mathematics progressions – common core
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DESCRIPTIONMathematics Progressions – Common Core. Elizabeth Gamino , Fresno Unified Public Schools Denise Walston, Council of the Great City Schools. Purpose. Review and take a closer look at the Mathematics Progressions Focus, Coherence, and Rigor Instructional implications Scaffolding - PowerPoint PPT Presentation
Mathematics Progressions Common CoreElizabeth Gamino, Fresno Unified Public SchoolsDenise Walston, Council of the Great City Schools
S1PurposeReview and take a closer look at the Mathematics Progressions Focus, Coherence, and RigorInstructional implicationsScaffoldingExplain their thinking; critique the reasoning of othersDeveloping mathematical argumentsReading and interpreting real-world problemsTechnical language of the discipline
2Instructional ShiftsMathematicsA Look Back
S3Mathematics Instructional ShiftsFocusCoherenceRigorConceptual understandingProcedural skill and fluencyApplications
4 FocusSignificantly narrow and deepen the way that time and energy is spent in the math classroom Communicate focus so that it is manageable in instruction; it is more than merely writing a standard a dayFocus deeply on those concepts emphasized in the standards
Provide the time for students to transfer mathematical skills and understanding across concepts and grade levelsDeep conceptual understandingConnect conceptual and procedural understandingTransitions from concretepictoriallanguageabstract
5CoherenceCoherence provides the opportunity for students to make connections between mathematical ideas and across content areasConnects the learning both within a grade and across gradesThinking across gradesEach standard is not a new event, but an extension of previous learningAllows students to see mathematics as inter-connected ideasMathematics instruction cannot be relegated to merely a checklist of topics to cover, but instead must be centered around a set of interrelated and powerful ideas, rather than a series of disconnected topics
6RigorConceptual UnderstandingInvolves more than getting the right answerAccess concepts from multiple perspectivesTransitions from concretepictoriallanguageabstractProcedural Skill and FluencyStudy algorithms as a way to see the structure of mathematics (organization, patterns, predictability) or apply a variety of appropriate procedure flexibly to solve problemsStudents are expected to achieve speed and accuracy with simple calculations (at specific grade levels)Fluent is used in the Standards to mean efficient and accurateClass time and/or homework should be structured for students to practice core functions such as single-digit multiplications ApplicationExpectation that students apply math and choose the appropriate concept for application, even when not prompted to do soApply math concepts in real-world situationsMathematical modelingWith equal intensity
Conceptual understanding students moving from the concrete -> representational -> abstract.
7Common Core State StandardsMathematicsCloser Look at FractionsS8Fraction Concepts Grade One:Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrase half of, fourth of, and quarter of. Describe the whole as two of or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.9Fraction Concepts Grade TwoPartition circles and rectangles into two,three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe thewhole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the sameshape.10Coherence strengthens foundationsProgression involving fractional concepts (conceptual understanding) and operations (multiplication and division of fractions):
Grade Three: Develop understanding of fractions as numbersDevelop understanding of fractions as part of a whole and as a number on the number lineExplain equivalence of fractions in special cases, and compare fractions by reasoning about their sizeGrade Four: Extend understanding of fraction equivalence and orderingBuild fractions from unit fractions by applying and extending previous understandings on whole numbers(decompose a fraction into a sum of fractions with the same denominator)11Coherence: Number and Operations - FractionsGrade Four: Multiply a fraction by a whole numberGrade Five:Multiply a fraction by a fractionDivide unit fractions by a whole number; and whole numbers by unit fractionsGrade Six:Interpret and compute quotients of fractions and solve word problems12The Mathematics ProgressionsDeveloping a deep understanding of focus, coherence, and rigor
S13Mathematics ProgressionsCommon Core State Standards in MathematicsInformed by research on childrens cognitive development and the logical structure of mathematics Narrative documentsProgression of concepts across several grade bands
The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics. These documents were spliced together and then sliced into grade level standards. From that point on the work focused on refining and revising the grade level standards. The early drafts of the progressions documents no longer correspond to the current state of the standards.
Closer Look Fraction Progression15
Fraction as number16Linking content to the Mathematical Practices
17District PerspectiveRead and analyze the progressions (book study, close read)Through the lens of focus, coherence and rigorNeed to look at the standards in their entirety K-12Determine what happens prior to students entering your door and where they need to go - beyond the grade level below and above. and In Fresno Unified:
Common Core training for us has been on the instructional shifts and the mathematical practices (the habits of mind for mathematically proficient students these are verbs what students will know and be able to do).
The writers of the common core wrote the progressions as a road map outlining the development (path students attend to in mastering) of the domains illustrating what it looks like in the early years and how it transcends grade levels and the complexity of the mathematics as students progress through the grade bands.
In the case of fractions we have had to illustrate for teachers in grades K -2 their role in developing fractional concepts though not explicitly stated in their grade level standards. In doing so teachers have been able to see how skills build upon each other (coherence ) and how fractions are developed in grades 3 -5 (focus and coherence) and how fractions as a big conceptual topic lends to students being fluent, making sense of how the math works and knowing when and where to use it in the real word (rigor).
Beginning in grade 4, students can use area models and number line diagrams to reason about equivalence. They see that the numerical process of multiplying the numerator and denominator of a fraction by the same number, n, corresponds physically to partitioning each unit fraction piece into n smaller pieces. The progressions shows the look both across grade levels and within. One of the hallmarks is the fact that it embeds the practices within the content standards. Gives a good picture of how this will look in the classroom.19Problems for considerationExploring the Fraction Progression using illustrative tasksS20
21Questions to considerWhat are some things that you had to attend to?What are some things that ELLs need to be aware of?What are the implications for planning for instruction?How might you assess student understanding specifically related to ELLs?What are the implications for professional development?22
The student who drew the lower incorrect picture does show some understanding of the fraction one half. Each side of the rectang;e has been cut in half, demonstrating an understanding of the fraction one half in the context of measuring a length. The mistake is that when both linear measurements are cut in half, the area if only one fourth the area of the original shape.24Online ResourcesFraction Professional Development ModuleS25Fraction Progression Online ModuleGoals/Purpose:Deepen educators content knowledge, specifically around the mathematics Fractions progression;Engage practicing educators in the development of CCSS-aligned professional development; andProvide consistent, high-quality professional development that can be used at large scale and online across PARCC states to inform elementary teachers and instructional leaders
Partnership with CGCS, Institute for Mathematics & Education, University of Arizona, and Achieve
Content rich but in discussions with teachers, we heard that these materials appeared to be long and very dense but they would appreciate someone guiding them through the progressions so that they could see how the concepts were developed across grade levels. CGCS important..
26Design within EdmodoFeatures of the moduleOnline and interactiveBased on a framework developed collaboratively with CGCSIllustrative of the Fractions Progression in the CCSSMSeven units anchored by a 3-5 minute video Three to four illustrative tasks associated with the progressionBuilt-in, interactive checks for understandingInteractive-check for understandings Quiz linked to the commentary Includes seven units anchored by a 3-5 minute video that introduces a major concept within the fraction progressionContains 3-4 illustrative tasks associated with the progression Includes built-in, interactive checks for understanding throughout the module that allow users to gauge their progress in understanding the mathematics associated with the progression and how it may be applied to instruction.Interactive check for understanding Quiz- is linked to commentary