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1780 West 49th Street, Hialeah, Florida 33012, USA Editorial Note: Polygon is MDC Hialeah's Academic Journal. It is a multi-disciplinary online publication whose purpose is to display the academic work produced by faculty and staff. We, the editorial committee of Polygon, are pleased to publish the 2017 Spring Issue Polygon which is the tenth consecutive issue of Polygon. It includes ten regular research articles, including a Foreword from the President of Our Hialeah Campus, Dr. Joaquin G. Martinez. We are pleased to present work from a diverse array of fields written by faculty from across the college. The editorial committee of Polygon is thankful to the Miami Dade College President, Dr. Eduardo J. Padrón, Miami Dade College District Board of Trustees, the Hialeah Campus President, Dr. Joaquin G. Martinez, Hialeah Campus Dean of Faculty and Student Services, Dr. Ramona Cox, Chairperson of Hialeah Campus Liberal Arts and Sciences, Dr. Caridad Castro, Chairperson of Hialeah Campus World Languages and Communication, Professor Liliana Cobas, Chairperson of Hialeah Campus Business, Engineering and Technology, Professor Juan J. Flores, Director of Hialeah Campus Administrative Services, Ms. Andrea M. Forero, Director of Hialeah Campus Network & Media Services, Mr. Juan Villegas, all staff and faculty of Hialeah Campus and Miami Dade College, in general, for their continued support and cooperation for the publication of Polygon. Sincerely, Editorial Committee of Polygon: Dr. M. Shakil (Mathematics), Dr. Jaime Bestard (Mathematics), and Professor Victor Calderin (English) Patrons: Dr. Joaquin G. Martinez, President, Hialeah Campus Dr. Ramona Cox, Dean of Faculty and Student Services Dr. Caridad Castro, Chair of Liberal Arts and Sciences Professor Liliana Cobas, Chair of World Languages and Communication Professor Juan J. Flores, Chair of Business, Engineering and Technology Ms. Andrea M. Forero, Director of Hialeah Campus Administrative Services Miami Dade College District Board of Trustees: Armando J. Bucelo, Jr., Chair Armando J. Olivera, Vice Chair Helen Aguirre Ferré Marili Cancio Benjamin León III Daniel Diaz Leyva Bernie Navarro Dr. Eduardo J. Padrón, College President Mission of Miami Dade College As democracy’s college, Miami Dade College changes lives through accessible, high-quality teaching and learning experiences. The College embraces its responsibility to serve as an economic, cultural and civic leader for the advancement of our diverse global community.


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Foreword By Dr. Joaquín G. Martínez Welcome to the tenth issue of Polygon, an interdisciplinary academic journal of Miami Dade College’s Hialeah Campus. This issue reflects a rich and vibrant campus tradition, exemplifying more than a decade of academic rigor and intellect originating from the distinguished faculty and staff members at the College. Their dedication to student success—anchored in the highest standards of scholarly inquiry—enriches the academic enterprise. As a result, generations of community college students continue to benefit from an unwavering commitment to explore ideas and transform lives through the opportunity of education. I invite you to delve into the hearts and minds of the journals' contributors, my colleagues and the driving force behind Miami Dade College's Polygon. Joaquín G. Martínez, Ph.D. President Miami Dade College, Hialeah Campus 1780 W.49th St. Rm 301 Hialeah, Fl. 33012


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CONTENTS ARTICLES / AUTHOR(S)

PAGES

The Study of Mathematics Has the Potential to Enhance Critical Thinking - Dr. Jack Alexander

1 -4

SOPHISTICATED COUNTING - Dr. Jack Alexander

5-7

MATHEMATICAL EXPERIMENTS IN THE CLASSROOM - Problems in Vibrations and Periodic Phenomena - Prof. Rene Barrientos

8- 30

CLASSROOM NOTES: BABY STEPS TOWARD THE FUNDAMENTAL THEOREM OF CALCULUS AND FIRST ORDER EQUATIONS - Prof. Rene Barrientos

31 - 36

Becoming Human: Building Bridges from Li to Logos - A comparative study of the Way toward the Good according to Confucius and Plato - Prof. Sarah Jacob

37 - 49

Students’ Statistics Research Projects in STA2023 using STATDISK - An Introduction - Dr. M. Shakil and Dr. J. Bestard

50 - 58

Using Sustainability Data to Teach MAC 2233 and MAC 2311 Courses via Maple and Mintab Software vis-a-vis Developing a Lesson Plan - Dr. M. Shakil

59 - 68

Survey of Students’ Opinion about “Service Learning - Math Mentorship Program” - A Statistical Analysis - Dr. M. Shakil

69 - 83

Survey of Students’ Attitudes towards Mathematics and Statistics - A Statistical Analysis

84 - 98

- Dr. M. Shakil THE DEMARCATION PROBLEM – Dr. Melissa Lammey Comments about Previous Issues of Polygon Previous Editions Link: https://issuu.com/mdc-polygon

Disclaimer: The views and perspectives of the authors presented in their respective articles published in Polygon do not represent those of Miami Dade College.

