Every teacher of mathematics, whether at the
elementary, middle, or high school level, has an individual goal to provide
students with the knowledge and understanding of the mathematics necessary to
function in a world very dependent upon the application of mathematics.
Instructionally, this goal translates into three components:
- conceptual understanding
- procedural fluency
- problem solving
Conceptual understanding consists of those relationships constructed
internally and connected to already existing ideas. It involves the
understanding of mathematical ideas and procedures and includes the knowledge
of basic arithmetic facts. Students use conceptual understanding of mathematics
when they identify and apply principles, know and apply facts and definitions,
and compare and contrast related concepts. Knowledge learned with understanding
provides a foundation for remembering or reconstructing mathematical facts and
methods, for solving new and unfamiliar problems, and for generating new
knowledge.
Procedural fluency is the skill in carrying out procedures
flexibly, accurately, efficiently, and appropriately. It includes, but is not
limited to, algorithms (the step-by-step routines needed to perform arithmetic
operations). Although the word procedural may imply an arithmetic procedure
to some, it also refers to being fluent with procedures from other branches of mathematics,
such as measuring the size of an angle using a protractor. The use of
calculators need not threaten the development of students’ computational
skills. On the contrary, calculators can enhance both understanding and
computing if used properly and effectively. Accuracy and efficiency with
procedures are important, but they should be developed through understanding. When
students learn procedures through understanding, they are more likely to
remember the procedures and less likely to make common computational errors.
Problem solving is the ability to formulate, represent, and
solve mathematical problems. Problems generally fall into three types:
- one-step problems
- multi-step problems
- process problems
Most problems that students will encounter in
the real world are multi-step or process problems. Solution of these problems
involves the integration of conceptual understanding and procedural knowledge.
Students need to have a broad range of strategies upon which to draw. Selection
of a strategy for finding the solution to a problem is often the most difficult
part of the solution. Therefore, mathematics instruction must include the
teaching of many strategies to empower all students to become successful
problem solvers. A concept or procedure in itself is not useful in problem
solving unless one recognizes when and where to use it as well as when and
where it does not apply. Many textbook problems are not typical of those that
students will meet in real life. Therefore, students need to be able to have a
general understanding of how to analyze a problem and how to choose the most
useful strategy for solving the problem.
Individually, each of these components (conceptual
understanding, procedural fluency, and problem solving) is necessary but not
sufficient for a student to be mathematically proficient. They are not,
however, independent of each other. They are integrally related, need to be
taught simultaneously, and should be a component of every lesson. The
mathematics standard presented in this document states that students will:
o
understand
the concepts of and become proficient with the skills of mathematics;
o
communicate
and reason mathematically;
o
become
problem solvers by using appropriate tools and strategies;
Through the integrated study of number sense and operations,
algebra, geometry, measurement, and statistics and probability. Mathematics
should be viewed as a whole body of knowledge, not as a set of individual
components. Therefore, local mathematics curriculum, instruction, and assessment
should be designed to support and sustain the components of this standard. New
York State’s yearly 3-8 mathematics assessments, as required by NCLB federal legislation,
will provide data measuring student progress toward obtaining mathematical proficiency.
Since the state assessments will measure conceptual understanding, procedural fluency,
and problem solving, local assessments should measure these components as well.
Thus, many schools may need to provide teachers with significant professional
staff development to assist them in developing local assessments. In this
document conceptual understanding, procedural fluency, and problem solving are represented
as process
strands and content
strands. These
strands help to define what students should know and be able to do as a result
of their engagement in the study of mathematics.
Process
Strands: The
process strands (Problem Solving, Reasoning and Proof, Communication,
Connections, and Representation) highlight ways of acquiring and using content
knowledge. These process strands help to give meaning to mathematics and help students
to see mathematics as a discipline rather than a set of isolated skills.
Student engagement in mathematical content is accomplished through these
process strands. Students will gain a better understanding of mathematics and
have longer retention of mathematical knowledge as they solve problems, reason
mathematically, prove mathematical relationships, participate in mathematical
discourse, make mathematical connections, and model and represent mathematical
ideas in a variety of ways.
Content
Strands: The
content strands (Number Sense and Operations, Algebra, Geometry, Measurement,
and Statistics and Probability) explicitly describe the content that students
should learn. Each school’s mathematics curriculum developed from these strands
should include a broad range of content. This broad range of content, taught in
an integrated fashion, allows students to see how various mathematics knowledge
is related, not only within mathematics, but also to other disciplines and the
real world as well. The performance indicators listed under each band within a
strand are intended to assist teachers in determining what the outcomes of
instruction should be. The instruction should engage students in the
construction of this knowledge and should integrate conceptual understanding
and problem solving with these performance indicators. The performance
indicators should not be viewed as a checklist of skills void of understanding
and application. Students will only become successful in mathematics if they
see mathematics as a whole, not as isolated skills and facts. As school
districts develop their own mathematics curriculum based upon the statements in
this standards document, attention must be given to both content and process
strands. Likewise, as teachers develop their instructional plans and their
assessment techniques, they also must give attention to the integration of
process and content. To do otherwise would produce students who have temporary
knowledge and who are unable to apply mathematics in realistic settings.
Curriculum, instruction, and assessment are intricately related and must be designed with this in mind. All
three domains must address conceptual understanding, procedural fluency, and
problem solving. If this is accomplished, school districts will produce
students who will (1) have mathematical knowledge, (2) have an understanding of
mathematical concepts, and (3) be able to apply mathematics in the solution of
problems.
School districts and individual teachers should
be aware that this document is a standards document that guides the development
of local curriculum. Local school districts remain responsible for developing
curriculum aligned to the New York State standards. In this document the
mathematics standard is succinctly stated. The standard outlines what students should
know and be able to do in mathematics. The content strands, consisting of bands
and performance indicators within each band, and the performance indicators of
the process strands help to define how the standard will be met. Each school
district’s mathematics curriculum should be developed to assure that all
students achieve the performance indicators for both the process and content
strands.
Helping all students become proficient in
mathematics is an imperative goal for every school. It is the hope that this
standards document will assist schools and individual teachers in meeting this
goal. For additional information visit the New York State Education Department mathematics
website http://www.emsc.nysed.gov/ciai/mst/math.html
.
