MODULE 1 PDF

MODULE 1: THE NATURE OF MATHEMATICS This module is an introductory part. I would be presenting the definition and nature of Mathematics. A video presentation “Math from a Biblical Worldview” @ https://www.youtube.com/watch?v=YcKi2t54d...

MODULE 1: THE NATURE OF MATHEMATICS This module is an introductory part. I would be presenting the definition and nature of Mathematics. A video presentation “Math from a Biblical Worldview” @ https://www.youtube.com/watch?v=YcKi2t54djk helps us out in looking at its aesthetic value. The target Learning outcomes of our Module 1, particularly for topics 1 and 2, are: a) Articulate the importance of mathematics in one’s life; b) Express appreciation for mathematics as a human endeavor. Pre-assessment Question: In 150 words explain the role of Mathematics in your field? UNIT 1: THE MEANING OF MATHEMATICS AND ITS CHARACTERISTICS PowerPoint 01: Definitions of Mathematics and Its characteristics. Mathematics may be viewed from different perspectives. In its wildest significance, it is the development of all types of formal deductive reasoning. Generally, it is said to be the science of calculation. Others view it as a science of numbers and space, and others say it is a science of measurement, quantity, and magnitude. Locke said, “Mathematics is a way to settle in the mind of children a habit of reasoning.” It is a discipline investigating “formal structures” (Bernays), it is the “science of orders” (Russell), it is the “science of order in progression” (Hamilton). Mathematics has also been seen as a logical construct based on many axioms of either set theory or number theory. Traditionally, mathematics is presented deductively at school. And is often perceived as well structured, and problems are algorithmically approached. Merriam dictionary defines mathematics as the science of numbers and their operations, interrelations, combinations, generalizations, abstractions, and space configurations of their structure, measurement, transformations, and generalizations. Mathematics is derived from the ancient word manthanein meaning "to learn." The Greek root mathesis means "knowledge" or its other form máthema meaning science, knowledge, or learning, and mathematikós or mathemata means "fond of learning." These might have been the notion of the early mathematicians and philosophers that is why they continue to seek knowledge and the truth. Mathematics could then be defined as a desire for a particular kind of knowing. Knowing that is self-contained on the individual or may be seen as autonomous thinking (Schaaf, 1963). Mathematics is described in so many ways that it fits within the area of human knowledge. Basically, it is seen as a study of patterns and relations. It is also a way of thinking. Property of and for the exclusive use of SLU. Reproduction, storing in a retrieval system, distributing, uploading or posting online, or transmitting in any form or by any 1 means, electronic, mechanical, photocopying, recording, or otherwise of any part of this document, without the prior written permission of SLU, is strictly prohibited.  Mathematics is seen as an art that is characterized by order and internal consistency. It is a language that uses carefully defined terms and symbols. Thus, mathematics is a tool (Reys, Lindquist, Lambdin, Smith, and Suydam, 2004). Mathematics has five essential characteristics’ namely: precision, definition, reasoning, coherence, and purposefulness. They are not independent of each other. It is precise in the sense that mathematical statements are clear and unambiguous. It is clear what is known and what is not known. Definitions abound in mathematics. It is the bedrock of mathematical structure and the platform that supports reasoning. Reasoning is the lifeblood of mathematics. It is the engine that drives proving and problem-solving. Its absence is the root cause of the learning by rote approach. Concepts and skills are interwoven in mathematics. And lastly, mathematics is goal-oriented, and for every concept or skill, there is a purpose for it. Hardy (1941) states that the beauty of mathematics resides in the fact that mathematics is all about not just patterns but patterns of ideas. Devlin defines mathematics as the “science of patterns” and then more fully as: “the science of order, patterns, structure, and logical relationships” (Devlin, 2001, p. 73). Mathematics has also often been described as the language of science. Since the mixture of symbols and words is so powerfully descriptive and communicative, perhaps the definition: “mathematics is the language of the science of order, patterns, structure, and logical relationships” may be considered. Whatever form or way one defines mathematics so long as it becomes meaningful to the user, it would be a definition for that user. UNIT 2: MATHEMATICS IN NATURE PowerPoint 02: Mathematics in Nature, Shapes This module would give you a perspective that mathematics is not numbers and operations. It is more than that. We have been emphasizing that God created the universe mathematically. Everything you see around you is Mathematics. The target learning outcomes of module 2 are: a) Identify patterns in nature and regularities in the world; b) Articulate the importance of mathematics in one’s life; c) Express appreciation for mathematics as a human endeavor To start with, lets watch this video entitled Nature’s Mathematics 1 and 2 https://www.youtube.com/watch?v=VE_RU0fNjt0 https://www.youtube.com/watch?v=n2WHNMfRmHE&t=643s Euclid said that "The laws of nature are but the mathematical thoughts of God." Galileo affirmed by stating that “Mathematics is the language in which God has written the Universe.” Mathematics is everywhere. It is seen anywhere in the universe. With the development of a formal system of thought for recognizing, classifying, and exploiting patterns, one could Property of and for the exclusive use of SLU. Reproduction, storing in a retrieval system, distributing, uploading or posting online, or transmitting in any form or by any 2 means, electronic, mechanical, photocopying, recording, or otherwise of any part of this document, without the prior written permission of SLU, is strictly prohibited.  