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THE IMPACT OF MATHEMATICS EDUCATION RESEARCH AND BRAINLEARNING RESEARCH ON STUDENT PERFORMANCE IN ALGEBRA I A Dissertation by CAMILLE MALONE Submitted to the Office of Graduate Studies Texas A&M UniversityCommerce in partial fulfillment of the requirements for the degree of DOCTOR OF EDUCATION August 2015THE IMPACT OF MATHEMATICS EDUCATION RESEARCH AND BRAINLEARNING RESEARCH ON STUDENT PERFORMANCE IN ALGEBRA I A Dissertation by CAMILLE MALONE Approved by: Advisor: Chuck Holt Committee: Art Borgemenke Katy Denson Head of Department: Glenda Holland Dean of the College: Timothy Letzring Dean of Graduate Studies: Arlene Horne iii Copyright © 2015 Camille Maloneiv ABSTRACT THE IMPACT OF MATHEMATICS EDUCATION RESEARCH AND BRAINLEARNING RESEARCH ON STUDENT PERFORMANCE IN ALGEBRA I Camille Malone, EdD Texas A&M UniversityCommerce, 2015 Advisor: Chuck Holt, EdD The first course in high school algebra is called a “gateway course” because of its importance for success in future mathematics coursework and for college and career opportunities. Unfortunately, student achievement in secondary mathematics in the United States, and in Algebra I in particular, is described as mediocre. However, a large gap exists between mathematics education research and brainlearning research and classroom instructional practice. The researchers conducted an ex post facto study on the implementation of a researchbased Algebra I curriculum built around highcognitive demand tasks and student performance in Algebra I. Two groups of students were selected for the study. One group enrolled in an Algebra I course that followed a typical textbookdriven curriculum and the second group enrolled in a course that followed a researchbased curriculum with highlevel tasks. The researcher used propensity score matching and multilevel modeling to compare the effects of the two curricula to determine whether student growth occurred. No significant difference existed in student performance between the traditional and researchbased curricula. Additionally, African v American and Hispanic student performance decreased and White student performance increased Grade 8 to Grade 9. However, differences were not significant for ethnicity between students using a researchbased Algebra I curriculum and those using a traditional curriculum. vi ACKNOWLEDGEMENTS I am grateful to Dr. Chuck Holt for his support and consistent encouragement. I am especially appreciative of the patience and guidance of Dr. Katy Denson, a teacher par excellence. vii DEDICATION This study is dedicated to the men and women of the Apollo Algebra I Project Accomplished Mathematicians Extraordinary Curriculum Writers & Dedicated Educators viii TABLE OF CONTENTS LIST OF TABLES ...................................................................................................................... xii CHAPTER 1. INTRODUCTION ......................................................................................................... 1 Student Performance in Algebra ............................................................................. 2 Attempts to Address the Problem ............................................................... 5 Mathematics Education Instructional Research: The Importance of Tasks 5 BrainLearning Research ............................................................................ 6 ResearchBased Algebra Curriculum ......................................................... 7 Consequences of Accountability ................................................................. 8 Statement of the Problem ........................................................................................ 9 Purpose of the Study ............................................................................................. 10 Research Questions ............................................................................................... 10 Null Hypotheses .................................................................................................... 11 Significance of the Study ...................................................................................... 11 Method of Procedure............................................................................................. 15 Selection of Sample .................................................................................. 16 Collection of Data ..................................................................................... 16 Treatment of Data ..................................................................................... 17 Definitions of Terms ............................................................................................. 18 Limitations and Delimitations ............................................................................... 19 Limitations ................................................................................................ 19 Delimitations ............................................................................................. 19 ix CHAPTER Assumptions .......................................................................................................... 20 Organization of Dissertation Chapters .................................................................. 20 2. LITERATURE REVIEW ............................................................................................ 21 The Tug of War Begins......................................................................................... 23 1900–1930: Expansion and Progressivism ........................................................... 27 1930–1950: The Great Depression and World War II .......................................... 31 1950s: Influence of Sputnik .................................................................................. 37 1960s: The New Math Era .................................................................................... 41 1970s: Back to the Basics ..................................................................................... 45 1980s: The Standards Movement Begins ............................................................. 50 1990s to the New Century: Debate Leads to War................................................. 56 The Math Wars ..................................................................................................... 61 A New Century ..................................................................................................... 64 Brain Research ...................................................................................................... 67 Summary ............................................................................................................... 69 3. METHOD OF PROCEDURE...................................................................................... 71 Research Questions ............................................................................................... 72 Null Hypotheses .................................................................................................... 72 Research Design.................................................................................................... 73 Propensity Score Matching ................................................................................... 75 Multilevel Modeling ............................................................................................. 77 Instrumentation ..................................................................................................... 78 x CHAPTER Reliability .................................................................................................. 79 Validity ..................................................................................................... 79 Sample Selection ................................................................................................... 80 Data Gathering ...................................................................................................... 81 Treatment of the Data ........................................................................................... 82 Propensity Score Matching ....................................................................... 82 Multilevel Modeling ................................................................................. 83 Research Question 1 ................................................................................. 83 Research Question 2 ................................................................................. 84 Summary ............................................................................................................... 84 4. PRESENTATION OF FINDINGS .............................................................................. 86 Construction of Data Set for Analysis .................................................................. 86 Multilevel Modeling ............................................................................................. 88 Model Levels ............................................................................................ 89 Descriptive Statistics for TAKS Math ...................................................... 90 No Predictors Model ................................................................................. 90 Model with Growth Rate .......................................................................... 92 Model with TimeVarying Treatment ....................................................... 93 Model with All Variables.......................................................................... 95 Summary of Model Variance ................................................................................ 97 Research Questions ............................................................................................... 99 Research Question 1 ................................................................................. 99 Research Question 2 ............................................................................... 100 xi CHAPTER 5. SUMMARY OF THE STUDY AND THE FINDINGS, CONCLUSIONS, IMPLICATIONS, AND RECOMMENDATIONS FOR FUTURE RESEARCH ......... 101 Summary and Findings ....................................................................................... 102 Research Question 1 ............................................................................... 103 Research Question 2 ............................................................................... 103 Conclusions ......................................................................................................... 104 Implications......................................................................................................... 107 Recommendations for Future Research .............................................................. 108 REFERENCES ........................................................................................................................... 111 VITA .......................................................................................................................................... 134 xii LIST OF TABLES TABLE 1. Student Characteristics of NinthGrade Algebra Students in Original and Matched Samples ............................................................................................................................. 88 2. TAKS Math Statistics for Each Measurement Occasion .................................................. 90 3. Null Model Estimates of Covariance Parameters – TAKS Math ...................................... 91 4. No Predictors Model Estimates of Fixed Effects .............................................................. 91 5. Estimates of Covariance Parameters: Growth Rate Model ............................................... 92 6. Estimates of Fixed Effects for Growth Rate Model .......................................................... 93 7. Estimates of Covariance Parameters: TimeVarying Treatment Model ........................... 94 8. Estimates of Fixed Effects for TimeVarying Treatment Model ...................................... 95 9. Estimates of Covariance Parameters: All Variables Model .............................................. 96 10. Estimates of Fixed Effects for All Variables Model ....................................................... 97 11. Percent of Variance Explained by Time, Model, and Source ......................................... 99 1 Chapter1 INTRODUCTION Educators and educational researchers, among others, commonly call high school algebra a gateway course for several reasons. The RAND Mathematics Study Panel reported that algebra proficiency primarily provides entrance into higherlevel secondary mathematics coursework (Educational Testing Service [ETS], 2009; Matthews & Farmer, 2008; National Council of Teachers of Mathematics [NCTM], 1989; RAND, 2003; U.S. Department of Education [DOE], 1997). According to the University of NebraskaLincoln (n.d.), “Students who successfully complete Algebra I often continue to pursue the study of high school mathematics that prepares them for college, while students who are unsuccessful in Algebra I find their path to success blocked” (p. 1). Additionally, the DOE (1997) stated that students who take algebra and geometry in high school attend college at much higher rates than those who do not take these courses. In fact, lowincome students who take algebra and geometry are almost three times as likely to go to college (ETS, 2009). Those who master algebra and attain higher levels of mathematics coursework also tend to have access to more and higher quality career opportunities (Stapel, 2009). According to RAND (2003), “Without proficiency in algebra, students cannot access a full range of educational and career options, and they have limited chances of success” (p. 47). RAND also reported that algebra is a gatekeeper that provides or denies access to academic and career success for all ethnic groups and socioeconomic statuses (SES) of society, which affects national culture as a whole. Lubienski (2007) stated, “Mathematics achievement is particularly important to our efforts to promote equity because it serves as a gatekeeper to highstatus occupations and 2 can provide a powerful ladder of mobility for lowSES students” (p. 55). The RAND report (2003) summarized: Lack of success in mathematics has significant consequences. Algebra, for example plays a significant gatekeeping role in determining who will have access to college and certain career opportunities. The ‘gates’ tend to be closed to the less advantaged, either by default—when the schools they attend simply do not offer advanced mathematics courses—or by discrimination—when low expectations for student performance lead to educational tracking that differentiates among students and therefore further limits students’ opportunities to develop math proficiency. (p. 11) Limitations of career opportunities for any section of the population have significant implications. RAND (2003) noted, Failure to learn algebra is widespread, and the consequences of this failure are that far too many students are disenfranchised. This curtailment of opportunity falls most directly upon groups that are already disadvantaged and exacerbates existing inequalities in our society. (p. xx) Regarding African American students, Moses and Cobb (2001) stated, “In similar fashion we believe that organizing around math literacy opens another path on which people can begin to transform their lives” (p. 5). They advocated for algebra as “the new civil right” (p. 5) that should be accessible to everyone. Student Performance in Algebra Because of its importance as a gatekeeper course, algebra is a requirement for many high schools in the United States, and in most, students must demonstrate proficiency in algebra to graduate (RAND, 2003). Unfortunately, poor student performance in algebra and secondary 3 mathematics in the United States is a significant national concern (Schoenfeld, 1992) expressed among teachers, parents, educational administrators, and national commentators that has persisted for several decades (RAND, 2003). Specifically, national and international assessments have indicated consistent poor math performance in upper grade levels (Haycock, 2002). In 1989, the National Research Council (NRC) reported that mathematics achievement of U.S. students was well below what was necessary to “sustain our nation’s leadership in a global technological society” (p. 1). In 1990, state governors nationwide joined then President Bush in establishing national educational goals. The group’s collective announcement echoed across the nation: “By the year 2000, United States students will be first in the world in mathematics and science achievement” (Haycock, 2002, p. 3). Since that time, the nation has received few encouraging reports. For example, in the 1990s, results of the National Assessment of Educational Progress (NAEP), commonly called the Nation’s Report Card, reported that elementary, middle, and high school students of every racial and ethnic group knew more mathematics than did students in the 1970s. In 2000, the College Board reported gains on the quantitative portion of the Scholastic Aptitude Test (SAT). However, in the early 2000s, results of the main NAEP report, which provides both national and state data, indicated that after steady improvement through 1996, high school mathematics performance declined from 1996 to 2000, and earlier improvements in Grade 12 were attributed to improvements during the elementary years (Haycock, 2002). In the latter part of the 1990s, high schools received students who were better prepared in mathematics; however, their high school coursework added less value to their ultimate levels of mathematics achievement (Haycock, 2002). Haycock (2002) reported that 35% of U.S. graduates left high school unable to perform basic math skills. She added, “While many of these 4 students have mastered basic computation, they cannot find the perimeter of a figure; their conceptual understanding of mathematics is limited; and they are unable to consistently see mathematical relationships” (p. 6). Further, analyses of other assessments, including the Third International Math Science Study (TIMSS), also indicated poor mathematics progress. Achievement levels on the TIMSS in the 1990s showed that by 12th grade when American students would have completed algebra, even those in the top 10%15% fell below their peers from 11 participating countries on international rankings (Haycock, 2002). It appeared that while American students were progressing, their counterparts in other countries were progressing faster and achieving more. In addition, gaps continued to exist between White students and those identified as lowincome, African American, and Latino (Haycock, 2002). According to Silver (1998), “In general the TIMSS results indicate a pervasive and intolerable mediocrity in mathematics teaching and learning in the middle grades and beyond” (p. 1). During the 1970s and 1980s, the United States made considerable progress in raising mathematics achievement of minority and lowSES students. Between 1973 and 1986, the gap between White and African American students in eighth grade narrowed by half, from 48 to 25 points. Additionally, the gap between White and Latino students narrowed from 35 to 20 points (Haycock, 2002). However, gaps increased in the 1990s, and at the turn of the century, African American and Latino students in 12th grade had math skills equivalent to White students in eighth grade (Haycock, 2002). In 2001, the Organization for Economic Cooperation and Development (OECD) published the results of an assessment administered to 250,000 15yearolds in 32 countries. American students ranked 19th among participating countries in achievement in reading, 5 mathematics, and science, and were relatively high on the size of achievement gaps between high and lowperforming students (OECD, 2001). Based on the ACT (2005) national readiness standards, only 40% of high school graduates in the U.S. are ready for college algebra. Attempts to Address the Problem The importance of algebra in secondary school curricula and the consistently poor performance of U.S. students in mathematics, particularly in algebra, has fueled persistent and urgent attempts to address the problem. Algebra textbooks, longused as the curriculum in most schools across the United States, have been transformed from the drab black and white publications of the mid20th century that had few or no illustrations, to bright and colorful resources with content aligned to NCTM standards (Donoghue, 2003). Teachers have relied on the enriched textbooks to assist in presenting mathematics content with narratives that place the student in the problem situation for relevancy, provide multiple examples with detailed stepbystep solutions, and include an abundance of practice for state examinations (Donoghue, 2003). Even with these changes, recent achievement results indicate that textbook improvements and other innovative strategies, including computer programmatic remedies, have had little effect. However, a significant amount of research published in recent decades may hold promise for greater success in teaching and learning algebra. Mathematics Education Instructional Research: The Importance of Tasks Cognitive scientists and educational researchers have published significant findings related to teaching and learning mathematics in the classroom, and many findings and recommendations focus on teaching strategies that call for student engagement in cognitively demanding mathematical tasks. Stein, Grover, and Henningsen (1996) stated that students must engage in mathematical thinking by doing mathematics through problemsolving tasks. 6 Henningsen and Stein (1997) described factors that support high levels of mathematical thinking or reasoning, including classroom activities that involve challenging problems. The NCTM (1991) recommended that to develop highlevel thinking in math classrooms, students must be provided opportunities to engage in dynamic, rich, and worthwhile mathematical tasks. BrainLearning Research Lakomski (2000) described the 21st century as “the century of the brain,” particularly because learning and brain research has characterized both educational and medical study landscapes. Brainlearning research since the late 1980s has grown exponentially (Caine & Caine, 2001). The study of how people learn as an outgrowth of neuroscience has significant instructional implications for the classroom (Bransford, Brown, & Cocking, 1999; Jensen, 2000; Willis, 2006). While skeptics feel that educators should ignore what they often called pseudoscience, the advent and use of medical technology has provided physiological support for the efficacy of certain strategies to facilitate learning (Bransford et al., 1999; Willis, 2006). Willis (2006) stated, Information obtained through brain imaging such as positron emission tomography (PET scans), functional magnetic resonance imaging (fMRI), and quantitative electroencephalography brain wave monitoring (qEEG) during the learning process have given us a science of education to add to our already powerful knowledge of the art of teaching. (p. vii) Many prominent brainlearning researchers have suggested specific pedagogy to ensure that content enters students’ longterm memories; however, such pedagogy is virtually unknown and untried in most educational settings (Caine & Caine, 2001; Erlauer, 2003; Jensen, 2000; Willis, 7 2006). According to Jensen (2008), “Brainbased education is the engagement of strategies based on principles derived from an understanding of the brain [and] this singular realization alone has fueled a massive and urgent movement worldwide to redesign learning” (p. 4). Caine and Caine (2001) synthesized the results of current brainlearning research by introducing learning principles. These principles included (a) the brain naturally searches for meaning that is best met by engagement in and making sense of mathematical tasks; (b) the brain responds to classroom activities that satisfy curiosity and hunger for novelty, discovery, and challenge; (c) the brain uses patterning as a way to search for meaning; and (d) the brain learns best through interactive experiences. Caine and Caine (1994) advocated for incorporating brainlearning principles into the curriculum. They stated, “All education can be enhanced when this type of embedding is adopted. That is the single most important element that the new brainbased theories of learning have in common” (p. 94). ResearchBased Algebra Curriculum In mathematics, particularly in Algebra I, a generalized description of curriculum and instruction that includes findings from cognitive science, mathematics education, and brainlearning research is best met by a curriculum and by teaching strategies that embody experiential learning. Caine and Caine (1994) stated, “One of the most important lessons to derive from brain research is that in a very important sense, all learning is experiential” (p. 113). Experiential learning is defined as that which occurs when the learner is actively engaged; therefore, curriculum based on mathematics and brainlearning research would include challenging, highthinking demand tasks experienced by students who practiced habits of mathematical thinking in group settings in the classroom (Caine & Caine, 2001; NCTM, 1989; Stein et al., 1996). However, rather than implementing this type of teaching pedagogy, school 8 districts have persisted in using rule memorization and skills practice, which are traditional teaching methods of past decades (Clements, 2003; Donoghue, 2003). Consequences of Accountability Although teachers and administrators may embrace the idea of teaching with rich, challenging algebra tasks, and they may envision classes where teachers provide more indepth and rigorous curriculum, they are hostages to state assessment systems (Boyd, 2008; Jones & Egley, 2007; Popham, 2007). Boyd (2008) stated, “With the passage of the Elementary and Secondary Education Act (ESEA) in 2002, all states implemented accountability systems to document student achievement in mathematics” (p. 1). Researchers have found that state tests have negatively influenced classroom teachers’ content instruction and assessment practices (Boyd, 2008; Nichols & Berliner, 2008; Popham, 2007). The penalties of low achievement have also caused resistance to innovation and have restricted students to classes of drill, repetition, and testtaking strategies (Popham, 2007; Wright, 2009). While students are taught how to work certain types of problems for future assessments, they have little opportunity to learn meaningful concepts (Abrams, Pedulla, & Madaus, 2003; Popham, 2007; Schmoker, 2009). Boyd (2008) reported that teachers react to tests in ways that contradict appropriate educational practices by spending more time (a) teaching to the test in a gamelike manner, (b) focusing on testtaking skills such as drills, (c) coaching for the test, and (d) practicing sample test items. In response to the pressures of accountability, assessmentlike practice problems only mimic authentic mathematics tasks that both cognitive mathematics and brainlearning research have advocated. However, educational administrators and teachers, among others, are reluctant to consider implementing an algebra curriculum with embedded tasks and brainlearning 9 recommendations without research findings that indicate whether such implementation supports an increase in algebra achievement. Statement of the Problem Educators in the United States understand the importance of and demand for increased algebra achievement, and they have responded by embracing a myriad of innovative, instructionalrelated resources, all with little effect on student outcomes (Schoenfeld, 1992). Concurrently, accountability systems that engender a culture of teaching to and for assessments have hijacked attempts to implement reformed, researchbased curriculum to teach algebra because educators are reluctant to follow recommendations for fear of unacceptable student scores (Schmoker, 2009). Adopting a researchbased curriculum to increase student conceptual understanding versus one that prepares students for accountability measures presents a conundrum for educators (Jones & Egley, 2007). Addressing recommendations for change, Davis (1992) stated, “If school systems are going to move far in this direction, however, research and development must create some reasonably explicit programs and demonstrate that they are capable of producing acceptable test scores” (p. 725). Chaikin (1989) noted that to address poor student achievement in algebra, research must be conducted on recommendations of cognitive and brainlearning research that includes measurements of student accountability outcomes. Mathematics education and brainlearning studies are at the forefront of educational research; however, literature on secondary mathematics education lacks information on the implementation of an algebra curriculum embedded with researchbased strategies. Limited research has been published on targeted brainlearning teaching strategies in mathematics that addresses specific content objectives. Additionally, Stein, Smith, Henningsen, and Silver (2009) 10 published an indepth treatise of case studies using mathematical tasks in middle school. However, to this researcher’s knowledge, no study has explored the relationship between the implementation of mathematics education using a brainlearning research algebra curriculum and student performance on state accountability measures. Purpose of the Study Secondary school administrators and mathematics teachers recognize the importance of firstyear algebra as a gateway mathematics course; however, low secondary mathematics achievement is problematic in all sectors (RAND, 2003). Further, educators are hostages to an accountability system that undermines the use of researchbased algebra curriculum and instruction in favor of testpreparation practices (Popham, 2007). The purpose of this study was to examine a meaningful, researchbased Algebra I curriculum and its implementation using cognitively demanding tasks to identify possible relationships to student performance as measured by state accountability assessments. Findings from this study will enable educators to make informed decisions regarding the use of researchbased curricula and teaching practices versus those that focus on test preparation. Research Questions The following research questions guided this study: 1. Does a statistically significant difference exist in the mathematical growth of students using a mathematics education and brainlearning researchbased Algebra I curriculum that includes implementation of highcognitive demand tasks compared to students using a traditional textbookdriven curriculum? 2. Does a statistically significant difference exist in the mathematical growth of students by ethnicity using a mathematics education and brainlearning researchbased 11 Algebra I curriculum that includes implementation of highcognitive demand tasks compared to students using a traditional textbookdriven curriculum? Null Hypotheses The researcher tested the following null hypotheses at the p < .05 level of significance. 1. No significant difference exists in the mathematical growth of students using a mathematics education and brainlearning researchbased Algebra I curriculum that includes implementation of highcognitive demand tasks compared to students using a traditional textbookdriven curriculum. 