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The Alignment of Early Numeracy Skills Bethel P-3 Professional Learning Community November 3, 2011 Julie Wagner Elementary Mathematics Specialist OSPI

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Goals Introduce participants to: Current research that supports a P-3 numeracy alignment The Common Core State Standards domains and alignment of P-3 Trajectories of learning Resources available

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The compelling basis for P-3 numeracy alignment

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A Meta-Analysis School Readiness and Later Achievement Duncan, et al, Developmental Psychology, The strongest predictors of later achievement are school-entry math, reading, and attention skills. Early math skills have the greatest predictive power. By contrast, measures of socio-emotional behaviors were generally insignificant predictors of later academic performance, even among children with relatively high levels of problem behaviors.

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Review of Research Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity National Research Council, The committee found that, although virtually all young children have the capability to learn and become competent in mathematics, for most the potential to learn mathematics in the early years of school is not currently realized. This stems from a lack of opportunities to learn mathematics either in early childhood settings or through everyday activities in homes and in communities.

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Article by NCTM President A Missed Opportunity: Mathematics in Early Childhood Henry Kepner, NCTM Summing Up, February Prior to kindergarten, many children have the interest and capacity to learn meaningful math and acquire considerable mathematical knowledge. Many early childhood programs do not extend children’s mathematical knowledge. Instead, they have these young students repeat the same tasks in varied settings without posing challenges that would push them to the next level.

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Social Policy Report Mathematics Education for Young Children: What It Is and How to Promote It Ginsburg, Lee, & Boyd, Society for Research in Child Development, 2008 Cognitive research shows that young children develop an extensive everyday mathematics and are capable of learning more and deeper mathematics than usually assumed. Typically, early childhood educators are poorly trained to teach mathematics, are afraid of it, feel it is not important to teach, and typically teach it badly or not at all.

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What does all this mean? Children can learn much about mathematics early in their lives. The numeracy skills children walk into the door with in kindergarten predicts later achievement in math and reading. Numeracy skills are often overlooked in child care settings through lack of precedence, neglect or fear. If kindergarten teachers had students who had numeracy skills, achievement would increase dramatically.

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The Common Core State Standards for Mathematics and a P-3 alignment

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The CCSS Document

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Design and Organization

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Standards for Mathematical Practice Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning

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Mathematical Practices Graphic 13

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Critical Areas of Focus 14 Insert a K-3 picture

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Overview Page

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Domains, Clusters, Standards 16

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Common Core State Standards for Mathematics - Domain Development

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Kindergarten Standards The Common Core State Standards in Mathematics at the kindergarten level include all of early numeracy…for a reason.

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Common Core State Standards for Mathematics - Domain Development

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Abridged Trajectories of Early Math Concepts

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Pre-Kindergarten Mathematics Standards

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Abridged Trajectories of Early Math Concepts

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Number and Operation What does it mean to count?

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Number and Operation Concepts in counting: Recognize counting words The sequence of numbers One-to-one correspondence Cardinality Reverse of cardinality

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Number and Operation The sequence of numbers: , 15 14,

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Number and Operation One-to-one correspondence Children have to know sequence of numbers and remember the sequence and where they are in the sequence as they count in order to master one-to- one correspondence – assigning one, and only one, number to each object in a group.

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Number and Operation Cardinality Child moves from just saying the number sequence to understanding that the last number stated answers the question, “How many?”

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Number and Operation Reversal of cardinality Child can be asked to, “Give me eight blocks,” and is able to count out the correct amount. Why would this be more difficult?

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Order of Counting Small numbers first In a line In a circle In a pattern Scrambled

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Counting Trajectory AgeDevelopmental Progression late 1Chants “sing-song” or sometimes indistinguishable number words. 2Verbally counts with number words, not necessarily in correct order above “five.” If knows more number words than number of objects, rattles them off quickly at end. If more objects, “recycles” number words. 3Verbally counts to ten with some correspondence Later 3Keeps one-to-one correspondence between counting words and objects for at least small groups. May recount if asked a second time, “How many?’ 4Accurately counts objects in a line to 5 and answers the “how many” question with the last number counted.

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Counting Trajectory AgeLearning Progression Later 4Counts arrangements of objects to 10. May be able to tell the number just after or just before another number, but only by counting from 1. Counts out objects to 5. 5Counts out objects to 10 and beyond. Keeps track of objects that have or haven’t been counted. Gives next number (usually to 20 or 30). Separates decade and one part of a number. Recognizes errors in other’s counting. Add 6 and maybe 7 and 8. Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009

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AgeLearning Progression 6Counts verbally and with objects from numbers other than 1 (but does not yet keep track of the number of counts). Skip counts by 10s to 100. Counts to 100, then 200. Understands place-value for 10s, 100s. 7Consistently conserves number even in the face of perceptual distractions. Recognizes that decades sequence mirrors single-digit sequence.

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Number and Operation Beginning components of operation Subitizing Comparison words Modeling

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Number and Operation Subitizing Knowing how many are in a collection without counting. How valuable is this skill? What is its role in operation?

