A journal of the Georgia Association of Teacher Educators ..... The development of online teacher surveys ...... of implementation: Implications for drug abuse.
GATEways to Teacher Education A journal of the Georgia Association of Teacher Educators
Teaching Singapore Math: Evaluating Measures to Effectively Teach and Implement a New Mathematics Curriculum in 21 Elementary Schools James Badger,
University of North Georgia
Fourth grade and eighth grade Singapore students’ first-place ranking in 1995 and 2003 – and second-place ranking in 2007 – in Trends in International Mathematics and Science Study has not gone unnoticed by policy makers and educators (Garelick, 2006; Hoven & Garelick, 2007). As a possible consequence of these strong international test results, a growing number of schools in the U.S. have adopted the Singapore Math curriculum (Cai, 2003; Gross & Merchlinsky, 2002; Hu, 2010; Leinwald & Ginsburg, 2007; Menon, 2000). This study evaluates the effectiveness in one U.S county that implemented the mathematics curriculum in 21 elementary schools in 2008. Two research questions framed the investigation: 1. How is Singapore Math implemented by elementary teachers? 2. Has the implementation of the Singapore Math curriculum resulted in increased teacher confidence communicating the curriculum and positively impacted student learning of mathematics? Answers to these questions will be informed first by describing the Singapore Math curriculum before surveying some of the literature related to elementary teachers’ competence in mathematics and curriculum implementation. Utilizing O’Donnell’s (2008) conceptual framework to determine fidelity of curriculum implementation, findings from interviews, achievement results, surveys, and
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classroom observations will determine fidelity of curriculum implementation along a strong or weak continuum. Conclusions advance recommendations for administrators and teachers to foster an effective curriculum implementation. Review of Literature Teaching Mathematics Literacy and numeracy are cornerstones in the U.S. Department of Education’s No Child Left Behind policy. Improving the quality of mathematics teaching and learning hinges on access to sound evidence regarding what teachers do with students to promote the development of learners’ mathematical proficiency and understanding, rather than concentrating instruction in activities such as recalling facts and applying well-rehearsed procedures to answer simple questions (Bransford et al., 2000; Silver et al., 2009; Slavin & Lake, 2008; Stigler & Heibert, 1999; U.S. DOE, 2008). In a review of effective instructional practices, Washaw and Anthony (2008) underscored the importance of classroom discourse as an effective pedagogy that contributes to students’ engagement and cognitive understanding of mathematics. Falkenberg (2011) identified a shift from theorizing about what teachers need to know to teach mathematics to the conceptualization of mathematical content knowledge and a focus on
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how mathematical content knowledge is established. Significant changes in teachers’ attitudes and beliefs can occur when teachers have the opportunity to explore new instructional strategies and ideas in the context of their own classroom practice (Borko et al., 1997). While identifying best teaching examples of mathematics, Silver et al. (2009) found that pedagogical approaches were not systematically used in ways that supported students’ engagement with cognitively demanding mathematical tasks. Similar results were found in the instruction of mathematics at the primary level in England (OFSTED, 2008). Slavin and Lake’s (2008) review of effective elementary mathematics programs in three categories – curricula, computer-assisted instruction, and instructional process – identified instructional strategies and cooperative student learning methods as the most effective in terms of student achievement. Muir’s (2008) synthesis of effective teachers’ numeracy practices identified a set of “principles of practices,” including making connections, challenging all learners, teaching for conceptual understanding, directing purposeful discussions, focusing on mathematics, and maintaining positive attitudes, that significantly affect student attainment in mathematics over a two-year period. Recent studies have identified the connection between teaching mathematics and the effective role manipulatives occupy to support content learning, especially for teaching low-achievers, students with learning disabilities, and English language learners (Boggan et al., 2010; Puchner et al., 2008; Uribe-Florez & Wilkins, 2010). Singapore Math The Singapore Math curriculum was created in Singapore in 1981 by the Curriculum Development Institute of Singapore and adapted for the U.S. in 2001. Singapore Math focuses on depth of student understanding of mathematical concepts instead of memorization of rules or repetitive drills. Students are expected to completely master fewer topics rather than be able to perform many tasks with mediocrity. More time is devoted to fewer topics in the curriculum, with a focus on problem solving, computational skills, and conceptual and
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strategic thinking processes. Concepts in Singapore Math are mastered through concrete manipulatives such as base ten blocks, number disks, and bar models. Lessons are organized so students have opportunities to make independent and meaningful discoveries using visual and hands-on aids such as blocks, cards, and bar charts. There is an emphasis in the curriculum on an analytical approach and conceptual grasp of mathematics, with a move to also make mathematics concepts relevant to the real world. The Singapore Math curriculum features a three-step learning process that moves from concrete to pictorial to abstract: a mathematical concept is introduced in a concrete representation before progressing to a pictorial demonstration and finally advancing to abstract conceptualization. The curriculum addresses a small number of topics in depth following a spiral organization in which topics presented at one grade level are covered in later grades. Further, the textbooks build students’ understanding of mathematical concepts through concrete illustrations that demonstrate how abstract mathematical concepts are used to solve problems from different perspectives. A comparison of U.S. mathematics with Singapore mathematics in the primary grades found differences in textbooks, assessments, and teacher training programs, with a large discrepancy identifying mastery of mathematics in Singapore students in contrast to U.S. students who mechanically apply mathematical procedures to solve routine problems (Cai, 2003; Ginsburg et al., 2005). Curriculum Implementation: A Conceptual Framework Investigating teachers’ implementation of a new curriculum is important to determine the effectiveness of a program in actual use and how close the program is implemented according to its original design or intention (Dobson & Shaw, 1988; Dusenbury et al., 2003; Ormundson et al., 2012). Efficacy studies of curriculum intervention often define, conceptualize, and measure fidelity to the extent to which participants understand and use the concepts in a curriculum as specified by curriculum developers (Gross & Merchlinsky, 2002; Johnson, 2000). O’Donnell’s (2008) extensive