2. No significant difference exists in the mathematical growth of students by ethnicity using a mathematics education and brainlearning researchbased Algebra I curriculum that includes implementation of highcognitive demand tasks compared to students using a traditional textbookdriven curriculum. Significance of the Study Mastery of high school algebra is critical for students to pursue advanced mathematics coursework, to gain access to more and higher quality college and career opportunities, and to facilitate equity among all students in society (RAND, 2003). While teaching methodology in algebra has changed little, achievement rates of U.S. students in algebra have been disappointingly poor for decades (Schoenfeld, 1992). Almost from the inception of mathematics education in the United States, two distinct camps of teaching methodologies for secondary mathematics have existed (Fey & Graber, 2003). As early as 1850, proponents in one camp have advocated direct instruction, also called synthetic or rote learning. This type of learning most often using a textbook and is accompanied by memorization, drill, and practice (Cohen, 2003). The second camp has advocated inductive reasoning through discovery or experiential learning, 12 with or without a textbook, characterized by student behaviors applied to solving special problems designed to facilitate student conceptual discovery and learning (Cohen, 2003). The two camps continued stringent debate into and throughout the 20th century. Influences such as national socioeconomic conditions and World War II during the first half of the century precluded calls for more studentcentered instruction in favor of vocationally oriented courses in which students memorized and practiced basic skills (Angus & Mirel, 2003). The launch of Sputnik and a heightened focus on conceptual understanding were also catalysts for reform that somewhat influenced teaching methodology during the second half of the century (FerriniMundy & Graham, 2003). Although research increased and expanded during the last few decades of the century, which contributed significantly to the mathematics education literature, the debate culminated in the Math Wars toward the end of the century (Angus & Mirel, 2003; Klein, 2003). More recently, educational literature has explored new factors that may influence teaching practice. Mathematics education researchers have begun examining how the brain learns and have added new insight into the centuriesold debate over issues of mathematics teaching methodologies (Erickson, 2002; Hiebert et al., 1997). Cognitive psychology has also contributed research into constructivism as the means for learning. In the late 20th and early 21st centuries, mathematics educators turned their attention toward the importance of cognitively demanding mathematics tasks to construct knowledge (Doyle, 1988; Hiebert et al., 1997). Stein et al. (2009) published a casebook of classroom practices built around the use and maintenance of highlevel tasks. In response to the need for U.S. students to improve their use of cognitively demanding skills as specified by the American Institutes for Research (Ginsburg, Cooke, Leinwand, Noell, & Pollock, 2005), Stein et al. argued for more opportunities for 13 students to engage in “tasks that require them to reason and make sense of the mathematics they are learning” (p. 150). Further, over the past two decades, brainlearning researchers have advocated the importance of experiential learning and problem solving, particularly in social settings (Caine & Caine, 2001; Jensen, 2008). Recently, the role of educational administrator has undergone profound change. The expectations placed on superintendents are both more challenging and more complex. In addition to community and school board responsibilities, superintendents have the added role of personal involvement to improve district student performance (Lambert et al., 2002). In fact, research indicates that reforms at the school district level can occur only when senior leadership takes the initiative to understand and support the instructional design of the district (Resnick & Glennan, 2002). For example, members of the Connecticut Superintendents’ Network recognized that teaching and learning is “job number one” (City, Elmore, Fiarman, & Teitel, 2009, p. x). In fact, the first objective of these superintendents was to develop the knowledge and skills necessary to lead a districtwide instructional improvement effort (City et al., 2009). Further, site management reforms have added more responsibility and authority to school principals. Henson (2006) stated, During the 1980s and 1990s the level of involvement of school administrators in curriculum and instruction reached heights unprecedented since the days of the oneroom school. Research on effective schools has made educators aware of the need for administrators—particularly building principals—to be at the center of instructional and curriculum planning. (p. 23) In addition, emphasis has increase concerning the link between the role of the principal as an instructional leader and student achievement (Fink & Resnick, 2001, Lambert et al., 2002). 14 Schlechty (2005) extended the definition of principal leadership to include support for teachers as “designers of engaging work for students” (p. 18). Thus, it is particularly significant that the roles of superintendents and building principals include the responsibility of algebra programmatic reform in their districts and schools. Administrators, although interested in developing and applying researchbased curriculum and pedagogy, may be too entrenched in accountability systems to change classroom practice (Popham, 2007). Curriculum has been narrowed to focus on the primary goal of raising test averages, and classroom instruction is often given as decontextualized practice for assessments (Nichols & Berliner, 2008). Administrators need data on student achievement outcomes as a result of researchbased teaching versus teaching for test preparation; however, there is currently a gap in the literature in this regard. That is, to this researcher’s knowledge, no study has explored the benefits, or lack thereof, of standardsbased curriculum and instruction in a firstyear algebra course based on mathematics education and brainlearning research conducted over the past two decades (Van de Walle, Karp, & BayWilliams, 2010). This study aimed to fill the gap in mathematics education literature by exploring the implementation of an Algebra I curriculum, developed and built around current mathematics education and brainlearning research. The researcher examined indications of student achievement growth to provide administrators and teachers with data on which to base decisions regarding the best type of curriculum and classroom instruction to increase students’ conceptual understanding of algebra. The results of this study contribute to the literature by providing educators with student achievement data related to the implementation of a researchbased algebra program. It is of particular importance to current and future generations of students 15 enrolling in algebra that a definitive exploration of an innovative researchbased algebra course be added to the literature. Method of Procedure It is not feasible to explore differences in student performance by randomly selecting some students to receive instruction based on one type of Algebra I curriculum and others to serve as a control group with another type of curriculum. In fact, experimental studies on the effects of student achievement using particular curricular programs are problematic because of the necessity and ethical considerations of providing one student group with a curriculum that is purported to be better for conceptual learning and withholding it from another group. However, the districtwide implementation of a textbookdriven curriculum in one school year and the districtwide implementation of a researchbased curriculum the following year provided a unique and excellent opportunity to explore outcomes. Gall, Gall, and Borg (2003) described nonexperimental research as that in which researchers study phenomena as they already exist without intervention. Nonexperimental research that explores previously occurring phenomena is called ex post facto. In 2007–2008, a large metropolitan public school district taught Algebra I using a curriculum based on an adopted textbook. In 2008–2009, an Algebra I curriculum was designed, built, and implemented around current mathematics education and brainlearning research. The new curriculum, exclusive of a textbook, was characterized by cognitively demanding mathematics tasks that students solved in socialized settings held approximately once every week. Additional curriculum topics addressed and expanded the study of taskrelated concepts. The researchers examined the effects of this researchbased, taskembedded curriculum versus the typical, textbookdriven Algebra I curriculum on student achievement. The Grade 9 16 mathematics Texas Assessment of Knowledge and Skills (TAKS) was used to examine student growth. Selection of Sample The researcher selected two groups of students from the same school district as samples for the study. One group included Grade 9 students who studied Algebra I using the textbookdriven course and who took the Grade 9 mathematics TAKS assessment in 2008. The second group included Grade 9 students who studied Algebra I using the mathematics education and brainlearning researchbased curriculum (which included highlevel tasks) and who took the Grade 9 mathematics TAKS assessment in 2009. Class rosters of students, identified by masked student identification numbers, who were enrolled in Algebra I were matched with their teachers whose identities were also masked. Only teachers who taught Grade 9 Algebra I in both 2007–2008 and 2008–2009 were included in the study. Additionally, only students who were in Grade 9 for the first time, who had both Grade 8 and Grade 9 TAKS assessment scores, and who had a first semester grade for Grade 9 Algebra I, were included. Students identified as having limited English proficiency (LEP) or special education students who received instruction separately from the regular student population were not included in the study. Finally, students whose teachers were not included in the study; that is, teachers who had not taught Algebra I during both the 2007–2008 and 2008–2009 school years were excluded. Collection of Data Data were obtained from the school district evaluation and accountability department and included ethnicity, gender, home language, SES (based on free and reduced lunch status), and Grade 8 and Grade 9 mathematics TAKS scale scores for the samples of Grade 9 students 17 enrolled in Algebra I. For those students who took Algebra I in 2008, TAKS math scale scores were obtained for 2007 and 2008. For students who took Algebra I in 2009, TAKS math scale scores were obtained for 2008 and 2009. Treatment of Data The researcher built a dataset that included students who were in Grade 9 in 2007–2008 and 2008–2009 who met the qualifications for participation. The dataset included student demographic and academic variables. A Statistical Package for the Social Sciences (SPSS) propensity score matching (PSM) procedure (Thoemmes, 2009) was used to calculate a propensity score for each student, which indicated the probability of receiving the treatment (researchbased curriculum). Propensity scores were then matched using a 1:1 nearest neighbor matching logistic regression algorithm with a .25 caliper on the pretest covariates of gender, ethnicity, SES (based on free and reduced lunch status), home language, and the pretest Grade 8 mathematics TAKS scale score to create a comparable sample and force exact matching on the masked teacher identification number to eliminate teacher and school effects. After matching, 767 students in each group were identified. The researcher checked data for differences no larger than .25 standard deviations. Statistical analyses were then computed on the outcomes of the treatment (researchbased curriculum) and control (traditional curriculum) groups. Research questions were answered using multilevel modeling (MLM) to examine data that were hierarchically structured as students nested within teachers, using both between and withinstudents analyses to assess students in the same classrooms with the same teachers. The null model was assessed to ensure sufficient variation to warrant an MLM analysis (Heck, Thomas, & Tabata, 2014). The researcher added growth rate to the model. To answer Research Question 1, timevarying treatment effects were included in the model and analyzed to evaluate 18 differences in outcomes between the two groups. To answer Research Question 2, the researcher computed dummy variables for the ethnicities of African American, Hispanic, and White. The researcher assessed the influence of ethnicity, gender, and SES. Definitions of Terms Algebra I. Algebra I is one of the five NCTM (2000) content standards. Algebra, typically the first high school mathematics course taken by Grade 9 students, includes a basic understanding of foundation concepts of Grades K8, algebraic thinking and symbolic reasoning, function concepts, relations between equations and functions, tools for algebraic thinking, and underlying mathematics processes (Texas Education Agency [TEA], 2006). While Algebra I may be offered as an advanced course in earlier grades, this study measured only student performance of Grade 9 enrollees. Highlevel tasks. Highlevel demand or highcognitive demand tasks are defined as challenging instructional tasks (mathematical problems) that “demand engagement with concepts and that stimulate students to make purposeful connections to meaning or relevant mathematical ideas” (Stein et al., 2009, p. 11) rather than to perform a memorized procedure in a routine manner. Highcognitive demand tasks may be realworld scenario problems that involve mathematical concepts or actual physical, experimental problems that require handson mathematical manipulation, both from which students must use information to extrapolate data, make conjectures, and formulate answers to problems based on the information given. By definition, highlevel tasks have multiple entry points and multiple solution paths. Mathematics education and brainlearning researchbased algebra curriculum. Algebra I researchbased curriculum is defined by highcognitive demand tasks (Stein et al., 2009). In this study, such tasks were implemented approximately once per week and addressed 19 by student groups in socialized settings. Additional curriculum, reinforcing, and extending concepts were included in the entire Algebra I course, exclusive of a textbook. Limitations and Delimitations Limitations In the last decade, brainlearning research has provided educators with a rich, extensive source of strategies to improve learning in the classroom. While this study was limited to one of those important strategies, it was a strategy that undergirds all others. Specifically, the strategy used provided learner experiences that enhanced conceptual development and understanding of the content. For this study, the brainbased learning methodology of experience was defined as collaborative participation in solving highthinking demand mathematical tasks. While many brainlearning strategies exist that may positively affect learning, this study was limited to one strategy that was tightly defined as a problemsolving task used in a firstyear high school algebra course. As such, the following limitations applied to this study: 1. Students included only Grade 9 Algebra I students in a large metropolitan public school district. 2. Teachers included in this study were limited to those who had taught Algebra I during both the 2007–2008 and 2008–2009 school years. Delimitations Delimitations of this study included both the sample size and the instrument used to measure growth. The sample was drawn from a Grade 9 regular education students enrolled in Algebra I in a large district that included over 3500 students and resulted in a matched sample of 767 students in each group. The instrument used for measurement was the TAKS, which has been recognized for both reliability and validity. 20 Assumptions Mathematics education and braincompatible learning in algebra may be defined by several parameters based on current research. This study measured the effects of a curriculum that included periodic (approximately weekly) mathematics tasks experienced by students in the classroom. The tasks were followed by curriculum activities built upon and around the concepts embedded in the task experiences. The researcher conducted this study and collected, analyzed, and interpreted the data based on the assumption that the Algebra I curriculum was available to every algebra teacher in the school district. Organization of Dissertation Chapters In this study, the researcher sought to contribute to empirical research regarding the implementation of teaching strategies based on mathematics education and brainlearning research. Chapter 2 includes a historical review of the literature on mathematics teaching strategies and the pendulous pathway that changes in pedagogy have followed. Chapter 3 includes a description of the methodology employed in the study. Chapter 4 includes the results of the study, and Chapter 5 includes a summary of findings and recommendations for future research. 21 Chapter2 LITERATURE REVIEW The study was designed to measure the effects, if any, of a firstyear high school algebra I course implemented with a curriculum characterized by strategies based on mathematics education cognitive research and brainlearning research of student achievement. This chapter provides (a) a historical review of literature regarding instructional methodologies for teaching secondary mathematics, (b) the context of instructional research as applied to mathematics, (c) pertinent research findings in teaching and learning mathematics, and (d) recommendations based on research over the last two decades in teaching methodology in mathematics. For the most part, literature on teaching and learning mathematics does not distinguish algebra from any other mathematics subject; however, it does focus on the commonalities in teaching and learning of all mathematics courses (Kieran, 1992). Fey and Graeber (2003) provided the following description of the pendulous pathway of teaching pedagogy of high school mathematics, including algebra, in the United States: Looking at events in education over a long period of time is a fascinating opportunity to chart a kind of intellectual, social, and political tugofwar in which the perspectives and theories of individuals and groups compete for influence on the goals and practices of school mathematics. The story of this struggle over the direction of curriculum and teaching in elementary and secondary school mathematics has a predictable rhythm of crisisreformreaction episodes. A prominent social, political, or professional group calls attention to serious problems in students’ performance and recommends action, only to find that reform initiatives ultimately run up against resistance from opposing views and the deeply conservative nature of educational institutions. The burst of concern and 22 energy sparked by crisisandreform rhetoric often settles down to a quieter pattern of business as usual, at best moderately perturbed by the energetic calls for change in standard practices. (p.521) An examination of teaching methodology in mathematics is characterized not by decades but centuries of contention over direct instruction versus discovery learning. In fact, this study is grounded in relatively recent literature from the past halfcentury and historical and anecdotal evidence of pedagogical advocacies in school mathematics. Perhaps Pythagoras provided the earliest recorded evidence of direct instruction through discovered stone engravings of circles, secants and sectors, geometric illustrations, and formulaic etchings, all of which may be inferred to be teaching resources that ancient mathematicians and educators used for instructional demonstration. Geometric figures were drawn in sand or etched on stone to illustrate concepts to students, which demonstrates direct instruction. On the other hand, in terms of the discovery teaching methodology, Socrates offers a premier example. His method included asking his disciple students questions to lead them to discover the truths he wanted them to know. Both direct instruction and the Socratic method have survived centuries as practiced pedagogies, more or less successfully, that many still advocate. While no ancient research exists that describes empirical data measuring the most successful pedagogic strategies in which to ground a study of current mathematics teaching methodology, particularly algebra, such data and descriptive evidence certainly exist in the United States. This evidence dates back to the early 19th century from public schools established by the colonists (Burton, 1850; Cajori, 1890; Cohen, 1999, 2003). Evidence culminating just after the Math Wars in the late 1990s and early 2000s continues to be published today (Star, 2005; Starr, 2002). Well over a century of 23 philosophical skirmishes between advocates of direct instruction and discovery learning are welldocumented from the mid19th century, throughout the 20th century, and into the 21st century. While peace may currently exist in the math community, it is an uneasy peace (English, 2007; Grouws & Cebulla, 2000). The Tug of War Begins When public schools were established in the United States, there was little concern regarding teaching methods as basic curriculum delivered using textbooks (Cohen, 2003). Children were schooled as a form of moral discipline with a regimen of memorization and recitation necessary for learning religious truths as well as grammar and multiplication facts (Reese, 2008). Elementary schools in colonial America routinely provided instruction in both reading and basic numeracy (Reese, 2008). Cohen (2003) stated, Whether taught in New England’s rural district day schools or in the urban feeforservice evening schools dotted in Boston, Newport, New York, Philadelphia, and Charleston, arithmetic was regarded as a vocational subject, a skill whose chief application was in the world of commerce. The appropriate pupil for such study was the twelve to fourteenyearold boy, judged to be mature enough to absorb the arcane techniques of computation as well as sufficiently competent in writing to create a permanent copybook. (p. 44) Burton (1850) provided a detailed description of his experience in school, including his entrance into formal arithmetic at age 12. His description inadvertently illustrated the math teaching methods of the time. A popular textbook of the early 1800s, Arithmetik (Pike, 1809), was a memorybased book that was used as a resource to provide explanations of how to solve problems using arithmetic operations (Cohen, 1999, 2003). Burton described transcribing the 24 text, rules, and examples that illustrated each rule, wordforword, chapterbychapter from his printed textbook, Adam’s Arithmetic (a memorybased text much like Pike’s), into a copybook that he maintained. He also depicted his difficulty understanding a particular procedure. Burton described, “Carrying tens in addition,” which he called “a mystery which that arithmetical oracle, our schoolmaster, did not see fit to explain” (p. 113). The clarity of his account allows researchers to infer that the strategy employed by young Burton’s schoolmaster was to have students copy a textbook as the instructional means to teach mathematics. The schoolmaster also relied on the textbook as a resource to illustrate math principles and provide his students with examples of problems and their solutions (Burton, 1850). According to Burton (1850), he copied each rule in his notebook and the process by which the answer was found: “Each rule, moreover, was, or rather was to be committed to memory, word for word, which was to me the most tedious and difficult job of the whole” (p. 113). In fact, his teacher used the same methodology that had been employed in American colonial schools for over a century (Cohen, 2003). When Burton was in school, educators were already discussing teaching methodology in math (Cohen, 1999). After 1820, many educational proponents advocated inductive reasoning as the method to teach mathematics rather than use memorybased textbooks (Cohen, 1999). From the 1820s to the 1860s, wellknown educators widely debated the merits of inductive reasoning versus rote learning and memorization. Warren Colburn, having served as a schoolmaster, superintendent, and author, “wanted to end children’s slavish reliance on rules and rote learning” and he advocated that students should discover mathematical rules by working specifically selected examples (as cited in Cohen, 2003, p. 59). 25 For a time, teachers put aside memorybased texts and used inductive reasoning with specially crafted problems (Cohen, 1999). However, by 1834, a review of several new textbooks questioned whether induction was adaptive as a teaching methodology, and by the mid19th century, opponents of Colburn’s discovery learning method voiced strong protests (Cohen, 2003). In 1851, Taylor Lewis, a professor at Union College in New York who strongly opposed inductive or discovery learning, called the methodology “quackery” and a practice that would “enfeeble the mind” (as cited in Cohen, 2003, p. 64). Lewis (1851) favored synthetic or rote learning; that is, direct instruction from authoritative teachers who explained and demonstrated content using the examples they worked. Lewis (1851) was also adamant that the discovery method undermined authority because it fostered independence in students. In summary, he held fast to his bottom line: rote learning has its place in the math classroom. According to Cohen (2003), “From the 1850s forward, the scales tipped back toward synthetic instruction” (p.65). In fact, in 1870, a New Jersey school superintendent described inductive methodology as “math taught backward” (Cohen, 2003, p. 65). Direct instruction had become solidly entrenched as the method used to teach school mathematics, and Cohen (2003) stated, “By the early 20th century, Warren Colburn was barely a distant memory” (p. 65). At this time in the United States, algebra was a relatively young mathematics course compared to arithmetic. However, while Colburn and Lewis debated the efficacy of inductive reasoning through discovery versus synthetic rote learning, particularly in arithmetic instruction, secondary mathematics gained a foothold in American high schools (Boyer, 1968). In 1820, Harvard made algebra a requirement for admittance, which insured that algebra would become a common secondary mathematics course in the years that followed (Rachlin, 1989). By 1880, 26 algebra was indeed entrenched as the firstyear mathematics curriculum for high school, although only 1 in 10 young people attended high school (Kieran & Wagner, 1989). Like elementary children who were schooled as a form of moral discipline, with a regimen of memorization and recitation, secondary students also memorized significant amounts of material from textbooks and recited it back to their teachers (Reese, 2008). Perhaps dissatisfied with the memorization method, like their elementary peers, secondary mathematics teachers became vocal in the pedagogy debate. In 1890, mathematical historian Florian Cajori (1890) wrote that mathematical teaching of the prior 10 years indicated a “rupture” with antiquated traditional methods and an alignment with the “march of modern thought” (p. 293). Cajori’s (1890) survey of algebra teachers indicated that they were opposed to an overemphasis on procedural skills. They called for educators to teach algebra in a meaningful way to facilitate better student understanding (Parshall, 2003; Rachlin, 1989). However, Cajori (1890) also stated that the alignment with modern thought had barely begun. McLellan and Dewey (1895) joined the continuing debate strongly critical of drilling a student on procedures and forgetting that he should actually use the ideas or concepts as well, a practice that is detrimental to learning. They stated, There is no attention, or too little attention, paid to the essential process of discrimination when it is taken for granted that definite ideas of number will be formed from the hearing and memorizing of numerical tables...apart from the child’s own activity in conceiving a whole of parts and relating parts in a definite whole. (p. 30) Finally, Dewey (1916) decried the treatment of students as pieces of “registering apparatus” that acquire and store information without experiencing the learning or understanding its purpose (p. 147). 27 1900–1930: Expansion and Progressivism From 1900 to the mid20th century, the debate continued regarding pedagogy in school mathematics (Kilpatrick, 1992). At the turn of the 20th century, several factors influenced the methodology for teaching mathematics in general and algebra in particular. First, public schooling had gained greater legitimacy for children of multiple ages; as a result, more children attended school (Reese, 2008). Simultaneously, university professors, education leaders, and mathematicians were weighing both the value of teaching mathematics and the types of mathematics courses that should be taught, if any (Klein, 2003). Finally, several significant societal influences played a role in mathematics education (Kieran & Wagner, 1989). The 20th century witnessed the continual expansion of the power and authority of public schools in the lives of children, and high schools that had served relatively few students expanded dramatically (Reese, 2008). As late as 1890, approximately 5% of all adolescents were enrolled in public schools. While most students were girls who later became school teachers, school attendance became increasingly universal, particularly with the assimilation of massive numbers of immigrants arriving in the 1890s (Reese, 2008). As a result, between 1890 and 1910, the number of high school students quadrupled. Compared to the 1 in 10 who enrolled in 1890, by 1910, 1 in 3 teenagers were enrolled in high school (Ballew, 2009). Along with the significant increase in the number of students enrolling in public schools, enrollment in mathematics courses also increased. By the end of the 19th century, boys much younger than Burton’s 12 years were learning elementary arithmetic (Cohen, 2003). In fact, by 1910, teachers routinely taught arithmetic to 6yearold boys and girls and algebra to male and female students in Grade 9 (Brookman, 1910; Cohen, 2003). 