GE 103 Mathematics in the Modern World Module PDF

D COLLEGE OF TEACHER EDUCATION BSEd A L U B H...

D COLLEGE OF TEACHER EDUCATION BSEd A L U B H A S COURSE MODULE A A N N G GE 103 L U N S O MATHEMATICS IN THE MODERN D N WORLD G S A N P A B L O Prepared by: Mr. Allan Melvin B. Comia, LPT GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U B GUIDE ON HOW TO USE THE MODULE H For Faculty A S A The primary objective of this module is to address the learning of the A students in Blended learning approach. The author aims that through N this module the school will meet the expectation of the learners. N G To the faculty of DLSP, you can use this module to supplement the learning of the students. However, do not use this module to gain L money from the students or sell this to other instructors. U N S You can also use this module as a reference in preparing your O instructional materials like PowerPoint presentation and others, for D synchronous/asynchronous learning or for preparing the same N module. G For Learners S A This module is designed to engage you as a learner, and to guide you N as you finish this subject Mathematics in the Modern World. P A In using this module, read every topic with comprehension. Make sure B that you have understand every lesson, because there are activities L for you to answer after every topic and all of the questions are O answerable based on what you have understood in the topic. You are expected to answer all the activities in this module because it is one way of assessing yourself if you really understand the lesson. You may answer your activities on another sheet of paper and submit the hard copy or answer it on your computer and send the output to your instructor. Strictly no searching for answer online, your answer must be the reflection of your learning about the lesson. Furthermore, read carefully the instruction, especially in online quiz, because it is automatically checked by the system. You might be marked incorrect if you don’t follow the instruction. You can optionally share this module to your classmates or upload this online, just make sure that it is for educational purposes and not for selling, and the credits must belong to the author. GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U FOREWORD B H In these trying times, we are all striving to cope up and continue with our A S lives. As instructors and students, we both continue to learn as we are all A forever students who keeps on learning inside or outside the school, with or A without books, face-to-face or distant. With the current situation going N around, all of us are affected, and I as an instructor understands that the struggles, we all face may affect our learning. It may be a challenge for all N G of us, but with active participation and collaboration of school, instructors, students and other stakeholders, learning will not stop. L U As we go through this semester, let us help each other to build an N encouraging online learning environment. I, as your instructor will take every S O condition into consideration, and you as students, are encouraged to voice D out your opinions and suggestions, and speak up about your concerns and conditions. N G This module is prepared as your guide as you study your lessons at home. S However, information and knowledge should not be limited within this A module. You are highly encouraged to maximize the use of different N resources, because most of you will be learning in your own pace. If other resources are difficult to access, the instructor will make sure that the P A provided materials are enough and easy to access. B L Remember that all of these are new to us. We are still in the phase of entering O the new normal. I hope instructors will be considerate, and students will be patient. Let us be understanding at all times. This module contains lectures that you can read and activities that you are required to do. Learning objectives and content are limited as we have to choose only the most important learning outcomes. However, this module is made to help students still gain knowledge and skills to achieve the expected learning outcomes while considering students’ access to resources. May we all continue to learn in spite of this pandemic. God bless us all. GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U B Unit 1: The Nature of Mathematics H A Mathematics reveals hidden patterns that help us understand the world around us. Now S much more than arithmetic and geometry, mathematics today is a diverse discipline that deals A A with data, measurements, and observations from science; with inference, deduction, and proof; N and with mathematical models of natural phenomena, of human behaviour, and of social systems. N As a practical G matter, mathematics is a science of pattern and L order. Its domain is not U N molecules or cells, but S numbers, chance, form, O algorithms, and change. As D a science of abstract N objects, mathematics relies G on logic rather than on observation