99 - 114 115 - 116


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Solicitation of Articles for the 2018 Issue (11th Issue) of Polygon: The editorial committee would also like to cordially invite the MDC community to submit their articles for consideration for the 2018 Issue (11th Issue) of Polygon. Guidelines for Submission POLYGON “Many Corners, Many Faces (POMM)” A premier professional refereed multi-disciplinary electronic journal of scholarly works, feature articles and papers on descriptions of Innovations at Work, higher education, and discipline related knowledge for the campus, college and service community to improve and increase information dissemination, published by MDC Hialeah Campus Liberal Arts and Sciences Department (LAS). Editorial Board: Dr. Mohammad Shakil (Mathematics) (Editor-in-Chief) Dr. Jaime Bestard (Mathematics) Prof. Victor Calderin (English) Manuscript Submission Guidelines: Welcome from the POLYGON Editorial Team: The Department of Liberal Arts and Sciences at the Miami Dade College–Hialeah Campus and the members of editorial Committee - Dr. Mohammad Shakil, Dr. Jaime Bestard, and Professor Victor Calderin – would like to welcome you and encourage your rigorous, engaging, and thoughtful submissions of scholarly works, feature articles and papers on descriptions of Innovations at Work, higher education, and discipline related knowledge for the campus, college and service community to improve and increase information dissemination. We are pleased to have the opportunity to continue the publication of the POLYGON, which will be bi-anually during the Fall & Spring terms of each academic year. We look forward to hearing from you. General articles and research manuscripts: Potential authors are invited to submit papers for the next issues of the POLYGON. All manuscripts must be submitted electronically (via e-mail) to one of the editors at mshakil@mdc.edu, or jbestard@mdc.edu, or vcalderi@mdc.edu. This system will permit the new editors to keep the submission and review process as efficient as possible. Typing: Acceptable formats for electronic submission are MSWord, and PDF. All text, including title, headings, references, quotations, figure captions, and tables, must be typed, with 1 1/2 line spacing, and one-inch margins all around. Please employ a minimum font size of 11. Please see the attached template for the preparation of the manuscripts. Length: A manuscript, including all references, tables, and figures, should not exceed 7,800 words (or at most 20 pages). Submissions grossly exceeding this limit may not be accepted for review. Authors should keep tables and figures to a minimum and include them at the end of the text. Style: For writing and editorial style, authors must follow guidelines in the Publication Manual of the American Psychological Association (5th edition, 2001). The editors request that all text pages be numbered.


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You may also please refer to the attached template for the preparation of the manuscripts. Abstract and keywords: All general and research manuscripts must include an abstract and a few keywords. Abstracts describing the essence of the manuscript must be 150 words or less. The keywords will be used by readers to search for your article after it is published. Book reviews: POLYGON accepts unsolicited reviews of current scholarly books on topics related to research, policy, or practice in higher education, Innovations at Work, and discipline related knowledge for the campus, college and service community to improve and increase information dissemination. Book reviews may be submitted to either themed or open-topic issues of the journal. Book review essays should not exceed 1,900 words. Please include, at the beginning of the text, city, state, publisher, and the year of the book’s publication. An abstract of 150 words or less and keywords are required for book review essays. Notice to Authors of Joint Works (articles with more than one author). This journal uses a transfer of copyright agreement that requires just one author (the Corresponding Author) to sign on behalf of all authors. Please identify the Corresponding Author for your work when submitting your manuscript for review. The Corresponding Author will be responsible for the following: 

ensuring that all authors are identified on the copyright agreement, and notifying the editorial office of any changes to the authorship.  securing written permission (via email) from each co-author to sign the copyright agreement on the co-author’s behalf.  warranting and indemnifying the journal owner and publisher on behalf of all coauthors. Although such instances are very rare, you should be aware that in the event a co-author has included content in their portion of the article that infringes the copyright of another or is otherwise in violation of any other warranty listed in the agreement, you will be the sole author indemnifying the publisher and the editor of the journal against such violation. Please contact the editorial office if you have any questions or if you prefer to use a copyright agreement for all coauthors to sign. Instructions for the Preparation of Manuscripts for the Polygon: (THE TITLE IS HERE) (12 pt, bold, 32 pt above) NAME IS HERE (11 pt16 pt above, 32 pt below) ABSTRACT is here, not exceeding 160 words. It must contain main facts of the work. (11 pt) Key words and phrases (11 pt) Introduction (11 pt, bold, 24 pt above, 12 pt below) Main Body of the Article


vi Discussion Conclusion Acknowledgements REFERENCES (11 pt, 30 pt above, 12 pt below) [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1970. [2] J. Galambos and I. Simonelli, Products of Random Variables – Applications to Problems of Physics and to Arithmetical Functions, CRC Press, Boca Raton / Atlanta, 2005. [3] S. Momani, Non-perturbative analytical solutions of the space- and time-fractional Burgers equations. Chaos, Solitons & Fractals, 28(4) (2006), 930-937. [4] Z. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlin. Sci. Numer. Simulat. 1(7) (2006), 15-27. (11 pt) XXXX YYYYY. Received his Master’s/Ph. D. Degree in Physics from the University of ZZZZZ (Country) in 1987 under the direction of Dr. M. N. OPQR. Since 1989, he has been at CCCC College in Hawaii, USA. His research interests focus on the Fractals, Solitons, Undergraduate Teaching of Physics, and Curriculum Development. (11 pt) Department of Liberal Arts & Sciences (Physics Program), CCCC College, P. O. Box 7777, Honolulu, Hawaii, USA. e-mail: xxyy@ccc (11 pt)


1 The Study of Mathematics Has the Potential to Enhance Critical Thinking By Dr. Jack Alexander, Ph.D. Mathematics Department Miami Dade College, North Campus

ABSTRACT Some people seem to have little trouble learning mathematics, physics, chemistry and other “hard” subjects. Others really struggle and seem to never grasp the basics of such academic areas. There have been attempts to explain away this dichotomy by suggesting that there are people with right brain dominance (math types), while others have left brain dominance (art types). This paper makes the case that both sides of the brain can be developed by a balanced approach to the study of mathematics through a critical thinking as well as a full immersion in the arts. Key Words: Estimation.

