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All students will be provided grade-appropriate curriculum and instruction; some students will receive additional support and services, as needed
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All students will complete the CCSS by grade 11, for a high probability of success on the PARCC end-of-course exams and an even higher probability of college and career readiness for high school graduates
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All seniors will have opportunities for college-level coursework, preparation for calculus, or continued work at the college- and career-ready level, depending upon individual interests
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Interested and motivated students will have opportunities to complete college-level calculus while in high school
Note: The Worthington mathematics program was already well poised for such revisioning. Because most Worthington students have been taking Algebra 1 in eighth grade (or earlier) much of the right content was already there. Furthermore, the UCSMP course sequence helped fend off the “faster is better” mentality …
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Integrated mathematics is low-level mathematics. In Ohio and across the country, there is an unfortunate history of using the phrase “integrated mathematics” to describe low-level courses designed for students deemed not ready for grade-level mathematics. Most districts abandoned such courses because they were largely ineffective. The phrase “integrated mathematics” in fact describes any high school mathematics course that explicitly attends to content across algebra, geometry, and statistics. An integrated course sequence for the
Common Core State Standards should complete the same mathematics content in the same amount of time as a sequence of traditionally-named courses.
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Algebra/Geometry/Algebra (AGA) is the natural way to sequence mathematics. The vast majority of the world teaches algebra, geometry, and statistics every year in secondary school.
The AGA sequence is an historical artifact. It developed (one course at a time) over the 19 th century because of increasing university admissions requirements, particularly in the Ivy League.
Around the turn of the 20 th century, Felix Klein recommended that the secondary curriculum be unified (and hence integrated) around the concept of function; many countries followed that advice. Perhaps because of the lack of centralized curricular decision making, the AGA course sequence persisted in most of the U.S. for most of the 20 th century. … despite the fact that the sequence was not designed for all students, and even more students enrolled in more courses of the sequence.
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Colleges won’t approve. There are anecdotes from the 1990s about colleges and the NCAA questioning the high-school appropriateness of integrated courses from individual districts. But given the then-common practice of teaching middle grades mathematics to high school

students, they were right to raise the questions. But both college admissions offices and the
NCAA know that course names are only very rough approximations of what was taught. o In almost every state, some high schools have integrated mathematics o Colleges figure this out for home school students o College admissions officers ask questions o College admissions officers pay attention to test scores, recommendations, and school reputations o ACT/SAT scores and the PARCC end-of-course exam scores will be provide evidence of
CCSS-appropriateness for college admissions. And for highly selective colleges, AP exam scores will be more important that the names of the courses on a transcript.
Furthermore, because so many
Independent of the names of the courses students take, the CCSS requires all high school students to develop integrated understandings of algebra, geometry, and data analysis, where concepts, skills, and representations in each content strand support concepts, skills, problem solving, and reasoning in the other strands. Of course, this goal requires that students be given instructional opportunities to develop such understandings, which implies that all courses should include problems that build connections among the content strands.
There are many effective ways to organize high school mathematics into course sequences that include
A2E. The different organizations are not easily compared, because what matters is not the names of the courses but what happens in the class among the teacher, the students, and the mathematical ideas.
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Connections will be required for CCSS learning. Geometry to support algebra and statistics and vice versa.
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Because of the wholeness of it. Seeing mathematics as a coherent whole
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Required curriculum and instruction for either pathway.
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STEM as integration. More natural opportunities for mathematics in science and project lead the way. Modeling.
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Connections course at Phoenix
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Algebra every year. Not a one year vacation from algebra
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Opportunities for connections across geometry and algebra units. Enhanced in the integrated approach
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Concept of function permeated throughout.
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Much more natural to design curriculum and instruction in an integrated way, so that geometry, statistics, and algebra progress every year.
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Much more likely to reach mastery of these advanced standards in the first three years of high school.
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Traditionally-named courses will have to be integrated to some extent to incorporate the standards for statistics and modeling.

