Learning Pathways In Numeracy - OSPI
Learning Pathways in Numeracy:Addressing Early Numeracy SkillsExcept where otherwise noted, content in this document is licensed by the Office of Superintendent of Public Instruction under a CreativeCommons Attribution-NonCommercial-NoDerivatives 4.0 International License. GOLD is the copyrighted work and registered trademark ofTeaching Strategies, LLC. All rights reserved. Any use of their materials without prior approval is strictly prohibited.
Learning Pathways in NumeracyAddressing Early Numeracy SkillsWhy we created this document. “Numeracy” is a term that refers to all the mathematics that young students learn including number, operations, andgeometry and measurement concepts. This Learning Pathways in Numeracy document was created by the Office of Superintendent of Public Instruction(OSPI) primarily as a tool to help teachers and parents to understand the role of progressions in developing numeracy skills in children. Having knowledge ofhow children progress in their early development of numeracy concepts helps teachers and parents to select and use activities that will intentionally buildnumeracy skills in children. Building these skills is foundational for children as they progress through their study of mathematics.This document is designed to facilitate discussions related to progression pathways and to indicate the body of concepts that can be developed in youngchildren. A study within a professional learning community of the resource documents (see page 10) used to create these pathways would be arecommended way to deepen understanding of each pathway. An informed teacher could then use this document as a quick reference to isolate children’slearning along a particular pathway. Determining the next steps in learning and the associated instructional tasks to accomplish that learning would movechildren forward along a particular pathway.Why mastery of math concepts is crucial at an early age. Research has shown that children are capable of learning math concepts at a much earlier age thanhas been previously recognized. A study also has indicated that when controlling for IQ, family income, gender, temperament, type of previous educationalexperience, and whether children came from single or two parent families, the mastery of early math concepts upon school entry was the strongest predictorof future academic success.¹ In 2012, Washington State began collecting observational data through the Washington Kindergarten Inventory of DevelopingSkills (WaKIDS) around early math concepts on large cohorts of kindergarten students within the first months of school entry. The results showed that manyWashington State children are entering kindergarten with limited numeracy skills. With this knowledge, there was a growing need for OSPI to develop ways tohelp teachers and parents improve the numeracy skills of pre-school children.How this document is connected to the Common Core. The Learning Pathways in Numeracy was written using the Common Core State Standards (CCSS)domain titles (Counting and Cardinality, Operations and Algebraic Thinking, etc.). This was done to show that foundational concepts that support CommonCore kindergarten standards and beyond can be learned at an early age. While the Learning Pathways for Numeracy gives some key benchmarks for a givenage-range, there are intermediary stages between these benchmarks. These intermediary stages are treated in greater depth in the resources that were usedto develop the pathways (see page 10). In addition, the pathways for kindergarten through third grade in this document follow directly from the CCSS. The CCSSwere built from progressions of learning that can be found at http://ime.math.arizona.edu/progressions/. OSPI considers these progressions required readingfor teachers at each grade level.It should be noted that while the Learning Pathways in Numeracy is divided into mathematical domains, there is much cross-over between these domains.This linear model does not accurately represent those connections or the coherence found between domains. Reading the listed resources and the CCSSprogressions will help teachers make those connections.2