Critical Thinking, Goals, Introspection, Problem Solving, Point of View,

2010 Mathematics Subject Classifications: 97C40, 97C70, 97D40, 97D50. NARRATIVE: The “balance” suggested in the abstract can be promoted by focusing on the interconnections between mathematics and the arts. For example, musical scores and rhythms have recurring mathematical patterns. Moreover, projective geometry is often employed in sketches, drawings and paintings. Revealing these mathematical underpinnings to learners promotes a more “Critical” understanding and appreciation for the arts. Just what is critical thinking? A workable definition is: Critical Thinking enables thinkers proficient in it to better produce and assess intellectual work as well as to act more reasonably and effectively in the world of affairs and personal life. Since one of the major tasks of mathematics is to parameterize the world in which we live, understanding of definitions, functions, formulas, designs and models has the potential to enhance appreciation of the world around us. So, how does a person improve their critical thinking, and, in the process, enhance the learning of mathematics? First of all, it must be understood that each individual has already developed a certain degree or level of critical thinking. Some (the math types) have a high level, while others have a low level. A low level of critical thinking is a bad state of affairs because every part of our lives is influenced by how well we reason. This applies to making good choices at the grocery store to who we vote for. The understanding of any subject implies the ability to apply classroom learning to specific situations. Fortunately, there are ways to improve our critical thinking and thus improve the quality of our lives. This enhancement process begins with goal setting since reasoning is always done to fulfill some need. This implies that once a goal has been set, it is critical to work at separating out those things that are inconsistent with the attainment of the goal. For example, if


2 your goal is to maintain a high grade point average, it would not be wise to take too many demanding courses in the same term. Taking chemistry and physics in the same term may mean an extremely large time commitment. These courses typically have extensive laboratory hours. Goals must be consistent. If you have goals to make good grades, carry a full time job and have an extensive night life, you may find that all three cannot be accomplished at the same time. Once a goal has been set, self examination (introspection) needs to be done. It is difficult to know where you are going, if you do not know where you are at the present time. This introspection will help you determine your strengths and weaknesses. Most people who have trouble in a calculus course have weak algebra skills. Therefore, if your goal is to do well in calculus, it would be a good idea to review algebra before taking the calculus courses. Another building block in the enhancement of critical thinking is the development of positive point of view. The fact is that if you think that you cannot learn or do something, this is a self-fulfilling prophecy. In addition, your view should not be too narrow, too biased or contain contradictions. Most importantly, a healthy and positive point of view will make it more likely that honest interpretations will be made from empirical data. Courses in mathematics that have the potential to assist in the development of a confident, positive, and healthy point of view as well as improve your ability to make relevant inferences are: set theory, logic, probability and statistics.

Finally, one cannot shy away from the practice of problem solving. In mathematics, as well as other courses, being able to solve problems builds the desired understanding of the subject. Moreover, problems tend to fit into categories. This means that solving a lot of problems will help to address other new problems as they surface. The great professional basketball player, Larry Bird said “Once you have practiced enough to learn the fundamentals, you need to practice your skills to the point that you can execute them without thinking�. Below is a set of critical thinking steps that will assist in solving problems. 1. Make sure you understand what the problem is asking. It makes no sense to do correct calculations on the wrong problem. 2. Assign a variable (letter) to represent what you are looking for, and express any remaining unknown quantities in terms of this variable. 3. Make a list of all known facts. Develop a table or chart if there is a lot of information. 4. Build a model with formulas or equations. Solve the equations. 5. Check your answer for reasonability. If you determine a man to be of age 325 years old or a probability to be -2.3, you did something wrong. This last step also implies that one should work at developing critically good estimation and approximation skills, The example below illustrates how such skills can be used for ones own financial self protection.


3 EXAMPLE 1 Below is a list of grocery prices along with “rounded” estimates. Note that the sum of the estimates gives a very close approximation of the actual total. Don’t think that you will always be charged the correct amount just because the cashier uses a computer scanner. PRICE Bread Sandwich Apple Juice Eggs Cheese

$ 2.60 $ 4.25 $ 0.45 $ 3.75 $ 3.25 $ 5.50

ESTIMATE $ 3.00 $ 4.00 $ 0.00 $ 4.00 $ 3.00 $ 6,00

Totals $19.80 $ 20.00 To summarize, we should follow the advice of Dr. William Porter of Elizabeth City State University. He said: “Set your goals, but concentrate on focus”. Chris Evert, the championship tennis player concurs with this contention. She said: “Dedication implies setting goals, and pursuing them. I never allowed my self to be distracted. I needed to be single-minded and have tunnel vision.” Lastly, we are fortunate that we are living in a time of high level computational science technology. There are many interactive computer software packages available that can be used to modify and adjust models that we design to address problematic situations. To improve logic skills that are essential to critical thinking, it would also be worth while to learn at least one computer programming language; such as JAVA, C++, FORTRAN, or BASIC. Below is a “Critical Thinking” Inventory. This inventory can be employed to start a person on the way to becoming a more effective critical thinker. In addition, it may very well help to reduce the mathematics anxiety that many suffer from and thus improve the chances of success in analytic courses like physics, chemistry, biology and sciences courses in general as well as mathematics. Critical Thinking Inventory: Your success in any scientific or mathematics course is directly related to your ability to think critically. The following inventory will start you on the way to becoming a more effective and critical thinker. This exercise will help you in all intellectual endeavors. INVENTORY: 1.

