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In the United States, the prevailing high school mathematics course sequence begins with a year of Algebra I, followed by a year of geometry and a year of Algebra II. Educators and others have raised concerns about the extent to which this sequence, which prioritizes the mastery of algebra, is appropriate for the longer-term education and career goals of students who do not intend to pursue STEM degrees in college. These concerns have impelled educators and policymakers to reexamine the prominence of algebra in high school mathematics curricula and to consider new approaches that provide students with more mathematics course options better aligned with their academic and career goals. In this paper, we explore existing approaches to high school mathematics curricula as well as new developments in the field. To start, we discuss a range of high school mathematics course sequences that are currently offered across the country and look at some of the systemic challenges embedded within the traditional paradigm. Then we explore federal and state changes to the provision of high school mathematics in the early 21st century, which we follow with a look at the influence of postsecondary institutions on high school math curricula. We then introduce short case studies of innovative high school math reforms that are occurring in five states. We conclude the paper by considering the Charles A. Dana Center's new initiative, Launch Years, and how this project works to reimagine high school mathematics and its relationship to postsecondary education and careers.

We derived an index of high school academic rigor by optimizing the prediction of first-year college GPA based on high school courses taken, grades, and indicators of advanced coursework. Using a large data set (n~108,000) and nominal parameterization of high school course outcomes, the high school academic rigor (HSAR) index capitalizes on differential contributions across courses and nonlinear relationships between course grades and first-year college GPA (FYGPA). Test scores from 8th grade were incorporated in the model to isolate the effect of HSAR. High school courses with the largest contributions to FYGPA were English 11, English 10, Chemistry, and Algebra 2. Participation in AP, accelerated, or honors courses increased HSAR. The correlation of the HSAR index and FYGPA was 0.50 and 0.49 in two cross-validation samples. While the HSAR index was the strongest predictor of FYGPA, it only led to a modest improvement in overall prediction when combined with high school GPA (HSGPA) and ACT Composite score. The predictive strength of the HSAR index was consistent across different types of high schools and colleges, and subgroup differences in the HSAR index were smaller than subgroup differences in ACT Composite score. Implications for high school counselors, researchers, and postsecondary student service personnel are discussed.

We investigate the relationship between teacher licensure test scores and student test achievement and high school course-taking. We focus on three subject/grade combinations--middle school math, ninth-grade algebra and geometry, and ninth-grade biology--and find evidence that a teacher's basic skills test scores are modestly predictive of student achievement in middle and high school math and highly predictive of student achievement in high school biology. A teacher's subject-specific licensure test scores are a consistent and statistically significant predictor of student achievement only in high school biology. Finally, we find little evidence that students assigned to middle school teachers with higher basic-skills test scores are more likely to take advanced math and science courses in high school.

There is increased policy interest in extending test-based evaluations in K-12 education to include student achievement in high school. High school achievement is typically measured by performance on end-of-course exams (EOCs), which test course-specific standards in a variety of subjects. However, unlike standardized tests in the early grades, students take EOCs at different points in their schooling careers. The timing of the test is a choice variable presumably determined by input from administrators, students and parents. Recent research indicates that school and district policies that determine when students take particular courses can have important consequences for achievement and subsequent outcomes like advanced course taking. We develop an approach for modeling EOC test performance that disentangles the influence of school and district policies regarding the timing of course taking from other factors. After separating out the timing issue, better measures of the quality of instruction provided by districts, schools and teachers can be obtained. Our approach also offers diagnostic value because it separates out the influence of school and district course-timing policies from other factors that determine student achievement.

The Common Core State Standards for Mathematics (CCSS-M) set rigorous standards for each of grades 6, 7 and 8. Strategic Education Research Partnership (SERP) has been working with two school districts, Oakland Unified School District and San Francisco Unified School District, to evaluate extant policies and practices and formulate new policies based on the CCSS-M shifts that promise to work better for students. This work has culminated in school board adoptions of new policies in both districts. This report focuses on the problems, solutions and designs that that have been found in the CCSS-M and can be of immediate use to other districts and states.

