Principles and Standards for School Mathematics 1561666 213553318 2008-05-19T21:04:12Z Puceron 5792803 /* Content standards */ '''Principles and Standards for School Mathematics''' is a book produced by the [[National Council of Teachers of Mathematics]] (NCTM) in 2000 to set forth a national vision for precollege [[mathematics education]] in the [[United States|US]] and [[Canada]]. It is the primary model for '''standards-based mathematics'''. The NCTM employed a consensus process that involved classroom teachers, [[mathematician]]s, and educational researchers. The resulting document sets forth a set of six principles (Equity, Curriculum, Teaching, Learning, Assessment, and Technology) that describe NCTM's recommended framework for mathematics programs, and ten general strands or standards that cut across the school mathematics curriculum. These strands are divided into mathematics content (Number and Operations, Algebra, Geometry, Measurement, and Data Analysis and Probability) and processes (Problem Solving, Reasoning and Proof, Communication, Connections, and Representation). Specific expectations for student learning are described for ranges of [[grade]]s ([[preschool]] to 2, 3 to 5, 6 to 8, and 9 to 12). == Origins == The original Principles and Standards for School Mathematics was developed by the NCTM. The NCTM's stated intent was to improve mathematics education. The contents were based on surveys of existing curriculum materials, curricula and policies from many countries, educational research publications, and government agencies such as the U.S. [[National Science Foundation]].<ref name="titlePrinciples and Standards - Standards 2000 Project" /> The original draft was widely reviewed at the end of 1998 and revised in response to hundreds of suggestions from teachers. The PSSM is intended to be "a single resource that can be used to improve mathematics curricula, teaching, and assessment."<ref name="titlePrinciples and Standards - Standards 2000 Project">{{cite web |url=http://standards.nctm.org/document/prepost/project.htm |title=Principles and Standards - Standards 2000 Project |accessdate=2008-03-08 |format= |work=}}</ref> The latest update was published in 2000. The PSSM replaces three prior publications by NCTM:<ref name="titlePrinciples and Standards - Standards 2000 Project" /> * ''Curriculum and Evaluation Standards for School Mathematics'' (1989), which was the first such publication by an independent professional organization instead of a government agency and outlined what students should learn and how to measure their learning. * ''Professional Standards for Teaching Mathematics'' (1991), which added information about best practices for teaching mathematics. * ''Assessment Standards for School Mathematics'' (1995), which focused on the use of accurate assessment methods. == Six principles == * ''Equity:'' The NCTM standards for equity, as outlined in the PSSM, encourage equal access to mathematics for all students, "especially students who are poor, not native speakers of English, disabled, female, or members of minority groups."<ref name="titleThe Equity Principle">{{cite web |url=http://standards.nctm.org/document/chapter2/equity.htm |title=The Equity Principle |accessdate=2008-03-08 |format= |work=}}</ref> The PSSM makes explicit the goal that all students should learn higher level mathematics, particularly underserved groups such as minorities and women. This principle encourages provision of extra help to students who are struggling and advocates high expectations and excellent teaching for all students.<ref name="titleThe Equity Principle"/> * ''Curriculum:'' In the PSSM's curriculum section, the NCTM promotes a "coherent" curriculum, in which an orderly and logical progression increases students' understanding of mathematics and avoids wasting time with unnecessary repetition.<ref name="titleThe Curriculum Principle">{{cite web |url=http://standards.nctm.org/document/chapter2/curric.htm |title=The Curriculum Principle |accessdate=2008-03-08 |format= |work=}}</ref> They acknowledge that the relative importance of some specific topics changes over time.<ref name="titleThe Curriculum Principle"/> For example, a basic understanding of [[iteration]] is important to students who are learning computer programming, and is almost absent from 19th century textbooks. Similarly, older American math textbooks included lessons that are no longer considered important, such as rules for calculating the number of [[bushel]]s of [[hay]] that could be stored in a bin of stated dimensions, because this skill was useful to [[farmer]]s at that time.<ref>Raub, Albert N. ''[http://books.google.com/books?id=Rk2xQJaz5pAC Complete Arithmetic.]'' Porter and Coates, 1877. See "Measurements of grain and hay" on page 313.