Orchard Software and the NCTM Principles and …siboneylg.com/docs/NCTMPrinciplesStandardsPositionPaper.pdfOrchard Software and the NCTM Principles and Standards for School Mathematics

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Orchard Software and the NCTM Principles and Standards for School Mathematics

Orchard Software and theNCTM Principles and Standards for School Mathematics

A Position Paper

Author: Greg Koenig, Marketing Communications Manager, Siboney Learning Group

Executive Summary

In 2000, the National Council of Teachers of Mathematics (NCTM) released Principles and Standards for School Mathematics, which updated NCTM’s seminal �989 statement, Curriculum and Evaluation Standards for School Mathematics. The earlier document was foundational in the development of Orchard Software’s Skill Trees. Orchard is shown also to be aligned with the principles and standards in the 2000 iteration. Since Orchard Software is consistent with Principles and Standards, it is also aligned with NCTM’s Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics (2006).

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Introduction

In 2000, the National Council of Teachers of Math-ematics (NCTM) published Principles and Stan-dards for School Mathematics, a revision and update of the association’s �989 document, Curriculum and Evaluation Standards for School Mathematics. Both sets of guidelines were developed to help “im-prove mathematics education in the United States and Canada… and [provide] focus, coherence, and new ideas in mathematics education” (Realizing the Vision, 2000). Among NCTM’s aims are for Prin-ciples and Standards to

• “Set forth a comprehensive and coherent set of learning goals for mathematics for all students from prekindergarten through grade �2… and• “Guide the development of curriculum frame- works, assessments, and instructional materials” (Executive Summary).

Principles and Standards, along with NCTM’s follow-up document, Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics (2006), extends and refines the guidelines estab-lished in Curriculum and Evaluation Standards, par-ticularly by identifying the underlying principles for school mathematics and clarifying and elaborating on the standards in the earlier document (Curriculum Focal Points, p.�).

The Skill Tree programs in Orchard Software’s Math bundles follow several basic assumptions that guided NCTM’s �989 Curriculum and Evalu-ation Standards for School Mathematics. That is,

Orchard’s Skill Trees reflect a balanced instructional approach that provides tutorials and practice in basic computational and procedural skills as well as instruction that emphasizes conceptual orienta-tion—through which students build meaning from problem-solving situations they can connect with individual experience. This affinity continues with Principles and Standards as well as Curriculum Focal Points; it can be seen especially in Orchard’s assessments, skill-building lessons, and real-world word problem activities.

The three components of Principles and Standards that this position paper will be concerned with are the Principles, the Content Standards, and the Pro-cess Standards sections. The statements in each sec-tion will be examined to demonstrate how Orchard Math software remains consistently aligned with the NCTM ideals as they evolve.

Principles

NCTM added the Principles for School Mathemat-ics in Principles and Standards to provide “sound professional guidance” for teachers, administrators and other education professionals who must make decisions about the “content and character of school mathematics” (Principles and Standards, p. ��). While NCTM’s Standards describe the content and processes students should learn, the Principles de-scribe the ideal at which the Standards are aimed—the “particular features,” that is, “of high-quality mathematics education” (Principles and Standards, p. ��). These principles upheld by NCTM are listed

Orchard Software and theNCTM Principles and Standards for School Mathematics

A Position Paper

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below; following each principle is a brief statement connecting the principle to the ideals that are the foundation of Orchard Software.

• Equity Principle. Excellence in mathematics education requires equity—high expectations and strong support for all students.

Orchard Software is built to the highest educational standards for each state. Orchard’s flexibil- ity makes it ideal for use with a wide variety of learning skills. Voiced tutorial content and vocabulary provide reinforcement for ELL students and students with disabilities. Orchard’s SkillBuilder Skill Trees provide a step-by-step process that gradually builds mastery in strug- gling students. Orchard may be used in a variety of settings—for example, in a computer lab or classroom center for individualized learning; with small peer-tutoring groups or project groups; or with an interactive whiteboard for whole-class learning.

• Curriculum Principle. A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades. Orchard’s curriculum is designed to reflect the standards published by the educational authori- ties in each state. In addition, the scope and se- quence of Orchard instruction demonstrates a coherent structure that moves logically from basic through intermediate to advanced mathematics knowledge and skill levels.

• Teaching Principle. Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.

Orchard Software is adaptive: it has been designed to provide instruction, tutorial content, and formative assessments that meet student needs progressively; that is, as the student’s responses on assessments show higher levels of mastery, Orchard presents content and assessments that are incrementally more sophisticated and more difficult. Finally, since there is no

substitute for a highly qualified professional educator in a student’s development, Orchard offers a robust reporting component to keep the teacher abreast of student progress. The result: a student with solid skills as well as confidence with complex concepts.