This paper examines the effects of policies that increase the number of students who take the first course in algebra in 8th grade, rather than waiting until 9th grade. Extending previous research that focused on the Charlotte-Mecklenburg school system, we use data for the 10 largest districts in North Carolina. We identify the effects of accelerating the timetable for taking algebra by using data on multiple cohorts grouped by decile of prior achievement and exploiting the fact that policy-induced shifts in the timing of algebra occur at different times in different districts to different deciles of students. The expanded data make it possible to examine heterogeneity across students in the effect of taking algebra early. We find negative effects among students in the bottom 60% of the prior achievement distribution. In addition, we find other sources of heterogeneity in effects.

This study investigated middle school students' conceptual understanding of algebraic equations. Participants in the study--257 sixth- and seventh-grade students--were asked to solve one set of algebraic equations and to generate story problems corresponding with another set of equations. Structural aspects of the equations, including the number of operations and the position of the unknown, influenced students' performance on both tasks. On the story-writing task, students' performance on two-operator equations was poorer than would be expected based on their performance on one-operator equations. Students made a wide variety of errors on the story-writing task, including (a) providing story contexts that reflected operations different from those in the given equations, (b) failing to provide a story context for some component of the given equations, (c) failing to include mathematical content from the given equations in their stories, and (d) including mathematical content in their stories that was not present in the given equations. The nature of students' story-writing errors suggests two main gaps in students' conceptual understanding. First, students lacked a robust understanding of the operation of multiplication. Second, students demonstrated difficulty combining multiple mathematical relationships into coherent stories. (Contains 4 tables and 2 figures.) [The research reported in this paper was supported by the McDonnell Foundation (Grant No.JSMF 97-56), the Interagency Education Research Initiative (Grant No. REC 0115661), and the Wisconsin Center for Education Research, School of Education, University of Wisconsin--Madison.]

The term "numeracy" is used in the adult education community to include an array of mathematically related proficiencies that are evident in adults' lives and worthy of attention in adult education settings. This occasional paper attempts to describe the complex nature of numeracy as it exists today. All stakeholders--including policymakers, program directors, educators, professional developers and curriculum designers--need a full understanding of numeracy to know how to provide adults with effective numeracy instruction. While there are large-scale assessments, standards documents, and position papers, there has not been a field- and research-based synthesis of the components required for adults to be numerate, to act numerately, and to acquire numeracy skills. This paper attempts to identify and clarify the nature of these components with the hope that such identification and clarification will guide instruction, contribute to the design of assessments, frame research, and inform policy. Appended are: (1) Reviewed Frameworks and Standards Documents; (2) Context in Adult-Focused Documents; (3) Mathematical Content Strands in Selected Frameworks; and (4) Cognitive and Affective References in Selected Frameworks. (Contains 4 figures.)

In this paper, I explore two beliefs about learning and teaching that I argue are prevalent among educators. The first is the belief that subject matter expertise is sufficient for success in teaching. The second belief is that to learn a specific content area, one needs to begin with an understanding of the formal structure and abstract principles that underlie the conceptual framework of the content area before one can effectively apply that knowledge. Together, these beliefs exert pressures on formal education that may not be for the betterment of learning and teaching for all students. The foundations of these beliefs are discussed, as are implications for classroom instruction, curriculum design, and education policy. (Contains 1 table.) [This paper was supported by the Interagency Education Research Initiative.]

To reflect on the mathematics achievement of children engaging "Algebra and Analogies for Kids," the teacher researcher will explore the following inquiry: What happens when kids display analogical thinking in an algebra project? Specifically, the proposed study will examine the reasoning of five children in portfolios of best works and selected performance assessments. Their intellectual products will engage replicated Piagetian tasks for analogical reasoning in mathematics vis a vis his reflecting abstraction model. Fourth in a series of occasional papers that explore thinking and learning, this document includes a 28 item annotated bibliography.

