From Wikipedia, the free encyclopedia
Principles and Standards for School Mathematics
are guidelines produced by the National
Council of Teachers of Mathematics (NCTM) in 2000, setting
forth recommendations for mathematics educators.^{[1]} They
form a national vision for preschool through twelfth grade mathematics education in the US and Canada. It is the primary model
for standards-based mathematics.
The NCTM employed a consensus process that involved classroom
teachers, mathematicians, and educational
researchers. The resulting document sets forth a set of six
principles (Equity, Curriculum, Teaching, Learning, Assessment, and
Technology) that describe NCTM's recommended framework for
mathematics programs, and ten general strands or standards that cut
across the school mathematics curriculum. These strands are divided
into mathematics content (Number and Operations, Algebra, Geometry,
Measurement, and Data Analysis and Probability) and processes
(Problem Solving, Reasoning and Proof, Communication, Connections,
and Representation). Specific expectations for student learning are
described for ranges of grades (preschool to 2, 3 to 5, 6
to 8, and 9 to 12).
Origins
The original Principles and Standards for School Mathematics was
developed by the NCTM. The NCTM's stated intent was to improve
mathematics education. The contents were based on surveys of
existing curriculum materials, curricula and policies from many
countries, educational research publications, and government
agencies such as the U.S. National Science
Foundation.^{
[2]} The original draft was
widely reviewed at the end of 1998 and revised in response to
hundreds of suggestions from teachers.
The PSSM is intended to be "a single resource that can be used
to improve mathematics curricula, teaching, and assessment."^{
[2]} The latest update was
published in 2000. The PSSM is available as a book, and in
hypertext format on the NCTM web site.
The PSSM replaces three prior publications by NCTM:^{
[2]}
- Curriculum and Evaluation Standards for School
Mathematics (1989), which was the first such publication by an
independent professional organization instead of a government
agency and outlined what students should learn and how to measure
their learning.
- Professional Standards for Teaching Mathematics
(1991), which added information about best practices for teaching
mathematics.
- Assessment Standards for School Mathematics (1995),
which focused on the use of accurate assessment methods.
Six
principles
- Equity: The NCTM standards for equity, as outlined in
the PSSM, encourage equal access to mathematics for all students,
"especially students who are poor, not native speakers of English,
disabled, female, or members of minority groups."^{[3]}
The PSSM makes explicit the goal that all students should learn
higher level mathematics, particularly underserved groups such as
minorities and women. This principle encourages provision of extra
help to students who are struggling and advocates high expectations
and excellent teaching for all students.^{[3]}
- Curriculum: In the PSSM's curriculum section, the NCTM
promotes a "coherent" curriculum, in which an orderly and logical
progression increases students' understanding of mathematics and
avoids wasting time with unnecessary repetition.^{[4
]} They acknowledge that the relative importance of
some specific topics changes over time.^{[4
]} For example, a basic understanding of iteration is important to
students who are learning computer programming, and is almost
absent from 19th century textbooks. Similarly, older American math
textbooks included lessons that are no longer considered important,
such as rules for calculating the number of bushels of hay
that could be stored in a bin of stated dimensions, because this
skill was useful to farmers at
that time.^{[5]} The
NCTM proposes that mathematics taught in modern classrooms be the
skills that are most important to the students' lives and
careers.
- Teaching: In the PSSM, the NCTM promotes
sound teaching methods, without prescribing a one-size-fits-all
approach.^{[6
]} The NCTM wants teachers to be able to use
their professional judgment in choosing teaching techniques. They
favor professional development opportunities in both mathematics
(content) and in effective teaching techniques (methods).
- Learning: According to the PSSM, a
combination of "factual knowledge, procedural facility, and
conceptual understanding" is necessary for students to use
mathematics.^{[7
]} While they state that 'Learning the
"basics" is important,'^{[7
]} the NCTM does not consider the most simplistic
forms of memorization by repetition to be sufficient
achievement in mathematics. A good student not only
understands how and when to use facts, procedures, and concepts,
but he or she also wants to figure things out and perseveres in the
face of challenge. The NCTM particularly deprecates attitudes
in schools that suggest only certain students are capable of
mastering math.