What is your educational goal?

2.

List all of the scientific and mathematics courses you have taken.

3.

Which of the courses listed in number 2 did you like the most? Rate these from 1 to 10, with 1 being the most liked.

4.

Which of the courses listed in number 2 did you like the least? Rate these from 1 to 10, with 10 being the least liked.


4

5.

In what academic area are you most interested?

6.

In what academic area are you least interested?

7.

List the five professions that you feel generate the highest income..

8.

When taking any new course, list your expectations for the course.

REFERENCES Blitzer, R. 2014. Thinking Mathematically, 6th Edition, Pages 1 – 37: Person Education, Inc., Boston Garfunkel, S. 1994. For All Practical Purposes, 3rd Edition, W. H. Freeman and Company, New York Eves, H. and Newsom, C. V. 1958. An Introduction to the Foundations & Fundamental Concepts of Mathematics, Rinehart & Company, Inc., New York Watnik, S. M. 2001. The Nature of Mathematics, 9th Edition, BROOKS/COLE, Pacific Grove, CA


5 SOPHISTICATED COUNTING By Dr. Jack Alexander, Ph.D. Mathematics Department Miami Dade College, North Campus ABSTRACT In order to correctly calculate the probability of an event, one must be able to determine the size of the associated Sample Space. While in some instances this is an easy task, in other instances it can be quite complicated. Detailed here are three Sophisticated method of counting that will assist in determining sample space size for simple and complicated probabilistic events. The three methods are: The Fundamental Counting Principle (F.C.P.), Combinations, and Permutations. KEYWORDS:

Sample Space, Probability, Factorials, Counting, Combinations, Permutations.

2010 Mathematics Subject Classifications: 62 – 07, 97C40, 97C70, 97D40, 97D50. DEFINITION: The Fundamental Counting Principle (F.C.P.) If you can choose one item from a group of M items and a second item from a group of N items, then the total number of two-item choices is M x N. Example 1: A Restaurant offers 7 appetizers and 13 main courses. In how many ways can a person order a two-course meal? Answer:

The total number of two-course meals is: 7 x 13 = 91.

Example 2: F.C.P. is not restricted to just two items. For example, if you own 20 pairs of jeans, 25 T-shirts, and 10 pairs of sneakers, you have: 20 x 25 x 10 = 5000 choices for your wardrobe. Example 3: A Social Security number has nine digits. Each digit can be any number from 0 to 9 inclusive. If a thief is found using your social security number and claims that he guessed the number. What is the probability that he is telling the truth? Answer: The Fundamental Counting Principle would yield 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 109 = 1,000,000,000. Therefore the probability that the thief is telling the truth is 1 chance in one billion. DEFINITION: Factorial Notation (n!) If n is a positive integer, the notation n! (read “n factorial”) is the product of all positive integers from n down through 1. That is: n! = n(n – 1)(n – 2) ٠٠٠(3)(2)(1). 0! (zero factorial) is defined to be 1. (0! = 1).


6 This is still the Fundamental Counting Principle. However, you lose a choice as you continue multiplying down to 1. Factorials become large very quickly. For example, 4! = 4 x 3 x 3 x 2 x 1 = 24. However, 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720. And furthermore, 10!

= 3,628,000.

Here is an interesting probability example. Suppose Martha, Lee, Nancy, Paul and Armando have all been invited to a dinner party. They arrive randomly and each person arrives at a different time. Find the probability that Martha arrives first and Armando last. Answer: Since there are five people, there are 5! ways they could arrive (5! = 5 x 4 x 3 x 2 x 1 = 120). If Martha arrives first and Armando arrives last, there are 3! ways the other three people could arrive. 3! =3 x 2 x 1 = 6. Hence, the desired probability is given by 6/120 = .05. DEFINITION: Combinations: The number of combination of n things taken r at a time is given by the formula: nCr = n!/((n – r)!r!) Example 4: A three-person committee is to be formed from a club with eight members. How many possible committees could be formed? Answer: From the above formula, we can calculate: 8C3 = 8!/((8 – 3)!3!) = 8!/(5!3!) = (8(7)(6)(5)(4)(3)(2)(1))/((5)(4)(3)(2)(1)(3)(2)(1)) = (8(7)(6))/(3(2)(1)) = (4)(7)(2) = 56. A more interesting example is given by an analysis of how lotteries work. To win the Florida lottery, one must correctly select 6 numbers from 1 to 53 inclusive. What is the probability of winning? Answer: Since the winning number can be given in any order, any “combination” of the correct 6 numbers will win the jackpot. Using the combination formula given above indicates that the number of possible number of possibilities is 53C6 = 22,957,480. Which, of course means that the chances of winning is 1 chance in 22,957.480. While this clearly suggest that people are fool hearty to participate in a situation with such a ridiculously low probability of winning, the “Power Ball” lottery is even worse. Winning the jackpot requires that a person select the correct five numbers between 1and 59 and, in a separate drawing, must also select the correct number between 1 and 39. To calculate the probability of winning, we first need to calculate the combination of 59 thing taken 5 a time. (59C5 = 5,006,386). This result must then be multiplied by 39. This gives 195,249,054. In other words, the probability of winning power ball is 1 chance in 195,249,054. In the view of this writer, this is what I call “Sophisticated Steeling”.