Developmental progressions are a guide, not an absolute. Although the resources used for this document agree that there is a progression of learning forearly numeracy concepts, the period when a particular concept can be developed may be slightly different from resource to resource. It needs to beunderstood that the Learning Pathways for Numeracy is not meant to be definitive about the age in which these concepts can be developed and so there canbe discrepancies between documents. The general progression should be what is considered and not necessarily the age given. Children develop theircognitive knowledge at different rates and the suggested age ranges are only that.Our hope is that this Pathways document will be the start of a discussion and then a movement to help children develop their interest and capacity to learnmeaningful mathematics in their early years.¹ School readiness and later achievement.Duncan, Greg J.; Dowsett, Chantelle J.; Claessens, Amy; Magnuson, Katherine; Huston, Aletha C.; Klebanov, Pamela; Pagani, Linda S.; Feinstein, Leon; Engel, Mimi; Brooks-Gunn, Jeanne; Sexton, Holly; Duckworth, Kathryn;Japel, CristaDevelopmental Psychology, Vol 43(6), Nov 2007, 1428-1446. http://dx.doi.org/10.1037/0012-1649.43.6.14282014. Developed by Julie Wagner in coordination with the Mathematics Learning and Teaching Department at the Office of Superintendent of PublicInstruction (OSPI). OSPI would like to thank Douglas Clements, Ph.D., University of Denver, for reviewing a draft version of this document3
Learning Pathways for Counting and CardinalityCounting and CardinalityCounting0 – 36months 3–4years 4–5years Imitates rote counting using somenames of numbers.Verbally counts to 5, may beincorrect beyond this.Verbally counts to 10.Keeps 1-1 correspondence for 5 orless objects in a line. Beyond this,may counts quickly at the end ifknows more numbers than objects,or recycles words if number ofobjects is greater than numbersknown by rote. ₃Counts verbally to 10 or beyond.Understands cardinality in counting,for at least 5 objects.Counts out 5 objects.ELGGOLDTM9–18 m20a OY16–36 m20a YG16–36 m20a YGSubitizing/Early Operations 3–4 yr20a YG3–4 yr20b GB20a YGCounts verbally to 20 and beyond.Counts 1020 objects accurately.Gives next number in sequence (1 10).Counts out 10 objects.Identifies numerals 110.Writes some numerals and connectseach to counted objects.4–5 yr20a BP4–5 yr20a BP4–5 yr20a BP4–5 yr20c BPCounts verbally to 100 by ones andtens. *Counts forward from a givennumber.*Writes and represents numbers to 20and beyond.*Counts to tell “How many?” to atleast 20. *Counts outout objectsobjects toto 20.20. **Counts5–K20cBP 16–36 m 20b OYInstantly recognizes andnames the number of itemsin a set of three or fourMakes a small collectionwith the same number asanother collection3–4 yrInstantly recognizes andnames the number of itemsin a set of five.Makes a collection with thesame number as anothercollection.Quickly names parts of anywhole up to 5, or the wholegiven the parts. ₃4–5 yr20aP20cP5–K20aPGOLDTMDemonstratesunderstanding of theconcepts of one, two, andmore.Recognizes and names thenumber of items in a set oftwo or three.Adds and subtracts verysmall collections (up to 3)nonverbally. ₃20aP5–KELG4Comparing and Ordering 20b GB20b GB20b GB 20b GB Understands the idea of “more”related to food or play.Puts objects in 1-to-1 or 1-to-many inprovoked correspondence. ₃Knows more/less for very smallcollections of items, or with bigdifferences in number of items.Uses comparison wordsCompares collections of 1 – 4 items ifthe collections are made up of thesame objects. ₃Uses gestures or words to makecomparisonsCompares groups of 1 -5 by matchingor counting when objects in eachgroup are about the same size.Accurately counts two equalcollections, but when asked, says thecollection of larger objects has more. ₃Uses comparative language (more,less, same) to compare collections upto 10 by counting, even when thecollection with the larger quantity ofobjects is made up of smaller objects.Orders three to six objects by oneattribute.ELGGOLDTM20b OY9–18 m16–36 m 22 OYG3–4 yr22GB4–5 yr20b BP4–5 yr22 GB
Counting and CardinalityCounting Grade 1 Counts to 120 starting from anynumber. *Reads and writes numerals to 120.* Represents a number of objects with awritten numeral.*ELG1st G1st GGOLDTMSubitizing/Early Operations Uses conceptual subitizingto compose anddecompose numbers, andto understand place valueand operations.ELGGOLDTMComparing and Ordering ELGGOLDTMDetermines “how many more” or“how many less” by addition orsubtraction in comparingsituations.Key: (a)* Aligned to CCSS; (b) ₃ third resource in list on page 10; (c) ELG Early Learning Guidelines; (d) GOLDTM Teaching Strategies GOLD In the GOLDTM column, the numbers and lower case lettersrepresent the objectives/dimensions in Teaching Strategies GOLD . Capital letters represent the associated color bands in Teaching Strategies GOLD (O orange, Y yellow, G green, B blue, P purple)5