systematize and organize these ideas of patterns. It would be here that we could discover great secrets of nature’s patterns. They are not just there to be admired; they are vital clues to the rules that govern the natural process. Analyzing thoroughly, having the essentials of mathematics as our basis, we could further discover mathematics in our world and unravel the mystery of the universe. The majority of our knowledge of mathematics and modern science is strictly based and supported by our environmental observations. What was once seen as the randomness of nature is now distinguished as the intricate applications of mathematics and illustrates the complexities of our natural world. Here are a very few properties of mathematics that are depicted in nature. A. SHAPES Geometry is the branch of mathematics that basically describes shapes and establishes the relationships between them. Figures with regular shapes are categorized as polygons. Polygons are fascinating, especially when they are approximated in nature. When looking carefully, one can see them all around us. Spatial patterns can be represented by a fairly small collection of fundamental geometrical shapes and relationships that have corresponding symbolic representations. The human mind relies heavily on its perception of shapes and patterns to make sense of the world. The artifacts around us (such as buildings, vehicles, toys, utensils, and basic things we use in life) and the familiar forms we see in nature (such as animals, leaves, stones, flowers, and the moon and sun) can often be characterized in terms of geometric form. Some of the ideas and terms of geometry have become part of everyday language. Although real objects never perfectly match a geometric figure, they more or less approximate them. The properties and characteristics of geometric figures and relationships can be associated with objects. For many purposes, it is sufficient to be familiar with points, lines, planes; triangles, rectangles, squares, circles, and ellipses; rectangular solids and spheres; relationships of similarity and congruence; relationships of convex, concave, intersecting, and tangent; angles between lines or planes; parallel and perpendicular relationships between lines and planes; forms of symmetry such as displacement, reflection, and rotation; and the Pythagorean theorem. Both shape and measurement (magnitude) or scale can have important consequences for the performance of systems. For example, triangular connections maximize rigidity, smooth surfaces minimize turbulence, and a spherical container minimizes surface area for any given mass or volume. Changing the size of objects while keeping the same shape can have profound effects due to scaling geometry: Area varies as the square of linear dimensions, and volume varies as the cube. On the other hand, some fascinating kinds of patterns known as fractals look very similar to one another when observed at any scale whatever—and some natural phenomena (such as the shapes of clouds, mountains, and coastlines) seem to be like that. Property of and for the exclusive use of SLU. Reproduction, storing in a retrieval system, distributing, uploading or posting online, or transmitting in any form or by any 3 means, electronic, mechanical, photocopying, recording, or otherwise of any part of this document, without the prior written permission of SLU, is strictly prohibited.  Some Common Shapes Visible in Nature Sphere. A sphere is a perfectly round geometrical object in three- dimensional space, such as the shape of a round ball. The shape of the Earth is very close to that of an oblate spheroid, a sphere flattened along the axis from pole to pole such that there is a bulge around the equator. Hexagons. A hexagon is a Two-dimensional six-sided closed polygon. For a beehive, close packing is important to maximize the use of space. Hexagons fit most closely together without any gaps; so hexagonal wax cells are what bees create to store their eggs and larvae. Cones. A cone is a three-dimensional geometric shape that tapers smoothly from a flat, usually circular base to a point called the apex or vertex. Volcanoes 2.5 form cones, the steepness and height of which depends on the runniness (viscosity) of the lava. Fast, runny lava forms flatter cones; thick, viscous lava forms steep-sided cones. Parallel lines. In mathematics, parallel lines stretch to infinity, neither converging or diverging. The parallel dunes in the Australian desert aren't perfect - the physical world rarely is. Interesting Figures in Nature Fractals. Like other figures, fractals are also geometric figures. Fractals involve dilation. They are objects with fractional dimensions and most have self-similarity. Self-similarity is when small parts of objects, when magnified resemble the same figure. The boundaries are of infinite length and are not differentiable anywhere (never smooth enough to have a tangent at a point). Thus, fractals have basic components that are similar to the whole. This means that you can find similar shapes even if you zoom the figure. However, fractals involve a complex process because it goes through an infinite number of iterations. Fractals can also be observed in nature such as trees, flowers, clouds, ocean waves, etc. Even the human body also has a fractal structure. Property of and for the exclusive use of SLU. Reproduction, storing in a retrieval system, distributing, uploading or posting online, or transmitting in any form or by any 4 means, electronic, mechanical, photocopying, recording, or otherwise of any part of this document, without the prior written permission of SLU, is strictly prohibited.  