The Nation's Report Card With 2014 approaching as the deadline by which states must get their all their students up to “proficient” levels on state tests, the U.S. Department of Education’s top statistics agency released data today suggesting that some states may have lowered student-proficiency standards on such tests in recent years. For the 47-state study, researchers for the National Center for Education Statistics used student test scores to figure out where the proficiency levels on various state tests would lie on the National Assessment of Educational Progress. Their results suggest that between 2005 and 2007, various states made their standards less rigorous in one or more grade levels or subjects in at least 26 instances. In 12 instances, particular states appeared to make their standards more stringent in one or more grade levels or subjects. Under the federal No Child Left Behind Act, adopted nearly eight years ago, states are required to show that all students have attained proficient levels of performance in reading and mathematics by the end of the 2013-14 school year, and most states are far from reaching that goal. “I think as 2014 looms, ... clearly what a lot of states are doing is changing the bar so that a lot more students will become proficient,” said Mark S. Schneider, who served as the NCES commissioner from 2005 to November 2008. In 4th grade reading, for example, the federal study found that 31 states set proficiency scores that were lower than the cutoff for “basic” performance on NAEP. At the 8th grade level, 15 states set standards judged to be lower than the basic level on the national assessment for that grade. But the study uses a methodology that is controversial among some testing experts. They caution that the standardized exams that states use and the more rigorous NAEP—the congressionally mandated program known as “the nation’s report card”—are too different to put on the same scale. “If two tests don’t measure the same thing, then mapping is misleading,” said Andrew D. Ho, an assistant professor of education at the Harvard Graduate School of Education. “You can map anything onto NAEP.” The results, nonetheless, are expected to figure in growing efforts to develop common academic standards for what K-12 students should know and be able to do. So far, 48 states are taking part in a push to craft such standards, while federal education officials are at the same time making plans to award $350 million in grants to help states plan common assessments. ("Duncan Unveils Details on Race to the Top Aid," June 15, 2009.) The Education Department’s statistics agency, for its part, has been developing and improving its test-mapping method since 2003. According to Peggy Carr, the associate commissioner of the NCES, the aim was to develop a common metric that state policymakers could use to assess how high their states have set the bar for proficiency. It’s a challenging task: While the national assessment is intended to measure students’ progress against what they ought to know, state tests may have different goals, such as measuring whether students are demonstrating grade-level work. NAEP also draws on nationally representative samples of students from participating states; state tests, in theory, are designed to assess learning for every child. The new analysis draws on testing data in reading and mathematics for grades 4 and 8. The study does not include California, Nebraska, Utah, and the District of Columbia. Both the new report and a 2007 study that grew out of the same project show that states vary widely in what qualifies a student as proficient. A 4th grade student judged to be proficient in math in Colorado or Tennessee, for example, could conceivably test at the “basic” level in Massachusetts or Missouri, where the standards were judged to be most rigorous, according to the study. “This is great fodder for why we need common standards that are of high quality,” said Amber M. Winkler, the research director for the Thomas B. Fordham Institute, a Washington think tank that supports common academic standards. In collaboration with the Northwest Evaluation Association, a testing group based in Lake Oswego, Ore., Fordham in 2007 published a report that mapped state NAEP data with state-assessment data from the Northwest group and found variations similar to those in the NCES study across states. ("Report Pans How States Set the Bar," Oct. 10, 2007.) “I think the intent of NCLB was to let states make that decision” on where to set the proficiency bar, added Ms. Carr, “but I don’t think anyone conceived that we would be this discrepant or this disparate in our definitions about what proficient meant.” The new study is the first by the NCES to analyze whether states had changed their testing standards, either implicitly or explicitly, over time. The report stops short, however, of alleging that states may be changing their standards to meet their NCLB targets for raising students to proficiency in reading and math. Under the law, students are tested annually in those subjects in grades 3-8 and once during high school. “There are different ways states could’ve gotten to this point,” Ms. Carr said. “Some states just kept their assessment exactly the way it was and just ratcheted the cut scores up or down. Others made many changes in their testing systems so, of course, they had to have new cut points on top of their new assessments.” In all, the study notes, between 12 and 18 states, depending on the grade and subject, made changes in their state testing systems between 2005 and 2007. One was Wyoming, where the proficient bar appeared to have slipped in both reading and math and at both grade levels studied. “There are really no surprises in Wyoming for this report,” said William Donkersgoed, the state’s national-assesssmen t coordinator. “There was a change in the assessment