*, HueyLien Kao1 1Graduate Institute of Mathematics and Science Education, National Pingtung University of Education, Pingtung, Taiwan, China 2Department of Computer Science, National Pingtung University of Education, Pingtung, Taiwan, China Email: *clin@mail.npue.edu.tw Received August 2nd, 2013; revised September 2nd, 2013; accepted September 9th, 2013 Copyright © 2013 WeiMin Hsu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This study aims to probe into the teaching performance and the effects of implementation of culturebased mathematics instruction by two indigenous teachers. By case study, this study treats two Paiwan elemen tary school teachers as the subjects and collects data by the design of teaching plans, instructional obser vations, video recordings, and mathematical cognitive tests. The researcher thus explores their cul turebased curriculum design, instructional implementation, and the effect on Grade 5 and Grade 6 Pai wan students’ learning performance of mathematics. The findings demonstrate that prior to implementa tion of culturebased mathematics instruction, mathematics learning performance of the students of the two teachers was behind those of other counties, cities, and schools. The two teachers adopt three types of instructional design, namely, Paiwan culture and festivals, stories and traditional art, and practice mathe matics questions upon cultural situations by teacher demonstration, individual problemsolving, and group discussion. After the teachers practice 23 and 31 units of culturebased mathematics instruction, the re searcher finds that the gap of learning performance between Paiwan students and those in other cities, counties, and schools is reduced, which demonstrates that culturebased mathematics instruction can en hance Paiwan students’ learning performance of mathematics. Keywords: CultureBased; Indigenous Students; Mathematics Instruction; Paiwan Research Background and Purpose Minority students’ mathematics learning is the focus of the governments and related researchers in many countries. They not only suggest that “all students” should be successful in mathematics learning (Artzt & ArmourThomas, 2002; Na tional Council of Teachers of Mathematics (NCTM), 2000; Rodriguez, 2005), but also determine better instructional strate gies in mathematics for minority students (Boaler & Staples, 2008; Gutstein, 2003; Huang, 2006) in order to accomplish educational equity. Mathematics knowledge means more than the students’ subject learned in schools; it also plays the role as a critical filter and is the most important indicator of individual success (Ernest, 1998). Moreover, mathematics is one of the most difficult subjects for minority students (Ensign, 2005; Rodriguez, 2005). In Taiwan, although many researches have investigated indigenous people’s mathematics learning, most focused on “the comprehension” of indigenous students’ learn ing performance of mathematics, and attempted to interpret indigenous students’ inferior learning performance of mathe matics per individual, level of school education, and environ ment (Chi, 2001; Hsu & Yang, 2009; Li, 2006; Tan & Lin, 2002), but rarely focused on the effects of mathematics cur riculum and instruction of students’ mathematics learning. However, teachers’ mathematics instruction is the key that directly influences students’ mathematics learning (Gutstein, 2003; Henningsen & Stein, 1997; Hiebert & Grouws, 2007; Hsu, 2011; Stein, Remillard, & Smith, 2007). In societies of multiple cultures, two principal objectives of mathematics instruction are “teaching for understanding” and “teaching for diversity” (Rodriguez, 2005). In order to fulfill the instructional goals of “teaching for understanding”, it is necessary to connect the students’ new learning content with previous experience when designing curriculum (Wiske, Franz, & Breit, 2005). Thus, the students will have meaningful learn ing and be able to comprehend mathematics concepts; in order to accomplish “teaching for diversity”, it is important to value the students’ language and culture in curriculum planning, as knowledge is part of culture and each student is a unique indi vidual with different cultural backgrounds (Barnes, M. B. & Barnes, L. W., 2005). In other words, in order to fulfill the goals of mathematics instruction under multiple cultures, curriculum must include the students’ life experience and culture in cur riculum design in order to match the equity emphasized in mathematics instruction. This view meets many suggestions for minority learning. For instance, Cummins (1986) suggested that including the students’ language and culture in curriculum design *Corresponding author.