28 Concurrent with the everincreasing number of students entering high school and enrolling in mathematics courses, was the continuing pedagogical debate in the mathematics educational community regarding inductive reasoning and discovery learning versus synthetic and rote learning that had begun in the late 1800s (Cohen, 2003). By the turn of the century, more students were enrolling in algebra, and as evidenced in the Cajori survey of 1890, algebra teachers had clearly called for a new teaching method (Parshall, 2003). Robinson (2010) stated that the early 1900s saw a movement toward pure mathematics and problemsolving methodology defined as the ability to manipulate equations algebraically to solve problems. This methodology was described as the “plug and chug” method (Robinson, 2010, p. 1). Teachers taught students that to solve a problem they needed to find the correct formula, plug it into the problem, and work through the mathematical operations (chug) to the solution. As such, the pure math movement was clearly an advocacy for procedural, rote learning of skills. The debate grew stronger in some educational communities and became secondary to a more significant issue in others (Klein, 2003). Early in the 1900s, much of the educational focus shifted dramatically from the best methodology for teaching algebra, and mathematics in general, to the very value of teaching mathematics (Klein, 2003). In the late 1800s, it was still generally accepted that mathematics be taught as a form of mental discipline; however, in 1901, psychologist Edward Thorndike conducted a series of experiments. Based on his findings, he challenged the justification of teaching mathematics to discipline the mind (Klein, 2003). That challenge contributed to the view that all mathematics should be taught and learned for utilitarian purposes only (Klein, 2003). Thorndike championed the progressive stance that in high school, algebra should be restricted to select students, and rather than emphasizing the debate on the methods of teaching 29 algebra, the focus should be on who should take the course and for what reason (Angus & Mirel, 2003; Klein, 2003). William Heard Kilpatrick, a significant voice of the progressive movement, particularly in mathematics, agreed with Thorndike and rejected the notion that the study of mathematics contributed to mental discipline. He advocated that math should be taught to students only for its practical value (Klein, 2003; Kliebard & Franklin, 2003). In fact, Kilpatrick carried the argument further by advocating that algebra and geometry in high school be discontinued “except as an intellectual luxury” (as cited in Klein, 2003, p. 3). Educators supported Kilpatrick, and in 1915, the Commission on the Reorganization of Secondary Education asked him to chair a committee of educators to study the problem of teaching mathematics in high school (Klein, 2003). Kilpatrick’s 1915 report, The Problem of Mathematics in Secondary Education, challenged the use of mathematics to promote mental discipline (Kliebard & Franklin, 2003). He recommended that only content of probative value be taught, and he called to restrict high school math to a very select group of students (Kliebard & Franklin, 2003). Prominent mathematicians, none of whom served on the committee, stringently opposed Kilpatrick’s report as an attack on the field of mathematics itself, and they attempted to stop the report from being published (Klein, 2003). Anticipating the publication, the Mathematical Association of America (MAA) responded vigorously by convening the National Committee on Mathematical Requirements, a group that included prestigious mathematicians and wellknown teachers and principals from secondary schools (Klein, 2003). Eventually, the Kilpatrick report was published in 1920, and in 1923, the MAA national committee published its own extensive report that advocated the importance of algebra for every 30 educated person (Klein, 2003). In addition, the MAA urged the formation of the National Council of Teachers of Mathematics (NCTM) in 1920 to counter arguments of progressive educators; however, throughout the 1920s and 1930s, Kilpatrick’s progressivism report was more influential than the 1923 MAA report (Klein, 2003). In fact, the influence of Thorndike, Kilpatrick, and the Progressive Movement in general from 1900–1950 was a catalyst for several changes in mathematics education, not the least of which was a dramatic increase in the number of vocational courses and a significant decline in the enrollment in secondary algebra (Angus & Mirel, 2003; Ballew, 2009). The limited number of research on algebra education from 1900 to 1930 dealt mainly with student performance in solving equations and curriculum rather than teaching methodology (Kieran & Wagner, 1989; Thorndike, 1922). In the 1920s, Kilpatrick and the progressives’ call for a move away from pure mathematics to a more practical, vocationally oriented curriculum, included algebra content as well (Angus & Mirel, 2003). Addressing the methodology for teaching algebra, Thorndike (1922) highlighted the importance of understanding algebraic formulas; however, he clearly called for prescriptive manipulation exercises and spoke to the amount of practice students should engage (Kieran & Wagner, 1989). Progressive sentiments included debate over who should take algebra. These sentiments, together with dwindling algebra enrollment, indirectly influenced teaching methodology for the course (Kieran & Wagner, 1989). The main influence on teaching algebra, however, was the socioeconomic condition of the nation produced by the Great Depression leading to World War II (WWII; Kieran & Wagner, 1989). Further, the loud cry from society for courses that taught practical and vocational mathematics rather than algebra, together with a rapidly growing interest in psychology, 31 contributed to an increasingly childcentered focus in the classroom (Kieran & Wagner, 1989). The innovative, studentcentered approach to teaching included a more active role for students identified as learning by doing (Reese, 2008). As such, limiting math content to its probable value, that which was directly applicable to ordinary living, significantly reduced the number of topics in the curriculum, limited academic content, and reduced the rigor of the coursework (Ballew, 2009). The decreasing number of students enrolled in advanced mathematics courses and the reduced number of topics taught justified the slow pace of teaching studentcentered algebra using a discovery approach (Klein, 2003). However, teachers often ignored pleas for a more studentcentered pedagogy, and they maintained the pedagogy of memorization and practice (Reese, 2008). Progressive educators despaired over the traditional ways of high schools, where subject matter and teacher authority had long reigned supreme. Critics discovered that even shop teachers lectured or read to pupils out of textbooks (Reese, 2008). The movement at the end of the 1920s toward vocational mathematics did not change, and traditional methods of teaching businessrelated practical problems using basic skills remained intact (Klein, 2003; Kliebard & Franklin, 2003). 1930–1950: The Great Depression and World War II From 1930 to the mid20th century, several significant factors directly influenced mathematics education and teaching methodology. Angus and Mirel (2003) stated, “In the late twenties, the bottom fell out of the teenage job market” (p. 460). The rapid decline of job opportunities for adolescents in cities because of the Great Depression caused significant increases in the number of young people entering high school and provided a strong incentive for them to remain there (Angus & Mirel, 2003; Kliebard & Franklin, 2003). Angus and Mirel 32 (2003) reported, “The high schools experienced the largest enrollment increase relative to the fourteentoseventeen year old population in history” (p. 460). To keep students in school and out of the job market for as long as possible, compulsory attendance became universal. In response to the burgeoning number of students, the childcentered focus in the classroom increased and high schools became more custodial in order to educate everyone (Angus & Mirel, 2003; Kieran & Wagner, 1989; Reese, 2008). The Great Depression was a catalyst for continued and even heightened attacks on education in general, and on mathematics in particular, as being too remote from everyday life, a sentiment that added significant support to the progressive platform (Kieran & Wagner, 1989). Education journals, university courses for administrators, teachers, and textbooks advocated major themes of progressivism that proposed that the school curriculum be determined by students’ vocational needs as determined by professional educators, rather than academic subjects (Klein, 2003). From 1930–1950, an era characterized by some as the LifeSkills Movement, preoccupation with issues of existence and survival led to deeper entrenchment of teaching practical and vocationally oriented mathematics, which provided greater influence on the way courses such as algebra were taught (Kieran & Wagner, 1989). Angus and Mirel (2003) stated, “Throughout the 1930s and beyond, the focus of curriculum reform was to expand the general track and to develop courses for this track that were interesting, undemanding, and closely related to the ‘immediate needs of youth’” (p. 460). Leading the progressive platform, Kilpatrick (1925) advocated that students should be actively involved in their learning, and he inspired what was called the Activity Movement of the 1930s. Kilpatrick also advocated for interdisciplinary curriculum of all subjects versus standalone math courses. While little disagreement existed in 33 elementary school, high school teachers were unwilling to abandon their courses in which they had specialized knowledge and expertise (Klein, 2003). Nevertheless, mathematics courses, including algebra, became even more focused on utilitarian applications (Kliebard & Franklin, 2003). Overall, secondary education was characterized by more electives, fewer graduation requirements, declining enrollment in traditional math courses, and curriculum written for commercial math (Angus & Mirel, 2003; Donoghue, 2003; Kieran & Wagner, 1989; Klein, 2003; Kliebard & Franklin, 2003). Regardless of curriculum issues, the debate over teaching methodology maintained its vigor. Thorndike’s advocacy in the 1920s for drill and practice was met in the following two decades with repercussions (Kieran & Wagner, 1989; Resnick & Ford, 1981). Thorndike had advocated that practice led to understanding; however, in the 1930s and later, theorists advocated that meaningful learning or understanding should be accomplished before practice (Kieran & Wagner, 1989). Brownell (1945) led the strong advocacy for meaningful mathematical learning, and he emphasized the importance of student opportunities for understanding the structure of mathematics (as cited in English, 2007; Hiebert & Carpenter, 1992). Adamantly opposed to the wideacceptance of drill, a result of published investigational studies, Brownell and Chazal (1935) accused researchers of being more concerned with the length and type of drill than with its effects on learning and not considering the place of drill in a total mathematics program. Brownell (1935) stated, In more recent years, the large number of investigations on drill have been less concerned with its effects upon learning than with such related matters as the length of the drill period, the comparative merits of mixed and isolated drill organization and the like. (p. 17) 34 Addressing the history on the effects of teaching for meaning dating from the 1940s, Grouws and Cebulla (2000) stated, “Investigations have consistently shown that an emphasis on teaching for meaning has positive effects on student learning” (p. 1). Several wellknown educators led the resistance to meaningful learning in the 1930s and 1940s. They voiced complaints of the military services in which an inordinate number of recruits had to be tutored in basic skills of mathematics that they should have learned in school (Klein, 2003). The Life Adjustment Movement in the mid1940s was based on teaching basic skills. Proponents complained that secondary schools were too devoted to an academic curriculum and students should be taught traditional math skills for daily living (Klein, 2003). Hiebert and Carpenter (1992) characterized the contention as an issue of the importance of conceptual versus procedural knowledge or understanding versus skill. They pointed out that the prevailing view had seesawed back and forth depending on the persuasiveness of the spokesperson of each particular position. Clements (2003) stated that of the most contentious topics among math educators was the role of drill and practice and the ways to achieve meaningful learning. Advocates of rote learning would often speak to the need to use drill with care so they also included meaning (Sobel, 1970). However, in practice, teachers rarely felt the need to distinguish among the meaning of drill practice and recurring experience (Clements, 2003). Throughout the 20th century, educators accepted drill as a useful tool in teaching and learning mathematics (Clements, 2003; Sueltz, 1953). Referring to Cajori’s survey findings, Rachlin (1989) stated, These calls have been echoed in waves of reform documents from that time (Cajori, 1890) until today—with proposals ranging from laboratory approaches that would 35 encourage an inductive learning of algebra to reallife applications that would demonstrate the relevance of algebra. (p. 257258) Algebra textbooks of the first half of the 20th century included narratives about the importance of learning “why” and “how,” but “the emphasis in the texts was on computational algebra” (Rachlin, 1989, p. 258). Osborne and Crosswhite (1970) identified that the common sequence for instruction was definition, illustration, rule and example, drill, review, and speed tests through direct instruction from the teacher. Research on teaching methodology has gone through several periods of reform, and it is generally categorized into loosely chronological phases. Little educational research was published in the first half of the 20th century (Rachlin, 1989). Statistics indicate only a gradual increase in the number of studies from 1892 to 1924. In fact, the average annual frequency of educational studies prior to 1930 was less than 25 (Kilpatrick, 1992). This slow growth of educational research persisted until 1960. According to Nisbet (1985), “Prior to 1960, research was mainly a sparetime amateur affair, unorganized and often ignored” (p. 12). Before the mid20th century, rather than examining pedagogy, the limited number of mathematics education studies concentrated on teacher characteristics or attributes in isolation (Koehler & Grouws, 1992). The first approach has been described as the behaviorist tradition that identified and analyzed components of teachers’ behaviors so that others could emulate effective practices (Nickson, 1992). Teachers were evaluated by measures such as math coursework completed and years of experience (Koehler & Grouws, 1992; Medley, 1979; Rosenshine, 1979). Personality traits such as enthusiasm, magnetism, and considerateness were also included in the studies, and they were often rated on good judgment, loyalty, and appearance (Barr & Emans, 1930; Charters & Waples, 1929). Effective teachers were identified by opinions 36 and ratings of principals, supervisors, and sometimes students, which provided the basis by which the researchers identified whether a particular trait correlated to successful teachers (Koehler & Grouws, 1992). Later research sometimes examined isolated teaching events (Medley, 1979; Rosenshine, 1979). Koehler and Grouws (1992) stated, “Although student outcomes were considered, the narrow focus of these studies and their lack of attention to the quality of teaching was limiting” (p. 116). Little, if any, attention was given to other factors, including the quality of teaching methodology. In summary, the research focused on the teacher rather than on the teaching (Koehler & Grouws, 1992). Although some prominent studies addressed other questions, curriculum concerns continued to be voiced, including the question of whether mathematics courses would, or should, be offered in high school, particularly algebra and geometry (Kliebard & Franklin, 2003; Rachlin, 1989). Rosenshine (1979) identified a second cycle of research that focused on teacherstudent interactions and a third that focused on students, their attention in the classroom, and the content they mastered. Other studies addressed various issues related to teaching and learning mathematics in North American schools from 1900 to 1965, which was accompanied by much debate (Kilpatrick, 1992). Clements identified two of the most contentious topics as the role of drill and practice and the methods to facilitate meaningful learning (Clements, 2003). The question of how to teach mathematics resided in the ongoing debate in the 19th century of whether it was more effective to tell students content directly or to facilitate their discovery of the content (Clements, 2003). Near the close of the halfcentury, Schoenfeld (1987) stated, “Nineteenforty five was both a banner year and a year of great chaos for problemsolving” (p. 283). Mathematics instruction in schools was mostly drill and practice, based on Thorndike’s behaviorist theory, but 37 drill and practice was under attack (Schoenfeld, 1987). The European Gestalt view, somewhat popular in the United States, claimed that rote instruction was of little value because students failed to gain an understanding of concepts and structures when they simply memorized and reproduced mathematical procedures (De Corte, 2010; Schoenfeld, 1987). In turn, the behaviorists counterattacked the Gestaltists (De Corte, 2010; Schoenfeld, 1987). The pendulum remained on the side of traditional instruction although the first half of the century closed with the same contention regarding teaching methodology as that with which it began. 1950s: Influence of Sputnik By the second half of the century, nearly 90% of all teens were enrolled in public school, and the technological nature of the war had highlighted the importance of mathematics in school curriculum; however, barely onehalf of teenagers were taking mathematics courses (Angus & Mirel, 2003; Garrett & Davis, 2003; Reese, 2008). During the years that research seemingly focused on a myriad of factors other than pedagogy, discussions continued regarding the best way to teach mathematics. By the 1950s, projects and programs evidenced the strength of the disagreement (Davis, 2003; Lester & Lambdin, 2003). However, using textbooks to explain, provide examples, and practice remained the most prevalent method, and a significant amount of research focused on textbooks (Seymour & Davidson, 2003). After WWII, American schools were under attack as they faced significant criticism from the military, business, education, and public sectors claiming that graduates were not equipped mathematically for work or further study (Kilpatrick, 1992; Klein, 2003; Lester & Lambdin, 2003). In fact, teachers pointed to scientific advances in technology, economic growth, and political changes that accompanied WWII as significant indicators of the need for new educational programs. In response, great efforts were made across the United States to improve 38 math and science content for students (Davis, 2003; FerriniMundy & Graham, 2003). Psychologists and some curriculum theorists recommended new teaching methods whereby students engaged in exploration and discovery (FerriniMundy & Graham, 2003; Fey & Graeber, 2003). Beginning in 1950, the requirements of industry, engineering, the sciences, and mathematics escalated dramatically, and in 1957, two events spurred significant change in mathematics education: the launch of Sputnik by the USSR in October and the establishment of the Madison Project at Syracuse University (Davis, 2003; Fey & Graeber, 2003; Kelly, 2003). FerriniMundy and Graham (2003) stated, “The launching of Sputnik was a watershed event that jolted the United States public, government and scientific communities into taking more concerted action in the areas of mathematics and science education” ( p. 1198). Fey and Graeber (2003) stated, “The military threat of Soviet space science and technology prompted a variety of political, business, and social groups to urge critical examination of American mathematical, scientific, and technical educations” (p. 521). By the end of WWII, deep concern existed about national security in the Cold War world, and while some educational reform activity had begun before the Sputnik launch, the military threat it posed heightened concerns and became the catalyst and sustaining force to upgrade mathematics and science in schools (FerriniMundy & Graham, 2003; Fey & Graeber, 2003; Schoenfeld, 1992). The new sentiments of mathematicians and mathematics educators, among others, resulted in a concerted challenge to traditional rote learning and classroom drill and practice (Kelly, 2003). Instead, educators favored an approach focused on higherorder thinking and conceptual understanding (Kelly, 2003). In fact, discovery teaching was widely advocated immediately following the war (FerriniMundy & Graham, 2003). In response to concerns regarding school mathematics, the Madison Project, led by mathematics professor Robert B. 39 Davis, was established in the post WWII era when teachers, schools, and parents were willing to explore new ideas in curriculum and teaching methodology (Davis, 2003). Accordingly, the highly innovative Madison Project, arguably the most significant program of the time, focused on the discovery approach to mathematics instruction, a teaching methodology that was among the most controversial aspects of the project (Davis, 2003; Kelly, 2003). A century earlier, Colburn had advocated for presenting students with carefully selected problems to discover the appropriate mathematical principles necessary to solve them (Cohen, 1999). With Madison, the pendulum moved again toward Colburn’s favored type of methodology, discovery teaching, and Madison Project teachers began by asking questions or posing selected or created tasks (Cohen, 1999; Davis, 2003). Davis (2003) stated, “The thing that came first was the problem. The relevant mathematical ideas were invented by the students in order to solve the problem” (p. 628). Davis illustrated the new methodology as follows: Discovery teaching requires a substantial change both in the role of the teacher and in the role of the student. The teacher creates a challenge, and it becomes the responsibility of students to invent methods of solution...This process is virtually a reversal of much common school practice, where the teacher presents a way of dealing with some class of problems, the student imitates the teacher, and the teacher evaluates the result. In the traditional teaching I encountered, students often knew what they were expected to do (because they had been shown and had practiced it) but had no idea why they were doing it or when (outside the lesson) it might be useful...Stimulus and response were inextricably wedded, with each having a quite different nature from most usual school tasks. Stimulus now meant a novel problem and the response was what the students invented in order to accomplish their goal. (p. 629) 40 Unfortunately, little to no research supports the discovery methodology used in the project because administrators regarded research studies as problematic (Davis, 2003). Madison overseers believed that different people have different learning experiences that result in different ideas; as such, comparative studies would not provide results that indicated the benefits of the discovery method (Davis, 2003). Mathematicians and researchers questioned the lack of appropriate comparative studies as evidence of a response to severe opposition (Davis, 2003). While project managers made films and videotapes to prove the value of discovery, the work at Syracuse did not accurately reflect the real world of most public schools (Davis, 2003). In summary, no quantitative research exists to substantiate the claims of the Madison Project in favor of discovery learning (Davis, 2003). Davis (2003) stated, “Whatever the reasons, opposition to discovery teaching was a serious obstacle to wider acceptance of project innovations (p. 633). In 1959, the Commission on Mathematics of the College Entrance Examination Board (CEEB) recommended that logic, modern algebra, concepts of relation and function, and probability and statistics be included in the school mathematics curriculum so secondary math students could study both pure and applied topics (CEEB, 1959; Fey & Graeber, 2003). Proponents also advocated that processes of deductive reasoning and the search for patterns would help students develop powerful understanding, whereas the prevailing programs of the time provided only rote training on a litany of disconnected procedural routines that most students quickly forgot (Fey & Graeber, 2003). The pendulum moved toward exploring teaching strategies as the 1959 CEEB report defined the new math agenda for the 1960s. 41 1960s: The New Math Era Fey and Graeber (2003) stated, The period from the start of the ‘new math’ movement in the mid1950s to the publication of An Agenda for Action by the National Council of Teachers of Mathematics (NCTM) in 1980 has all the ingredients of a typical era in the evolution of mathematics education. (p. 521) In other words, great advocacy for reform was followed by great resistance in favor of maintaining a traditional agenda. The pendulum continued its movement, but perhaps in the 1960s, it swung a little further in each direction. The goal of school mathematics in the 1930s and 1940s was to provide a mostly vocational, universal education for everyone based on the individual needs of each student; a 1960–1961 survey indicated that practical, consumerrelated courses in mathematics continued to be popular (Angus & Mirel, 2003). However, the childcentered focus prior to and during the war years shifted with events of the 1950s, including Sputnik. During this time, the goal became one of educating an elite group of college preparatory students to develop a scientifically and mathematically strong workforce that could facilitate United States competition with the Soviet Union (Becker & Perl, 2003; FerriniMundy & Graham, 2003; Fey & Graeber, 2003; Kieran & Wagner, 1989). To produce knowledgeable students quickly, interest increased in developing new approaches to make math meaningful, and controversy arose over teaching methods once again (Payne, 2003; Resnick & Ford, 1981). Mathematicians and mathematics educators, among others, advocated replacing rote skills and drill instruction with conceptual teaching and the New Math Movement began in the late 1950s and early 1960s (Kelly, 2003; Resnick & Ford, 1981). 42 At the same time, there was a great deal of interest in the students’ mental processing, and as the field of cognitive psychology was born, conceptual teaching for meaningful learning gained impetus during the 1960s (Bransford et al., 1999; Resnick & Ford, 1981). Research prior to the 1960s was amateurish. However, because of the shifting focus, research in mathematics education proliferated during the 1960s (Kilpatrick, 1992; Lester & Lambdin, 2003; Nisbet, 1985). Nisbet (1985) observed, “It is only within the past 25 years that research in education has received public funding on any substantial scale” (p. 12). Public support of educational research in the 1960s grew very rapidly, and expenditures doubled each year from 1964–1967 (Nisbet, 1985). In the late 1950s and early 1960s, dozens of national, regional, and local mathematics education projects and programs were established to develop curricula and teaching methodology that reflected the reform agenda (Angus & Mirel, 2003). Much of the research was based on efforts to measure the effectiveness of the new math programs (Fey & Graeber, 2003; Lester & Lambdin, 2003). In 1960, prominent psychologist and researcher, Jerome Bruner, published The Process of Education. Bruner had become interested in the cognitive processes of humans as they learned, particularly during classroom activities in mathematics (as cited in Resnick & Ford, 1981). Perhaps the bestknown educational psychologist of the decade, Bruner advocated that students could better understand and apply mathematics if teachers focused on the conceptual understanding of the structures of mathematics (Bruner, 1960; Clements, 2003; Fey & Graeber, 2003). Contrary to a vocational orientation for mathematics curriculum, Bruner proposed teaching math problems that were interesting to students, thereby providing a stimulus for learning (Smith, 2002). He also advocated a teaching methodology that facilitated student acquisition of the habits of thinking like a professional mathematician (Bruner, 1960). For the 43 most part, the mathematics community agreed that the main objective of math instruction was student understanding rather than rote skill practice, memorization, and recitation (Kieran & Wagner, 1989). Psychological and cognitive issues were included in reform perspectives of the new math era. Fey and Graeber (2003) stated, For example, one of the persistent tensions in teaching was finding the most productive balance between telling and asking: Should a teacher provide clear and convincing exposition of ideas and techniques or stimulating questions and problems whose solution by students will reveal important concepts and procedures. (p. 526) The same disagreements persisted among teachers, as many believed that students must be told specifically how to work problems and others believed the opposite. Just as it had a century earlier, the disagreement in the 1960s fostered many controversies over “discovery learning” (Davis, 1992, p. 725). In addition to the debates over telling versus asking, issues of the student’s role in learning were widely discussed (Fey & Graeber, 2003). Based on new insights of the time into how students learn math, prominent psychologists recommended a reformed teaching pedagogy that included “emphasizing the importance of students’ engagement in exploration and discovery through developmentally appropriate activities” (Fey & Graeber, 2003). Fey and Graeber (2003) stated of the 1960s, If there was a dominant pedagogical principle underlying many new math innovations, it was that students acquire understanding and skill most effectively through classroom activities that help them discover mathematics themselves. Thus new math reformers 44 explored a variety of strategies for helping teachers conduct Socratic dialogues with their students, leading the students to the discovery of major concepts and principles. (p. 526) The methodology of discovery teaching included lessons in which students hypothesized, investigated, and explored mathematical solutions to problems, which allowed them to experience the thinking and activities of professional mathematicians as Bruner (1960) advocated (Fey & Graeber, 2003). Discovery was also consistent with new cognitive theories that advocated learning tasks for students that challenged their conceptions and lead them to new understanding (Fey & Graeber, 2003). Fey and Graeber (2003) stated, “For an exciting period of at least ten years in the decade of the1960s, the reform of school mathematics was frontpage news around the world” (p. 522). Unfortunately, many mathematics educators found it much easier to recommend discovery teaching, but much more difficult to prepare teachers who could deliver the instruction in everyday classrooms (Fey & Graeber, 2003). Teachers were skeptical of the new math teaching agenda, and they did not exhibit the enthusiasm of the math community at large (Fey & Graeber, 2003). Even with the multitude of studies, programs and proposals for reformed curricula, and classroom teaching techniques, teachers relied almost exclusively on textbooks for both curriculum and teaching methodology (Seymour & Davidson, 2003). During the reform decade, typical classrooms were described as plain and rather sterile, and teachers used only textbooks and chalkboards to teach mathematics where the dominant activity was the same traditional memorization and computation seen in the reform era and where realworld application problems were limited (Seymour & Davidson, 2003). While many different reform programs and projects were characterized as new math, some called for teachers to guide student learning, as in a discovery lesson, while others retained 45 traditional characteristics such as starting a lesson with formal definitions (Davis 2003). While all projects advocated for the replacement of traditional curriculum, before widespread implementation of these recommendations occurred, many different groups and individuals criticized discovery approaches (Davis, 2003; Fey & Graeber, 2003). Loud calls were issued for the return to traditional curriculum and teaching methodologies based on behavioral psychology, including instructional approaches that involved procedural lessons and calculation (Davis, 2003; Fey & Graeber, 2003). Fey and Graeber (2003) stated, “For a time this more traditional view of school mathematics curricula and teaching gained the upper hand in debates over elementary and secondary school programs” (p. 522). The new math reform was ending. Public memory of new math was an image of failures and mistakes, and the term new math is now part of educational rhetoric that connotes a failed reform (Fey & Graeber, 2003; Payne, 2003). Once again, after all the fervor in the 1960s regarding new math, the pendulum gained momentum as it moved like clockwork in the opposite direction back toward traditionalism. 