Permutations: The number of permutations of n things taken r at a time is given by the formula: nPr = n!/(n – r)! The basic difference between combinations and permutations is that permutations count all the arrangements as well as the number of combinations. In the example above about the


7 Florida lottery, we have a combination situation. They do not require that the winning number be in a prescribed order. If this was required, this would be a permutation situation. The number of combinations (22,957,480) would have to be multiplied by 6! = 720. Thus, the probability of winning would be I chance in 16.53 billion. That probability of winning is so small that it would be unlikely than anyone would ever win. This would be so discouraging that people would eventually stop playing. Example 5: The club from Example 4 that has eight members needs to elect a president, a vice-president and a secretary. In how many ways can this be done? Answer:

From the above formula, we can calculate: 8P3 = 8!/(8 – 3)! = (8(7)(6)(5)(4)(3)(2)(1))/(5(4)(3)(2)(1) = 8(7)(6) = 336.

Note: The permutations of 8 things 3 at a time (336) is much larger than the combinations of 8 things 3 at a time (56). CONCLUSION: Centuries ago mathematical thinker discovered that the intuition could not be trusted to evaluate and estimate situations where “large” numbers are concerned. Thus, such thinkers saw the need to develop techniques, strategies and formulas that could determine “exactly” how large some seemingly simple situation could become. For example, even if a person is told that there are 9 digits in a social security number, few would believe that there are 1 billion possible such numbers. Moreover, those hundreds of people who play the lottery have no idea that the chances of winning when you must get the correct combination of 6 numbers from 1 to 53 is as large as 22,957,480. The illustrations of examples using the Fundamental Counting Principle (F.C.P.), Combinations and Permutations in this paper hopefully demonstrate that these techniques and formulas allow us to bring reality to situations that would otherwise be outside of our reach.

REFERENCES Blitzer, R. 2011. Thinking Mathematically, 6th Edition, Pages 688 – 706; Person Education, Inc., Boston Mendenhall, W., Reinmultb, J. E. and Beaver, R. J. 1989. Statistics for Management and Economics, 7th Edition, Pages 119 – 124; PWS-KENT Publishing Company, Belmont California Sobecki, D. and Bluman, A. G. 2015. Math in Our World, 3rd Edition, Pages 622 – 634; McGrawHill, New York Triola, M. F. 2018. Elementary Statistics, 13th Edition, Pages 169 – 174; Person Education, Inc., Boston


8 MATHEMATICAL EXPERIMENTS IN THE CLASSROOM Problems in Vibrations and Periodic Phenomena Prof. Rene Barrientos Department of Mathematics, Miami Dade College | Hialeah Campus FL 33012 rbarrien@mdc.edu Abstract: A first course in differential equations provides fertile ground in which to link mathematics and the physical sciences while avoiding the trap of a “cookbook� approach to the subject. These classroom notes were written for a 9-week summer course taught at the Kendall Campus of Miami Dade College. During summer terms, the class time is longer and therefore one can engage in experimental activities and demonstrations in differential equations. These notes were used in the summer of 2014. Key Words: linear spring, periodic motion, mathematical modeling, differential equations, small angle approximation, damping, energy methods AMS Subject Classification: 02 Introduction In a typical introductory course to differential equations, students learn about vibrations by studying systems as the ones shown below: Spring-mass system

Shock absorber system

đ?‘š

đ?‘˜

P

R

Simple Pendulum

đ?œƒ

Electric circuit L

đ?‘š E

F

(a)

(b)

đ?‘Ł

đ?‘šđ?‘”

(c)

C

(d)

Figure 1 Although these are very different systems, they are members of the same family because they obey the same second order linear differential equation đ?‘Ž2 đ?‘Œ ′′ + đ?‘Ž1 đ?‘Œ ′ + đ?‘Ž0 đ?‘Œ = đ?‘“(đ?‘Ą); đ?‘Œ(đ?‘Ą0 ) = đ?‘Ś0 , đ?‘Œ ′ (đ?‘Ą0 ) = đ?‘Ś1 The typical introductory problem involves a mass-spring system (as the ones shown in Figures 1(a) or 1(b) above), where it is assumed that the coil1 obeys Hooke’s Law đ??šđ?‘ = −đ?‘˜âˆ†đ?‘™ where ∆đ?‘™ denotes the amount by which the coil has been compressed of stretched from its rest length and the negative sign is used to reminds us that the force exerted by coil always acts to restore it to its rest length. The spring constant đ?‘˜ varies from one coil to another and it is a measure of its stiffness.

1

“Coil� and “spring� will hereinafter mean the same thing.


