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AMATYC
STANDARDS for the Teaching and Learning of Mathematics by Joseph Stearns 6/3/03 I.
Philosophical basis: A. All students should grow in their knowledge
of mathematics while in college. B. The mathematics that students study should
be meaningful and relevant. C. Mathematics must be taught as a laboratory
discipline. D. The use of technology is an essential part
of an up-to-date curriculum. E. Students will acquire mathematics through a
carefully balanced educational program that emphasizes the content and
instructional strategies recommended in the standards along with the
viable components of traditional instruction. F. Introductory college mathematics should
significantly increase students' options in educational and career
choices. G. Increased participation by all students in
mathematics and in careers using mathematics is a critical goal in our
heterogeneous society. II.
Framework for Mathematics Standards
A. Standards for Intellectual Development 1. Problem Solving: Students will engage in
substantial problem
solving. 2. Modeling: Students will learn mathematics
through modeling real-world situations. 3. Reasoning: Students will expand their
mathematical reasoning skills as they develop convincing mathematical
arguments. 4. Connecting with Other Disciplines: Students will
develop the view that mathematics is a growing discipline, interrelated
with human culture, and understand its connections to other
disciplines. 5. Communicating: Students will acquire the
ability to read, write, listen to, and speak mathematics. 6. Using Technology: Students will use appropriate
technology to enhance their mathematical thinking and understanding and to
solve mathematical problems and judge the reasonableness of their
results. 7. Developing Mathematical Power: Students will
engage in rich experiences that encourage independent, nontrivial
exploration in mathematics, develop and reinforce tenacity and confidence
in their abilities to use mathematics and inspire them to pursue the study
of mathematics and related disciplines. B. Standards for Content 1.
Students will perform arithmetic
operations, as well as reason and draw conclusions from numerical
information. 2. Students will translate problem situations
into their symbolic representations and use those representations to solve
problems. 3. Students will develop a spatial and
measurement sense. 4. Students will demonstrate understanding of
the concept of function by several means (verbally, numerically,
graphically, and symbolically) and incorporate it as a central theme into
their use of mathematics. 5. Students will use discrete mathematical
algorithms and develop combinatorial abilities in order to solve problems
of finite character and enumerate sets without direct
counting. 6. Students will analyze data and use
probability and statistical models to make inferences about real-world
situations. 7. Students will appreciate the deductive
nature of mathematics as an identifying characteristic of the discipline,
recognize the roles of definitions, axioms, and theorems, and identify and
construct valid deductive arguments. C. Standards for Pedagogy 1.
Mathematics faculty will model the use
of appropriate technology in the teaching of mathematics so that students
can benefit from the opportunities it presents as a medium of
instruction. 2. Mathematics faculty will foster interactive
learning through student writing, reading, speaking and collaborative
activities so that students can learn to work effectively in groups and
communicate about mathematics both orally and in writing. 3. Mathematics faculty will actively involve
students in meaningful mathematics problems that build upon their
experiences, focus on broad mathematical themes, and build connections
within branches of mathematics and between mathematics and other
disciplines so that students will view mathematics as a connected whole
relevant to their lives. 4. Mathematics faculty will model the use of
multiple approaches-numerical, graphical, symbolic, and verbal-to help
students learn a variety of techniques for solving problems. 5. Mathematics faculty will provide learning
activities, including projects and apprenticeships that promote
independent thinking and require sustained effort and time so that
students will have the confidence to access and use needed mathematics and
other technical information independently, to form conjectures from an
array of specific examples, and to draw conclusions from general
principles. III. Goals for types of
courses
A. The Foundation: 1.
Goals a Help students develop mathematical intuition along
with a relevant base of mathematical knowledge. b Integrate numeric, symbolic, functional, and
spatial concepts. c. Provide students with experiences that
connect classroom learning and real-world applications. d. Efficiently, but thoroughly, prepare students for
additional college experiences in mathematics. e. Prepare students to work in groups and
independently. f. Enable students to construct their knowledge of
mathematics through meaningful applications and explorations as well as
techniques of reasoning, regardless of their level of
preparation. g. Provide multiple entry points to meet the needs of
students who enter college mathematics at different levels of mathematical
sophistication. h. Challenge students, but at the same time foster
positive attitudes and build confidence in their abilities to learn and
use mathematics. 2. Content: a. Number Sense: Number sense involves the
intuitive understanding of the properties of numbers and the ability to
solve realistic arithmetic problems using appropriate mathematical tools.