system, and it was like comparing apples and oranges from 2005 to 2007.” Mr. Donkersgoed said the state’s previous system, like NAEP, was an “aspirational” system. It was developed so that schools could measure their progress against a high standard, and there were no consequences for schools that fell short. The new system, in contrast, was devised in part by panels of teachers examining what grade-level performance should look like. Rising testing standards in Montana, on the other hand, did not seem to be linked to any dramatic, or intentional, changes in the state testing system, according to Mike Chapman, the NAEP coordinator for that state. He said state officials surmise that the change resulted from instituting tests in additional grades and offering teachers more professional development—two actions that may have had indirect ramifications for the level of that state’s 4th and 8th grade cutoff scores, which are recalibrated from year to year. “In most states, the actual cut scores haven’t changed,” said Mr. Ho of Harvard. “What’s happening is that a change on one test is outpacing the change on another.” Just because some states may be drifting away from the NAEP tests, he said, it doesn’t necessarily mean they are either toughening or loosening their testing standards. Besides measuring the changes in state standards over time, the NCES researchers looked to see whether achievement gains on state tests tracked with improvements in state students’ performance on the national assessment. For states that had not overhauled their testing systems, that was the case 60 percent to 90 percent of the time, said Ms. Carr. “That’s good,” she added, “because that is what NAEP was intended to do.” Vol. 29, Issue 10 The CT Coalition For World Class Math is raising a warning flag now, in order to avoid a reopening of the math wars. The AOA curriculum is modeled after the most extreme theories of conceptual understanding alone. CT should not be the last state to embrace the resolution of the math wars. It is time to put those battles behind us and agree, conceptual understanding and computational fluency are mutually reinforcing and neither should be given pre-eminence in mathematics education. Enduring Understandings Introduction The Connecticut Algebra One for All Course Overview establishes eight Enduring Understandings, or Big Ideas, which set forth a framework for establishing curricular priorities for the Algebra I Model Curriculum that the CT Academy of Education has been commissioned to develop. By definition, the Enduring Understandings must have value that lasts beyond the classroom, reside in the heart of the discipline of mathematics, uncover abstract or often misunderstood ideas, and offer potential for engaging students. (Wiggins and McTighe, 1998) The position of the CT Alliance4 World Class Math is that these Enduring Understandings fail to meet these necessary objectives because they are neither clear nor appropriate and because they fail to represent the Big Ideas in Algebra. The Enduring Understandings and Big Ideas must be in alignment with state and national mathematics standards that are internationally benchmarked against the top-performing nations and those states with exemplary mathematics standards (MA, IN, CA) in keeping with the recommendations of NCTM’s Curricular Focal Points and the National Mathematics Advisory Panel (NMAP) Final Report. Furthermore, the proposed Enduring Understandings fail to take into account the research and recommendations of the Achieve American Diploma Project (ADP) despite the fact that, as a member state and “critical friend”, Connecticut has access to Achieve’s benchmarked standards, assessments, and existing model curricula developed by mathematicians, educators and experts in pedagogy. What is the objective of the Algebra I Model Curriculum? As stated in the CSDE Algebra I Model Curriculum Grant, “The purpose of these model curricula is to ensure common standards and consistency in the content of core courses throughout the state.” Consequently there should be consistency such that adhering to the Enduring Understanding/Big Ideas would prepare Connecticut students for college and career in a cohesive progression of mathematics education PreK-20. However, upon comparing the Enduring Understandings to the NMP Major Topics of Algebra, the topics covered by the Accuplacer assessment used by Connecticut state universities for proper mathematics placement, and the American Diploma Project’s Algebra I Content Standards, it becomes clear that the Enduring Understandings fail to represent exemplary state and national standards such that the objectives of the curriculum grant, and by extension of the Connecticut Plan, may be met. (See Algebra Comparison Chart) If the objective of the Algebra One for All project is to “reduce the student achievement gap and impose greater rigor, student engagement, and 21st century learning expectations” as stated in the Response to the State Board of Education and Commissioner’s Secondary School Reform Initiative of May 30, 2008, then we must identify the desired results, determine what constitutes acceptable evidence of competency in the outcomes and results (assessment) in order to be able to plan instructional strategies and learning experiences that bring students to these competency levels. (Wiggins and McTighe, 1998) What are the desired results? We must do a better job of preparing Connecticut students for college and career. “All districts will ensure that every student, by no later than grade eight, is prepared to succeed in an authentic algebra course as identified in the National Mathematics Advisory Panel’s March 13, 2008, Report.” (Response to the State Board of Education and Commissioner's Secondary School Reform Initiative, 2008) Having identified this as our objective, we must work backwards from Pre-K through 