9 Thus, If â&#x2C6;&#x2020;đ?&#x2018;&#x2122; > 0 then the coil is stretched and đ?&#x2018;­ will act to compress it, and If â&#x2C6;&#x2020;đ?&#x2018;&#x2122; < 0, then the coil is compressed and the force will act in opposition to restore it to its natural length. đ?&#x2018;&#x2122;0 đ?&#x2018;­đ?&#x2019;&#x201D;

đ?&#x2018;&#x2122;0

â&#x2C6;&#x2020;đ?&#x2018;&#x2122;

We will assume that our spring obeys Hookeâ&#x20AC;&#x2122;s Law. Spring-mass Systems A spring-mass system consists of a block of mass đ?&#x2018;&#x161; attached to a secured spring (coil) and free to move either vertically or horizontally, as shown below. The differential equation governing either system is the same so either one serves the purpose of illustrating the effect of a restoring force on the motion of the block. Vertical spring-mass system

Horizontal spring-mass system

In practice, unless one has access to a frictionless track, the vertical set up works better in the classroom provided that one is careful when initially displacing the block; the earthâ&#x20AC;&#x2122;s rotation will eventually disturb the blockâ&#x20AC;&#x2122;s periodic motion, but for a reasonable amount of time this motion will be fairly stable. The forces acting on the block are identified in the figure below (friction and drag are not shown): đ?&#x2018;ľ đ?&#x2018;­đ?&#x2019;&#x201D; đ?&#x2018;­đ?&#x2019;&#x201D; đ?&#x2018;&#x161;đ?&#x2019;&#x2C6; đ?&#x2018;&#x161;đ?&#x2019;&#x2C6;

The free-body force diagram on the left show that the gravitational force đ?&#x2018;&#x161;đ?&#x2019;&#x2C6; does not come into play for the horizontal system because the normal force đ?&#x2018;ľ cancels it out. Therefore, the only forces that accelerate the block are the restoring force đ?&#x2018;­đ?&#x2019;&#x201D; due to the spring, frictional and drag forces, and perhaps other external agents. On the other hand, the vertical block is accelerated by the combined effect of the force of gravity and the restoring force. At first glance it is not intuitively clear that the same differential equation should govern both the vertical and the horizontal systems. After all, gravity plays an active role in the vertical system


10 whereas it is cancelled by the normal force đ?&#x2018;ľ (see figure) exerted by the table on the block in the horizontal one. As we will find out, these two systems are mathematically identical. It is instructive for the student to go over the details of the derivation and understand where the equations come from in both the vertical and horizontal set up. Friction Friction and drag are the other two important forces that act on moving objects. We make the simplifying assumption that these resistive forces can be represented by a single force đ?&#x2018;Ť such that đ?&#x2018;Ť = â&#x2C6;&#x2019;đ?&#x2018;?đ?&#x2019;&#x2014; where đ?&#x2019;&#x2014; is the velocity of the block and đ?&#x2018;? a positive constant. Working with these models also gives the instructor the opportunity to stress the importance of physical units and make connections with what students are seeing in their science courses. In the SI system, we use the kilogram (Kg) for mass, the meter (đ?&#x2019;&#x17D;) for distance, and the second (sec) for time. Near the earthâ&#x20AC;&#x2122;s surface, objects are subjected to gravityâ&#x20AC;&#x2122;s force which causes them to accelerate with an acceleration of approximate magnitude 9.8 m/sec2. Sometimes we will use 10 instead of 9.8 in order to simplify calculations in which precision is not essential. In the English system we use the â&#x20AC;&#x153;slugâ&#x20AC;? for mass, the foot (ft) for distance, and the second (sec) for time. The acceleration due to Earthâ&#x20AC;&#x2122;s gravitational force in these units is 32 ft/sec2. It is important for students to understand the difference between weight and mass: the â&#x20AC;&#x153;weightâ&#x20AC;? of an object is the product đ?&#x2018;&#x161;đ?&#x2018;&#x201D;. Thus, an object weighing 120 lb. has a mass of

120 32

= 3.75 slug

Free Undamped Motion A block of mass đ?&#x2018;&#x161; is attached to a spring2 with spring constant đ?&#x2018;&#x2DC;, as shown below. Assume that friction and all other dissipative forces can be neglected. If the block is displaced from equilibrium and set it in motion with some initial velocity đ?&#x2019;&#x2014;đ?&#x;&#x17D; [if the object is simply released, we take đ?&#x2019;&#x2014;0 = đ?&#x;&#x17D;]. Then it will oscillate periodically to and fro with some frequency đ?&#x2018;&#x201C;. We are interested in describing this periodic motion; that is, to determine the objectâ&#x20AC;&#x2122;s position function, frequency of oscillation, and other dynamical variables. đ?&#x2018;ľ đ?&#x2018;­đ?&#x2019;&#x201D; đ?&#x2018;Ľ đ?&#x2018;Ľ=0

đ?&#x2019;&#x160;Ě&#x201A;

The figure shows the situation when the object is to the right of its equilibrium position đ?&#x2018;Ľ = 0. We take this to be the positive direction. đ?&#x2018;&#x2013;Ě&#x201A; is the unit vector in the x-direction.

đ?&#x2018;Ľ đ?&#x2018;&#x161;đ?&#x2019;&#x2C6;

Identifying the Forces and Applying Newtonâ&#x20AC;&#x2122;s laws The gravitational force đ?&#x2018;&#x161;đ?&#x2019;&#x2C6; and the normal contact force đ?&#x2018;ľ cancel out since the block is constrained to move horizontally only. We apply Newtonâ&#x20AC;&#x2122;s Second Law of Motion: 2

This is a spring that has both the capacity to be stretched and compressed, i.e., a coil.