As the Better Math Teaching Network (BMTN) matured, different strategies for sharing network learning emerged. These strategies became more formalized and intentional as BMTN members developed tools and routines to spread the learning. Our multi-year evaluation sought to understand the affordances and constraints of each strategy. In the sections below, we present the five strategies that reflect this developmental trajectory. We begin by providing examples of each strategy. We then identify the resources necessary for implementation and reflect on the affordances and constraints of the strategy. To understand the possibilities for scaling the learning of the BMTN--and other improvement networks--we analyze each strategy using Coburn's four dimensions of scale: widespread use, depth of implementation, shift in reform ownership, and sustainability.

This study investigates middle school mathematics students' views and interpretations of graphical representations as they use graphs to answer algebraic questions--specifically, questions that require them to extrapolate information from graphs. From data gathered in videotaped interviews, students' verbal responses were analyzed as well as any gestures they used during the solution or answer phases. From the analysis of both response modalities, the constructs of bounded and unbounded views of graphs developed. The effects of these constructs and students' metaphors for the graphs were tested against the students' performance on far prediction tasks. Gestures provided a particularly useful window into students' views and predicted student performance on complex tasks more effectively than speech. (Contains 4 tables and 1 figure.)

The purpose of this study was to examine rural and non-rural students in order to understand similarities and differences between their learning styles and the ease with which they learned Algebra with a graphic calculator. Two samples of students, one from a rural high school and one from a non-rural one, answered a survey asking about their use of graphic calculators. The students also were administered a Myers-Briggs Type Indicator (MBTI), a popular instrument for analyzing personality types, including learning styles. No significant differences were found between the two samples, either in their comfort with using graphic calculators to learn Algebra or in their Myers-Briggs learning-style types. For the non-rural students, the predominant Myers-Briggs type was ENFP (extroverts, intuitive, feelers and perceivers), accounting for 52.7% of the sample. Among the rural students, ENFP was also the most frequently identified Myers-Briggs type, 33.3% of the sample, followed by ESFP (extroverts, sensing, feelers and perceivers ), with 20%. The modal type was ENF. Study findings did not show any evidence that students in the rural sample achieved less academically than those in the non-rural sample. (Contains 11 tables.)

This paper describes the content, organization and rigor of the f2f and online summer algebra courses that were delivered in summers 2011 and 2012. Examining the content of both types of courses is important because research suggests that algebra courses with certain features may be better than others in promoting success for struggling students. One key finding from the literature is that algebra students should have ongoing opportunities to develop procedural fluency and conceptual understanding and engage in meaningful problem solving opportunities, rather than focusing exclusively on skill development and symbolic manipulation. Another reason it is important to examine the content of summer credit recovery courses, in particular, is due to the perception that these courses may get "watered down," rewarding students who show up for summer school but who may not have mastered the material. More specifically, the paper will address the following research questions: (1) How did the online and f2f Algebra IB courses compare in terms of the difficulty of the content? (e.g. what proportion of time in each type of course was devoted to second semester algebra, first semester algebra and pre-algebra topics?); (2) How did the online and f2f Algebra IB courses compare in terms of the nature of the content? (e.g. developing procedural skills, conceptual understanding and problem solving); (3) How did the online and f2f Algebra IB courses compare in terms of the coherence and sequencing of topics?; and (4) How did the online and f2f Algebra IB courses compare in terms of grading expectations? [What proportion of online and f2f students' grades were based on assessments (quizzes, tests) and other criteria (effort, participation, behavior, etc.)]? The authors will draw from several different sources of data that were collected in both conditions in both summers to answer the paper's research questions. These include archival data generated from the online course, course materials (syllabi, annotated tables of contents), and teacher surveys. The initial results suggest that the online course (in both summers), in comparison to the f2f courses, was more rigorous in terms of the algebra content that students were expected to learn, more coherent in terms of how topics were sequenced, and more demanding in terms of the criteria used to calculate grades. More specifically, the online course content was considered typical of second semester algebra and included a fixed set of topics that were organized sequentially within and across 5 units.