1970s: Back to the Basics Fey and Graeber (2003) spoke of the perpetual “tugofwar” in education for intellectual, social, and political influence. They stated that the struggle in mathematics has a “predictable rhythm of crisisreformreaction episodes” (p. 521). Payne (2003) stated, “If the 1960s were the zenith decade, then the 1970s were the nadir.” He described the period as the “20th century’s worse decade” (p. 590.) The school mathematics community characterized the 1970s as calmer than the “turbulent new math era” of the 1960s (Fey & Graeber, 2003, p. 539). Professional publications at the time documented continued advocacy for progressivism and described the ongoing debate regarding the most effective teaching pedagogy (Fey & Graeber, 2003). Certain 46 groups in the math community referred to successes from the 1960s to support their stances on the issue (Fey & Graeber, 2003). However, new math reformers in the educational community encountered several significant difficulties, and by 1978, the 10 most widely used mathematics programs included traditional rather than reform materials (Fey & Graeber, 2003). First, school personnel were skeptical of the new ideas, and it was often very difficult to prepare teachers to implement classroom methodology to use discovery learning (Fey & Graeber, 2003). Second, while many programs and projects were based on reform recommendations of the 1960s, the textbooks and other materials developed for reform teaching were actually very conventional and included explanations of math content, examples, and homework (Fey & Graeber, 2003). Third, anxious parents reacted with great concern toward the new math content and teaching methodology their children experienced, particularly because it was so different from their own experiences in school mathematics (Fey & Graeber, 2003). Public perception was that new math failed because students could do nothing with accuracy (Payne, 2003). Payne (2003) stated, “Certainly, they could not compute, and practical skills had all but been abandoned” (p. 590). Public conviction against reform was a catalyst for press reports that were critical of the movement, and the fervor sparked recommendations to abandon reforms and return to the basics (Fey & Graeber, 2003). Along with calls for traditional curriculum, the public wanted traditional teaching as well (Fey & Graeber, 2003). Specifically, the call was to abandon discovery learning in favor of traditional teaching practices that emphasized direct instruction of procedural skills (Fey & Graeber, 2003). Opposition to the 1960s reform swelled in the early 1970s, and significant backlash against the New Math Movement occurred (Kieran & Wagner, 1989). The 1970s became 47 commonly described as the back to basics era (Fey & Graeber, 2003; Schoenfeld, 1992). Fey and Graeber (2003) stated, “The phrase suggested renewed emphasis on developing skills in arithmetic and algebraic calculation through instruction that features teacher exposition and student practice” (p. 538). In other words, the pedagogical pendulum responded to the uproar and gained speed in its swing back toward traditional teaching methodology. Two events occurred in the 1970s that sealed the return to traditionalism in mathematics teaching methodology. First, in 1975, the National Advisory Committee on Mathematical Education (NACOME) published an analysis of math education in the United States (FerriniMundy & Graham, 2003). The NACOME report suggested that in spite of advocacy in the 1960s for new math discovery, some had significant doubts that teachers were using the methodology. In fact, evidence existed that teachers had difficulty figuring out how and when to implement the various new pedagogies (Fey & Graeber, 2003). Forbes (1970) stated that most U.S. high school teachers depended heavily on textbooks with little deviation from the scope and sequence. The NACOME report described the median classroom in mathematics as one in which teachers closely followed the textbook, “for the most part [teachers were] teaching the way they had been taught” (FerriniMundy & Graham, 2003). Perhaps most detrimental to reform, the report indicated a lack of evidence that suggested any particular method or pattern of instruction was superior to any other, and more information was needed (FerriniMundy & Graham, 2003). The National Science Foundation (NSF) addressed the need for more information on classroom instruction in mathematics in the late 1970s (McLeod, 2003). The NSF funded three large survey projects, and pertinent to this study, the third project included a set of detailed case studies on classroom instruction in math and science (McLeod, 2003; Stake & Easley, 1978). 48 These case studies were particularly influential in substantiating the view that teachers were still teaching traditional mathematics, with its emphasis on rules and procedures, and that the new math of the 1960s had not produced lasting change in classroom instruction (McLeod, 2003). Second, in 1977, the CEEB issued a report that provided data on a 10year decline in scores on the Scholastic Aptitude Test (SAT); an additional catalyst for abandoning new math reforms (Fey & Graeber, 2003). McLeod (2003) stated, From the information published in the NACOME report in 1975 and the data from other studies in the late 1970s, it was clear that the dreams of the new math era prior to 1970 had been dashed against the rocky reality of traditional mathematics classrooms. (p. 758) Enthusiasm for new math was replaced by fervor for instruction based on behavioral psychology principles (Fey & Graeber, 2003). McLeod (2003) stated, “The emphasis in the 1970s on drill and practice in mathematics curriculum had a long history in support from behaviorist psychology” (p. 807). Behavioral psychology had dominated school practice for most of the 20th century prior to the brief new math era of the 1960s. Behaviorist theories were in control of mathematics instruction again with the return to traditionalism, thus, dominating the backtobasics movement in the 1970s (Fey & Graeber, 2003). Relying on behavioral research and theories, many influential educators argued that classroom instruction in mathematics should emphasize mastery of objectives on a systematic path, and goals should be defined in terms of explicitly observing student performance (Fey & Graeber, 2003). Parents and politicians continued to express concerns about student competence in the basics, and NCTM reform leaders continued attempts to convince the public of the necessity to move away from the emphasis on computational skills (McLeod, 2003). In 1977, the National Council of Supervisors of Mathematics (NCSM), in support of reform, responded to the 49 traditional, backtobasics movement by redefining basic skills to include problem solving and application skills rather than just computational prowess. In 1978, leaders in the mathematics community brought researchers from mathematics and cognitive science together in response to concerns that more educational research could not produce relevant studies to benefit classroom teaching (Kilpatrick, 1992). Researchers were encouraged to consider the practitioner’s point of view (Sowder, 1989). Deficiencies in traditional behaviorism became more apparent in the 1970s, and as the field of cognitive study became popular in mathematics education and educational psychology, researchers published a number of cognitive studies (De Corte, 2010; Lester & Lambdin, 2003; McLeod, 2003). Research in the 1950s and 1960s examined only one aspect of a teaching session at a time (e.g., time allocation on various lesson activities); however, research in the 1970s was marked by several classroom observations of teacher and student behaviors that yielded detailed results about instruction (Koehler & Grouws, 1992). Often called processproduct research (measures of teacher behaviorprocess and measures of student achievementproduct), the methodology documented behaviors such as the frequency of teacher and student interactions and sometimes included the types of examples and questions posed by the teacher, the length of students’ responses, and the amount of practice and review in the lesson, among other variables (Koehler & Grouws, 1992). Student achievement outcomes were correlated to the frequencies of observed teacher behaviors to see which might result in performance gains (Koehler & Grouws, 1992). The assumption was that teacher behaviors influenced student behaviors, and the quality of teaching was based more so on the frequency of particular behaviors than on the quality of the teaching methodology itself (Koehler & Grouws, 1992). 50 However, Doyle (1975) addressed what he called the failure of teacher effectiveness studies, and he stated that the processproduct paradigm assumed that teacher effects on student achievement were stable and generalizable; however, a lack of evidence supported either assumption (Doyle, 1975). Doyle concluded that little reason existed to expect teacher behavior variables as being strongly related to student achievement gains. He was also concerned about the neglect of student behaviors in processproduct studies. Doyle later focused on the importance of studentlearning tasks in the lesson and the role these tasks played in influencing outcomes. According to Schoenfeld (1992): By the end of the 1970s, it became clear that the backtobasics movement was a failure. A decade focused on rote mechanical skills produced students who performed dismally on measures of thinking and problem solving. Further, they were no better at basics than students being taught a reform curriculum. (p. 336) Educational researchers were viewed as unable to demonstrate that they had made advances that could be translated into practical benefits for the classroom (Kilpatrick, 1992). Unhappy with aspects of backtobasics, leading math educators planned reforms in different directions (Fey & Graeber, 2003). Despite continued reform efforts, by the end of the 1970s, the math pedagogical pendulum had swung once again toward traditionalist teaching via direct instruction. 1980s: The Standards Movement Begins FerriniMundy and Graham (2003) reported that the 1980s began with a number of reports that highlighted concerns and provided potential pathways for the continued improvement of science and mathematics education. Research on mathematics teaching and teacher education prior to the 1980s was sparse, and few studies addressed high school algebra or 51 the role of the algebra teacher in classroom instruction (Booth, 1989; Cooney, 1980; FerriniMundy & Graham, 2003; Kieran, 1992). Graubard (1981) addressed problems of American schools at that time and stated that it might be beneficial for educators to acknowledge that the early 1980s were the first days of educational research. In fact, researchers characterized the mathematics educational research effort as “still in its infancy” (Brophy, 1986, p. 125; Cooney, 1994, p. 613). The disparity between theoretical research and onsight classroom research had also recently come to light. Kilpatrick (1992) stated, The 1980s began with the promise of a more fruitful integration of research and practice than at any previous time in the history of mathematics education. More and more, research in mathematics education was moving out of the library and laboratory, and into the classroom and school. (p. 31) Although there was consensus that the quality of mathematics education was deteriorating, advocacy for reform grew slowly in the 1970s and 1980s (Klein, 2003; McLeod, 2003). After the failure of reforms in the 1960s and 1970s, a continuing issue for NCTM leaders was convincing the public of the necessity for change. Although politically passive in previous decades, the organization took a more active role in 1980 with An Agenda for Action: Recommendations for School Mathematics of the 1980s (Fey & Graeber, 2003; NCTM, 1980; McLeod, 2003). The agenda made recommendations, among others, that problemsolving should be the focus of school mathematics, that “basic skills” should be defined more extensively than calculation, and that teachers should decrease the emphasis on isolated drill and practice (FerriniMundy & Graham, 2003; Fey & Graeber, 2003; McLeod, 2003). Together with the clear failure of backtobasics, agenda recommendations prompted a philosophical move toward problem solving, which became the theme of the 1980s (Schoenfeld, 1992). 52 The 1981 NCTM study, Priorities in School Mathematics, reported that nearly 60% of mathematics teachers in the United States believed that more than half of classroom instructional time should be spent on drill and practice, a clear indication that math teachers were still using memorization and drill techniques, and teaching procedural skills rather than practicing pedagogy to develop conceptual understanding (NCTM, 1981). According to Kieran (1992), research findings in algebra instruction indicated that algebra teachers, like their counterparts in other math classes, made their first priority classroom management and covering the curriculum. Two critical reports in 1983, Educating Americans for the 21st Century by the National Science Board Commission (NSBC) on Precollege Education in Mathematics, Science and Technology of the NSF and A Nation at Risk: The Imperative for Educational Reform by the National Commission on Excellence in Education (NCEE) described mathematics and science education in the United States as being in crisis (Fey & Graeber, 2003; NCEE, 1983; NSBC, 1983). Fey and Graeber (2003) stated, “Those policy documents marked the beginning of a period of intense study and reform activity that continued through the end of the century” (p. 553554). The instinct of educational leaders, among others, for progressive reform resurfaced prominently in response to publicized descriptions of mathematics and science education being in deplorable states, particularly because nearly a decade of declining test scores was evidence that backtobasics had failed (FerriniMundy & Graham, 2003; Fey & Graeber, 2003). The fervor for reform after Sputnik was based on an urgent need related to scientific readiness, while that in the early 1980s was linked to public concern regarding national economic and technical growth (FerriniMundy & Graham, 2003). The goal changed from creating a scientifically elite group of students to educating everyone, and the “mathematics for all” agenda was born (Ferrini53 Mundy & Graham, 2003). This concern was the precursor to the next era of publicized educational goals and recommendations, and a new educational movement began that was not unlike that seen during the new math era of the 1960s (FerriniMundy & Graham, 2003). The education community and the public at large widely cite and discuss A Nation at Risk (NCEE, 1983). This report declared that education in the United States was in a state of mediocrity, and in terms of mathematics, the report listed declining achievement scores, lack of knowledge and skills of graduating seniors, and the need for remedial mathematics courses in higher education venues as evidence for the crisis (NCEE, 1983). In response to A Nation at Risk and to provide benchmarks for then mathematics community, mathematics education leaders called for the development of mathematics standards for all school grade levels that included standards for curriculum, teaching, and evaluation (McLeod, 2003). To reduce the scope, teaching standards were postponed, and work began in the mid1980s on another publication that would provide a guide for educators on what students should know and be able to do at each grade level (McLeod, 2003). The NCTM Curriculum and Evaluation Standards published in 1989 and built on recommendations of the Agenda for Action published nearly a decade earlier, was recognized as a significant contributor to mathematics reform, both in and outside the math community. This report was also hailed as a symbol of education reform in general (McLeod, 2003; NCTM, 1989). The NCTM Agenda for Action focused on math as a personally constructed, internal set of knowledge (Clements, 2003; Dossey, 1992), and researchers in the 1980s explored teacher and student behaviors in the classroom from a constructivist viewpoint. According to Lester (1982), “During the past decade there had been increasing interaction among those interested in cognitive research and those interested in mathematics instruction (Lester, 1982)” (Mayer, 1985, 54 p. 124). Math educators themselves were increasingly interested in theories of cognitive psychology. Silver (1985) stated, “Thus the moment seems opportune for the mathematics education and cognitive science communities to benefit from one another” (p. vii). Reformers and researchers began discussing constructivist view of learners during the 1980s (McLeod, 2003). The constructivist viewpoint was a link between research in teaching and research in learning (Koehler & Grouws, 1992). Koehler and Grouws (1992) stated that the constructivist view advocated that teachers no longer prepared lessons or activities in a way for students to receive knowledge, but in a way for students to engage in mathematical problem solving. Rather than looking at teacherdirected instruction in the classroom, the constructivist theory of learning focused attention on the learner (Brooks & Brooks, 1999). Resnick and Ford (1981) stated that a fundamental assumption of cognitive learning psychology is that students do not simply add new ideas to their storehouses, but that new information must be attached to existing structures and new relationships constructed among those structures. Koehler and Grouws (1992) stated, “In the constructivist approach, teaching behavior is examined from the viewpoint of how much it encourages or facilitates learner construction of knowledge” (p. 123). Thus, teaching was considered a means to facilitate students by providing appropriate math activities and engaging students in mathematical discussion regarding the various aspects of working through the activities (Koehler & Grouws, 1992). In addition, social interactions were identified as a critical part of knowledge construction (Koehler & Grouws, 1992; Schoenfeld, 1992). Reform teachers served in a role of “coexplorers” with students, and in that role, they were to “ask more openended questions, engage in more problemposing, and be less tied to the textbook” (Koehler & Grouws, 1992). Yackel, Cobb, Wood, Wheatley, and Merkel (1990) pointed out that when students are given 55 opportunities in a constructivist classroom to interact with oneanother and with the teacher, they can “verbalize their thinking, explain or justify their solutions, and ask for clarifications” (p. 19). Consistent with the transition to a cognitive approach, according to research publications as well as the agenda, mathematics knowledge was equated with “doing,” a result of activities in which students participated, and during which the teacher functioned as a guide rather than a dispenser of knowledge (Dossey, 1992). Stein et al. (1996) stated, “Increased emphasis is being placed not only on students’ capacity to understand the substance of mathematics but also on their capacity to ‘do mathematics’” (p. 456). The recommendation was that students should experience the practice of math by experimenting, conjecturing, discovering, and generalizing in the process of learning mathematics, not by receiving a welldeveloped communication of content (Dossey, 1992). Schoenfeld (1988) spoke to the interaction of cognitive and social factors in school that contribute to the perception of mathematics by students. He stated that for math to “make sense,” students must do math by analyzing, hypothesizing, conjecturing, and synthesizing, among other behaviors. They must practice the habits of thinking like a mathematician as Bruner argued in 1960. By the mid1980s, researchers considered the constructivist perspective as wellgrounded and widely accepted (Schoenfeld, 1992). Researchers began challenging decades of earlier research and constructivism and examining student thinking led to a focus on students’ classroom work behaviors. Doyle (1988) stated that descriptions of the instructional methods used, the frequencies of a teacher’s behavior, or student timeontask provided an incomplete account of student learning. He stated, “What is missing is a description of the work students were required to do. This missing element is important because work creates a context for students to interpret information during class sessions and to think about subject matter” (p. 168). 56 Kilpatrick (1985) stated, “Something the last 25 years have given us is the math problem as task” (p. 3). He also spoke to the developing perspectives in the mid1980s of the mathematical problem, one of which was a task given to students in the social context of the classroom. He summarized, “We do not have a final version of what problem solving is and how to teach it, but we are much more keenly aware of the complexity of both” (p. 13). Doyle (1983) proposed, “Tasks influence learners by directing their attention to particular aspects of content and by specifying ways of processing information” (p. 161). Doyle (1988) later stated, “A fundamental premise of this theory is that the work students do, which is defined in large measure by the tasks teachers assign, determines how they think about a curriculum domain and come to understand its meaning” (p. 167). In the summer of 1989, the first draft of the Professional Teaching Standards was developed (McLeod, 2003; NCTM, 1989). During the process, one writing team produced teaching vignettes to better illustrate significant aspects of teaching, a seeming precursor of future instructional research in mathematics. From those vignettes came the first section of the NCTM standards that discussed classroom tasks and classroom discourse of mathematical concepts (McLeod, 2003). The pedagogical pendulum reached its traditional instructional zenith and moved somewhat rapidly toward standardsbased instruction. 1990s to the New Century: Debate Leads to War Research on mathematics teacher education and pedagogy, as well as the standards movement as a whole, grew, expanded, and gained momentum during the 1990s (Clements, 2003; FerriniMundy & Graham, 2003). Specifically, stakeholders in mathematics education, including mathematicians, professors of teacher education, and the public, among others, focused on both mathematics teachers’ education and pedagogy (FerriniMundy & Graham, 2003). 57 According to Ferrini Mundy and Graham (2003), “Questions about teacher knowledge became more visible, and the field struggled to find effective ways to help teachers be effective in their practice” (p. 1289). Addressing mathematics pedagogy, Klein (2003) pointed out that the American education establishment had long supported and promoted a progressive education agenda. He stated, “Throughout the 20th century the professional students of education have militated for childcentered discovery learning and against systematic practice and teacher directed instruction” (p. 2). During the last two decades of the century, the ongoing conflict between “telling” and “facilitating discovery” (Fey & Graeber, 2003, p. 526) was ever present in the minds of reformers. FerriniMundy and Graham (2003) stated, “In the mid1990s, the debate intensified” (p. 1271). Davis (1992) provided an indepth description of the stances taken by both camps: Those who defend the assumption that we must tell students how to solve mathematics problems usually argue that if we do not do so, some (or many) students will be lost and will quickly become demoralized. By contrast, those who argue against this assumption usually claim that teaching based on it sends students the message that nobody can solve a math problem unless someone tells them how to do it. Hence, students quickly give up the habit of trying to think for themselves, and they adopt the strategy of merely trying to remember what the teacher has said. Anyone who interviews students extensively will find many who say, at least in effect, ‘I couldn’t possibly do this problem, because you haven’t told me how to do it.’ The opponents of ‘telling’ argue, first, that this attitude on the part of students renders them almost incapable of making good progress in mathematics, and second, that the attitude is not inevitable, but rather was taught to the students by implicit messages repeatedly sent by teachers and textbooks. (p. 725) 58 The 1989 NCTM publication, Curriculum and Evaluation Standards for School Mathematics, was an additional catalyst for progressive change in mathematics education (FerriniMundy & Graham, 2003, NCTM, 1989). Continuing its pivotal role in the standards movement, NCTM published Professional Standards for Teaching Mathematics in 1991. Although the publication did not have the same impact as the Curriculum and Evaluation Standards, it offered recommendations on teaching methodologies to implement the goals of the 1989 standards, and the NCTM considered it a major contributor to the reform movement (Clements, 2003; FerriniMundy & Graham, 2003; McLeod, 2003). FerriniMundy and Graham (2003) stated, “In particular, the pedagogical recommendations were tied directly to the various aspects of a teacher’s role in preparing and implementing mathematics instruction” (p. 1291). The innovative teaching recommendations emphasized problem solving and application, a definite change from traditional curriculum, and included choosing appropriate mathematics tasks to be used in a lesson and facilitating classroom discussion of mathematical concepts as the lesson was being studied (FerriniMundy & Graham, 2003; McLeod, 2003). Recommendations also called for reducing the procedural emphasis of past years such as paper and pencil computation, and increasing opportunities for students to make and defend deductive arguments like those use by mathematicians, which was reminiscent Bruner’s work in the 1960s (as cited in McLeod, 2003). The standards also recommended more focus on realworld problems. Although the recommendation was never to focus on only realworld problems, the standards were often misinterpreted to mean that (McLeod, 2003). A popular slogan of progressive reformers who favored reform strategies was that the teacher should be a “guide on the side and not a sage on the stage” (Klein, 2003, p. 2). 59 In the early 1980s, researchers had argued against “strict drill practice” in the classroom as “an influence that habituates students to an unthinking response” (Resnick & Ford, 1981, p. 19). Those who defended traditional instruction supported the role of drill and practice, which was still a contentious topic among teachers and one that many, if not most, educators considered vital in teaching and learning math at all levels (Clements, 2003). However, in the 1990s, support of the importance of conceptual teaching and learning grew rapidly (Clements, 2003; Fey & Graeber, 2003; McLeod, 2003). Hiebert and Carpenter (1992) stated, In summary, research efforts now being directed toward uncovering relationships between conceptual and procedural knowledge now appear to be more useful than earlier attempts to establish the importance of one over the other. At this point, both theory and available data favor stressing understanding before skill proficiency. (p. 79) The idea and practice of discovery teaching and learning was born from the educational theory of constructivism (Brooks & Brooks, 1999; Lambert et al., 2002). In fact, constructivism was the driving force of the standards movement and a prominent theme in mathematics education in the 1990s (Clements, 2003). Hiebert and Carpenter (1992) stated, “It is now well accepted that students construct their own mathematical knowledge rather than receiving it finished from the teacher or a textbook” (p. 74). They added that it seemed evident from research that procedures and concepts should not be taught as isolated bits of information. Further, they identified the goal as “an attempt to teach students to make the same kinds of connections observed in experts” (Hiebert & Carpenter, 1992, p. 81). Rather than taking a singular focus, as in previous studies, researchers in the 1990s focused on pairing research on teaching with research on learning (Koehler & Grouws, 1992). Consistent with the themes of constructivism, Koehler and Grouws (1992) identified research 60 from the 1980s and early 1990s as a model whose outcomes of learning were based on students’ personal actions or behaviors. They advocated that mathematics teaching and learning research perspectives in the 1990s varied. Koehler and Grouws reviewed constructivist theories and compared and contrasted those of several wellknown researchers, including Leinhardt (1989), Lampert (1990), Hiebert and Wearne (1988), and Cobb et al. (1991). They revealed important similarities and differences among the models. However, in conclusion, they stated, “All these perspectives accepted the premise that students are not passive absorbers of information, but rather have an active part in the acquisition of knowledge and strategies” (Koehler & Grouws, 1992, p. 123). Whatever the means by which students construct knowledge, Koehler and Grouws (1992) believed, “In most cases, these actions are influenced largely by what the teacher does or says within the classroom” (p. 117118). The strong support for innovative teaching standards at the beginning of the 1990s and the research published in the years that followed—all advocating reformed classrooms—had unexpected outcomes, and in fact, the standards publications became the focus of significant controversy (FerriniMundy & Graham, 2003). Clements (2003) stated that NCTM called for reformoriented teachers to create constructivists classrooms; that was the rhetoric, but the reality was quite different. Teachers were often unaware of recommendations for standardsbased curriculum and instruction or they chose not to implement them (Clements, 2003). McLeod stated that both advocates and detractors of reform quickly claimed that reform policies of the NCTM standa
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Title  The Impact of Mathematics Education Research and BrainLearning Research on Student Performance in Algebra I 
Author  Malone, Camille 
Subject  Educational administration 
