184.108.40.206.1.2Foundational PrinciplesWhat Children Bring to School:Prior Knowledge and DispositionHow and What ChildrenShould LearnAssessment in aComprehensive,Standards-Based Curriculum2.2Design ofEveryday Mathematics2.2.1 The Spiral: How EverydayMathematics DistributesLearning2.2.2 Everyday MathematicsInstructional Design2.2.3 The Development Process: ACommitment to IterativeImprovementReferencesContentsThe principles that informed the original design and the continuing iterativedevelopment of Everyday Mathematics are founded in the learning sciences,authoritative recommendations, and the authors’ experience and judgment.They underlie an instructional approach that maximizes student learning whilekeeping teachers’ work manageable.Section 2Copyright McGraw-Hill Education. Permission is granted to reproduce for classroom use.2.1.3Guiding Principles for theDesign and Developmentof Everyday Mathematics2.1 Foundational PrinciplesThe foundational principles that guide Everyday Mathematics’ developmentaddress what children know when they come to school, how they learn best,what they should learn, and the role of problem solving and assessment inthe curriculum.2.1.1 What Children Bring to School:Prior Knowledge and DispositionA body of research, including research by the original developers of EverydayMathematics, Max and Jean Bell, has established new understandings aboutthe informal mathematical knowledge of young children. These understandingscontinue to inform the development of the curriculum. Children construct mathematical understandings and problem-solvingstrategies by building on their own knowledge and experiences. Mostchildren begin school knowing a great deal about numbers, measurement,and geometry. Children have abundant common sense and knowledge oftheir everyday worlds and come to school with inquisitive dispositions andGuiding Principles for the Design and Development of Everyday Mathematics0019 0030 EM4 T IG GK6 S2 140777.indd 19Program: Everyday Math 4Component: Implementation GuideVendor: MPSGrades: K–61901/04/2015 10:06 amPDF Pass
positive attitudes. Curricula can aim higher than in the past by building onthese early dispositions, attitudes, and knowledge. Children’s social environments, including peers, parents, teachers, and otheradults, are also critically important to their development. Teachers’ workinvolves connecting children’s experiences with the discipline of mathematics.The curriculum should foster strong home-school connections. Children learnbest when home and school work in partnership. Children should expect mathematics to make sense. They should learn thatthings are true in mathematics not simply because the teacher says so butbecause mathematics is a coherent network of ideas that fit together.Mathematics should be woven into daily classroom routines so that itbecomes a habitual way of making sense of the world. Hard work is more important than talent for success in mathematics.2.1.2 How and What Children Should LearnEveryday Mathematics incorporates findings from more than 50 years ofresearch in mathematics education. This edition integrates established findingswith more recent research about how best to teach specific content. The elementary school curriculum should help children progress fromintuitions and concrete operations to abstractions and symbolicmanipulations, while at the same time building new intuitions that willmature in middle school and beyond. The development of skills and concepts works best if spread overrelatively long time periods, sometimes over months or years, with earlyintroduction, multiple exposures with increasing sophistication, and spacedpractice. Concepts, skills, and applications should be interwoven over time. All children can and should learn mathematics. All children can masterthe Common State Standards for Mathematics (CCSS-M) and becomeproficient with the practice standards, if they receive high-qualityinstruction and work hard. Children should be doing the thinking in mathematics class. Whole-classdiscussions, small-group explorations, individual and group practice,problem-solving activities, and guided instruction all have a place in abalanced curriculum. Varied modes of instruction with rich, high cognitivedemand content facilitate many opportunities for differentiation. Children should have a voice in mathematics class; they should explain,compare, and discuss problems and solutions. A classroom with lots of “mathtalk” helps make children’s thinking visible, which informs teacher decisionmaking during a lesson. Reasoning, conjecture, and proof are closelyconnected to communication and discourse. The beginnings of mathematicalproof are in children’s explanations and justifications of their ideas.Copyright McGraw-Hill Education. Permission is granted to reproduce for classroom use.Section 2How Children Learn BestWhat Children Should Learn 20The Common Core provides a framework that guides curriculumdevelopment, but coherent instructional materials are needed to translatestandards into implementable classroom reality. In such instructionalGuiding Principles for the Design and Development of Everyday Mathematics0019 0030 EM4 T IG GK6 S2 140777.indd 2001/04/2015 10:06 amProgram: Everyday Math 4Component: Implementation GuideVendor: MPSGrades: K–6PDF Pass