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Clearly coherence and rigor are enhanced when the courses are integrated. Why not place integration up front? Why not be explicit?
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Rigor, relevance, and relationships…
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Fluidity and flexibility
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Strengthen the program for high-achieving students
Some of these districts are taking an approach of minimal change, moving a few topics in and out of their current course offerings, believing that such changes will be sufficient. Frankly, many of these districts [or, rather, a critical mass of teachers and administrators] have not yet realized that successful
CCSS implementation will require major instructional changes inside each of their mathematics courses.
Many of these districts do not realize that the status quo is not working for significant populations of students. And many of these districts have not yet embraced the goal of ALL students reaching college and career readiness, preferring instead to continue practices of slowing students down when they are already behind.
Among the districts that understand the scope and magnitude of required changes to curriculum and instruction, one sees two possible approaches toward instructional change. [Essentially, we have two theories of change.] Some districts believe that they are most likely to promote instructional improvement from within the framework of familiar course names. Other districts believe, in contrast, that explicitly new courses will be useful catalysts for instructional change. Since it is teachers who must enact these instructional changes, teacher consensus is the most important criterion in this decision making.
[Because these two approaches are found throughout the country, in various states of progress and success, the CCSS writers were unwilling to choose one approach over the other, opting instead to leave such decisions to states and districts.]
Each approach has both benefits and challenges. Furthermore the benefits and challenges vary considerably across contexts, depending upon the particular teachers and the local community. Both approaches, if they are to be successful, will require serious work, new materials, and new habits of collaboration among teachers. The mathematics teachers in Worthington see implementation of the
CCSS as an opportunity to re-vision (redesign and reframe) the mathematics programs fundamentally, in order to embrace the ambitious goal of reaching all students. Furthermore, they believe that such changes will enhance the already high quality of the courses for high achieving and advanced students.
The Worthington teachers understand that new course names will bring the challenge of explaining the new courses to students and their parents, and they believe that students and parents will understand the benefits for all students. Motivated students will still have high-quality opportunities to take college-level calculus in high school as well as new opportunities for college-level statistics. All students will be more likely to see mathematics as a coherent whole, more likely to see connections across the
STEM disciplines, and more likely to see mathematics and relevant and worthwhile. Furthermore, more students will be likely to graduate high school ready for college and careers.

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Georgia adopted integrated high school courses statewide in 2004 (check date). They allowed district choice in 2011. The rhetoric is that their new CCSS high school courses are a blend of
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 integrated and traditional courses.
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In Delaware, 16 of 17 districts (check) began rolling out integrated high school courses in 2010
(check date), before CCSS adoption.
Utah and West Virginia have opted for integrated high school courses.
All but a few states are allowing district choice of integrated or traditional courses.
Throughout the country, there are districts with a long history of integrated mathematics. Most of these districts are likely to stay with integrated programs.
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In Ohio, Bay Village, Aurora, and others have integrated programs
Perception of fewer materials available. But in fact there are few materials currently available for both course sequences.
Perception that colleges will not understand.
Perception that parents will not understand. More communication is necessary.
First, note that this is not a new problem. Districts have always struggled with determining appropriate mathematics placements for transfer students because of the great variety in mathematics teaching and learning throughout the country—even when the course names are the same. After all, the course
“Algebra 1” on a transcript can mean many different things, depending upon the district’s program and the particular teacher, not to mention the level of engagement of the student in the course.
With the CCSS, the great majority of transfer students will be more easily organized into two groups: those coming from integrated courses and those coming from traditionally-named courses.
Based upon current draft documents regarding the end-of-course exams, the third-year courses
(Algebra II and Mathematics 3) are very close to the same course. Furthermore, the two-year sequences
(Algebra I and Geometry; Mathematics 1 and Mathematics 2) are very similar in aggregate. Thus, the transition is most challenging in the middle of this sequence…. [Get notes from Gahanna.]
Felix Klein recommended (c. 1900) that the secondary curriculum be unified (and hence integrated) around the concept of function; many countries followed that advice
In my view, the “layer cake” approach to high school mathematics that currently dominates so many secondary school mathematics programs—built on course sequences such as algebra I, geometry, algebra II, or algebra I, algebra II, geometry—is an outmoded approach in a 21st-century educational