• Learning Principle. Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.

Orchard provides SkillBuilder Skill Tree curriculum modules to supplement the instruction in the Math Concepts Skill Trees, as well as Learning Games Skill Trees that reinforce concepts that have been presented. Orchard students engage in actively learning skills and in building the capacity to apply increasingly complex concepts.

• Assessment Principle. Assessment should support the learning of important mathematics and furnish useful information to both teachers and students.

Orchard Software has been developed to assess students’ mastery of the content standards set by each state under the guidance of the No Child Left Behind Act. Orchard’s assessments identify areas that need strengthening, after which Orchard devises a customized learning plan for each student to help him or her improve in those critical areas.

• Technology Principle. Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning.

Siboney Learning Group, the publisher of Orchard Software, is committed to providing educational technology products that excel at engaging students and increasing the rewards of achievement at all levels.

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Standards

Content Standards

• Number and Operations. Instructional programs from prekindergarten through grade �2 should enable all students to

• understand numbers, ways of representing numbers, relationships among numbers, and number systems. • understand meanings of operations and how they relate to one another; • compute fluently and make reasonable esti- mates.

Orchard Alignment

Through the Number Sense Skill Tree in Orchard’s Math Concepts series, students learn counting, � to � correspondence, ordering and comparing numbers, estimating, and rounding. Orchard’s Numeration K-3 Skill Tree introduces students to the concepts involved in performing mathematical operations; students develop skill computing addition and subtraction problems, and learn fundamental multiplication and division facts. In the Basic Operations and Order of Operations (�-6) Skill Tree, students receive in-depth instruction in the operations of addition, subtraction, multiplication and division, as well as in the relationships among the operations. Students learn to perform computations in addition, subtraction, multiplication, and division.

• Algebra. Instructional programs from prekindergarten through grade �2 should enable all students to

• understand patterns, relations, and functions; • represent and analyze mathematical situations and structures using algebraic symbols; • use mathematical models to represent and understand quantitative relationships; • analyze change in various contexts.

Orchard Alignment

In the Patterning and Algebra K-3 Skill Tree of Orchard’s Math Concepts Series, students identify and describe patterns, including counting patterns and multiples in hundred charts. Students extend their understanding of patterning and algebraic concepts in the Algebra, Patterning & Logic Skill Tree (�-6): they use data tables, graphs, and equations to represent a single relationship; they also discover variables in functions and learn how changing one variable results in changes to others. Orchard’s Patterning and Algebra Concepts (7-8) Skill Tree challenges students to investigate, experiment with, learn to identify, write, and evaluate variables and expressions, linear equations and inequalities, as well as add, subtract, multiply, and factor poly- nomials.

Number and Operation: Orchard’s Numeration K-3 Skill Tree

Algebra: Patterning and Algebra K-3 Skill Tree

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• Geometry. Instructional programs from prekindergarten through grade �2 should enable all students to

• analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; • specify locations and describe spatial relationships using coordinate geometry and other representational systems; • apply transformations and use symmetry to analyze mathematical situations; • use visualization, spatial reasoning, and geometric modeling to solve problems.

Orchard Alignment

Orchard’s Math Concepts series includes Geometry K-3, Geometry and Spatial Sense (�-6), Geometry Concepts (7-8), and Functional, Geometric & Statistical Relationships (7-8) Skill Trees. Geometry K-3 introduces students to basic geometric concepts such as identifying, sorting, and classifying 2-D and �-D objects, as well as to identifying and performing transformations and determining lines of symmetry. Geometry and Spatial Sense (�-6) focuses on the vocabulary of geometry and a variety of the foundational concepts involved in working with shapes and space. Through Geometry Concepts (7-8), students investigate lines and angles, as

well as properties and formulas associated with triangles, quadrilaterals, and circles; students also receive advanced instruction about transformations—particularly reflections, rotations, and translations.

• Measurement. Instructional programs from prekindergarten through grade �2 should enable all students to

• understand measurable attributes of objects and the units, systems, and processes of measurement; • apply appropriate techniques, tools, and formulas to determine measurements.

Orchard Alignment

The Measurement K-1 and 2-3 Skill Trees of Orchard’s Math Concepts series provide instruction in estimating, perimeter, area, volume, time, money, and temperature using standard and non- standard units. In the Measurement Skill Tree of the Orchard Math �-6 bundle, students solve a variety of problems involving measurement of mass, area, perimeter, volume, and length. Students deepen their understanding in the Measurement Concepts Skill Tree (7-8) by cal- culating the length, area, surface area, and the volume of various shapes; the program also provides instruction about the practical uses of measurement in everyday life.

