Strengths and Challenges of Applying Singapore Mathematics in United States High School

This study examines the strengths and limitations of applying Singapore math techniques with high school students in a private school geometry class. A qualitative method with a constructivism framework was used to collect the data from surveys and interviews. The students were then introduced to the Singapore math's bar modeling techniques through solving a word problem activity. The students were all visibly impressed and full of praise of Singapore math's bar modeling techniques. Singapore math has influenced my teaching style that appeals to all of the students visually. My visually inclined teaching style will be used continually to engage my students in math. Singapore math's bar modeling techniques should have a place in high schools because it can help increase students' confidence in math and improve students' level of critical thinking and problem solving skills. Teacher training in Singapore math and choosing an appropriate Singapore math textbook are challenges. More studies are still needed to implement Singapore math successfully at the secondary level in the U.S. However, it is not possible to completely transfer everything about Singapore math over to solve the problems of U.S. educational system.

Singapore Mathematics in U.S. High School 1 STRENGTHS AND CHALLENGES OF APPLYING SINGAPORE MATHEMATICS IN UNITED STATES HIGH SCHOOL by Jonathan Hsu A thesis submitted in partial fulfillment of the requirement for the degree of Master of Education Westminster College Salt Lake City, Utah April 2013 Singapore Mathematics in U.S. High School 2 APPROVAL of a thesis/project submitted by Author's Name __________________________________________________________ School Department _______________________________________________________ Title of Thesis/Project _____________________________________________________ The above named master's thesis/project has been read by each member of the supervisory committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to the Westminster College Library. ______________________ ________________________________________________ Date Marilee Coles-­‐Ritchie, Chairperson ______________________ ________________________________________________ Date Nancy Garrison, Member Approved for the School ______________________ ________________________________________________ Date Dean, School Singapore Mathematics in U.S. High School 3 STATEMENT OF PERMISSION TO DUPLICATE THESIS & DEPOSIT/DISPLAY IN THE INSTITUTIONAL REPOSITORY Name of Author(s) ___________________________________________________________________________________ School/Department __________________________________________________________________________________ Title of Thesis/Project _______________________________________________________________________________ _________________________________________________________________________________________________________ With permission from the author(s), on the basis of an occasional and individual request, the staff of the Giovale Library of Westminster College has the right to make a copy of the above named thesis/project. The Giovale Library staff also has the right to mail or otherwise disseminate a copy to the requesting party and to be reimbursed by the requesting party for the cost of duplicating and mailing the thesis/project. I hereby give my permission to the staff of the Giovale Library of Westminster College to duplicate as described the above named thesis/project. Signature of Author(s) Date With permission from the author(s), the staff of the Giovale Library of Westminster College has the right to deposit and display an electronic copy of the above named thesis/project in its Institutional Repository for educational purposes only. I hereby give my permission to the staff of the Giovale Library of Westminster College to deposit and display as described the above named thesis/project. I retain owership rights to my work, including the right to use it in future works such as articles or a book. Signature of Author(s) Date The above duplication and deposit rights may be terminated by the author(s) at any time by notifying the Director of the Giovale Library in writing that permission is withdrawn. Singapore Mathematics in U.S. High School 4 Abstract This study examines the strengths and limitations of applying Singapore math techniques with high school students in a private school geometry class. A qualitative method with a constructivism framework was used to collect the data from surveys and interviews. The students were then introduced to the Singapore math's bar modeling techniques through solving a word problem activity. The students were all visibly impressed and full of praise of Singapore math's bar modeling techniques. Singapore math has influenced my teaching style that appeals to all of the students visually. My visually inclined teaching style will be used continually to engage my students in math. Singapore math's bar modeling techniques should have a place in high schools because it can help increase students' confidence in math and improve students' level of critical thinking and problem solving skills. Teacher training in Singapore math and choosing an appropriate Singapore math textbook are challenges. More studies are still needed to implement Singapore math successfully at the secondary level in the U.S. However, it is not possible to completely transfer everything about Singapore math over to solve the problems of U.S. educational system. Singapore Mathematics in U.S. High School 5 Table of Content Chapter 1: Introduction..………………………………………………………………………………………8 Vignette.……………………………………………………………………………………………………8 Statement of Topic and Purpose…………………………………………………………………9 Framework and Research Question………………………………………………………….10 Statement of Researcher………………………………………………………………………….12 Potential Significance and Limitations……………………………………………………...14 Chapter 2: Literature Review……………………………………………………………………………...19 Singapore Educational System and Mathematics………………………………………20 Singapore mathematics curriculum revisions and bar modeling……..23 International Mathematics Comparisons…………………………………………………..25 American Mathematics…………………………………………………………………………….26 Challenges in Applying Singapore Math in U.S. School Settings…………………..27 Textbooks…………………………………………………………………………………….27 Teachers………………………………………………………………………………………28 Difference between SMCF and NCTM……………………………………………………….29 Strengths and Challenges of Applying Singapore Math in High School……….30 Chapter 3: Methods……………………………………………………………………………………………33 Overall Approach…………………………………………………………………………………….33 Setting…………………………………………………………………………………………………….35 Participants…………………………………………………………………………………………….36 Data Gathering Methods and Rationale…………………………………………………….37 Data Analysis…………………………………………………………………………………………..40 Singapore Mathematics in U.S. High School 6 Validity and Trustworthiness…………………………………………………………………40 Ethical Considerations……………………………………………………………………………41 Conclusions……………………………………………………………………………………………44 Chapter 4: Data Analysis……………………………………………………………………………………45 Introduction…………………………………………………………………………………………..45 Influence of Singapore Math on Teaching and Learning Styles in Math……..45 Impact of Singapore Math on Students' Confidence and Abilities in Math…47 Visual Appeal of Singapore Math's Bar Modeling Technique…………………….53 Training Requirements on Teachers for Teaching Singapore Math…………..59 Summary………………………………………………………………………………………………61 Chapter 5: Implications and Recommendations for Applying Singapore Math….…63 Offer Visually Stimulating Math……………………………………………………….……..63 Improve Critical Thinking and Problem Solving Skills……………………….….…64 Future Research Recommendations………………………………………………….……66 Interviewing a whole class, a whole school and other schools…….…66 Measure pre-­‐ and post math tests using Singapore math………………66 Radical change in United States' Math Education…………………………………….67 Summary…………………………………………………………………………………………..…..68 References…………………………………………………………………………………………………….…70 Appendix A: SurveyMonkey Online Questions………………………………………………...…72 Appendix B: One-­‐on-­‐One Interview Questions……………………………...………………...…73 Appendix C: Transcripts of Interviews……………………………………………………..…...……75 Appendix D: Students' Work on Word Problems Activity……………………………..……129 Singapore Mathematics in U.S. High School 7 Appendix E: SurveyMonkey Results…………………………………………………………………135 Appendix F: IRB Form……………………………………………………………………………………..138 Singapore Mathematics in U.S. High School 8 Chapter 1 Introduction Vignette "Hello, may I speak to Dr. Rossi please?" "Yes, I am Dr. Rossi. Who is this?" "Hi Dr. Rossi. My name is Jonathan Hsu. I learned from the local newspapers that you are the person fighting to get the bill passed to use Singapore Mathematics in Utah." "I was born, raised and educated in Singapore. I have been teaching mathematics in a private junior high school for the past eight years. The school is closing down and I am desperately looking for a mathematics teaching position in a local high school. Do you know of any schools that are trying to hire mathematics teachers to teach Singapore Mathematics? Can you help?" "In fact, I think I can help you. There is a workshop on Singapore Mathematics that we are organizing in a few weeks. Would you come to the workshop? I can try to introduce to you some of the human resources personnel of various districts attending the workshops." "Excellent! Thanks. I look forward to it." This was the first phone conversation I had with Dr. Hugo Rossi, a Mathematics professor from the University of Utah. I was first re-­‐introduced to Singapore Mathematics from the local newspaper, Salt Lake Tribune and found Dr. Hugo Rossi's name on the article. The article: Singapore math bill gets go-­‐ahead by Singapore Mathematics in U.S. High School 9 Lisa Schencker was published on February 6, 2009. The article reported that a Utah senate committee approved a bill, SB 159 that would allow districts and charter schools to apply for grants to use the Singapore method to teach math. SB 159 would offer competitive grants to districts that come up with plans for teaching Singapore math in kindergarten through sixth grade and some secondary school classes. The bill would also require districts to train teachers in Singapore math and offer grants to colleges and other groups to train mathematicians to be teachers (Schencker, 2009). I remembered thinking to myself that this will be my ticket to find a mathematic teaching position in the Utah public schools. After the conversation, I paused and asked myself if I actually knew what Singapore Mathematics is, and more importantly, if I knew how to teach it to American students. Growing up in Singapore in the 70s and 80s, school did not always come easy for me. Amidst stressful examinations and the fear of failures in a competitive, meritocratic society like Singapore, mathematics is the only subject that I know well and did well. It was only after coming to the United States and teaching math for about eight years when I learned that Singapore Mathematics has become a highly successful mathematics program used by some U.S. schools. Why is Singapore Mathematics so successful? Statement of Topic and Purpose I have often heard students lamenting over how much they dislike mathematics. They would reason that mathematics is hard. Many would agree that mathematics is not fun and boring. Historically, most students do not enjoy Singapore Mathematics in U.S. High School 10 mathematics because of its tedious homework and complicated algebraic reasoning and steps. However, it is a critical skill that is essential in our daily lives. Not to mention that it is required in most colleges. Mathematics helps to develop our logical thinking, critical thinking and analytical skills. Math is such an important skill because it gives us the capacity for higher-­‐level thinking. Fields in science, technology, engineering and mathematics (STEM) are advancing at a very fast rate. There is a huge push to advance STEM education in order to replenish the pool of scientists, engineers and mathematicians who will lead space exploration in the 21st century. Hence, we are now preparing students for jobs that have not been created yet. There is a very strong need to improve the United States mathematics curriculum so as to keep up with the rest of the world when it comes to mathematics achievement for young students. Singapore math is successful because the program uses a focused, coherent syllabus that integrates concepts and skills in a concrete to pictorial to abstract way, all while emphasizing problem solving (Math in Focus - Research Base, p.2). Since Singapore Mathematics has been so successful in Singapore, other parts of Asia and, to some extend, some parts of the United States, I feel that it is critical and beneficial for me to reconnect with Singapore Mathematics; and examine the possibilities of applying it to enrich my students and hopefully to raise the mathematics level both locally as well as nationally. Framework and Research Question Singapore Mathematics in U.S. High School 11 The theoretical framework of this particular action research project would best fit within the realm of qualitative constructivism. Typically whenever one thinks of teaching of mathematics, one