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Number and Operation Comparison words Bigger, smaller Longer, shorter Less, more Lighter, heavier

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Comparison Trajectory AgeDevelopmental Progression 1Puts objects, words, actions in one-to-one or one- to-many correspondence. 2Implicitly sensitive to the relation of “more than/less than” involving very small numbers. Compares collections that are quite different in size. If same size, numbers must be small (one or two). 3Compares collections of 1-4 items verbally or nonverbally. Has to be same item. 4Compares groups of 1-6 by matching. Doesn’t need the same object. Accurately counts two equal groups, but more spread out, or larger will be more. Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009

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Comparison Trajectory AgeDevelopmental Progression 5Compares with counting, even when larger collection’s objects are smaller. Later, figures out how many more or less. Accurately counts two equal collections and says they are the same number, even if one collection has larger blocks. Names a small number for sets that cover little space and a “big number.” Names a big number for sets that are spread out. 6Uses internal images and knowledge of number relationship to determine relative size and position. Orders numerals, and collection. Orders lengths marked into units. Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009

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Comparison Trajectory AgeDevelopmental Progression 7Compares numbers with place value understanding. Uses internal images and knowledge of number relationships to determine relative size and position to 100s. 8Uses internal images and knowledge of number relationships, including place value, to determine relative size and position to 1000s. Estimating- subitizing is used to quantify a subset and repeated addition or multiplication used to produce an estimate. Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009

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Operation (+/-) Trajectory AgeDevelopmental Progression 1Sensitivity to adding and subtracting perceptually combined groups. No formal adding 2 - 3Adds and subtracts very small collections nonverbally. 4Finds sums for joining problems up to by counting –all with objects. 4-5Finds sums for joining and part-part-whole by direct modeling, counting-all with objects. Solves take-away problems by separating with objects. 5-6Finds sums for joining and part-part-whole problems with finger patterns and/or by counting on Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009

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Operation (+/-) Trajectory AgeDevelopmental Progression 6Has initial part-whole understanding. Solve all previous problems types using flexible strategies. Sometimes can do start unknown, but only by trial and error. 6-7Recognizes when a number is part of a whole and can keep the part and whole in mind simultaneously; Solves simple cases of multi-digit addition by incrementing tens and/or ones. 7Solves all types of single-digit problems, with flexible strategies and known combinations. Multi-digit may be solved by incrementing or combining tens and ones Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009

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Shape (Geometry) Recognition of basic shapes Application to world around us Classification and sorting

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Trajectory for Shapes AgeLearning Progression 0-2Compares real-world objects and says whether they are the same or different Matches familiar shapes Matches familiar shapes with different sizes Matches familiar shapes with different orientations 3Recognizes and names squares and circles, less often triangles May rotate shape to mentally match to a prototype Judges two shapes the same if they are more visually similar than different. Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009

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Trajectory for Shapes AgeLearning Progression 3-4Matches a wider variety of shapes with same size and orientation. Matches a wider variety of shapes with different sizes and orientations. 4Recognizes some less typical triangles, some rectangles but not rhombuses Says two shapes are the same after matching one side on each. Looks for differences in attributes, but may examine only part of a shape. 4-5Recognizes more objects. Looks for differences in attributes, but may ignore some spatial relaitonships. Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009

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Trajectory for Shapes AgeLearning Progression 5Recognizes most familiar shapes and typical examples of other shapes, such as hexagon, rhombus, and trapezoid. 6Names most common shapes without making mistakes such as calling ovals circles. Recognizes right angles. 7Identifies shapes in terms of their components. Determines congruence by comparing all attributes and all spatial relationships. 8Refers to geometric properties and explains with transformations (moves on top of each other to show congruence). Represents various angle contexts as two lines and, at least implicitly, the size of the angle as the rotation between these lines.

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Measurement Assigns a number to a measureable attribute of an object, usually length, weight, capacity or mass In the CCSS, connections between measurement and whole number operations and number line

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Length Trajectory AgeLearning Progression 2Does not identify length as attribute 3Identifies length/distance as attribute. May understand length as an absolute descriptor (e.g., all adults are tall), but not as a comparative. 4Physically aligns two objects to determine which is longer or if they are the same length. Compares the length of two objects by representing them with a third. 5Order lengths, marked in 1 to 6 units 6Lays units end to end. May not recognize the need for equal length units. Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009

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Length Trajectory AgeLearning Progression 7Measures by repeated use of a unit. Relates size and number of units explicitly. Can add up two lengths to obtain the length of a whole. 8Considers the length of a bent path as the sum of its parts. Measures, knowing need for identical units, parts of unit, and zero point on rulers. Possesses and “internal” measurement tool. Mentally moves along an object, segmenting it, and counting the segments. Estimates with accuracy. Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009

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A look at available resources

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Resources OSPI Website - Common Core State Standards Common Core State Standards Transition Documents Learning Progressions (trajectories) Arizona examples Learning and Teaching Early Math: The Learning Trajectories Approach, Clements and Sarama (2009) Early learning trajectories, pre-k standards Sources of your own

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Reflections and Questions

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Thank you for undertaking this important work! Office of Superintendent of Public Instruction

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