12th grade if students are to be successful. One hundred twenty thousand students are misplaced in their eighth-grade math classes. They have not been prepared to learn the mathematics that they are expected to learn. This unfortunate situation arose from good intentions and the worthy objective of raising expectations for all American students. Two groups of students pay a price. The misplaced eighth graders waste a year of mathematics, lost in a curriculum of advanced math when they have not yet learned elementary arithmetic. They should be taught whole number and fraction arithmetic so that they can then move on to successfully learn advanced mathematics. No element of this story is educationally sound. It arose from good intentions: to democratize advanced math courses by assigning students to Algebra I, Geometry, and Algebra II who were once locked out of such courses. But this is false democratization. No social benefit is produced by placing students in classes for which they are unprepared. Indeed, it is difficult to imagine any educational benefit accruing to these students. They do not possess the family or school resources to overcome problems arising from taking inappropriate courses. Let us not forget the hundreds of thousands of well-prepared students—who are also predominantly black, Hispanic, or poor—sitting in the same classrooms as the misplaced students and equally deserving of a good education. Well-prepared students need a real algebra class, not a fake one teaching elementary school mathematics. Any teacher who stops to teach misplaced students fractions shortchanges the well-prepared students who sit in that algebra class. (Loveless, September 2008) We must do a better job of teaching and assessing prerequisite skills. Top-performing countries prepare their students for Algebra by focusing on the critical foundations, aligning instruction throughout a child’s education, and teaching critical topics to mastery. Proficiency on these fundamental mathematical topics needs to be acquired before entry to algebra. Indeed, in a 2008 study of students in San Diego, Zau and Betts found that fourth-grade math scores were as good at predicting success on the California high school exit exam as ninth-grade scores. This finding suggests that elementary mathematics is essential and failure to learn it has long term consequences. (Loveless, September 2008) The National Mathematics Advisory Panel (NMAP) in the Critical Foundations of Algebra, considers the following guideposts of concepts and skills foundational for formal algebra coursework: 1) Fluency with whole numbers; 2) Fluency with fractions; and 3) Particular aspects of geometry and measurement. Proficiency with whole numbers, fractions, and particular aspects of geometry and measurement are the Critical Foundation of Algebra. Emphasis on these essential concepts and skills must be provided at the elementary- and middle-grade levels. The coherence and sequential nature of mathematics dictate the foundational skills that are necessary for the learning of algebra. By the nature of algebra, the most important foundational skill is proficiency with fractions (including decimals, percent, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in Algebra can be expected. (The National Mathematics Advisory Panel, 2008) See Benchmarks for the Critical Foundations, pages 3-xii-3-xiii of the Report of the Task Group on Conceptual Knowledge and Skills of the National Mathematics Advisory Panel Final Report 2008. “Connecticut’s secondary school reform alone cannot ensure higher student achievement. Consequently, state education agencies and PreK-20 educational systems, together with parents and other stakeholders, must move forward in partnership to achieve the Plan’s goals.” (Response to the SBE and Commissioner's SSR Initiative, 2008) We must focus on developing a coherent progression of instruction that emphasizes the key topics in elementary and middle school mathematics as the high-achieving nations do. “To effectively adapt and implement international standards, education leaders and policymakers should align curricula from pre-kindergarten through higher education and work force development.” (Education Commission of the States, 2009) “Every CT student upon graduation should be thoroughly prepared to enter 21st Century technical certification and/or apprentice programs, two- or four-year higher education studies, or immediate work opportunities dependent upon the student’s interests, strengths and curiosity.” (Response to the SBE and Commissioner's SSR Initiative, 2008) Algebra is widely recognized as a “gateway” course and as the single strongest predictor of completion of a bachelor’s degree. Completing a course beyond advanced algebra, such as pre-calculus or statistics, more than doubles the chance that a student entering college will complete a degree and is the single greatest predictor of lifetime earning potential, cutting across gender and other demographic groups (Adelman, 1994). As such, we will reduce the achievement gap when we adequately prepare our students for authentic algebra as defined by NMAP. A recent report from McKinsey & Company indicates that “when large variations in performance exist among similar operations, relentless efforts to benchmark and implement what works can lift performance substantially.” (McKinsey & Company, Social Sector Office, April 2009) In order to reduce the achievement gap we must benchmark the performance of our students against their international peers. As it stands, Connecticut participates in the American Diploma Project Network , but has not fully utilized the benchmarking services or participated in Achieve’s Alignment Institutes. Furthermore, there must be a plan in place to address the needs of remedial students if we are to succeed in closing the achievement gap. “Research shows that