ISBE EMAG DRAFT 11/21/11
Math Standards for Middle Grade 6-8 Teachers
Math is a continuum of knowledge that requires teachers to have content knowledge beyond their
defined domain. The standards are based on the belief that mathematics must be approached as a
unified whole. Its concepts, procedures, and intellectual processes are so interrelated that, in a
significant sense, its “whole is greater than the sum of the parts.” In order to develop this
expertise in prepared teachers, programs must integrate stakeholders including content knowledge
specialists, pedagogical specialists, experienced teachers in the field, administrators and
instructional leaders.
Teachers’ comfort with, and confidence in, their knowledge of mathematics affects both what they
teach and how they teach it. Knowing mathematics includes understanding specific concepts and
procedures and the mathematical practices as defined by Common Core State Standards.
Well prepared middle math teachers demonstrate enthusiasm and show proficiency and fluency in
the processes and application of mathematics. They will demonstrate a mastery of the
Kindergarten through 12th grade Common Core State Standards in both the content standards and
mathematical practices. In addition to the CCSS, middle school math teachers must demonstrate
mastery of portions of calculus and discrete mathematics as defined in the document.
Middle Grade teachers of mathematics should have knowledge of and ability to use and evaluate
instructional strategies. These include but are not limited to classroom organizational models,
ways to represent mathematical concepts and procedures, instructional materials and resources,
ways to promote discourse, and means of assessing student understanding.
Dispositions:
The middle grades math teacher will possess and demonstrate the following dispositions to
promote effective student learning:
1. Shows fair-mindedness, empathy and ethical behavior.
2. Exhibits a comfort with and confidence in their knowledge of mathematics.
3. Exhibits a view that all students can learn mathematics but recognizes each student as an
individual learner.
4. Supports all learners and holds all learners to high expectations of academic achievement.
5. Shares/exhibits a genuine interest in mathematical concepts and connections.
6. Demonstrates a persistence with finding solutions to problems
7. Demonstrates a willingness to consider multiple processes and multiple solutions to the
same problem.
8. Exhibits an appreciation for mathematics-related applications across disciplines such as
art, music, architecture, geography, demographics and technology.
9. Embraces inquiry and investigation during the teaching and learning of mathematics.
10. Understands the importance of connections and applications of mathematics to model
phenomenon and real-world situations.
11. Demonstrates an understands the importance of staying abreast of current research with
regard to the science of mathematics, trends in mathematical pedagogical content
knowledge and the practice of teaching in general.
12. Commits to and pursues on-going professional growth opportunities.
13. Understands the need to balance procedural, conceptual and factual mathematical
knowledge as well as formal and informal learning environments.
14. Exhibits an habitual inclination to see mathematics as sensible, useful, and worthwhile.
ISBE EMAG DRAFT 11/21/11
15. Demonstrates a belief in diligence and one’s own efficacy in doing mathematics.
16. Develops collaborative and respectful relationships with families, colleagues and
community members to support students’ reading and writing.
Standard 1 - Foundational Knowledge and Skills.
Problem Solving
The competent middle grades mathematics teacher knows, understands, and applies the process of
mathematical problem solving and problem posing.
Knowledge Indicators
A. Understands that there are multiple problem-solving strategies for any problem.
B. Understands that some math problems have more than one solution.
C. Understands that problem solving is a multistep process.
D. Knows how to model real world situations with mathematics.
Performance Indicators
A. Makes sense of quantities and their relationships in problem situations.
B. Applies, adapts and connects a variety of appropriate strategies to solve problems.
C. Solves problems that arise in mathematics and the real world.
D. Builds new mathematical knowledge through problem solving.
E. Monitors and reflects on the process of mathematical problem solving.
F. Determines the meaning of the problem, analyzes available information for relevance,
determines solution paths, attends to precision, and reflects on the reasonableness and
meaning of the solution.