Abstract  THE IMPACT OF MATHEMATICS EDUCATION RESEARCH AND BRAINLEARNING RESEARCH ON STUDENT PERFORMANCE IN ALGEBRA I A Dissertation by CAMILLE MALONE Submitted to the Office of Graduate Studies Texas A&M UniversityCommerce in partial fulfillment of the requirements for the degree of DOCTOR OF EDUCATION August 2015THE IMPACT OF MATHEMATICS EDUCATION RESEARCH AND BRAINLEARNING RESEARCH ON STUDENT PERFORMANCE IN ALGEBRA I A Dissertation by CAMILLE MALONE Approved by: Advisor: Chuck Holt Committee: Art Borgemenke Katy Denson Head of Department: Glenda Holland Dean of the College: Timothy Letzring Dean of Graduate Studies: Arlene Horne iii Copyright © 2015 Camille Maloneiv ABSTRACT THE IMPACT OF MATHEMATICS EDUCATION RESEARCH AND BRAINLEARNING RESEARCH ON STUDENT PERFORMANCE IN ALGEBRA I Camille Malone, EdD Texas A&M UniversityCommerce, 2015 Advisor: Chuck Holt, EdD The first course in high school algebra is called a “gateway course” because of its importance for success in future mathematics coursework and for college and career opportunities. Unfortunately, student achievement in secondary mathematics in the United States, and in Algebra I in particular, is described as mediocre. However, a large gap exists between mathematics education research and brainlearning research and classroom instructional practice. The researchers conducted an ex post facto study on the implementation of a researchbased Algebra I curriculum built around highcognitive demand tasks and student performance in Algebra I. Two groups of students were selected for the study. One group enrolled in an Algebra I course that followed a typical textbookdriven curriculum and the second group enrolled in a course that followed a researchbased curriculum with highlevel tasks. The researcher used propensity score matching and multilevel modeling to compare the effects of the two curricula to determine whether student growth occurred. No significant difference existed in student performance between the traditional and researchbased curricula. Additionally, African v American and Hispanic student performance decreased and White student performance increased Grade 8 to Grade 9. However, differences were not significant for ethnicity between students using a researchbased Algebra I curriculum and those using a traditional curriculum. vi ACKNOWLEDGEMENTS I am grateful to Dr. Chuck Holt for his support and consistent encouragement. I am especially appreciative of the patience and guidance of Dr. Katy Denson, a teacher par excellence. vii DEDICATION This study is dedicated to the men and women of the Apollo Algebra I Project Accomplished Mathematicians Extraordinary Curriculum Writers & Dedicated Educators viii TABLE OF CONTENTS LIST OF TABLES ...................................................................................................................... xii CHAPTER 1. INTRODUCTION ......................................................................................................... 1 Student Performance in Algebra ............................................................................. 2 Attempts to Address the Problem ............................................................... 5 Mathematics Education Instructional Research: The Importance of Tasks 5 BrainLearning Research ............................................................................ 6 ResearchBased Algebra Curriculum ......................................................... 7 Consequences of Accountability ................................................................. 8 Statement of the Problem ........................................................................................ 9 Purpose of the Study ............................................................................................. 10 Research Questions ............................................................................................... 10 Null Hypotheses .................................................................................................... 11 Significance of the Study ...................................................................................... 11 Method of Procedure............................................................................................. 15 Selection of Sample .................................................................................. 16 Collection of Data ..................................................................................... 16 Treatment of Data ..................................................................................... 17 Definitions of Terms ............................................................................................. 18 Limitations and Delimitations ............................................................................... 19 Limitations ................................................................................................ 19 Delimitations ............................................................................................. 19 ix CHAPTER Assumptions .......................................................................................................... 20 Organization of Dissertation Chapters .................................................................. 20 2. LITERATURE REVIEW ............................................................................................ 21 The Tug of War Begins......................................................................................... 23 1900–1930: Expansion and Progressivism ........................................................... 27 1930–1950: The Great Depression and World War II .......................................... 31 1950s: Influence of Sputnik .................................................................................. 37 1960s: The New Math Era .................................................................................... 41 1970s: Back to the Basics ..................................................................................... 45 1980s: The Standards Movement Begins ............................................................. 50 1990s to the New Century: Debate Leads to War................................................. 56 The Math Wars ..................................................................................................... 61 A New Century ..................................................................................................... 64 Brain Research ...................................................................................................... 67 Summary ............................................................................................................... 69 3. METHOD OF PROCEDURE...................................................................................... 71 Research Questions ............................................................................................... 72 Null Hypotheses .................................................................................................... 72 Research Design.................................................................................................... 73 Propensity Score Matching ................................................................................... 75 Multilevel Modeling ............................................................................................. 77 Instrumentation ..................................................................................................... 78 x CHAPTER Reliability .................................................................................................. 79 Validity ..................................................................................................... 79 Sample Selection ................................................................................................... 80 Data Gathering ...................................................................................................... 81 Treatment of the Data ........................................................................................... 82 Propensity Score Matching ....................................................................... 82 Multilevel Modeling ................................................................................. 83 Research Question 1 ................................................................................. 83 Research Question 2 ................................................................................. 84 Summary ............................................................................................................... 84 4. PRESENTATION OF FINDINGS .............................................................................. 86 Construction of Data Set for Analysis .................................................................. 86 Multilevel Modeling ............................................................................................. 88 Model Levels ............................................................................................ 89 Descriptive Statistics for TAKS Math ...................................................... 90 No Predictors Model ................................................................................. 90 Model with Growth Rate .......................................................................... 92 Model with TimeVarying Treatment ....................................................... 93 Model with All Variables.......................................................................... 95 Summary of Model Variance ................................................................................ 97 Research Questions ............................................................................................... 99 Research Question 1 ................................................................................. 99 Research Question 2 ............................................................................... 100 xi CHAPTER 5. SUMMARY OF THE STUDY AND THE FINDINGS, CONCLUSIONS, IMPLICATIONS, AND RECOMMENDATIONS FOR FUTURE RESEARCH ......... 101 Summary and Findings ....................................................................................... 102 Research Question 1 ............................................................................... 103 Research Question 2 ............................................................................... 103 Conclusions ......................................................................................................... 104 Implications......................................................................................................... 107 Recommendations for Future Research .............................................................. 108 REFERENCES ........................................................................................................................... 111 VITA .......................................................................................................................................... 134 xii LIST OF TABLES TABLE 1. Student Characteristics of NinthGrade Algebra Students in Original and Matched Samples ............................................................................................................................. 88 2. TAKS Math Statistics for Each Measurement Occasion .................................................. 90 3. Null Model Estimates of Covariance Parameters – TAKS Math ...................................... 91 4. No Predictors Model Estimates of Fixed Effects .............................................................. 91 5. Estimates of Covariance Parameters: Growth Rate Model ............................................... 92 6. Estimates of Fixed Effects for Growth Rate Model .......................................................... 93 7. Estimates of Covariance Parameters: TimeVarying Treatment Model ........................... 94 8. Estimates of Fixed Effects for TimeVarying Treatment Model ...................................... 95 9. Estimates of Covariance Parameters: All Variables Model .............................................. 96 10. Estimates of Fixed Effects for All Variables Model ....................................................... 97 11. Percent of Variance Explained by Time, Model, and Source ......................................... 99 1 Chapter1 INTRODUCTION Educators and educational researchers, among others, commonly call high school algebra a gateway course for several reasons. The RAND Mathematics Study Panel reported that algebra proficiency primarily provides entrance into higherlevel secondary mathematics coursework (Educational Testing Service [ETS], 2009; Matthews & Farmer, 2008; National Council of Teachers of Mathematics [NCTM], 1989; RAND, 2003; U.S. Department of Education [DOE], 1997). According to the University of NebraskaLincoln (n.d.), “Students who successfully complete Algebra I often continue to pursue the study of high school mathematics that prepares them for college, while students who are unsuccessful in Algebra I find their path to success blocked” (p. 1). Additionally, the DOE (1997) stated that students who take algebra and geometry in high school attend college at much higher rates than those who do not take these courses. In fact, lowincome students who take algebra and geometry are almost three times as likely to go to college (ETS, 2009). Those who master algebra and attain higher levels of mathematics coursework also tend to have access to more and higher quality career opportunities (Stapel, 2009). According to RAND (2003), “Without proficiency in algebra, students cannot access a full range of educational and career options, and they have limited chances of success” (p. 47). RAND also reported that algebra is a gatekeeper that provides or denies access to academic and career success for all ethnic groups and socioeconomic statuses (SES) of society, which affects national culture as a whole. Lubienski (2007) stated, “Mathematics achievement is particularly important to our efforts to promote equity because it serves as a gatekeeper to highstatus occupations and 2 can provide a powerful ladder of mobility for lowSES students” (p. 55). The RAND report (2003) summarized: Lack of success in mathematics has significant consequences. Algebra, for example plays a significant gatekeeping role in determining who will have access to college and certain career opportunities. The ‘gates’ tend to be closed to the less advantaged, either by default—when the schools they attend simply do not offer advanced mathematics courses—or by discrimination—when low expectations for student performance lead to educational tracking that differentiates among students and therefore further limits students’ opportunities to develop math proficiency. (p. 11) Limitations of career opportunities for any section of the population have significant implications. RAND (2003) noted, Failure to learn algebra is widespread, and the consequences of this failure are that far too many students are disenfranchised. This curtailment of opportunity falls most directly upon groups that are already disadvantaged and exacerbates existing inequalities in our society. (p. xx) Regarding African American students, Moses and Cobb (2001) stated, “In similar fashion we believe that organizing around math literacy opens another path on which people can begin to transform their lives” (p. 5). They advocated for algebra as “the new civil right” (p. 5) that should be accessible to everyone. Student Performance in Algebra Because of its importance as a gatekeeper course, algebra is a requirement for many high schools in the United States, and in most, students must demonstrate proficiency in algebra to graduate (RAND, 2003). Unfortunately, poor student performance in algebra and secondary 3 mathematics in the United States is a significant national concern (Schoenfeld, 1992) expressed among teachers, parents, educational administrators, and national commentators that has persisted for several decades (RAND, 2003). Specifically, national and international assessments have indicated consistent poor math performance in upper grade levels (Haycock, 2002). In 1989, the National Research Council (NRC) reported that mathematics achievement of U.S. students was well below what was necessary to “sustain our nation’s leadership in a global technological society” (p. 1). In 1990, state governors nationwide joined then President Bush in establishing national educational goals. The group’s collective announcement echoed across the nation: “By the year 2000, United States students will be first in the world in mathematics and science achievement” (Haycock, 2002, p. 3). Since that time, the nation has received few encouraging reports. For example, in the 1990s, results of the National Assessment of Educational Progress (NAEP), commonly called the Nation’s Report Card, reported that elementary, middle, and high school students of every racial and ethnic group knew more mathematics than did students in the 1970s. In 2000, the College Board reported gains on the quantitative portion of the Scholastic Aptitude Test (SAT). However, in the early 2000s, results of the main NAEP report, which provides both national and state data, indicated that after steady improvement through 1996, high school mathematics performance declined from 1996 to 2000, and earlier improvements in Grade 12 were attributed to improvements during the elementary years (Haycock, 2002). In the latter part of the 1990s, high schools received students who were better prepared in mathematics; however, their high school coursework added less value to their ultimate levels of mathematics achievement (Haycock, 2002). Haycock (2002) reported that 35% of U.S. graduates left high school unable to perform basic math skills. She added, “While many of these 4 students have mastered basic computation, they cannot find the perimeter of a figure; their conceptual understanding of mathematics is limited; and they are unable to consistently see mathematical relationships” (p. 6). Further, analyses of other assessments, including the Third International Math Science Study (TIMSS), also indicated poor mathematics progress. Achievement levels on the TIMSS in the 1990s showed that by 12th grade when American students would have completed algebra, even those in the top 10%15% fell below their peers from 11 participating countries on international rankings (Haycock, 2002). It appeared that while American students were progressing, their counterparts in other countries were progressing faster and achieving more. In addition, gaps continued to exist between White students and those identified as lowincome, African American, and Latino (Haycock, 2002). According to Silver (1998), “In general the TIMSS results indicate a pervasive and intolerable mediocrity in mathematics teaching and learning in the middle grades and beyond” (p. 1). During the 1970s and 1980s, the United States made considerable progress in raising mathematics achievement of minority and lowSES students. Between 1973 and 1986, the gap between White and African American students in eighth grade narrowed by half, from 48 to 25 points. Additionally, the gap between White and Latino students narrowed from 35 to 20 points (Haycock, 2002). However, gaps increased in the 1990s, and at the turn of the century, African American and Latino students in 12th grade had math skills equivalent to White students in eighth grade (Haycock, 2002). In 2001, the Organization for Economic Cooperation and Development (OECD) published the results of an assessment administered to 250,000 15yearolds in 32 countries. American students ranked 19th among participating countries in achievement in reading, 5 mathematics, and science, and were relatively high on the size of achievement gaps between high and lowperforming students (OECD, 2001). Based on the ACT (2005) national readiness standards, only 40% of high school graduates in the U.S. are ready for college algebra. Attempts to Address the Problem The importance of algebra in secondary school curricula and the consistently poor performance of U.S. students in mathematics, particularly in algebra, has fueled persistent and urgent attempts to address the problem. Algebra textbooks, longused as the curriculum in most schools across the United States, have been transformed from the drab black and white publications of the mid20th century that had few or no illustrations, to bright and colorful resources with content aligned to NCTM standards (Donoghue, 2003). Teachers have relied on the enriched textbooks to assist in presenting mathematics content with narratives that place the student in the problem situation for relevancy, provide multiple examples with detailed stepbystep solutions, and include an abundance of practice for state examinations (Donoghue, 2003). Even with these changes, recent achievement results indicate that textbook improvements and other innovative strategies, including computer programmatic remedies, have had little effect. However, a significant amount of research published in recent decades may hold promise for greater success in teaching and learning algebra. Mathematics Education Instructional Research: The Importance of Tasks Cognitive scientists and educational researchers have published significant findings related to teaching and learning mathematics in the classroom, and many findings and recommendations focus on teaching strategies that call for student engagement in cognitively demanding mathematical tasks. Stein, Grover, and Henningsen (1996) stated that students must engage in mathematical thinking by doing mathematics through problemsolving tasks. 6 Henningsen and Stein (1997) described factors that support high levels of mathematical thinking or reasoning, including classroom activities that involve challenging problems. The NCTM (1991) recommended that to develop highlevel thinking in math classrooms, students must be provided opportunities to engage in dynamic, rich, and worthwhile mathematical tasks. BrainLearning Research Lakomski (2000) described the 21st century as “the century of the brain,” particularly because learning and brain research has characterized both educational and medical study landscapes. Brainlearning research since the late 1980s has grown exponentially (Caine & Caine, 2001). The study of how people learn as an outgrowth of neuroscience has significant instructional implications for the classroom (Bransford, Brown, & Cocking, 1999; Jensen, 2000; Willis, 2006). While skeptics feel that educators should ignore what they often called pseudoscience, the advent and use of medical technology has provided physiological support for the efficacy of certain strategies to facilitate learning (Bransford et al., 1999; Willis, 2006). Willis (2006) stated, Information obtained through brain imaging such as positron emission tomography (PET scans), functional magnetic resonance imaging (fMRI), and quantitative electroencephalography brain wave monitoring (qEEG) during the learning process have given us a science of education to add to our already powerful knowledge of the art of teaching. (p. vii) Many prominent brainlearning researchers have suggested specific pedagogy to ensure that content enters students’ longterm memories; however, such pedagogy is virtually unknown and untried in most educational settings (Caine & Caine, 2001; Erlauer, 2003; Jensen, 2000; Willis, 7 2006). According to Jensen (2008), “Brainbased education is the engagement of strategies based on principles derived from an understanding of the brain [and] this singular realization alone has fueled a massive and urgent movement worldwide to redesign learning” (p. 4). Caine and Caine (2001) synthesized the results of current brainlearning research by introducing learning principles. These principles included (a) the brain naturally searches for meaning that is best met by engagement in and making sense of mathematical tasks; (b) the brain responds to classroom activities that satisfy curiosity and hunger for novelty, discovery, and challenge; (c) the brain uses patterning as a way to search for meaning; and (d) the brain learns best through interactive experiences. Caine and Caine (1994) advocated for incorporating brainlearning principles into the curriculum. They stated, “All education can be enhanced when this type of embedding is adopted. That is the single most important element that the new brainbased theories of learning have in common” (p. 94). ResearchBased Algebra Curriculum In mathematics, particularly in Algebra I, a generalized description of curriculum and instruction that includes findings from cognitive science, mathematics education, and brainlearning research is best met by a curriculum and by teaching strategies that embody experiential learning. Caine and Caine (1994) stated, “One of the most important lessons to derive from brain research is that in a very important sense, all learning is experiential” (p. 113). Experiential learning is defined as that which occurs when the learner is actively engaged; therefore, curriculum based on mathematics and brainlearning research would include challenging, highthinking demand tasks experienced by students who practiced habits of mathematical thinking in group settings in the classroom (Caine & Caine, 2001; NCTM, 1989; Stein et al., 1996). However, rather than implementing this type of teaching pedagogy, school 8 districts have persisted in using rule memorization and skills practice, which are traditional teaching methods of past decades (Clements, 2003; Donoghue, 2003). Consequences of Accountability Although teachers and administrators may embrace the idea of teaching with rich, challenging algebra tasks, and they may envision classes where teachers provide more indepth and rigorous curriculum, they are hostages to state assessment systems (Boyd, 2008; Jones & Egley, 2007; Popham, 2007). Boyd (2008) stated, “With the passage of the Elementary and Secondary Education Act (ESEA) in 2002, all states implemented accountability systems to document student achievement in mathematics” (p. 1). Researchers have found that state tests have negatively influenced classroom teachers’ content instruction and assessment practices (Boyd, 2008; Nichols & Berliner, 2008; Popham, 2007). The penalties of low achievement have also caused resistance to innovation and have restricted students to classes of drill, repetition, and testtaking strategies (Popham, 2007; Wright, 2009). While students are taught how to work certain types of problems for future assessments, they have little opportunity to learn meaningful concepts (Abrams, Pedulla, & Madaus, 2003; Popham, 2007; Schmoker, 2009). Boyd (2008) reported that teachers react to tests in ways that contradict appropriate educational practices by spending more time (a) teaching to the test in a gamelike manner, (b) focusing on testtaking skills such as drills, (c) coaching for the test, and (d) practicing sample test items. In response to the pressures of accountability, assessmentlike practice problems only mimic authentic mathematics tasks that both cognitive mathematics and brainlearning research have advocated. However, educational administrators and teachers, among others, are reluctant to consider implementing an algebra curriculum with embedded tasks and brainlearning 9 recommendations without research findings that indicate whether such implementation supports an increase in algebra achievement. Statement of the Problem Educators in the United States understand the importance of and demand for increased algebra achievement, and they have responded by embracing a myriad of innovative, instructionalrelated resources, all with little effect on student outcomes (Schoenfeld, 1992). Concurrently, accountability systems that engender a culture of teaching to and for assessments have hijacked attempts to implement reformed, researchbased curriculum to teach algebra because educators are reluctant to follow recommendations for fear of unacceptable student scores (Schmoker, 2009). Adopting a researchbased curriculum to increase student conceptual understanding versus one that prepares students for accountability measures presents a conundrum for educators (Jones & Egley, 2007). Addressing recommendations for change, Davis (1992) stated, “If school systems are going to move far in this direction, however, research and development must create some reasonably explicit programs and demonstrate that they are capable of producing acceptable test scores” (p. 725). Chaikin (1989) noted that to address poor student achievement in algebra, research must be conducted on recommendations of cognitive and brainlearning research that includes measurements of student accountability outcomes. Mathematics education and brainlearning studies are at the forefront of educational research; however, literature on secondary mathematics education lacks information on the implementation of an algebra curriculum embedded with researchbased strategies. Limited research has been published on targeted brainlearning teaching strategies in mathematics that addresses specific content objectives. Additionally, Stein, Smith, Henningsen, and Silver (2009) 10 published an indepth treatise of case studies using mathematical tasks in middle school. However, to this researcher’s knowledge, no study has explored the relationship between the implementation of mathematics education using a brainlearning research algebra curriculum and student performance on state accountability measures. Purpose of the Study Secondary school administrators and mathematics teachers recognize the importance of firstyear algebra as a gateway mathematics course; however, low secondary mathematics achievement is problematic in all sectors (RAND, 2003). Further, educators are hostages to an accountability system that undermines the use of researchbased algebra curriculum and instruction in favor of testpreparation practices (Popham, 2007). The purpose of this study was to examine a meaningful, researchbased Algebra I curriculum and its implementation using cognitively demanding tasks to identify possible relationships to student performance as measured by state accountability assessments. Findings from this study will enable educators to make informed decisions regarding the use of researchbased curricula and teaching practices versus those that focus on test preparation. Research Questions The following research questions guided this study: 1. Does a statistically significant difference exist in the mathematical growth of students using a mathematics education and brainlearning researchbased Algebra I curriculum that includes implementation of highcognitive demand tasks compared to students using a traditional textbookdriven curriculum? 2. Does a statistically significant difference exist in the mathematical growth of students by ethnicity using a mathematics education and brainlearning researchbased 11 Algebra I curriculum that includes implementation of highcognitive demand tasks compared to students using a traditional textbookdriven curriculum? Null Hypotheses The researcher tested the following null hypotheses at the p < .05 level of significance. 1. No significant difference exists in the mathematical growth of students using a mathematics education and brainlearning researchbased Algebra I curriculum that includes implementation of highcognitive demand tasks compared to students using a traditional textbookdriven curriculum. 