thinks of a teacher standing in front of the classroom instructing the class with the PowerPoint in a positivistic manner in which teachers basically teach from the textbook in the traditional approach such as spoon feeding, explicit instruction and giving students way too many problems to solve (Garelick, 2006). However, Singapore Mathematics is a proponent of constructivist methods of teaching and learning. Vygotsky's theories of constructivism in social learning, his ideas of zone of proximal development (ZPD) and taking control of one's learning are demonstrated heavily in Singapore Mathematics (Lu & Lee, 2011). The constructivist learning theory focuses on how people make sense of their experiences. Social learning is an inherently interpersonal experience where students learn from their teachers, peers and environmental interactions. Under the meta-­‐cognition or "thinking about thinking" component in the mathematics framework of the Singapore Mathematics curriculum, Singapore Mathematics focused on teaching the students to take control of their own learning. The awareness of and the ability to control one's thinking processes, in particular the selection and use of problem-­‐solving strategies align very well with Vygotsky's theory of social learning. The progress of learning mathematics is based heavily on previous or past-­‐taught concepts that need to be reviewed constantly and expanded on. This scaffolding process is demonstrated in Vygotsky's philosophy of ZPD. Therefore, it is not difficult to see that the theoretical framework of Singapore Mathematics is deeply rooted in constructivism depicted by Singapore Mathematics in U.S. High School 12 social learning and ZPD. It is with this framework that this action research project is being conducted. This research is designed to answer the following questions: How Singapore math help high school students become more confident in math? How Singapore math help students develop better problem solving and critical thinking skills? How the visual aspects of Singapore math's bar modeling techniques appeal to high school students? What are the strengths and challenges of applying Singapore Mathematics in United States high school? I am always curious to find out what makes Singapore Mathematics so powerful and whether it is suitable for use in the United States. I hope this research will help to answer the question whether is a hybrid of Singapore math and United States' math the way to improve the future of United States' math programs. As a math educator, I hope to identify and establish some core best practices that can be applied to my mathematics classes to help my students enjoy mathematics and be successful in it. The areas that allow me to play and apply into my class are the areas of strategies, methods and techniques. Although I am not fully trained in Singapore Mathematics as a teacher, I think I can still isolate some individual aspects of Singapore Mathematics such as bar modeling and concrete to pictorial to abstract representation and apply them into my Geometry class to see the impact and beauty of Singapore Mathematics at work. Statement of Researcher Singapore Mathematics in U.S. High School 13 My journey as a mathematics educator began in 2000 when I became a substitute teacher in my former high school at Deyi Secondary School in Singapore after graduating with a Mechanical and Production Engineering degree from Nanyang Technological University (NTU). Early in my career, I taught for nine years in a private Montessori method junior high school in Bountiful, Utah. Incidentally, that was also my first full time job. It was during my teaching experiences there that I decided to get my second bachelor's degree in mathematics so as to convince myself that I know enough mathematics to teach mathematics. I was also motivated by my successes at teaching and I knew that I needed to get a teaching certification in mathematics, if this is something I wanted to pursue seriously. I was intrigued by some of the positive comments and feedback that my students are giving me because of my teaching styles and the way I made mathematics fun and applicable. I immediately attributed the success to my Singapore mathematics background. I wanted to know what more from Singapore Mathematics could help my students be successful in mathematics and learn mathematics in a fun and applicable way? Students are constantly encouraged to consciously take ownership of their own learning. I can think of no better way of teaching Geometry as it contains visually demanding content that is extremely applicable in our daily lives. The progress of learning mathematics is based heavily on previous or past-­‐taught concepts that need to be reviewed constantly and expanded on. I often view ZPD as an elastic rubber band that contains all the old and new mathematics concepts around the individual student which is constantly being pushed and pulled as one grow with new concepts and apply old knowledge to complete more and more Singapore Mathematics in U.S. High School 14 difficult tasks. Incidentally, this constructivist philosophy towards education aligns very well with how Westminster College teaches its students. In a recent MED 679 class at Westminster College, my group was discussing and revisiting the idea of theoretical frameworks. After a bit of soul searching, I reached a temporary conclusion for my own personal theoretical framework for this project. I had a difficult time trying to "fit" exclusively into a single paradigm that would best align with my research topic. On one hand, the field of math propels me towards the positivism paradigm where I would expect a steady, predictable, practical, straightforward, objective and rational view within my mathematic classes. After discussing my research topic with the project supervisor, and deciding to focus on the effectiveness of one or two significant features of Singapore Mathematics, I found myself jumping on the positivism wagon immediately because the methods and experimental testing, data collection and conclusion will be pretty objective and analytical. I also cannot completely isolate myself from the descriptive interpretivism paradigm because of the constructivistic approach of my teaching style. However, I cannot ignore the fact that there are some elements of critical humanism in the research because of the race, ethnicity and culture cards at play. Potential Significance and Limitations I have considered the political, theoretical and practical significance of my study. Mathematics teachers have always been under fire for under performing nationally as well as locally. The National Council of Teachers of Mathematics Singapore Mathematics in U.S. High School 15 (NCTM) standards have been under intense scrutiny (Ahuja, 2006; Dindyal, n.d.). This study contributes to the research that has already been completed in teaching Singapore Mathematics in American schools. Research has been done to determine whether Singapore Mathematic is the solution to the weak mathematics achievements by the American students compared to other countries. This project has the potential of adding new unique information and fresh insights to the current debate of the strengths and challenges of applying Singapore Mathematics in the United States. It is more practical than other studies in that it is exploring specific techniques, rather than the entire system which previous research indicates is complex to authentically deliver in the U.S. education setting (N. Garrison, personal communication, April 8, 2013). It also has the potential to influence policy makers and district administrators when they use the results of this research together with others to consider passing education legislation and reform and allocate more funding towards using Singapore Mathematics in local educational institutions, especially in kindergarten and elementary schools. This project involves some work that combines international relations between Singapore and US. The introduction of Singapore math, in my opinion, is quite radical as we are instituting changes at an individual level for the students in terms of learning content from somewhat of a different approach, but hopefully with better results. With political and economical implications of my research, I hope to bring to the table some positive findings to convince the lawmakers of the usefulness of Singapore Mathematics and why we need it in the American educational system. Singapore Mathematics in U.S. High School 16 Can a mathematics program from a tiny country like Singapore solve the woes of the mathematics curriculum of one of the most powerful countries in the world? This paper is designed to critically examine the characteristics of Singapore Mathematics and carefully explore the strengths and challenges of using certain elements in the United States. The bottom line is, if I can help make mathematics more interesting, fun and applicable for my students, I am willing to try anything. I am already having successes reaching some of my current students with what I am able to give them. However, I am not happy with just some of my students doing well. Ultimately, I want to reach all of my students. If I can use Singapore Mathematics to reach more students, then it will benefit me by strengthening my teaching to help my students be more successful. It may also let them take mathematics outside of the classroom and be lifelong learners. I have very strong beliefs that some of the Singapore Mathematics strategies and techniques will benefit my immediate students who can hopefully develop good number sense for higher-­‐level mathematics. I have witnessed some successes in the past when I used some of my preferred methods or strategies that I learned in Singapore. Students' confidence will be increased and they will experience more successes and fun in mathematics. This is a small but definite step towards a better mathematics curriculum in my classes, then I hope to extend that to all the mathematics classes in the school. With the success from this research, I hope to bring about attention the importance and positive impact of Singapore Mathematics to other schools in the States and then perhaps nationally. Singapore Mathematics in U.S. High School 17 Being educated both in Singapore first with a Mechanical Engineering degree and then later in the United States with a Mathematics degree, I cannot deny the fact that the mathematics standards from the United States are weaker than that from Singapore. This poor mathematics performance is demonstrated by the recent Trends in International Mathematics and Science Study (TIMSS) results (Dindyal, n.d.; Ezarik, 2005; Forsten, 2004; Garelick, 2006). I am not saying that the mathematics curriculum in the United States is terrible, but there is definitely room for improvement if the United States wishes to raise its mathematics scores. It is my hope to improve the poor mathematical achievement that the United States is suffering. As with any research, there are limitations in this study. While we are examining the effectiveness of Singapore Mathematics, we should not be so blind to just limit ourselves to one mathematic program, as there may be other cutting edge mathematic programs that are worthy of research as well. The small classroom sample size does not lend itself to broad applications. The setting of conducting such a project in a geometry class within a private high school may not represent a typical population to generalize the results. We need to keep in mind that the objective of this work is not to replace nor disregard America's mathematics system. Perhaps a hybrid of systems may just be the answer to the United States' declining mathematics performances. There are huge debates about the flaws of the mathematics curriculum in the United States. Authorities need to look at Singapore Mathematics objectively and carefully. We need to take the good out of Singapore Mathematics and combine with Singapore Mathematics in U.S. High School 18 the strengths of the mathematics in America to come up with a new hybrid curriculum of mathematics that would work to improve student success. Following this introduction chapter, a literature review will be offered. Next, the method chapter will focus on the setting, data gathering methods and rationale, participants, data analysis, trustworthiness and ethical considerations of the project. Chapter four will detail findings and my analysis. Finally, recommendations will be provided in the final chapter. Singapore Mathematics in U.S. High School 19 Chapter 2 Literature Review This literature review provides a necessary background regarding my research questions. Here I explore the Singapore educational system pertaining to mathematics. Next, different international mathematics systems are compared. America's mathematics curriculum is examined, followed by the potential challenges that may occur when applying Singapore Mathematics in United States school settings with regards to textbooks, quality of teachers and the difference between Singapore Mathematics Curriculum Framework (SMCF) and the National Council of Teachers of Mathematics (NCTM). Currently, Singapore Mathematics is mostly introduced at the Elementary level. This research is an eye opener as I learn about Singapore