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INTRODUCTION
Model Content Standards for
Mathematics
Responsible and productive members
of today's technological society need to have
a broad, connected, and useful knowledge of
mathematics. The St. Philomena School Model
Content Standards for Mathematics are designed
to serve as a guide that will enable every student
to develop the mathematical literacy needed
for citizenship and employment in the 21st century.
| "Today's students will
live and work in the 21st century, in an
era dominated by computers, by worldwide
communication, and by a global economy.
Jobs that contribute to this economy will
require workers who are prepared to absorb
new ideas, to perceive patterns, and to
solve unconventional problems. Mathematics
is the key to opportunity for these jobs."1
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Mathematics is not simply a collection of facts
and procedures, and doing mathematics is not
simply recalling these facts, nor performing
memorized procedures. Mathematics is a coherent
and useful discipline that has expanded dramatically
in the last 25 years. The mathematics students
study in school must reflect these changes,
and the ways students study mathematics must
capitalize on the growth in our understanding
of how students learn.
| "There has been a mentality
that you have to be ... special to be successful
in mathematics, that you have to be the
best and the brightest. Well, we are demystifying
mathematics. We can no longer say that there
is any segment of society that doesn't need
mathematics."2
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Three questions have guided the development
of the St. Philomena School Model Content Standards
for Mathematics: What is mathematics? What does
it mean to know, use, and do mathematics? What
mathematics should every student learn?
Responses to these questions have
resulted in six goals, adapted from those of
the National Council of Teachers of Mathematics3,
that serve as the framework for the St. Philomena
School Model Content Standards for Mathematics.
The six goals that students should reach are
stated on the following page.
1
L. Steen, (1989), " Teaching
Mathematics for Tomorrow's World", Educational
Leadership, 47: 18-22.
2
Quote by Iris Carl found in A.
Wheelock, (1992), Crossing the Tracks,
(New York: The New Press).
3
National Council of Teachers
of Mathematics, (1989), Curriculum and Evaluation
Standards for School Mathematics, (Reston, VA:
author).
Model Content Standards
Mathematics
1. Students develop number
sense and use numbers and number relationships
in problem-solving situations and communicate
the reasoning used in solving these problems.
2. Students use algebraic methods
to explore, model, and describe patterns and
functions involving numbers, shapes, data, and
graphs in problem-solving situations and communicate
the reasoning used in solving these problems.
3. Students use data collection
and analysis, statistics, and probability in
problem-solving situations and communicate the
reasoning used in solving these problems.
4. Students use geometric concepts,
properties, and relationships in problem-solving
situations and communicate the reasoning used
in solving these problems.
5. Students use a variety
of tools and techniques to measure, apply the
results in problem-solving situations, and communicate
the reasoning used in solving these problems.
6. Students link concepts
and procedures as they develop and use computational
techniques, including estimation, mental arithmetic,
paper-and-pencil, calculators, and computers,
in problem-solving situations and communicate
the reasoning used in solving these problems.
Six Goals for Students of
Mathematics
- Become mathematical problem solvers.
To be problem solvers, students need to
know how to find ways to reach a goal when
no routine path is apparent. To develop the
flexibility, perseverance, and wealth of strategies
that are characteristic of good problem solvers,
students need to be challenged frequently
and regularly with non-routine problems, including
those they pose themselves.
- Learn to communicate mathematically.
The development of students' power to use
mathematics involves learning the signs, symbols,
and terms of mathematics. This is best accomplished
in problem situations where students have
an opportunity to read, write, and discuss
ideas in the language of mathematics. As students
communicate their ideas, they learn to clarify,
refine, and consolidate their thinking.
- Learn to reason mathematically.
Students who reason mathematically gather
data, make conjectures, assemble evidence,
and build an argument to support or refute
these conjectures. Such processes are fundamental
to doing mathematics.
- Make mathematical connections.
The study of mathematics should provide
students with many opportunities to make connections
among mathematical ideas (for example, the
connection between geometric and algebraic
concepts) and among mathematics and other
disciplines (for example, art, music, psychology,
science, business). The curriculum should
portray mathematics as an integrated whole
that permeates activities both in and out
of school. These connections make mathematics
meaningful and useful to each student.