materials, the CCSS-M Standards for Mathematical Practice should beembedded in every lesson.Fluency with basic facts is essential for building number sense, estimationskills, and flexibility in problem solving. In early work, the curriculum shouldemphasize strategies. Later, it should include practical routines to help buildfact fluency. Games can be especially useful for building fluency. Instruction should connect children’s common sense with formalmathematics. By connecting children’s everyday experience withmathematical concepts and procedures, teachers can help children learnto use the mathematics they are learning. Realistic applications andmathematical modeling should be at the core of school mathematics. Conceptual understanding supports learning of both content andprocedures. Understanding includes knowing how to carry out aprocedure, why that procedure works, how that procedure can be used tosolve problems, how mathematical ideas can be represented in variousways, and how concepts, procedures, and representations are connected.Copyright McGraw-Hill Education. Permission is granted to reproduce for classroom use.Learning Through and About Problem Solving Children can and should devise their own methods for solving problems.If children are to learn how to solve problems, it is not enough to drill themon procedures. Cognitive demand should be maintained at a high level. Children shouldlearn that not every problem can be solved in three minutes or less. Someproblems should be novel, not routine, and should take longer, sometimesmuch longer. Children should learn and come to expect that there are multiple strategiesfor solving problems. They should make sense of others’ strategies andsolutions, which deepens their understanding of the content and practicesaddressed in the problem. Discussions should compare strategies in termsof ease of use, understandability, and efficiency.For more information onproblem solving and theStandards for MathematicalPractices, see Section 1.2Standards for MathematicalPractice.Section 2 2.1.3 Assessment in a Comprehensive,Standards-Based Curriculum Assessment should be ongoing, should reflect the types of activities inwhich students are engaged, and should use data from a variety of sources.Assessment tasks should themselves be valuable learning experiences.Most importantly, assessment should provide actionable information thatteachers can use to make decisions about instruction. The curriculum should prepare students to perform well on high-qualityassessments, including those from PARCC and SBAC. The best preparationfor such tests is a proven, research-based curriculum that is aligned withthe content of those tests and is coherent within and across grades toallow for continuous, cumulative learning.2.2 Design of Everyday MathematicsThe work of curriculum developers is to translate general principles intopractical tools that teachers can use. The Everyday Mathematics authors usedthe principles outlined above to develop a comprehensive and cohesiveeducational experience for students and their teachers.Guiding Principles for the Design and Development of Everyday Mathematics0019 0030 EM4 T IG GK6 S2 140777.indd 21Program: Everyday Math 4Component: Implementation GuideVendor: MPSGrades: K–62102/04/2015 10:22 amPDF Pass
2.2.1 The Spiral: How Everyday MathematicsDistributes LearningIn a spiral curriculum, learning is spread over time rather than concentratedin shorter periods. Content is revisited repeatedly over months and acrossgrades. The “spacing effect”—the learning boost from distributing rather thanmassing learning and practice—has been verified by many researchers fordecades: “Space learning over time” is the first research-basedrecommendation in a practice guide from the U. S. Department of Education’sInstitute of Educational Sciences (Pashler et al., 2007). In a recent literaturereview, Son and Simon write, “On the whole, both in the laboratory and theclassroom, both in adults and in children, and in the cognitive and motorlearning domains, spacing leads to better performance than massing” (2012).The spiraling of instruction and practice has been a defining characteristic ofEveryday Mathematics since its inception and has proven