Mathematics. This unique opportunity of teaching mathematics as someone who was educated both in Singapore and United States enables me to experience some encouraging successes at the Elementary and Secondary levels. These experiences give me rare insights while I instinctively apply Singapore Mathematics strategies, methods and techniques in my teaching in my previous and current schools. Hence, it comes as no surprises that it is something close to my heart when I want to research more into the strengths and weaknesses of applying Singapore Mathematics in my high school. The theoretical framework for this research project is one of Vygotsky's theories of constructivism and social learning. Incidentally, my own personal theoretical framework also branched from Vygotsky's philosophies of constructivism and social learning with emphasis on ZPD. Singapore Mathematics in U.S. High School 20 In order to understand what others have previously researched on Singapore Mathematics, a thorough review of the literature was conducted. This chapter presents when and how Singapore Mathematics evolved. Later an overview of the Singapore educational system in which the mathematics is taught and a discussion of the similarities and differences between U.S. mathematics standards and those of Singapore is offered. Finally, a review of the strengths and challenges of applying Singapore Mathematics in U.S. high school settings will be detailed. Singapore Educational System and Mathematics Singapore has devised a very successful system of mass education that is both free and universal. The public education system is highly centralized with curriculum and standards that are uniform across all schools. National high-­‐stakes assessment examinations measure achievement in the 6th, 10th, and 12th year of education (Ahuja, 2006; Dindyal, n.d.; Kho, Yeo & Lim, 2009). The centralized authority - the Ministry of Education (MOE) - is responsible for formulating and implementing educational policies, developing national curriculum frameworks and guidelines and administrating national examinations in collaboration with the Cambridge General and Certificate of Education (Kho, Yeo & Lim, 2009). Education begins at a young age in Singapore. Typically, children attend two to three years of "Kindergarten" instruction, beginning as early as age three. The kindergarten years are serviced privately, while compulsory public education begins at about age five or six when students enter "primary" school. At the completion of six years of primary education, students take the Primary School Singapore Mathematics in U.S. High School 21 Leaving Examination (PSLE). This assesses students' achievement levels and determines their suitability for "secondary" education (Kho, Yeo & Lim, 2009). Singapore's secondary level entails four to five years of education roughly equivalent to that of U.S. grades seven to ten. Based on their PSLE performance, students enter one of four secondary streams: Special, Express, Normal Academic, or Normal Technical. After four years, both the Special and Express Streams take the Singapore - Cambridge General Certificate Exam, Level O (GCE ‘O'). Normal Stream students who perform well in the GCE ‘N' may continue with the program for a 5th year, moving on to take the GCE ‘O'. Each secondary offers all streams, and students are able to move from one stream to another, based on merit (Kho, Yeo & Lim, 2009). After four to five years of secondary study and successful completion of the GCE ‘O' at age 16 or older, students are presented with a variety of options to pursue an academic course of study or a "professional centric" course of study that focuses on professional-­‐level technical education. As determined by their GCE ‘O' scores, individual students continue their education either in pre-­‐university or post-­‐ secondary institutions. In addition, all students must take the GCE ‘A' Level exam for entrance into the university (Kho, Yeo & Lim, 2009). However, Singapore Mathematics was not always so successful. Singapore, a former British colony, has a system of education modeled largely on the traditional British system. After independence in 1965, Singapore continued her collaboration with the University of Cambridge Local Examinations Syndicate (UCLES) to develop a curriculum more suited to her local needs. By the end of the 1980s changes in the Singapore Mathematics in U.S. High School 22 school mathematics curriculum had roughly followed a ten-­‐year cycle. It was the right time for Singapore to launch a major change in school mathematics education - the Singapore Mathematics Curriculum Framework (SMCF) also called the Pentagon Model (Kho, Yeo & Lim, 2009). The Pentagon Model of the SMCF (Figure 1) was first proposed in 1990. At the core of the model is problem solving and the five sides forming the pentagon are: attitudes, skills, concepts, processes and metacognition (Dindyal, n.d). Singapore Mathematics believes that mathematical problem solving is central to mathematical learning. The development of such a mathematical problem-­‐solving ability is dependent upon: Concepts - Numerical, Algebraic, Statistical, Probabilistic, Geometric, Analytical; Skills - Calculation, Estimation, Spatial Visualization, Algebraic Manipulation; Processes - Thinking Skills, Reasoning and Application; Attitudes - Math Confidence; Metacognition - Monitoring One's Own Thinking. Figure 1: The Singapore Mathematics Curriculum Framework - Pentagon Model (SMCF, 2000) Singapore Mathematics in U.S. High School 23 Singapore mathematics curriculum revisions and bar modeling. The mathematics curriculum in Singapore has been under revisions. The last revision introduced applications and modeling to be part of the teaching and learning of mathematics. Bar modeling has always been an integral part of Singapore Mathematics. This method is a learning approach where students create diagrams (models) to represent problems and concepts with bars. Drawing these types of models helps students to visualize strategies for problem solving and to make algebraic concepts more concrete. Model drawing can help children solve simple and complex word problems, develop algebraic thinking, help students visualize the part-­‐whole structure of the problem, develop students' operational sense and foster proportional reasoning (Math in Focus - Research Base, (n.d.)). Hoven & Garelick (2007) describe bar modeling as a pictorial way of representing a real world problem and solving it without algebra. Bar modeling is a specific variant of the common Draw a Picture problem strategy. Because Singapore Math uses this one variant consistently, students know what kind of picture to draw. That's an advantage if the bar model is versatile enough to apply to many complex problems - and it is. It is especially useful for problems that involve comparisons, part-­‐whole calculations, ratios, proportions, and rates of change. It communicates graphically and instantly the information that the learner already knows, and it shows the student how to use that information to solve the problem. (p.28) One of the main objectives of mathematics education in Singapore is to enable students to develop their abilities in problem solving. The model-­‐drawing method Singapore Mathematics in U.S. High School 24 requires the construction of pictorial models, namely the part-­‐whole model and the comparison model, to help students visualize abstract mathematical relationships and various problem structures through pictorial representations. It is a power visual aid for solving complex problems involving fraction, ratio and percentage. Above all, it is closely related to the algebraic method for solving algebra word problems (Kho, Yeo & Lim, 2009). Figure 2: An example of a mathematic problem using bar modeling. (Retrieved: December 4, 2013, from: www.utahsmathfuture.com/example.cfm.) Singapore Mathematics in U.S. High School 25 International Mathematics Comparisons Singapore was ranked first in the world in mathematics achievement in the Trends in International Mathematics and Science Study in 1995, 1999 and 2003. U.S. mathematics achievement is still lagging far below world-­‐class standards in terms of mathematics achievement (Garelick, 2006; Kho, Yeo & Lim, 2009). United States 8th graders did not even make the top ten in the 2003 round; they ranked 16th. Scores for American students were among the lowest of all industrialized countries. There has been a big decline in the numbers of U.S. trained scientists and engineers, compared with the increasing numbers of those trained in Asian countries (Garelick, 2006). If the trend continues, the U.S. universities may find themselves hard pressed to get sufficient number of American students for their challenging programs in mathematics and science (Ahuja, 2006; Garelick, 2006). Ahuja (2006) stated that the difference in performances between Singapore and the U.S. "lie in deep-­‐rooted complex aspects of teaching, learning process (such as thinking skills and heuristics), certain special skills (such as estimation, approximation, mental calculation, communication, arithmetic and algebraic manipulation), mathematics curricula, textbooks, student attitudes, culture, and parental support" (p.228). It is very important that we address quickly the issue of why high school students are not doing well and losing confidence in mathematics. It is my belief that Singapore Mathematics may have some benefits which we can explore closely and use them to increase the level of mathematics abilities and interests. The reason Singapore math works is simple - the program has a consistent and strong emphasis on problem solving. Other elements that contribute to Singapore Mathematics in U.S. High School 26 the program's success include the program's focus on and support for building skills, concepts, and processes and its attention to developing students' metacognition and positive attitudes towards mathematics. Students are given opportunities to reflect on their thinking, communication, and problem solving so that they can apply these skills to varied problem solving activities. (Math in Focus - Research Base, p.2) American Mathematics American Institutes for Research (AIR), funded by the U.S. Department of Education, seek to find out what is happening in Singapore such that this small country is able to come out on top for student achievement in a well-­‐known international study (Ezarik, 2005; Garelick, 2006). AIR found out that while the U.S. math program is weaker than Singapore's in most respects, it is stronger in some areas. The U.S. mathematics curriculum is not a complete failure. U.S. mathematics framework gives greater emphasis to developing 21st century mathematical skills such as representation, reasoning, making connections and communication compared to Singapore. U.S. frameworks and textbooks also place greater emphasis on applied mathematics, including statistics and probability. (Ezarik, 2005) Dindyal (n.d.) considers, explores and describes the similarities and differences between the Singapore Mathematics Curriculum Framework (SMCF) and the National Council of Teachers of Mathematics (NCTM) Standards published in the U.S. There appears to be more differences than similarities between the two curriculums. Singapore Mathematics model is one that strongly supports Singapore Mathematics in U.S. High School 27 constructivism. It focuses on concrete to pictorial to abstract presentation. Concrete manipulatives are used to explain abstract mathematical concepts. Pictures, visual models, and diagrams are used to present examples with solutions. Numerals, mathematical notation, and symbols are used once students are familiar with the abstract representation (Chan, 2009). Challenges in Applying Singapore Math in U.S. School Settings Textbooks. Several U.S. states have already adopted Singapore mathematics with reasonable successes. Garelick (2006) highlighted several successes and challenges of using Singapore mathematics in some U.S. elementary schools. One of the Singapore Mathematics' strength is also a challenge -­‐ textbooks. Hiebert et al. (2003) and Stigler and Hiebert (2004) argued that people could learn about the advantages and disadvantages of the textbooks of their own country through international comparative studies. A huge aspect of the success of Singapore math is the design and development of the Singapore math textbooks by the Ministry of Education. Singapore's textbooks build deep understanding of mathematical concepts while traditional U.S. textbooks rarely get beyond definitions and formulas (Ezarik, 2005). Textbooks from the U.S. are always a lot thicker compared to those in Asia. Thicker textbooks means that the books are a lot more heavy to carry and costly to buy. As such, students are less incline to bring heavy textbooks to class or to school their textbooks to follow the lessons. Also they did not want to run the risk of losing their textbooks (Yang, Reys & Wu, 2010). Singapore Mathematics in U.S. High School 28 Studies have showed the importance and effectiveness of using Singapore math textbooks (Fan, 2007; VanTassel-­‐Baska, 2008; Yang, 2010). The Asian textbooks emphasize procedures while the U.S. textbooks concentrated more on conceptual understanding and less on procedural methods. It was also brought to light