Learning Pathways for Number and Base TenNumber and Operations in Base Ten/FractionsPlace ValueKindergarten Grade 1 Grade 2 Grade 3 Composes and decomposesnumbers 11-19 into tens andones.*Understands that 2-digitnumbers represent amountsof tens and ones. *Compares two two-digitnumbers based on meaningsof tens and ones using , ,and .*Understands place value for3-digit numbers.*Counts within 1000; skipcounts by 5s, 10s, and100s.*Reads and writes numbersto 1000.*Compares two three-digitnumbers based on meaningsof hundreds, tens, and onesdigits using , , and .*Uses place value to roundwhole numbers to nearer 10or 100.*ELG1st GGOLDTMPlace Value and Operations 2nd G Adds within 100 includingadding a 2-digit number and a1-digit number, and adding a 2digit number and a multiple of10.*Finds 10 more or less than anumber, explainingreasoning.*Fluently adds and subtractswithin 100 usingstrategies.*Using concrete models, addsor subtracts within 1000.*Mentally adds or subtracts 10or 100 from any given number100- 900.*Explains why addition andsubtraction strategieswork.*Fluently adds or subtractswithin 1000.*Multiplies 1-digit numbersby multiples of 10 from 1090.*ELG1st GGOLDTMFractions 2nd G 2nd G Partitions circles and rectanglesinto two and four equal sharesdescribing the whole and parts bynumber of shares.*Understands that decomposinginto more shares creates smallershares.* (Found in Geometrydomain)ELG1st GRecognizes that equal shares ofidentical wholes need not havethe same shape.*Partitions circles and rectanglesinto two, three, or four equalshares and names them.*(Found in Geometry domain)2nd G 3rd G Develops understanding offractions as numbers on anumber line.*Understands unit fractions.*Understands, recognizes,and generates simpleequivalent fractions.*Compares two fractions with thesame numerator ordenominator.*Recognizes that comparisons arevalid only when referring to thesame whole.*Partitions shapes into parts withequal area and expresses eachpart as a unit fraction.*3rd GKey: (a)* Aligned to CCSS; (b) ₃ third resource in list on page 10; (c) ELG Early Learning Guidelines; (d) GOLDTM Teaching Strategies GOLD In the GOLDTM column, the numbers and lower case lettersrepresent the objectives/dimensions in Teaching Strategies GOLD . Capital letters represent the associated color bands in Teaching Strategies GOLD (O orange, Y yellow, G green, B blue, P purple).6GOLDTM
Learning Pathways for Operations and Algebraic ThinkingOperations and Algebraic ThinkingOperations0 – 36months3 – 4 years 4 – 5 years 5Kindergarten Grade 1 Grade 2 Grade 3 Uses gestures to ask for more.Understands getting more or having less.9-18mKnows that a whole is bigger than its parts.Finds the total sum of small groups of items.Combines and separates up to five objects and describes theparts.Finds results for joining and take-away by using concreteobjects and counting all.3-4 yrUses counting-on and counting-up-to strategies to findresults.Fluently adds and subtracts within 5.*Represents addition and subtraction within 10 with objects.*Makes sets of 6 – 10 objects and describes the parts,identifying which part has more, less, or the same.Represents and solves all addition and subtraction situations(result unknown, change unknown, start unknown) within 20.*Adds and subtracts within 20 using a variety of informal andintuitive strategies and can describe the strategies used.*Demonstrates fluency within 10.*Works with addition and subtraction equations with theunknown in all positions.*Solves one- and two-step word problems within 100 withunknowns in any situation and subtype.*Describes thinking when solving a word problem.Adds and subtracts within 20, knowing facts by end of year.*Represents and solves problems involving multiplication anddivision.*Multiplies and divides within 100.*Multiplies a single digit by a multiple of 10.*Solves problems involving the four operations.*GOLDTMELGAlgebraic ThinkingELG20b O Intuitively has understanding that when adding twonumbers, it does not matter which number “comes first.” ₃ Intuitively has understanding that when adding threenumbers it does not matter which two you add first. ₃ Makes algebraic generalizations (such as, subtracting zerofrom any number gives that number, or subtracting anumber from itself gives zero), with guidance. ₃ Understands and applies properties of operations and therelationship between addition and subtraction.