education systems in the United States tend to give disadvantaged and low-achieving students a watered down curriculum and place them in larger classes taught by less qualified teachers—exactly the opposite of the educational practices of high-performing countries.” (National Governors Association, 2008) Consequently, we must closely monitor individual student progress and intervene early to prevent students from falling behind. What constitutes acceptable evidence of competency? The Connecticut Plan references national curriculum initiatives including the America Diploma Project, of which the state of Connecticut is a member. Acceptable evidence of competency will be attained when Connecticut students are able to meet international benchmarking criteria as established by objective measures that compare them to their peers in other states, and ultimately in the top-performing nations. The American Diploma Project has developed and made available End-of-Course Exams for both Algebra I and Algebra II that would demonstrate the type of competency that has meaning. Additionally, Achieve has developed a Model Course Sequence for the traditional track of Algebra I, Geometry, and Algebra II as well as an Integrated Course sequence that provides a roadmap to achieve competency. Placing our model curricula in line with ADP and respecting the coherence of the Major Topics in School Algebra and Critical Foundations of Algebra as set forth by NMAP would set our state on course to meet international benchmarks thereby empowering our students to succeed in a global economy. Moreover, we must respect the state college math requirements as we consider PreK-20 curricula alignment. States with performance standards for college admission in place have seen a long-term decline in the proportion of students who need remediation. (Abraham & Creech, 2002) The current need for mathematics remediation in Connecticut colleges is unacceptable. In order to bring an end to this distressing decline, we must respect the assessment guidelines already put in place by our post-secondary institutions. For example, most Connecticut state colleges require incoming students to take an Accuplacer exam which covers the following sample topics: Elementary Algebra operations with integers and rational numbers computations with integers and negative rationals the use of absolute values, and ordering operations with algebraic expressions adding, subtracting, multiplying, and dividing polynomials simplifying exponents simplifying algebraic fractions factoring translating written phrases into algebraic expressions solving equations, inequalities, quadratic equations solving verbal problems presented in an algebraic context College-Level Mathematics simplifying rational algebraic expressions factoring and expanding polynomials manipulating roots and exponents solving linear, quadratic, and inequalities system of equations trigonometric, logarithmic and exponential functions practical problems “Mathematics, science and technology education should be perceived as a continuum of student proficiencies and teaching expectations from elementary school through post secondary education.” (Response to the State Board of Education and Commissioner's Secondary School Reform Initiative, 2008) Acceptable evidence of competency would include passing scores (as determined by the colleges) on the Accuplacer exam. We cannot arbitrarily create assessments of competency; we must respect those in existence so that the door to the real world of college and career remains open to all students. The primary component for a system of effective learning is a guaranteed and viable curriculum. This means having a Pre K-12 course of study that gives all students the opportunity to learn mathematics focused on a number of cohesive and effectively taught topics each year so they are well prepared for college and career. Such a system requires all its parts to be in alignment. This means that the standards must be supported by instructional practices that enable all students to achieve the standards. This in turn requires that all teachers understand and be prepared to deliver such instruction. It also requires that all instructional materials, including textbooks, support and align with the standards. In a world class system, assessment must also be aligned with the standards, not only in terms of topics, but by having a balance of skills, concepts, and applications, demanding that students demonstrate reasoning, problem solving, and critical thinking. Furthermore, assessments must be objective measures that are internationally benchmarked. Finally, teachers at all levels in a world class system must participate in ongoing reflection and renewal in order to prepare them to support powerful learning for all students. With all these components in place – standards that are focused, coherent, with a small number of attainable topics at each grade, teachers who are well prepared and effective in their instructional practices, and an entire system of standards, textbooks, tests, teacher preparation, and ongoing professional development that is aligned – Connecticut will produce students who are on a path to rise to world class status and achievement and graduates who will be ready to compete in a global economy. Conclusion The Enduring Understandings fail on various levels. First, they do not have value that lasts beyond the classroom because they do not address the Major Topics of Algebra as set forth by NMAP, they are not benchmarked against national and international standards of excellence, and fail to set a “common baseline” determined by the mathematics departments of our universities. Second, they do not respect the discipline of mathematics by failing to “ensure that students are proficient in computational procedures, can reason logically and clearly, and can formulate and