that Singapore textbooks introduced and developed fractions the earliest. As such, Singapore students are learning the content taught from Singapore textbooks one grade before their counterparts from the United States. (Yang, Reys & Wu, 2010). It is very interesting to note the difference in driving forces of changes in math textbooks in different countries. The Ministry of Education in Singapore controls the math curriculum but the U.S. has no national math curriculum. U.S. math textbooks are influenced by recommendations by the National Council of Teachers of Mathematics (NCTM) and these powers of recommendations varies greatly among states (Reys & Reys, 2006). Teachers. Another strength of Singapore Mathematics is that teachers are well trained by the MOE; and teachers know their content well. Elementary school teachers in Singapore are required to demonstrate mathematics skills superior to those of their U.S. counterparts before beginning paid college training to become teachers (Ezarik, 2005). The study by VanTassel-­‐Baska, Feng, MacFarlane, Heng, Teo, Wong, et al. (2008) showed that Singapore teachers demonstrate a higher level of effectiveness than American teachers regarding behaviors and differentiation strategies. The level of instructional effectiveness appeared to be positively related proportionally to the number of years of teaching experience and training in differentiation practices for the gifted. The numbers from the study demonstrated Singapore Mathematics in U.S. High School 29 that despite the lack of advanced degrees, the Singapore teachers are better prepared in both content knowledge and gifted education compared to their U.S. counterparts. This is partly due to the strong support and professional training that the MOE gives to its teachers. The Ministry of Education in Singapore pays for the minimum number of courses necessary to satisfy the degree requirements for potential teachers to get their teaching degree. However, if sponsored teachers fail or withdraw from the courses, they assume responsibility for the full cost. After completion of the master's program, graduates must remain employed with the MOE as teachers for at least one year (Wang-­‐Iverson, Myers & Lim, 2009). This method has tremendous pull to entice potential teachers into the teaching field and a clever way of identifying good teachers. It would require a lot of funds and resources for the United States to follow and implement such a method but I feel that it will create a lot of opportunities to identify and hopefully safe keep good teachers. Difference Between SMCF and NCTM After scrutinizing the Singapore Mathematics Curriculum Framework (SMCF) and the National Council of Teachers of Mathematics (NCTM) Standards, there are several differences between the two curriculums. Dindyal (n.d.) considers, explores and describes the similarities and differences between the SMCF and the NCTM Standards published in the U.S. There appears to be more differences than similarities between the two curriculums. He discovered that the SMCF is more unified, consistent, transparent, standardized and nationally uniform in terms of Singapore Mathematics in U.S. High School 30 setting up the mathematics curriculum for all schools in the primary and secondary levels as compared to the various interpretations of the U.S. standards by the different States and districts Teachers from both cultures appear to embrace strikingly similar beliefs about what the best teaching practices are and what constitutes exemplary teachers. Such results suggest that the educational research community might be able to identify and establish an array of core best practices that can be universally applied to selective secondary schools worldwide. (VanTassel-­‐ Baska, p.357) Strengths and Challenges of Applying Singapore Math in High School There are various aspects of Singapore Mathematics methodology that are tricky to implement in U.S. settings. Singapore Mathematics strategies emphasizes number sense, mental mathematics skills, deep understanding of place value, mathematics literacy and mathematics confidence. Applying Singapore Mathematics strategies and techniques by focusing on word problems, using Singapore textbooks, bar modeling methods, increasing the depth of the mathematics curriculum but not compromising the content, consciously slowing down pace to minimize the amount of review, promote and develop good number sense and mental math in a secondary school setting will be challenging. Firstly, using Singapore Mathematics strategies and techniques effectively using word problems and bar modeling requires years of proper Singapore Mathematics training with the teachers, whose own standards of proficiencies and content knowledge must improved, needs to be handled with Singapore Mathematics in U.S. High School 31 careful planning and purpose (Chan, 2009; Ezarik, 2005; Fan, 2007; Forsten, 2004; Lu, 2011; Kho, Yeo & Lim, 2009; Vantassel-­‐Baska, 2008). Secondly, the Singapore textbooks are costly and funding has been difficult. Most elementary schools that are using Singapore math are currently using textbooks published by Singapore's Marshall Cavendish Education. Although Utah State Senate passed S.B. 179, the math Education Initiative in 2011, the funding of $1.813 million appropriated for the initiative was scaled down because of high cost and demand from the school districts and charter schools. Finding the appropriate Singapore Mathematics textbooks for U.S. high schools will be a challenge because there is no Singapore math high school textbook written especially for high school math students available at the moment. Typically in most U.S. schools, textbooks are only changed once every 10 years. Thus, schools are less inclined to adopt the Singapore math curriculum due to their unwillingness to invest in huge cost in changing textbooks, work needed to prepare for the new textbook and the teacher training needed to use the textbook. (Fan, 2007; Garelick, 2006; Vantassel-­‐Baska, 2008; Wang-­‐Iverson, 2009; Yang, 2010). It is important to understand that it is not possible to completely bring over to the United States the things that make Singapore Mathematics successful and apply them directly in the United States (Wang-­‐Iverson, Myers & Lim, 2009). At the U.S. high school level, mathematics courses are taught usually in the following sequence yearly from Grade 9 to Grade 12: Algebra I, Geometry, Algebra II, Pre-­‐Calculus, Calculus Honors. In Utah, only Calculus Honors is available instead of regular Calculus class. In order for the U.S. to achieve a comparable K-­‐12 Singapore Mathematics in U.S. High School 32 mathematics program with Singapore, all the states in the country must successfully develop world-­‐class standards in mathematics education and all other key educational policies: salaries, teacher preparation and development, accountability, textbooks, graduation requirements, etc, must be examined together with those standards. (Ahuja, 2006) The literature points to the opportunities and challenges of improving US math standards when implemented effectively in terms of curriculum, using Singapore math textbooks, Singapore math trained teachers, word problems solving, etc. A part of this research is to observe and understand the behavior of the students. Another goal is to assess the students' performances on solving word problem to display their level of critical thinking and problem solving skills. Surveys will be conducted to collect data on students' level of confidence after learning and using Singapore math. The following chapter details my method for conducting this study. Singapore Mathematics in U.S. High School 33 Chapter 3 Methods The debate on mathematics is not whether we should study mathematics to pass examinations, but how we can give our students a deeper understanding of mathematics and help them be more successful using a good mathematics program. I believe with all of my heart that mathematics can be fun and that deeper understanding of the subject leads to success both in school and life. It is with this tremendous conviction that I am conducting a study to see if elements of Singapore Mathematics can be applied successfully into my current high school mathematics classes to enrich my geometry students. It was my hope to use some of my Singapore math techniques to make math fun and appealing to high school student. I wanted to create a stronger mathematics program to raise the level of mathematics achievement for future students so that they can think critically and have good problem solving skills. In this chapter, I will explain my approach and rationale for conducting this action research project. The type of setting and participants selected for this study will be discussed. The rationale for the methods that I have used to collect the data will also be included. Next, I will indicate how I analyzed the research data and how I proved that the conclusions are valid and trustworthy. Lastly, I will conclude by highlighting some of the ethical considerations that were faced during the research. Overall Approach Singapore Mathematics in U.S. High School 34 In the world of education, we as teachers need to constantly justify why we choose one program over another to help our students. This action research project is based on a qualitative approach and the design. The study was the result of my strong desire to understand more about the effectiveness of using Singapore Mathematics in my teaching environment as well as to self-­‐evaluate my abilities in teaching mathematics with Singapore Mathematics influences in order to improve my teaching practice. Fifteen years ago I would not have dreamed or thought I would answer the call of teaching. However I know now that teaching is what I want to pursue. The hype about Singapore Mathematics is really growing, especially at the elementary level, and I wanted to know if it really works in high school. I wanted to know if I have enough of the Singapore Mathematics techniques to blend with what I have learned and taught up to this point. I hoped this research would help to answer the question of whether a hybrid of Singapore math and United States' math was the way to improve the future of United States' math programs. My research question involves understanding the workings of Singapore Mathematics in my students' learning of mathematics. I think this is difficult for the students to articulate and explain. Hence, the way I can truly evaluate the effectiveness of Singapore Mathematics is through the students' responses on the surveys and interviews and their performances on solving word problems. In addition, I wanted to explore a link between Singapore Mathematics and an increase in my students' level of confidence as well as their level of critical thinking and problem solving skills. A big part of this research was to obtain the actual Singapore Mathematics in U.S. High School 35 experiences and reactions of the students as they went through the process of learning Singapore Mathematics. Setting Currently, I am a mathematics teacher at a Catholic High School in Utah. My research was conducted under the supervision of the Catholic schools of the Diocese of Salt Lake City. The school is a 9th to 12th grade college preparatory private high school with about 770 students. The high school is situated in downtown Salt Lake City. The students are mostly middle to high socio-­‐economic status population. The school consists of students of all faiths (one-­‐third of our students are not Catholic). Student demographics for the school is as follows: 74% are Caucasian, 14% are Hispanic, 6% are Asian, 3% are African Americans, 2% are Pacific Islanders, and 1% are American Indians. Incoming students have a diverse range of abilities and come from a variety of backgrounds in math; consequently, the mathematics department offers a diversified program to fit the needs of the students. Students will develop problem-­‐ solving skills, learn to be critical and creative thinkers, and be life-­‐long learners in the context of the abilities of each student. The mathematics department provides the opportunity for every student to take a plethora of college-­‐recommended mathematics courses. The school offers many mathematics programs such as Algebra I, Geometry/Algebra II Honors, Elementary Geometry, Geometry, Advanced Geometry Honors, Algebra II, Advanced Algebra II Honors, Advanced Algebra/Trigonometry, Pre-­‐Calculus, Pre-­‐Calculus Honors, Calculus Honors, AP Singapore Mathematics in U.S. High School 36 Calculus AB and AP Calculus BC. We have five different programs or tracks for students: elementary, Pre-­‐Calculus, Calculus, Honors, AP. Courses are taken sequentially depending on where the students qualify when they enter the school. Students are required to have at least three years of mathematics but almost every student will opt for four years of mathematics as this will look very good on their high school transcripts. The mathematics department prides itself for saying that we teach to the abilities of our students and not based on their chronological age. Traditionally, my education institution does a good job in maintaining high mathematics test scores when compared locally as well as nationally. Participants The participants involved in this research