</ref> The NCTM proposes that mathematics taught in modern classrooms be the skills that are most important to the students' lives and careers. * ''Teaching:''  In the PSSM, the NCTM promotes sound teaching methods, without prescribing a one-size-fits-all approach.<ref name="titleThe Teaching Principle">{{cite web |url=http://standards.nctm.org/document/chapter2/teach.htm |title=The Teaching Principle |accessdate=2008-03-10 |format= |work=}}</ref>  The NCTM wants teachers to be able to use their professional judgment in choosing teaching techniques. They favor professional development opportunities in both mathematics (content) and in effective teaching techniques (methods). * ''Learning:''  According to the PSSM, a combination of "factual knowledge, procedural facility, and conceptual understanding" is necessary for students to use mathematics.<ref name="titleThe Learning Principle">{{cite web |url=http://standards.nctm.org/document/chapter2/learn.htm |title=The Learning Principle |accessdate=2008-03-10 |format= |work=}}</ref>  While they state that 'Learning the "basics" is important,'<ref name="titleThe Learning Principle"/> the NCTM does not consider [[rote learning|the most simplistic forms of memorization by repetition]] to be ''sufficient'' achievement in mathematics.  A good student not only understands how and when to use facts, procedures, and concepts, but he or she also wants to figure things out and perseveres in the face of challenge.  The NCTM particularly deprecates attitudes in schools that suggest only certain students are capable of mastering math. * Assessment * Technology == Standards == Ten general strands or standards of mathematics content and processes were defined that cut across the school mathematics curriculum. Specific expectations for student learning, derived from the philosophy of [[outcome-based education]], are described for ranges of [[grade]]s ([[preschool]] to 2, 3 to 5, 6 to 8, and 9 to 12). These standards were made an integral part of nearly all outcome-based education and later [[standards-based education reform]] programs that were widely adopted across the United States by the 2000s. === Content standards === * ''Number and Operations:'' Number and operations is the fundamental basis of all mathematics, and teaching this critical area is the first content standard. All students must be taught to "understand numbers, ways of representing numbers, relationships among numbers, and number systems; understand meanings of operations and how they relate to one another; [and] compute fluently and make reasonable estimates."<ref name="titleStandards for School Mathematics: Number and Operations">{{cite web |url=http://standards.nctm.org/document/chapter3/numb.htm |title=Standards for School Mathematics: Number and Operations |accessdate=2008-03-10 |format= |work=}}</ref> The ability to perform mental calculations and to calculate answers on paper is "essential."<ref name="titleStandards for School Mathematics: Number and Operations"/> * ''Algebra:'' The PSSM names four skills related to algebra that should be taught to 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; [and] analyze change in various contexts."<ref name="titleStandards for School Mathematics: Algebra">{{cite web |url=http://standards.nctm.org/document/chapter3/alg.htm |title=Standards for School Mathematics: Algebra |accessdate=2008-03-10 |format= |work=}}</ref> Very simple algebra skills are often taught to young children. For example, a student might convert an addition equation such as 19+15=? into a simpler equation, 20+14=? for easy calculation. Formally, this is described in algebraic notation like this: (19+1) + (15−1) = x, but even a young student might use this technique without calling it algebra. The PSSM recommends that all students complete pre-algebra coursework by the end of eighth grade and take an algebra class during high school.<ref name="titleStandards for School Mathematics: Algebra"/> * ''Geometry:'' The overall goals for learning geometry are 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; [and] use visualization, spatial reasoning, and geometric modeling to solve problems."<ref name="titleStandards for School Mathematics: Geometry">{{cite web |url=http://standards.nctm.org/document/chapter3/geom.htm |title=Standards for School Mathematics: Geometry |accessdate=2008-03-10 |format= |work=}}</ref> Some geometry skills are used in many everyday tasks, such as reading a map, describing the shape of an object, arranging furniture so that it fits in a room, or determining the amount of fabric or construction materials needed for a project. Teaching should be appropriate to students' developmental level: Young students should be able to explain the difference between a [[rectangle]] and a [[square (geometry)|square]], while older students should be able to express more complex reasoning, including simple [[mathematical proof]]s.