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

Description

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. CONTENTS SECTION I: The Uses of Multi-Valent Materials: Models for Mathematical Relationships A. Cuisenaire Rods 1 . Free play and qualitative work 2. Staircases 3. Trains a) many-colored trains: introduction to addition and subtraction b) single-colored trains: introduction to multiplication, division and fractions 4. Towers a) crosses and towers: factors and products b) single-color towers: exponents 5. The study of fractions 6. Other uses of the rods: factoring, HCF and LCM, bases of numeration, sum of integers B. Product Chart and Product Cards C. Cubes and Prisms D. Geoboards 1. Empirical work; classifications 2. Study of areas and perimeters 3. Study of some basic theorems; Pick's theorem; Pythagorean theorem; diagonals of a polygon; altitudes and medians of a triangle 4. Geometry on the circle; the equilateral triangle E. Attribute Blocks/Relationshapes F. Other Manipulative Materials SECTION II: Alternative Teaching Approaches: Learning-Based Instructional Activities A. "A Lot from a Little": specific techniques to increase learning efficiency 1. Number array 2. Complements 3. Math tic-tac-toe 4. Halving and doubling; extensions 5. ~ activities 6. Bases of numeration; modular arithmetic 7. "Quickies": rapid mental calculations 8. Other techniques B. Basic Mathematical Principles 1. Algebra 2. Transformations 3. Patterns 4. Families of equivalence c. Algorithms 1. Alternatives for addition 2. Alternatives for subtraction 3. Alternatives for multiplication 4. Alternatives for division 5. Equivalence A: addition and subtraction of fractions 6 . Equivalence M: mul tiplication and division of fractions 7. Decimal fractions, decimals, percents SECTION III: Independent Student Work: Practice Without Repetition A. Open- Ended Worksheets B. Making Practice More Interesting: palindromic and "quicksand" games; broken typewriter game c. Creative Writing in Mathematics D. Research Problems SECTION IV: Applied Mathematics A. Measurements 1. Units; conversion of units 2 . The Metric system B. Word Problems SECTION V: Classroom Management A. Teacher-Led Activities B. Blackboard Lessons C. Grouping Students D. Use of Activity Cards E. Learning Centers F. Peer Teaching Techniques G. Use of the Textbook H. Effective Use of Instructional Aides I. Productive Homework Assignments SECTION VI: Theoretical Considerations: The Heuristic Approach to the Teaching of Mathematics SECTION VII: Testing and Evaluation SECTION VIII: Bibliography and Cross-Referenced Index