were math students from one of my three geometry classes. I purposely picked a geometry class instead of a Calculus Honors class because I felt that it would be easier to administer Singapore Mathematics to low-­‐level mathematics students rather than high-­‐level mathematics students. Out of three geometry classes, I picked the one where I have a better rapport. It is my hope that the class will believe that I have their interests at heart and that I want them to be successful. The size of the geometry class selected is 25. There are 15 boys and 10 girls. The class consists of mixed ages: four 8th graders from another school, 17 9th graders and four 10th graders. It was nice to have a wide spectrum of students with varying level of ages and abilities. With much anticipation and curiosity, I determined students' interests and possible initial responses from an online Singapore Mathematics in U.S. High School 37 questionnaires made on a free website called the SurveyMonkey which can be found on http://www.surveymonkey.com/. From a class of 25 students, only ten students responded to the online survey. I had hoped to interview all ten students who had volunteered from the survey. However, of the ten students, only five volunteered their time after school to meet with me for the one-­‐on-­‐one interviews. The five students consist of four boys and one girl. They are of various academic abilities. For confidentiality reasons, their real identities have been withheld and they shall be identified as A.C., L.D., C.J., V.C. and C.H.. Data Gathering Methods and Rationale The conceptual framework of collecting and viewing my data is one of constructivism. Collection was done in a constructivist fashion to observe students learn in a cooperative setting as well as in a scaffolded environment where I helped and supported students learn by modeling. Research data were collected from several areas such as online surveys/questionnaires, interviews, video tape of problem solving activity as well as field notes/journals. I realized right from the beginning that being a passive observer with a researcher note pad may not be sufficient to capture all the necessary information needed to come up with a comprehensive understanding. With my engineering and math background, I am compelled to study and quantify the performances of students based on assessments and tests scores. This information is quintessential in determining the strengths and weaknesses of Singapore Mathematics when the tests scores are being compared among all three Singapore Mathematics in U.S. High School 38 geometry classes. The way the students answer and show the work for the word problem exercise will provide valuable information about the students' critical thinking skills and their level of understanding. However, I very quickly realized that it was going to be difficult to get any strong data through assessment and test scores. I could not find a way to teach the class the Singapore math's bar modeling technique without stopping my current math curriculum to accommodate that. Hence, I discovered that conducting this research project through a qualitative analysis lens was the best way to proceed. My first piece of data came from the online survey. A free online survey website like Survey Monkey was a hassle-­‐free way to post and obtain some basic questions and answers to find out how the students think about the mathematics concepts that they were learning. There are only 10 simple questions on the survey (See Appendix A). The objective of the survey was to find out the level of the students' interest in math, their learning style, level of confidence related to mathematics, level of basic algebra skills and more importantly their willingness to allow a one-­‐on-­‐one interview with me. As I looked for the positives and the negatives behind the students' experiences and development of mathematics knowledge, the use of individual interviews is a major avenue to gather valuable information and feedback. Individual interviews were the most effective way of collecting honest opinions because of their tremendous potential for open discussions, flow of honest feedback and intimate setting where they could say things with no pretenses; as well as the opportunity to explain their feelings in details. Through this process, I better Singapore Mathematics in U.S. High School 39 understood my students and was able to weigh the strengths and shortcomings of Singapore Mathematics to see if it has a place among other high school mathematics programs. The feedback provided from the online survey is a good spring board for me to create a list of interview questions to be given to selected students to delve deeper into how Singapore Mathematics has impact their learning of mathematics. Accordingly, I designed the interview questions as an extension to the questions from the previous survey in the hope of getting more details from their elaborated answers during the discussions (See Appendix B). Through the individual interviews, I obtained more insights into the strengths and weaknesses of how Singapore Mathematics is being used in my classes. The interviews were video recorded using my iPhone with permission from the participants. I thought that video camera phones would be less intimidating to students than traditional videoing equipment. I conducted the interviews in an open environment where the students could be candid and have honest conversations about specific questions with regards to the authentic effectiveness of Singapore Mathematics. The subtle nonverbal cues were priceless and good indicators to how the students felt. The interviews were on average about 15 minutes long. The quality and sound were good. I then transcribed and analyzed the five videos with the help from a critical friend. As part of my one-­‐on-­‐one interview, I included a word problem activity where I asked the student to work out a word problem on a piece of paper. I chose the word problem from the book, The Singapore Model Method for learning Mathematics, an integral part of my literature review. The word problem is as Singapore Mathematics in U.S. High School 40 follows: Aidan had 1750 stamps. Cole had 480 fewer stamps than Aidan. Aidan gave some stamps to Cole. Now Cole has 3 times as many stamps as Aidan. How many stamps did Cole have at first? How many stamps does Aidan have now? I chose this question because it is a two-­‐part word problem. I was confident that the students should be able to handle the first part of the word problem without any problem. However, it is the second part where it would test the students' critical thinking and problem solving skills. The students' work on the word problem is available in the Appendix D. Data Analysis The online survey input from Survey Monkey were summarized and tabulated. Key questions were formulated in a more refined questionnaire after the first round of online survey. All interviews were transcribed and analyzed. The student work, responses given during the word problem activity and my observation notes were analyzed and summarized numerically. From my interpretations and comparison of the data, the data were coded and grouped in various themes. After analyzing the themes, the results were summarized. Research in mathematics will not be complete if the collected data are not put in tables and charts to help summarize the conclusions. Validity and Trustworthiness I strengthened the validity and trustworthiness of my research by member checking and obtaining the research conclusions through triangulation of all of my Singapore Mathematics in U.S. High School 41 data sources. To ensure accuracy, credibility, authenticity and validity of my conclusions, the data were collected from multiple interviews, online surveys, questionnaires and classroom observations. Then the data were synthesized and summarized. The students' responses and work on the word problem were also analyzed and summarized thoroughly to check for critical thinking and confidence. The findings were later presented back to the participants as well as to my immediate community of practice and critical colleagues in order to verify my conclusions and test my thinking. Through this process I conducted a valid and trustworthy study that allowed for the most honest analysis of the data. Ethical Considerations Ethical considerations for my study included confidentiality, the principal of do no harm, selection criteria for participation, burden of classroom demands and managing parents' perceptions of the study. Before conducting the research, I discovered that it was a hard decision choosing which of my three classes would I do my research with. My initial feeling was to choose the weakest class to do this research because I assumed that will be the class that could benefit from Singapore Mathematics the most. I felt that the top class would continue to do well in spite of Singapore Mathematics. However I did not want to introduce Singapore Mathematics to weak mathematics students and make it harder on them to try a modified mathematics teaching format and techniques. I did not want to confuse them further because it would be harder for weaker students. Students learning Singapore math would require some time to fully understand bar modeling Singapore Mathematics in U.S. High School 42 techniques and effectively use it on mathematics problems to see the problems transformed from concrete concepts to visual ideas. The weaker students sometimes already struggle to keep up with regular mathematics. It would do more harm than good if the results were not favorable. On the other hand, I thought the stronger students could appreciate and understand the idea of model drawing a lot quicker and appreciate the Singapore Mathematics techniques better. In other words, I had to make sure that my students exposed to Singapore Mathematics must be able to withstand the intervention. I was also very much aware of the possible risks on my students. Those possible risks that my students would experience included frustration with new strategies, lack of confidence in basic math, regression of skills due to uncertainties. I keenly watched out for any of these behaviors right at the beginning to see how they were handling the new strategies. If I sensed the slightest frustration in lack of confidence or regression of skills, I would either slow the process down, give those students plenty of opportunities to come in for help or have weekend review sessions to help them. I was also quite concerned not to have complaints from parents about why I did not choose all three geometry classes to test my research. This related to the fact I was enriching those students with Singapore Mathematics that has a good reputation in the academic world. I did not want the parents nor the students to feel left out or that I was shortchanging the other two geometry classes. I did not want to give the impression that I was not giving my best to all of my classes and that my intention was to make one class more superior than the other two. I was also aware Singapore Mathematics in U.S. High School 43 that parents might worry that they may not be equipped to help their child without prior knowledge of Singapore Mathematics. Ethical considerations for my research also included various classroom and professional burdens like principal of do no harm and maintaining State standards. I created a safe classroom environment so that my students would not feel threatened or embarrassed for making mistakes in learning a new technique in Singapore math. Another issue I faced was my need to constantly remind myself that I had to make sure that I have covered State standards in my mathematics curriculum while using Singapore Mathematics techniques and methods. Issues of confidentiality and accuracy were a major concern particularly because the participants were minors. To maintain the rapport, support and trust my students offered me, I guaranteed to my participants that the feedback they shared with me was for my research only. I would be unbiased in treating all positive as well as negative feedback. I was sensitive to their responses and feedback. I did not wish the students to tell me things that I wanted to hear as their teacher, but rather their true objective feelings and opinions. I did not wish the students to just give me the good feedback but I also wanted to hear the bad ones as well. The students' responses on the surveys and one-­‐on-­‐one interviews and their performances based on word problem activity were analyzed carefully. The findings were reported but no names were shared to school administrators or listed in the report. All participants signed a consent form stating that their identities would be kept confidential. Singapore Mathematics in U.S. High School 44 Conclusions Using a qualitative based research study, I gathered and analyzed the data to put all the cards on the table with regards to the integration and effectiveness of Singapore Mathematics elements for my participant group. I conducted my research with the highest level of integrity and objectivity. I scrutinized the data inside out to present both the positives and negatives of Singapore Mathematics with clarity and honestly to enhance the credibility of my study. The privacy and confidentiality of my participants