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review of the literature from K-12 schools and public health settings identified levels of fidelity during implementation that impact effectiveness. Fidelity of implementation was found to have multiple but similar definitions of fidelity for K12 core curriculum interventions with frequent measures connected to instructional quality. In a review of the literature concerned with fidelity of implementation in K-12 settings, O’Donnell (2008) identified four constructs that overlap: teaching, curriculum potential, curriculum-in-use or perceived curriculum, and adaptation. According to O’Donnell (2008), while fidelity of implementation is a relatively recent construct in K-12 curriculum intervention research, fidelity of implementation has been defined as the determination of how well an intervention is implemented in comparison with the original program design during an efficacy and/or effectiveness study. Measures of fidelity should indicate how an intervention is maintained consistently across multiple groups over time “or describe the parameters under which variations in the implementation may occur” (p. 54). K-12 curriculum implementation was found to encompass adherence and integrity to efficacy – “focusing on whether a program is implemented at all and to what degree” (p. 41) – and effectiveness – “interpreting evidence of effectiveness of the program for generalizability . . . and observing the implementation of the program in the field” (p. 42). Explicated as guiding questions, the four constructs below inform issues concerning fidelity of curriculum implementation and provide a framework for answering the first research question: 1. Teaching: How does one distinguish good teaching and fidelity of implementation to good teaching practices prompted by the curriculum material? 2. Curriculum profile: What are the critical components of the curriculum? What ranges of variations are acceptable? What does it mean to implement the program with fidelity as defined by school administrators and county supervisors? 3. Curriculum-in-use or perceived curriculum: How are the curriculum and
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the perceived curriculum viewed and implemented by teachers? How are curriculum materials and instruction mutually supportive and reinforcing? 4. Adaptation: Does the curriculum promote variation and adaptation of curriculum implementation (O’Donnell, 2008)? Taken together, these four constructs provide a framework through which to evaluate the implementation of a new curriculum. A County’s Implementation of the Singapore Math Curriculum The adoption in 2008 of the Singapore Math curriculum was mapped to the state’s standards in 21 elementary schools in a county in Georgia. Eleven schools in the county qualified for Title I status, reflective of settings where more than 70% of students attending a school are from low-income households. A school that qualifies as Title I receives federal funds to support the improvement of student test scores, prevent student drop-out, and provide free- and reduced-lunch programs with general school improvement. Thirteen of the twenty-one schools studied were identified Title I; fifteen of the schools reflected a growing English language learner (ELL) population of largely Latino students. In one school studied, 94% of the students were eligible for the free-andreduced lunch program. Prior to implementation of the new mathematics curriculum, teachers and school administrators participated in a two-day, districtwide training session that introduced bar modeling, number discs, abstract conceptualization, number bonds, place value, and manipulatives. In the first year of implementation, two representatives from the county were employed full-time to facilitate the rollout of the Singapore Math curriculum and regularly visited each school to conduct classroom observations, deliver mini-workshops, and answer curriculum questions. The county created a password-accessible website for teachers to access Singapore Math resources and videos to facilitate instruction and learning. In addition, a teacher representative in each school was selected to occupy the role of liaison between the school and county to address
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curriculum- and instruction-specific questions. For funding reasons, only one county representative was employed in the second year to facilitate the implementation of the curriculum. While students in Kindergarten and Grade 1 would move into higher grade levels with a foundation of Singapore Math in subsequent years, the rollout of the new curriculum also occurred in grades 2, 3, and 4 in the first year of implementation. As a result, the Singapore Math curriculum was introduced as a new approach for students learning mathematics in the higher grade levels for whom a previous curriculum was familiar; that is, students in the higher grade levels would not have a spiraling Singapore Math curriculum as the lower grade levels would experience. Methodology Data Collection Determining the fidelity of the Singapore Math curriculum implementation was measured through multiple measures rather than relying on a single indicator (Ormundson et al., 2012): teacher surveys and journals, teacher and student interviews, end-of-year state criterion-referenced test (CRT) and Iowa Test of Basic Skills (ITBS) student achievement scores, and classroom observations. One teacher in each of the 21 schools from Kindergarten to Grade 4 volunteered to participate in the first year of the study after receiving a description by the principal or assistant principal in a staff meeting. There were, therefore, 105 teachers in the first year. Participating teachers each received a $100.00 stipend at the conclusion of the school year for their contributions to the study. The educators agreed to journal four times over the course of the school year and allow classroom observations. The first-year teachers were invited to return to participate at the start of the second year, and others volunteered to replace five teachers who did not to return in the second year. Five teachers withdrew after the first year because of pregnancy, change in work assignment, growing school commitments, and inability to maintain the journaling and classroom video-recording requirements. In the second year of the study, Grade 5 was
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implemented by the county and also included in the research design, increasing the number to 127 participating teachers. There were no design changes in the second year of the study. Surveys The development of online teacher surveys was informed by the research questions of this investigation, pilot-testing the surveys in 2008, and a review of literature on teachers’ perceptions of mathematics (Luppescu & Hart, 2005; Tapia & Marsh, 2004; Wisconsin Center for Education Research, 2004). Results collected from the same survey disseminated over two years were produced in an Excel file and data disaggregated by grade level and school. Journals Teachers’ journals were submitted at the conclusion of four data-collection periods: September-October, November-December, January-February, and March-April. To increase participants’ response rate, teachers received email reminders requesting the completion of a journal reflection at the conclusion of each data collection period. Teachers answered guiding questions related to the delivery of the mathematics curriculum and student learning: What has been particularly successful for you when teaching mathematics (in this unit)? What has been confusing or frustrating for you when teaching mathematics (in this unit)? What math activities in this unit have your students liked the best? What math activities in this unit do you feel have most helped your students to learn mathematics? What has been confusing or frustrating in this unit for your students as they learn mathematics? Interviews Structured interviews were conducted for 30 to 40 minutes with participating teachers, principals, and assistant principals who selfselected themselves in schools where classroom observations were conducted. The experience of interviewed educators ranged from 2-20 years, with five educators possessing a masters’ degree in education and the same number with ESOL endorsement. Three educators were career changers. Interviews probed their perceptions, delivery, and concerns about the new curriculum. Interviews with participating
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teachers were commonly conducted over a lunch break or after school (Rubin & Rubin, 2005; Stecher et al., 2006). Student focus-group interviews with Kindergarten to Grade 4 students in participating teachers’ classrooms asked learners to reflect on their likes or dislikes in mathematics, favorite activities, and how best they learned mathematics. No more than four students were randomly selected for a focus-group interview from the returned group of parent/guardian letters of consent. In total, 60 students participated in focus-group interviews during the first year and 52 students during the second year. Interviews were conducted in the school library, a hallway, or in a classroom.