Standards
Ten general strands or standards of mathematics content and
processes were defined that cut across the school mathematics
curriculum. Specific expectations for student learning, derived
from the philosophy of outcome-based education, are
described for ranges of grades
(preschool to 2, 3 to 5, 6 to 8, and 9 to
12). These standards were made an integral part of nearly all
outcome-based education and later standards-based education
reform programs that were widely adopted across the United
States by the 2000s.
Content
standards
- Number and Operations: Number and operations is the
fundamental basis of all mathematics, and teaching this critical
area is the first content standard. All students must be taught to
"understand numbers, ways of representing numbers, relationships
among numbers, and number systems; understand meanings of
operations and how they relate to one another; [and] compute
fluently and make reasonable estimates."^{
[8]} The ability to perform
mental calculations and to calculate answers on paper is
"essential."^{
[8]}
- Algebra: The PSSM names four skills related to algebra
that should be taught to all students: to "understand patterns,
relations, and functions; represent and analyze mathematical
situations and structures using algebraic symbols; use mathematical
models to represent and understand quantitative relationships;
[and] analyze change in various contexts."^{
[9]} Very simple algebra skills are
often taught to young children. For example, a student might
convert an addition equation such as 19+15=? into a simpler
equation, 20+14=? for easy calculation. Formally, this is described
in algebraic notation like this: (19+1) + (15−1) = x, but even a
young student might use this technique without calling it algebra.
The PSSM recommends that all students complete pre-algebra
coursework by the end of eighth grade and take an algebra class
during high school.^{
[9]}
- Geometry: The overall goals for learning geometry are
to "analyze characteristics and properties of two- and
three-dimensional geometric shapes and develop mathematical
arguments about geometric relationships; specify locations and
describe spatial relationships using coordinate geometry and other
representational systems; apply transformations and use symmetry to
analyze mathematical situations; [and] use visualization, spatial
reasoning, and geometric modeling to solve problems."^{
[10]} Some geometry skills are
used in many everyday tasks, such as reading a map, describing the
shape of an object, arranging furniture so that it fits in a room,
or determining the amount of fabric or construction materials
needed for a project. Teaching should be appropriate to students'
developmental level: Young students should be able to explain the
difference between a rectangle and a square, while older students should
be able to express more complex reasoning, including simple mathematical
proofs.^{
[10]} (See van Hiele
model.) The PSSM promotes the appropriate use of physical
objects, drawings, and computer software for teaching geometry.^{
[10]}
- Measurement: Measurement skills have many practical
applications, as well as providing opportunities for advancing
mathematical understand and for practicing other mathematical
skills, especially number operations (e.g., addition or
subtraction) and geometry. Students should "understand measurable
attributes of objects and the units, systems, and processes of
measurement; [and] apply appropriate techniques, tools, and
formulas to determine measurements."^{
[11]} Unlike more abstract
skills, the practical importance of measurement is readily apparent
to students and parents.
- Data analysis and probability: The PSSM says that all
students should learn to "formulate questions that can be addressed
with data and collect, organize, and display relevant data to
answer them; select and use appropriate statistical methods to
analyze data; develop and evaluate inferences and predictions that
are based on data; [and] understand and apply basic concepts of
probability."^{
[12]} These skills allow
students to make sense of critical information, such medical
statistics and the results of political surveys. These skills are
increasingly important as statistical data are used selectively by
manufacturers to promote products. While young students learn
simple skills such as ways to represent the number of pets
belonging to their classmates,^{[13]}
or traditional skills such as calculating the arithmetic mean
of several numbers, older students might learn concepts that were
traditionally neglected, such as the difference between the
occasionally dramatic relative risk reduction figures
and the more concrete absolute risk reduction, or why
political pollsters report the margin of error with their survey
results.
Process
standards
- Problem Solving
- Reasoning and Proof
- Communication
- Connections
- Representation
Curriculum Focal Points
In 2006, NCTM issued a document called "Curriculum Focal Points"
that presented the most critical mathematical topics for each grade
in elementary and middle schools. American mathematics instruction
tends to be diffuse and is criticized for including too many topics
each year. In part, this publication is intended to assist teachers
in identifying the most critical content for targeted attention.