W.M. HSU ET AL. can enhance their learning performance; Leonard and Dantley (2005) proposed “culturally relevant teaching”, and indicated that students should be empowered through instruction, and their culture and life experience could be included in curricu lum design in order to lead to the success of mathematics learning for minority students; according to Ensign (2005), for all students, when culture both in and out of schools is closely connected, they were more likely to achieve successful mathe matics learning; by planning and implementation of mathemat ics curriculum, based upon life experience, and Gutstein (2003) effectively enhanced minority students’ learning performance of mathematics. The objectives of mathematics instruction in multiple cul tures, and the views related to minority students’ mathematics learning, demonstrate the importance of “culturebased” in struction for the students’ mathematics learning. In Taiwan, although there exists research on indigenous students’ mathe matics learning, as well as the factors of their learning per formance in mathematics (Chi, 2001; Hsu & Yang, 2009), few studies investigated the influence of culturebased mathematics instruction on indigenous students’ mathematics learning. Al though Paiwan is one of the main tribes in southern Taiwan, almost no research has focused on the effect of Paiwan culture based instruction on Paiwan students’ mathematics learning. Thus, this study treats two Paiwan elementary school teachers as the subjects, and by case study, probes into the teachers’ implementation of culturebased mathematics instruction and effect of culturebased instruction on Paiwan students’ mathe matics learning in order to enrich our comprehension of mathe matics instruction and learning of indigenous students. Literature Review Relationship between Culture and Mathematics Learning Mathematics has been regarded as cultural neutral or value neutral knowledge system. For instance, the “angle sum of a triangle is 180 degrees”, is a universal principle (Bishop, 1988); however, anthropologists’ research gradually overthrew this idea, and they suggested that mathematics was the product of human cultural development. They studied the development of mathematics knowledge from the perspective of culture, and proposed the concept “ethnomathematics” (D’Ambrosio, 1985). In studies on the importance of mathematics knowledge and its contribution to life, suggested that mathematics knowledge was part of the human culture (Barton, 2009; Orton & Frobisher, 1996). In many countries, mathematics history has been gradu ally included in textbooks (Fauvel & Van Maanen, 2000), which allows students to recognize the evolution of mathematics con cepts, and it suggests, “mathematics is a human cultural prod uct”, which changes previous views of mathematics knowledge. Bishop (1988) elaborated on the relationship between mathe matics and culture and suggested that different nations develop their own mathematics knowledge through six common activi ties of humans, including counting, locating, measuring, de signing, playing, and explaining. Through such activities, they can create many important concepts related to mathematics knowledge. Anthropologists suggest that mathematics knowl edge is produced through “exploration, invention, use of proper symbols, symbol normalization, transmission, and sharing within the culture” (Gerdes, 1996). Barton (2009) suggested that mathematics knowledge is the knowledge system of hu mans, which aims at understanding numbers, relationships, and spaces encountered in daily life, thus, mathematics knowledge is closely related to social culture. In addition, many empirical studies demonstrate that, in Tai wan, the reason for indigenous students’ inferior mathematics learning performance can be attributed to “cultural differences”. For instance, in traditional indigenous cultures, the people have language instead of words, thus, they cannot develop logic and reasoning by words. When encountering complicated mathe matical problems, they will not be able to comprehend the meanings of the questions and solve the problems (Chien, 1998); indigenous tribes emphasize sharing rather than compe tition (Chien, 1998), and they use “approximation” rather than requiring precise numbers (Huang, 2002), which is different from severely competitive study environments and precision of mathematics. Thus, Chi (2001) suggested that, due to cultural differences, there was a gap between indigenous students’ selfconcepts constructed in daily life and the science concept, as instructed in schools. Mathematics Curriculum and Instruction Suitable for the Minority According to the relationship between mathematics and cul ture, as well as related research findings, effective instruction must allow students to learn mathematics based on familiar culture and life experience. Thus, students can acquire the mathematics concepts from their original knowledge and ex perience. During instruction, teachers should reduce the gap between the students’ original reasoning model and important mathematics concepts (Anderson, 2003), as “cultural” differ ence might result in indigenous students’ inferior learning per formance of mathematics. Based on past research and literature, this study attempts of determine mathematics curriculum and instruction suitable for minority students. Requirements of CultureBased and Life Experience Curriculum Rodriguez (2005) proposed two major goals of mathematics instruction for multiple cultures: “teaching for understanding” and “teaching for diversity”, which match “studentcentered” and “understandingbased” purposes in the current reform of mathematics instruction (Hudson & Miller, 2006; Kilpatrick & Silver, 2000). In order to accomplish “teaching for understand ing”, the design of curriculum based upon students’ life ex periences (Anderson, 2003), connect mathematics concepts with students’ previous experiences and allow students to en gage in active exploration and open discussion. Thus, students can become involved in meaningful learning and gain an un derstanding of course content (Wiske et al., 2005). “Teaching for diversity” should include students’ language and culture content in courses and curriculum design (Barton, 2009; Cum mins, 1986), as this combination can lead to minority students’ successful mathematics learning (Ensign, 2005). Thus, includ ing students’ life experiences and culture in mathematics cur riculum is the first step to enhance minority students’ learning performance of mathematics. Instruction upon Empower ment and Exploration After the inclusion of culture and life experience in mathe matics curriculum, how to implement instruction that effec tively enhances minority students’ mathematics learning per Copyright © 2013 SciRes. 664