were respected and observed. All ethnical considerations explored and addressed appropriately. Singapore Mathematics in U.S. High School 45 Chapter 4 Data Analysis Introduction In this data analysis chapter, I analyzed and synthesized data from questionnaires, one-­‐on-­‐one interviews, journals and critical friends to evaluate the strengths and limitations of applying Singapore math in a private high school setting. The conclusion to the data analysis will help to shed light on the following questions: How Singapore math help high school students become more confident in math? How Singapore math help students develop better problem solving and critical thinking skills? How the visual aspects of Singapore math's bar modeling techniques appeal to high school students? If the answer to the above questions is a yes, then is it possible to implement Singapore math at the high school level in the United States? I organized the following chapter into several phases. The first phase is to analyze the significance and implications of the data. After much analysis, I discovered that the data revealed the following important themes, categories and patterns: influence of Singapore math on teaching and learning styles in math, impact of Singapore math on students' confidence and abilities in math, visual appeal of Singapore math's bar modeling technique and training requirements on teachers for teaching Singapore math. Influence of Singapore Math on Teaching and Learning Styles in Math Singapore Mathematics in U.S. High School 46 It is no mystery that no two students learn the same way. The same thing can also be said for teachers teaching their content area. Different teachers have their own unique style of teaching that reach different students. I have always been very curious to learn how my students evaluate my teaching style. Do I have a different teaching style compared to other math teachers because of my Singapore math background? All five interviewees indicated they liked my teaching style. Students illustrated their sentiments often, "Yeah. I like your teaching style. Because you know when you are showing the slides on the board. Once you have showed us the slides, you might show us a visual or a diagram. "(L.D., One-­‐on-­‐One Interview, January 23, 2013) Another student went on to say, I like that you give us the notes and we write down or draw the pictures after you have explained or drawn on the board because that helps us understand. The worksheets help because it is homework related and if a problem comes up on the homework, you know, like a way that we did on the worksheet, then you can use the knowledge from the worksheet to help you with that problem. (C.H., One-­‐on-­‐One Interview, January 25, 2013) They like the fact that I am visually inclined when it comes to teaching math concepts. They also commented that I tend to teach by modeling. They like the fact that I do the examples on the worksheets with them in class after I have taught them the concepts from the lessons. Then they would go home and practice the concepts through the homework problems assigned from the textbook. That affirmed my teaching philosophy in constructivism, especially Vygotsky's philosophy of zone of Singapore Mathematics in U.S. High School 47 proximal development (Lu & Lee, 2011), where the students are given enough scaffolding to practice the math concepts with and without the help of the teacher. I attribute this inclination to teach math concepts visually with scaffolding to my Singapore math background. Singapore math is visually oriented, particularly with bar modeling techniques. When I started learning math in Singapore from my math teachers, I always learned the following techniques: draw diagrams first based on the problem, secondly set up the equations based on the diagrams and finally use basic algebra skills to solve the problem at hand. In reflecting my years of teaching, I always try to teach using a lot of visual aids and modeling to help students hear, see and do. One of my favorite quote sums up my teaching style and philosophy aptly. If I hear, I will forget. If I hear and see, I will remember. If I hear, see and do, then I will understand. Now I realize that I teach the way that I would like to be taught personally. Singapore math has conditioned me since young to visualize the problem in terms of diagrams and numbers. In the case of Singapore math's bar modeling technique, I have been taught to think critically and problem solve using visual aids like the bar models to represent what I have been given and what I need to solve. The visual aids that I present to my students in my daily lessons have been well received based on the responses from the interviewees. I am very fortunate that this approach and style have worked for my students and I. I will no doubt continue to impress upon my students the importance of drawing diagrams to connect to the problem. Impact of Singapore Math on Students' Confidence and Abilities in Math Singapore Mathematics in U.S. High School 48 More than half of the students said they liked math (See figure 1). I know those ten students rather well. Hence, this is a rather surprising finding. This is because with the exception of two students, A.C. and L.D., the other eight students often struggled and complained in class about how hard math is. Math is the one class that has homework assigned everyday and most students in my classes do not like to do homework. I expected my interviewees to respond that they do not like math and that the reason why they do not like math is because it has so much homework. Figure 1 shows the online responses from the 10 participants when they were asked the question of whether they like math. Figure 1: Do you like math? (SurveyMonkey) During my 13 years of teaching math and classroom observation in the United States, many students do not like math. This is primarily because of the subject's requirement of constant homework load and tedious mathematical computation with countless rules and formulas. I understand why math is not popular. Students have complained constantly that math is too difficult. However when I tried to support them by giving the students extra time in class to complete the homework assignment or to help them with the homework, they would rather save it to do it at home. I believe that the pace of any regular math curriculum is Singapore Mathematics in U.S. High School 49 rather rigorous in terms of introducing new concepts in each successive class lesson and the only way to make sure the students understand the concepts and retain the material is through repetition of practice and drills at home. However, the students these days are bombarded by many after school activities like sports, family commitments and other activities that constitute a more rounded education. They do not have time to do homework. They struggle when they come to class unprepared and then they dig themselves into a big hole trying to get caught up with the class. Some students are academically strong enough to recover and some, unfortunately, are not. The glaring thing that I am witnessing now is that students are progressing through one math class after another without fully understanding critical basic math fundamentals and they continue to struggle throughout high school math. Figure 2 shows the online responses from the 10 participants when they were asked the question of how they would rate their own individual basic algebra skills. Figure 2: How would you rate your basic algebra skills? (SurveyMonkey) All 10 volunteers rated their basic algebra skills to be average, good or very good (Figure 2). This is an encouraging piece of data but I was a bit surprised with their responses. I have taught this group of students for almost three quarters of a year in my geometry class. In my opinion, there are some students who are very Singapore Mathematics in U.S. High School 50 weak in their basic algebra skills because they rely too much on their calculators. They are pretty good in punching the numbers to get the answers but they will struggle doing manual computations on paper without a calculator. It is interesting to find out that the students actually feel that their basic algebra skills are decent even though they depend on a calculating device that limits the use of their thinking abilities. During the interviews, some of the students struggled doing simple arithmetic computations by hand. They asked to use a calculator, but I told them to work on the problem without one. This suggested that perhaps the students have a skewed impression of how good their algebra skills really are. This could be the result of false and excessive praising and rewarding students for mediocre performances. There are significant differences between the U.S. and Singapore education systems that might explain the students' self-­‐perceptions. Having gone through Singapore's and the United States' educational system, I believe that the typical American educational practice of rewarding effort rather than focusing solely on academic performance may explain this phenomenon. This sharp contrast compared to Singapore's demanding and unforgiving educational standards may play a significant role in student confidence. However, student confidence is not consistent. When asked if they feel confident about math in general, their answers were not quite so positive. This is yet another interesting observation from figure 3 below because I would expect students who have good basic math skills to be confident in math. I believe that Singapore Mathematics in U.S. High School 51 although the students have decent basic arithmetic skills of adding, subtracting, multiplying and dividing, the students often struggled at computing higher-­‐level math problems that require them to think critically and put their problem solving skills to the test. With each level of math class, the students are realizing that they are able to do the math when they are taught how to do it through practice and drills. However, if the questions do not look familiar (similar to those that they have tried before), they become lost as they lack the critical thinking and problem solving skills. That would explain why they feel confident in their basic math skills, but not as confident in dealing with math in general as showed in the next figure. Figure 3 shows that out of five interviewees, only one student felt good in both algebra skills and is confident in math. Figure 3 shows the online responses from the 10 participants when they were asked the question of what their level of confidence in math before learning Singapore math. Figure 3: How is your level of confidence in math before using Singapore math? (SurveyMonkey) Students confirmed their survey responses during the one-­‐on-­‐one interviews. When asked on a scale of 1 to 10 (1 being no confidence and 10 be strongest confidence), how would they rate their basic algebra skills one student Singapore Mathematics in U.S. High School 52 indicated, "Maybe 7, no, 8 or 9 maybe." (C.J., One-­‐on-­‐One Interview, January 23, 2013) Another said, "About an 8. The reason being is because last year I did not take algebra 1. I took it online." (A.C., One-­‐on-­‐One Interview, January 23, 2013) While a third noted, "On a scale of 1 to 10, probably a 9." (L.D., One-­‐on-­‐One Interview, January 23, 2013) A less confident student said, "A little above average."