solve problems”—objectives set forth by NMAP. Third, they fail to effectively uncover abstract or often misunderstood ideas due to a lack of clarity or coherence and because they fail to focus on the Major Topics of School Algebra. Finally, the potential for engaging students is artificial at best, as it emphasizes entertainment value and dependence on technology at the expense of authentic rigor and genuine engagement that is only earned as the result of true understanding gained through content mastery and consistent effort. Abraham, A., & Creech, J. (2002). Reducing remedial education: What progress are states making? Atlanta, GA: Southern Regional Education Board. Adelman, C. (1991). Answers in the toolbox: Academic Intensity, Attendance Patterns, and Bachelor’s Degree Attainment. Washington, D.C.: U.S. Department of Education. Adelman, C. (1994). Lessons of a Generation. San Francisco, Jossey-Bass. Minnesota Office of Higher Education, 2006. “ACT Scores.” Adelman, C. (2006). The toolbox revisited: Paths to degree completion from high school through college. Washington, DC: U.S. Department of Education. Alliance for Excellent Education. (August 2006). Paying Double: Inadequate High Schools and Community College Remediation. Achieve, Inc., American Diploma Project Network. (October 2008). Achieve ADP Algebra I End-of-Course Exam Content Standards with Comments & Examples. Achieve, Inc., American Diploma Project Network. (January 2008). Achieve Data Profile: Connecticut. Achieve, Inc., American Diploma Project Network. (May 2008). High School Traditional Plus Model Course Sequence. Algebra I, Geometry, Algebra II. Achieve, Inc., American Diploma Project Network. (March 2008). Connecticut Data Fact Sheet. Education Commission of the States. (2009). An International Benchmarking Blueprint. California Department of Education. (June 2008). Mathematics Content Standards. Connecticut Academy of Education. (May 30, 2008). The CT Algebra I Model Curriculum Proposal. Response to the State Board of Education and Commissioner's Secondary School Reform Initiative. Connecticut Department of Education. (September 2005). 2005 Connectciut Mathematics Curriculum Famework: A Guide for the Development of PreK-12 Mathematics Understanding. Connecticut Department of Education. (2008). Connecticut’s Plan for Secondary School Reform brochure. Connecticut Department of Education. (2008) The Connecticut Plan: Academic and Personal Success for Every Middle and High School Student. Connecticut Department of Education. (September 2005). 2005 Connecticut Mathematics Curriculum Framework: Grades PreK through 12 Matrix. Loveless, T. (September 2008). The Misplaced Math Student. The Brookings Institution. Massachusetts Department of Education. (November 2000). Massachusetts Mathematics Curriculum Framework. McKinsey & Company, Social Sector Office. (April 2009). The Economic Impact of the Achievement Gap in America's Schools. National Governors Association. (2008). Ensuring U.S. Students Receive a World-Class Education. NGA, CCSSO, and Acheive. The National Mathematics Advisory Panel. (2008). Final Report. The National Mathematics Advisory Panel. (2008) Report of the Task Groups and Subcommittees. Wiggins and McTighe. (1998). Understanding by Design. Zau, A. and J. Betts. (2003). Determinants of Student Achievement: New Evidence from San Diego. Public Policy Institute of California. Response to the Academy's Guiding Principles Response to Course Overview’s Introduction and Guiding Principles Introduction. The Course Overview fails to draw on the best research and scholarship available to establish a coherent theoretical approach that could be expected to produce a model algebra 1 curriculum that is consistent with or linked to state and national math standards. Right from the very start, the Course Overview lets the reader know that a radical departure from mainstream algebra instruction, research and scholarship will be pursued. The problem with the governing philosophy of the Algebra One for All (AOA) program is that, like all radical departures, we have to take a great deal on faith. No program like the AOA has ever been attempted, and certainly not on the scale that is being envisioned as part of high school reform in the State of Connecticut, which may impact the many thousands of students attending public school. The ill-conceived AOA program lacks any evidentiary-base to support its effectiveness. It is irresponsible and arrogant to use the students of Connecticut, who are compelled to attend school, to be guinea pigs in this experiment. Nonetheless, small fixes and tinkering around the edges are almost certainly not the answer to solving the crisis of math achievement in Connecticut. Major changes are needed. In order for all students to achieve success in algebra, the drafters of AOA would do well to consider how the highest levels of success were achieved in other States (California, Indiana, and Massachusetts) and the highest performing nations (Singapore, Japan, South Korea, Hong Kong, Flemish Belgium, and the Czech Republic). Internationally benchmarked standards are an essential first step. Just as an arrow can only reach its mark if it is aimed at a target, so it is with math -- set algebra standards at a high level so that curriculum and instruction can be aimed accordingly at those standards. Connecticut has neglected to set algebra standards. This oversight is glaringly apparent in the draft AOA curriculum that is now before us. To compound the problem, no attempt was made by the drafters to consider algebra standards that have been established in the larger mathematics community. Setting high algebra standards will have a secondary benefit (other than giving the curriculum designers something to aim for). To the extent schools and teachers can clearly identify what will be expected of students when they reach algebra, instruction will be improved in grades K-7/8. Connecticut students must be better prepared in the elementary and middle grades. The problems of high school are a direct consequence of poor preparation in earlier years. If the draft AOA assumes little or no math knowledge prior to high school, an opportunity will be squandered to force better math instruction and curriculum in earlier grades. 