program to analyze quantitative and qualitative data, text, and multimedia. Using grounded theory as a framework to systematically group the coded data into similar concepts or themes, recurring themes were identified (Marshall & Rossman, 1995; Miles & Huberman, 1994; Rubin & Rubin, 2005; Saldana, 2009). Teacher Surveys The experience level of surveyed teachers is shown in Table 1:
Classroom observations Classroom observations were conducted by three researchers in three, three-week periods in four randomly selected schools in 2009 and six schools in 2010. Half of the schools selected in 2009 and 2010 were Title I. One teacher from Kindergarten to Grade 4 was observed at least six times. Classroom observations supplemented journal data by focusing on teachers’ explanations and modeling of the new curriculum together with the dialogue and interactions between the teacher and students about mathematical concepts, examples, or activities (Mercer, 2001; Muijs, 2006). Student mathematics achievement The criterion-referenced competency (CRT) is administered each April for all grade levels. The CRT is a standardized, state assessment that tests students’ knowledge in grades 1 through 8 of mathematics, language arts, and reading, with grades 3 through 8 also assessed in science and social studies. The Iowa Test of Basic Skills (ITBS) is a standardized, norm-referenced test administered to Kindergarten to Grade 5 students every September. Student achievement results from the CRT and ITBS from 2006 to 2010 were provided by the county to the researchers. Data Analysis The sections below analyze the gathered data. Journal entries, teacher interviews, and classroom observation data were coded with MAXQDA, a computer-assisted software
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GATEways to Teacher Education A journal of the Georgia Association of Teacher Educators
Table 1 Years of Teaching Experience, 2008-2010 Years of Experience (All Teachers) – 2008
1-3 27%
Over 12 31%
4-6 16%
10-12 11% 7-9 15%
Years of Experience (All Teachers) – 2009
1-3 24% Over 12 36%
4-6 15% 10-12 11%
7-9 14%
Years of Experience (All Teachers) – 2010
Series1, Over 12, 43.4%, 43%
Series1, 10-12, 10.2%, 10%
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Series1, 1-3, 16.6%, 17% Series1, 4-6, 18.5%, 19% Series1, 7-9, 11.2%, 11%
Surveyed teachers reported that they received sufficient training and resources from school and county administration to effectively teach the new curriculum. Survey data revealed that the training and mentoring initiatives that accompanied the implementation of the Singapore Math curriculum were appreciated by many teachers in response to the survey statement ‘I believe I have the training and resources to effectively teach math.’ Further, elementary teachers’ attitudes toward mathematics rose slightly rather than dipped when the new curriculum was introduced. Teachers of all grade levels were considered as a single group to the survey item ‘I like teaching mathematics,’ suggesting that teachers’ affinity for teaching mathematics increased slightly from 2008 to 2010. While survey data found teachers’ affinity for teaching mathematics increased slightly between 2008 and 2010, there was no significant change in teachers’ understanding of or confidence in mathematical concepts, either for teachers at any grade level or for all surveyed teachers considered as a whole group. In 2008, 40% of surveyed teachers in grades 3 and 4 reported that they understood mathematics concepts covered at K-2 levels to the survey statement, ‘I am confident that I effectively teach mathematics concepts covered at K-2 only, K-5 only, K-8 only, K-10 only, K-12 only, college level.’ More than 70% of surveyed teachers agreed or strongly agreed to the following survey statements, with no significant change over time while implementing the new mathematics curriculum, either for teachers at any individual grade level or for all teachers considered as a whole group: ‘I believe developing problem-solving skills is an important component for success in learning mathematics’ and ‘I believe that in math class, students can learn to be creative and discover concepts independently.’ The affirmative response to these questions may be indicative of the teachers’ dispositions that math content is accessible to all students and that all are capable of learning mathematics. Surveyed teachers identified the training and resources provided by the district prior to and throughout the first and second year of
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implementation as essential for their understanding and instruction of the new curriculum. Findings from the survey data revealed that the Singapore Math curriculum did not significantly impact elementary educators’ knowledge of mathematics. While lingering questions remain from the survey data concerning elementary teachers’ reported low levels of mathematical understanding, conceptualizing elementary teachers’ receptive attitudes toward mathematics through O’Donnell’s curriculum profile and curriculumin-use indicate the potential for fidelity of implementation that was moderately strong. Teacher Journals Teachers’ journals, which were submitted each quarter of the school year, reflected benefits and challenges associated with delivering the new mathematics curriculum. Coding of the journal data identified a number of themes. In the first year of implementation, entries by teachers identified the utility of manipulatives, place-value disks, and number bonds as effective tools to communicate mathematical concepts, especially for learners who required more time to process mathematical ideas. The use of manipulatives occupied a central role in both the instruction and learning of a mathematical concept. Reflecting after a taught class or unit of instruction, teachers noted that interactive, hands-on activities were important for facilitating students’ learning of demanding mathematical concepts, and that the same engagement stimulated the learners’ interest and cultivated their interpersonal social skills. Elementary educators also valued the curriculum’s problems and questions that connected mathematical concepts to real-life examples. While elementary teachers claimed that mathematical problems and discovery activities instilled more engagement and interest in students, others expressed a desire for an even greater number of real-life examples in the curriculum and struggled to identify strategies to accommodate students with special needs. Still other educators identified a poor alignment of the curriculum, at times, with state standards. During the first year of implementation, educators expressed frustration that the county’s
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pacing guide curtailed instruction and reduced some students’ deeper understanding of a concept. Elementary educators expressed a desire for a more flexible pacing guide to allow for longer instruction time to facilitate learning in all students. A second theme that emerged from the journal data was the teachers’ observations of a vocabulary-rich curriculum that also contained challenging assessment instruments. In schools with a majority of English language learners, elementary educators reflected on strategies to effectively differentiate the new curriculum not only for English language learners but also advanced students who “got bored easily once the concept was clear,” wrote one teacher. Teachers noted that additional instruction was needed to describe a term or phrase found to be confusing for students. One example was the difference between the “value of the digit” and the “place of the digit” in the Grade 4 curriculum. These and other content-specific nomenclatures were particularly troublesome for English language learners. One teacher claimed that if students had been exposed to the Singapore Math vocabulary and bar-modeling strategies in Kindergarten, the terminology might be less confusing, thereby providing more time for teachers in upper grade levels to scaffold conceptual problems. A third theme that arose from the journal data was an observation that concrete exercises progressed too quickly to pictorial examples, and the same rapid progression outlined in the student textbook and teacher guide occurred when presenting material in pictorial representations that moved to complicated abstract concepts. In response, teachers claimed they supplemented the curriculum with additional problems at each stage to facilitate students’ learning. A related challenge was the observation that the Singapore Math assessments tended to “add questions containing problems that have never been taught,” according to one teacher. The inclusion of such problems may reflect the focus on problem solving and applying concepts, challenging learners to be creative and discover solutions by using and extending what students learned in a unit rather