W.M. HSU ET AL. formance and accomplishes the goals of mathematics instruc tion becomes the issue of focus. Cummins (1986) suggested that instruction should be based upon interaction and collabora tive learning. Thus, minority students are provided with oppor tunities in active participation, thinking, exploration, discussion, and opinion sharing. In class, students change from passive listening and receiving to active thinking and problemsolving in order to be successful in learning. Wiske et al. (2005) sug gested that, besides the students’ life experience, instruction should allow students to comprehend concepts by multiple representations, such as objects, images, and symbols. In addi tion, through exploration, students can comprehend concepts by continuous construction and feedback during interactions with others. Through the design of interesting openended questions, students can participate in discussions from different perspec tives, thus, increasing learning. According to M. B. Barnes and L. W. Barnes (2005), teachers should allow students to explore and help them to recognize the gap between the world they perceive and the world of science through the construction of bridges. In the process, the teachers should be sensitive and continuously review students’ comprehension and modify the instruction in order to provide the students with different assis tance and guidance. The above statements emphasize empow erment and exploration oriented mathematics instruction. Thus, students can explore the relationships of mathematics concepts, and through cooperation and dialogue, their mathematics learning will be enhanced (Feldman, 2003; Fuson et al., 2000). Statements related to minority students’ mathematics instruc tion reflect the current trends of mathematics instruction, plac ing emphasize on active participation, exploration, and con struction through comprehension (Artzt & ArmourThomas, 2002; Hudson & Miller, 2006; NCTM, 2000). Inspiration from Related Studies : Ef fec tiv eness of CultureBased Mathematics Instruction Mathematics concepts are originated from the daily life ex perience model (White & Mitchelmore, 2010), which is a knowledge system developed to solve the problems related to numbers, relationships, and spaces in daily life (Barton, 2009). Therefore, design of a mathematics curriculum and instruction must include the students’ cultural experiences in order to ac complish the goal of equity in mathematics learning. From the perspective of a mathematics curriculum, Gutstein (2003) probed into the effect on students’ mathematics learning by treating the students of immigrants from Latin America, lowincome work class, and studies in urban schools as the subjects. By curriculum and instructional design, he intended to allow students to learn the world through mathematics, culti vate mathematics ability, and change students’ attitudes toward mathematics. In order to connect mathematics learning and students’ life experiences, Gutstein not only selected the text book “Mathematics in Context”, but also designed mathematics questions matching reality. After implementation for two years, students could demonstrate their problem solving abilities by varied methods, and with effective communication. Boaler and Staples (2008) treated three high schools, with students of different backgrounds as the subjects, and con ducted a fiveyear study. They found that the students of the Railside high school were consisted of varied ethic groups, were mostly from lowincome families, and their mathematics performance was significantly lower than two high schools. Through cooperation and interaction, the teachers conducted instruction by openended questions regarding concepts with high cognitive demands (in the other two high schools, the teachers adopted more traditional and closedended questions). Testing after implementation for one year demonstrated that mathematics performance of the students in Railside was the same as those in the other two high schools. After implementa tion for two years, the mathematics performance of the students in Railside was significantly higher than those in the other two high schools. Moreover, the gap of mathematics performance among the students in different ethic groups in Railside was the least among the three schools, most students intended to study more advanced mathematics courses, and they were more posi tive toward mathematics learning. The findings of above two researches not only match the theoretical suggestion of minority students’ mathematics cur riculum and instruction, but also show the effects of enhance ment of minority students’ mathematics learning performance (including cognition and affection). In Taiwan, although there are 14 different indigenous tribes, few studies probe into the effects of culturebased curriculum or instruction on the stu dents’ mathematics learning. The research of Hsu and Yang (2009) demonstrated that when indigenous students were used to solving mathematics questions with regular processes and teachers’ direct demonstrations, the design of culturebased mathematics questions and group discussions would be resisted by students, and thus, would influence the effectiveness of in structional implementation. The findings of Hsu and Yang (2009) did not totally match the research results of Gutstein (2003), or Boaler and Staples (2008). This not only demon strates the complicated factors of indigenous students’ mathe matics learning, but also suggests that it is worthy to probe into culturebased mathematics instruction in Taiwan. Research Method, Subjects, and Field Research Method In Taiwan, there exist only primary studies on the enhancement of indigenous students’ learning performance of mathematics by culturebased mathematics instruction. In addition, teachers’ mathematics instructions are a highly personalized result with varied factors (Artzt & ArmourThomas, 2002; Hsu, 2011). Case study is suitable for both primary study and complicated issues for exploring the process and effectiveness of the teach ers’ implementation of culturebased mathematics instruction. Research Subjects Two teachers, Wei and Ling (pseudonym), who participate in this study, are Paiwan people. Currently they teach Grades 5 and 6 in the Happy Elementary School (pseudonym, HES) of the Paiwan tribe in Pingtung County, Taiwan. They have taught for 3 and 5 years, respectively. Before participating in the re search project, Wei and Ling tended to conduct traditional lec tures in mathematics instruction. After an explanation, they allowed the students to practice with questions. When solving the problems, Wei usually cooperated with the students through questions and responses. The questions were mostly close ended and with fixed answers. Ling asked the students to the stage only when encountering special questions. After integrat ing culturebased instruction, both teachers decided to change their teaching methods from the traditional means. Wei and Ling asked students to discuss and cooperate in class rather Copyright © 2013 SciRes. 665
W.M. HSU ET AL. than the traditional means, lecturing. They provided more op portunities for students to discuss their thinking and ideas on problem solving. Therefore, more student interaction and em powerment were found in their classrooms. Research Field HES is located in the indigenous tribe of Pingtung County, and is a small school, with one class for each grade. There are more than 60 students in the school, and they are Paiwan people. The school is at a distance of about 20 minutes drive from other towns in the neighborhood. Most of the students are nurtured by grandparents, and young adults mostly work out of town. Although the tribe is an independent community, it is not iso lated from the external world. There are two internet cafes and most of the students have cable TV at home. Common student hobbies after school are attending the internet cafes or watching TV. On holidays, the tribe hosts activities, thus, the students in HES approach both traditional culture and external information. Data Collection Instructional Observations and Video Recordings In order to probe into the two teachers’ implementations of culturebased mathematics instruction, the researcher collects the data of one semester by instructional observations and video recordings (From March to June 2011). Wei and Ling recorded 23 and 31 lessons in one semester. The focuses of observation and video recording were based on the suggestions of Stein et al. (2007) and Hsu (2011), and were the content and imple mentation of mathematics questions. Interviews and Documents Interviews were conducted immediately following the two teachers’ instruction, and lasted 10  20 minutes. The purpose was to learn the teachers’ aims to design the teaching plans, thoughts after implementation, students’ reactions, and future improvements. Throughout the interviews, the researcher in tended to probe into the teachers’ instructional implementation and determine the effects of the students’ reactions on teachers’ instructional implementation, and learn the teachers’ feelings regarding the implementation and objectives of mathematics instruction. Cognitive Test of Mathematics According to the specifications of the national curriculum outline of mathematics in 2003 (Ministry of Education, 2003), the researcher plans cognitive testing of mathematics. Specifi cations in Taiwan not only emphasize mathematics concept, but also indicate the cultivation of mathematics abilities. Thus, mathematics concepts and mathematics abilities become two critical dimensions in the design of cognitive testing. Regarding mathematics concepts, this study analyzes the concepts of mathematics learning required by students upon reaching the first semesters of Grades 5, 6, and 7 as the criteria to design tests. Mathematics abilities include “conceptual comprehen sion”, “calculation fluency”, and “problem solving” based on the ability indicators of the curriculum outline. Data Analysis Qualitative Data Qualitative data includes instructional observations and video recordings, as well as teachers’ interview records and docu ments. Regarding video recordings of instructions, the re searcher transcribes the instructional processes, treats the mathematics questions as units, and analyzes the content and implementation of mathematics questions in order to learn the teachers’ implementation of culturebased mathematics instruc tion. Regarding the content of mathematics questions, the re searcher reviews the suggestion of Stein et al. (2007) and Hsu (2011) on the analysis of mathematics instruction, as well as past literature related to minority’s mathematics instruction (Barton, 2009; Cummins, 1986). The researcher analyzes the sources and situations of mathematics questions. The sources include the total matches between the questions and the text books, partial revisions, and selfdesign (Llody, 2008). Ac cording to the sources, the researcher can recognize the teach ers’ roles and the role of textbooks in culturebased mathemat ics instruction. The situations of the questions, according to the relationship with Paiwan culture, are divided into cultural situa tions, common situations, and pure calculation. The analysis of the situations shows the culturebased degree and focus of the teachers’ instruction. Interviews and records will be transcribed and encoded. The encoding is based on “datecategory” and the date is shown by “monthday”. Quantitative Data Analysis of quantitative data refers to the analysis of cogni tive testing. After designing mathematics cognitive tests, the researcher invites two experienced mathematics education re searchers to conduct expert testing and review the formation of the twoway specification table, classification of tests, and con tents of tests in order to share opinions regarding the revision of the tests. After expert review and revision of the tests, the re searcher examines the quality of tests by the pretest. Regarding “from Grades 4 to 5”, there are 177 participants, in 7 classes, in 4 schools, in the pretest. Average difficulty is .58 and average discrimination is .54. Internal consistency reliability α is .89. Regarding “Grades 5 to 6”, there