Mathematics in the Modern World PDF

Mathematics in the Modern World The Nature of Mathematics Perceptions of the nature and role of mathematics held by our society have a major influence on the development of mathematics curriculum, instruction, and research. Diverse views of nature of mathematics have a pronounced impa...

Mathematics in the Modern World The Nature of Mathematics Perceptions of the nature and role of mathematics held by our society have a major influence on the development of mathematics curriculum, instruction, and research. Diverse views of nature of mathematics have a pronounced impact on the ways in which our society conceives of mathematics and reacts to its ever- widening influence on our daily lives. Learning Objectives: At the end of the lesson, the student should be able to:  Explain the concept of mathematics  Define the characteristics of mathematics  Identify the different types of patterns in mathematics  Appreciate the different aspects of mathematics; and  Understand how simple mathematics models are constructed to solve problems. Introduction of Mathematics  “The abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra” – Oxford English Dictionary  “The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols” – American Heritage Dictionary  “The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects” Three leading types of definition of mathematics: Logicist – “All mathematics is symbolic logic” Intuitionist – “Mathematics is the mental activity which consists in carrying out constructs one after the other” Formalist – “Identify mathematics with its symbols and the rules for operating on them” Nature of Mathematics  The science of measures  An intellectual game  The art of drawing conclusions  A tool subject  A system of logical processes  An intuitive method Characteristics of Mathematics  Logical sequence  Structure  Precision and accuracy  Abstractness  Mathematical Language and Symbolism  Applicability  Generalization and classification  Mathematical systems  Rigor and logic  Simplicity and complexity  Mathematics reveals hidden patterns that help us understand the world around us.  Mathematics today is a diverse discipline that deals with data, measurements and observations from science, with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems. By studying patterns in nature, we gain an appreciation and understanding of the world in which we live and how everything is connected. By engaging Nature, we acquire a deeper connection with our spiritual self. We are surrounded by a kaleidoscope of visual patterns – both living and non-living. “Math is about patterns … and patterns are what life is about.” - K. Devlin Mathematics as a Science of Patterns  Logic Patterns  Number Patterns  Word Patterns  Patterns in Nature Mathematics on Our World Mathematical patterns and forms exist everywhere in nature. Let’s explore our nature and things that convey various mathematical concepts. Mathematics on Our World These mathematical concepts include:  Patterning  Shapes and Geometry  and other mathematical principles.  Patterns are referred to as visible consistencies found in nature. There are several types of patterns including symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks, and stripes. While the scientific explanation for how each of these is formed - and why they are significant in the natural world is amazing - the visual result is equally amazing. Symmetry - includes two types of patterns: radial and bilateral. Radial symmetry references the numerical symmetry referred to as the Fibonacci sequence (1, 2, 3, 5, 8, 13, 21, 34, 55, 89...) If you counted the seeds within a sunflower, you would find the number of seeds is equal to a Fibonacci number. Bilateral (or mirror) symmetry, meaning they could be split into two matching halves, much like the plant and sea life Trees/Fractal are patterns formed from chaotic equations and form self similar patterns of complexity increasing with magnification. If you divide it into parts, you will get a nearly identical copy of the whole. Infinite iteration is not possible in nature, so all fractal patterns are approximate. Spiral patterns are attributed to complicated mathematical algorithms, sequences and equations - and are common in plants and some animals like the fern and desert big horn sheep. Meanders are represented by bends in rivers and channels but can also be seen in other forms throughout the natural environment. For example, the salt pans of the desert and pattern within the kelp leaves contain meanders. Waves are disturbances that carry energy as they move. Wind waves are created as wind passes over a large body of water, creating patterns or ripples. When wind passes over land, it creates dunes. Dunes may form a range of patterns as well. Foams are typically referred to as a mass of bubbles, but other types of foams can be seen within the patterns of certain animal species such as the leopard, giraffe, and tortoises. Tessellations are repeating tiles over a surface commonly seen in reptiles like snakes and alligators. Cracks are linear openings that form in materials to relieve stress. When a material fails in all directions it results in cracks. The patterns created reveal if the material is elastic or not. Stripe The stripe pattern is evolutionary in that it increases the chances of survival through camouflage. It is most commonly known in zebras, but other species contain stripes - even butterflies. Patterns and Numbers in Nature and the World  SEASHELLS/ or simply SHELLS Some shells have natural patterns. Chambered Nautilus Shell The shape process is called “LOGARITHMIC SPIRAL” https://www.google.com/url?sa=i&url=https%3A%2F%2Fsomanautiko.com%2Fembracing-the-sacred-symbol-of-the-chambered-nautilus%2F&psig=AOvVaw3Np85k49L_RIU2Ax- CloBT&ust=1594960468572000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCLiCsJHW0eoCFQAAAAAdAAAAABAD Patterns and Numbers in Nature and the World  Geometric Patterns Spider Web It illustrates wonderful geometric pattern  Spider creates this structure by performing simple, distinctive steps. https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.amnh.org%2Fexplore%2Fology%2Fbiodiversity%2Fwhat-is-biodiversity&psig=AOvVaw31RuodV4kVys- TzdWRFL8X&ust=1594966241132000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCOCYo_rS0eoCFQAAAAAdAAAAABAD Patterns and Numbers in Nature and the World  Geometric Patterns Snowflakes Frozen rain A crystal of snow, having approximate hexagonal symmetry. https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.vectorstock.com%2Froyalty-free-vector%2F3d-snowflakes-vector- 698527&psig=AOvVaw0Pt5QemMZSuEsVh0EHgbsH&ust=1594966894846000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCIj47NHS0eoCFQAAAAAdAAAAABAD  Patterns and Numbers in Nature and the World  Zebras Several theories have been proposed for the function of these distinctive black-and-white striped coats of zebra, with most evidence supporting them as a form of protection from biting flies. The general pattern is a dorsal line that extends from the forehead to the tail. Stripes may provide particularly good camouflage at nighttime. https://www.google.com/url?sa=i&url=https%3A%2F%2Fabcnews.go.com%2FTechnology%2Fzebras-stripes-staying- cool%2Fstory%3Fid%3D63685771&psig=AOvVaw0byDeNySwZbXuBEwxgEXZy&ust=1594985621928000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCOj Zp87W0eoCFQAAAAAdAAAAABAD Grevy’s zebra Plains zebra Mountain zebra Patterns and Numbers in Nature and the World  Honeycomb is a mass of regular hexagonal wax cells built by honey bees in their nests to contain their larvae and stores honey and pollen.  Why might a hexagon be a suitable shape for storing honey? Could other shapes, such as circles, triangles, or squares, work just as well? https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.npr.org%2Fsections%2Fkrulwich%2F2013%2F05%2F13%2F183704091%2Fwhat-is-it-about-bees-and- Patterns and Numbers in Nature and the World  Dendrochronology is the scientific method of dating tree rings (also called growth rings) to the exact year they were formed. https://www.google.com/url?sa=i&url=http%3A%2F%2Fwww.pbs.org%2Ftime-team%2Fexperience-archaeology%2Fdendrochronology%2F&psig=AOvVaw39EDJq4XK4Aeu8OQ4Z- YbV&ust=1594981042461000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCOCwuMTF0eoCFQAAAAAdAAAAABAZ  Other Examples: Tiger’s stripe https://www.google.com/url?sa=i&url=https%3A%2F%2Finspirechange27.wordpress.com%2F2018%2F11%2F18%2Fthe-tiger-as-the-national- animal%2F&psig=AOvVaw24hwmRr19t16lLFbOlVUlI&ust=1595049227191000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCJizj8jD0-oCFQAAAAAdAAAAABAV Other Examples: Hyena’s spots https://www.google.com/url?sa=i&url=https%3A%2F%2Fideas.ted.com%2Feverything-you-know-about-hyenas-is-wrong-these-animals-are-fierce-social-and-incredibly- smart%2F&psig=AOvVaw1OJ24GVvdcxPF-6ZN2c8nv&ust=1595048917615000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCMDdh7XC0-oCFQAAAAAdAAAAABAO Other Examples: Sunflowers’ petals and seeds https://www.google.com/url?sa=i&url=https%3A%2F%2Fgilmour.com%2Fgrowing-sunflowers&psig=AOvVaw1hTbLocDMoaQTsINEAxln- &ust=1595048747107000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCNjIq-XB0-oCFQAAAAAdAAAAABAD Other Examples: World’s Population https://www.google.com/url?sa=i&url=https%3A%2F%2Fnationbuilder.com%2Fpeoples_vote&psig=AOvVaw260Gfp7RFC7x- https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.theatlantic.com%2Fphoto%2F2019%2F02%2Fmasks-we-wear%2F583837%2F&psig=AOvVaw34nTYFA03T7nxhFnU- mPbIsO26H&ust=1595049412947000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCIiZwKHE0-oCFQAAAAAdAAAAABAd 17Tt&ust=1595050516534000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCNiqm6_I0-oCFQAAAAAdAAAAABAk Other Examples: Weather Other Examples: Tidal waves https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.b2bmarketing.net%2Fen-gb%2Fresources%2Fblog%2Fwhy-b2b-marketers-must-ride-tidal-wave- change&psig=AOvVaw1b04NPdHwtMB7N4MoIO5DP&ust=1595047925560000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCLjaleK-0-oCFQAAAAAdAAAAABAD Other Examples: Hurricane https://www.google.com/url?sa=i&url=https%3A%2F%2Fnews.miami.edu%2Fstories%2F2020%2F06%2Fscientists-examine-why-hurricane-wind-speeds-vary-within-urban- settings.html&psig=AOvVaw3BNEWb6Ph-mr5Qe7rF9AC_&ust=1595048332178000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCLDdo6nA0-oCFQAAAAAdAAAAABAV Other Examples: Core idea of the lesson: Mathematics is a useful way to think about nature and our world. Activity for the day What is your disposition towards Math? Do you like it or not? Why? References:  Aufmann, R.,Lockwood, J., Nation, R. and Clegg, D. (2013). Mathematical Excursions(3rd ed). Brookes/Cole, Cengage Learning  3G E-Learning LLC, Mathematics in the Modern World (2018)  Calingasan, R., Martin, M. and Yambao, E.(2018) Mathematics in the Modern World C&E Publishing, Inc.  Miller, C. Heeren, V. Hornsby, J.(2008). Mathematical Ideas (11th ed).Pearson Education Inc.  https://ecstep.com/natural-patterns/