b. Symbolism and Algebra: The study of algebra in the
Foundation must focus on modeling real phenomena via mathematical
relationships. Within this context the student should develop an
understanding of the abstract versions of basic number properties and
learn how to apply these properties. Students should develop
reasonable facility in simplifying the most common and useful types of
algebraic expressions, recognizing equivalent expressions and equations,
and understanding and applying principles for solving simple
equations. c. Geometry and Measurement: Geometry will include
the study of basic properties of angles, polygons, and circles and the
concepts of perimeter, area, and volume for basic plane and solid
figures. Geometry may also be used as a vehicle to acquaint students
with the study of logic and to provide an awareness of valid and invalid
forms of argument. Right triangle trigonometry should be included
with the study of geometry in the foundation. The topic provides a
context to connect arithmetic operations, algebraic formulas, and
geometric properties. d. Functions: During the students study of
functions they will be able to calculate values for, plot and interpret a
variety of basic functions. The student should be able to create and
identify a variety of functions that are based on patterns in collected
data. Students will develop their analytical skills by examining a
function for periodicity, minimum and maximum values, positive and
negative slope, domain and range, and the average rate of change.
Students should be able to see the relationship between the parameters of
a function and the behavior of the function. The use of functions to
model real world relationships is highly recommended. e. Discrete Mathematics: The use of discrete
mathematics to present traditional topics from a different viewpoint can
spark interest in the variety of mathematical concepts and tools available
to those versed in mathematics. Topics such as tree diagrams, Venn
diagrams, permutations, combinations, recursion and difference equations,
matrices are all available to the instructor. f. Probability and Statistics: The student in a
foundation course should be able to collect, organize, and display data
that enables reasonable conclusions to be drawn. An awareness of the
measures of central tendency and dispersion along with the basics of
probability and simulations will enable the student to solve problems
involving random events. g. Deductive proof: Formal deductive proof does not
belong in the curriculum of the foundations. However, informal
argument can give the student an awareness of correct and fallacious
mathematical arguments. 3. The Pedagogy: a. Mathematics faculty who teach the foundation
material should make appropriate use of technology as a routine part of
instruction. However, paper and pencil algorithms should be applied
to basic computations.
b. Use cooperative learning strategies such as group work and
teaming. c. Foundation faculty need to show compassion to
their students to help them work through the frustrations but enough tough
love to encourage them to become independent thinkers, and that sustained
effort is requited to master the material. d. Faculty should use manipulative and other concrete
models of mathematics phenomena to help students make the transition from
concrete to abstract thinking. e. The foundation need not be tied to traditional
course structures. Remedial courses maybe replaced by courses that
introduce students to new areas of mathematics such as probability and
statistics, linear programming, game theory, and symbolic logic.
Skills can be introduced and practiced as needed. Another model
deals with students who have previously mastered the material but need to
review previously learned skills and techniques. Place the students
in a pre-calculus, technical, or statistics course, then as needed special
instruction and assignments as needed. The strategies needed
for the foundation must accommodate students with
disabilities. B. Technical Programs
1. The Content: a. Mathematics courses for technical students should
include realistic problem solving, extended projects, collaborative work,
and portfolios. b. The most important mathematics must be decided
upon with professionals in other disciplines and representatives from
business and industry. c. Central to the mathematics education
of the technical student is the development of the ability to design and
use algorithmic procedures for solving problems. d. Mathematics courses should not only be
designed to meet the immediate needs of the technical student.