*With guidance, makes algebraic generalizations, such as anynumber added to and then subtracted from x, leaves x. ₃3-4 yr20b GB4-5 yr20b GB5–K20b BP5-K20b BP1st G20b P 1st G2nd G Works with equal groups of objects to gain foundations formultiplication.* Understands properties of multiplication and therelationship between multiplication and division.*Identifies and explains patterns in arithmetic.*2nd G2nd G3rd G3rd G 3rd G3rd GKey: (a)* Aligned to CCSS; (b) ₃ third resource in list on page 10; (c) ELG Early Learning Guidelines; (d) GOLDTM Teaching Strategies GOLD In the GOLDTM column, the numbers and lower case lettersrepresent the objectives/dimensions in Teaching Strategies GOLD . Capital letters represent the associated color bands in Teaching Strategies GOLD (O orange, Y yellow, G green, B blue, P purple).7GOLDTM
Learning Pathways for Measurement and Data (Measurement)Measurement and Data0 – 36months3 – 4 yearsMeasurementELG Attends to overall appearance of size, labeling as big/little.Explores measuring tools, such as measuring cups, or a ruler.9-18m Compares size by sight, feel, and comparing to hands, feet, etc.Compares and orders a small set of objects as appropriateaccording to size, length, weight, area, volume.Knows usual sequence of basic daily events.Knows a few ordinal numbers. 4 – 5 years 5 andKindergarten Physically aligns two objects to determine which is longer or ifsame length. ₃May be able to measure with ruler, but often lacks understandingor skill. ₃Describes measureable attributes of objects.*Directly compares two objects.*Uses ordinal numbers from first to tenth.₃Uses measurement words and some standard measurement toolsaccurately.GOLDTM22 OYG3-4 yr3-4 yrGeometric Measurement Manipulates shapes individually, but does notcombine to compose a larger shape. Fills simple pattern block puzzles using trial and error.₃ Puts several shapes together to make one part ofa picture. ₃ Analyzes and compares two-dimensional and threedimensional shapes in different sizes andorientations.*Models shapes by building them fromcomponent parts.*Composes simple shapes to make larger shapes.* Grade 2Grade 3 Measures lengths indirectly and by iterating length units withoutgaps or overlaps. *Tells and writes time in hours and half-hours.*Measures and estimates lengths in standard units.*Relates addition and subtraction to length and the number line.*Tells time to the nearest 5 minutes.*Solves problems involving money.*Recognizes that comparisons are valid only when referring to thesame unit.*Tells and writes time to the nearest minute and measures timeintervals.*Solves addition and subtraction time interval problems on anumber line.*Measures and estimates liquid volumes and masses of objectsusing grams, kilograms, and liters.*Solves problems of all types that involve measurement.*GOLDTM22 GB22 GB22 GB4-5 yr22 BP22 P22 P Grade 1ELG1st G2nd G Composes two-dimensional and threedimensional shapes to create a compositeshape.* Partitions a rectangle into rows and columns ofsame size squares and count to find total.* Understands concepts of area and relates areato multiplication and to addition.*Recognizes perimeter as an attribute of planefigures and solves area and perimeter problems.*21b BP21b P21b P1st G2nd G 3rd G3rd GKey: (a)* Aligned to CCSS; (b) ₃ third resource in list on page 10; (c) ELG Early Learning Guidelines; (d) GOLDTM Teaching Strategies GOLD In the GOLDTM column, the numbers and lower case lettersrepresent the objectives/dimensions in Teaching Strategies GOLD . Capital letters represent the associated color bands in Teaching Strategies GOLD (O orange, Y yellow, G green, B blue, P purple).8
Learning Pathways for Measurement and Data (Data)Measurement and DataCategorical Data3 – 4 years4 – 5 years5 and Kindergarten ELGGOLDTMSorts and/or describes objects by a non-geometric attribute(size, color) or by shape.3-4 yr Sorts objects by one attribute.4-5 yr13 GB Sorts and classifies objects into more than one category andcounts the number in each.*Resorts objects into new categories. ₃Organizes, represents and interprets data with up to threecategories.*Draws a bar and/or picture graph and answers questionsrelated to them.*5-K13 BPGrade 1 Grade 2 Measurement Data GOLDTM13 GB13 P Grade 3ELG Draws scaled picture and bar graphs to represent a data setand solves problems.*Generates measurement data by measuring length tothe nearest whole unit.