11 đ?&#x2018;&#x161;đ?&#x2019;&#x201A; = đ?&#x2018;&#x2020;đ?&#x2018;˘đ?&#x2018;&#x161; đ?&#x2018;&#x153;đ?&#x2018;&#x201C; đ?&#x2018;&#x17D;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122; đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x;đ?&#x2018;?đ?&#x2018;&#x2019;đ?&#x2018; Since the only force acting on the block along the direction of motion is the force due to the spring, we have đ?&#x2018;&#x161;đ?&#x2019;&#x201A; = đ?&#x2018;­đ?&#x2019;&#x201D; Letting đ??ą(đ?&#x2018;Ą) = đ?&#x2018;Ľ(đ?&#x2018;Ą)đ?&#x2019;&#x160; be the position vector, the previous equation says đ?&#x2018;&#x161;đ??ąâ&#x20AC;˛â&#x20AC;˛(đ?&#x2018;Ą) = đ?&#x2018;­đ?&#x2019;&#x201D; The spring force is given by đ?&#x2018;­đ?&#x2019;&#x201D; = â&#x2C6;&#x2019;đ?&#x2018;&#x2DC;đ?&#x2018;Ľ(đ?&#x2018;Ą)đ?&#x2019;&#x160;Ě&#x201A;. Thus, đ?&#x2018;&#x161;

đ?&#x2018;&#x2018;2 đ?&#x2018;Ľ đ?&#x2019;&#x160;Ě&#x201A; = â&#x2C6;&#x2019;đ?&#x2018;&#x2DC;đ?&#x2018;Ľ(đ?&#x2018;Ą)đ?&#x2019;&#x160;Ě&#x201A; đ?&#x2018;&#x2018;đ?&#x2018;Ą 2

or đ?&#x2018;&#x161;

đ?&#x2018;&#x2018;2 đ?&#x2018;Ľ = â&#x2C6;&#x2019;đ?&#x2018;&#x2DC;đ?&#x2018;Ľ đ?&#x2018;&#x2018;đ?&#x2018;Ą 2

and we have the following linear second order differential equation: đ?&#x2018;&#x2018;2đ?&#x2018;Ľ đ?&#x2018;&#x2DC; + đ?&#x2018;Ľ=0 đ?&#x2018;&#x2018;đ?&#x2018;Ą 2 đ?&#x2018;&#x161;

(1)

Equation (1) has characteristic polynomial đ?&#x2018;?(đ?&#x153;&#x2020;) = đ?&#x153;&#x2020;2 + đ?&#x2018;&#x2DC;/đ?&#x2018;&#x161; whose roots are đ?&#x153;&#x2020; = đ?&#x2018;&#x2013;â&#x2C6;&#x161;đ?&#x2018;&#x2DC;/đ?&#x2018;&#x161; and đ?&#x153;&#x2020; = â&#x2C6;&#x2019;đ?&#x2018;&#x2013;â&#x2C6;&#x161;đ?&#x2018;&#x2DC;/đ?&#x2018;&#x161; Thus, Using Eulerâ&#x20AC;&#x2122;s identity,

đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ?&#x2018;?1 đ?&#x2018;&#x2019; đ?&#x2018;&#x2013;â&#x2C6;&#x161;đ?&#x2018;&#x2DC;/đ?&#x2018;&#x161;đ?&#x2018;Ą + đ?&#x2018;?2 đ?&#x2018;&#x2019; â&#x2C6;&#x2019;đ?&#x2018;&#x2013;â&#x2C6;&#x161;đ?&#x2018;&#x2DC;/đ?&#x2018;&#x161;đ?&#x2018;Ą đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ??´ cos â&#x2C6;&#x161;đ?&#x2018;&#x2DC;/đ?&#x2018;&#x161; đ?&#x2018;Ą + đ??ľ sin â&#x2C6;&#x161;đ?&#x2018;&#x2DC;/đ?&#x2018;&#x161; đ?&#x2018;Ą

The units of â&#x2C6;&#x161;đ?&#x2018;&#x2DC;/đ?&#x2018;&#x161; are the radian per second â&#x20AC;&#x201C; the same as those of angular velocity. For that reason, we call this quantity the angular (or circular) frequency and also denote it by đ?&#x153;&#x201D;: đ??&#x17D; = â&#x2C6;&#x161;đ?&#x2019;&#x152;/đ?&#x2019;&#x17D; đ?&#x2019;&#x2122;(đ?&#x2019;&#x2022;) = đ?&#x2018;¨ đ??&#x153;đ??¨đ??Ź đ??&#x17D;đ?&#x2019;&#x2022; + đ?&#x2018;Š đ??Źđ??˘đ??§ đ??&#x17D;đ?&#x2019;&#x2022;

(2)

The relation đ?&#x2018;&#x201C; = đ?&#x153;&#x201D;/2đ?&#x153;&#x2039; allows us to obtain the frequency of oscillation in cycles per second. Its reciprocal đ?&#x2018;&#x2021; = 1/đ?&#x2018;&#x201C; gives us the period of oscillation. Equation (2) is the standard solution of equation (1) and it is a reasonable answer: we expect the motion to be periodic when there are no frictional losses. The constants đ??´ and đ??ľ are obtained from the initial conditions đ?&#x2018;Ľ(0) = đ?&#x2018;Ľ0 , đ?&#x2018;Ł(0) = 0 at đ?&#x2018;Ą = 0. For example, we may be given that:


