Section 21.150 Mathematics Standards for
Mathematics Teachers in the Middle Grades
In addition to the standards set forth in Subpart B of this
Part, each mathematics teacher in the middle grades shall possess the knowledge
and skills articulated in this Section.
a) Core Content Area
Knowledge
1) Calculus
Effective middle grade mathematics
teachers:
A) demonstrate
knowledge of properties and notation of real numbers, properties of exponents
and radicals, factoring techniques, solving polynomial equations and operations
with rational expressions;
B) on the
Cartesian Plane, graph polynomial, rational and radical functions and circles,
and find horizontal and vertical asymptotes, and points of intersection of
curves;
C) define
function, domain, range, inverse functions, operate on functions, and use
functional notation;
D) define
one-sided, general and at infinity limits, and evaluate them by using the
properties of limits;
E) define
and apply the properties of continuous functions and determine discontinuities;
F) define
first-order and higher-order derivatives and evaluate them using constant
power, constant multiple, product, quotient and chain rules and by implicit
differentiation;
G) apply
the rules of derivatives to find tangent line, slope, rate of change, velocity
and acceleration, marginal analysis, increasing and decreasing functions, curve
sketching with maxima and minima and concavity, and solving optimization
problems;
H) demonstrate
knowledge of properties of exponential and logarithmic functions and their
derivatives;
I) demonstrate
knowledge of basic anti-derivatives, explore integration using the notion of
"area under the curve" to determine definite integrals and understand
the "Fundamental Theorem of Calculus" as a tool to evaluate definite
integrals and relate integration and differentiation; and
J) apply
the above knowledge and skills to applications from natural, physical and
social sciences.
2) Statistics
Effective middle grade mathematics
teachers:
A) construct,
identify and interpret frequency distributions, histograms, cumulative
frequency tables, ogives and box plots;
B) identify,
calculate and interpret measures of central tendency and dispersion;
C) identify,
calculate and apply the methods of counting;
D) identify,
calculate and interpret probabilities and expected value;
E) define
random variables and analyze and interpret the probability distributions they
generate;
F) identify
and describe the sampling distribution of sample means and sample proportions;
G) create
and interpret confidence intervals for single population means and proportions;
H) identify,
analyze and perform formal tests of hypotheses concerning single population
means and single population proportions; and
I) identify,
calculate and interpret the correlation coefficient and regression equations.
b) The Mathematics
Curriculum
Effective middle grade mathematics
teachers:
1) understand
the Illinois Learning Standards for Mathematics (see 23 Ill. Adm. Code
1.Appendix D), their organization, progressions and the interconnections among
the domains; and
2) know
the developmental sequence of mathematics skills, along with age-level or
grade-level benchmarks of development.
c) Foundational Knowledge
1) Standards for
Mathematical Practice
Effective middle grade mathematics
teachers enable students to acquire the skills necessary for strong
mathematical practice in that they are able to:
A) make sense of problems
and persevere in solving them;
B) reason abstractly and
quantitatively;
C) construct viable
arguments and critique the reasoning of others;
D) model with mathematics;
E) use appropriate tools
strategically;
F) attend to precision;
G) look for and make use of
structure; and
H) look for and express
regularity in repeated reasoning.
2) Ratio and Proportional
Relationships
Effective middle grade mathematics
teachers are prepared to develop student proficiency and address common
misconceptions related to ratio and proportional relationships and:
A) understand and apply fractions as numbers that can be modeled
from a length perspective (number line), an area perspective (pattern blocks,
geoboards, etc.), and a discrete perspective (set of dots or circles);
B) understand and apply the concept of unit fractions, benchmark
fractions and the whole (referent unit) as defined in the Illinois Learning
Standards for Mathematics;
C) extend the associated meanings of the properties of operations
from whole numbers to fractions;
D) understand and use equivalent fractions, including those of
whole numbers, to reveal new information and as a tool for comparison or to
perform operational procedures;
E) understand and apply the connection between fractions and
division, and how fractions, ratios and rates are connected via unit rates, and
solve problems and formulate equations for proportional relationships;
F) describe the relationship between fractions and terminating,
periodic and delayed-periodic decimals;
G) reason about how quantities vary together in a proportional
relationship, using tables, double number lines and tape diagrams as supports;
H) distinguish proportional relationships from other
relationships, such as additive relationships and inversely proportional
relationships; and
I) understand the connection between a proportional relationship
and a linear relationship.
3) The Number System
Effective middle grade mathematics
teachers are prepared to develop student proficiency and address common
misconceptions related to the number system and:
A) understand how the place value system relies on repeated
groupings of any fixed natural number quantity (including ten) and can show how
to use objects, drawings, layered place value cards and numerical expressions
to help reveal place value structure, and extend place value system knowledge
to negative, rational, irrational and real numbers;
B) efficiently use place value computation methods for addition,
subtraction, multiplication and division with an understanding of composing and
decomposing numbers using the commutative, associative and distributive
properties, and, using multiple models, explain why rules for multiplying and
dividing with negative numbers make sense;
C) derive various (multiple) algorithms and recognize these as
summaries of reasoning, rather than rules, and distinguish between and
understand the appropriate use of computation strategies and computation
algorithms as defined in the Illinois Learning Standards for Mathematics,
recognizing the importance of "mental math";
D) understand and explain methods of calculating products and
quotients of fraction, by using area models, tape diagrams and double number
lines, and by reading relationships of quantities from equations;
E) understand the concepts of greatest common factor, least common
multiple, units, scale, origin, quantities, integer exponents, rational
exponents, irrational numbers, complex numbers and radicals; and
F) understand the connections between fractions and decimals,
particularly with regard to decimal computations.
4) Expressions and
Equations
Effective middle grade mathematics
teachers are prepared to develop student proficiency and address common
misconceptions related to expressions and equations and:
A) understand operations and their associated inverses, and use
properties of operations to rewrite polynomial expressions to reveal new
information and to solve equations;
B) illustrate the meaning of 0 and why division by 0 leads to an
undefined answer;
C) explain each step in solving an equation as following from the
equality asserted at the previous step, while using the equal sign
appropriately;
D) create and solve, using multiple representations, one-variable and
two-variable equations and inequalities with letters representing an unknown
quantity, defining constraints as necessary, and understand and illustrate what
it means to be a solution of one‑variable and two-variable equations and
inequalities;
E) use the structure of an expression to identify ways to rewrite
it, and choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression;
F) strategically use algebraic tools, such as tape diagrams,
number lines, double number lines, graphing calculators and computer algebra
systems, to solve problems and connect the strategy for the solution to
standard algebraic techniques;
G) validate or dismiss the chains of reasoning used to solve
equations and systems of equations;
H) understand proportional relationships and arithmetic sequences
as special cases of linear relationships;
I) derive and justify multiple forms for the equations of
non-vertical lines; and
J) understand and apply properties of integer exponents and
radicals to generate equivalent numerical expressions and solve problems.
5) Geometry
Effective middle grade mathematics
teachers are prepared to develop student proficiency and address common
misconceptions related to geometry and:
A) compose and decompose shapes, classify shapes into categories
and justify the relationships within and between the categories, and summarize
and illustrate the progression from visual to descriptive to analytic to
abstract characterizations of shapes;
B) use multiple models to informally explain and prove geometric
theorems about angles, angle relationships, parallel and perpendicular lines,
circles, parallelograms and triangles, including the Pythagorean theorem and
its converse;
C) describe the connections (relationships) between geometric
properties and arithmetic and algebraic properties, including proportional
relationships, and adapt a problem in one domain to be solved in the other
domain;
D) use the coordinate plane to reason about spatial locations,
graph shapes and solve problems;
E) derive area formulas, such as the formulas for areas of
triangles and parallelograms, considering the different height and base cases,
including oblique cases;
F) demonstrate an understanding of dilations, translations,
rotations and reflections, and combinations of these using dynamic geometry
software and constructions;
G) understand congruence in terms of translations, rotations and
reflections; understand similarity in terms of translations, rotations, reflections
and dilations; solve problems involving congruence and similarity in multiple
ways; and explain the criteria for triangle congruence
and apply the congruence properties to prove geometric theorems and
properties; and
H) understand area and volume, and give rationales for area and
volume formulas that can be obtained by compositions and decompositions of unit
squares or unit cubes, and solve real-world problems involving area, volume and
surface area of any two-dimensional or three-dimensional shape.
6) Statistics and
Probability
Effective middle grade mathematics
teachers are prepared to develop student proficiency and address common
misconceptions related to statistics and probability and:
A) use data displays to ask and answer questions about data in
real-life situations and demonstrate an understanding of measures used to
summarize data, including but not limited to, shape, center, mean, median,
interquartile range, mean absolute deviation, spread and standard deviation;
B) examine the distinction between categorical and numerical data,
reason about data displays and recognize the connection to statistical
variability and distributions;
C) develop an understanding of statistical variability and its
sources, and the role of randomness in statistical inference;
D) explore and explain relationships between two variables by
studying patterns in bivariate data and two-way frequency tables;
E) use technology, including calculators, spreadsheets and tables,
to create scatter plots, linear models, dot plots, histograms and box plots, as
well as calculate correlation coefficients of data; and
F) calculate theoretical and experimental probabilities of simple
and compound events, and understand why their values may differ for a given
event in a particular experimental situation.
7) Functions
Effective middle grade mathematics
teachers are prepared to develop student proficiency and address common
misconceptions related to functions and:
A) define and use appropriately the concepts of function, input,
output, domain, range, rate of change, intercept, interval, end behavior,
function notation, relative maximum and minimum, symmetry, zeros, graphical
transformation, recursive formula, explicit formula, arithmetic and geometric
sequence.