<ref name="titleStandards for School Mathematics: Geometry"/> The PSSM promotes the appropriate use of physical objects, drawings, and computer software for teaching geometry.<ref name="titleStandards for School Mathematics: Geometry"/> * ''Measurement:'' Measurement skills have many practical applications, as well as providing opportunities for advancing mathematical understand and for practicing other mathematical skills, especially number operations (e.g., addition or subtraction) and geometry. Students should "understand measurable attributes of objects and the units, systems, and processes of measurement; [and] apply appropriate techniques, tools, and formulas to determine measurements."<ref name="titleStandards for School Mathematics: Measurement">{{cite web |url=http://standards.nctm.org/document/chapter3/meas.htm |title=Standards for School Mathematics: Measurement |accessdate=2008-03-10 |format= |work=}}</ref> Unlike more abstract skills, the practical importance of measurement is readily apparent to students and parents. * ''Data analysis and probability:'' The PSSM says that all students should learn 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; [and] understand and apply basic concepts of probability."<ref name="titleStandards for School Mathematics: Data Analysis and Probability">{{cite web |url=http://standards.nctm.org/document/chapter3/data.htm |title=Standards for School Mathematics: Data Analysis and Probability |accessdate=2008-03-10 |format= |work=}}</ref> These skills allow students to make sense of critical information, such medical statistics and the results of political surveys. These skills are increasingly important as statistical data are used selectively by manufacturers to promote products. While young students learn simple skills such as ways to represent the number of pets belonging to their classmates,<ref name="titleFeldman_Norton">{{cite web |url=http://www.cbmsweb.org/NationalSummit/WG_Speakers/feldman_norton.htm |title=Feldman_Norton |accessdate=2008-03-10 |format= |work=}}</ref> or traditional skills such as calculating the [[arithmetic mean]] of several numbers, older students might learn concepts that were traditionally neglected, such as the difference between the occasionally dramatic [[relative risk reduction]] figures and the more concrete [[absolute risk reduction]], or why political pollsters report the [[margin of error]] with their survey results. === Process standards === * Problem Solving * Reasoning and Proof * Communication * Connections * Representation == Curriculum Focal Points == In 2006, NCTM issued a document called "Curriculum Focal Points" that presented the most critical mathematical topics for each grade in elementary and middle schools. American mathematics instruction tends to be diffuse and is criticized for including too many topics each year. In part, this publication is intended to assist teachers in identifying the most critical content for targeted attention. More such publications are planned. NCTM stated that "Focal Points" was a step in the implementation of the Standards, not a reversal of its position on teaching students to learn foundational topics with conceptual understanding.<ref name="titleHow Do the Curriculum Focal Points Relate to Principles and Standards for School Mathematics?">{{cite web |url=http://www.nctm.org/standards/content.aspx?id=268 |title=How Do the Curriculum Focal Points Relate to Principles and Standards for School Mathematics? |accessdate=2008-03-24 |format= |work=}}</ref> Contrary to the expectation of many textbook publishers and educational progressives, the 2006 Curriculum Focal Points strongly emphasized the importance of basic arithmetic skills in lower and middle grades. Because of this, the "Curriculum Focal Points" was perceived by the media<ref>[http://www.nytimes.com/2006/09/13/education/13math.html?_r=1&ref=us] Report Urges Changes in the Teaching of Math in U.S. Schools by TAMAR LEWIN New York Times September 13, 2006</ref><ref>[http://www.suntimes.com/output/commentary/cst-edt-edits13b.html] Chicago Sun Times "Fuzzy teaching ideas never added up" September 13, 2006</ref> as an admission that the PSSM had originally recommended, or at least had been interpreted as recommending, reduced instruction in basic arithmetic facts. The 2006 Curriculum Focal Points identifies three critical areas at each grade level from for pre-kindergarten through Grade 8.<ref name="titleHow Do the Curriculum Focal Points Relate to Principles and Standards for School Mathematics?"