as its standard S of truth, yet employs observation, simulation, and even experimentation as means of discovering A truth. N The special role of mathematics in education is a consequence of its universal P A applicability. The results of mathematics – theorems and theories – are both significant and useful; B the best results are also elegant and deep. Through its theorems, mathematics offers science both L a foundation of truth and a standard of certainty. O Aside from theorems and theories, mathematics offers distinctive modes of thought which are both versatile and powerful, including modelling, abstraction, optimization, inference from data, logical analysis, and use of symbols. Experience with mathematical modes of thought builds mathematical power – a capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to assess risk, and to suggest alternatives. Mathematics empowers us to understand better the information-laden world we live in. Unit outcomes By the end of this unit, you should be able to: Identify patterns in nature and regularities in the world. Articulate the importance of mathematics in one’s life. Argue about the nature of mathematics, what it is, how it is expressed, represented, andused. Express appreciation for mathematics as a human endeavour. GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U B Lesson 1: The Nature of Mathematics H Without mathematics, there's nothing you can do. Everything around you is mathematics. A S Everything around you is numbers. A –Shakuntala Devi A N Introduction N The body of knowledge and practice known as mathematics is derived from the G contributions of thinkers throughout the ages and across the globe. It gives us a way to understand L patterns, to quantify relationships, and to predict the future. Math helps us understand the world U — and we use the world to understand math. In this lesson, you will be able to characterize N mathematics and its nature. You will be able to also name the use and importance of mathematics S to your daily life and to the modern world. O D Content is adapted from: N Mathematics in the Modern World, Calingasan et. al., Chapter 1 G Lesson Learning Outcomes S A By the end of this unit you should be able to: N ❖ explain the nature and characteristics of mathematics P A ❖ enumerate important applications of mathematics in the modern world B ❖ value mathematics as an essential human endeavor L O Mathematics: An Overview According to John William Navin Sullivan, a famous science writer, "Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self- consciousness. The significance of mathematics resides precisely in the fact that it is an art: by informing us of the nature of our own minds, it informs us of much that depends on our minds." True to what Sullivan said, mathematics is at the center of history and development of culture, Mathematics is evident in ancient arts, Egyptian architecture, and music. In fact, Pythagoras (570-495 BC), the "Father of Numbers," discovered numerical reasons in musical harmony. Mathematics is very beneficial to every human being in his or her day-to-day live. It helps an individual develop reasoning, analytical thinking and critical thinking. Analytical thinking helps person investigate and determine the truth around him or her. Mathematics can explain how things work and phenomena can be best understood through it. It is evident that mathematics has played a central role in the physical sciences for centuries, und currently it is being used by life scientists in analyzing patterns present in plants, animals, and humans. GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U The things and benefits that mathematics can do and provide are countless. A knowledge B of the basic concepts of mathematics will enable people to explore its applications and in what H A phenomena and activities in life this field can be helpful. S To learn more, watch the following video on https://youtu.be/64643Op6WJo. A A N Definition of Mathematics N The word mathematics comes from the Greek μάθημα (máthēma), which, in the ancient G Greek language, means that which is learnt." or "lesson" in modern Greek. Mathema is derived from "manthano" while the modern Greek equivalent is "mathaino" which means to learn. L U The definition of mathematics does not seem as readily available as other fields, like N science. Life defines biology as chemicals do chemistry, and matter does physics. Math lacks an S empirical component. The fundamental units of mathematics are elements and sets, which given O D their abstract rather than tangible nature, seems to suggest that mathematics defines itself. Math, as a pure subject, aligns more closely with philosophy than it does with science because N mathematics does not intend to talk about the universe, but rather about an imaginary universe G where only axioms exist. More than any other discipline, mathematics concerns itself with logic and truth. By definition, math is the abstract study of how the structures