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than memorizing specific algorithms to be applied to all questions. While some issues in the second year overlapped with the first, journal narratives in the second year were more nuanced, reflecting, in part, teachers’ growing familiarity with the curriculum and students’ knowledge of Singapore Math from the previous year of study. Lessons and activities that integrated manipulatives, number bonds, and number disks continued to be perceived as effective mediums to develop students’ conceptual understanding of mathematics. Teachers valued problems that connected concepts to real-life examples for instilling student interest and fostering discovery and experiential learning. The outcomes from adopting these approaches positively impacted student learning and confidence, as noted by a Grade 2 teacher: “My favorite part of this unit is when I see those struggling students look at me and say, ‘Look, Miss. I did it right!’” In the second year, educators claimed they had higher expectations of students and could plan lessons more effectively. The teachers’ written narratives revealed a perspective of the new curriculum that was perceived by some as being poorly aligned with state standards, presenting linguistic challenges for English language learners, and using assessment instruments that did not reflect a stronger connection to a taught unit. Alongside these observations, however, were equally detailed observations underscoring students’ interest in mathematics and the rich source of interactive activities. Teachers highlighted the value of number bonds, place value, and manipulatives that fostered students’ deeper understanding of mathematics. As reflected through journal data, teachers’ implementation of the new mathematics curriculum conceptualized through O’Donnell’s concepts of curriculum profile and adaptation intimated a marginally strong fidelity. Interviews Teacher interviews The most frequently occurring theme that emerged from the teacher interviews in 2009 related to the training and support delivered by the county. Teachers stated that the county’s workshops, Singapore Math website, and
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devoted county- and school-based Singapore Math representatives facilitated a deeper understanding of the new mathematics curriculum. As a consequence of the county’s introduction of Singapore Math and ongoing professional development over two years, interviewed teachers stated there was sustained support and interest in the new curriculum. When asked to contrast Singapore Math with the preceding curriculum, Harcourt Math, teachers claimed that they possessed greater confidence delivering a curriculum that was described as more engaging, challenging, and creative than the previous. Some teachers distinguished the new curriculum from the prior one as more structured and teacher-centered that required mastery of skill, greater emphasis on student literacy to understand the mathematics curriculum, and development of higher-order thinking skills in students: I think it’s very evident this year more than in the past how much language impacts learning when it comes to mathematics. (Grade 1 Teacher) I have found that even my lowest math students – who before would really struggle – know they can understand the concepts better because you have the hands-on with the disc which makes it much more concrete: you say what the number looks like and how you can manipulate it and how you can change the number. I feel that has helped a lot, especially with the lower (ability) kids. (Grade 2 Teacher) Singapore Math was described by teachers as an approach to learning mathematics that moved from a “drill and memorization skill” to an interactive, rigorous approach of learning and instruction. Further, the activity-oriented approach to learning was viewed as communicating deeper conceptual depth of mathematical concepts and fostering more critical and higher-order thinking skills in students. Teachers also stated that they had greater expectations of students and noticed higher formative test results as consequence of the new approach to learning mathematics. Teachers claimed that students learned Singapore Math through hands-on activities. The
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integration of manipulatives was a way to model and demonstrate concepts as well as an approach that fostered student learning in the concrete and pictorial stages of development: I’m more of a facilitator. We have a lot of time for the kids to practice and use the manipulatives. I introduce all the concepts and we try to work in a lot of time for the kids to work in small groups and to do centers and activities with what they have learned and also do a lot of practice, especially when you get to the algorithms for additions and subtraction - they just need a lot of practice with that. Give them what they need, introduce the concepts, and give them time to use it. (Grade 2 Teacher) Rather than memorize concepts, teachers stated that bar modeling, number bonds, and manipulatives fostered students’ conceptual understanding of mathematics through model drawing and challenging activities. In contrast to the previous curriculum, teachers described Singapore Math as a “rigorous program” that demanded strong reading strategies from students and sought learners to think about numbers in their head so “they’ll be stronger with the understanding of number sense and kind of help them concentrate.” In the process, a few teachers claimed that they asked more “why” questions to deepen the students’ understanding of mathematical strategies, such as “Why do you think we are doing it this way?” or “Why are we doing this?” The rationale for adopting this form of interaction was to instill higher-level thinking skills and independent learning: We’re more focused on having students explain their reasoning and getting into the higher level thinking skills rather than just rote getting math facts. (Grade 1 Teacher) We are getting them to see for themselves what the patterns are and what the rules are. Yes, I’m going to tell them, but I want them to start trying to figure it out for themselves as well. (Grade 3 Teacher) One teacher observed that the ability to competently communicate the Singapore Math curriculum to learners depended upon a
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teacher’s patience, flexibility, strong understanding, and preparation of Singapore Math’s central concepts and strategies: I feel like with Singapore you have to understand the math and the thinking behind how you’re going to teach them the math if that makes sense. And if you can’t think that way, then you’re not going to be able to teach them to think that way. (Grade 1 Teacher) The new curriculum was not without its challenges and limitations. In the first and second year of implementation, some teachers reported difficulties with effectively teaching bar modeling and illustrating word problems, which sometimes confused student learning. Echoing some of the themes in the journal data, particularly concerning for teachers, were the assessment instruments, paucity of real-life activities in the curriculum, and struggles to differentiate the new curriculum for ELLs: Yes, they may be more rigorous because it [test] allows you to have higher level thinking to see if you can apply the material but, as a teacher, I need to know what I taught was effective and so I was a little concerned with that. So we had those types of questions, like what do we do with these tests here? We’re finding that when we give a test, all of a sudden there’s this particular question that was not even part of our lesson. (Grade 1 Teacher) Teachers in the first and second year of implementation observed that the Singapore Math materials tended to, according to one teacher, ‘‘add questions containing problems that have never been taught.” Some educators perceived a disconnect between the instruction and the test questions that would often be, according to one teacher, “different than what was taught.” A few of the interviewed teachers remarked that the Singapore Math curriculum involved a lot more from the teacher modeling concepts than previous mathematic curriculums, thus emphasizing teacher-centered instruction at the expense of integrating student-focused math centers. Further, some teachers claimed the county’s pacing guide restricted their ability to effectively deliver a lesson and did not allow