are 295 participants, from 12 classes, in 6 schools, in the pretest. Average difficulty is .62, average discrimination is .49, and internal consistency reliabil ity α is .86. Regarding “Grades 6 to 7”, there are 191 partici pants, in 7 classrooms, in 4 schools, in the pretest. Average difficulty is .53, average discrimination is .49, and internal consistency reliability α is .86. The pretest was conducted in October 2010. In pretesting, according to the percentages of the schools, through purposive stratified sampling, the researcher conducted pretesting on schools of different scales in Pingtung County and Kaohsiung City. By the end of June 2011, the posttesting was randomly conducted in the schools from pre testing. Reliability and Validity of Data Analysis Regarding qualitative data, the classifications (situations and implementations of mathematics questions) are not only based on the suggestions of Stein et al. (2007) and Hsu (2011), but also rely on three researchers to construct score reliability. First, the researchers explained the definitions of the categories to three researchers, and tested the analysis by the recordings of one class period. The results showed that, regarding the classi fication of situations and implementations of the questions, the three researchers’ analytical results are consistent. The sources of questions are compared with mathematics questions in text books in order to allocate them into “entirely the same as the Copyright © 2013 SciRes. 666
W.M. HSU ET AL. textbooks”, “partially the same as the textbooks”, or “entirely different from the textbooks”. Regarding quantitative data, according to the specifications and objectives of national cur riculum, each cognitive test of mathematics is turned into a twoway specification table. After editing the tests, the re searcher invited two experienced mathematics education re searchers to revise the tests, and then modify the pretest, the items, and construct proper difficulty, discrimination, and in ternal consistency reliability. Research Results Mathematics Learning Performance of the Students in HES before CultureBased Mathematics Instruction There are 2 large schools (at least 24 classes) in Pingtung County (PC), with 3 classes of participants in each school; 4 medium schools (12  23 classes), with 2 classes of participants in each school; and 6 small schools (under 11 classes) (includ ing HES), with one class of participants in each school. There are 12 schools, 20 classes, 466 Grade 5, and 475 Grade 6 stu dents in the pretest. Regarding Kaohsiung City (KC), there are 4 large schools, with 3 classes of participants in each school; and 2 medium schools with 2 classes of participants in each school. A total of 16 classes, 459 Grade 5 and 477 Grade 6 students participated in the pretest. The number of selected schools is based on the percentages of the schools in different counties and cities. Regarding the locations of the schools (Table 1), in the pre test, Grades 5 and 6 students in the HES demonstrate an aver age number of items with correct responses, lower than those in PC and KC. Since the numbers of the samples in the pretest are extremely different, the researcher conducted the analysis by the methods of Post hoc Comparison. Grade 5 students are analyzed by the Post hoc Comparison of Scheffe and LSD. It demonstrates that only KC and PC reveal significant differ ences (p < .001); Grade 6 students are analyzed by the Scheffe method, which shows that KC and PC reveal significant differ ences (p = .046). Through LSD, KC & PC (p < .001) and KC & HES (p = .023) show significant differences. Regarding the school scale (Table 2), Grade 5 students are analyzed by the Post hoc Comparison of the Scheffe and LSD method. The researcher demonstrates that there are significant differences between large and medium schools (p < .001), and between large and small schools (p < .001). Moreover, through LSD, large schools and HES reveal no significant differences (p = .06). Grade 6 students are analyzed by the Scheffe and LSD method, which shows that there are significant differences be tween large and small schools (p < .001), and between medium and small schools (p < .001). And through LSD, the researcher realizes that there are also significant differences between large schools and the HES (p = .049). Teachers’ Implementation of CultureBased Mathematics Instruction Instructional Design upon Cultural Situations and Conceptual Frameworks of the Textbooks Regarding lesson plan design before the instruction, the two teachers focused on the content in the textbooks (regarding the concern of national curriculum), and the design of each unit is based on one cultural theme or story. In the design of four units, Table 1. Mean and standard deviation of students’ pretest performance from different school locations. GradeLocation# of SubjectsMean SD Df F Sig. HES 9 22.56 2.10 PC 457 23.63 .40 Grade 5 KC 459 26.14 .32 2 12.90<.001*** HES 10 19.80 2.24 PC 465 22.00 .28 Grade 6 KC 477 24.10 .26 2 16.40.003** Note: **p < .01, ***p < .001. Table 2. Mean and standard deviation of students’ pretest performance from different school scale. GradeSchool Scale# of Subjects Mean SD Df F Sig. HES 9 22.56 2.10 Large 571 27.09 .25 Medium 253 21.14 .59 Grade 5 Small 92 21.51 .76 3 48.18<.001*** HES 10 19.80 2.24 Large 578 23.51 .24 Medium 269 23.24 .36 Grade 6 Small 95 19.86 .62 3 11.46<.001*** Note: ***p < .001. Wei used one type of culture and festivals (including a Paiwan wedding and a harvest ritual). In six units, Ling adopted three methods: culture and festivals (hunting season and wedding), stories (legend of the origin of Paiwan and Kua’s and Tai’s participation in harvest ritual), and traditional art (Paiwan handicraft and traditional art fair). Using line symmetry units (Paiwan handicrafts) designed by Ling as an example, the unit includes three important concepts: introduction of figures of line symmetry, introduction of symmetrical points, symmetrical edges, and symmetrical angles and drawing of line symmetry figures. Regarding each concept, Ling used related traditional handicraft figures as the teaching materials. For instance, in the introduction of line symmetry, Ling allows the students to ob serve line symmetry in the Figure 1. In culture and festivals, or stories in the design of teaching plans, in the first section of the unit, the teacher describes the situations related to the festivals or mythology, and designs mathematics questions for students’ problemsolving according to the situations. Before culturebased instruction, the two teachers would analyze the contents of the textbook, and extract the main con cepts of mathematics in the culturebased unit. Then they would consider cultural customs, festivals, stories, or traditional art as the base of the unit. Thus, in cultural situations, the students can learn the mathematics concepts in the textbook. Copyright © 2013 SciRes. 667