Rather, they should also be broad-based and rich in content in order to
meet the students employment and personal needs now and in the
future. e. Mathematical content should be introduced in
the context of real problem solving situations. 2. Pedagogy: For mathematics
courses for technical students, the instructional strategies should
include: a. Interactive learning through reading,
writing, and collaborative activities. b. Projects and apprenticeship opportunities
that encourage independent thinking and require sustained
effort. c. The use of multiple approaches (numerical,
graphical, symbolic, and verbal) to solve meaningful problems. d. The use of interactive and multimedia
technology. C. Mathemathics-Intensive Programs
1. The content: The topics outlined below
are basic to the modeling and problem-solving standards that should form
the heart of the precalculus program. a. Mathematics-intensive programs should
include the study of linear, power, polynomial, rational, algebraic,
exponential, logarithmic, trigonometric, and inverse trigonometric
functions. b. Although the rectangular form for functions should
be emphasized, parametric and polar representations should also be
studied. c. In discrete mathematics the
recognition and use of patterns, including those dealing with aspects of
the very large and the very small, are essential to problem solving in
mathematics. d. Matrices are very powerful mathematical tools that
are often overlooked in introductory college mathematics. e. In statistics, data analysis is especially
important for students in mathematics-intensive programs.
2. The pedagogy a. The standards advocate building connections with
other fields and approaching problem solving with a variety of
strategies.
b. The pedagogy standards also place emphasis on cooperative
learning and the use of technology to encourage student
investigation, discovery, and insight. D. Liberal Arts Programs: The goals of
the course are to develop, as fully as possible, the mathematical and
quantitative capabilities of the students; to enable them to understand a
variety of applications of mathematics; to prepare them to think logically
in subsequent courses and situations in which mathematics occurs; and to
increase their confidence in their ability to reason
mathematically.
1. The content: a. The course is a part of a broad quantitative
literacy program aimed at developing capabilities in thought, analysis,
and perspective that is beyond what in normally studied in high
school. b. The curriculum should enable the student to see
mathematics in a few specific contexts studied in depth. 2. The pedagogy: The role of the teacher
must change from a sage who hands down knowledge to a coach who provides
guidance and support. E. Prospective Teachers Programs: 1. The content: The mathematics studied
by preservice teachers must help them develop an understanding of the
subject that goes beyond what they will be expected to teach.
Prospective teachers should learn to;
a. View mathematics as a system of interrelated principles;
b. Communicate mathematics accurately, both orally and in
writing;
c. Understand the elements of mathematical modeling; d. Use calculators and computers appropriately in the
teaching and learning of mathematics; and
e. Appreciate the historical and cultural development of
mathematics, 2 The pedagogy:
a. Follow new professional recommendations on teaching
strategies; b. Gain a better understanding of the mathematical
needs of future elementary, middle, and high school teachers; c. Rethink their teaching to promote in-depth
understanding as well as a broad vision of the “big ideas” of k-12
mathematics, and d. Keep abreast of the research on how students learn
mathematics and adjust their teaching accordingly. A CHECKLIST FOR AMATYC STANDARDS
Will students be engaged in substantial mathematical
problem solving? Will students be learning mathematics through
modeling real-world situations? Will students expand their mathematical reasoning
skills as they develop convincing mathematical arguments? Will students acquire the ability to read, write,
listen to, and speak mathematics? Will students use appropriate technology to enhance
their mathematical thinking and understanding to solve mathematical
problems and judge the reasonableness of their result? Have I modeled the appropriate use of technology in
the teaching of mathematics? Have I given my students the opportunity to work in
groups and develop collaborative learning techniques? Have I explained the concept verbally, with a graph,
with a table of values, and analytically? Have I involved my students in meaningful mathematics
problems that build upon their experiences, focus on broad mathematical
themes and build connections between mathematics and other disciplines so
that students will view mathematics as a connected whole relevant to their
lives? Have I provided learning activities that promote
independent thinking and that require sustained effort and time so that
students will have the confidence to access and use needed mathematics and
other technical information independently, to form conjectures from an
array of specific examples, and to draw conclusions from general
principles? |