G. Classifies problem solving tasks in terms of underlying concepts, cognitive challenges,
and complexity.
H. Solves multi-step problems.
Reasoning and Proof
The competent middle grades mathematics teacher reasons, constructs, and evaluates
mathematical arguments and develops an appreciation for mathematical rigor and inquiry.
Knowledge Indicators
A. Recognizes types of deductions and inductions as fundamental aspects of mathematics.
B. Knows relevant mathematical theorems, rules, concepts, and properties as defined in the
content standards Number & Operation, Algebra & Algebraic Thinking, Geometry,
Statistics & Probability and Measurement & Data and understands their limitations.
ISBE EMAG DRAFT 11/21/11
Performance Indicators
A. Uses givens, definitions, and mathematical relationships to argue mathematically.
B. Constructs viable arguments and critiques the reasoning of others.
C. Makes and investigates mathematical conjectures.
D. Selects and uses various types of reasoning and methods of proof.
E. Makes generalizations.
Mathematical Communication
The competent middle grades mathematics teacher communicates mathematical thinking
verbally, graphically, and in written form to students, peers, faculty, and others.
Knowledge Indicators
A. Knows the vocabulary of mathematics and the connections to mathematical concepts.
B. Knows the symbolic representation of mathematics.
C. Knows and recognizes different types of graphical representations of quantitative and
spatial relationships.
Performance Indicators
A. Uses the language of mathematics to express ideas precisely.
B. Communicates using symbolic mathematical representations.
C. Applies and interprets graphical representations appropriately.
D. Communicates mathematical thinking coherently and clearly.
E. Analyzes and evaluates the mathematical thinking and strategies of others.
Mathematical Connections
The competent middle grades mathematics teacher recognizes, uses, and makes connections
between and among mathematical ideas, in real world contexts, and other content areas to build
mathematical understanding.
Knowledge Indicators
A. Understands the developmental nature of mathematical concepts.
B. Understands how mathematical concepts, algorithms, and representations interconnect and
build on one another to produce a coherent whole.
C. Understands that there are connections within and among mathematical disciplines.
ISBE EMAG DRAFT 11/21/11
D. Recognizes mathematics in real world contexts and other content areas outside of
mathematics
E. Recognizes connections among mathematical concepts, algorithms, and representations.
Performance Indicators
A. Demonstrates how mathematical concepts, algorithms, and representations interconnect
and build on one another to produce a coherent whole.
B. Uses mathematics in real world contexts and other content areas outside of mathematics
C. Uses connections among mathematical disciplines, concepts, algorithms, and
representations.
Mathematical Representation
The competent middle grades mathematics teacher uses varied representations of
mathematical ideas to support and deepen students’ mathematical understanding.
Knowledge Indicators
A. Knows that mathematics can be represented concretely, verbally, numerically,
symbolically, and graphically.
B. Understands mathematical models as they apply to various contexts.
C. Understands that a single mathematical concept can be represented in multiple ways.
D. Knows that mathematical concepts can be presented in a progression from concrete to
abstract and understands the connections between those representations.
E. Recognizes the expansion of symbolic mathematical notations that middle grade students
must learn and use.
Performance Indicators
A. Creates and uses representations to organize, record, and communicate mathematical
ideas
B. Uses multiple representations to model and interpret physical, social, and mathematical
phenomena.
C. Selects and applies various mathematical representations in the process of solving
problems.
D. Demonstrates the progression of mathematical concepts from concrete to abstract
representations.
Modeling with Mathematics
Knowledge
ISBE EMAG DRAFT 11/21/11
Performance
Number Systems and Quantity
The competent middle grades mathematics teacher demonstrates computational proficiency,
including a conceptual understanding of numbers, ways of representing numbers, relationships
among numbers and number systems, and the meanings of operations.
Knowledge Indicators
A. Knows and understands the concepts and learning progression of number and operation as
it occurs in grades K-8.