More such publications are planned.
NCTM stated that "Focal Points" was a step in the implementation
of the Standards, not a reversal of its position on teaching
students to learn foundational topics with conceptual
understanding.^{
[14]} Contrary to the
expectation of many textbook publishers and educational
progressives, the 2006 Curriculum Focal Points strongly emphasized
the importance of basic arithmetic skills in lower and middle
grades. Because of this, the "Curriculum Focal Points" was
perceived by the media^{[15]}^{[16]} as an
admission that the PSSM had originally recommended, or at least had
been interpreted as recommending, reduced instruction in basic
arithmetic facts.
The 2006 Curriculum Focal Points identifies three critical areas
at each grade level for pre-kindergarten through Grade 8.^{
[14]} Samples of the specific
focal points for three grades are below. (Note that the Simple
Examples below are not quotes from the Focal Points, but are based
on the descriptions of activities found in the Focal Points.)
Focal
points |
Related
content standard |
Simple
Example |
Pre-Kindergarten Focal Points^{[17]}
(student age: 4 or 5 years old) |
Developing an understanding of whole numbers |
Number and Operations |
How many blue pencils are on the table? |
Identifying shapes and describing spatial relationships |
Geometry |
Can you find something that is round? |
Identifying measurable attributes and comparing objects by
using these attributes |
Measurement |
Which one is longer? |
Fourth Grade Focal Points^{[18]}
(student age: 9 or 10 years old) |
Developing quick recall of multiplication facts and related
division facts and fluency with whole number multiplication |
Number and Operations, Algebra |
An auditorium has 26 rows of 89 seats. How many seats are
there? |
Developing an understanding of decimals, including the
connections between fractions and decimals |
Number and Operations |
Draw a picture of 0.2. What fraction is this? |
Developing an understanding of area and determining the areas
of two-dimensional shapes |
Measurement |
How could we find the area of this L-shaped room? |
Eighth Grade Focal Points^{[19]}
(student age: 13 or 14 years old) |
Analyzing and representing linear functions and solving linear
equations and systems of linear
equations |
Algebra |
The equation y = 4x + 4 shows the cost y of washing x windows.
How much more will it cost each time I add 2 more windows to the
job? |
Analyzing two- and three-dimensional space and figures by using
distance and angle |
Geometry, Measurement |
Use the Pythagorean theorem to find the
distance between the two points on the opposite corners of this
rectangle. |
Analyzing and summarizing data sets |
Data Analysis, Number and Operations, Algebra |
What is the median price
in this list? Does the median change if I lower the most expensive
price? |
The Focal Points define not only the recommended curriculum
emphases, but also the ways in which students should learn them, as
in the PSSM. An example of a complete description of one focal
point is the following for fourth grade:
Number and Operations and Algebra:
Developing quick recall of multiplication facts and related
division facts and fluency with whole number
multiplication
Students use understandings of multiplication to develop quick
recall of the basic multiplication facts and related division
facts. They apply their understanding of models for multiplication
(i.e., equal-sized groups, arrays, area models, equal intervals on
the number line), place value, and properties of operations (in
particular, the distributive property) as they develop, discuss,
and use efficient, accurate, and generalizable methods to multiply
multidigit whole numbers. They select appropriate methods and apply
them accurately to estimate products or calculate them mentally,
depending on the context and numbers involved. They develop fluency
with efficient procedures, including the standard algorithm, for
multiplying whole numbers, understand why the procedures work (on
the basis of place value and properties of operations), and use
them to solve problems.
Controversy
Because most education agencies in the United States have
adopted the NCTM recommendations to varying degrees, many textbook
publishers promote their products as being compliant with the
publishers' interpretations of the PSSM.^{[20
]}^{[21
]}^{[22]}^{[23
]} However, the NCTM does not endorse, approve, or
recommend any textbooks or other products and has never agreed that
any textbook accurately represents their goals.^{[24
]}
See also
References
External
links