Fractals possess the following characteristics: self-similarity, fractional dimension, and formation by iteration. Natural objects exhibit scaling symmetry, but only over a limited range of scales. They also tend to be approximately self-similar, appearing more or less the same at different scales of measurement. Sometimes this means that they are statistically self-similar; that is to say, they have a distribution of elements that are similar under magnification. Below are some manifestations of fractals in nature. Ferns Clouds Bacterial Colony Lightning Waterfalls Mountains The world around us seems to make up several distinct patterns, evolving various complex steps of formation. However, looking more deeply we see many similarities and resemblances. The numerous models explained above have no experimental proof and may not be correct, but they definitely show linkages between patterns formed under highly contrasting natural conditions e.g. (a zebra coat and sand dunes) and also show that the mechanisms between the formations of these patterns need not necessarily be complex. B. SYMMETRY PowerPoint 03: Symmetry Symmetry is a type of invariance: a property that something does not change under a set of transformations. It is a mapping of the object onto itself which preserves the structure. Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. Although these two meanings of "symmetry" can sometimes be told apart, they are related. Plainly, symmetry is when a figure has two sides that are mirror images of Property of and for the exclusive use of SLU. Reproduction, storing in a retrieval system, distributing, uploading or posting online, or transmitting in any form or by any 5 means, electronic, mechanical, photocopying, recording, or otherwise of any part of this document, without the prior written permission of SLU, is strictly prohibited.  one another. It would then be possible to draw a line through a picture of the object and along either side the image would look exactly the same. This line would be called a line of symmetry. There are Two Kinds of Symmetry One is bilateral symmetry in which an object has two sides that are mirror images of each other. The human body would be an excellent example of a living being that has bilateral symmetry. The other kind of symmetry is radial symmetry. This is where there is a center point and numerous lines of symmetry could be drawn. The most obvious geometric example would be a circle. C. PATTERNS PowerPoint 04: Patterns in Nature Though every living and non-living thing of the world may seem to follow a pattern of its own, looking deeply into the geometry and mechanism of the pattern formation can lead you to classify them into merely two categories broadly: 1. Self-organized patterns/ Inherent organization 2. Invoked organization Self-Organized patterns/ Inherent organization. A self-organizing pattern follows a simple set of rules, and they use only local information to determine how a particular subunit evolves. Successive patterns represent them. This pattern can be described as successive horizontal rows; the 'successor' pattern is just under its predecessor. A complex pattern develops when the basic rule just defined is applied to that row (the active row) and then to subsequent rows. Thus, self-organization is a process in which patterns at the global level of a system emerge solely from numerous interactions among the lower-level components of the system. Moreover, the rules specifying interactions among the system’s components are executed using only local information, without reference to the global pattern. In other words, the pattern is an emergent property of the system rather than a property imposed on the system by an external influence. Therefore, if rules are useful for understanding life patterns, such as the stripes on a zebra's coat, there must be a specific rule. The zebra's coat alternates in contrasting areas of light and dark pigmentation. Hence, the zebra's coat patterns reflect the early interaction of chemicals as they diffused through the embryonic skin. Property of and for the exclusive use of SLU. Reproduction, storing in a retrieval system, distributing, uploading or posting online, or transmitting in any form or by any 6 means, electronic, mechanical, photocopying, recording, or otherwise of any part of this document, without the prior written permission of SLU, is strictly prohibited.  Invoked Organization. Not all patterns that occur in nature arise through self- organization. A weaver bird uses its own body as a template to build the hemispherical egg chamber of the nest. A spider when creating a web follows a genetically determined recipe in relation to its sticky orb and the various radii and spirals it creates. A similar invoked organization is that of the honeycomb made by bees. In these cases, the structures are built by an architect that oversees and imposes order and pattern. There are no subunits that interact with one another to generate a pattern. Each animal acts like a stonemason or laborer, measuring, fitting, and moving pieces into place. This is an example of social insect architecture: the wasp nests, an example of an invoked organization in nature. More than this biological system existing on the patterns created in nature, great mathematics is embedded in it aside from the plain geometric figure. The geometry of the patterns could also be linked to mathematical numbers directly or indirectly. The series of numbers seem to have been forced on them. The Fibonacci numbers or sequence and the Golden ratio are but the primary example of them. Learning Reinforcement 1 Directions: Submit your output as a single pdf file in the submission bin for this activity in the Classroom. Search or take pictures of revelations of Mathematics in nature. 1 picture that exhibit Golden Ratio and another picture that exhibit Fibonacci Sequence. Explain each picture in 150 words how does each picture exhibit said pattern. Cite your references. Output must be placed in Property of and for the exclusive use of SLU. Reproduction, storing in a retrieval system, distributing, uploading or posting online, or transmitting in any form or by any 7 means, electronic, mechanical, photocopying, recording, or otherwise of any part of this document, without the prior written permission of SLU, is strictly prohibited.