- Become confident of their mathematical
abilities.
As a result of studying mathematics, students
need to view themselves as capable of using
their growing mathematical power to make sense
of new problem situations in the world around
them. School mathematics must endow all students
with a realization that doing mathematics
is a common human activity. Students learn
to trust their own mathematical thinking by
having numerous and varied experiences.
- Learn the value of mathematics.
In addition to providing the tools to solve
problems, mathematics provides a way of thinking
about and understanding the world around us.
Students should have numerous and varied opportunities
to think mathematically about their world.
They should also explore the cultural, historical,
and scientific evolution of mathematics so
that they can appreciate the role of mathematics
in the development of our contemporary society.
The following Model Content Standards for Mathematics
provide a new vision of the content students
should study in order to achieve these goals.
The standards reinforce the need for technical
skills, long a goal of school mathematics, and
also the need to know when to apply them and
why they work. They also broaden considerably
the context in which these technical skills
might be attained. Students who have a working
knowledge of the mathematics in each of these
standards will be better able to reason critically,
vote responsibly, and work productively in today's
complex world.
STANDARD
1:
Students develop number sense and use numbers
and number relationships in problem-solving
situations and communicate the reasoning used
in solving these problems.
In order to meet this standard, a student will
- Construct and interpret number meanings
through real-world experiences and the use
of hands-on materials;
- Represent and use numbers in a variety
of equivalent forms (for example, fractions,
decimals, percents, exponents, scientific
notations);
- Know the structure and properties of the
real number system (for example, primes,
factors, multiples, relationships among sets
of numbers); and
- Use number sense, including estimation
and mental arithmetic, to determine the reasonableness
of solutions.
RATIONALE:
Numbers play a vital role in our daily
lives, from cooking to reading the newspaper
to performing jobs. Because we use numbers to
measure, to count, to order, and to label, it
is important to understand the many uses of
numbers. These include knowing both the symbols
for and the meanings of various kinds of numbers,
including whole numbers, fractions, decimals,
percents, roots, exponents, and scientific notation.
Number sense is "common sense" about numbers.
Students with number sense recognize the relative
magnitudes of numbers and relationships between
numbers; for example, _ is equivalent to .5
and 50%. In addition, they have references for
measures of common objects and situations in
the environment. They know how much a million
is and how much a loaf of bread costs. Developing
number sense strengthens students' ability to
acquire basic facts, to solve problems, and
to determine the reasonableness of results.
GRADES K-4
In grades K-4, what students know and are able
to do includes
- Demonstrating meanings for whole numbers,
and commonly-used fractions and decimals (for
example, _ , _, 0.5, 0.75), and representing
equivalent forms of the same number through
the use of physical models, drawings, calculators,
and computers;
- Reading and writing whole numbers and knowing
place-value concepts and numeration through
their relationships to counting, ordering,
and grouping;
- Using numbers to count, to measure, to
label, and to indicate location;
- Developing, testing, and explaining conjectures
about properties of whole numbers, and commonly-used
fractions and decimals (for example, _,
_, 0.5, 0.75); and
- Using number sense to estimate and justify
the reasonableness of solutions to problems
involving whole numbers, and commonly-used
fractions and decimals (for example, _,
_, 0.5, 0.75).
GRADES 5-8
As students in grades 5-8 extend their knowledge,
what they know and are able to do includes
- Demonstrating meanings for integers, rational
numbers, percents, exponents, absolute value,
square roots, and pi (¼) using physical materials
and technology in problem-solving situations;
- Reading, writing, and ordering integers,
rational numbers, and common irrational numbers
such as ¯2, ¯5, and ¼;
- Applying number theory concepts (for
example, primes, factors, multiples) to
represent numbers in various ways;
- Using the relationships among fractions,
decimals, and percents, including the concepts
of ratio and proportion, in problem-solving
situations;
- Developing, testing, and explaining conjectures
about properties of integers and rational
numbers; and
- Using number sense to estimate and justify
the reasonableness of solutions to problems
involving integers, rational numbers, and
common irrational numbers such as ¯2,
¯5, and ¼.