*Shows length measurements on a line plot markedwith whole number units, connecting this to therepresentation of whole numbers on a number line.*Generates data by measuring lengths marked withwhole numbers, halves and fourths and shows on aline plot, connecting this to the representations offractions on a number line.*Key: (a)* Aligned to CCSS; (b) ₃ third resource in list on page 10; (c) ELG Early Learning Guidelines; (d) GOLDTM Teaching Strategies GOLD In the GOLDTM column, the numbers and lower case lettersrepresent the objectives/dimensions in Teaching Strategies GOLD . Capital letters represent the associated color bands in Teaching Strategies GOLD (O orange, Y yellow, G green, B blue, P purple).9
Learning Pathways for GeometryGeometryShapes0 - 36 months 3 – 4 years 4 – 5 years Puts things together, such as simple matching puzzles, nestingcups.Matches familiar shapes (circle, square, typical triangle) withthe same size and orientation.Recognizes and names circles and squares, maybe triangles.Matches shapes by rotating to prototype.Judges shapes the same if they are more visually similar thandifferent. ₃Recognizes more shapes in real-world, less typical triangleshapes, and some rectangles with same size and orientation.Recognizes some shapes with different sizes and orientation. ₃ELGGOLDTM9-18m13 OY16-36m21b OY16-36m21b GSpatial Relationships/Structuring Follows simple directions related to position (in, on,up, down). Follows simple directions related to proximity(behind, under, beside, next to, between). Identifies positions of objects in space by using words like,beside, inside, next to, above, below, under.Uses and responds to positional words. Uses and makes simples sketches to locate objects.Correctly uses position words to describe objects.21b BP20a OY3-4 yr21a
Learning Pathways in Numeracy: Addressing Early Numeracy Skills Except where otherwise noted, content in this document is licensed by the Office of Superintendent of Public Instruction under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Practical Numeracy is a course run from S1-S3. The Practical Numeracy course will help to develop the numeracy skills you will use in your practical STEM subjects. The numeracy skills you will use in Practical Numeracy are the same skills you will be using in all your other STEM subjects. These are called transferable skills.
level of proficiency in functional (a) literacy and (b) numeracy skills. The target age group for this indicator is the population of 15 years and older. This report recommends that the literacy and numeracy indicators be based upon the definitions of literacy and numeracy used in the OECD’s PIAAC adult skill assessment program.
Practise 4 Times Tables you wearing your hat by and/or Fitness Practise 11 Times Tables and/or Fitness Practise 3 Times Tables and/or Fitness Practise 7 Times Tables Numeracy Ninjas Session 1 Week 10 Numeracy Ninjas Week 10 Session 2 Numeracy Ninjas Week 10 Session 3 Wellbeing Numeracy Nin
Research Report No. 19 Transition from Preschool to Primary School Literacy and Numeracy for Learning and Life: The National Strategy to improve Literacy and Numeracy among Children and Young People 2011-2020 Literacy and Numeracy for Learning and Life is the national strategy launched by the Department of Education and Skills
Washington’s previous arts learningstandards addresseddance, music, theatre,and visual arts. When the National Core Arts Standards (NCAS) were published in June 2014, OSPI engaged a cadre of arts educators representing dance, music, theatre, and visual arts to review the
benchmarks. Facing the Future. June 30, 2008 Report to OSPI on ESE Standards for Washington Students . National Academy of Sciences, National Science Education Standards. OSPI, “Education for Environment and Sustainability Initiatives and EE Study,” . Learning and Teaching Scotland website, “Sustainable Development Education”
to our primary focus on academic content. 2. Prepare for Health and Safety in 2020–21 OSPI is committed to supporting learning environments that protect student and staff health and safety. To do this, OSPI will partner with
INTRODUCTION TO OPENFOAM open Field Operation And Manipulation C libraries Name. INTRODUCTION TO OPENFOAM open Field Operation And Manipulation C libraries Rita F. Carvalho, MARE, Department of Civil Engineering, University of Coimbra, Portugal OpenFOAM Equations Solvers How to use/code Examples Conclusions 3 25 26 33 46 49 50. SOLVE PARTIAL DIFFERENTIAL EQUATIONS (PDE .