/> Samples of the specific focal points for three grades are below: {|class="wikitable" ! style="background: #C0C0C0; text-align: center;" | Focal points !! style="background: #C0C0C0; text-align: center;" | Related content standard !! style="background: #C0C0C0; text-align: center;" | Example |- ! colspan="3" align="center"| Pre-Kindergarten Focal Points<ref name="titlePrekindergarten">{{cite web |url=http://www.nctm.org/standards/focalpoints.aspx?id=300 |title=Prekindergarten |accessdate=2008-03-24 |format= |work=}}</ref> (student age: 4 or 5 years old) |- | Developing an understanding of whole numbers || Number and Operations || How many blue pencils are on the table? |- | Identifying shapes and describing spatial relationships || Geometry || What is the name of this shape? |- | Identifying measurable attributes and comparing objects by using these attributes || Measurement || Which one is bigger? |- ! colspan="3" align="center"| Fourth Grade Focal Points<ref name="titleGrade 4">{{cite web |url=http://www.nctm.org/standards/focalpoints.aspx?id=332 |title=Grade 4 |accessdate=2008-03-24 |format= |work=}}</ref> (student age: 9 or 10 years old) |- | Developing quick recall of multiplication facts and related division facts and fluency with whole number multiplication || Number and Operations, Algebra || Fluently multiplying 26 by 89. |- | Developing an understanding of decimals, including the connections between fractions and decimals || Number and Operations || Write 0.25 as a fraction. |- | Developing an understanding of area and determining the areas of two-dimensional shapes || Measurement || What is the area of a rectangular room that measures eight feet by ten feet? |- ! colspan="3" align="center"| Eighth Grade Focal Points<ref name="titleGrade 8">{{cite web |url=http://www.nctm.org/standards/focalpoints.aspx?id=340 |title=Grade 8 |accessdate=2008-03-24 |format= |work=}}</ref> (student age: 13 or 14 years old) |- | Analyzing and representing linear functions and solving linear equations and systems of linear equations || Algebra || Solve simple linear algebra equations, such as 12 = 4x + 4. |- | Analyzing two- and three-dimensional space and figures by using distance and angle || Geometry, Measurement || Use the [[Pythagorean theorem]] to determine the length of one side of a [[right triangle]]. |- | Analyzing and summarizing data sets || Data Analysis, Number and Operations, Algebra || What is the [[median]] price in this list? |- |} == Controversy == {{Main|Math wars}} Because most education agencies in the United States have adopted the NCTM recommendations to varying degrees, many textbook publishers promote their products as being compliant with the publishers' interpretations of the PSSM.<ref name="titleGlencoe.com eCatalog">From the advertising materials: "Correlated to the NCTM Standards, they encourage students to understand the relationship..."{{cite web |url=http://www.glencoe.com/catalog/index.php/component?isbn=0078254493&c=1674&s=3324&p=5012&t=2 |title=Glencoe.com eCatalog |accessdate=2008-03-24 |format= |work=}}</ref><ref name="titleThinking Algebraically">From the advertising materials: "To address the call for “Algebra for All” from NCTM, this classroom-tested, standards-based program..."{{cite web |url=http://www.pearsonschool.com/index.cfm?locator=PSZ16u&PMDbSiteId=2781&PMDbSolutionId=6724&PMDbSubSolutionId=&PMDbCategoryId=&PMDbProgramId=1047&level=4 |title=Thinking Algebraically |accessdate=2008-03-24 |format= |work=}}</ref><ref name="titleBeyond Arithmetic">From the advertising materials: "''Beyond Arithmetic'' provides a philosophical framework that links the NCTM goals with what actually happens in classrooms..."{{cite web |url=http://www.pearsonschool.com/index.cfm?locator=PSZ16u&PMDbSiteId=2781&PMDbSolutionId=6724&PMDbSubSolutionId=&PMDbCategoryId=&PMDbProgramId=288&level=4 |title=Beyond Arithmetic |accessdate=2008-03-24 |format= |work=}}</ref><ref name="Saxon Math E02 Product Detail">From a brief description of the Saxon Math textbooks: "Correlated to the NCTM curriculum focal points." {{cite web |url=http://saxonpublishers.harcourtachieve.com/en-US/Products/Product+Detail.htm?CS_ProductID=1600324401&CS_Category=Saxon_E02_Math1_04&CS_Catalog=Harcourt+Achieve+Catalog |title=saxonpublishers Product Detail |accessdate=2008-03-24 |format= |work=}}</ref> However, the NCTM does not endorse, approve, or recommend any textbooks or other products and has never agreed that any textbook accurately represents their goals.<ref name="titleQuestions & Answers">{{cite web |url=http://www.nctm.org/standards/focalpoints.aspx?id=274 |title=Questions & Answers |accessdate=2008-03-24 |format= |work=}}</ref> ==See also== *[[Education in the United States]] *[[Reform mathematics]] ==References== {{reflist|2}} ==External links== *[http://standards.nctm.org/ Access to online version of Standards] [[Category:Mathematics books]] [[Category:Mathematics education]]