STANDARD
2:
Students use algebraic methods to explore,
model, and describe patterns and functions involving
numbers, shapes, data, and graphs in problem-solving
situations and communicate the reasoning used
in solving these problems.
In order to meet this standard, a student will
- Identify, describe, analyze, extend, and
create a wide variety of patterns in numbers,
shapes, and data;
- Describe patterns using mathematical language;
- Solve problems and model real-world situations
using patterns and functions;
- Compare and contrast different types of
functions; and
- Describe the connections among representations
of patterns and functions, including words,
tables, graphs, and symbols.
RATIONALE:
The study of patterns, functions, and algebra
helps learners to recognize and generalize patterns;
identify and clarify functional relationships;
and represent and manipulate these relationships
verbally, numerically, symbolically, and graphically.
Symbolic representation, including the many
interpretations of the concept of variable,
is important but only one of many ways to represent
patterns and functions. Students who are adept
at identifying and classifying patterns and
functional relationships are better able to
use these relationships in real situations,
both in school and out. The portrayal of functions
and algebra in this standard is broader, deeper,
more connected, and more useful to learners
than in the traditional high school algebra
curriculum.
Because the understandings developed through
this standard are critical to success in mathematics
and to the appropriate use of quantitative reasonings
in other disciplines, students should explore
and use the ideas of functions, patterns, and
algebra from kindergarten through 12th grade.
GRADES K-4
In grades K-4, what students know and are able
to do includes
- Reproducing, extending, creating, and describing
patterns and sequences using a variety of
materials (for example, beans, toothpicks,
pattern blocks, calculators, unifix cubes,
colored tiles);
- Describing patterns and other relationships
using tables, graphs, and open sentences;
- Recognizing when a pattern exists and using
that information to solve a problem; and
- Observing and explaining how a change in
one quantity can produce a change in another
(for example, the relationship between the
number of bicycles and the numbers of wheels).
GRADES 5-8
As students in grades 5-8 extend their knowledge,
what they know and are able to do includes
- Representing, describing, and analyzing
patterns and relationships using tables, graphs,
verbal rules, and standard algebraic notation;
- Describing patterns using variables, expressions,
equations, and inequalities in problem-solving
situations;
- Analyzing functional relationships to explain
how a change in one quantity results in a
change in another (for example, how the
area of a circle changes as the radius increases,
or how a person's height changes over time);
- Solving simple equations in problem-solving
situations using a variety of methods (informal,
formal, graphical).
STANDARD
3:
Students use data collection and analysis,
statistics, and probability in problem-solving
situations and communicate the reasoning used
in solving these problems.
In order to meet this standard, a student will
- Solve problems by systematically collecting,
organizing, describing, and analyzing data
using surveys, tables, charts, and graphs;
- Make valid inferences, decisions, and arguments
based on data analysis; and
- Use counting techniques, experimental probability,
or theoretical probability, as appropriate,
to represent and solve problems involving
uncertainty.
RATIONALE:
Statistics are used to understand how information
is processed and translated into usable knowledge.
Through the study of statistics, students learn
to collect, organize, and summarize information.
Students also need to know how to interpret
data and make decisions based on their interpretations.
Probability is part of this standard because
statistical data are often used to predict the
likelihood of future events and outcomes. Students
learn probability — the study of chance
— so that numerical data can be used to
predict future events as well as record the
past. A command of statistics and probability
is important in adult life.
GRADES K-4
In grades K-4, what students know and are able
to do includes
- Constructing, reading, and interpreting
displays of data including tables, charts,
pictographs, and bar graphs;
- Interpreting data using the concepts of
largest, smallest, most often, and middle;
- Generating, analyzing, and making predictions
based on data obtained from surveys and chance
devices; and
- Solving problems using various strategies
for making combinations (for example, determining
the number of different outfits that can be
made using two blouses and three skirts).