2. No significant difference exists in the mathematical growth of students by ethnicity using a mathematics education and brainlearning researchbased Algebra I curriculum that includes implementation of highcognitive demand tasks compared to students using a traditional textbookdriven curriculum. Significance of the Study Mastery of high school algebra is critical for students to pursue advanced mathematics coursework, to gain access to more and higher quality college and career opportunities, and to facilitate equity among all students in society (RAND, 2003). While teaching methodology in algebra has changed little, achievement rates of U.S. students in algebra have been disappointingly poor for decades (Schoenfeld, 1992). Almost from the inception of mathematics education in the United States, two distinct camps of teaching methodologies for secondary mathematics have existed (Fey & Graber, 2003). As early as 1850, proponents in one camp have advocated direct instruction, also called synthetic or rote learning. This type of learning most often using a textbook and is accompanied by memorization, drill, and practice (Cohen, 2003). The second camp has advocated inductive reasoning through discovery or experiential learning, 12 with or without a textbook, characterized by student behaviors applied to solving special problems designed to facilitate student conceptual discovery and learning (Cohen, 2003). The two camps continued stringent debate into and throughout the 20th century. Influences such as national socioeconomic conditions and World War II during the first half of the century precluded calls for more studentcentered instruction in favor of vocationally oriented courses in which students memorized and practiced basic skills (Angus & Mirel, 2003). The launch of Sputnik and a heightened focus on conceptual understanding were also catalysts for reform that somewhat influenced teaching methodology during the second half of the century (FerriniMundy & Graham, 2003). Although research increased and expanded during the last few decades of the century, which contributed significantly to the mathematics education literature, the debate culminated in the Math Wars toward the end of the century (Angus & Mirel, 2003; Klein, 2003). More recently, educational literature has explored new factors that may influence teaching practice. Mathematics education researchers have begun examining how the brain learns and have added new insight into the centuriesold debate over issues of mathematics teaching methodologies (Erickson, 2002; Hiebert et al., 1997). Cognitive psychology has also contributed research into constructivism as the means for learning. In the late 20th and early 21st centuries, mathematics educators turned their attention toward the importance of cognitively demanding mathematics tasks to construct knowledge (Doyle, 1988; Hiebert et al., 1997). Stein et al. (2009) published a casebook of classroom practices built around the use and maintenance of highlevel tasks. In response to the need for U.S. students to improve their use of cognitively demanding skills as specified by the American Institutes for Research (Ginsburg, Cooke, Leinwand, Noell, & Pollock, 2005), Stein et al. argued for more opportunities for 13 students to engage in “tasks that require them to reason and make sense of the mathematics they are learning” (p. 150). Further, over the past two decades, brainlearning researchers have advocated the importance of experiential learning and problem solving, particularly in social settings (Caine & Caine, 2001; Jensen, 2008). Recently, the role of educational administrator has undergone profound change. The expectations placed on superintendents are both more challenging and more complex. In addition to community and school board responsibilities, superintendents have the added role of personal involvement to improve district student performance (Lambert et al., 2002). In fact, research indicates that reforms at the school district level can occur only when senior leadership takes the initiative to understand and support the instructional design of the district (Resnick & Glennan, 2002). For example, members of the Connecticut Superintendents’ Network recognized that teaching and learning is “job number one” (City, Elmore, Fiarman, & Teitel, 2009, p. x). In fact, the first objective of these superintendents was to develop the knowledge and skills necessary to lead a districtwide instructional improvement effort (City et al., 2009). Further, site management reforms have added more responsibility and authority to school principals. Henson (2006) stated, During the 1980s and 1990s the level of involvement of school administrators in curriculum and instruction reached heights unprecedented since the days of the oneroom school. Research on effective schools has made educators aware of the need for administrators—particularly building principals—to be at the center of instructional and curriculum planning. (p. 23) In addition, emphasis has increase concerning the link between the role of the principal as an instructional leader and student achievement (Fink & Resnick, 2001, Lambert et al., 2002). 14 Schlechty (2005) extended the definition of principal leadership to include support for teachers as “designers of engaging work for students” (p. 18). Thus, it is particularly significant that the roles of superintendents and building principals include the responsibility of algebra programmatic reform in their districts and schools. Administrators, although interested in developing and applying researchbased curriculum and pedagogy, may be too entrenched in accountability systems to change classroom practice (Popham, 2007). Curriculum has been narrowed to focus on the primary goal of raising test averages, and classroom instruction is often given as decontextualized practice for assessments (Nichols & Berliner, 2008). Administrators need data on student achievement outcomes as a result of researchbased teaching versus teaching for test preparation; however, there is currently a gap in the literature in this regard. That is, to this researcher’s knowledge, no study has explored the benefits, or lack thereof, of standardsbased curriculum and instruction in a firstyear algebra course based on mathematics education and brainlearning research conducted over the past two decades (Van de Walle, Karp, & BayWilliams, 2010). This study aimed to fill the gap in mathematics education literature by exploring the implementation of an Algebra I curriculum, developed and built around current mathematics education and brainlearning research. The researcher examined indications of student achievement growth to provide administrators and teachers with data on which to base decisions regarding the best type of curriculum and classroom instruction to increase students’ conceptual understanding of algebra. The results of this study contribute to the literature by providing educators with student achievement data related to the implementation of a researchbased algebra program. It is of particular importance to current and future generations of students 15 enrolling in algebra that a definitive exploration of an innovative researchbased algebra course be added to the literature. Method of Procedure It is not feasible to explore differences in student performance by randomly selecting some students to receive instruction based on one type of Algebra I curriculum and others to serve as a control group with another type of curriculum. In fact, experimental studies on the effects of student achievement using particular curricular programs are problematic because of the necessity and ethical considerations of providing one student group with a curriculum that is purported to be better for conceptual learning and withholding it from another group. However, the districtwide implementation of a textbookdriven curriculum in one school year and the districtwide implementation of a researchbased curriculum the following year provided a unique and excellent opportunity to explore outcomes. Gall, Gall, and Borg (2003) described nonexperimental research as that in which researchers study phenomena as they already exist without intervention. Nonexperimental research that explores previously occurring phenomena is called ex post facto. In 2007–2008, a large metropolitan public school district taught Algebra I using a curriculum based on an adopted textbook. In 2008–2009, an Algebra I curriculum was designed, built, and implemented around current mathematics education and brainlearning research. The new curriculum, exclusive of a textbook, was characterized by cognitively demanding mathematics tasks that students solved in socialized settings held approximately once every week. Additional curriculum topics addressed and expanded the study of taskrelated concepts. The researchers examined the effects of this researchbased, taskembedded curriculum versus the typical, textbookdriven Algebra I curriculum on student achievement. The Grade 9 16 mathematics Texas Assessment of Knowledge and Skills (TAKS) was used to examine student growth. Selection of Sample The researcher selected two groups of students from the same school district as samples for the study. One group included Grade 9 students who studied Algebra I using the textbookdriven course and who took the Grade 9 mathematics TAKS assessment in 2008. The second group included Grade 9 students who studied Algebra I using the mathematics education and brainlearning researchbased curriculum (which included highlevel tasks) and who took the Grade 9 mathematics TAKS assessment in 2009. Class rosters of students, identified by masked student identification numbers, who were enrolled in Algebra I were matched with their teachers whose identities were also masked. Only teachers who taught Grade 9 Algebra I in both 2007–2008 and 2008–2009 were included in the study. Additionally, only students who were in Grade 9 for the first time, who had both Grade 8 and Grade 9 TAKS assessment scores, and who had a first semester grade for Grade 9 Algebra I, were included. Students identified as having limited English proficiency (LEP) or special education students who received instruction separately from the regular student population were not included in the study. Finally, students whose teachers were not included in the study; that is, teachers who had not taught Algebra I during both the 2007–2008 and 2008–2009 school years were excluded. Collection of Data Data were obtained from the school district evaluation and accountability department and included ethnicity, gender, home language, SES (based on free and reduced lunch status), and Grade 8 and Grade 9 mathematics TAKS scale scores for the samples of Grade 9 students 17 enrolled in Algebra I. For those students who took Algebra I in 2008, TAKS math scale scores were obtained for 2007 and 2008. For students who took Algebra I in 2009, TAKS math scale scores were obtained for 2008 and 2009. Treatment of Data The researcher built a dataset that included students who were in Grade 9 in 2007–2008 and 2008–2009 who met the qualifications for participation. The dataset included student demographic and academic variables. A Statistical Package for the Social Sciences (SPSS) propensity score matching (PSM) procedure (Thoemmes, 2009) was used to calculate a propensity score for each student, which indicated the probability of receiving the treatment (researchbased curriculum). Propensity scores were then matched using a 1:1 nearest neighbor matching logistic regression algorithm with a .25 caliper on the pretest covariates of gender, ethnicity, SES (based on free and reduced lunch status), home language, and the pretest Grade 8 mathematics TAKS scale score to create a comparable sample and force exact matching on the masked teacher identification number to eliminate teacher and school effects. After matching, 767 students in each group were identified. The researcher checked data for differences no larger than .25 standard deviations. Statistical analyses were then computed on the outcomes of the treatment (researchbased curriculum) and control (traditional curriculum) groups. Research questions were answered using multilevel modeling (MLM) to examine data that were hierarchically structured as students nested within teachers, using both between and withinstudents analyses to assess students in the same classrooms with the same teachers. The null model was assessed to ensure sufficient variation to warrant an MLM analysis (Heck, Thomas, & Tabata, 2014). The researcher added growth rate to the model. To answer Research Question 1, timevarying treatment effects were included in the model and analyzed to evaluate 18 differences in outcomes between the two groups. To answer Research Question 2, the researcher computed dummy variables for the ethnicities of African American, Hispanic, and White. The researcher assessed the influence of ethnicity, gender, and SES. Definitions of Terms Algebra I. Algebra I is one of the five NCTM (2000) content standards. Algebra, typically the first high school mathematics course taken by Grade 9 students, includes a basic understanding of foundation concepts of Grades K8, algebraic thinking and symbolic reasoning, function concepts, relations between equations and functions, tools for algebraic thinking, and underlying mathematics processes (Texas Education Agency [TEA], 2006). While Algebra I may be offered as an advanced course in earlier grades, this study measured only student performance of Grade 9 enrollees. Highlevel tasks. Highlevel demand or highcognitive demand tasks are defined as challenging instructional tasks (mathematical problems) that “demand engagement with concepts and that stimulate students to make purposeful connections to meaning or relevant mathematical ideas” (Stein et al., 2009, p. 11) rather than to perform a memorized procedure in a routine manner. Highcognitive demand tasks may be realworld scenario problems that involve mathematical concepts or actual physical, experimental problems that require handson mathematical manipulation, both from which students must use information to extrapolate data, make conjectures, and formulate answers to problems based on the information given. By definition, highlevel tasks have multiple entry points and multiple solution paths. Mathematics education and brainlearning researchbased algebra curriculum. Algebra I researchbased curriculum is defined by highcognitive demand tasks (Stein et al., 2009). In this study, such tasks were implemented approximately once per week and addressed 19 by student groups in socialized settings. Additional curriculum, reinforcing, and extending concepts were included in the entire Algebra I course, exclusive of a textbook. Limitations and Delimitations Limitations In the last decade, brainlearning research has provided educators with a rich, extensive source of strategies to improve learning in the classroom. While this study was limited to one of those important strategies, it was a strategy that undergirds all others. Specifically, the strategy used provided learner experiences that enhanced conceptual development and understanding of the content. For this study, the brainbased learning methodology of experience was defined as collaborative participation in solving highthinking demand mathematical tasks. While many brainlearning strategies exist that may positively affect learning, this study was limited to one strategy that was tightly defined as a problemsolving task used in a firstyear high school algebra course. As such, the following limitations applied to this study: 1. Students included only Grade 9 Algebra I students in a large metropolitan public school district. 2. Teachers included in this study were limited to those who had taught Algebra I during both the 2007–2008 and 2008–2009 school years. Delimitations Delimitations of this study included both the sample size and the instrument used to measure growth. The sample was drawn from a Grade 9 regular education students enrolled in Algebra I in a large district that included over 3500 students and resulted in a matched sample of 767 students in each group. The instrument used for measurement was the TAKS, which has been recognized for both reliability and validity. 20 Assumptions Mathematics education and braincompatible learning in algebra may be defined by several parameters based on current research. This study measured the effects of a curriculum that included periodic (approximately weekly) mathematics tasks experienced by students in the classroom. The tasks were followed by curriculum activities built upon and around the concepts embedded in the task experiences. The researcher conducted this study and collected, analyzed, and interpreted the data based on the assumption that the Algebra I curriculum was available to every algebra teacher in the school district. Organization of Dissertation Chapters In this study, the researcher sought to contribute to empirical research regarding the implementation of teaching strategies based on mathematics education and brainlearning research. Chapter 2 includes a historical review of the literature on mathematics teaching strategies and the pendulous pathway that changes in pedagogy have followed. Chapter 3 includes a description of the methodology employed in the study. Chapter 4 includes the results of the study, and Chapter 5 includes a summary of findings and recommendations for future research. 21 Chapter2 LITERATURE REVIEW The study was designed to measure the effects, if any, of a firstyear high school algebra I course implemented with a curriculum characterized by strategies based on mathematics education cognitive research and brainlearning research of student achievement. This chapter provides (a) a historical review of literature regarding instructional methodologies for teaching secondary mathematics, (b) the context of instructional research as applied to mathematics, (c) pertinent research findings in teaching and learning mathematics, and (d) recommendations based on research over the last two decades in teaching methodology in mathematics. For the most part, literature on teaching and learning mathematics does not distinguish algebra from any other mathematics subject; however, it does focus on the commonalities in teaching and learning of all mathematics courses (Kieran, 1992). Fey and Graeber (2003) provided the following description of the pendulous pathway of teaching pedagogy of high school mathematics, including algebra, in the United States: Looking at events in education over a long period of time is a fascinating opportunity to chart a kind of intellectual, social, and political tugofwar in which the perspectives and theories of individuals and groups compete for influence on the goals and practices of school mathematics. The story of this struggle over the direction of curriculum and teaching in elementary and secondary school mathematics has a predictable rhythm of crisisreformreaction episodes. A prominent social, political, or professional group calls attention to serious problems in students’ performance and recommends action, only to find that reform initiatives ultimately run up against resistance from opposing views and the deeply conservative nature of educational institutions. The burst of concern and 22 energy sparked by crisisandreform rhetoric often settles down to a quieter pattern of business as usual, at best moderately perturbed by the energetic calls for change in standard practices. (p.521) An examination of teaching methodology in mathematics is characterized not by decades but centuries of contention over direct instruction versus discovery learning. In fact, this study is grounded in relatively recent literature from the past halfcentury and historical and anecdotal evidence of pedagogical advocacies in school mathematics. Perhaps Pythagoras provided the earliest recorded evidence of direct instruction through discovered stone engravings of circles, secants and sectors, geometric illustrations, and formulaic etchings, all of which may be inferred to be teaching resources that ancient mathematicians and educators used for instructional demonstration. Geometric figures were drawn in sand or etched on stone to illustrate concepts to students, which demonstrates direct instruction. On the other hand, in terms of the discovery teaching methodology, Socrates offers a premier example. His method included asking his disciple students questions to lead them to discover the truths he wanted them to know. Both direct instruction and the Socratic method have survived centuries as practiced pedagogies, more or less successfully, that many still advocate. While no ancient research exists that describes empirical data measuring the most successful pedagogic strategies in which to ground a study of current mathematics teaching methodology, particularly algebra, such data and descriptive evidence certainly exist in the United States. This evidence dates back to the early 19th century from public schools established by the colonists (Burton, 1850; Cajori, 1890; Cohen, 1999, 2003). Evidence culminating just after the Math Wars in the late 1990s and early 2000s continues to be published today (Star, 2005; Starr, 2002). Well over a century of 23 philosophical skirmishes between advocates of direct instruction and discovery learning are welldocumented from the mid19th century, throughout the 20th century, and into the 21st century. While peace may currently exist in the math community, it is an uneasy peace (English, 2007; Grouws & Cebulla, 2000). The Tug of War Begins When public schools were established in the United States, there was little concern regarding teaching methods as basic curriculum delivered using textbooks (Cohen, 2003). Children were schooled as a form of moral discipline with a regimen of memorization and recitation necessary for learning religious truths as well as grammar and multiplication facts (Reese, 2008). Elementary schools in colonial America routinely provided instruction in both reading and basic numeracy (Reese, 2008). Cohen (2003) stated, Whether taught in New England’s rural district day schools or in the urban feeforservice evening schools dotted in Boston, Newport, New York, Philadelphia, and Charleston, arithmetic was regarded as a vocational subject, a skill whose chief application was in the world of commerce. The appropriate pupil for such study was the twelve to fourteenyearold boy, judged to be mature enough to absorb the arcane techniques of computation as well as sufficiently competent in writing to create a permanent copybook. (p. 44) Burton (1850) provided a detailed description of his experience in school, including his entrance into formal arithmetic at age 12. His description inadvertently illustrated the math teaching methods of the time. A popular textbook of the early 1800s, Arithmetik (Pike, 1809), was a memorybased book that was used as a resource to provide explanations of how to solve problems using arithmetic operations (Cohen, 1999, 2003). Burton described transcribing the 24 text, rules, and examples that illustrated each rule, wordforword, chapterbychapter from his printed textbook, Adam’s Arithmetic (a memorybased text much like Pike’s), into a copybook that he maintained. He also depicted his difficulty understanding a particular procedure. Burton described, “Carrying tens in addition,” which he called “a mystery which that arithmetical oracle, our schoolmaster, did not see fit to explain” (p. 113). The clarity of his account allows researchers to infer that the strategy employed by young Burton’s schoolmaster was to have students copy a textbook as the instructional means to teach mathematics. The schoolmaster also relied on the textbook as a resource to illustrate math principles and provide his students with examples of problems and their solutions (Burton, 1850). According to Burton (1850), he copied each rule in his notebook and the process by which the answer was found: “Each rule, moreover, was, or rather was to be committed to memory, word for word, which was to me the most tedious and difficult job of the whole” (p. 113). In fact, his teacher used the same methodology that had been employed in American colonial schools for over a century (Cohen, 2003). When Burton was in school, educators were already discussing teaching methodology in math (Cohen, 1999). After 1820, many educational proponents advocated inductive reasoning as the method to teach mathematics rather than use memorybased textbooks (Cohen, 1999). From the 1820s to the 1860s, wellknown educators widely debated the merits of inductive reasoning versus rote learning and memorization. Warren Colburn, having served as a schoolmaster, superintendent, and author, “wanted to end children’s slavish reliance on rules and rote learning” and he advocated that students should discover mathematical rules by working specifically selected examples (as cited in Cohen, 2003, p. 59). 25 For a time, teachers put aside memorybased texts and used inductive reasoning with specially crafted problems (Cohen, 1999). However, by 1834, a review of several new textbooks questioned whether induction was adaptive as a teaching methodology, and by the mid19th century, opponents of Colburn’s discovery learning method voiced strong protests (Cohen, 2003). In 1851, Taylor Lewis, a professor at Union College in New York who strongly opposed inductive or discovery learning, called the methodology “quackery” and a practice that would “enfeeble the mind” (as cited in Cohen, 2003, p. 64). Lewis (1851) favored synthetic or rote learning; that is, direct instruction from authoritative teachers who explained and demonstrated content using the examples they worked. Lewis (1851) was also adamant that the discovery method undermined authority because it fostered independence in students. In summary, he held fast to his bottom line: rote learning has its place in the math classroom. According to Cohen (2003), “From the 1850s forward, the scales tipped back toward synthetic instruction” (p.65). In fact, in 1870, a New Jersey school superintendent described inductive methodology as “math taught backward” (Cohen, 2003, p. 65). Direct instruction had become solidly entrenched as the method used to teach school mathematics, and Cohen (2003) stated, “By the early 20th century, Warren Colburn was barely a distant memory” (p. 65). At this time in the United States, algebra was a relatively young mathematics course compared to arithmetic. However, while Colburn and Lewis debated the efficacy of inductive reasoning through discovery versus synthetic rote learning, particularly in arithmetic instruction, secondary mathematics gained a foothold in American high schools (Boyer, 1968). In 1820, Harvard made algebra a requirement for admittance, which insured that algebra would become a common secondary mathematics course in the years that followed (Rachlin, 1989). By 1880, 26 algebra was indeed entrenched as the firstyear mathematics curriculum for high school, although only 1 in 10 young people attended high school (Kieran & Wagner, 1989). Like elementary children who were schooled as a form of moral discipline, with a regimen of memorization and recitation, secondary students also memorized significant amounts of material from textbooks and recited it back to their teachers (Reese, 2008). Perhaps dissatisfied with the memorization method, like their elementary peers, secondary mathematics teachers became vocal in the pedagogy debate. In 1890, mathematical historian Florian Cajori (1890) wrote that mathematical teaching of the prior 10 years indicated a “rupture” with antiquated traditional methods and an alignment with the “march of modern thought” (p. 293). Cajori’s (1890) survey of algebra teachers indicated that they were opposed to an overemphasis on procedural skills. They called for educators to teach algebra in a meaningful way to facilitate better student understanding (Parshall, 2003; Rachlin, 1989). However, Cajori (1890) also stated that the alignment with modern thought had barely begun. McLellan and Dewey (1895) joined the continuing debate strongly critical of drilling a student on procedures and forgetting that he should actually use the ideas or concepts as well, a practice that is detrimental to learning. They stated, There is no attention, or too little attention, paid to the essential process of discrimination when it is taken for granted that definite ideas of number will be formed from the hearing and memorizing of numerical tables...apart from the child’s own activity in conceiving a whole of parts and relating parts in a definite whole. (p. 30) Finally, Dewey (1916) decried the treatment of students as pieces of “registering apparatus” that acquire and store information without experiencing the learning or understanding its purpose (p. 147). 27 1900–1930: Expansion and Progressivism From 1900 to the mid20th century, the debate continued regarding pedagogy in school mathematics (Kilpatrick, 1992). At the turn of the 20th century, several factors influenced the methodology for teaching mathematics in general and algebra in particular. First, public schooling had gained greater legitimacy for children of multiple ages; as a result, more children attended school (Reese, 2008). Simultaneously, university professors, education leaders, and mathematicians were weighing both the value of teaching mathematics and the types of mathematics courses that should be taught, if any (Klein, 2003). Finally, several significant societal influences played a role in mathematics education (Kieran & Wagner, 1989). The 20th century witnessed the continual expansion of the power and authority of public schools in the lives of children, and high schools that had served relatively few students expanded dramatically (Reese, 2008). As late as 1890, approximately 5% of all adolescents were enrolled in public schools. While most students were girls who later became school teachers, school attendance became increasingly universal, particularly with the assimilation of massive numbers of immigrants arriving in the 1890s (Reese, 2008). As a result, between 1890 and 1910, the number of high school students quadrupled. Compared to the 1 in 10 who enrolled in 1890, by 1910, 1 in 3 teenagers were enrolled in high school (Ballew, 2009). Along with the significant increase in the number of students enrolling in public schools, enrollment in mathematics courses also increased. By the end of the 19th century, boys much younger than Burton’s 12 years were learning elementary arithmetic (Cohen, 2003). In fact, by 1910, teachers routinely taught arithmetic to 6yearold boys and girls and algebra to male and female students in Grade 9 (Brookman, 1910; Cohen, 2003). 