Geometry: Geometry Concepts 7-8 Skill TreeMeasurement: Orchard’s Measurement 2-3 Skill Tree

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• Data Analysis and Probability. Instructional programs from prekindergarten through grade �2 should enable all students to

• formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them; • select and use appropriate statistical methods to analyze data; • develop and evaluate inferences and predictions that are based on data; • understand and apply basic concepts of probability.

Orchard Alignment

Orchard Math Concepts’ Data Management and Probability Skill Tree (K-�) introduces the fundamental concepts of data management and probability: the structure and characteristics of graphs; conducting surveys, collecting and organizing data, and performing experiments. The Probability Skill Tree (�-6) provides students opportunities to apply and extend their understanding by making predictions based on decisions about patterns in events and about other data. In the Data Management Concepts and Probability Concepts Skill Trees (7-8), students learn data-collecting strategies; organize, analyze, and interpret data; and compute probability.

Process Standards

• Problem Solving. Instructional programs from prekindergarten through grade �2 should enable all students to

• build new mathematical knowledge through problem solving; • solve problems that arise in mathematics and in other contexts; • apply and adapt a variety of appropriate strategies to solve problems; • monitor and reflect on the process of mathematical problem solving.

Orchard Alignment

The Orchard Math Concepts series is designed to offer opportunities throughout for real-world application of mathematical concepts through word problems. Skill Trees that provide intensive instruction and opportunities to practice problem solving include Math Strategies 1-3; Math Word Problems: Beginning (�-�), Inter- mediate (�-6), and Advanced (7-8); Basic Operations and Order of Operations (�-6); Algebra, Patterning & Logic (�-6); Fraction Concepts (�-6); Percent Concepts (7-8); and Rate, Ratio, and Proportion Concepts (7-8).

Data Analysis and Probability: Orchard’s Data Management and Probability Concepts 7-8 Skill Tree

Problem Solving: Orchard’s Math Word Problems 1-3 Skill Tree

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• Reasoning and Proof. Instructional programs from prekindergarten through grade �2 should enable all students to

• recognize reasoning and proof as fundamental aspects of mathematics • make and investigate mathematical conjec- tures; • develop and evaluate mathematical arguments and proofs; • select and use various types of reasoning and methods of proof.

Orchard Alignment

Opportunities to learn and apply the concepts of reasoning and proof are an integral part of Orchard’s Math Concepts series. Skill Trees that focus particularly on these concepts include Math Strategies 1-3, 4-6, and 7-8; Data Management and Probability 1-3; Algebra, Patterning & Logic (�-6); Graphing and Managing Data (�-6); and Data Management and Probability Concepts.

• Communication. Instructional programs from prekindergarten through grade �2 should enable all students to

• organize and consolidate their mathematical thinking through communication;

• communicate their mathematical thinking coherently and clearly to peers, teachers, and others; • use the language of mathematics to express mathematical ideas precisely.

Orchard Alignment

Orchard’s Math Concepts series is rich in features that encourage students to learn and apply the verbal and non-verbal language used to express mathematical concepts and processes. Skill Trees that offer intensive instruction and practice in expressing ideas in mathematical terms include Number Sense 1-3; Measurement K-1 and 2-3; Data Management and Probability 2-3; Geometry and Spatial Sense (�-6); Graphing and Managing Data (�-6); Probability (�-6); Algebra, Patterning, and Logic (�-6); Algebraic Foundations (7-8); and Linear Relations (7-8). In addition, the glossary and voice components reinforce student understanding. The Journal and Explain Your Answer features in a number of Orchard Skill Trees provide opportunities for students to express their reasoning and thought processes verbally as well as in the language of mathematics.

Reasoning and Proof: Orchard’s Geometry Concepts 7-8 Skill Tree

Communication: Math Journal in various Orchard Skill Trees (shown: Fraction Concepts 4-6)

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• Connections. Instructional programs from prekindergarten through grade �2 should enable all students to

• recognize and use connections among mathematical ideas; • understand how mathematical ideas interconnect and build on one another to produce a coherent whole; • recognize and apply mathematics in contexts outside of mathematics.

Orchard Alignment

Orchard Skill Trees are designed with a number of specific program benefits in mind. Among these are the development and reinforcement of mathematical skills and knowledge; development of problem-solving skills; development of thinking skills; and support of skills required for future education and employment. Some specific Skill Trees that support student understanding of how mathematical concepts are connected include Number Sense K-3; Basic Operations and Order of Operations (�-6); Fraction Concepts (�-6 and 7-8); Patterning and Algebra Concepts (7-8); Percent Concepts (7-8); Rate, Ratio, and Proportion Concepts (7-8); and the Measurement Skill Trees.