GRADES 5-8
As students in grades 5-8 extend their knowledge,
what they know and are able to do includes
- Reading and constructing displays of data
using appropriate techniques (for example,
line graphs, circle graphs, scatter plots,
box plots, stem-and- leaf plots) and appropriate
technology;
- Displaying and using measures of central
tendency, such as mean, median, and mode,
and measures of variability, such as range
and quartiles;
- Evaluating arguments that are based on
statistical claims;
- Formulating hypotheses, drawing conclusions,
and making convincing arguments based on data
analysis;
- Determining probabilities through experiments
or simulations;
- Making predictions and comparing results
using both experimental and theoretical probability
drawn from real-world problems; and
- Using counting strategies to determine
all the possible outcomes from an experiment
(for example, the number of ways students
can line up to have their picture taken).
STANDARD
4:
Students use geometric concepts, properties,
and relationships in problem-solving situations
and communicate the reasoning used in solving
these problems.
In order to meet this standard, a student will
- Connect various physical objects with their
geometric representation;
- Connect mathematical concepts from across
the standards with their geometric representations;
- Recognize, draw, describe, and analyze
geometric shapes in one, two, and three dimensions;
- Make, investigate, and test conjectures
about geometric ideas; and
- Solve problems and model real-world situations
using geometric concepts.
RATIONALE:
Long before humans computed, they observed
that the full moon, the iris of an eye, and
circular ripples of water emanating from a cast
stone all have the same shape. Recording and
analyzing shapes and their properties eventually
gave us the branch of mathematics called geometry.
The process continues today as mathematicians
develop powerful models of our world. Students
who understand the concepts and language of
geometry are better prepared to learn number
and measurement ideas as well as other advanced
mathematical topics. Students' spatial capabilities
frequently exceed their numerical skills and
tapping these strengths can foster an interest
in mathematics and improve number understandings
and skills..
GRADES K-4
In grades K-4, what students know and are able
to do includes
- Recognizing shapes and their relationships
(for example, symmetry, congruence) using
a variety of materials (for example, pasta,
boxes, pattern blocks);
- Identifying, describing, drawing, comparing,
classifying, and building physical models
of geometric figures;
- Relating geometric ideas to measurement
and number sense;
- Solving problems using geometric relationships
and spatial reasoning.
GRADES 5-8
As students in grades 5-8 extend their knowledge,
what they know and are able to do includes
- Describing, analyzing, and reasoning informally
about the properties (for example, parallelism,
perpendicularity, congruence) of two-
and three-dimensional figures;
- Applying the concepts of ratio, proportion,
and similarity in problem-solving situations;
- Solving problems using coordinate geometry;
- Solving problems involving perimeter and
area in two dimensions, and involving surface
area and volume in three dimensions.
STANDARD
5:
Students use a variety of tools and techniques
to measure, apply the results in problem-solving
situations, and communicate the reasoning used
in solving these problems.
In order meet this standard, a student will
- Understand and apply the attributes of
length, capacity, weight, mass, time, temperature,
perimeter, area, volume, and angle measurement
in problem-solving situations;
- Make and use direct and indirect measurements
to describe and compare real-world phenomena;
- Understand the structure and use of systems
of measurements;
- Describe and use rates of change (for
example, temperature as it changes throughout
the day, or speed as the rate of change of
distance over time) and other derived
measures; and
- Select appropriate units, including metric
and US customary, and tools (for example,
rulers, protractors, compasses, thermometers)
to measure to the degree of accuracy required
to solve a given problem.
RATIONALE:
Every day, people measure to answer common
questions: How long will it take? How high is
it? How much will it hold? Using agreed-upon
units, such as inches, paper clips, kilograms,
heartbeats, paces, or degrees Celsius, we quantify
the world in which we live. Measurement is one
way to make numbers meaningful to students.
Naturally, measurement is closely allied with
geometry (for example, through angular, linear,
area, and volume measurements), but measurement
involves more than using a ruler and a protractor.
Measuring diverse quantities involves making
connections within mathematics and across the
curriculum.
GRADES K-4
In grades K-4, what students know and are able
to do includes
- Knowing, using, describing and estimating
measures of length, perimeter, capacity, weight,
time, and temperature;
- Demonstrating the process of measuring and
explaining the concepts related to units of
measurement;
- Comparing and ordering objects according
to measurable attributes (for example,
longest to shortest, lightest to heaviest);
- Using the approximate measures of familiar
objects (for example, the width of your
finger, the temperature of a room, the weight
of a gallon of milk) to develop a sense
of measurement; and
- Selecting and using appropriate standard
and non-standard units of measurement in problem-solving
situations.