system. Mike Shaughnessy, “An Opportune Time to Consider Integrated Mathematics,” NCTM
Presidential Message, March 2011. See http://www.nctm.org/about/content.aspx?id=28655
Quote from Cathy Seeley, former President NCTM
Quote David Connelly’s report, https://www.epiconline.org/files/pdf/ReachingtheGoal-FullReport.pdf
In all of the schools of Europe, algebra and geometry are studied simultaneously during a considerable number of years. The various mathematical subjects are more closely correlated than in this country.
(p. 173) The Reorganization of Mathematics in Secondary Education: A Report by the National
Committee on Mathematical Requirements, under the auspices of The Mathematical Association of
America, Inc. (1923)
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We must stop trying to teach mathematics by the stupid ‘water-tight compartment’ method long since abandoned or greatly modified by practically all other countries.
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[W]e should have four years of the best type of ‘integrated’ mathematics or some form of superior ‘general mathematics’ for our very best students. Unfortunately, the term ‘general mathematics’ has come to mean inferior or non-college mathematics…. This is the kind of mathematics taught in all countries outside the United States.
Shuster, C. N. (1948). A Call for Reform in High School Mathematics. The American Mathematical
Monthly, 55(8), 472-475.
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Our traditional ‘water-tight compartment’ method of teaching algebra, then geometry, then intermediate algebra, leads to a great deal of unnecessary repetition of subject matter that results in the loss of time and energy. (W. D. Reeve, 1947)
“It must be noted that we are the only developed country in the world that teaches a whole year of synthetic geometry, and nothing else, in the 10 th year, followed by a whole year of intermediate algebra in the 11 th year, and a more integrated course only in the grade 12. (SSMCIS, 1971, pp. 8-9) [Countries included in the book: Great Britain, The Netherlands, Sweden, Denmark, West Germany, Belgium,
France, Russia, Japan.]
By Jacob William Albert Young, 1907, Longmans, Green, and Co., New York.
That arithmetic, algebra, geometry, trigonometry should be taught side by side by the same teacher to the same pupils, each helping and illuminating the other, and not tandem, as is the custom in America, has long been urged. That the "water-tight compartments" be abolished, that mathematics be treated as one subject is one of the leading theses of the laboratory method. (p. 97)
Presidential address on “The Foundations of Mathematics.” Bulletin of the American Mathematical
Society, 1903. (E.H. Moore?)
We have in addition a conspicuous and easy reform to make that these other countries have long since accomplished, // namely, simultaneous instruction in algebra and geometry…. [T]he more difficult parts

of either subject are harder than the easier parts of the other, and each can be made of valuable help in the development of the other. (pp. 182-183)
The reasons for the failure to make the improvement seem to be those of inertia, of conservatism, rather than a conviction that the change proposed is not good or that the customary order is better. I know of no published defence (sic) of the traditional procedure, but still the old order, which finds no defenders in theory, persists in practice.
Writers, committees, associations, have recommended change. The time to act has come. Fortunately this is a change which can be effected in any single school without disturbing its relations with either the colleges and technical schools for which it prepares or the grade schools which furnish its pupils. The time is ripe for single schools, acting independently, to rearrange their own curriculum in mathematics.
The change can be made within the mathematical work alone without disturbing the rest of the curriculum. It consists merely in beginning both single schools algebra and geometry at the outset and carrying them side by side through the first two years. (p. 184)
Possible Reasons for Poor Achievement
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U.S. instruction focuses mostly on skills and procedures
The U.S. curriculum is redundant and repetitive o “A mile wide and an inch deep”
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U.S. tracking practices fail many students and condemn others to “terminal” courses
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The U.S. is the only country that separates mathematics into “water-tight compartments”
Possible Responses to Poor Achievement
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Balance skills, concepts, problem solving, and reasoning o Adding It Up: Helping Children Learn Mathematics (National Research Council, 2001)
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In each grade, teach fewer topics and teach them deeply o Curriculum Focal Points (National Council of Teachers of Mathematics, 2006)
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Recognize that both college and the workplace require significant mathematical preparation, through Algebra 2 or its equivalent o National Governor’s Association; Achieve, Inc.
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Teach mathematics as an integrated whole, regularly using previous mathematics in service of new ideas
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Rigor as precision in reasoning (appropriate mathematical depth)
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Sufficient access for concepts to make sense
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Using previous mathematics in service of new ideas o Extending and expanding prior mathematical understandings to develop new ones o Review is usually a waste of time

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Rule of four: explore concepts graphically, numerically, symbolically, and verbally (in context) o Interpretation in context
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Opportunities for exploration and data gathering
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Data analysis process: question, data collection, analysis, interpretation
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Using mathematics to reason and to solve “real” problems, interpreting solutions in context
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Rigor and balance: skills, concepts, and problem solving (NMP report, Adding It Up)
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Exact vs. approximate answers; significant digits and measurement error
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Standards for Mathematical Practice (and NCTM Process Standards (and Ohio’s process standards)
th
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Effective and accurate use of formal mathematical notation, vocabulary, and concepts
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Integrated understandings of algebra, geometry, number, and data analysis: Students are expected to tie things together
Connections and distinctions among functions, expressions, equations, and inequalities
Graphing technology assumed, especially for graphical solutions of equations
Symbolic techniques should emphasize structure and equivalent forms, should serve a purpose
(rather than merely drill), and the results should be interpreted in the problem context
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Families of functions and their application: quadratic, polynomial, rational, radical, piecewise, exponential
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Transformations of graphs of functions as transformational geometry o Graphs of functions have geometry
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Modeling, regression analysis, correlation vs. causation, identifying trends, drawing conclusions, making predictions
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Behavior of functions: domain, range, intervals of increase and decrease, rates of change, intercepts, zeros, number of roots, extreme values, end behavior, asymptotes, symmetry
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Sequences, recursion, and iteration; arithmetic and geometric sequences o Sequences as functions with whole-number domains
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Binomial theorem o Application of counting principles o A culmination/goal of symbol manipulation practice
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Number systems (attention to irrational and complex numbers) o Irrationals requires approximation. Notation is exact.
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Linear equations and inequalities
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Linear functions, direct and inverse proportions
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Coordinate geometry (plotting points, midpoint, distance, slope) o A function doesn’t have slope: the graph of a function does o Slope is an application of similar triangles o Distance is an application of the Pythagorean theorem
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Negative exponents and rules of exponents
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Counting techniques (multiplication and addition principles)
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Area, surface area, and volume