28 Concurrent with the everincreasing number of students entering high school and enrolling in mathematics courses, was the continuing pedagogical debate in the mathematics educational community regarding inductive reasoning and discovery learning versus synthetic and rote learning that had begun in the late 1800s (Cohen, 2003). By the turn of the century, more students were enrolling in algebra, and as evidenced in the Cajori survey of 1890, algebra teachers had clearly called for a new teaching method (Parshall, 2003). Robinson (2010) stated that the early 1900s saw a movement toward pure mathematics and problemsolving methodology defined as the ability to manipulate equations algebraically to solve problems. This methodology was described as the “plug and chug” method (Robinson, 2010, p. 1). Teachers taught students that to solve a problem they needed to find the correct formula, plug it into the problem, and work through the mathematical operations (chug) to the solution. As such, the pure math movement was clearly an advocacy for procedural, rote learning of skills. The debate grew stronger in some educational communities and became secondary to a more significant issue in others (Klein, 2003). Early in the 1900s, much of the educational focus shifted dramatically from the best methodology for teaching algebra, and mathematics in general, to the very value of teaching mathematics (Klein, 2003). In the late 1800s, it was still generally accepted that mathematics be taught as a form of mental discipline; however, in 1901, psychologist Edward Thorndike conducted a series of experiments. Based on his findings, he challenged the justification of teaching mathematics to discipline the mind (Klein, 2003). That challenge contributed to the view that all mathematics should be taught and learned for utilitarian purposes only (Klein, 2003). Thorndike championed the progressive stance that in high school, algebra should be restricted to select students, and rather than emphasizing the debate on the methods of teaching 29 algebra, the focus should be on who should take the course and for what reason (Angus & Mirel, 2003; Klein, 2003). William Heard Kilpatrick, a significant voice of the progressive movement, particularly in mathematics, agreed with Thorndike and rejected the notion that the study of mathematics contributed to mental discipline. He advocated that math should be taught to students only for its practical value (Klein, 2003; Kliebard & Franklin, 2003). In fact, Kilpatrick carried the argument further by advocating that algebra and geometry in high school be discontinued “except as an intellectual luxury” (as cited in Klein, 2003, p. 3). Educators supported Kilpatrick, and in 1915, the Commission on the Reorganization of Secondary Education asked him to chair a committee of educators to study the problem of teaching mathematics in high school (Klein, 2003). Kilpatrick’s 1915 report, The Problem of Mathematics in Secondary Education, challenged the use of mathematics to promote mental discipline (Kliebard & Franklin, 2003). He recommended that only content of probative value be taught, and he called to restrict high school math to a very select group of students (Kliebard & Franklin, 2003). Prominent mathematicians, none of whom served on the committee, stringently opposed Kilpatrick’s report as an attack on the field of mathematics itself, and they attempted to stop the report from being published (Klein, 2003). Anticipating the publication, the Mathematical Association of America (MAA) responded vigorously by convening the National Committee on Mathematical Requirements, a group that included prestigious mathematicians and wellknown teachers and principals from secondary schools (Klein, 2003). Eventually, the Kilpatrick report was published in 1920, and in 1923, the MAA national committee published its own extensive report that advocated the importance of algebra for every 30 educated person (Klein, 2003). In addition, the MAA urged the formation of the National Council of Teachers of Mathematics (NCTM) in 1920 to counter arguments of progressive educators; however, throughout the 1920s and 1930s, Kilpatrick’s progressivism report was more influential than the 1923 MAA report (Klein, 2003). In fact, the influence of Thorndike, Kilpatrick, and the Progressive Movement in general from 1900–1950 was a catalyst for several changes in mathematics education, not the least of which was a dramatic increase in the number of vocational courses and a significant decline in the enrollment in secondary algebra (Angus & Mirel, 2003; Ballew, 2009). The limited number of research on algebra education from 1900 to 1930 dealt mainly with student performance in solving equations and curriculum rather than teaching methodology (Kieran & Wagner, 1989; Thorndike, 1922). In the 1920s, Kilpatrick and the progressives’ call for a move away from pure mathematics to a more practical, vocationally oriented curriculum, included algebra content as well (Angus & Mirel, 2003). Addressing the methodology for teaching algebra, Thorndike (1922) highlighted the importance of understanding algebraic formulas; however, he clearly called for prescriptive manipulation exercises and spoke to the amount of practice students should engage (Kieran & Wagner, 1989). Progressive sentiments included debate over who should take algebra. These sentiments, together with dwindling algebra enrollment, indirectly influenced teaching methodology for the course (Kieran & Wagner, 1989). The main influence on teaching algebra, however, was the socioeconomic condition of the nation produced by the Great Depression leading to World War II (WWII; Kieran & Wagner, 1989). Further, the loud cry from society for courses that taught practical and vocational mathematics rather than algebra, together with a rapidly growing interest in psychology, 31 contributed to an increasingly childcentered focus in the classroom (Kieran & Wagner, 1989). The innovative, studentcentered approach to teaching included a more active role for students identified as learning by doing (Reese, 2008). As such, limiting math content to its probable value, that which was directly applicable to ordinary living, significantly reduced the number of topics in the curriculum, limited academic content, and reduced the rigor of the coursework (Ballew, 2009). The decreasing number of students enrolled in advanced mathematics courses and the reduced number of topics taught justified the slow pace of teaching studentcentered algebra using a discovery approach (Klein, 2003). However, teachers often ignored pleas for a more studentcentered pedagogy, and they maintained the pedagogy of memorization and practice (Reese, 2008). Progressive educators despaired over the traditional ways of high schools, where subject matter and teacher authority had long reigned supreme. Critics discovered that even shop teachers lectured or read to pupils out of textbooks (Reese, 2008). The movement at the end of the 1920s toward vocational mathematics did not change, and traditional methods of teaching businessrelated practical problems using basic skills remained intact (Klein, 2003; Kliebard & Franklin, 2003). 1930–1950: The Great Depression and World War II From 1930 to the mid20th century, several significant factors directly influenced mathematics education and teaching methodology. Angus and Mirel (2003) stated, “In the late twenties, the bottom fell out of the teenage job market” (p. 460). The rapid decline of job opportunities for adolescents in cities because of the Great Depression caused significant increases in the number of young people entering high school and provided a strong incentive for them to remain there (Angus & Mirel, 2003; Kliebard & Franklin, 2003). Angus and Mirel 32 (2003) reported, “The high schools experienced the largest enrollment increase relative to the fourteentoseventeen year old population in history” (p. 460). To keep students in school and out of the job market for as long as possible, compulsory attendance became universal. In response to the burgeoning number of students, the childcentered focus in the classroom increased and high schools became more custodial in order to educate everyone (Angus & Mirel, 2003; Kieran & Wagner, 1989; Reese, 2008). The Great Depression was a catalyst for continued and even heightened attacks on education in general, and on mathematics in particular, as being too remote from everyday life, a sentiment that added significant support to the progressive platform (Kieran & Wagner, 1989). Education journals, university courses for administrators, teachers, and textbooks advocated major themes of progressivism that proposed that the school curriculum be determined by students’ vocational needs as determined by professional educators, rather than academic subjects (Klein, 2003). From 1930–1950, an era characterized by some as the LifeSkills Movement, preoccupation with issues of existence and survival led to deeper entrenchment of teaching practical and vocationally oriented mathematics, which provided greater influence on the way courses such as algebra were taught (Kieran & Wagner, 1989). Angus and Mirel (2003) stated, “Throughout the 1930s and beyond, the focus of curriculum reform was to expand the general track and to develop courses for this track that were interesting, undemanding, and closely related to the ‘immediate needs of youth’” (p. 460). Leading the progressive platform, Kilpatrick (1925) advocated that students should be actively involved in their learning, and he inspired what was called the Activity Movement of the 1930s. Kilpatrick also advocated for interdisciplinary curriculum of all subjects versus standalone math courses. While little disagreement existed in 33 elementary school, high school teachers were unwilling to abandon their courses in which they had specialized knowledge and expertise (Klein, 2003). Nevertheless, mathematics courses, including algebra, became even more focused on utilitarian applications (Kliebard & Franklin, 2003). Overall, secondary education was characterized by more electives, fewer graduation requirements, declining enrollment in traditional math courses, and curriculum written for commercial math (Angus & Mirel, 2003; Donoghue, 2003; Kieran & Wagner, 1989; Klein, 2003; Kliebard & Franklin, 2003). Regardless of curriculum issues, the debate over teaching methodology maintained its vigor. Thorndike’s advocacy in the 1920s for drill and practice was met in the following two decades with repercussions (Kieran & Wagner, 1989; Resnick & Ford, 1981). Thorndike had advocated that practice led to understanding; however, in the 1930s and later, theorists advocated that meaningful learning or understanding should be accomplished before practice (Kieran & Wagner, 1989). Brownell (1945) led the strong advocacy for meaningful mathematical learning, and he emphasized the importance of student opportunities for understanding the structure of mathematics (as cited in English, 2007; Hiebert & Carpenter, 1992). Adamantly opposed to the wideacceptance of drill, a result of published investigational studies, Brownell and Chazal (1935) accused researchers of being more concerned with the length and type of drill than with its effects on learning and not considering the place of drill in a total mathematics program. Brownell (1935) stated, In more recent years, the large number of investigations on drill have been less concerned with its effects upon learning than with such related matters as the length of the drill period, the comparative merits of mixed and isolated drill organization and the like. (p. 17) 34 Addressing the history on the effects of teaching for meaning dating from the 1940s, Grouws and Cebulla (2000) stated, “Investigations have consistently shown that an emphasis on teaching for meaning has positive effects on student learning” (p. 1). Several wellknown educators led the resistance to meaningful learning in the 1930s and 1940s. They voiced complaints of the military services in which an inordinate number of recruits had to be tutored in basic skills of mathematics that they should have learned in school (Klein, 2003). The Life Adjustment Movement in the mid1940s was based on teaching basic skills. Proponents complained that secondary schools were too devoted to an academic curriculum and students should be taught traditional math skills for daily living (Klein, 2003). Hiebert and Carpenter (1992) characterized the contention as an issue of the importance of conceptual versus procedural knowledge or understanding versus skill. They pointed out that the prevailing view had seesawed back and forth depending on the persuasiveness of the spokesperson of each particular position. Clements (2003) stated that of the most contentious topics among math educators was the role of drill and practice and the ways to achieve meaningful learning. Advocates of rote learning would often speak to the need to use drill with care so they also included meaning (Sobel, 1970). However, in practice, teachers rarely felt the need to distinguish among the meaning of drill practice and recurring experience (Clements, 2003). Throughout the 20th century, educators accepted drill as a useful tool in teaching and learning mathematics (Clements, 2003; Sueltz, 1953). Referring to Cajori’s survey findings, Rachlin (1989) stated, These calls have been echoed in waves of reform documents from that time (Cajori, 1890) until today—with proposals ranging from laboratory approaches that would 35 encourage an inductive learning of algebra to reallife applications that would demonstrate the relevance of algebra. (p. 257258) Algebra textbooks of the first half of the 20th century included narratives about the importance of learning “why” and “how,” but “the emphasis in the texts was on computational algebra” (Rachlin, 1989, p. 258). Osborne and Crosswhite (1970) identified that the common sequence for instruction was definition, illustration, rule and example, drill, review, and speed tests through direct instruction from the teacher. Research on teaching methodology has gone through several periods of reform, and it is generally categorized into loosely chronological phases. Little educational research was published in the first half of the 20th century (Rachlin, 1989). Statistics indicate only a gradual increase in the number of studies from 1892 to 1924. In fact, the average annual frequency of educational studies prior to 1930 was less than 25 (Kilpatrick, 1992). This slow growth of educational research persisted until 1960. According to Nisbet (1985), “Prior to 1960, research was mainly a sparetime amateur affair, unorganized and often ignored” (p. 12). Before the mid20th century, rather than examining pedagogy, the limited number of mathematics education studies concentrated on teacher characteristics or attributes in isolation (Koehler & Grouws, 1992). The first approach has been described as the behaviorist tradition that identified and analyzed components of teachers’ behaviors so that others could emulate effective practices (Nickson, 1992). Teachers were evaluated by measures such as math coursework completed and years of experience (Koehler & Grouws, 1992; Medley, 1979; Rosenshine, 1979). Personality traits such as enthusiasm, magnetism, and considerateness were also included in the studies, and they were often rated on good judgment, loyalty, and appearance (Barr & Emans, 1930; Charters & Waples, 1929). Effective teachers were identified by opinions 36 and ratings of principals, supervisors, and sometimes students, which provided the basis by which the researchers identified whether a particular trait correlated to successful teachers (Koehler & Grouws, 1992). Later research sometimes examined isolated teaching events (Medley, 1979; Rosenshine, 1979). Koehler and Grouws (1992) stated, “Although student outcomes were considered, the narrow focus of these studies and their lack of attention to the quality of teaching was limiting” (p. 116). Little, if any, attention was given to other factors, including the quality of teaching methodology. In summary, the research focused on the teacher rather than on the teaching (Koehler & Grouws, 1992). Although some prominent studies addressed other questions, curriculum concerns continued to be voiced, including the question of whether mathematics courses would, or should, be offered in high school, particularly algebra and geometry (Kliebard & Franklin, 2003; Rachlin, 1989). Rosenshine (1979) identified a second cycle of research that focused on teacherstudent interactions and a third that focused on students, their attention in the classroom, and the content they mastered. Other studies addressed various issues related to teaching and learning mathematics in North American schools from 1900 to 1965, which was accompanied by much debate (Kilpatrick, 1992). Clements identified two of the most contentious topics as the role of drill and practice and the methods to facilitate meaningful learning (Clements, 2003). The question of how to teach mathematics resided in the ongoing debate in the 19th century of whether it was more effective to tell students content directly or to facilitate their discovery of the content (Clements, 2003). Near the close of the halfcentury, Schoenfeld (1987) stated, “Nineteenforty five was both a banner year and a year of great chaos for problemsolving” (p. 283). Mathematics instruction in schools was mostly drill and practice, based on Thorndike’s behaviorist theory, but 37 drill and practice was under attack (Schoenfeld, 1987). The European Gestalt view, somewhat popular in the United States, claimed that rote instruction was of little value because students failed to gain an understanding of concepts and structures when they simply memorized and reproduced mathematical procedures (De Corte, 2010; Schoenfeld, 1987). In turn, the behaviorists counterattacked the Gestaltists (De Corte, 2010; Schoenfeld, 1987). The pendulum remained on the side of traditional instruction although the first half of the century closed with the same contention regarding teaching methodology as that with which it began. 1950s: Influence of Sputnik By the second half of the century, nearly 90% of all teens were enrolled in public school, and the technological nature of the war had highlighted the importance of mathematics in school curriculum; however, barely onehalf of teenagers were taking mathematics courses (Angus & Mirel, 2003; Garrett & Davis, 2003; Reese, 2008). During the years that research seemingly focused on a myriad of factors other than pedagogy, discussions continued regarding the best way to teach mathematics. By the 1950s, projects and programs evidenced the strength of the disagreement (Davis, 2003; Lester & Lambdin, 2003). However, using textbooks to explain, provide examples, and practice remained the most prevalent method, and a significant amount of research focused on textbooks (Seymour & Davidson, 2003). After WWII, American schools were under attack as they faced significant criticism from the military, business, education, and public sectors claiming that graduates were not equipped mathematically for work or further study (Kilpatrick, 1992; Klein, 2003; Lester & Lambdin, 2003). In fact, teachers pointed to scientific advances in technology, economic growth, and political changes that accompanied WWII as significant indicators of the need for new educational programs. In response, great efforts were made across the United States to improve 38 math and science content for students (Davis, 2003; FerriniMundy & Graham, 2003). Psychologists and some curriculum theorists recommended new teaching methods whereby students engaged in exploration and discovery (FerriniMundy & Graham, 2003; Fey & Graeber, 2003). Beginning in 1950, the requirements of industry, engineering, the sciences, and mathematics escalated dramatically, and in 1957, two events spurred significant change in mathematics education: the launch of Sputnik by the USSR in October and the establishment of the Madison Project at Syracuse University (Davis, 2003; Fey & Graeber, 2003; Kelly, 2003). FerriniMundy and Graham (2003) stated, “The launching of Sputnik was a watershed event that jolted the United States public, government and scientific communities into taking more concerted action in the areas of mathematics and science education” ( p. 1198). Fey and Graeber (2003) stated, “The military threat of Soviet space science and technology prompted a variety of political, business, and social groups to urge critical examination of American mathematical, scientific, and technical educations” (p. 521). By the end of WWII, deep concern existed about national security in the Cold War world, and while some educational reform activity had begun before the Sputnik launch, the military threat it posed heightened concerns and became the catalyst and sustaining force to upgrade mathematics and science in schools (FerriniMundy & Graham, 2003; Fey & Graeber, 2003; Schoenfeld, 1992). The new sentiments of mathematicians and mathematics educators, among others, resulted in a concerted challenge to traditional rote learning and classroom drill and practice (Kelly, 2003). Instead, educators favored an approach focused on higherorder thinking and conceptual understanding (Kelly, 2003). In fact, discovery teaching was widely advocated immediately following the war (FerriniMundy & Graham, 2003). In response to concerns regarding school mathematics, the Madison Project, led by mathematics professor Robert B. 39 Davis, was established in the post WWII era when teachers, schools, and parents were willing to explore new ideas in curriculum and teaching methodology (Davis, 2003). Accordingly, the highly innovative Madison Project, arguably the most significant program of the time, focused on the discovery approach to mathematics instruction, a teaching methodology that was among the most controversial aspects of the project (Davis, 2003; Kelly, 2003). A century earlier, Colburn had advocated for presenting students with carefully selected problems to discover the appropriate mathematical principles necessary to solve them (Cohen, 1999). With Madison, the pendulum moved again toward Colburn’s favored type of methodology, discovery teaching, and Madison Project teachers began by asking questions or posing selected or created tasks (Cohen, 1999; Davis, 2003). Davis (2003) stated, “The thing that came first was the problem. The relevant mathematical ideas were invented by the students in order to solve the problem” (p. 628). Davis illustrated the new methodology as follows: Discovery teaching requires a substantial change both in the role of the teacher and in the role of the student. The teacher creates a challenge, and it becomes the responsibility of students to invent methods of solution...This process is virtually a reversal of much common school practice, where the teacher presents a way of dealing with some class of problems, the student imitates the teacher, and the teacher evaluates the result. In the traditional teaching I encountered, students often knew what they were expected to do (because they had been shown and had practiced it) but had no idea why they were doing it or when (outside the lesson) it might be useful...Stimulus and response were inextricably wedded, with each having a quite different nature from most usual school tasks. Stimulus now meant a novel problem and the response was what the students invented in order to accomplish their goal. (p. 629) 40 Unfortunately, little to no research supports the discovery methodology used in the project because administrators regarded research studies as problematic (Davis, 2003). Madison overseers believed that different people have different learning experiences that result in different ideas; as such, comparative studies would not provide results that indicated the benefits of the discovery method (Davis, 2003). Mathematicians and researchers questioned the lack of appropriate comparative studies as evidence of a response to severe opposition (Davis, 2003). While project managers made films and videotapes to prove the value of discovery, the work at Syracuse did not accurately reflect the real world of most public schools (Davis, 2003). In summary, no quantitative research exists to substantiate the claims of the Madison Project in favor of discovery learning (Davis, 2003). Davis (2003) stated, “Whatever the reasons, opposition to discovery teaching was a serious obstacle to wider acceptance of project innovations (p. 633). In 1959, the Commission on Mathematics of the College Entrance Examination Board (CEEB) recommended that logic, modern algebra, concepts of relation and function, and probability and statistics be included in the school mathematics curriculum so secondary math students could study both pure and applied topics (CEEB, 1959; Fey & Graeber, 2003). Proponents also advocated that processes of deductive reasoning and the search for patterns would help students develop powerful understanding, whereas the prevailing programs of the time provided only rote training on a litany of disconnected procedural routines that most students quickly forgot (Fey & Graeber, 2003). The pendulum moved toward exploring teaching strategies as the 1959 CEEB report defined the new math agenda for the 1960s. 41 1960s: The New Math Era Fey and Graeber (2003) stated, The period from the start of the ‘new math’ movement in the mid1950s to the publication of An Agenda for Action by the National Council of Teachers of Mathematics (NCTM) in 1980 has all the ingredients of a typical era in the evolution of mathematics education. (p. 521) In other words, great advocacy for reform was followed by great resistance in favor of maintaining a traditional agenda. The pendulum continued its movement, but perhaps in the 1960s, it swung a little further in each direction. The goal of school mathematics in the 1930s and 1940s was to provide a mostly vocational, universal education for everyone based on the individual needs of each student; a 1960–1961 survey indicated that practical, consumerrelated courses in mathematics continued to be popular (Angus & Mirel, 2003). However, the childcentered focus prior to and during the war years shifted with events of the 1950s, including Sputnik. During this time, the goal became one of educating an elite group of college preparatory students to develop a scientifically and mathematically strong workforce that could facilitate United States competition with the Soviet Union (Becker & Perl, 2003; FerriniMundy & Graham, 2003; Fey & Graeber, 2003; Kieran & Wagner, 1989). To produce knowledgeable students quickly, interest increased in developing new approaches to make math meaningful, and controversy arose over teaching methods once again (Payne, 2003; Resnick & Ford, 1981). Mathematicians and mathematics educators, among others, advocated replacing rote skills and drill instruction with conceptual teaching and the New Math Movement began in the late 1950s and early 1960s (Kelly, 2003; Resnick & Ford, 1981). 42 At the same time, there was a great deal of interest in the students’ mental processing, and as the field of cognitive psychology was born, conceptual teaching for meaningful learning gained impetus during the 1960s (Bransford et al., 1999; Resnick & Ford, 1981). Research prior to the 1960s was amateurish. However, because of the shifting focus, research in mathematics education proliferated during the 1960s (Kilpatrick, 1992; Lester & Lambdin, 2003; Nisbet, 1985). Nisbet (1985) observed, “It is only within the past 25 years that research in education has received public funding on any substantial scale” (p. 12). Public support of educational research in the 1960s grew very rapidly, and expenditures doubled each year from 1964–1967 (Nisbet, 1985). In the late 1950s and early 1960s, dozens of national, regional, and local mathematics education projects and programs were established to develop curricula and teaching methodology that reflected the reform agenda (Angus & Mirel, 2003). Much of the research was based on efforts to measure the effectiveness of the new math programs (Fey & Graeber, 2003; Lester & Lambdin, 2003). In 1960, prominent psychologist and researcher, Jerome Bruner, published The Process of Education. Bruner had become interested in the cognitive processes of humans as they learned, particularly during classroom activities in mathematics (as cited in Resnick & Ford, 1981). Perhaps the bestknown educational psychologist of the decade, Bruner advocated that students could better understand and apply mathematics if teachers focused on the conceptual understanding of the structures of mathematics (Bruner, 1960; Clements, 2003; Fey & Graeber, 2003). Contrary to a vocational orientation for mathematics curriculum, Bruner proposed teaching math problems that were interesting to students, thereby providing a stimulus for learning (Smith, 2002). He also advocated a teaching methodology that facilitated student acquisition of the habits of thinking like a professional mathematician (Bruner, 1960). For the 43 most part, the mathematics community agreed that the main objective of math instruction was student understanding rather than rote skill practice, memorization, and recitation (Kieran & Wagner, 1989). Psychological and cognitive issues were included in reform perspectives of the new math era. Fey and Graeber (2003) stated, For example, one of the persistent tensions in teaching was finding the most productive balance between telling and asking: Should a teacher provide clear and convincing exposition of ideas and techniques or stimulating questions and problems whose solution by students will reveal important concepts and procedures. (p. 526) The same disagreements persisted among teachers, as many believed that students must be told specifically how to work problems and others believed the opposite. Just as it had a century earlier, the disagreement in the 1960s fostered many controversies over “discovery learning” (Davis, 1992, p. 725). In addition to the debates over telling versus asking, issues of the student’s role in learning were widely discussed (Fey & Graeber, 2003). Based on new insights of the time into how students learn math, prominent psychologists recommended a reformed teaching pedagogy that included “emphasizing the importance of students’ engagement in exploration and discovery through developmentally appropriate activities” (Fey & Graeber, 2003). Fey and Graeber (2003) stated of the 1960s, If there was a dominant pedagogical principle underlying many new math innovations, it was that students acquire understanding and skill most effectively through classroom activities that help them discover mathematics themselves. Thus new math reformers 44 explored a variety of strategies for helping teachers conduct Socratic dialogues with their students, leading the students to the discovery of major concepts and principles. (p. 526) The methodology of discovery teaching included lessons in which students hypothesized, investigated, and explored mathematical solutions to problems, which allowed them to experience the thinking and activities of professional mathematicians as Bruner (1960) advocated (Fey & Graeber, 2003). Discovery was also consistent with new cognitive theories that advocated learning tasks for students that challenged their conceptions and lead them to new understanding (Fey & Graeber, 2003). Fey and Graeber (2003) stated, “For an exciting period of at least ten years in the decade of the1960s, the reform of school mathematics was frontpage news around the world” (p. 522). Unfortunately, many mathematics educators found it much easier to recommend discovery teaching, but much more difficult to prepare teachers who could deliver the instruction in everyday classrooms (Fey & Graeber, 2003). Teachers were skeptical of the new math teaching agenda, and they did not exhibit the enthusiasm of the math community at large (Fey & Graeber, 2003). Even with the multitude of studies, programs and proposals for reformed curricula, and classroom teaching techniques, teachers relied almost exclusively on textbooks for both curriculum and teaching methodology (Seymour & Davidson, 2003). During the reform decade, typical classrooms were described as plain and rather sterile, and teachers used only textbooks and chalkboards to teach mathematics where the dominant activity was the same traditional memorization and computation seen in the reform era and where realworld application problems were limited (Seymour & Davidson, 2003). While many different reform programs and projects were characterized as new math, some called for teachers to guide student learning, as in a discovery lesson, while others retained 45 traditional characteristics such as starting a lesson with formal definitions (Davis 2003). While all projects advocated for the replacement of traditional curriculum, before widespread implementation of these recommendations occurred, many different groups and individuals criticized discovery approaches (Davis, 2003; Fey & Graeber, 2003). Loud calls were issued for the return to traditional curriculum and teaching methodologies based on behavioral psychology, including instructional approaches that involved procedural lessons and calculation (Davis, 2003; Fey & Graeber, 2003). Fey and Graeber (2003) stated, “For a time this more traditional view of school mathematics curricula and teaching gained the upper hand in debates over elementary and secondary school programs” (p. 522). The new math reform was ending. Public memory of new math was an image of failures and mistakes, and the term new math is now part of educational rhetoric that connotes a failed reform (Fey & Graeber, 2003; Payne, 2003). Once again, after all the fervor in the 1960s regarding new math, the pendulum gained momentum as it moved like clockwork in the opposite direction back toward traditionalism. 