National+Council+of+Teacher+of+Mathematics

Programs for Initial Preparation of Mathematics TeachersStandards for Middle Level Mathematics Teachers
 * NCTM Standards (2003) – Middle Level Mathematics Teachers**


 * Process Standards (Standards 1-7)**

The process standards are based on the belief that mathematics must be approached as aunified whole. Its concepts, procedures, and intellectual processes are so interrelatedthat, in a significant sense, its “whole is greater than the sum of the parts.” This approachwould best be addressed by involvement of the mathematics content, mathematicseducation, education, and field experience faculty working together in developing thecandidates’ experiences.

Likewise, the response to the disposition standard will require total faculty input. Thisstandard addresses the candidates’ nature and temperament relative to being amathematician, an instructor, a facilitator of learning, a planner of lessons, a member of aprofessional community, and a communicator with learners and their families.

Standard 1: Knowledge of Mathematical Problem Solving Candidates know, understand, and apply the process of mathematical problem solving. Indicators 1.1 Apply and adapt a variety of appropriate strategies to solve problems.1.2 Solve problems that arise in mathematics and those involving mathematics in othercontexts.1.3 Build new mathematical knowledge through problem solving.1.4 Monitor and reflect on the process of mathematical problem solving.

Standard 2: Knowledge of Reasoning and ProofCandidates reason, construct, and evaluate mathematical arguments and develop anappreciation for mathematical rigor and inquiry. Indicators 2.1 Recognize reasoning and proof as fundamental aspects of mathematics.2.2 Make and investigate mathematical conjectures.2.3 Develop and evaluate mathematical arguments and proofs.2.4 Select and use various types of reasoning and methods of proof.

Standard 3: Knowledge of Mathematical Communication Candidates communicate their mathematical thinking orally and in writing to peers,faculty, and others.

Indicators 3.1 Communicate their mathematical thinking coherently and clearly to peers, faculty,and others.3.2 Use the language of mathematics to express ideas precisely.3.3 Organize mathematical thinking through communication.3.4 Analyze and evaluate the mathematical thinking and strategies of others.

Standard 4: Knowledge of Mathematical Connections Candidates recognize, use, and make connections between and among mathematical ideasand in contexts outside mathematics to build mathematical understanding.

Indicators 4.1 Recognize and use connections among mathematical ideas.4.2 Recognize and apply mathematics in contexts outside of mathematics.4.3 Demonstrate how mathematical ideas interconnect and build on one another toproduce a coherent whole.

Standard 5: Knowledge of Mathematical Representation Candidates use varied representations of mathematical ideas to support and deepenstudents’ mathematical understanding. Indicators 5.1 Use representations to model and interpret physical, social, and mathematicalphenomena.5.2 Create and use representations to organize, record, and communicate mathematicalideas.5.3 Select, apply, and translate among mathematical representations to solve problems.

Standard 6: Knowledge of TechnologyCandidates embrace technology as an essential tool for teaching and learningmathematics. Indicator 6.1 Use knowledge of mathematics to select and use appropriate technological tools, suchas but not limited to, spreadsheets, dynamic graphing tools, computer algebra systems,dynamic statistical packages, graphing calculators, data-collection devices, and presentation software.

Standard 7: DispositionsCandidates support a positive disposition toward mathematical processes andmathematical learning. Indicators 7.1 Attention to equity7.2 Use of stimulating curricula7.3 Effective teaching7.4 Commitment to learning with understanding7.5 Use of various assessments7.6 Use of various teaching tools including technology

In addition to knowing students as learners, mathematics teacher candidates shoulddevelop knowledge of and ability to use and evaluate instructional strategies andclassroom organizational models, ways to represent mathematical concepts andprocedures, instructional materials and resources, ways to promote discourse, and meansof assessing student understanding. This section on pedagogy is to address thisknowledge and skill.
 * Pedagogy (Standard 8)**

Standard 8: Knowledge of Mathematics PedagogyCandidates possess a deep understanding of how students learn mathematics and of thepedagogical knowledge specific to mathematics teaching and learning. Indicators 8.1 Selects, uses, and determines suitability of the wide variety of available mathematicscurricula and teaching materials for all students including those with special needs suchas the gifted, challenged and speakers of other languages.8.2 Selects and uses appropriate concrete materials for learning mathematics.8.3 Uses multiple strategies, including listening to and understanding the ways studentsthink about mathematics, to assess students’ mathematical knowledge.8.4 Plans lessons, units and courses that address appropriate learning goals, includingthose that address local, state, and national mathematics standards and legislativemandates.8.5 Participates in professional mathematics organizations and uses their print and on-lineresources.8.6 Demonstrates knowledge of research results in the teaching and learning ofmathematics.8.7 Uses knowledge of different types of instructional strategies in planning mathematicslessons.8.8 Demonstrates the ability to lead classes in mathematical problem solving and indeveloping in-depth conceptual understanding, and to help students develop and testgeneralizations. 8.9 Develop lessons that use technology’s potential for building understanding ofmathematical concepts and developing important mathematical ideas.