B. Understands the meaning and multiple representations of arithmetic operations as they
apply to real and complex numbers.
C. Understands integers, rational, and irrational numbers, their relative sizes, and how
operations with whole numbers extend to them.
D. Understands the theories of number fields, identities, reciprocals, opposites, and the
implication of these ideas to the operations.
E. Understands the properties of addition, multiplication, and equality.
F. Knows the historical development of numbers and number systems including
contributions from diverse cultures.
G. Understands when and how to use proportional reasoning.
Performance Indicators
A. Uses place value in representing, comparing, and ordering whole numbers and finite
decimals.
B. Demonstrates proficiency in computation of integers, rational, and irrational
numbers using a variety of algorithms, mental mathematics, and computational
estimation.
C. Analyzes integers, rational, and irrational numbers, their relative size, and how
operations with whole numbers extend to them.
D. Uses proportional reasoning to explore relationships among quantities and to solve
problems beyond matching of two ratios and cross multiplying.
Algebra and Algebraic Thinking
The competent middle grades mathematics teacher emphasizes relationships among
quantities including functions, ways of representing mathematical relationships, and the
analysis of change.
ISBE EMAG DRAFT 11/21/11
Knowledge Indicators
A. Recognizes that every operation has an inverse operation
B. Understands the properties of equality and inequality.
C. Understands the difference between variable and placeholder.
D. Understands that there are linear and nonlinear relationships.
E. Understands equivalence of various forms of expressions.
F. Understands the effect of changing various parameters in an equation or inequality.
G. Understands algebraic sequence and series
H. Recognizes the contextual significance of the components of various forms of an equation.
I. Knows the historical development of algebra including contributions from diverse
cultures.
Performance Indicators
A. Explores and analyzes patterns, relations, and functions.
B. Applies inverse operations and the properties of equality to solve for an unknown
value.
C. Creates and manipulates mathematical models to represent quantitative relationships using
symbols and numbers.
D. Analyzes how changing one variable affects another.
E. Derive a formula and use to solve real world applications
F. Demonstrate the ability to define the relationship between two quantities and create a
linear, quadratic or exponential equation or inequality that represents this relationship.
G. Creates, solves and graphs logarithmic, parametric, trigonometric, polynomial, rational,
radical, and absolute value relations.
H. Transforms expressions into alternate forms to evaluate equivalency.
Geometry
The competent middle grades mathematics teacher uses spatial visualization and geometric
modeling to explore and analyze geometric shapes, structures, and their properties.
Knowledge Indicators
A. Understands the definitions and properties of two and three dimensional geometrical
shapes and drawings.
B. Understands the relationships between and among geometric shapes.
ISBE EMAG DRAFT 11/21/11
C. Understands the basic trigonometric concepts and their applications to geometry and the
real world.
D. Knows the proper vocabulary terms associated with geometrical shapes, their components,
and their properties.
E. Knows the origin and use of formulas for perimeter, area, surface area, and volume.
F. Knows the construction and use of two and three dimensional coordinate grids.
G. Recognizes that geometric and algebraic concepts are inextricably related.
H. Knows the historical development of Euclidean and non-Euclidean geometries including
contributions from diverse cultures.
I. Understands transformations, symmetry, congruence and similarity.
Performance Indicators
A. Classifies geometric shapes using definitions and properties.
B. Explores and describes relationships among and between two and three dimensional
shapes.
C. Uses right triangle trigonometry and laws of sines and cosines to solve real world
problems.
D. Constructs and manipulates representations of two- and three-dimensional objects using
concrete models and drawings.
E. Perform calculations involving various geometric relationships in order to solve for an
unknown.
F. Uses precise mathematical language to compare and contrast the properties of two and
three dimensional shapes according to their attributes.
G. Describes spatial relationships among points on a coordinate graph.
H. Applies transformations and uses symmetry, congruence, and similarity.
Statistics and Probability
The competent middle grades mathematics teacher demonstrates an understanding of
concepts and practices related to data analysis, statistics, and probability.