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student mastery of particular mathematical concepts as a consequence of following a prescribed time period to complete a unit. It was noted, however, that modifications to the pacing guide were integrated in the second year and teachers were permitted by the school district to supplement specific units of the curriculum that did not meet state standards. Some educators remarked that parents’ reactions to the new mathematics curriculum ranged from enthusiasm to frustration, resulting in e-mail requests to teachers querying the terminology and concepts of Singapore Math as well as for information packets and modifications to better assist their child with assignments. Administrator interviews County and school administrators described the Singapore Math as a curriculum that would not only improve student learning but would develop deeper mathematical skills in elementary teachers who were described as possessing stronger literacy skills than mathematical abilities. For county administrators, a consequence for an elementary educator who possessed greater confidence in the knowledge of mathematics would be an improved ability to deliver the content. Principals claimed that increased elementary teachers’ competence in mathematics and higher student achievement results were desirous in the medium- rather than short-term: For the students, we’ve got to have our teachers better educated and better math teachers. So that’s definitely a goal to have better math teachers and stronger math teachers. They need to become stronger math teachers, and they need to understand how to teach students to develop this level of thinking and problem solving. (Elementary Principal 1) County administrators claimed that over time the new mathematics curriculum would deepen educators’ content-knowledge, positively impact students’ critical thinking and problem solving skills, and foster the students’ interest and mathematical understanding beyond the primary years extending into middle and secondary schooling:
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We want [teachers] to develop an understanding of how this process works on what numbers are and [students] know how to work with numbers. So that’s the goal: to develop the understanding. It takes more than just the teacher writing it on the board and the kids copying it on their paper. . . . What the very engaged classroom doesn’t necessarily mean is that the students are busy with the manipulatives all the time, but there needs to be a lot of interaction going on between the teacher and the students as far as [teachers] hearing that the students can process: “Did you get that answer?” or “Tell me, how did you arrive at that solution?” That sort of thing – good strong teacher instruction. (County Administrator 1) According to interviewed school and county administrators, these skills were cultivated over years rather than months, with teachers actively engaged in the students’ discovery process of learning and cognitive development: The rigor part is not going to come overnight. It takes a while to adjust to a different computation style. We’re hoping down the road that this will make a big difference for our children as far as their critical thinking skills and higher-order thinking skills. You just can’t evaluate [those abilities] overnight. To me, the children have to develop that year-in and year-out. (Elementary Principal 2) We want the children to think over and above what they’ve thought before, and this program offers that. We not only want them to problem solve in math but throughout their lives. We want them to have the skills they need to just think through things and start developing a pattern of “How do I deal with this problem?” and “How do I solve this problem?” (County Administrator 2) To successfully attain the interconnected goals of elevated teachers’ content-knowledge and increased students’ learning, administrators emphasized the importance of teacher training and continual county- and school-level support. Monitoring an effective and sustained
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implementation was determined through initial and succeeding teacher training workshops, installation of a county representative to address curriculum and pedagogical concerns, and regular classroom observations from school administrators. One principal stated that higher student expectations and improved tests results were contingent upon effective teaching “that we can sometimes control,” while student attitude toward the subject could not be influenced because it “sometimes originates from parents.“ Principals claimed that parent buy-in of the new curriculum depended upon student interest and academic success in mathematics. Based on classroom observations of and discussions with teachers and senior school staff, county administrators and principals claimed they were conscious of some of the difficulties and obstacles classroom teachers faced delivering the new curriculum. Some of the emergent challenges included differentiating the curriculum, responding to the needs of ELLs, modifying the Singapore Math assessment instruments, and developing teacher-training workshops. Principals recognized that classroom teachers were struggling to effectively integrate some math concepts and content-specific vocabulary as well as attaining master-level results in the assessment instruments. Principals described the challenges, on the one hand, of meeting adequate yearly progress and pressures preparing students for the high-stakes end-ofyear tests, with, on the other hand, a desire to cultivate student interest, learning, and critical thinking skills in mathematics. Student interviews Focus-group interviews revealed that Kindergarten to Grade 4 students were interested in mathematics and claimed they were not adverse to the challenges, questions, activities, or word problems expressed in their Singapore Math workbook or presented by their teacher. Interviewed students recognized the importance of mathematics, claiming that mathematics was not only used by their parents in their places of employment, but was important for future employment and valued for being able to answer questions without pen and paper when purchasing items in a store. When asked to
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identify whether the students preferred teacherled instruction or working together in studentcentered stations to answer math questions, students from Kindergarten to Grade 4 favored both learning stations and teacher-centered instruction. The teachers’, county administrators’, principals’, and students’ reflections of the Singapore Math curriculum contributed to a deeper understanding of how the curriculum was perceived coupled with tensions that imbue the delivery. Teachers’ ability to differentiate the mathematics curriculum for English language learners, adherence to the county’s pacing guide, and professed disconnect between the curriculum’s assessment instruments and unit of instruction presented lingering challenges to effectively deliver the curriculum. These limitations were overshadowed, however, by the teachers’ stated higher student expectations and the county’s ongoing training and support of the new curriculum. School and county administrators claimed to adopt a broader view of the new mathematics curriculum: the curriculum would not only positively impact student achievement scores, but foster in educators a deeper knowledge of mathematics that would positively impact the classroom instruction and cultivate students’ mathematical knowledge and problem-solving skills. Conceptualizing these themes through O’Donnell’s curriculum profile and adaptation constructs, the teachers’ and administrators’ positive perceptions of Singapore Math suggested a moderately strong implementation of the new approach to instruction and student learning. Classroom Observations Classroom observations provided a perspective of how teachers implemented the Singapore Math curriculum through O’Donnell’s curriculum-in-use category. Participant classroom observations collected data on teachers’ delivery of the Singapore Math program. Evident in classroom observations were bar modeling strategies and number bonds to review and explicate mathematical problems as a whole class. Kindergarten to Grade 4 teachers modeled mathematical examples at the concrete and pictorial level using magnetic
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place-value disks on a white board, and periodically explained some vocabulary-rich terms. Mental math strategies were also infused in many lessons. It was not uncommon for a teacher in the lower grade levels to ask students to open their workbook to “the page with seven tens and three ones [page 73].” With few exceptions in the first year of observations, all teachers held – or placed on a table nearby – a copy of the Singapore Math Teacher’s Edition to consult while delivering a lesson. Teachers stated that this practice was permitted by school administrators to facilitate accurate delivery of the new curriculum. Classroom observations reflected a preponderance of teacher-centered instruction instead of providing more occasions for studentcentered learning. There were not a lot of opportunities for students to experience the concrete manipulatives themselves, to collaborate on workbook exercises, or progress to the abstract level from the concrete and pictorial stages. The majority of teachers tended to emphasize low-level questioning and did not ask students to draw associations of a mathematical concept to real-world contexts. In one lesson, a teacher drew pictures of number disks (manipulatives) on a whiteboard, but neither used number disks herself nor permitted the students to employ them. In another case, a teacher demonstrated a mathematical operation with some manipulatives but did not allow the students to perform the same operations with the manipulatives. Teachers in grades 1, 2, and 3 integrated dry-erase whiteboards for each student to display their answer to a question during a teacher demonstration or review of a mathematical concept. One teacher engaged her Grade 2 students with individual clocks, whereby each student manipulated a time piece to set the correct time announced by the teacher, such as 20 after 6:00, 5 minutes before 8:00, or three-quarters after 9:00. Another Grade 2 teacher integrated number bonds to reinforce the principle of subtraction using the ones and tens place values. In the Grade 2 example that follows, the strategy of bar modeling was incorporated; however, the teacher provided few opportunities for the students to apply the model