GRADES 5-8
As students in grades 5-8 extend their knowledge,
what they know and are able to do includes
- Estimating, using and describing measures
of distance, perimeter, area, volume, capacity,
weight, mass, and angle comparison;
- Reading and interpreting various scales
including those based on number lines, graphs,
and maps;
- Developing and using formulas and procedures
to solve problems involving measurement; and
- Selecting and using appropriate units and
tools to measure to the degree of accuracy
required in a particular problem-solving situation.
STANDARD
6:
Students link concepts and procedures as
they develop and use computational techniques,
including estimation, mental arithmetic, paper-and-pencil,
calculators, and computers, in problem-solving
situations and communicate the reasoning used
in solving these problems.
In order to meet this standard, a student will
- Model, explain, and use the four basic
operations — addition, subtraction, multiplication,
and division — in problem-solving situations;
- Develop, use, and analyze algorithms; and
- Select and apply appropriate computational
techniques to solve a variety of problems
and determine whether the results are reasonable.
RATIONALE:
Computation is an indispensable part of
mathematics and our daily lives. We use it to
balance our checkbooks, figure our taxes, and
make business decisions. The basic facts of
addition, subtraction, multiplication, and division
are similarly indispensable. Today's students
must be able to use a variety of computational
tools and techniques including estimation, mental
arithmetic, paper-and-pencil, calculators, and
computers. Estimation and mental arithmetic
serve a practical function in our daily lives,
and help students develop meaning for numbers
and understanding of number relationships. The
use of calculators and computers is not intended
to replace proficiency with basic facts. Appropriate
uses of calculators and computers include solving
real-world problems that may involve tedious
or time-consuming computations or exploring
number patterns to develop understanding of
numbers and number relationships. Proficiency
with basic facts is essential for knowing when
and how to use each of these tools and techniques.
Computational skill is related to "operation
sense". Students with operation sense know when
and how to use addition, subtraction, multiplication,
and division, and are able to apply them to
solve real-world problems. Students build operation
sense by modeling their understanding of number
operations and their properties, by describing
how number operations are related to one another,
and by seeing how the use of a particular operation
changes the value of the numbers involved.
Computational skill and operation sense
go hand in hand with number sense. When children
have a well-developed sense of number and operations,
they can more easily evaluate the reasonableness
of their solutions. The ability to apply computational
skills and operation sense will extend students'
mathematical power by giving them confidence
in their ability to work with numbers and to
solve problems in a variety of situations.
GRADES K-4
In grades K-4, what students know and are able
to do includes
- Demonstrating conceptual meanings for the
four basic arithmetic operations of addition,
subtraction, multiplication, and division;
- Comparing, adding and subtracting commonly-used
fractions and decimals (for example, _,
_, 0.5, 0.75);
- Demonstrating understanding of and proficiency
with basic addition, subtraction, multiplication,
and division facts without the use of a calculator;
- Constructing, using, and explaining procedures
to compute and estimate with whole numbers;
and
- Selecting and using appropriate methods
for computing with whole numbers in problem-solving
situations from among mental arithmetic, estimation,
paper-and-pencil, calculator, and computer
methods.
GRADES 5-8
As students in grades 5-8 extend their knowledge,
what they know and are able to do includes
- Explaining how ratios, proportions, and
percents can be used to solve real-world problems;
- Constructing, using, and explaining procedures
to compute and estimate with whole numbers,
fractions, decimals, and integers;
- Developing, applying, and explaining a
variety of different estimation strategies
in problem-solving situations, and explaining
why an estimate may be acceptable in place
of an exact answer; and
- Selecting and using appropriate methods
for computing with commonly used fractions
and decimals, percents, and integers in problem-solving
situations form among mental arithmetic, estimation,
paper-and-pencil, calculator, and computer
methods, and determining whether the results
are reasonable.
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