effective: EMstudents outscore comparable non-EM students on assessments of long-termlearning, such as end-of-year standardized tests. Spiraling leads to better longterm mastery of facts, skills, and concepts.The Spiral in Everyday MathematicsUsing the design principles that have made previous editions successful, thisedition of Everyday Mathematics (EM) has been rebuilt from the ground up tosupport the Common Core State Standards for Mathematics (CCSS-M). Inparticular, the content defined by the standards was carefully sequenced andincorporated into the spiral to distribute both instruction and practice ofconcepts and skills to maximize long-term learning.Section 2For information on the CCSScontent standards and EverydayMathematics’ GMCs, seeSection 1.1.5 Unpacking theContent Standards.Common Core State StandardsStandards for Mathematical ContentDomain Operations and AlgebraicThinking 1.OAEveryday MathematicsGoals for Mathematical ContentCluster Represent and solve problems involving addition and subtraction.1.OA.1 Use addition and subtraction within 20 to solve wordproblems involving situations of adding to, taking from, puttingtogether, taking apart, and comparing, with unknowns in allpositions, e.g., by using objects, drawings, and equations with asymbol for the unknown number to represent the problem.the first standardunder this DomainGMCGMCSolving number stories byadding and subtracting.Model parts-and-total, change,and comparison situations.program goals forfiner-grained tracking ofstudent progressCopyright McGraw-Hill Education. Permission is granted to reproduce for classroom use.Each grade’s Common Core content standards were unpacked into 45 to 80Everyday Mathematics Goals for Mathematical Content (GMC). The standardsand the corresponding GMCs are listed in the back of the Teacher’s LessonGuide for each grade. See excerpt below.Excerpt from the table listing the Common Core State Standards andcorresponding GMCs, Grade 1 Teacher’s Lesson Guide22Guiding Principles for the Design and Development of Everyday Mathematics0019 0030 EM4 T IG GK6 S2 140777.indd 2201/04/2015 10:06 amProgram: Everyday Math 4Component: Implementation GuideVendor: MPSGrades: K–6PDF Pass
The Common Core standards and corresponding GMCs were carefullyintegrated into the unit structure of the curriculum in three phases:Introductory Instruction, Developing Instruction, and Concluding Instruction.For each GMC within a content standard, Introductory Instruction beginsthrough exploration or focus activities in one or more lessons in a unit. Initialpractice also begins in this unit. Developing Instruction continues in succeedingunits that include additional activities that teach the concept or skill andprovide continuing practice and applications. This phase may include “pauses”during which little or no instruction or practice related to the concept or skilltakes place. In the final phase, Concluding Instruction, work narrowly focusedon the concept or skill is completed, as is most practice. Instruction during thisphase involves practice for long-term retention, applications in more complexproblem situations, occasional review, and generalization and transfer.For more information onassessment, see Section 9.3Assessment Opportunities.Spiral TransparencyA major goal of Everyday Mathematics 4 is to make the spiral moretransparent to teachers and district leaders. Teachers will be able to seewhere content is introduced, developed, and concluded within a grade; whereit is assessed; and when students are expected to master curricular goals andstandards. This information will help teachers know “when to worry” and when(and how) to intervene with students who struggle to meet expectations for agiven curricular goal or standard. Teachers will also know when “watchfulwaiting” is appropriate and intervention may be premature.For more information onassessment and trackingtools, see Section 9.4Assessment Tools.The Spiral Snapshot is a lesson-level feature that sketches previous and futureexperiences with one of the content goals that is a focus of the lesson. TheSpiral Snapshot is intended to help teachers connect each lesson’s content toprevious and upcoming lessons.Section 2Copyright McGraw-Hill Education. Permission is granted to reproduce for classroom use.Three features of Everyday Mathematics 4 are particularly useful for makingthe spiral more transparent
Instruction should connect children’s common sense with formal mathematics. By connecting children’s everyday experience with mathematical concepts and procedures, teachers can help children learn to use the mathematics they are learning. Realistic applications and mathematical modeling should be at the core of school mathematics.
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