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and did not question the students’ thinking to determine how an answer was determined. The Grade 2 teacher read a word problem from the textbook, then: T: What did the sentence just tell you? Who are we talking about? How much money did she have? S: She had $6.80 for a watermelon. T: OK. And $2.40 to buy beans. Draw a bar: $6.80 $2.40 [watermelon] [beans] Are we finding the sum or subtracting? S: Sum. T: Quickly try to find the answer. Make sure the decimals are on top of each other. Remember we are not just finding the answer, but write your answer in a sentence: part-part-whole. Set-up the alignment: 6.80 [part] + 2.40 [part] [whole] On the count of three, show me your answer. Students displayed their answers by raising dry-erase whiteboards over their heads for the teacher to view their answer. The teacher responded with either “good” or “check your answer” to each displayed white board. Individually rather than in pairs or in groups and either alone or in silence, students applied mathematical concepts described by a teacher during the last 5-10 minutes of a 50minute class. When student collaboration or discussion occurred at the end of a lesson, there was little time for the teacher to probe the students’ reasons for arriving at an answer. Classroom observations in Kindergarten to Grade 4 revealed teachers integrating manipulatives in their instruction as well as incorporating bar modeling strategies and explicating Singapore Math vocabulary. Taken together, classroom observations data revealed a moderately strong implementation of the Singapore Mathematics framed through
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O’Donnell’s teaching and curriculum-in-use constructs. Standardized Achievement Results The introduction of the Singapore Math curriculum did not negatively impact either the CRT test outcomes or ITBS achievement results from 2006-2009. Instead, CRT results from the 2007-2008 to the 2009-2010 academic year revealed that at least 65% of schools increased the percentage of students meeting minimum state CRT requirements in mathematics at every grade level: 65% of schools increased CRT achievement in mathematics in Grade 1, 75% of schools at Grade 2 and Grade 3, 85% of schools at Grade 4, and 80% of schools at Grade 5. ITBS results from the 2007-2008 to the 2009-2010 academic year found 95% of schools increased average student percentile rankings on the mathematics portion of the ITBS for Grade 3 and Grade 4, 75% of schools increased for Grade 5, and 65% of schools increased for Grade 2. The largest increase in average percentile ranking on the ITBS was 34.91 points at one school; this increase occurred at Grade 3. The largest decrease was 10.79 points at one school; this decrease occurred in Grade 5. Table 2 shows the changes in ITBS percentile ranking in mathematics from 2006-2009:
Table 2 Change in ITBS Mean Percentile Ranking in Math, 2006-2009 Change in ITBS Mean Percentile Ranking in Math Grade 2 2006-2009 Decrease 5%
Increase 95%
Change in ITBS Mean Percentile Ranking in Math Grade 3 2006-2009 Decrease 0%
Increase 100%
Change in ITBS Mean Percentile Ranking in Math Grade 4 2006-2009 Decrease 5%
Increase 95%
Change in ITBS Mean Percentile Ranking in Math Grade 5 2006-2009 Decrease 15%
Increase 85%
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Similar to the CRT results, the implementation of the Singapore Math curriculum did not negatively affect the ITBS achievement scores. Table 3 reveals the elevating ITBS mean percentile ranking in mathematics for grades 2-5 from 2006-2009: Table 3 ITBA Percentile Ranking, 2006-2009
Mean ITBS Percentile Ranking Among All Schools
mathematics but does not, however, provide a more nuanced understanding into the research question concerning teachers’ confidence, practices, and perceptions implementing the new mathematics curriculum. O’Donnell’s (2008) framework offers a more meaningful analysis of the teachers’ successes and struggles delivering the new mathematics program by providing multiple constructs for evaluating the effective implementation of a new curriculum. Taken together, applying O’Donnell’s four constructs to the collected data in Table 4, reveals a curriculum that was implemented with moderately strong fidelity between 2008 and 2010: Table 4 Implementation of the Singapore Math Curriculum, 2008-2010
Grade 2
Grade 3
Grade 4
Grade 5
Relating the positive CRT and ITBS student achievement results to O’Donnell’s curriculum-in-use construct suggests the implementation of the new curriculum may have contributed to an increase in standardized test results. Determining the measures that may have contributed to these academic results as well as confounding factors implementing the new curriculum identified through surveys, journals, interviews, and observation data provide deeper insights into teachers’ perceptions and practices delivering the new mathematics curriculum. Findings A strong claim can be made that the Singapore Math curriculum positively impacts student learning after the first and second year as revealed in the CRT and ITBS tests results. This finding begins to answer the second research question related to student learning of
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DATA
COMPONENTS OF O’DONNELL’S MODEL (2008)
FIDELITY OF CURRICULUM IMPLEMENTATION (FCI)
Teacher Surveys
Curriculum profile & curriculum-inuse Curriculum profile & adaptation Curriculum profile & adaptation Teaching & curriculum-inuse Curriculum-inuse
Moderately strong fidelity (+FCI)
Teacher Journals Teacher & Administrator Interviews Classroom Observations Student Achievement Results
Marginally strong fidelity (-/+FCI) Moderately strong fidelity (+FCI) Marginally strong fidelity (-/+FCI) Strong fidelity (++FCI)
An equivocal response is advanced to the research question concerning the influence of Singapore Math curriculum contributing to elementary teachers’ confidence and competence delivering mathematics. Survey data found no significant change in educators’ understanding of or confidence in mathematical concepts, either for educators at any individual grade level or for teachers considered as a whole group. Interview data collected from teachers and students as well as teachers’ journal reflections suggest that there is a growing interest in Singapore Math as it pertains to the
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integration of manipulatives, conceptual development through bar modeling and number disks, hands-on activities, and real-life applications. Teachers’ journals revealed instructional areas that county and school administrators could target through training and ongoing support to cultivate instruction and strategies to foster student learning. Principals as well as teachers expressed interest in the integration of manipulatives, bar modeling strategies, number disks, and real life applications of Singapore Math, a small number of elementary educators were not receptive to the curriculum or strategies for teaching mathematics. Classroom observations revealed a large percentage of time devoted to teachercentered instruction, possibly reflective of teachers’ developing understanding of the new curriculum. Founded on an extensive review of efficacy and effectiveness studies, O’Donnell’s (2008) conceptualization of fidelity of K-12 core curriculum implementation acknowledges student achievement results as well as curriculum potential, teaching, curriculum-inuse, and adaptation. Qualitative data provided insights into the first research question concerning how the new mathematics curriculum was implemented by elementary teachers. Participating teachers identified workshops and professional development sessions as important occasions that provided knowledge and strategies when delivering the curriculum in the first and second year of implementation. Classroom observations in Kindergarten to Grade 5 revealed elementary teachers delivering the Singapore Math curriculum through extensive explanations of mathematical concepts, describing curriculumspecific vocabulary, and reviewing problem-sets as a class. There was a discrepancy, however, between teachers’ stated claims during interviews of how lessons were communicated and actual classroom practices observed. Classroom teachers’ focus on the pictorial and concrete, with intermittent attention devoted to student-centered application of abstract mathematical concepts and extension to realworld problems, raise questions related to fidelity of implementation.