1. Scope -- the scope should refer specifically to the essential topics of school algebra as stated by the National Mathematics Advisory Panel. The scope should also refer to internationally benchmarked standards. Student appreciation of math is too vague a principle to be set out as a guiding principle. Student preparation for higher mathematics study, specifically geometry, trigonometry and calculus, should be articulated. 2. Connections Principle -- in addition to math curriculum being "explicit" and "coherent" it should be clearly stated that mathematics is undeniably cumulative. This guiding principle must acknowledge that skills and knowledge build upon each other. Deep conceptual knowledge is achieved through mastery of cumulative foundational skills. 3. Context Principle -- This principle probably overstates its own significance. Too much "real world" and "contextual" foundation can be a distraction to students and interfere with their ability to gain a deeper, more abstract, understanding. Algebra is, at heart, an abstract mathematical discipline. Purging abstraction from algebra in the name of creating a context, waters down content and leaves out important topics. To the extent context can be used to introduce or illustrate a concept, without distracting pupils with lengthy extraneous projects and information, it may be helpful. 5. Teaching principle – this principle should be stated in the reverse. Best practices and evidence-based instructional strategies should support the curriculum; curriculum should not be designed to support teachers. 6. Differentiation: this principle is vague and ill-defined. If students shared a strong foundational skill set from their K-8 years, less differentiation would be necessary to compensate for the poor or uneven prior math education; students struggling with concepts must be given sufficient practice with solving problems and gaining fluency; some students may need more repetitions and more practice, while others may be able to move on with less practice; differentiation should explicitly encourage teachers to assign more practice problems for students that may need more support and repetition to gain mastery of important algebra topics 7. Assessment principle -- Assessments, above all else, should be fair and objective. 8. Technology principle -- unless evidence based research supports the use of technology for a particular application, graphing calculators and computers should have a limited role in the classroom. Technology may well distract or prevent the mastery of computational and procedural fluency if introduced too soon or too frequently. 9. Mastery principle -- mastery of algebraic concepts must be an essential principle. Students should be given sufficient practice so that they can reasonably achieve mastery. Teachers should expect that students will master the material in order to build cumulative knowledge and skills. Chart Comparing Algebra Standards “Big Ideas” of Algebra · Operations and inverses · Equivalence (symbolic & representational) and properties · Equations and inequalities 1. The fundamental structure of algebra provides a systematic method for identifying, describing, extending, analyzing and generalizing patterns. 2. Information may be represented by physical models, diagrams, data tables, graphs and symbolic expressions, and choosing a particular representation may give different insights into the solution of a problem. 3. Algebra provides a way to understand and manipulate the parameters of any function and thus explore the affects of variables in real life situations. 4. Algebra is a process of conjecturing about the relationships among quantities and measures and enables us to estimate and describe correlations and make predictions, including extrapolation and interpolation of data. 5. Algebra provides the underlying structure to make connections among all branches of m 6. Innov 7. Algebra provides ways to describe and classify relationships and functions and use the classifications to derive models that have practical applications. 8. The practice of mathematics includes making conjectures, reducing the complexities of data sets, justifying claims, using symbolic notation efficiently and making generalizations through inductive and deductive reasoning. Algebraic Reasoning and Patterns 1.1 Understand and describe patterns and functional relationships. 1.2 Represent and analyze quantitative relationships in a variety of ways. 1.3 Use operations, properties and algebraic symbols to determine equivalence and solve problems. Numerical and Proportional Reasoning 2.1 Understand that a variety of numerical representations can be used to describe quantitative relationships. 2.2 Use numbers and their properties to compute flexibly and fluently, and to reasonably estimate measure and quantities. Geometry and Measurement 3.1 Use properties and characteristics of two- and three-dimensional shapes and geometric theorems to describe relationships, communicate ides and solve problems. 3.2 Use spatial reasoning, location and geometric relationships to solve problems. 3.3 Develop and apply units, systems, formulas and appropriate tools to estimate and measure. Working with Data: Probability and Statistics 4.1 Collect, organize and display data using appropriate statistical and graphical methods. 4.2 Analyze data sets to form hypotheses and make predictions. 