W.M. HSU ET AL. Figure 1. The figures of Paiwan traditional handicrafts. Mathematics Questions Are Mostly from the Textbooks, However, They Are Revised According to the Situations of Paiwan Cul ture 1) Sources of Mathematics Questions When designing culturebased lesson plans, the two teachers followed the conceptual framework of the textbooks, and adopted and designed the units according to related cultural situations. Thus, the sources of mathematics questions are mostly from the textbook. In 23 lessons, Wei used 105 mathe matics questions, for an average of 4.6 questions adopted in each lesson. In 31 lessons, Ling used 271 mathematics ques tions, for an average of 8.7 questions adopted in each lesson. The researcher compared mathematics questions adopted by the two teachers with the questions in the textbook, and realized that, from among the 105 mathematics questions of Wei, only 3 are entirely the same as those in the textbooks. The teacher describes the definitions and relationships according to the questions in the textbooks: 1) question is used in the unit “equa tion and problemsolving”. The question introduces the equa tion of the circumference of a rectangle; 2) questions are used in “problem solving”, and one describes the meanings of base and comparison. One question describes the definition of “price = cost × (1 + profit)”. The remaining 102 questions are “par tially the same” as the textbooks. Some questions are adopted into the situations of Paiwan culture, some add situations re garding students’ levels (nonPaiwan culture), and some are the questions of pure calculation. However, the types of the above three types of questions are the same as the textbooks. Among 271 mathematics questions, Ling used 130 questions entirely the same as the textbooks, 139 questions that are partially the same and 2 questions entirely different. Ling used many ques tions from the textbooks, as she adopts the practices in the textbooks. In addition, after finishing culturebased instruction, she mostly reviewed the questions in the textbooks with the students (she reviewed 27 out of 31 lessons). Thus, she used more mathematics questions than Wei. Ling suggests that there are key points in the questions or activities designed in the textbooks, and the editors have rich knowledge and experience. In order to allow the students to have complete learning and attempt to determine if the students learn the related mathemat ics concepts in culturebased situations, the teacher reviewed the questions in the textbooks, with the focus on the practices (0503 interviews). Regarding the questions that are partially the same, the types are the same as those in the textbooks; however, the numbers or situations are adopted. Regarding the 2 ques tions that are entirely different, they are the questions of review for Ling before implementing the units. For example, in “deci mal fraction multiplication and estimation”, the teacher first reviews addition and subtraction of decimal fraction (0315 observation); in “time calculation”, before the multiplication and division of time, the teacher first reviews addition (0411 observation). The sources of the questions used by the two teachers in the instruction are as shown in Table 3. 2) Situations of Questions Upon the situations of the questions, the researcher analyzes the types of mathematics questions adopted by the two teachers, and demonstrates that the types can be allocated into three categories: Paiwan culture situations, common situations, and pure calculation. Wei used culturebased questions the most, while Ling used three types of questions. The statistics of the types of mathematics questions used by the two teachers are shown in Table 4. Chisquare testing on the difference of the types of questions used by the teachers shows that Wei signifi cantly focused on Paiwan culture, while Ling does not demon strate significant differences (χ2 values of Wei and Ling are 55.6 and .49, respectively, and p value is .78). Implementation of the Questions is Based on Individual Problem Solving, and ProblemSolving Relies on Teacher Demonstrations The researcher studied two teachers’ implementation through mathematics questions as the units, and demonstrates that the two teachers both adopt three methods: teachers’ demonstration, the students’ individual problem solving, and group discussion. Teachers’ demonstration means after introducing mathematics questions, the two teachers directly explain the meanings of the questions and demonstrate the methods and results, or directly inform the students of the problemsolving method and proc esses for their calculations and do not allow them to have the opportunity to think, discuss, or share opinions. The students’ individual problem solving and group discussion mean after introducing the questions, the teachers ask the students to solve the problems individually or by group discussion. After the students complete the work, individually or in groups, the two teachers enact three types of actions: including providing the answers without description and explanation (the students can compare their answers with the correct ones), direct demonstra tion (sometimes they invite the students to write their solution on the board and the teachers directly explain the solution), and asking students to share their solutions on the board. Wei di rectly demonstrates 20 mathematics questions (19%), the stu dents’ solve 44 questions individually (41.9%), and solve 41 questions in group discussion (39.1%); Ling directly demon strates 29 questions (10.7%), the students solve 161 questions individually (59.4%), and solve 81 questions in group discus sion (29.9%). The two teachers mostly allow the students to solve the problems individually and rarely demonstrate the solution. However, Wei, in comparison to Ling, adopted group discussions more frequently. Statistics of the implementation of the two teachers’ mathematics questions are as shown in Table 5 . Copyright © 2013 SciRes. 668