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Accountability pressures, time constraints, and elementary teachers’ developing understanding of the new mathematics curriculum may explain why the curriculum was delivered with a prevalence of teacher-centered instruction rather than integrating more opportunities for student-centered learning as well as a preponderance of concrete and pictorial concepts at the expense of abstract concepts. Addressing these inconsistent instructional approaches could be taken up through targeted training sessions to differentiate the new curriculum for English language learners. If teachers can learn to leverage these teaching strategies by encouraging students to explore the less familiar exercises and by guiding students’ thinking through appropriate questioning strategies, educators may be more effective cultivating students’ independent thinking and conceptual understanding of mathematics. Permitting students to develop the important skill of problem solving requires less teachercentered instruction and more independent student effort or collaboration to discover solutions to mathematical questions. Further, when educators allow, in a supportive environment, all learners – including underachieving students – to get frustrated by mathematics, it can lead to dramatically improved achievement and test scores as well as increased student confidence and engagement (Stigler & Hiebert, 1999; Yeung, 2009). Discussion Findings from this research identify multiple areas that both positively impact and restrict the teachers’ ability to deliver the Singapore Math curriculum with fidelity. Results from this study provide insights for those embarking on implementing a new curriculum, including the Common Core State Standards (CCSS), which is comparable to the Singapore Math curriculum (Achieve, 2010). Teachers expressed an appreciation of and desire for professional development workshops to deepen their understanding of and strategies to deliver the Singapore Math curriculum. The importance of teacher workshops in the first and second year of implementation to cultivate the efficacy and effectiveness of curriculum implementation is underscored, reflecting
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educators’ need to comprehend not only subtle elements of the curriculum but to revisit essential concepts, principles, and strategies that positively impact student learning. Facilitating the implementation of a new curriculum can be fostered through teacherdriven workshops delivered throughout the school year that target, for example, particular gaps or problems in teachers’ understanding or delivery of the curriculum (Bantwini & KingMcKenzie, 2011; Boone et al., 2004; Obara & Sloan, 2009). Such an approach may reduce elementary educators’ anxiety about teaching the new curriculum (Hadley & Dorward, 2011). In conjunction, peer observations could foster collaborative learning that encourages the adoption of an approach to thinking through teachers’ development of a shared vision, personal mastery, mental models, and team learning – influencing teachers’ perceptions of a curriculum over time (Senge, 2007). Further, school administrators and faculty could conceptualize the implementation of a new curriculum through the problem-solving coherence framework (Childress & Marietta, 2008). Alternatively, faculty could construct a checklist – employed in other professions such as medicine to ensure rigor and fidelity when rolling out a new approach (Gawande, 2008) – to be utilized in classroom observations that may be modified from one quarter to the next and informed by issues raised in grade level meetings, peer observations, and targeted areas such as higher-order questioning strategies, vocabulary learning approaches for English language learners, strategies to differentiate the curriculum, integration of abstract mathematical problems, and opportunities for student-centered activities (Louis et al., 2010). The introduction of a new curriculum extends to the provision, support, and mentoring of neophyte educators and teachers transferring into a system to resolve instruction and assessment issues that may be encountered. To be an effective educator, elementary teachers need to possess a deeper understanding of and confidence in mathematics well beyond the grade level they teach. Implementation also extends to the endorsement and support of parents/guardians through periodic newsletters,
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e-mails, website postings, and parent/guardianteacher meetings. If professional development is not afforded, teachers may modify a new curriculum and resort to previous teaching strategies or implement a new curriculum less effectively than it could if they received initial training and ongoing support. The implementation of a new curriculum involves the coordination of multiple stakeholders and is sometimes fraught with internal and external pressures confronting teachers and school administrators. Reflective of curriculum changes that educators and administrators face, teacher education programs should continue to develop in teacher candidates the dispositions of openness, flexibility, and creativity in preparation for their future profession of change (Darling-Hammond, 2010). Preparing teacher candidates involves not only the knowledge of a subject-discipline, skills to manage a classroom of diverse learners, and ability to differentiate instruction, but the openness and flexibility to integrate a new curriculum, ability to adopt to a new schedule or pacing guide, and participation in peer observations. It is reasonable to expect that as teachers become more acquainted with a new curriculum, teacher confidence and competency will increase and student test scores may also improve. Future studies evaluating the fidelity of implementation would want to assess the impact of teachertraining workshops on teachers’ instruction, carry out in-depth case studies of teachers’ attitudes and pedagogical practices related to a new curriculum, and conduct longitudinal studies tracking students’ interest and academic results in mathematics from the primary years to the middle years of schooling. References Achieve. (2010). Comparing the common core state standards and Singapore’s mathematics syllabus. Washington, DC. Retrieved from http://www.achieve.org/files/CCSSandSingapor e.pdf. Bantwini. B.D., & King-McKenzie, E.L. (2011). District officials’ assumptions about teacher learning and change: Hindering factors to curriculum reform implementation in South Africa. International Journal of Education, 3 (1), 1-25.
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Bickman, L. et al. (2009). Approaches to measuring implementation fidelity in school-based program evaluations. Journal of Research in Character Education, 7(2), 75–101. Bransford, J.D., Brown, A.L., & Cocking, R.R. (2000). How People Learn: Brain, Mind, Experience, and School. Washington, D.C.: National Academy Press. Boggan, M., Harper, S., & Whitmire, A. (2010). Using manipulatives to teach elementary mathematics. Journal of Instructional Pedagogies, 3, 1-6. Boone, W.J., D’Ambrosio, B.S., & Harkness, S.S. (2004). Planning district-wide professional development: insights gained from teachers and students regarding mathematics teaching in a large urban district. School Science and Mathematics, 104(1), 5-15. Borko, H., Mayfield, V., Marion, S., Flexer, R. J., & Cumbo, K. (1997). Teachers' developing ideas and practices about mathematics performance assessment: Successes, stumbling blocks and implications for professional development. Los Angeles: National Center for Research on Evaluation, Standards, and Student Testing (CRESST). Cai, J. (2003). Singaporean students’ mathematical thinking in problem solving and problem posing: an exploratory study. International Journal of Mathematical Education in Science and Technology, 34(5), 710-737. Childress, S., & Marietta, G. (2008). A problemsolving approach to designing and implementing a strategy to improve performance. Public Education Leadership Project at Harvard University, PEL-056, 1-16. Darling-Hammond, L. (2010). The flat world and education: How America’s commitment to equity will determine our future. New York and London: Teachers College Press. Dobson, K. S., & Shaw, B. F. (1988). The use of treatment manuals in cognitive therapy: Experience and issues. Journal of Counseling and Clinical Psychology, 56(5), 673-680. Dusenbury, L., Brannigan, R., Falco, M., & Hansen, W. B. (2003). A review of research on fidelity of implementation: Implications for drug abuse prevention in school settings. Health Education Research: Theory and Practice, 18(2), 237-256. Falkenberg, T. (2011). Knowing mathematics-forteaching: The case of planning learning activities. Problems of Education in the 21st Century, 35, 58-69. Garelick, B. (2006). Miracle math. Education Next, 6(4), 39-45.