of systems relate and S A operate. In simpler terms we can define mathematics as the science of patterns and relationship. N Drill 1 P Use the acrostics below to describe or define mathematics in your own words. A B L O Origin of Mathematics To understand the complex relationship between humans and mathematics, we must understand how we first used it. In its early forms, mathematics helped us quantify time, make measurements, and take records. Rudimentary mathematics was especially useful during the development of agriculture when surpluses in food allowed trade. Mathematics satisfied the Figure 1. The number systems of different need to keep accurate records and perform basic civilizations GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U calculations. Symbols for representing quantities ranged from everyday objects, to geometric B shapes. H A As mathematical techniques evolved, so did our perspective on this new system. Ancient S Greeks believed that numbers were both living entities and universal principles; numbers were A active agents in nature. Plato pioneered the study of the ontology of mathematical objects, and A N Aristotle studied logic and issues related to infinity. Philosophers quickly realized that numbers and their operations were very useful in describing our world. The profound convergence of N diverse aspects of mathematics, and existent theories in physics, caused many leading minds to G ponder the dilemma of invention versus discovery. L Discovery seems to imply that the thing in question was there beforehand, while invention U implies an original concoction. Two school of thoughts have opposing views on the matter. N According to realism, math exists objectively and independent of human thought. Mathematical S concepts are disembodied in the universe and available for us to uncover and bring into practical O D use. On the other hand, Mathematical anti-realism or idealism holds that mathematics is a product of the human imagination and is carefully engineered to make formal statements about N nature in order to aid our understanding of the behavior of the universe. G To learn more, watch the following video on https://youtu.be/X_xR5Kes4Rs. S A Drill 2 N Why do realists believe that mathematics is discovered? P A B Why do anti-realists believe that mathematics is invented? L O Importance of Mathematics People often wonder what relevance mathematicians serve today. In a modern world, math such as applied mathematics is not only relevant, it's crucial. Applied mathematics is the branch of mathematics that are involved in the study of the physical, biological, or sociological world. The idea of applied math is to create a group of methods that solve problems Figure 2. Mathematics enabled us to develop technology that we use in in science. Modern areas of the modern world. applied math include mathematical physics, mathematical biology, control theory, aerospace engineering, and math GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U finance. Not only does applied math solve problems, but it also discovers new problems or B develops new engineering disciplines. Applied mathematicians require expertise in many areas of H A math and science, physical intuition, common sense, and collaboration. The common approach in S applied math is to build a mathematical model of a phenomenon, solve the model, and develop A recommendations for performance improvement. A N While not necessarily an opposite to applied mathematics, pure mathematics is driven by abstract problems, rather than real world problems. Much of what's pursued by pure N mathematicians can have their roots in concrete physical problems, but a deeper understanding of G these phenomena brings about problems and technicalities. These abstract problems and technicalities are what pure mathematics attempts to solve, and these attempts have led to major L U discoveries for mankind, including the Universal Turing Machine, theorized by Alan Turing in N 1937. The Universal Turing Machine, which began as an abstract idea, later laid the groundwork S for the development of the modern computer. Pure mathematics is abstract and based in theory, O and is thus not constrained by the limitations of the physical world. D According to one pure mathematician, pure mathematicians prove theorems, and applied N mathematicians construct theories. Pure and applied are not mutually exclusive, but they are G rooted in different areas of math and problem solving. Though the complex math involved in pure S and applied mathematics is beyond the understanding of most people, the solutions developed A from the processes have affected and improved the lives of all. N To learn more, watch the following video on https://youtu.be/OmJ-4B-mS-Y. P Drill 2 A B Why is applied mathematics important to the modern world? L O Why is pure mathematics important to the modern world? You and Mathematics While it may seem like math problems have no real use in life, this couldn’t be farther from