This paper describes the Milwaukee Public Schools' involvement in Equity 2000, a standards-based reform initiative to enhance mathematics education and achievement among students of color, thereby increasing their likelihood of college enrollment and completion. The study highlights efforts to support and sustain a key component of Equity 2000: districtwide policy change to end tracking and raise academic standards for all students, beginning with the requirement that all students complete algebra by 9th grade and geometry by 10th grade, and including curriculum reform to reflect National Council of Teachers of Mathematics standards. Section 1, "Introduction," presents key themes. Section 2 discusses "Equity 2000: Policy Rationale and Implementation in Milwaukee." Section 3 discusses "Increasing Student Achievement in Mathematics." Section 4, "Barriers to Student Achievement in Mathematics," discusses student attendance and preparation in mathematics, large class size and meeting diverse learning needs, and limitations on building teacher capacity. Section 5, "Getting to the Right Algebra," discusses algebra curriculum content, when algebra should be taught, and how algebra should be taught. Section 6 discusses "The Legacy of Equity 2000 in Milwaukee." Two appendixes offer the context of reform in Milwaukee's schools and other professional development math initiatives in Milwaukee Public Schools. (Contains 58 references.) (SM)

The purpose of this study was to determine the more appropriate strategy for improving eighth-grade portfolio entries. There was a comparison between two specific types of revising/correcting conferences, self-assessment and peer conferences. Most of the research concerning the writing process focuses on peer conferencing; however, it was speculated that in math, self-assessment may prove to be more appropriate. The reason behind this speculation rests in the fact that students might simply copy each others' solutions and claim it as their own. The study consisted of a total of 52 eighth-grade students enrolled in Pre-Algebra. Each was taught how to use self-assessment and peer conferencing forms with the intention of improving portfolio entries. The study examined a random sample of 32 papers revised and/or corrected using both types of forms. A seven category checklist was prepared and used as a tally sheet to record the actual changes made between first drafts and revised/corrected drafts. The findings of this study showed that a substantially higher percentage of improvements were made to original portfolio entries when students used a peer conference. The findings did not indicate that the students copied from one another nor simply copied another student's solution to an entry. This study supports the recognition of using a writing process approach to enhance written work in the subject of mathematics. It also acknowledges that eighth-grade math students can, in fact, learn from one another. The peer conferences appeared to be more beneficial than the self-assessment technique for students who used the information to improve their portfolio entries. (Author)

This document features the experiences of three science teachers who video and audiotaped classroom proceedings in order to go back over what was said in classroom events, to explore the intentions and ideas that were perhaps not fully understood at the time. Mary DiSchino, who teaches 3rd and 4th graders, teaches a science curriculum which she organizes around questions her students ask about the world. She tells the story of her students' investigation of one question, "Why do bees sting and why do they die afterwards?" Laura Sylvan has searched for ways to introduce "authentic" discussion into her 7th and 8th grade science classroom, discussion in which students are both respectful and genuinely engaged. She presents two discussions which she felt were particularly fruitful and explores what the students were saying and what she learned about them from their talk. Chris Whitbeck, who teaches middle-school science, describes an investigation he did with four students from his class on how bicycles work; he explores the deeper view he gained on how the students thought about bicycles and on what experiences influenced their thinking. (AA)

This publication describes several different teaching strategies employed to enhance students' understanding of mathematics, such as working in cooperative learning groups, giving presentations based on problems that were solved, and writing about solutions that were found. This innovative approach to teaching mathematics resulted in questioning whether traditional methods of assessment were sufficient for students who had been exposed to this teaching method and led to the development of a performance assessment as a final exam for a high school Algebra II class. Organization of the exam and how students prepared for it are discussed. Results of this innovative assessment indicate that: (1) some students performed surprisingly well and were able to explain concepts in detail; (2) if a question was rephrased, students were able to do very well explaining a problem in contrast to drawing a blank on a traditional test; and (3) others did not do as well as expected, demonstrating an inability to explain concepts, but instead, relying on memorized facts and simple computation. Suggestions for test administration and an evaluation sheet for the assessment of student discussions are appended. (Author/LL)