(V.C., One-­‐on-­‐One Interview, January 24, 2013) And finally one student noted, "Ok, out of 10, probably 8." (C.H., One-­‐on-­‐One Interview, January 25, 2013) The five students with one exception gave themselves a high score on their basic math skills. However, most of them gave themselves a low score on their level of math confidence. One student responded, "Like a 8 to 8.5." (A.C., One-­‐on-­‐One Interview, January 23, 2013) while another added, "Well, at my old school, it would probably be a 8 or 9. But here, it is probably a 7 or 8." (C.J., One-­‐on-­‐One Interview, January 23, 2013) One even went to explain, "Like a 6 because I had to have my mother helped me." (C.H., One-­‐on-­‐One Interview, January 25, 2013) A hesitant student said, "A little above average." (V.C., One-­‐on-­‐One Interview, January 24, 2013) With deep thought, a student replied, "Thought I was good at math. But I don't know. Sometimes I am good at math and sometimes I think not." (L.D., One-­‐on-­‐ One Interview, January 23, 2013) All five students struggled on the word problem that I presented to them. I could sense that they were not confident before starting on the word problem. Many students are usually apprehensive when dealing with word problems. It will be interesting to find out how and why students are having a low level of confidence despite believing that they have good algebra skills in math. While I observed and Singapore Mathematics in U.S. High School 53 helped the students during the problem solving activity in the interviews, I also noticed that there is a distinct lack of critical thinking skills and problem solving abilities. I watched as all five students struggled to put together and use the information given in the word problem to set up the right diagram or equation to solve the question in their own method. I do realize that there is more than one way to solve the word problem, but none of the students came close to fully understanding the question and setting up the math problem in a cognitive manner. These findings are consistent with recent trends and U.S. comparative performance. It is important to highlight the recent TIMSS (Trends In International Mathematics and Science Study) results released in December, 2011 that United States math students have improved in general, but they are still very weak compared to many other developed and developing countries. The results also revealed that United States math students are not good problem solvers and are weak in critical thinking skills (TIMMS, 2012). My students seem to match this study's findings. Visual Appeal of Singapore Math's Bar Modeling Technique The most prominent Singapore math technique is the bar model. It is a wonderful tool that I feel all math students should learn and apply in whatever level of math they are doing. As part of their interview, the five students were given a simple two-­‐part word problem. They all could do the first part without any trouble using their basic algebra skills. However, they all struggled on the second part and none of them could figure out how to solve the word problem. I did not rush them as Singapore Mathematics in U.S. High School 54 I give them plenty of time to work on the problem. When I felt it was appropriate to help them, I read the problem out loud and confirmed with them if they understood what has been asked of them from the word problem. They all understood what was being asked but did not have the critical thinking and problem solving skills to solve the question. I proceeded to show them how to solve the problem using the bar modeling technique. This is an interesting short conversation taken from one of the one-­‐on-­‐one interview. Hsu: OK, good. You need to do that. Now have you heard of Singapore math? C.J.: No. Hsu: So you don't know anything about it? If I were to tell you that this bar modeling method of Singapore math is quite famous, would you believe me? C.J.: No. Hsu: No problem, no worry. The bar modeling method is a very integral part of Singapore math. Singapore math can dissect a word problem and then pictorially represent it and then solve it. It is like solving algebra problems without using algebra. How neat is that? C.J.: (Looking rather confused!) That is confusing. Hsu: Really? Why would that be confusing? C.J.: How can you possibly solve algebra problems without using algebra? Singapore Mathematics in U.S. High School 55 Hsu: Well, I understand that the concept is hard to fathom right now but wouldn't it be cool to be able to solve algebra problems without using algebra? C.J.: Yes! Yes! After allowing C.J. to work on the word problem and helping him to reach a possible solution, the discussion continued. Hsu: (In a very calm and reassuring voice) Did I use any algebra to solve this? C.J.: No, that is crazy! (Shaking his head) Hsu: I know. (Nodding his head). Can you see how powerful this bar modeling technique is? C.J.: (Still shaking his head) Yes." (C.J., One-­‐on-­‐One Interview, January 23, 2013) C.J. was not alone when he felt amazed at how effective and powerful the Singapore math's bar modeling technique is. All five interviewees were visibly fascinated and intrigued when I showed them how to solve the word problem with the bar modeling method. After showing the five interviewees how I would solve the word problem using Singapore math's bar modeling technique, I asked them if they would feel more confident in their math ability to solve word problems if they know Singapore math. I was very relieved and happy to hear one student say, "Probably a 9.5!" (L.D., One-­‐on-­‐One Interview, January 23, 2013) Visibly impressed, one student acknowledged, "Much more confident!" (A.C., One-­‐on-­‐One Interview, January 23, 2013) After thinking for a brief time, one answered, "Like if I practiced it enough, I Singapore Mathematics in U.S. High School 56 would probably be pretty confident." (C.J., One-­‐on-­‐One Interview, January 23, 2013) One student did not even think twice before he replied, "I would be pretty confident." (V.C., One-­‐on-­‐One Interview, January 24, 2013) Another student exclaimed, "More, a lot more because I could not do it before and now I can do it with practice." (C.H., One-­‐on-­‐One Interview, January 25, 2013) These students perceived a great value from Singapore math's bar modeling techniques. As indicated by my students' responses to the impact of Singapore math's influence on their abilities as well as their confidence, all of them agreed that if they know Singapore math, they will be more equipped to handle word problems and that would have a huge boost on their level of confidence in math. In my opinion, the disconnect between the observations from the survey responses (figures 2 and 3), along with student performance in the word problem exercise in one on one interviews could be the result that the students may have the basic math skills, but do not have the necessary problem solving and critical thinking skills to develop confidence to problem solve. All five interviewees were astounded by the ease at which the word problem which they struggled to answer could be solved so easily with the bar modeling technique without using any algebra. Singapore math appeals to the students of this current generation because the bar modeling technique has a very visual approach. The current generation of students has been conditioned by their access to technology and the frequency and duration of its use. I think we have allowed math students to rely too much on the calculators because now they are too dependent on it. I saw that when the students were trying to do the word problem during the interview. They struggled to do Singapore Mathematics in U.S. High School 57 simple arithmetic computation and asked to use a calculator. We see young people spending an abundance of time in front of screens for leisure and schoolwork. These days, students are more inclined to use their phones to take pictures of notes or information on the board instead of writing things down on their books. An observation that I have made in my own classroom is that most of my students prefer to learn through visual aids now more. A picture is truly worth a thousand words as people are learning and teaching more and more through visual aids. I believe that Singapore math has an important place in U.S. educational system to help teach math successfully to students who are drawn to visual presentations and simulations. My students even proposed its use in lower grades: I definitely think it does have a place in elementary schools. In high school…hmm… I think would so, like in a foundation class, just to get a foundation of what you are going to learn in the next four years. As that is a great way as it builds a foundation early. It is not confusing. (A.C., Interview, January 23, 2013) Singapore's educational system is well known to be unforgiving and competitive. I remembered I was struggling in math when I was about 10 years old because I was not a good student. I played too many sports. My mother then forced me to study math with a tutor. I met with the tutor twice a week. I did not have a fond memory of those tutoring session as I was always given a lot of word problems for homework. We would work on those word problems several times using different approaches and bar modeling techniques. Then one day, after about three years of tutoring, she told my mother that she is not able to tutor me anymore. She Singapore Mathematics in U.S. High School 58 explained that there was nothing more she could help me in math and that I was good enough to do the math on my own. I was very surprised by that. I did not realize that the tutoring had helped me tremendously in developing my critical thinking and problem solving skills. In retrospect, Singapore math's bar modeling techniques had really made a big difference in helping me master and appreciate math as an elementary student. The students that I have interviewed are genuinely interested and intrigued by the bar modeling method. While I was conducting the interview with L.D. and was just about to show this student the way to solve the word problem after she had struggled like the rest, A.C. walked into my classroom. A.C. then asked to watch me use bar modeling technique again to solve the word problem (I had already conducted the interview with him just the day before). His interest in returning to the classroom to observe showed that this student wanted to learn more about the bar modeling method if given a chance. There is a big potential to push for the use of the bar modeling technique because students are fascinated by the visual impact and usefulness of the bar modeling technique. All five interviewees agree that Singapore math's bar modeling technique can help with improving basic algebra skills and that this type of problem solving tool will appeal to visual learners. Every participant also strongly called for the use of Singapore math's bar modeling technique at the high school level because they feel that it has a place in high school. To back up this claim, I would use some of the responses provided by the interviewees when asked if Singapore math has a place in high school. One student declared, "Yes. I think so because it helps a lot." (L.D., Singapore Mathematics in U.S. High School 59 One-­‐on-­‐One Interview, January 23, 2013) One simply replied, "Yes." (C.J., One-­‐on-­‐ One Interview, January 23, 2013) With quite a bit of conviction, one said, "Yes I do. It could definitely." (V.C., One-­‐on-­‐One Interview, January 24, 2013) Looking rather seriously, one student reasoned, "Yes, as long as they have a lot less problems because you are going to fill up two pages of that." (C.H., One-­‐on-­‐One Interview, January 25, 2013) One student gave an interesting reply and remarked, I definitely think it does in elementary school. In high school, I think so, like in a foundation class, just to get a foundation of what you are going to learn in the next four years. As that is a great way as it builds a foundation early. It is not confusing. (A.C., One-­‐on-­‐One Interview, January 23, 2013) Training Requirements on Teachers for Teaching Singapore Math Teacher training and confidence are also an issue in implementing bar modeling in a U.S. high school setting. I found myself second-­‐guessing myself when I first showed the bar modeling technique to A.C. who was my first interviewee. However, I became more confident with each successive interview. I can only wonder how many years of training that Singapore teachers have to go through in order to be proficient in teaching Singapore math. The years of training Singapore teachers have to go through must not be overlooked. "And the thing is that I don't dare to teach it to the class because typically Singapore math is taught at the elementary level in the United States. I can understand your disbelief and confusion right now because you have not been properly introduced to and taught in Singapore math. If I was to Singapore Mathematics in U.S. High School 60 introduce the bar modeling concept to high school students abruptly, it is very difficult for the kids to actually grasp the concept." (Hsu, One-­‐on-­‐one Interview, January 23, 2013) I know I struggled a bit with teaching the bar modeling techniques to the five students. I was explaining how to solve the problem rather than teaching them the method. I would need a lot of teacher training on Singapore math in order for me to teach it proficiently. The study by VanTassel-­‐Baska, Feng, MacFarlane, Heng, Teo, Wong, et al. (2008) showed that Singapore teachers demonstrate a higher level of effectiveness than American teachers regarding behaviors and differentiation strategies. I believe that math teachers would be willing to be trained in Singapore math knowing that Singapore teachers are better prepared in content knowledge. The ability to apply Singapore math gives students tremendous confidence in problem solving and thinking critically. It is like riding a bike, once you learn it, you will always know how to do it. The same thing is true with learning Singapore math's bar modeling; once you have learned it and understand the workings of the techniques, the powerful number sense that it develops, the high confidence it instills to tackle word problems and its ability to be a powerful tool of critical thinking skills, it will always be available for use. Another challenge in implementing Singapore math has to do with teaching materials. Teacher training on Singapore math aside, I think a big part of why Singapore math is so successful in the United States is the use of Singapore math textbooks, which is an essential tool for teachers teaching the techniques and curriculum. Although Singapore textbooks are easily available to the United States Singapore Mathematics in U.S. High School 61 and used primarily at the elementary level, we must not take it for granted that it would be an easy fix to improving math achievement by just adopting the textbooks (Wang-­‐Iverson, Myers & Lim, 2009). To really understand the impact and feasibility of using Singapore math textbooks at the high school level, more research is necessary. Now that I have acknowledged that my teaching style is different and