the truth! Math is incredibly important in our lives and, without realizing it, we use mathematical concepts, as well as the skills we learn from doing math problems every day. The laws of mathematics govern everything around us, and without a good understanding of them, one can encounter significant difficulties in life. 1. Learning math is good for your brain. Research conducted by Dr. Tanya Evans of Stanford University indicates that children who know math can recruit certain brain regions more reliably, and have higher gray matter volume in those regions, than those who perform more poorly in math. The brain regions involved in higher math skills in high-performing children were associated with various cognitive tasks involving visual attention and decision-making. While correlation may not imply causation, this study GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U indicates that the same brain regions that help you do math are recruited in decision-making and B attentional processes. H A 2. Math helps you with your finances. S A Math can be helpful for balancing your budget because you will have a good understanding A of how to make sure that your costs are less than the money you have. Balancing one’s bank N account, for example, is an important life skill that requires math in order to subtract balances. People who know math are therefore less likely to go into debt because they did not know how N much money they had versus how much money they spent. G 3. Math helps us have better problem-solving skills. L U Math helps us think analytically and have better reasoning abilities. Analytical thinking refers N to the ability to think critically about the world around us. Reasoning is our ability to think S logically about a situation. Analytical and reasoning skills are essential because they help us solve O problems and look for solutions. The skills that you use in framing math problems, identifying D the knowns and unknowns, and taking steps to solve the problem can be a very important strategy N that is applicable to other issues in life. G 4. Practically every career uses math in some way. S Obviously, mathematicians and scientists rely on mathematical principles to do the most basic A aspects of their work, such as test hypotheses. While scientific careers famously involve math, N they are not the only careers to do so. Even operating a cash register requires that one understands P basic arithmetic. People working in a factory must be able to do mental arithmetic to keep track A of the parts on the assembly line and must, in some cases, manipulate fabrication software utilizing B geometric properties (such as the dimensions of a part) in order to build their products. Any job L requires math because you must know how to interpret your salary and balance your budget. O 5. Math is all around us and helps us understand the world better. To live in a mathematically-driven world and not know math is like walking through an art museum with your eyes closed. Learning and appreciating math can help you appreciate things that you would not otherwise notice about the world. You will learn in the next lessons where to find them. 6. Math is a universal language. Sure, it’s mostly equations, numbers, and some Greek letters, but math is understood the same virtually all over the world (and who knows, maybe all over the universe)! A math equation doesn’t need to be translated to another language to be understood by someone on the other side of the planet. A mathematical law doesn’t change because someone has a different religion than you or speaks a different language from you. 2 + 2 = 4 in every single place on planet Earth. The universality of math is one of the many things that makes it such a powerful tool and, indeed, essential life skill. In summary, math is not only important for success in life; it is all around us. While many may complain that math is boring or complicated, the truth is that a life devoid of math means that we go around experiencing the world on a much less interesting level than we could. GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U To learn more, watch the following video on https://youtu.be/KX608C2RZek. B H A S Drill 4 A How do you use mathematics in your life? A N N G Answer the following questions about the lesson you read. L 1. Why is mathematics difficult to define? U N S O D N 2. Which between the mathematical realist and anti-realist do you believe more? Why? G S A N 3. How would life in the modern world be different if mathematics did not exist? P A B L O 4. Should mathematics be removed as a school subject? Why or why not? 5. Why do you think many people find math difficult? What do you think should be done so students can learn math better? GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U Lesson Exercises B H A. Encircle the number that closely applies to you on each statement. Find the total of the A results and compare it with the interpretation below. S A Strongly Neither Agree Strongly Statement Disagree Agree A Disagree nor Disagree Agree N 1. I cringe when I have to go to math class. 