B) examine and reason about functional relationships represented
using tables, graphs, equations and descriptions of functions in words, and
translate between representations of graphs, tables, real-life situations or
equations; and
C) examine the patterns of change in proportional, linear,
inversely proportional, quadratic and exponential functions, and the types of
real-world relationships these functions can model, and write expressions,
equations and/or functions based on these patterns.
d) Using High-Leverage
Instructional Practices
Effective middle grade mathematics
teachers:
1) choose
and use mathematical tasks that entail complex mathematical work, build basic
skills and allow for multiple answers or methods;
2) teach
and use the content-specific language of mathematics;
3) lead
whole-class math discussions (e.g., math talks) that engage all learners;
4) respond
productively to student "errors" by probing the underlying thinking
and providing targeted feedback;
5) appraise,
choose and modify tasks and texts for a specific learning goal;
6) use
specific mathematically focused positive reinforcement;
7) use
public recording (e.g., posters, whiteboard) to collect and probe mathematical
thinking (e.g., demonstrating multiple answers and methods; exploring when an
algorithm may be the best solution and when another approach may provide a more
efficient solution);
8) diagnose
common (and not so common) patterns of student thinking; and
9) assess
students' mathematical proficiency and teach responsively.
e) Using Materials, Tools
and Technology
Effective middle
grade mathematics teachers:
1) apply
mathematical content and pedagogical knowledge to select and use instructional
tools, such as manipulatives and physical models, drawings, virtual
environments, spreadsheets, presentation tools, websites and
mathematics-specific technologies (e.g., graphing tools, interactive geometry
software), recognizing both the insight to be gained and any limitations;
2) empower
students to make sound decisions about the appropriate use of mathematical
tools;
3) when
making mathematical models, recognize that technology can enable one to
visualize the results of varying assumptions, explore consequences, examine
characteristics and compare predictions with data;
4) select
mathematical examples that address the interests, backgrounds and learning
needs of each student; and
5) evaluate
curricular materials for appropriate level and depth of content, focus on and
relevance to required learning goals and incorporation of the standards set
forth in subsection (c)(1) of this Section.
f) Monitoring Student
Learning through Assessment
Effective middle
grade mathematics teachers:
1) engage
in purposeful classroom assessment aligned to appropriate learning expectations
for every student and monitor student progress in meeting developmental
benchmarks in mathematics;
2) provide
a variety of well-designed one-step, two-step, and complex multi-step
assessment items and performance tasks that incorporate real-life situations,
to allow students to demonstrate their learning;
3) ensure
that assessments are responsive to, and respectful of, cultural and linguistic
diversity and exceptionalities, and are not influenced by factors unrelated to
the intended purposes of the assessment;
4) guide
students in developing the skills and strategies for them to assess their work
and set appropriate goals for their progress as mathematicians;
5) analyze
student work to determine misunderstandings, misconceptions, predispositions
and newly developing understandings, and use the results of this analysis to
guide instruction and provide meaningful feedback; and
6) communicate
the purposes, uses and results of assessments appropriately and accurately to
students, parents and colleagues.
g) Meeting the Needs of
Diverse Learners
Effective middle
grade mathematics teachers:
1) understand
the impact of cultural, linguistic, cognitive, academic, physical, social and
emotional differences on mathematics development and progression of knowledge;
2) plan
and implement mathematics instruction that capitalizes on strengths and is
responsive to the needs of each student;
3) use a
variety of approaches and classroom-based intervention strategies to respond to
the needs of each student, particularly those who are struggling or advanced;
4) seek
appropriate assistance and support for struggling and/or advanced learners;
5) collaborate
and plan with other professionals to deliver a consistent, sequenced and
supportive instructional program for each student;
6) differentiate
strategies, materials, pace and levels of cognitive complexity to introduce
concepts and skills to meet the learning needs of each student; and
7) make
content accessible in appropriate ways to English language learners and students
with exceptionalities.
h) Constructing a
Supportive Mathematics Environment
Effective middle
grade mathematics teachers:
1) create
an environment that empowers every student to engage in the practices set forth
in subsection (d) of this Part;
2) motivate
and engage students by designing learning experiences that build
self-direction, perseverance and ownership of mathematics;
3) guide
students to work productively and collaboratively with each other to achieve
mathematics learning goals by using a strategic combination of individual,
group and whole class instruction to meet the learning needs of each student
efficiently and effectively;
4) provide tools that are
accessible and developmentally appropriate;
5) establish
norms and routines for classroom discourse that allow for the respectful
analysis of mistakes and the use of mathematical reasoning for mindful critique
and argument; and
6) create
opportunities and expectations that all students, including English language
learners and students with exceptionalities, use appropriate written and oral
mathematical language.
i) Professionalism,
Communication and Collaboration
Effective middle grade mathematics
teachers:
1) continually
engage in intensive, ongoing professional growth opportunities that serve to
increase mathematical knowledge for teaching, such as lesson study or
continuing coursework;
2) use
self-reflection to analyze instruction and make improvements and make use of
strategies such as journal writing, video self-analysis and peer observation;
3) communicate
and collaborate with other professionals, such as within a professional
learning community, to plan teaching, discuss student needs, secure special
services for students and manage school policies;
4) communicate
and collaborate with families to support student needs and discuss student
progress; and
5) maintain
professional connections to improve mathematics instruction at local, State,
regional and national levels.