Knowledge Indicators
A. Understands the value of probability and statistics as a field of mathematics.
B. Understands the concept of classification as organization of data in mutually exclusive
sets.
C. Understands key concepts associated with sampling and making inferences about data.
ISBE EMAG DRAFT 11/21/11
D. Understands the utilities of different measures of central tendency and variability.
E. Knows the common misconceptions of statistics.
F. Knows the historical development of probability and statistics including contributions
from diverse cultures.
Performance Indicators
A. Demonstrates various methods of data collection and organization.
B. Calculates the various measures of central tendency and describes the advantages and
disadvantages of each.
C. Describes different forms of distribution and how it relates to variability.
D. Identifies pertinent questions to explore and the most relevant statistical process for
analyzing the results.
E. Identifies and analyzes differing interpretations of the same set of data.
Measurement and Data
The competent middle grades mathematics teacher applies and uses measurement concepts and
tools in a variety of situations and across mathematical disciplines.
Knowledge Indicators
A. Understands the relationship between units of measurement for length, area and volume.
B. Knows the relationship between units of measurement for velocity and acceleration.
C. Knows the historical development of measurement and measurement systems including
contributions from diverse cultures.
D. Understands sets and relationships among sets.
Performance Indicators
A. Demonstrates how operations allow calculations of length, area and volume.
B. Make connections between the arithmetical operations of addition and multiplication and
the concepts of perimeter, area, surface area and volume.
C. Demonstrates the measurement of angles using different instruments.
D. Describes and interprets data using graphs and pictorial representations.
E. Perform simulations to determine experimental probability and compare the results to the
theoretical probability. Designs investigations that can be addressed by creating data sets
and collecting, organizing, and displaying relevant data.
F. Interprets and uses scatter plots, regression lines and correlation for bivariate data.
ISBE EMAG DRAFT 11/21/11
Calculus
Candidates demonstrate a conceptual understanding of limit, continuity, differentiation, and
integration and a thorough background in the techniques and application of the calculus.
Knowledge Indicators
A. Understands the idea of limits with respect to relationships observed over time and
relationships that exhibit constant and variable rate of change.
B. Understands the concept of differentials.
C. Understands the concept of derivatives as a measure of small and instantaneous change.
D. Understands the concept of integration.
E. Understands the concepts of infinite series.
F. Knows the Fundamental Theorum of Calculus.
G. Demonstrates knowledge of the historical development of calculus including contributions
from diverse cultures.
Performance Indicators
A. Demonstrates the ability to determine limits of sequences, series and functions.
B. Demonstrates the ability to use derivatives to sketch graphs of polynomial functions.
C. Applies the concept of intergration using areas to determine definite integrals
D. Connects calculus to real world problems and scenarious
E. Demonstrates the ability to use calculus for modeling and optimization applications.
Discrete Mathematics
Candidates apply the fundamental ideas of discrete mathematics in the formulation and solution
of problems.
Knowledge Indicators
A. Understands classes of graphs such as paths, cycles, wheels and complete graphs
B. Understands various properties of graphs such as vertices, edges and degrees.
C. Understands how to apply discrete mathematics definitions to construct solid logical
mathematical arguments including proofs by contradiction.
D. Understands the connection between discrete mathematics and programming.
E. Understands finite graphs, trees and combinatoric concepts.
ISBE EMAG DRAFT 11/21/11
Performance Indicators
A. Models permutations and combinations in real world applications.
B. Demonstrates knowledge of the historical development of discrete mathematics
including contributions from diverse cultures.
C. Reasons mathematically and constructs various types of graphs and charts
D. Demonstrates appropriate use of various classes of graphs to model real world data.
E. Uses graph theory to model map coloring.
Standard 2 – Instructional Strategies
Mathematics Pedagogy
The competent middle grades mathematics teacher possesses a deep understanding of how
students learn mathematics and the pedagogical knowledge specific to mathematics teaching
and learning.
