AIMS Plans Pattern-Based Math/Science Curriculum

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The AIMS Education Foundation has launched a major program to develop a pattern-based math/science curriculum. Extensive analysis provides substantial evidence that patterns are the most powerful integrative force for making connections both within mathematics as well as between mathematics, science, and the real world. Because patterns are the essence of both mathematics and science, AIMS has determined that the time has come for a concerted effort to explore the nature and potential of a pattern-based curriculum.

The basis for this effort has been forming for the past several years. Numerous articles and activities involving the study of patterns have appeared in the AIMS magazine. Dr. Richard Thiessen has written extensively about the nature and role of patterns in mathematics. Wilbert and Luetta Reimer have authored two series of books published by AIMS dealing with a broad range of patterns. A series of articles, Mathematics as the Study of Patterns, has appeared in AIMS since March, 1994. All of these contributions will be incorporated into this new effort.

The study of patterns opens wide the door for implementing NCTM Standards such as ³actively involving students individually and in groups in exploring, conjecturing, analyzing, and applying mathematics in both mathematical and real-world contexts,² helping students to ³recognize and apply deductive and inductive reasoning,² and ³make and evaluate mathematical conjectures and arguments.² The Standards emphasize that ³conjecturing and demonstrating the logical validity of conjectures are the essence of the creative act of doing mathematics.² All of these are an integral part of the study of patterns.

Fifteen years ago AIMS made a commitment to provide leadership in integrating the study of science and mathematics. Over the course of these years that decision has impacted the nation. Now, AIMS is making a parallel commitment to provide the same energetic leadership in developing a pattern-based math/science curriculum. This, too, holds high promise of making a national impact.

Up to the present, most AIMS efforts to integrate mathematics and science have originated in science content, represented by the right-hand circle in the AIMS Venn diagram.

The primary, but not exclusive, emphasis has been on infusing scientific methods into the integrative process. This program will not only continue unabated, but expand to respond to the broadening demand for the AIMS curriculum.

The AIMS pattern-based math/science curriculum will originate primarily in mathematics content, represented by the left-hand or mathematics circle.

Mathematical methods will play a predominant role in forging integration. Thus, this two-pronged strategy of originating activities in science content on the one hand and in mathematics content on the other will provide complimentary approaches to the development of the comprehensive AIMS K-9 program.

The rationale for and advantages of a pattern-based curriculum will be examined in this series of articles. Examples drawn from work in progress will be used to illustrate this refreshingly different approach. A separate series entitled Patterns, Problem Solving, and Practice begins as a regular feature with this issue of the AIMS magazine.

The AIMS Venn diagram continues to serve as a guide for the AIMS research and development program. The force of its fundamental ideas helps AIMS Research Fellows shape new activities. The diagram itself continues to evolve to accurately reflect the current thinking and philosophy of AIMS. The intersection representing the integration of science and mathematics has recently been enlarged to reflect the greater scope of integration envisioned as a result of adding the pattern-based approach to AIMS curriculum development.

AIMS continues to evolve as the result of significant influences. In l991, the School Science and Mathematics Association sponsored the Wingspread Conference on ³Integrated Science and Mathematics Teaching and Learning.² Fifty selected science and mathematics educators, including this writer, engaged in a rigorous analysis of why, how, and when the integration of mathematics and science best occurs. The conclusion of one study group was particularly appropriate and is reflected in the AIMS Venn: ³Integration infuses mathematical methods into science and scientific methods into mathematics so that it becomes indistinguishable as to whether it is mathematics or science.² This supports the decision of AIMS to devote full energy to this two-fold approach.

During the conference, John Bansford stressed that "the integration of science and mathematics is highly desirable, not as an end to itself, but to help students experience the excitement and importance of mathematical and scientific inquiry, to realize it is within their potential to engage in such inquiry, and to offer them the kinds of experiences that will set the stage for lifelong learning." Since science is the study of patterns in nature and the subject of mathematics is patterns, the study of patterns is essential if students are to ³experience the excitment and importance² of mathematical and scientific inquiry.