• Representation. Instructional programs from prekindergarten through grade �2 should enable all students to

• create and use representations to organize, record, and communicate mathematical ideas; • select, apply, and translate among mathematical representations to solve problems; • use representations to model and interpret physical, social, and mathematical phenomena.

Orchard Alignment

Students are taught and encouraged to represent mathematical concepts in a variety of ways beginning in the earliest Skill Trees of the Math Concepts series. In Numeration K-3, students demonstrate addition and subtraction using manipulatives. Throughout the series, student activities include building models to demonstrate patterns; visualize fractions and integers; depict geometric and spatial relationships; organize, display, and interpret data; and show linear relationships.

Conclusion

In an age of information, technology, and rapid change, the need to provide mathematics educa-tion that meets rigorous standards and expectations

Connections: Orchard’s Measurement 2-3 Skill TreeRepresentation: Orchard’s Patterning and Algebra 4-6

Skill Tree

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is critical. The bar set by NTCM’s Principles and Standards is challenging but attainable, in terms of teaching both basic computational and procedural skills and conceptual orientation. Orchard Software is a supplementary curriculum tool that continues to support NCTM’s ideals and provide comprehen-sive, targeted instruction that can be used in support of a strategy that incorporates NCTM’s guidelines.

References

National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindertarten through grade 8 mathematics: a quest for coher-ence. Reston, VA: National Council of Teachers of Mathematics.

---. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

---. (2000). Realizing the vision – Standards 2000 project. Retrieved July ��, 2007 at http://standards.nctm.org/document/prepost/project.htm

Appendix

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence

In late 2006, NCTM published Curriculum Focal Points. The purpose of the document was to bring into focus a large and complex set of mathematics curriculum expectations and to identify the most important concepts and processes to be taught at each grade level.

Below are the Focal Points established by NCTM, arranged by grade level from pre-kindergarten through grade 8. Note that the subcategories cor-respond to the category listings under Content Stan-dards in Principles and Standards—it may be in-ferred that since Orchard curriculum is aligned to Principles and Standards, it is also aligned to Cur-riculum Focal Points. Additionally, the recently published Principles of Math Skill Tree offers inten-sive instruction in fundamental math concepts such as correspondence, cardinality, and comparison.

Prekindergarten

• Number and Operations: Developing an understanding of whole numbers, including concepts of correspondence, counting, cardinality, and comparison

• Geometry: Identifying shapes and describing spatial relationships

• Measurement: Identifying measurable attributes and comparing objects by using these attributes

Kindergarten

• Number and Operations: Representing, comparing, and ordering whole numbers and joining and separating sets

• Geometry: Describing shapes and space

• Measurement: Ordering objects by measurable attributes

Grade 1

• Number and Operations and Algebra: De- veloping an understanding of addition and subtraction and strategies for basic addition facts and related subtraction facts

• Number and Operations: Developing an understanding of whole number relationships, including grouping in tens and ones

• Geometry: composing and decomposing geometric shapes

Grade 2

• Number and Operations: Developing an understanding of the base-ten numeration system and place-value concepts

• Number and Operations and Algebra: De- veloping quick recall of addition facts and related subtraction facts and fluency with multi- digit addition and subtraction

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Grade 2 (cont.)

• Measurement: Developing an understanding of linear measurement and facility in measuring lengths

Grade 3

• Number and Operations and Algebra: De- veloping understandings of multiplication and division and strategies for basic multiplication facts and related division facts

• Number and Operations: Developing an understanding of fractions and fraction equivalence

• Geometry: Describing and analyzing properties of two-dimensional shapes

Grade 4

• Number and Operations and Algebra: De- veloping quick recall of multiplication facts and related division facts and fluency with whole number multiplication

• Number and Operations: Developing an understanding of decimals, including the connections between fractions and decimals

• Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes

Grade 5

• Number and Operations and Algebra: De- veloping an understanding of and fluency with division of whole numbers

• Number and Operations: Developing an understanding of and fluency with additioin and subtraction of fractions and decimals

• Geometry and Measurement and Algebra: Describing three-dimensional shapes and analyzing their properties, including volume and surface area

Grade 6

• Number and Operations: Developing an understanding of and fluency with multiplication and division of fractions and decimals

• Number and Operations: Connecting ratio and rate to multiplication and division

• Algebra: Writing, interpreting, and using mathematical expressions and equations

Grade 7

• Number and Operations and Algebra and Geometry: Developing an understanding of and applying proportionality, including similarity

• Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areas and volumes of three-dimensional shapes

• Number and Operations and Algebra: De- veloping an understanding of operations on all rational numbers and solving linear equations

Grade 8

• Algebra: Analyzing and representing linear functions and solving linear equations and systems of linear equations

• Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle

• Data Analysis and Number and Operations and Algebra: Analyzing and summarizing data sets

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