4.3 Understand and apply basic concepts of probability. NMAP- Major Topics Symbols and Expressions Linear Equations Quadratic Equations Functions Algebra of Polynomials Combinatorics and Finite Probability CT State U- placement req’s Computations with integers and negative rationals The use of absolute values, and ordering Operations with algebraic expressions Adding, subtracting, multiplying, and dividing polynomials Simplifying exponents Simplifying algebraic fractions Factoring Translating written phrases into algebraic expressions Solving equations, inequalities, quadratic equations Solving verbal problems presented in an algebraic context College-Level Mathematics Factoring and expanding polynomials Manipulating roots and exponents Solving linear, quadratic, and inequalities System of equations Trigonometric, logarithmic and exponential functions Practical problems Foerster Expressions & Equations Operations with Negative Numbers Distributing: Axioms & Other Properties Equations Operations w/Polynomials and Radicals Quadratic Equations Expressions & Equations w/Two Variables Linear Functions, Scattered Data, and Probability Properties of Exponents Polynomial Operations (GCF, Factoring) Rational Algebraic Expressions Radical Algebraic Expressions Inequalities Functions and Advanced Topics Introduction to Algebra Working with Real Numbers Solving Equations and Problems Polynomials Factoring Polynomials Fractions Applying Fractions Introduction to Functions (Brown & Dolciani continued) Systems of Linear Equations Inequalities Rational and Irrational Numbers Our Press Release April 29, 2009: Media Contacts: CT Coalition4 Phone (860) 371-9871 email ctmath@sbcglobal.net Connecticut Coalition4 World Class Math Press Release Local Group Concerned About Proposed Algebra Curriculum CT Academy for Education says “NO” to guidelines established by the National Mathematics Advisory Panel and ACHIEVE’s American Diploma Project despite call to benchmark standards Hartford, CT, April 29, 2009: The CT Coalition4 World Class Math expressed shock and disappointment at the Model Algebra I curriculum being developed under a quarter of a million dollar grant from the Connecticut Department of Education. The new Algebra course is the first step in Commissioner Mark McQuillan's ambitious high school reform effort to raise achievement across Connecticut schools. The model curriculum, even at this draft stage of design, has already provoked serious concerns about its suitability for Connecticut students. Critics complain that the model Algebra course would leave out important topics that most mathematicians agree are essential to preparing students for college and career. “Connecticut has not had a successful program in mathematics for at least the last 10 years,” said Stanford University’s James Milgram, Professor Emeritus Algebraic Topology. “There is now overwhelming evidence that these mathematics curricula do not work.” Milgram predicts Connecticut will “continue its decline in math outcomes relative to the U.S. and even more dramatically, relative to the rest of the world.” · A staggering 40% of incoming college freshman at Connecticut colleges and universities need remedial math courses · Only 5% of college students take higher level mathematic classes necessary in fields such as engineering and finance · The high remediation rate comes with a steep price as well. It is estimated the state would save $12.5 million annually and students would earn an additional $16.4 million if the remediation rate were lowered Algebra is considered a gateway course for students, as data by the College Board found a close correlation between completion of Algebra in high school and the ability of students to earn a degree. Nevertheless, students are arriving to our colleges and universities unprepared and in need of remediation. The Accuplacer Exam, developed by the College Board to determine placement in college level courses including mathematics, covers topics that are in keeping with the recommendations of the National Mathematics Advisory Panel and those of ACHIEVE’s American Diploma Project. The state of Connecticut is a network member of ACHIEVE, an organization created in 1996 by the nation’s governors and corporate leaders, whose goal is to improve the rigor and clarity of the process of standard-setting and testing. The CT Coalition4 World Class Math is disquieted by the drafters’ (CT Academy for Education) dismissal of ACHIEVE and the American Diploma Project. Achievement in the state as measured by critical indicators (CAPT, CMT, NAEP) is stagnant or declining. “There is clearly something wrong,” says Sandra Stotsky, Professor of Education Reform at the University of Arkansas. “Large percentages of students report taking advanced mathematics courses, such as Algebra II, trigonometry, and pre-calculus. But scores are flat. What is being taught in these courses? Are they being watered down?” questions Stotsky. Commissioner McQuillan sought to address the poor performance of many Connecticut students in his high school reform proposal known as The Connecticut Plan. “I fear that the disastrous model Algebra course could doom the Commissioner’s entire high school reform effort,” said spokesperson Laura Troidle. “Connecticut citizens are counting on the Department of Education to get this first step right. It will serve as a model for future courses and this could endanger the Commissioner’s desperately needed high school reform effort.” If you are concerned with the curriculum developed by the Academy For Education let them hear from you!
CT National Assessment of Educational Progress (NAEP) Results for Math 2009
Only 46% of CT's 4th graders and 40% of CT's 8th graders are proficient in math based on 2009 scores.
http://nces.ed.gov/nationsreportcard/states/profile.asp
NCES Finds States Lowered 'Proficiency' Bar
Common Metric Sought
Changing Standards
Did you know that CT does not have Algebra 1 curriculum standards?
