دانلود کتاب Handbook of Activities for the Teaching for Mathematics at the Elementary School

This Guide embodies a comprehensive set of suggestions,
instructional techniques and pedagogical considerations for the
use of a heuristic approach to the teaching of elementary
mathematics. It is designed to give teachers in grades K to 6
the detailed knowhows and understandings they need to put into
practice an instructional program that respects the individualism, mental powers and experiences of their students.
As such, it is neither a manual for "new" math nor for
"old" math. For the distinctions between the two, and the
current debate over which is "better", have focussed principally
on a single aspect of the currieulum -- its content -- and have
ignored other fundamental considerations.
In contrast, this Guide is concerned with the totality
of the classroom situation, and seeks to transform the study of
mathematics into an inquiry, self- educating (that is, heuristic)
process, in which students are enabled to make sense of the
material under study on their own terms, and to develop their
own generalizations and skills according to their own criteria.
Translated into classroom terms, this approach to
learning is based on the premise that s~udents learn best when
they are given the opportunity to be respon~le for their own
learning. Rather than being taught at, they are explicitly
enabled to acquire their own understandings of mathematics.
Thus, the educational perspective is shifted away from
the concept of teachers (and materials) as purveyors of knowledge, and of students as passive entities entrusted mostly
with the retention of that knowledge.
Instead, teachers are asked to concern themselves,
primarily and continuously, with the dynamics of learning. Who
the students are, what mental capacities and experiences they
bring with them to school, how they learn, what the learning
of mathematics entails in terms of what they already possess,
and how explicit connections are made between where they are
and where they could be -- all these questions become of paramount significance to teachers engaged in the application of
a heuristic approach to their teaching.
To be used effectively, this approach requires that
teachers re-appraise their attitudes, beliefs and concepts at
three levels:
1. Mathematics: Mathematics must come to be
viewed not as a branch of knowledge dealing
with numbers, formulae and proofs or
even "logical'' thinking -- but as an activity
of the mind aware of its own dynamics. These
include the faculties of perception, recognition, intuition, a ''sense" of truth, classification and categorization, the ability to
substitute virtual actions for actual ones,
and the awareness of relationships through
various transformations and equivalences.
2. The Learning of Mathematics : This, in turn,
can no longer be limited to the memorization
of a given body of knowledge , but must be
replaced by the gradual and dynamic development of the mind's awareness of relationships
and of the possibilities that these relationships permit. As the learners' awareness of
mathematical relationships increases, they
enlarge his power over the algebra of the
situations under study, acquiring more and
more certainty over the range of available
equivalences , transformations, connections
and extensions.
3. The Teaching of Mathematics : The teaching
mode must also undergo changes if it is to
enable learners to achieve their own conquest
of mathematics. Teachers are asked to relinquish their traditional roles as central
figures in the classroom, as setters of learning
content, pace and style, as authorities making
absolute decisions about who learns what and
when, and as final arbiters of what is right
and what is wrong .
Instead, they must become "facilitators" who,
by subordinating teaching to the students'
learning, enable them to develop their mathematical awarenesses as a function of their
own experience, motivation and mental power.
By restoring the responsibility for learning
to the students, teachers not only come to
preserve the spontaneous ways in which
children learn , but also ensure the greatest
possible levels of involvement, effectiveness and enjoyment in the classroom.
The implications of a heuristic approach to teaching
include:
the recognition that indi vidualism is not a
method but a fact about students. Each is
unique and largely unknowable externally by
a nyone else . As a consequence we are compelled to take each into account , and to
allow the f r eedom of his learning mode , his
contributions, his growth - - in a word his
universe of experience . Thus it is truly an
individualized approach
the realization that, in a sense, every
student has to discover everything for himself .
Merely to absorb what others have already
structured does not give him any opportunity
to know anything . . . to be self-educated
the understanding that the learning of every
human being must be through experiences to
which he relates, of which he can make sense,
always progressing as his meanings are sustained and deepened. Thus it is a selfawareness and inquiry approach
the perception that human beings need selfrespect and also respect from others. A
large measure of self-respect results from
an awareness of one's own progress and conquest of difficulties. Respect from others
is a function of one's recognition of joint
membership to a group
To be implemented successfully, this approach requires
the use of specific materials, techniques and classroom organizational patterns.
The materials must:
be multi-valent; that is, they must lend
themselves to the study of large numbers of
mathematical notions, fostering maximum
economy and efficiency of learning
be models for patterns and relationships; that
is they must enable the detailed perceptions on
the part of students of the algebra of given
situations: the more is perceived, the more is
"mathematized"
be able to "talk"; that is they must allow
students to enter into dialogues with the
situations they present; from such on-going
dialogues between the materials and the learners'
perceptions of dynamics, knowledge of mathematics
grows and is refined
be models for equivalences and transformations;
that is, as sets of similarities and differences ,
possibilities and restrictions within given situations become better identified and studied, the
learning that evolves is structured in terms of
the algebraic properties of mathematical operations
Classroom techniques and organizational patterns, in
turn, must foster student autonomy and self-directiveness.
Large group instruction is de-emphasized and replaced with a
variety of techniques and settings which take into account the
students' constantly changing needs. These include:

-- small-group, teacher-led learning sessions,
in which new topics are presented and explored,
promising areas of investigation developed,
and preliminary understandings and insights into
the dynamics of given situations reached by the
learners

-- learning centers, open-ended worksheets and peer teaching settings, which enable learners to pursue and refine their investigations, acquire more complete and generalized understandings, and ultimately test their own mastery over given topics to their satisfaction

-- progress assessment through criterion-referenced evaluation of the level of mathematical awareness reached by learners, not merely the ability to provide the right answer to a problem

-- effective planning and management techniques, through which teachers are helped to organize the variety of simultaneous activities that respect the multiple demands of their students

In sum, the various facets of a heuristic approach, when used comprehensively, restore to learners the full measure
of their humanity . Mathematics comes to be seen by them for what it is: an example of human development accessible to
all precisely because it is man-made, that is, the product of minds in contact with their own functionings.

And if such a learner-centered approach is, initially at least, more demanding of teachers, it is also far more
rewarding. Students are literally seen to blossom in the classroom. Because they are treated as intellectual equals, they respond with trust, and their view of the world (it is often forgotten that the classroom is a large part of their world) becomes positive, open, hopeful.

The classroom ceases to be a polarized, alienating environment. Instead an atmosphere of anticipation and purposeful cooperativeness inspires teachers and learners alike. And challenges come to be expected -- and even relished -- by both, in the knowledge that they are surmountable.

The sections that follow, it is hoped, will afford teachers opportunities for that most exciting of educational functions: entry into students' worlds on their own terms and, from it, a new insight into the remarkably efficient learning capabilities of young minds.