W.M. HSU ET AL. Table 3. Sources of mathematics questions used by two teachers. Wei Ling Teachers Question sources Number (ratio) Number (ratio) Entirely the same 3 (2.9%) 130 (48%) Partially the same 102 (97.1%) 139 (51.3%) Entirely different 0 (0%) 2 (0.7%) Total 105 (100%) 271 (100%) Table 4. Types of mathematics questions used by two teachers. Wei Ling Teachers Type of questions Number (ratio) Number (ratio) Paiwan cultural situations 71 (67.6%) 94 (34.7%) Common situations 16 (15.2%) 92 (33.9%) Pure calculation 18 (17.1%) 85 (31.4%) Total 105 (100%) 271 (100%) Table 5. Statistics of the two teachers’ practices of mathematics questions. Wei Ling Teachers Implementation Number (ratio) Number (ratio) Teachers’ demonstration 20 (19.0%) 29 (10.7%) Only giving the answers9 (8.6%) 59 (21.8%) Teachers’ demonstration25 (23.8%) 71 (26.2%) Individual problem solving Students’ sharing 10 (9.5%) 31 (11.4%) Only giving the answers3 (2.9%) 19 (7.0%) Teachers’ demonstration27 (25.7%) 48 (17.7%) Group discussion Students’ sharing 11 (10.5%) 14 (5.2%) Total 105 (100%) 271 (100%) However, according to the problemsolving methods, Wei demonstrates 68.5% mathematics questions, only gives the answers to 11.5% of the questions, and asks the students to share 20% of the questions; Ling demonstrates 54.6% of the mathematics questions, only gives answers to 28.8% of the questions, and asks the students to share 16.6% of the questions. Thus, the two teachers mostly demonstrated and explained the questions, and only 20% of the mathematics questions are solved and shared by the students. Chisquare testing on the number of questions solved by the three methods adopted by the two teachers demonstrated that, the two teachers signifi cantly focused on teachers’ demonstrations (χ2 values of Wei and Ling are 59.83 and 61.25, respectively, and p values are 0). It means that although the two teachers adopted culturebased mathematics instruction, they tend to lead and demonstrate in problemsolving. Effectiveness of the Teachers’ Implementation of CultureBased Mathematics Instruction After the two teachers finished culturebased mathematics instruction, the researcher conducted posttests on the same classes that participated in the pretest: 2 large, 2 medium, and 3 small schools (including HES; among 3 schools, only 1 school has Grade 6 participants) in PC. One class was randomly se lected in each school. There were 143 Grade 5 students in 6 classes, and 140 Grade 6 students in 7 classes, from 7 schools participating in the posttest. Among the 4 large and 2 medium schools in KC, 1 class was randomly selected in each school. There were a total of 157 Grade 5 students and 168 Grade 6 students in 6 classes participating in the posttest. Tables 6 and 7 show the comparison of mathematics learning performances between the students in the HES, those in nonindigenous areas, and those in the schools of different scales in the posttest. According to Table 6, the researcher demonstrates that after culturebased mathematics instruction, Grades 5 and 6 students in the HES show mathematics learning performance no differ ent from those in PC or KC (p values are .36 and .27, respec tively). Grade 5 students’ average number of items with correct answers in the posttest is even higher than that of the students in PC and KC, while Grade 6 students’ performance is inferior to those in other cities and counties. However, in general, after culturebased mathematics instruction, Grades 5 and 6 students in the HES demonstrate similar mathematics learning perform ance to those in PC and KC. According to Table 7, after cul turebased mathematics instruction, variance analysis shows that mathematics learning performance of Grade 5 students in the HES is not different from those in the schools of different scales (p = .68), and their performance is even better than those in the schools of different scales. Regarding Grade 6 students, mathematics learning performance is significantly different (p = .04). The Scheffe method does not show the significance. According to LSD, the researcher realizes that the learning performances of the students in large and medium schools are significantly different (p = .01). However, there is no difference between HES and the schools of different scales. According to the average number of items with correct answers, the students’ performance in the HES remains inferior to those in the schools of other scales. Discussion In contemporary societies of multiple cultures, educational equity is an important objective of mathematics instruction. In order to accomplish educational equity, it must value student diversity and allow “all students” the opportunity to be suc cessful in mathematics learning (Gutstein, 2003; NCTM, 2000; Rodriguez, 2005). The proposal of educational equity resulted in minority students’ mathematics learning to gain notice. Many scholars have suggested culturebased and life experi ence curriculum and instruction, which is based upon interac tion and empowerment, in order to help minority students ap proach mathematics (Barnes, M. B. & Barnes, L. W., 2005; Cummins, 1986; Ensign, 2005). According to scholars’ suggestions regarding minority stu dents’ mathematics curriculum and instruction, the researcher realizes that in the two teachers’ culturebased mathematics instruction for Paiwan students, they tended to adopt culture based mathematics curriculum and mathematics questions. Copyright © 2013 SciRes. 669