Candidates’ comfort with, and confidence in, their knowledge of mathematics affectsboth what they teach and how they teach it. Knowing mathematics includesunderstanding specific concepts and procedures as well as the process of doingmathematics. That knowledge is the subject of the following standards.
 * Content (Standards 9-15)**

Standard 9: Knowledge of Number and OperationCandidates demonstrate computational proficiency, including a conceptual understandingof numbers, ways of representing number, relationships among number and numbersystems, and meanings of operations. Indicators 9.1 Develop the mathematics that underlies the procedures used for operations involvingwhole numbers, integers, and rational numbers.9.2 Use properties involving number and operations, mental computation, andcomputational estimation.9.3 Provide equivalent representations of fractions, decimals, and percents.9.4 Create, solve, and apply proportions.9.5 Apply the fundamental ideas of number theory.9.6 Make sense of large and small numbers and use scientific notation.9.7 Analyze and explain the distinctions among whole numbers, integers, rationalnumbers, and real numbers and whether or not the field axioms hold.9.8 Demonstrate knowledge of the historical development of number and number systemsincluding contributions from diverse cultures.

Standard 10: Knowledge of Different Perspectives on AlgebraCandidates emphasize relationships among quantities including functions, ways ofrepresenting mathematical relationships, and the analysis of change. Indicators 10.1 Explore, analyze, and represent patterns, relations, and functions.10.2 Represent and analyze mathematical structures.10.3 Investigate equality, equations, and proportional relationships.10.4 Use mathematical models to represent quantitative relationships.10.5 Analyze change in various contexts.10.6 Demonstrate knowledge of the historical development of algebra includingcontributions from diverse cultures.

Standard 11: Knowledge of GeometriesCandidates use spatial visualization and geometric modeling to explore and analyzegeometric shapes, structures, and their properties. Indicators 11.1 Demonstrate knowledge of core concepts and principles of Euclidean geometry intwo and three dimensions.11.2 Exhibit knowledge of informal proof.11.3 Build and manipulate representations of two- and three-dimensional objects andperceive an object from different perspectives.11.4 Specify locations and describe spatial relationships using coordinate geometry.11.5 Analyze properties and relationships of geometric shapes and structures.11.6 Apply transformation and use congruence, similarity, and line or rotationalsymmetry.11.7 Demonstrate knowledge of the historical development of Euclidean and non-Euclidean geometries including contributions from diverse cultures.

Standard 12: Knowledge of CalculusCandidates demonstrate a conceptual understanding of limit, continuity, differentiation,and integration and a thorough background in the techniques and application of thecalculus. Indicators 12.1 Demonstrate a conceptual understanding of basic calculus concepts.12.2 Demonstrate knowledge of the historical development of calculus includingcontributions from diverse cultures.

Standard 13: Knowledge of Discrete MathematicsCandidates apply the fundamental ideas of discrete mathematics in the formulation andsolution of problems. Indicators 13.1 Demonstrate a conceptual understanding of the fundamental ideas of discretemathematics such as finite graphs, trees and combinatorics.13.2 Use technological tools to apply the fundamental concepts of discrete mathematics.13.3 Demonstrate knowledge of the historical development of discrete mathematicsincluding contributions from diverse cultures.

Standard 14: Knowledge of Data Analysis, Statistics, and ProbabilityCandidates demonstrate an understanding of concepts and practices related to data analysis, statistics, and probability. Indicators 14.1 Design investigations, collect data through random sampling or random assignmentto treatments, and use a variety of ways to display the data and interpret datarepresentations.14.2 Draw conclusions involving uncertainty by using hands-on and computer-basedsimulation for estimating probabilities and gathering data to make inferences anddecisions.14.3 Identify misuses of statistics and invalid conclusions from probability.14.4 Use appropriate statistical methods and technological tools to analyze data anddescribe shape, spread, and center.14.5 Investigate, interpret, and construct representations for conditional probability,geometric probability, and for bivariate data.14.6 Demonstrate knowledge of the historical development of probability and statisticsincluding contributions from diverse cultures.

Standard 15: Knowledge of MeasurementCandidates apply and use measurement concepts and tools. Indicators 15.1 Recognize measurement attributes and their effect on the choice of appropriate toolsand units.15.2 Apply techniques, tools, and formulas to determine measurements.15.3 Employ estimation as a way of understanding measurement units and processes.15.4 Completes error analysis through determining the reliability of the numbers obtainedfrom measurement.15.5 Demonstrate knowledge of the historical development of measurement andmeasurement systems including contributions from diverse cultures.

The development of mathematics teacher candidates should include opportunities toexamine the nature of mathematics, how it should be taught and how students learnmathematics; observe and analyze a range of approaches to mathematics teaching andlearning, focusing on the tasks, discourse, environment and assessment; and work with adiverse range of students individually, in small groups, and in large class settings. Standard 16: Field-Based ExperiencesCandidates complete field-based experiences in mathematics classrooms. Indicators 16.1 Engage in a sequence of planned opportunities prior to student teaching that includesobserving and participating in middle grades mathematics classrooms under thesupervision of experienced and highly qualified teachers.16.2 Experience full-time student teaching in middle grades mathematics that issupervised by an experienced and highly qualified teacher and a university or collegesupervisor with middle grades mathematics teaching experience.16.3 Demonstrate the ability to increase students’ knowledge of mathematics.
 * Field-Based Experiences (Standard 16)**