1 2 3 4 5 N 2. I am uneasy about going to the board in a math G 1 2 3 4 5 class. L 3. I am afraid to ask questions in math class. 1 2 3 4 5 U N 4. I am always worried about being called on in 1 2 3 4 5 S math class. O 5. I understand math now, but I worry that it's D 1 2 3 4 5 going to get really difficult soon. N 6. I tend to zone out in math class. 1 2 3 4 5 G 7. I fear math tests more than any other kind. 1 2 3 4 5 S A 8. I don't know how to study for math tests. 1 2 3 4 5 N 9. It's clear to me in math class, but when I go 1 2 3 4 5 P home it's like I was never there. A 10. I'm afraid I won't be able to keep up with the B 1 2 3 4 5 L rest of the class. O TOTAL = ________ 40-50 You have a very high chance of having math anxiety. 30-39 You have a high chance of having math anxiety. 20-29 You have a moderately high chance of having math anxiety. 10-19 You have no math anxiety and possibly even a math enthusiast Mathematics Anxiety Mathematics anxiety is a serious problem that can affect anyone. Approximately 93% of adults can potentially be affected by some form of mathematics anxiety (Blazer, 2011). About 17% can have a relatively high degree of mathematics anxiety (Ashcraft & Moore, 2009). Mathematics anxiety is defined as an adverse emotional reaction to anything that involves anything mathematical whether it is in a regular situation or one in an academic setting. Trying to solve mathematical problems and dealing with numerical tasks can induce feelings of tension as well as anxiety that can affect functioning in one’s life (Nuffield Foundation, 2020). People suffering from mathematics anxiety have an irrational belief that they are unable to fulfill any task involving GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U mathematical ability. Some physiological effects of mathematics anxiety include unusual B nervousness and tension, elevated heart rate, sweaty hands, light-headedness, and distressed H A stomach (Nisbet, 2019). S Do you agree with the result you got? How does this information help you? A A N B. List at least five ways on how mathematics can be of use to you in your future career. N 1. G 2. L 3. U N 4. S O 5. D Lesson Applications N C. Fill out the following timeline with significant events in your life involving mathematics. You G may put good memories (winning a math contest/high exam score) or bad memories (failing S a math exam/ failing to answer math problem on the board) that you had of mathematics. A N P A B L O References: Ashcraft, M. H., & Moore, A. M. (2009). Mathematics Anxiety and the Affective Drop in Performance. Journal of Psychological Assessment, 197-205. Blazer, C. (2011, September). Strategies for Reducing Math Anxiety. Retrieved from ERIC Institute of Educational Sciences: https://files.eric.ed.gov/fulltext/ED536509.pdf Calingasan, R. M., Martin, M. C., & Yambao, E. M. (2018). Mathematics in the Modern World. Quezon City: C & E Publishing, Inc.,. Commission on Higher Education. (2017, October 6). Mathematics in the Modern World Preliminaries. Retrieved from CHED: https://ched.gov.ph/wp-content/uploads/2017/10/KWF- Mathematics-in-the-Modern-World.pdf GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U Freedman, E. (2017, August 31). Do You Have Math Anxiety? Retrieved from Math Power: B http://www.mathpower.com/anxtest.htm H A Hom, E. J. (2013, August 16). What is Mathematics? Retrieved from Live Science: S https://www.livescience.com/38936- A mathematics.html#:~:text=Mathematics%20is%20the%20science%20that,%2C%20engineering A %2C%20and%20even%20sports. N Lessel, M. (2016). About the Origin: Is Mathematics Discovered or Invented? Retrieved from N Lehigh University: https://preserve.lehigh.edu/cgi/viewcontent.cgi?article=1038&context=cas- G lehighreview-vol-24 Mitchener, W. G. (1996, September 3). The Nature of Mathematics. Retrieved from Duke L University Department of Mathematics: U https://services.math.duke.edu/undergraduate/Handbook96_97/node5.html N S Mometrix. (2020, August 22). Why is Math Important in Life. Retrieved from Pi Day: O https://www.piday.org/10-reasons-why-math-is-important-in- D life/#:~:text=Math%20helps%20us%20have%20better,and%20have%20better%20reasoning% 20abilities.&text=Reasoning%20is%20our%20ability%20to,problems%20and%20look%20for N %20solutions. G Nisbet, J. (2019, March 9). Overcoming Math Anxiety: 12 Evidence-Based Tips That Work. S Retrieved from Prodigy: https://www.prodigygame.com/main-en/blog/math-anxiety A Nuffield Foundation. (2020, August 17). What is Mathematics Anxiety? Retrieved from Centre N for Neuroscience in Education: https://www.cne.psychol.cam.ac.uk/math-memory/what-is- mathematics- P A anxiety#:~:text=Mathematics%20Anxiety%20is%20a%20negative,and%20academic%20situati B ons1%2C%202%20. L Sirug, W. S. (2018). Mathematics in the Modern World. Manila: Mindshapers Co., Inc. O GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd