Section 20.120 Mathematics Standards for Elementary
Teachers
a) Core Content Area
Knowledge
1) College Algebra
Effective elementary teachers:
A) identify,
solve and apply linear and absolute value equations and inequalities;
B) identify
and interpret the domain, inverse (if it exists) and graph polynomial,
rational, exponential and logarithmic equations;
C) identify
the sum, difference, quotient, product of two functions and the resulting
domain;
D) identify
the composition of two functions and the resulting domain;
E) identify
and solve polynomial, rational, exponential and logarithmic equations and
inequalities, and apply these methods in solving word problems;
F) solve
systems of linear equations using the augmented matrix method;
G) recognize and graph
conic sections; and
H) input
and interpret data and use technology to find the appropriate regression.
2) Statistics
Effective elementary 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 as well as 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 elementary 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 elementary 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) Counting and
Cardinality
Elementary teachers are prepared
to develop student proficiency and address common misconceptions related to
counting and cardinality and:
A) Demonstrate
an understanding of the intricacy of learning to count, assisting students to:
i) know
the names of numbers and orally present them in order, starting from the
numeral 1 and from various other numbers; being able to recognize written
numerals and the quantity each represents; and knowing the names of numbers,
starting with eleven, with special attention paid to helping students
understand the differences between numbers ending in "teen" and those
ending in "ty";
ii) count
the number of objects using one-to-one correspondence, regardless of the way in
which the object is arranged, and understand cardinality (connecting number
name to quantity, the last number of the count, and nesting of numbers) to
counting out a given number of objects; and
iii) compare
numbers by matching quantity represented with objects or pictures or written
numerals; and
B) recognize
the role of ten and the difficulties English language learners face because the
base-ten structure is not evident in all of the English words for numbers.
3) Operations and
Algebraic Thinking
Elementary teachers are prepared
to develop student proficiency and address common misconceptions related to
operations and algebraic thinking and:
A) solve addition, subtraction, multiplication and division
problems with unknowns in any position;
B) demonstrate an understanding of addition and subtraction
relationships and multiplication and division relationships, including the use
of properties of operations (i.e., the field axioms);
C) demonstrate an understanding of the equal sign as meaning "the
same amount as" rather than "calculate the answer";
D) demonstrate an understanding of the meaning of 0 and why
division by 0 leads to an undefined answer;
E) understand and apply the meaning and uses of remainders,
factors, multiples, parentheses and prime and composite numbers;
F) recognize the following strategies when using the operations
of addition and subtraction: counting all, counting on and converting to an
easier problem by composing or decomposing ten;
G) recognize extensions of the strategies enumerated in subsection
(c)(3)(F) of this Section in multiplication, division and beginning work in
expressions and equations;
H) strategically use algebraic tools, such as tape diagrams,
number lines, bar models, math racks and double number lines;
I) extend understanding of arithmetic and operations to algebraic
expressions and equations, and solve one-step and two-step equations and
inequalities; and
J) view numerical and algebraic expressions as "calculation
recipes", describing them in words, parsing them into their component
parts, and interpreting the components in terms of a context.
4) Numbers and Operations
in Base Ten
Elementary teachers are prepared
to develop student proficiency and address common misconceptions related to
numbers and operations in base ten and:
A) understand how the place value system relies on repeated
groupings of any fixed natural number quantity (including ten) and can
demonstrate how to use oral counting, objects, drawings, layered place value
cards and numerical expressions to help reveal place value structure;
B) understand how to compare numbers, fractions and decimals using
the symbols for "greater than", "less than" and "equal
to";
C) understand composing and decomposing numbers using the
commutative, associative and distributive properties to efficiently use place
value methods for addition, subtraction, multiplication and division;
D) extend place value system knowledge to decimals and view
decimals as numbers that can be placed on number lines and explain the
rationale for decimal computation methods;
E) understand and distinguish between the appropriate use of
computation strategies and computation algorithms, as defined in the Illinois
Learning Standards for Mathematics, recognizing the importance of "mental
math" and derive various algorithms and recognize these as summaries of
reasoning, rather than rules;
F) extend place value system knowledge to negative, rational and
irrational numbers; and
G) use mathematical drawings, manipulative materials or
mathematical properties to reveal, discuss and explain the rationale behind, as
well as validate or dismiss, any computational algorithm that a student might
present.
5) Number and Operations –
Fractions
Elementary teachers are prepared
to develop student proficiency and address common misconceptions related to
numbers and operations involving fractions 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 demonstrate how fractions, ratios and rates are connected via
unit rates;
F) demonstrate an understanding of decimal notation for
fractions, and compare decimal fractions;
G) represent ratios and equivalent ratios as an application of
equivalent fractions, and solve ratio and rate problems using tables, tape
diagrams, number lines and double number lines;
H) understand the connection between a proportional relationship
and a linear relationship, and recognize the connection between an inversely
proportional relationship and a reciprocal relationship;
I) defend the ordering of a list of fractions using common
denominators, using common numerators, comparing to benchmark fractions or
using reasoning; and
J) understand the connection between fractions and decimals,
particularly with regard to decimal computations.
6) Measurement and Data
Elementary teachers should be
prepared to develop student proficiency and address common misconceptions
related to measurement and data and:
A) understand and apply the general principles of measurement;
that is, measurement requires a choice of measurable attribute, that
measurement is a comparison with a unit and how the size of a unit affects
measurements, and the iteration, additivity and invariance used in determining
measurement;
B) recognize and demonstrate the relationship of different units;
C) connect
the number line to measurement;
D) demonstrate an understanding of 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;
E) use data displays to ask and answer questions about data;
F) understand the measures used to summarize data, including the
mean, median, interquartile range and mean absolute deviation, and use these
measures to compare data sets;
G) examine the distinction between categorical and numerical data
and reason about data displays; and
H) recognize the connection of categorical and measurement data to
statistical variability and distributions.
7) Geometry
Elementary teachers should be prepared
to develop student proficiency and address common misconceptions related to
geometry and:
A) compose and decompose shapes and classify shapes into
categories, and justify the relationships within and between the categories;
B) understand geometric concepts of angle, parallel and
perpendicular, and use them to describe and define shapes;
C) describe and reason about spatial locations (including the
coordinate plane);
D) reason about proportional relationships in scaling shapes up
and down;
E) describe the connections (relationships) between geometric
properties and arithmetic and algebraic properties, and adapt a problem in one
domain to be solved in the other domain;
F) summarize and illustrate the progression from visual to
descriptive to analytic to abstract characterizations of shapes; and
G) use
the coordinate plane to graph shapes and solve problems.
d) Using High-Leverage
Instructional Practices
Effective
elementary 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., number talks) that engage all learners;
4) respond
productively to students' "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 (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 an
easier 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
elementary 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 and 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 Illinois
Learning Standards for Mathematics.
f) Monitoring Student
Learning through Assessment
Effective elementary 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, incorporating 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 to assess their work and set
appropriately ambitious 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 elementary 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 struggling and/or advanced learners;
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 elementary teachers:
1) create
an environment that empowers every student to engage in the practice set forth
in subsection (c)(1) of this Section;
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 use appropriate written and
oral mathematical language, including English language learners and students
with exceptionalities.
i) Professionalism,
Communication and Collaboration
Effective elementary 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) analyze
instruction for the purpose of self-reflection and making 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; and
4) communicate and
collaborate with families to support student needs and discuss student
progress.