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Gawande, A. (2008). The checklist manifesto. New York: Metropolitan Books. Ginsburg, A., Leinwand, S., Anstrom, T., & Pollock, E. (2005). What the United States can learn from Singapore’s world-class mathematics system. Washington, D.C.: American Institutes for Research. Gross, S., & Merchlinsky, S. (2002). Evaluation of the Singapore Math pilot program: Year 1 report of findings, Montgomery County Public Schools. Rockville, Maryland: Office of Shared Accountability. Hadley, K.M., & Dorward, J. (2011). The relationship among elementary teachers’ mathematics anxiety, mathematics instructional practices, and student mathematics achievement. Journal of Curriculum and Instruction, 5(2), 27-44. Hogan, B. (2004). Singapore Math: A problemsolving approach. Principal, 84(2), 22-23. Hoven, J., & Garelick, B. (2007). Singapore Math: Simple or complex? Educational Leadership, 65(3), 28-31. Hu, W. (2010). Making math as easy as 1, pause, 2, pause… New York Times. Retrieved from http://www.nytimes.com/2010/10/01/education/ 01math.html?_r=2&src=tp. Johnson, M.H. (2000). A district-wide agenda to improve teaching and learning in mathematics. Journal of Classroom Interactions, 35, 1-7. Lee, J.G., & Dian, G. (2007). The nation’s report card: Mathematics 2007. National Center for Education Statistics, Institute of Education Sciences, U.S. Department of Education, Washington, D.C. Leinwand, S., & Ginsburg, A.L. (2007). Learning from Singapore Math. Educational Leadership, 65(3), 32-36. Louis, K.S., Leithwood, K., Wahlstrom, K.L., & Anderson, S.E. (2010). Learning from leadership: Investigating the links to improved student learning. University of Minnesota: Center for Applied Research and Educational Improvement. Luppescu, S., & Hart, H. (2005). Sample Elementary 2005 – Details of student and teacher responses improving Chicago’s schools, 2005 survey report. Consortium on Chicago School Research, Chicago IL. Retrieved from www.consortium-chicago.org. Marshall, C., & Rossman, G.B. (1995). Designing qualitative research, second edition. London: Sage Publications. Menon, R. (2000). Should the United States emulate Singapore’s education system to achieve
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Singapore’s success in the TIMSS? Mathematics Teaching in the Middle School, 5(6), 345-347. Mercer, N. (2002). Developing dialogues, In Wells, G. & Claxton, G. (Eds), Learning for Life in the 21st Century. Malden, MA: Blackwell, pp. 141153. Miles, M.B., & Huberman, M.A. (1994). Qualitative data analysis, second edition. London: Sage Publishers. Muijis, D. (2006). Measuring teacher effectiveness: some methodological reflections. Educational Research and Education, 12(1), 53-74. Muir, T. (2008). Principles of practice and teacher actions: Influences on effective teaching of numeracy. Mathematics Education Research Journal, 20(3), 78-101. Newton, X. (2007). Reflections on math reforms in the U.S.: a cross-national perspective. Phi Delta Kappa, 88(9), 681-685. Obara, S., & Sloan, M. (2009). Classroom experiences with new curriculum materials during the implementation of performance standards in mathematics: A case study of teachers coping with change. International Journal of Science and Mathematics Education, 8(2), 349-372. O’Donnell, C.L. (2008). Defining, conceptualizing, and measuring fidelity of implementation and its relationship to outcomes in K-12 curriculum intervention research. Review of Educational Research, 78(1), 33-84. OFSTED (Office for Standards in Education) (2009). Mathematics: understanding the score (Improving practice in mathematics teaching at primary level). London, England. Osmundson, E., Herman, J., Ringstaff, C., Dai, Y., & Timms, M. (2012). Measuring fidelity of Implementation- Methodological and conceptual issues and challenges. (CRESST Report 811). Los Angeles, CA: University of California, National Center for Research on Evaluation, Standards, and Student Testing (CRESST). Patton, M.Q. (2002). Utilization-focused evaluation (U-FE) checklist. Western Michigan University, Evaluation Checklists Project, Retrieved from http://web.idrc.ca/uploads/userS/10905198311Utilization_Focused_Evaluation .pdf. Penuel, W.R., Fishman, B.J., Yamaguchi, R., & Gallagher, L.P. (2007). What makes professional development effective? Strategies that foster curriculum implementation.
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American Educational Research Journal, 44(4), 921-958. Puchner, L., Taylor, A., O’Donnell, B., & Fick, K. (2008). Teacher learning and mathematics manipulatives: A collective case study about teacher use of manipulatives in elementary and middle school mathematics lessons. School Science & Mathematics, 108(7), 313-325. Reimer, K., & Moyer, P.S. (2005). Third-graders learn about fractions using virtual manipulatives: a classroom study. Journal of Computers in Mathematics and Science Teaching, 24(1), 5-25. Rubin, H.J., & Rubin, I.S. (2005). Qualitative interviewing: The art of hearing data, second edition. Thousand Lakes, CA: Sage. Saldana, J. (2009). The coding manual for qualitative researchers. Thousand Oaks, California: Sage. Senge, P.M. (2006). The fifth discipline: The art and practice of the learning organization. New York, NY: Doubleday. Silver, E.A., Mesa, V.M., Morris, K.A., Star, J.R., & Benken, B.M. (2009). Teaching mathematics for understanding: an analysis of lesson submitted by teachers seeking NBPTS certification. American Educational Research Journal, 46(2), 502-531. Slavin, R.E., & Lake, C. (2008). Effective programs in elementary mathematics: a best-evidence synthesis. Review of Educational Research, 78(3), 427-515. Stecher, B., Le, V., Hamilton, L., Ryan, G., & Lockwood, J.R. (2006). Using structured classroom vignettes to measure instructional practices in mathematics. Educational Evaluation and Policy Analysis, 28(2), 101-130. Stigler, J.W., & Hiebert, J. (1999). The Teaching Gap. New York: Free Press. Tapia, M., & Marsh, G.E. (2004). An instrument to measure mathematics attitude. Academic Exchange Quarterly, 8(2), 1-9. Uribe-Florez, L.J., & Wilkins, J.L.M. (2010). Elementary school teachers' manipulative use. School Science & Mathematics, 110(7), 363371. U.S. Department of Education (2008). Foundations for success: The final report of the national mathematics advisory panel. Washington, D.C.: National Mathematics Advisory Panel. Walshaw, M., & Anthony, G. (2008). The teacher’s role in classroom discourse: a review of recent research into mathematics classrooms. Review of Educational Research, 78(3), 516-551. Wisconsin Center for Education Research, (2004). Survey of Instructional Practices: Teacher
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Survey Grades K-12, Mathematics. A joint project of the Council of Chief State School Officers and the Wisconsin Center for Education Research, Surveys of Enacted Curriculum. Retrieved from http://seconline.wceruw.org/Reference/K12mth Survey.pdf. Yeung, B. (2009). Kids master mathematics when they’re challenged but supported. Edutopia. Retrieved from http://www.edutopia.org/mathunderachieving-mathnext-rutgers-newark.
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About the Author James Badger, PhD James Badger is an assistant professor of research and assessment in the College of Education at the University of North Georgia. His research areas include professional development of pre-service teachers, program evaluation, values in education, qualitative research, and charter schools.
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