Knowledge Indicators
A. Knows the importance of identifying prerequisite mathematical skills.
B. Discerns appropriate and effective instructional strategies that are research based.
C. Recognizes common and uncommon student misconceptions and procedural errors.
D. Knows research-based instructional strategies in mathematics.
E. Knows appropriate scaffolding techniques.
F. Knows that mathematical concepts can be presented in a progression from concrete to
abstract.
G. Understands the parameters of appropriate pacing and sequencing of mathematical
concepts.
H. Understands the practice and values of metacognition in respect to learning mathematics.
I. Recognizes the cognitive challenge facing students with the expansion of the symbolic
mathematical notations that middle grade students face.
J. Understands the cognitive challenge facing students that algebra is generalized arithmetic.
K. Math and literacy strategies…
L. Math and RtI strategies…
M. Knows the importance of developing students’ habits of mind to view mathematics as
sensible, useful and worthwhile.
N. Understands the importance of developing students’ sense of mathematical efficacy.
ISBE EMAG DRAFT 11/21/11
Performance Indicators
A. Identifies and diagnoses skills prerequisite to the instruction of mathematical concepts.
B. Analyzes student errors to determine their sources in terms of procedural and conceptual
knowledge and adjust instruction accordingly.
C. Uses multiple strategies, including listening to and understanding the ways students think
about mathematics, to assess students’ mathematical knowledge.
D. Uses scaffolding techniques to build conceptual understanding and to move students from
concrete operations to abstract thinking.
E. Makes informed decisions about instructional pace, sequence, and topic priorities.
F. Makes metacognition an explicit component of mathematical instruction.
G. Develops assessment tools that effectively target conceptual development, mastery, and
fluency in mathematics.
H. Demonstrates an informed transition to the effective use of the symbolic mathematical
notations.
I. Demonstrates an informed transition from arithmetic to algebraic thinking through a
process of decontextualization and recontextualization.
J. Enables students to build mathematical knowledge through problem solving.
K. Motivates students to study mathematics.
L. Creates engaging and challenging tasks for students of different learning styles and levels.
Standard 3 – Materials, Text and Technology
The competent middle grades mathematics teacher applies current technologies and materials in
the mathematics classroom.
Knowledge Indicators
A. Understands various units of measurement and their relationships and applications.
B. Recognizes measurable attributes of objects and events.
C. Knows different measurement tools.
D. Knows how to use physical and virtual manipulatives.
E. Understands that technologies such as but not limited to spreadsheets, calculators,
dynamic software, data collection devices, computer algebra systems, dynamic statistical
packages, and graphing calculators can be used as a tool to aid instruction and assessment.
F. Knows how to use technology in conjunction with the standards and to attain instructional
goals.
G. Knows a variety of mathematical literature for instructional purposes.
ISBE EMAG DRAFT 11/21/11
H. Knows a variety of research literature pertinent to 6-8 mathematics education.
Performance Indicators
A. Selects and uses appropriate measurement units, techniques, and tools.
B. Converts units within and between systems of measurement.
C. Employs estimation as a way of understanding measurement units and processes.
D. Uses current mathematical literature and references
E. Demonstrates the use of physical and virtual manipulatives.
F. Uses mathematical knowledge to choose and use appropriate technologies such as but not
limited to spreadsheets, calculators, dynamic software, data collection devices, computer
algebra systems, dynamic statistical packages, and graphing calculators can be used as a
tool to aid instruction and assessment.
G. Demonstrates the ability to make informed decisions in the selection and use of technology
and materials.
H. Uses technology effectively to explore and learn math.
Standard 4 – Assessment and Evaluation
Standards 5 – Differentiation for Diverse Learners
Standard 6 – Mathematical Environment
EMAG Draft
– Educator & School Development – ISBE – November 2011