The CT Academy of Education (www.ctacad.org) is in the process of drafting an Algebra 1 curriculum together with:
CT Association of Teachers of Mathematics
The CT Council of Leaders of Mathematics
The CT Basic Skills Committee of CT
Math professors from 2-year colleges
Project to Increase Mastery of Mathematics and Science
CT Academy of Education, (860) 346-1177, www.ctacad.org
If you are concerned with the curriculum developed by the Academy For Education let them hear from you!
A must see video of testimony by Michael DeBell, Director of Seattle School Board of Education
http://www.youtube.com/watch?v=6ywxLqte6lc
The CT SDOE Grant: Request for Proposal #981
CT Academy For Education: grant proposal
Our Responses To the Algebra Curriculum Developed by the CT Academy For Education:
Response to the Academy's Enduring Understandings
How do we assure that our students are prepared for algebra in eighth grade?
How do we reduce the achievement gap?
How do we plan instructional strategies and learning experiences that bring students to these competency levels?
Works Cited
Turning to the Guiding Principles
(per Algebra One for All project narrative)
AOA – Enduring Understandings
-Polynomial expressions
-Rational expressions
-Arithmetic & finite geometric series
-Real numbers as points on the # line
-Linear equations and their graphs
-Solving problems w/ linear equations
-Linear inequalities & their graphs
-Graphing & solving systems of simultaneous linear equations
-Factors and factoring of quadratic polynomials w/integer coefficients
-Completing the square in quadratic expressions
-Quadratic formula and factoring of general quadratic polynomials
-Using the quadratic formula to solve equations
-Linear functions
-Quadratic functions—word problems involving quadratic functions
-Graphs of quadratic functions and completing the square
-Polynomial functions (including graphs of basic functions)
-Simple nonlinear functions (e.g., square and cube root functions; absolute value; rational functions; step functions)
-Rational exponents, radical expressions, and exponential functions
-Fitting simple mathematical models to data
-Roots and factorizations of polynomials
-Complex numbers and operations
-Fundamental theorem of algebra
-Binomial coefficients (and Pascal’s Triangle)
-Mathematical induction and the binomial theorem
-Combinations & permutations, as applications of the binomial theorem and Pascal’s Triangle
Elementary Algebra
Operations with integers and rational numbers
Simplifying rational algebraic expressions
Operations on Numbers and Expressions
Number Sense and Operations
-Reasoning with real numbers
-Using rations, rates, and proportions
-Using numeral exponential expressions
Algebraic Expressions
-Using algebraic exponential expressions
-Operating with polynomial expressions
-Factoring polynomial expressions
-Using algebraic radical expressions
Linear Relationships
Linear Functions
-Representing linear functions in multiple ways
-Analyzing linear functions
-Graphing linear functions involving absolute value
-Using linear models
Linear Equations and Inequalities
-Solving linear equations and inequalities
-Solving equations involving absolute value
-Graphing linear inequalities
-Solving systems of linear equations
-Modeling with single variable linear equations, one- or two-variable inequalities or systems of equations
Non-linear Relationships
Non-linear Functions
-Representing quadratic functions in multiple ways
-Distinguishing between function types
-Using quadratic models
Non-linear Equations
-Solving literal equations
-Solving quadratic equations
Data, Statistics and Probability
Data and Statistical Analysis
-Interpreting linear trends in data
-Comparing data using summary statistics
-Evaluating data-based reports in the media
Probability
-Using counting principles
-Determining probability
-like terms
-distributing
-variables
-decimals
- literal equations
-Trigonometric
-Quadratic Functions
-completing the square
Brown, Dolciani, et al
-variables & equations
-applications
-numbers on a line
-addition and subtraction
-Multiplication
-Division
-Transforming equations
-Solving Problems
-Extending problem solving skills
-addition and subtraction
-Multiplication
-problem solving
-quotients and factoring
-products and factors
-factoring patterns
-general factoring and its application
-algebraic fractions
-adding and subtracting fractions
-polynomial division
-Ratio and proportion
-fractional equations
-percent problems
-mixture and work problems
-problems involving exponents
-Using two variables
-Linear equations
-functions
-variation
-solving systems of linear equations
-application
-inequalities in one variable
-combining open sentences
-inequalities in two variables
-rational numbers
-irrational numbers
-radical expressions
Quadratic Functions
-quadratic equations
-Using quadratic equations
Lisa Lopez
Lynn Guezlow
Laura Troidle
World Class Math