Note that the AIMS Venn includes the fundamental cycle of mathematical processes recommended in Science for All Americans. This cycle begins with a real-world event from which students abstract data by counting or measuring; the data is then subjected to symbolic manipulation and transformation using the processes of mathematics; finally, it is applied back into the real world to test its validity. Thus, the cycle begins and ends in the real world. AIMS users are well aware that this cycle is amplified and utilized in the AIMS Five-Star Learning Model. This AIMS model delineates the four environments in which the teaching/learning process takes place, the special processes peculiar to each, and the need to be able to translate from any one to any of the others. The circle denotes the real world. It is made accessible through the senses. As an event in the real world is studied, one or more of its properties are quantified through the process of counting or measuring. Then in symbolic form, represented by the triangle, the numbers are studied for patterns and manipulated using appropriate mathematical processes. Frequently, quantitative information is clarified or induced to yield new information and patterns by translating it into pictorial form such as graphs, diagrams, maps, etc. The square denotes this pictorial representation. The goal with any pattern in number or nature, is to discover the generalization that encapsulates it, represented by the hexagon. Finally, it is essential to test the generalization in the real-world event or situation from which it originated to determine its validity.

What is the rationale for a pattern-based mathematics/science curriculum? It is essential that we examine the nature of mathematics, the nature of science, and the nature of the learner to answer the question.

Readers of this magazine are well aware of the fact that Dr. Richard Thiessen has written extensively on mathematics as the science of patterns. His articles contain numerous quotations from eminent mathematicians that deal with the foundational role of patterns in mathematics. This illustration appeared in the April, 1994 issue of AIMS.

Because of their relevance to the rationale for a pattern-based math/science curriculum, the key statements made by these representative mathematicians and in key publications are included here.

G. H. Hardy says, ³A mathematician is a maker of patterns.² Lynn Steen, in On the Shoulders of Giants writes that ³mathematics is the science of patterns.² Alfred North Whitehead describes mathematics as ³the most powerful technique for the understanding of patterns.² Descartes stated that ³mathematics is the science of order.²

In Everybody Counts, mathematics is described as a science of pattern and order. In the National Research Council publication Reshaping School Mathematics, a Philosophy and Framework for Curriculum we read, ³Since mathematics is both the language of science and a science of patterns, the special links are far more than just those between theory and applications. The methodology of mathematical inquiry shares with the scientific method a focus on exploration, investigation, conjecture, evidence and reasoning. Firmer school ties between science and mathematics should especially help student´s grasp of both fields.² This statement makes clear the integrative nature of exploration, investigation, conjecture, evidence, and reasoning. Patterns constitute a natural subject for these processes.

There is universal agreement that mathematics is the science and study of patterns and that this is central to understanding mathematics. Only when students gain this understanding of mathematics through the priority we give to the study of patterns can we consider ourselves successful as their mentors!

Rutherford and Ahlgren in Science for All Americans write: ³Science and mathematics are both trying to discover general patterns and relationships, and in sense they are part of the same endeavor.² The history of science is replete with discoveries of patterns in nature. Cycles, geometric designs, cause and effect, physical laws, etc., are but a beginning of countless examples that could be listed.

As Thiessen pointed out in an earlier series, Johann Kepler spent more than 20 years pouring over masses of data, searching for a pattern of the planets´ movements through the sky before formulating the three laws of planetary motion. Newton searched the astronomical data of his day and discovered the patterns that led to his formulation of the law of gravitation. It allowed him to not only explain but to derive Kepler´s laws of motion.

What about the learner? Our own introspection will convince us of the crucial role patterns play in our lives. Most of what we do we do by patterns. All of us have a natural tendency to order our lives and thinking by patterns. We are so inclined to shape our lives by patterns that interruptions cause us discomfort. In such instances, disruptions are like discrepant events and cause disequilibrium.

Dr. Richard Thiessen points out that ³we are seekers, noticers, finders, and recognizers of patterns.²

Douglas Hofstadter, in his book Fluid Concepts and Creative Analogies, writes, ³...as a math major, one thing I had become convinced of was that pattern-finding was close to the core, if not the core of intelligenceŠit seems a trivial thing to assert that analogy making lies at the heart of pattern perception and extrapolation. And when thisŠis put together with my earlier claim that pattern-making is the core of intelligence, the implication is clear: analogy-making lies at the heart of intelligence.²

Thus, the nature of mathematics, the nature of science, and the nature of the learner converge in a pattern-based curriculum! Given that mathematics is the science of patterns, science is the study of patterns in nature, and the learner has a proclivity for patterns, it is logical to conclude that a pattern-based curriculum holds unlimited promise to unlock the door of success by advancing the integration of mathematics and science and making their study appealing and comprehensible to the learner!

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