it has to do with my Singapore math experience. I realize that I cannot execute the method the way my Singapore teachers' did and use it here in the United States. I recognize I still need a lot of training on how to blend both my Singapore math influenced teaching style and classroom management skills necessary for United States. On block days like today, students did not appreciate the classroom time I gave them after the lesson was over to work on their homework in class. Some students took the opportunity to finish up the easy homework. However, there are several students who were not so grateful for such opportunity. They opted to chitchat and visit with each other. This is most frustrating. But I do get it. I probably would do the same. When I looked at them, I get frustrated because when I was taking math in high school back in Singapore, I was never given the chance to do my homework in class. The math teacher would pack the class time completely full with examples and questions. (Journal, January 15, 2013) Summary Singapore Mathematics in U.S. High School 62 Both the literature and my findings support the fact that Singapore math, particularly the bar modeling technique of Singapore math, can indeed increase students' confidence in math and improve students' level of critical thinking and problem solving skills at the high school level. In order for me to apply Singapore math's bar modeling techniques in my own math classes in the near future, more work, research and training are necessary from both my administration and me. I am willing to go through retraining either in Singapore or workshops in the United States to get more proficient in Singapore math's bar modeling technique and curriculum. More time and research is needed to look into the appropriate Singapore math textbook for the high school level. For the rest of the year, I am hoping to throw in a few bar modeling techniques into my math lessons and obtain positive results from my high school students. However, I must carefully put into action some more research from myself and other Singapore math teachers and approval from administration before I can proceed. Singapore Mathematics in U.S. High School 63 Chapter 5 Implications and Recommendations for Applying Singapore Math As stated in my results, I believe that Singapore math has a place in high school math curriculum in terms of increasing students' confidence in math and improving students' level of critical thinking and problem solving skills. The conclusions from the data analysis are very compelling and interesting. I found four themes from analyzing the data. They are firstly the influence of Singapore math on teaching and learning styles in math, impact of Singapore math on students' confidence and abilities in math, visual appeal of Singapore math's bar modeling techniques and the teacher training requirements for Singapore math. Offer Visually Stimulating Math I want to use Singapore math, especially bar modeling methods, to excite my students about math and to help my students see math visually through my renewed passion and confidence in Singapore math. The influence of Singapore math on my learning and teaching style has been huge. In Singapore, I have always learned math visually and personally that is how I learn best. In my opinion, Singapore math appeals to the young students to learn and understand math in depth through visual aids. I am grateful that Singapore math has shaped my teaching philosophies in Vygotsky's constructivism and scaffolding in ZPD. From the interviews in this study, the students like my visually inclined teaching style because I tend to use many visual aids, diagrams and manipulative and I like to model the work in class. Knowing that my students have connected to my style of Singapore Mathematics in U.S. High School 64 teaching is a huge boost to my teaching confidence and a validation to my pedagogical abilities. I will continue with my visually inclined teaching philosophy of using Vygotsky's constructivism and scaffolding through modeling to help students master and enjoy math the way I did. From now on, I will spare no effort in adding more visual aids in terms of manipulative and drawing diagrams in my lessons. It is my hope that the administration will see the benefits of applying Singapore math by requiring all math teachers to be Singapore math trained and committing as much resources as possible to provide the necessary manipulative and visual aids. Improve Critical Thinking and Problem Solving Skills Implementing Singapore math at the elementary and high school level will help prepare students visually to understand math better in terms of developing better critical thinking and problem solving skills. If a young learner can learn to appreciate and master math visually with the help of Singapore math's bar modeling techniques, those skills will help students think critically as well as learn to use the tools they have mastered to problem solve. There is a big disparity between students' impression of their own basic math skills and their level of confidence in solving word problems. Although the students feel that they have the skills to do basic math, they do not feel confident in solving problem due to the lack of critical thinking and problem solving skills. This was very evident during the interviews when I asked them to solve a word problem and all five interviewees struggled in solving the word problem completely. A possible explanation of why the students in Singapore Mathematics in U.S. High School 65 this study are confident in their basic math skills but that confidence did not translate to the problem solving activity is that the students may have a skewed opinion of how good their basic arithmetic skills really are. However, it does not mask the serious problem of students' lack of critical thinking and problem solving skills. I believe that the students have the tools to do the math but they do not possess the skills to know which tool to use to problem solve. This issue was definitely highlighted in 2011's TIMSS study where United States ranked 10th for grade 4 math and 9th for grade 8 math (TIMSS, December, 2011). Internationally, United States' math curriculum is still behind many other countries in terms of critical thinking and problem solving skills. Students in the study were amazed at the powerful visual impact and ease of solving algebra problems using Singapore math's bar modeling technique because it does not involve setting up the algebraic equations. They were really intrigued and visibly very interested in the simple but yet effective use of the bar modeling technique. The bar modeling method appealed to the five interviewees who are all visual learners. All five students strongly called for the use of Singapore math's bar modeling technique at the high school level because they feel that Singapore math can help give them the necessary critical thinking and problem solving skills. They also agreed unanimously that the techniques of Singapore math could improve their confidence in problem solving. Singapore math has an important place in the United States' educational system now more than ever to help teach math successfully to students who are constantly drawn to visual presentations and simulations. Singapore Mathematics in U.S. High School 66 Future Research Recommendations Interviewing a whole class, a whole school and other schools. One of the things that I would do differently if I get more time to do this project again or my advice to those that would like to investigate deeper into this research is to use a larger sample size. I would recommend interviewing the whole, entire class, then the data collected on students' confidence and abilities would make this research more statistically significant. A focus group may be a good thing to do for the entire group to discuss their issues with their individual confidence, abilities and fears in math. Perhaps through this mean, the solution of the lack of critical thinking and problem solving skills can be bettered pinpointed out and addressed. Future researchers can also extend this research to a bigger scale by conducting the research on math students for the entire school and then proceed on to conducting this research with several other high schools instead of using students from only one high school. Measure pre-­ and post math tests using Singapore math. To obtain some form of quantitative data, future researchers could teach or use Singapore math techniques, particularly the bar modeling method, exclusively on a particular and appropriate math topic or chapter to measure the pre-­‐ and post-­‐ test results. The numbers would definitely tell an interesting story in showing how well high school students could learn and apply Singapore math at the high school level to learn to think critically and problem solve. I had initially wanted to do this with my geometry class but I struggled to find an appropriate way and time of introducing Singapore Mathematics in U.S. High School 67 the bar modeling technique in my current geometry curriculum. I was hopeful to teach the bar modeling technique to one of my three geometry classes and see if they outscored the other two classes on the relevant chapter test. If there was time, I would recommend giving all the math students a standard primary six math examination papers from Singapore where they will have to use their basic math skills to solve a number of algebra and geometry word problems. This examination is equivalent to about the United States' high school Algebra 1. This could be used as a pre-­‐test. Then the classes will undergo a series of Singapore math lessons where they learn how to solve word problems using the bar modeling techniques of Singapore math. Let them take the tests again and examine the results. I am very confident that there will be a significant improvement. Radical Change in United States' Math Education Based on my project, I think it is still quite premature to determine completely whether Singapore math will be 100% successful when applied at the high school level and whether Singapore math appeals to all the students. I believe more research is needed to fully determine how Singapore math can increase all students' level of confidence in math through improving their critical thinking and problem solving skills. If the results are positive, then that will provide tremendous evidence to present to the school district or policy makers to bring about a radical change in the U.S. math education in terms of curriculum and pedagogy. However, the challenge will be to introduce bar modeling technique to high school students in a systematic manner with steady progression so that the students Singapore Mathematics in U.S. High School 68 can become comfortable and not be confused by it. A lot of time and resources will have to be provided to train teachers how to teach Singapore math and choose an appropriate Singapore math textbook. At the time of this project, no appropriate Singapore math textbook for high school is available. A pilot program would have to be conducted in a school and then to a group of schools where all the math teachers will have to be trained properly in Singapore math and an appropriate Singapore math textbook is used. More research data will be needed to convince people that Singapore math is the answer to the problems of the U.S. math education. Summary I am indeed very fortunate to be educated in Singapore and to have found a passion in math with the help of Singapore math's bar modeling techniques. I will continue to use my visually inclined teaching style to engage my students in math. I truly believe that Singapore math can help high school students improve their critical thinking abilities and problem solving skills. It will also help raise students' level of confidence in math. Hence, Singapore math's bar modeling techniques should have a place in high school because its visual appeal is undeniable. However, more research needs to happen before the real impact of Singapore math can be measured and assessed. There is a lot of data on Singapore math having success at the elementary level. It will be very interesting to see if the success of Singapore math at the elementary level can be transferred to the high school level. More studies need to be conducted to see if elementary students going on to high school can be just as successful in math and how Singapore math can be fully implemented Singapore Mathematics in U.S. High School 69 in high schools where students do not have prior knowledge of Singapore math. There are several challenges in terms of teacher training and use of appropriate Singapore math textbooks. In my professional opinion, if the United States hopes to raise its math achievement levels to be globally competitive, radical change in the United States' math educational system is necessary. Implementing Singapore math may be a good solution. Singapore Mathematics in U.S. High School 70 References Ahuja, O. P. (2006). World-­‐class high quality mathematics education for all K-­‐12 American students. The Montana Mathematics Enthusiast, 3(2), 223-­‐248. Chan, C. M. E. (2009). Mathematical modeling as problem solving for children in the Singapore mathematics classrooms. 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