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Data representation and measures of center and spread
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Right triangle trigonometry
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Pythagorean theorem and similarity
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Angle sums
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Synthetic geometry, constructions, and the language of proof
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Assessment that identifies what students DO know
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Problem solving for all kids
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A full year of (separate) geometry (U.S. is the outlier)
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Connect to science, STEM, international benchmarking; math as a tool/language for science and technology (to participate in the discussion)
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Promoting integration within mathematics (before science)
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Fight for math literacy (even STEM literacy). Everyone needs it.
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Involving the public; getting the message to decision makers
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What does integration have to do with national standards?
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Better achievement, real-world problem solving, incorporate technology, other countries
(integrate mathematics first); our students need to compete internationally; social justice
Our definition: The connection of ten mathematics standards developed over time during every year of study. These standards should be examined, taught, developed, and deepened over time. The integrated approach will help students make sense of mathematics, and is founded in the evidence of how people learn. Furthermore, this presentation of material is supported by the curricula of highperforming nations. There must be a balance between application, skill, and theory. Students must be able to not only make sense of mathematics, but to see the relevance to their lives and the connection to their future (in mathematics) education.
Coherent mathematics integration promotes higher K-12 mathematics achievement/understanding by connecting mathematics concepts and thoughts across algebra, geometry, and data analysis.
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Consistent vocabulary defined across mathematics curricula
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Problem solving in realistic environments
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Consistent practice
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Multiple pathways to solutions
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Integrating technology in resolving and exploring practical mathematics
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Integrating teacher learning and team processes in learning (student where appropriate)

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Not just resequencing—truly blended—topics
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Requires motivation through problems
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Builds on modeling and connection in the Common Core
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Better than applications within each standard
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What do we mean by integration? Need a clear description
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Real-world problems require integration of knowledge
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Significant needs for resources and professional development
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Is resequencing topics a reasonable first step? (layering)
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Integration is necessary for the mathematical practices
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The current system is not working—even for the best students
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Need new words: blended, connected, coherent, interwoven
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All teachers recognize that they need to learn
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Don’t organize h.s. content into grade-level courses
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Pay attention to trajectories, learning progressions
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More than one way to organize curriculum
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Algebra courses need to include geometry and data analysis
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Frame connections in the beginning and highlight more
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Three dimensions: connections, progression, applications
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Content motivates content
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Simple language to convince the public
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Progressions (keep minimal), connections, language
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Artificial divisions in high school
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Reasoning is central
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People learn through rich conceptual frameworks
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Concepts and skills together, multiple sources of knowledge
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High School Geometry: A full year is inappropriate
In this statement, “mathematics” includes statistics. Reasoning and sense-making should be the underlying structure of school mathematics. In that light, the content topics outlined in the Core
Standards document needs to be organized in a way that stresses: the connections between and among topics that portrays mathematics as a coherent whole, that provides a format to actuate the
Mathematical Practices Standard, that provides rich opportunities for the Mathematical Modeling
Standard.
There are many ways to blend and sequence mathematical content topics to accomplish this.
Therefore, the recommendation is to provide only trajectories of the content standards from the Core
Standards document—especially at the high school level, instead of bucketing topics by grade.
Rationale for emphasizing connections across standards

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Cognitive science evidence regarding the importance of “rich conceptual frameworks”
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Pulling things together
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Learning concepts & skills together
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Connections in brain
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Able to employ logical deduction when you have more routes to understand
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International competitiveness. A separate geometry course overemphasizes the content, compared to high-achieving peers
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Motivation, by applications, content to content, across algebra/geometry/data, need to
 understand this
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