W.M. HSU ET AL. Table 6. Mean and standard deviation of students’ posttest performance from different school locations. Grade School scale # of subjects Mean SD Df F Sig. HES 9 23.78 1.51 PC 134 22.18 .54 Grade 5 KC 157 23.12 .47 2 1.03.36 HES 10 17.70 1.69 PC 130 20.14 .58 Grade 6 KC 168 20.89 .54 2 1.32.27 Table 7. Mean and standard deviation of students’ posttest performance from different school scale. Grade School scale # of subjects Mean SD Df F Sig. HES 9 23.78 1.51 Large 182 22.99 .43 Medium 92 22.18 .65 Grade 5 Small 17 22.18 1.80 3 .5.68 HES 10 17.70 1.69 Large 158 21.51 .52 Medium 117 19.38 .64 Grade 6 Small 23 20.00 1.49 3 2.91.04* Note: *p < .05. Without changing the original conceptual framework and order of the textbooks, the teachers adopted units and questions into Paiwan culture related situations, including customs, festivals, legends, and traditional art, in order to trigger students’ learning motivation, where students can learn mathematics from the cultural situations. According to Paiwan students’ performance in pretests and posttests, in comparison to students of other schools, culturebased instruction and implementation should enhance Paiwan students’ mathematics learning performance. However, regarding instruction methods, although the two teachers adopted teacher demonstrations, individual problem solving, and group discussions, they tended to lead the prob lemsolving processes and results, and failed to adopt interac tion, exploration, and empowerment, as suggested by other scholars. The main reason is that the two teachers were adopt ing culturebased instruction for the first time; however, it is difficult to change one’s original teaching habits or patterns, particularly when theory suggests that the teachers’ past in struction or learning experiences are different (Rodriguez, 2005; Stuart & Thurlow, 2000). Therefore, it is understandable that the two teachers mostly demonstrated the lessons; however, if the two teachers were to adopt interaction and empowerment oriented instruction, as suggested by theory, would they effec tively enhance Paiwan students’ mathematics learning? The answer to this question will rely on further study. Both the teachers used the design and implementation of culturebased mathematics instruction. However, why does it seem that Grade 5 students’ learning effectiveness is more sig nificant than Grade 6 students? The reason can be that before the instruction, Grade 5 students’ learning performance is not significantly different from those in other areas. In addition, it can be because Ling was more involved and provided more opportunities of mathematics question practice. Regarding the design of unit plans, 6 culturebased unit plans are designed by Ling. Among the unit plans designed, Wei designed only one 1 independently (Mayi’s harvest ritual). Mayi’s wedding is the adoption of a unit plan designed by Ling. Teachers’ personal commitment and involvement are the keys to students’ learning effectiveness, as demonstrated in the research of Boaler and Staples (2008), and Gutstein (2003). However, how do the teachers’ involvement and offerings of practice opportunities influence indigenous students’ mathematics learning? This is an issue for further exploration by future research. Conclusions and Suggestions Conclusions This study treats two Paiwan teachers as subjects and probes into their implementation of culturebased mathematics instruc tion, and the possible effects on Grades 5 and 6 indigenous students’ mathematics learning. Regarding the implementation of culturebased mathematics instruction, this study demon strates that when two teachers design culturebased mathemat ics instruction, they rely on the conceptual framework in text books, and consider situations that include Paiwan culture, such as culture and festivals, stories, and traditional art of Paiwan; although most of the mathematics questions in the instruction are from the textbooks, they turn the situations in the questions into those related to Paiwan culture, while the types and struc tures remain the same. Ling uses more mathematics questions that are entirely the same as in the textbooks. The reason is that after implementing culturebased mathematics instruction, she was concerned about the representative nature of the content in the textbooks and reviewed the students’ learning effectiveness. Thus, she guides the students to review mathematics questions in the textbooks, which is why she used more mathematics questions than Wei (by almost two times). Regarding the situa tions of mathematics questions, Wei significantly used ques tions with cultural situations, whereas Ling focused on ques tions related to cultural situations, common situations, and pure calculation. Regarding the implementation of mathematics questions, although the two teachers use demonstrations, indi vidual problem solving, and group discussions, according to the problem solving methods and results, the two teachers signifi cantly relied on demonstrations, and rarely allowed the students to think and discuss. Although the two teachers’ culturebased instructional implementation is different from the views of theoretical argument in the literatures, particularly the empow erment of students’ learning, after their implementation of cul turebased mathematics instruction, the gap of mathematics learning performance between the students in the school and those in other regions, and with medium and large scales, is reduced. Grade 5 students’ performance is similar to those in the schools of Pingtung County and Kaohsiung City; moreover, their average number of items with correct responses is higher than those in schools of Pingtung County, Kaohsiung City, and of different scales. In addition to culturebased mathematics instruction, success can be related to the active involvement of Ling, and more practices with mathematics questions. However, the effect of these two factors should be further studied. Copyright © 2013 SciRes. 670
W.M. HSU ET AL. Suggestions Regarding the findings, this study proposes the suggestions below for future researchers or teachers. First, regarding the design of culturebased mathematics instruction, this study demonstrates that the teachers’ design is based on the concep tual framework in the textbooks and situations of Paiwan cul ture. Can we include the unique learning style of the indigenous students (particularly Paiwan students) in the design? How to combine them? What about effectiveness after implementation? Such questions will rely on further study to enhance the sig nificance of culturebased instruction. In addition, regarding implementation of instruction, Wei used more questions of cul tural situations, whereas Ling used questions of cultural situa tions, common situations, and pure calculation. How do we distribute different types of questions in order to enhance in digenous students’ mathematics learning? 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