1970s: Back to the Basics Fey and Graeber (2003) spoke of the perpetual “tugofwar” in education for intellectual, social, and political influence. They stated that the struggle in mathematics has a “predictable rhythm of crisisreformreaction episodes” (p. 521). Payne (2003) stated, “If the 1960s were the zenith decade, then the 1970s were the nadir.” He described the period as the “20th century’s worse decade” (p. 590.) The school mathematics community characterized the 1970s as calmer than the “turbulent new math era” of the 1960s (Fey & Graeber, 2003, p. 539). Professional publications at the time documented continued advocacy for progressivism and described the ongoing debate regarding the most effective teaching pedagogy (Fey & Graeber, 2003). Certain 46 groups in the math community referred to successes from the 1960s to support their stances on the issue (Fey & Graeber, 2003). However, new math reformers in the educational community encountered several significant difficulties, and by 1978, the 10 most widely used mathematics programs included traditional rather than reform materials (Fey & Graeber, 2003). First, school personnel were skeptical of the new ideas, and it was often very difficult to prepare teachers to implement classroom methodology to use discovery learning (Fey & Graeber, 2003). Second, while many programs and projects were based on reform recommendations of the 1960s, the textbooks and other materials developed for reform teaching were actually very conventional and included explanations of math content, examples, and homework (Fey & Graeber, 2003). Third, anxious parents reacted with great concern toward the new math content and teaching methodology their children experienced, particularly because it was so different from their own experiences in school mathematics (Fey & Graeber, 2003). Public perception was that new math failed because students could do nothing with accuracy (Payne, 2003). Payne (2003) stated, “Certainly, they could not compute, and practical skills had all but been abandoned” (p. 590). Public conviction against reform was a catalyst for press reports that were critical of the movement, and the fervor sparked recommendations to abandon reforms and return to the basics (Fey & Graeber, 2003). Along with calls for traditional curriculum, the public wanted traditional teaching as well (Fey & Graeber, 2003). Specifically, the call was to abandon discovery learning in favor of traditional teaching practices that emphasized direct instruction of procedural skills (Fey & Graeber, 2003). Opposition to the 1960s reform swelled in the early 1970s, and significant backlash against the New Math Movement occurred (Kieran & Wagner, 1989). The 1970s became 47 commonly described as the back to basics era (Fey & Graeber, 2003; Schoenfeld, 1992). Fey and Graeber (2003) stated, “The phrase suggested renewed emphasis on developing skills in arithmetic and algebraic calculation through instruction that features teacher exposition and student practice” (p. 538). In other words, the pedagogical pendulum responded to the uproar and gained speed in its swing back toward traditional teaching methodology. Two events occurred in the 1970s that sealed the return to traditionalism in mathematics teaching methodology. First, in 1975, the National Advisory Committee on Mathematical Education (NACOME) published an analysis of math education in the United States (FerriniMundy & Graham, 2003). The NACOME report suggested that in spite of advocacy in the 1960s for new math discovery, some had significant doubts that teachers were using the methodology. In fact, evidence existed that teachers had difficulty figuring out how and when to implement the various new pedagogies (Fey & Graeber, 2003). Forbes (1970) stated that most U.S. high school teachers depended heavily on textbooks with little deviation from the scope and sequence. The NACOME report described the median classroom in mathematics as one in which teachers closely followed the textbook, “for the most part [teachers were] teaching the way they had been taught” (FerriniMundy & Graham, 2003). Perhaps most detrimental to reform, the report indicated a lack of evidence that suggested any particular method or pattern of instruction was superior to any other, and more information was needed (FerriniMundy & Graham, 2003). The National Science Foundation (NSF) addressed the need for more information on classroom instruction in mathematics in the late 1970s (McLeod, 2003). The NSF funded three large survey projects, and pertinent to this study, the third project included a set of detailed case studies on classroom instruction in math and science (McLeod, 2003; Stake & Easley, 1978). 48 These case studies were particularly influential in substantiating the view that teachers were still teaching traditional mathematics, with its emphasis on rules and procedures, and that the new math of the 1960s had not produced lasting change in classroom instruction (McLeod, 2003). Second, in 1977, the CEEB issued a report that provided data on a 10year decline in scores on the Scholastic Aptitude Test (SAT); an additional catalyst for abandoning new math reforms (Fey & Graeber, 2003). McLeod (2003) stated, From the information published in the NACOME report in 1975 and the data from other studies in the late 1970s, it was clear that the dreams of the new math era prior to 1970 had been dashed against the rocky reality of traditional mathematics classrooms. (p. 758) Enthusiasm for new math was replaced by fervor for instruction based on behavioral psychology principles (Fey & Graeber, 2003). McLeod (2003) stated, “The emphasis in the 1970s on drill and practice in mathematics curriculum had a long history in support from behaviorist psychology” (p. 807). Behavioral psychology had dominated school practice for most of the 20th century prior to the brief new math era of the 1960s. Behaviorist theories were in control of mathematics instruction again with the return to traditionalism, thus, dominating the backtobasics movement in the 1970s (Fey & Graeber, 2003). Relying on behavioral research and theories, many influential educators argued that classroom instruction in mathematics should emphasize mastery of objectives on a systematic path, and goals should be defined in terms of explicitly observing student performance (Fey & Graeber, 2003). Parents and politicians continued to express concerns about student competence in the basics, and NCTM reform leaders continued attempts to convince the public of the necessity to move away from the emphasis on computational skills (McLeod, 2003). In 1977, the National Council of Supervisors of Mathematics (NCSM), in support of reform, responded to the 49 traditional, backtobasics movement by redefining basic skills to include problem solving and application skills rather than just computational prowess. In 1978, leaders in the mathematics community brought researchers from mathematics and cognitive science together in response to concerns that more educational research could not produce relevant studies to benefit classroom teaching (Kilpatrick, 1992). Researchers were encouraged to consider the practitioner’s point of view (Sowder, 1989). Deficiencies in traditional behaviorism became more apparent in the 1970s, and as the field of cognitive study became popular in mathematics education and educational psychology, researchers published a number of cognitive studies (De Corte, 2010; Lester & Lambdin, 2003; McLeod, 2003). Research in the 1950s and 1960s examined only one aspect of a teaching session at a time (e.g., time allocation on various lesson activities); however, research in the 1970s was marked by several classroom observations of teacher and student behaviors that yielded detailed results about instruction (Koehler & Grouws, 1992). Often called processproduct research (measures of teacher behaviorprocess and measures of student achievementproduct), the methodology documented behaviors such as the frequency of teacher and student interactions and sometimes included the types of examples and questions posed by the teacher, the length of students’ responses, and the amount of practice and review in the lesson, among other variables (Koehler & Grouws, 1992). Student achievement outcomes were correlated to the frequencies of observed teacher behaviors to see which might result in performance gains (Koehler & Grouws, 1992). The assumption was that teacher behaviors influenced student behaviors, and the quality of teaching was based more so on the frequency of particular behaviors than on the quality of the teaching methodology itself (Koehler & Grouws, 1992). 50 However, Doyle (1975) addressed what he called the failure of teacher effectiveness studies, and he stated that the processproduct paradigm assumed that teacher effects on student achievement were stable and generalizable; however, a lack of evidence supported either assumption (Doyle, 1975). Doyle concluded that little reason existed to expect teacher behavior variables as being strongly related to student achievement gains. He was also concerned about the neglect of student behaviors in processproduct studies. Doyle later focused on the importance of studentlearning tasks in the lesson and the role these tasks played in influencing outcomes. According to Schoenfeld (1992): By the end of the 1970s, it became clear that the backtobasics movement was a failure. A decade focused on rote mechanical skills produced students who performed dismally on measures of thinking and problem solving. Further, they were no better at basics than students being taught a reform curriculum. (p. 336) Educational researchers were viewed as unable to demonstrate that they had made advances that could be translated into practical benefits for the classroom (Kilpatrick, 1992). Unhappy with aspects of backtobasics, leading math educators planned reforms in different directions (Fey & Graeber, 2003). Despite continued reform efforts, by the end of the 1970s, the math pedagogical pendulum had swung once again toward traditionalist teaching via direct instruction. 1980s: The Standards Movement Begins FerriniMundy and Graham (2003) reported that the 1980s began with a number of reports that highlighted concerns and provided potential pathways for the continued improvement of science and mathematics education. Research on mathematics teaching and teacher education prior to the 1980s was sparse, and few studies addressed high school algebra or 51 the role of the algebra teacher in classroom instruction (Booth, 1989; Cooney, 1980; FerriniMundy & Graham, 2003; Kieran, 1992). Graubard (1981) addressed problems of American schools at that time and stated that it might be beneficial for educators to acknowledge that the early 1980s were the first days of educational research. In fact, researchers characterized the mathematics educational research effort as “still in its infancy” (Brophy, 1986, p. 125; Cooney, 1994, p. 613). The disparity between theoretical research and onsight classroom research had also recently come to light. Kilpatrick (1992) stated, The 1980s began with the promise of a more fruitful integration of research and practice than at any previous time in the history of mathematics education. More and more, research in mathematics education was moving out of the library and laboratory, and into the classroom and school. (p. 31) Although there was consensus that the quality of mathematics education was deteriorating, advocacy for reform grew slowly in the 1970s and 1980s (Klein, 2003; McLeod, 2003). After the failure of reforms in the 1960s and 1970s, a continuing issue for NCTM leaders was convincing the public of the necessity for change. Although politically passive in previous decades, the organization took a more active role in 1980 with An Agenda for Action: Recommendations for School Mathematics of the 1980s (Fey & Graeber, 2003; NCTM, 1980; McLeod, 2003). The agenda made recommendations, among others, that problemsolving should be the focus of school mathematics, that “basic skills” should be defined more extensively than calculation, and that teachers should decrease the emphasis on isolated drill and practice (FerriniMundy & Graham, 2003; Fey & Graeber, 2003; McLeod, 2003). Together with the clear failure of backtobasics, agenda recommendations prompted a philosophical move toward problem solving, which became the theme of the 1980s (Schoenfeld, 1992). 52 The 1981 NCTM study, Priorities in School Mathematics, reported that nearly 60% of mathematics teachers in the United States believed that more than half of classroom instructional time should be spent on drill and practice, a clear indication that math teachers were still using memorization and drill techniques, and teaching procedural skills rather than practicing pedagogy to develop conceptual understanding (NCTM, 1981). According to Kieran (1992), research findings in algebra instruction indicated that algebra teachers, like their counterparts in other math classes, made their first priority classroom management and covering the curriculum. Two critical reports in 1983, Educating Americans for the 21st Century by the National Science Board Commission (NSBC) on Precollege Education in Mathematics, Science and Technology of the NSF and A Nation at Risk: The Imperative for Educational Reform by the National Commission on Excellence in Education (NCEE) described mathematics and science education in the United States as being in crisis (Fey & Graeber, 2003; NCEE, 1983; NSBC, 1983). Fey and Graeber (2003) stated, “Those policy documents marked the beginning of a period of intense study and reform activity that continued through the end of the century” (p. 553554). The instinct of educational leaders, among others, for progressive reform resurfaced prominently in response to publicized descriptions of mathematics and science education being in deplorable states, particularly because nearly a decade of declining test scores was evidence that backtobasics had failed (FerriniMundy & Graham, 2003; Fey & Graeber, 2003). The fervor for reform after Sputnik was based on an urgent need related to scientific readiness, while that in the early 1980s was linked to public concern regarding national economic and technical growth (FerriniMundy & Graham, 2003). The goal changed from creating a scientifically elite group of students to educating everyone, and the “mathematics for all” agenda was born (Ferrini53 Mundy & Graham, 2003). This concern was the precursor to the next era of publicized educational goals and recommendations, and a new educational movement began that was not unlike that seen during the new math era of the 1960s (FerriniMundy & Graham, 2003). The education community and the public at large widely cite and discuss A Nation at Risk (NCEE, 1983). This report declared that education in the United States was in a state of mediocrity, and in terms of mathematics, the report listed declining achievement scores, lack of knowledge and skills of graduating seniors, and the need for remedial mathematics courses in higher education venues as evidence for the crisis (NCEE, 1983). In response to A Nation at Risk and to provide benchmarks for then mathematics community, mathematics education leaders called for the development of mathematics standards for all school grade levels that included standards for curriculum, teaching, and evaluation (McLeod, 2003). To reduce the scope, teaching standards were postponed, and work began in the mid1980s on another publication that would provide a guide for educators on what students should know and be able to do at each grade level (McLeod, 2003). The NCTM Curriculum and Evaluation Standards published in 1989 and built on recommendations of the Agenda for Action published nearly a decade earlier, was recognized as a significant contributor to mathematics reform, both in and outside the math community. This report was also hailed as a symbol of education reform in general (McLeod, 2003; NCTM, 1989). The NCTM Agenda for Action focused on math as a personally constructed, internal set of knowledge (Clements, 2003; Dossey, 1992), and researchers in the 1980s explored teacher and student behaviors in the classroom from a constructivist viewpoint. According to Lester (1982), “During the past decade there had been increasing interaction among those interested in cognitive research and those interested in mathematics instruction (Lester, 1982)” (Mayer, 1985, 54 p. 124). Math educators themselves were increasingly interested in theories of cognitive psychology. Silver (1985) stated, “Thus the moment seems opportune for the mathematics education and cognitive science communities to benefit from one another” (p. vii). Reformers and researchers began discussing constructivist view of learners during the 1980s (McLeod, 2003). The constructivist viewpoint was a link between research in teaching and research in learning (Koehler & Grouws, 1992). Koehler and Grouws (1992) stated that the constructivist view advocated that teachers no longer prepared lessons or activities in a way for students to receive knowledge, but in a way for students to engage in mathematical problem solving. Rather than looking at teacherdirected instruction in the classroom, the constructivist theory of learning focused attention on the learner (Brooks & Brooks, 1999). Resnick and Ford (1981) stated that a fundamental assumption of cognitive learning psychology is that students do not simply add new ideas to their storehouses, but that new information must be attached to existing structures and new relationships constructed among those structures. Koehler and Grouws (1992) stated, “In the constructivist approach, teaching behavior is examined from the viewpoint of how much it encourages or facilitates learner construction of knowledge” (p. 123). Thus, teaching was considered a means to facilitate students by providing appropriate math activities and engaging students in mathematical discussion regarding the various aspects of working through the activities (Koehler & Grouws, 1992). In addition, social interactions were identified as a critical part of knowledge construction (Koehler & Grouws, 1992; Schoenfeld, 1992). Reform teachers served in a role of “coexplorers” with students, and in that role, they were to “ask more openended questions, engage in more problemposing, and be less tied to the textbook” (Koehler & Grouws, 1992). Yackel, Cobb, Wood, Wheatley, and Merkel (1990) pointed out that when students are given 55 opportunities in a constructivist classroom to interact with oneanother and with the teacher, they can “verbalize their thinking, explain or justify their solutions, and ask for clarifications” (p. 19). Consistent with the transition to a cognitive approach, according to research publications as well as the agenda, mathematics knowledge was equated with “doing,” a result of activities in which students participated, and during which the teacher functioned as a guide rather than a dispenser of knowledge (Dossey, 1992). Stein et al. (1996) stated, “Increased emphasis is being placed not only on students’ capacity to understand the substance of mathematics but also on their capacity to ‘do mathematics’” (p. 456). The recommendation was that students should experience the practice of math by experimenting, conjecturing, discovering, and generalizing in the process of learning mathematics, not by receiving a welldeveloped communication of content (Dossey, 1992). Schoenfeld (1988) spoke to the interaction of cognitive and social factors in school that contribute to the perception of mathematics by students. He stated that for math to “make sense,” students must do math by analyzing, hypothesizing, conjecturing, and synthesizing, among other behaviors. They must practice the habits of thinking like a mathematician as Bruner argued in 1960. By the mid1980s, researchers considered the constructivist perspective as wellgrounded and widely accepted (Schoenfeld, 1992). Researchers began challenging decades of earlier research and constructivism and examining student thinking led to a focus on students’ classroom work behaviors. Doyle (1988) stated that descriptions of the instructional methods used, the frequencies of a teacher’s behavior, or student timeontask provided an incomplete account of student learning. He stated, “What is missing is a description of the work students were required to do. This missing element is important because work creates a context for students to interpret information during class sessions and to think about subject matter” (p. 168). 56 Kilpatrick (1985) stated, “Something the last 25 years have given us is the math problem as task” (p. 3). He also spoke to the developing perspectives in the mid1980s of the mathematical problem, one of which was a task given to students in the social context of the classroom. He summarized, “We do not have a final version of what problem solving is and how to teach it, but we are much more keenly aware of the complexity of both” (p. 13). Doyle (1983) proposed, “Tasks influence learners by directing their attention to particular aspects of content and by specifying ways of processing information” (p. 161). Doyle (1988) later stated, “A fundamental premise of this theory is that the work students do, which is defined in large measure by the tasks teachers assign, determines how they think about a curriculum domain and come to understand its meaning” (p. 167). In the summer of 1989, the first draft of the Professional Teaching Standards was developed (McLeod, 2003; NCTM, 1989). During the process, one writing team produced teaching vignettes to better illustrate significant aspects of teaching, a seeming precursor of future instructional research in mathematics. From those vignettes came the first section of the NCTM standards that discussed classroom tasks and classroom discourse of mathematical concepts (McLeod, 2003). The pedagogical pendulum reached its traditional instructional zenith and moved somewhat rapidly toward standardsbased instruction. 1990s to the New Century: Debate Leads to War Research on mathematics teacher education and pedagogy, as well as the standards movement as a whole, grew, expanded, and gained momentum during the 1990s (Clements, 2003; FerriniMundy & Graham, 2003). Specifically, stakeholders in mathematics education, including mathematicians, professors of teacher education, and the public, among others, focused on both mathematics teachers’ education and pedagogy (FerriniMundy & Graham, 2003). 57 According to Ferrini Mundy and Graham (2003), “Questions about teacher knowledge became more visible, and the field struggled to find effective ways to help teachers be effective in their practice” (p. 1289). Addressing mathematics pedagogy, Klein (2003) pointed out that the American education establishment had long supported and promoted a progressive education agenda. He stated, “Throughout the 20th century the professional students of education have militated for childcentered discovery learning and against systematic practice and teacher directed instruction” (p. 2). During the last two decades of the century, the ongoing conflict between “telling” and “facilitating discovery” (Fey & Graeber, 2003, p. 526) was ever present in the minds of reformers. FerriniMundy and Graham (2003) stated, “In the mid1990s, the debate intensified” (p. 1271). Davis (1992) provided an indepth description of the stances taken by both camps: Those who defend the assumption that we must tell students how to solve mathematics problems usually argue that if we do not do so, some (or many) students will be lost and will quickly become demoralized. By contrast, those who argue against this assumption usually claim that teaching based on it sends students the message that nobody can solve a math problem unless someone tells them how to do it. Hence, students quickly give up the habit of trying to think for themselves, and they adopt the strategy of merely trying to remember what the teacher has said. Anyone who interviews students extensively will find many who say, at least in effect, ‘I couldn’t possibly do this problem, because you haven’t told me how to do it.’ The opponents of ‘telling’ argue, first, that this attitude on the part of students renders them almost incapable of making good progress in mathematics, and second, that the attitude is not inevitable, but rather was taught to the students by implicit messages repeatedly sent by teachers and textbooks. (p. 725) 58 The 1989 NCTM publication, Curriculum and Evaluation Standards for School Mathematics, was an additional catalyst for progressive change in mathematics education (FerriniMundy & Graham, 2003, NCTM, 1989). Continuing its pivotal role in the standards movement, NCTM published Professional Standards for Teaching Mathematics in 1991. Although the publication did not have the same impact as the Curriculum and Evaluation Standards, it offered recommendations on teaching methodologies to implement the goals of the 1989 standards, and the NCTM considered it a major contributor to the reform movement (Clements, 2003; FerriniMundy & Graham, 2003; McLeod, 2003). FerriniMundy and Graham (2003) stated, “In particular, the pedagogical recommendations were tied directly to the various aspects of a teacher’s role in preparing and implementing mathematics instruction” (p. 1291). The innovative teaching recommendations emphasized problem solving and application, a definite change from traditional curriculum, and included choosing appropriate mathematics tasks to be used in a lesson and facilitating classroom discussion of mathematical concepts as the lesson was being studied (FerriniMundy & Graham, 2003; McLeod, 2003). Recommendations also called for reducing the procedural emphasis of past years such as paper and pencil computation, and increasing opportunities for students to make and defend deductive arguments like those use by mathematicians, which was reminiscent Bruner’s work in the 1960s (as cited in McLeod, 2003). The standards also recommended more focus on realworld problems. Although the recommendation was never to focus on only realworld problems, the standards were often misinterpreted to mean that (McLeod, 2003). A popular slogan of progressive reformers who favored reform strategies was that the teacher should be a “guide on the side and not a sage on the stage” (Klein, 2003, p. 2). 59 In the early 1980s, researchers had argued against “strict drill practice” in the classroom as “an influence that habituates students to an unthinking response” (Resnick & Ford, 1981, p. 19). Those who defended traditional instruction supported the role of drill and practice, which was still a contentious topic among teachers and one that many, if not most, educators considered vital in teaching and learning math at all levels (Clements, 2003). However, in the 1990s, support of the importance of conceptual teaching and learning grew rapidly (Clements, 2003; Fey & Graeber, 2003; McLeod, 2003). Hiebert and Carpenter (1992) stated, In summary, research efforts now being directed toward uncovering relationships between conceptual and procedural knowledge now appear to be more useful than earlier attempts to establish the importance of one over the other. At this point, both theory and available data favor stressing understanding before skill proficiency. (p. 79) The idea and practice of discovery teaching and learning was born from the educational theory of constructivism (Brooks & Brooks, 1999; Lambert et al., 2002). In fact, constructivism was the driving force of the standards movement and a prominent theme in mathematics education in the 1990s (Clements, 2003). Hiebert and Carpenter (1992) stated, “It is now well accepted that students construct their own mathematical knowledge rather than receiving it finished from the teacher or a textbook” (p. 74). They added that it seemed evident from research that procedures and concepts should not be taught as isolated bits of information. Further, they identified the goal as “an attempt to teach students to make the same kinds of connections observed in experts” (Hiebert & Carpenter, 1992, p. 81). Rather than taking a singular focus, as in previous studies, researchers in the 1990s focused on pairing research on teaching with research on learning (Koehler & Grouws, 1992). Consistent with the themes of constructivism, Koehler and Grouws (1992) identified research 60 from the 1980s and early 1990s as a model whose outcomes of learning were based on students’ personal actions or behaviors. They advocated that mathematics teaching and learning research perspectives in the 1990s varied. Koehler and Grouws reviewed constructivist theories and compared and contrasted those of several wellknown researchers, including Leinhardt (1989), Lampert (1990), Hiebert and Wearne (1988), and Cobb et al. (1991). They revealed important similarities and differences among the models. However, in conclusion, they stated, “All these perspectives accepted the premise that students are not passive absorbers of information, but rather have an active part in the acquisition of knowledge and strategies” (Koehler & Grouws, 1992, p. 123). Whatever the means by which students construct knowledge, Koehler and Grouws (1992) believed, “In most cases, these actions are influenced largely by what the teacher does or says within the classroom” (p. 117118). The strong support for innovative teaching standards at the beginning of the 1990s and the research published in the years that followed—all advocating reformed classrooms—had unexpected outcomes, and in fact, the standards publications became the focus of significant controversy (FerriniMundy & Graham, 2003). Clements (2003) stated that NCTM called for reformoriented teachers to create constructivists classrooms; that was the rhetoric, but the reality was quite different. Teachers were often unaware of recommendations for standardsbased curriculum and instruction or they chose not to implement them (Clements, 2003). McLeod stated that both advocates and detractors of reform quickly claimed that reform policies of the NCTM standa 
Date  2015 
Faculty Advisor  Holt, Charles 
Committee Members 
Borgemenke, Art Denson, Kathleen 
University Affiliation  Texas A&M UniversityCommerce 
Department  EdD Educational Administration 
Degree Awarded  Ed.D. 
Pages  146 
Type  Text 
Format  
Language  eng 
Rights  All rights reserved. 



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