12 1) The object is displaced to the right (đ?&#x2018;Ľ0 > 0) and released. orâ&#x20AC;Ś 2) The object is pulled to the left (đ?&#x2018;Ľ0 < 0) and given an additional impulse to the right (đ?&#x2018;Ł0 > 0). It is important for the student to develop a qualitative understanding of what the solutions of this equation will look like given the initial condition. For example, if the block starts to the right of equilibrium (đ?&#x2018;Ľ0 > 0) and is given impulse toward the right (đ?&#x2018;Ł0 > 0), then the student should expect a solution whose graph looks like this: đ?&#x2018;Ł0 > 0

đ?&#x2018;Ľ(đ?&#x2018;Ą) đ?&#x2018;Ľ(đ?&#x2018;Ą) > 0 (right of equilibrium)

1.0

đ?&#x2018;Ą

0.5

đ?&#x2018;Ľ(đ?&#x2018;Ą) < 0 (left of equilibrium)

1

2

3

4

5

6

0.5

1.0

It is also important to stress that the actual motion of the block is in one dimension. The graph of đ?&#x2018;Ľ(đ?&#x2018;Ą) merely illustrates how the blockâ&#x20AC;&#x2122;s position changes with respect to time. Block Hanging Vertically If the block hangs vertically, the general analysis is as follows: set up the spring-mass system as shown below. After the object attains its equilibrium position (and the spring has been stretched by an amount Î&#x201D;đ?&#x2018;&#x2122;), we disturb it, say by pulling it down and releasing it. We want to describe the up-and-down motion that ensues by means of its position function đ?&#x2018;Ś(đ?&#x2018;Ą) as we did in the horizontal case. đ?&#x2018;Ą =0 đ?&#x2019;&#x160;Ě&#x201A;

đ?&#x2018;Ą >0

đ?&#x2019;?đ?&#x;&#x17D; Î&#x201D;đ?&#x2018;&#x2122;

Fs

â&#x2C6;&#x2019;đ?&#x2018;&#x2DC;Î&#x201D;đ?&#x2018;&#x2122;đ?&#x2019;&#x160;Ě&#x201A;

đ?&#x2018;Ś = đ?&#x2018;Ś(đ?&#x2018;Ą)

đ?&#x2018;&#x161;đ?&#x2018;&#x201D;đ?&#x2019;&#x160;Ě&#x201A;

+ đ?&#x2018;Ś direction

đ?&#x2018;Ś=0

đ?&#x2018;&#x161;đ?&#x2018;&#x201D;đ?&#x2019;&#x160;Ě&#x201A;

It is important for students to appreciate the value of an appropriate choice of coordinate system. Although the motion will be the same, the equations that are obtained with a wrong choice of coordinate systems can be unnecessarily complicated. Taking the â&#x20AC;&#x153;positive đ?&#x2018;Ś directionâ&#x20AC;? as indicated in the figure, the force of gravity will have a positive component. Applying Newtonâ&#x20AC;&#x2122;s Second Law: = đ?&#x2018;&#x161;đ?&#x2018;&#x201D;đ?&#x2019;&#x160;Ě&#x201A; + đ?&#x2018;­đ?&#x2019;&#x201D;

đ?&#x2018;&#x161;đ?&#x2019;&#x201A;

(3)

When the block is located at some position đ?&#x2018;Ś relative to equilibrium, the springâ&#x20AC;&#x2122;s restoring force is given by đ?&#x2018;­đ?&#x2019;&#x201D; = â&#x2C6;&#x2019;đ?&#x2018;&#x2DC;(đ?&#x2018;Ś + Î&#x201D;đ?&#x2018;&#x2122;)đ?&#x2019;&#x160;Ě&#x201A;, thus:


13 đ?&#x2018;&#x161;đ?&#x2018;Śâ&#x20AC;˛â&#x20AC;˛(đ?&#x2018;Ą)đ?&#x2019;&#x160;Ě&#x201A; = đ?&#x2018;&#x161;đ?&#x2018;&#x201D;đ?&#x2019;&#x160;Ě&#x201A; â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;(đ?&#x2018;Ś + Î&#x201D;đ?&#x2018;&#x2122;)đ?&#x2019;&#x160;Ě&#x201A; or đ?&#x2018;&#x161;đ?&#x2018;Śâ&#x20AC;˛â&#x20AC;˛ = đ?&#x2018;&#x161;đ?&#x2018;&#x201D; â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;(y + Î&#x201D;đ?&#x2018;&#x2122;) = đ?&#x2018;&#x161;đ?&#x2018;&#x201D; â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;Î&#x201D;đ?&#x2018;&#x2122; â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;đ?&#x2018;Ś But đ?&#x2018;&#x2DC;Î&#x201D;đ?&#x2018;&#x2122; = đ?&#x2018;&#x161;đ?&#x2018;&#x201D; [because when the object just hangs vertically, the spring force and the force due to gravity cancel]. Therefore, đ?&#x2018;&#x161;

đ?&#x2018;&#x2018;2 đ?&#x2018;Ś = â&#x2C6;&#x2019;đ?&#x2018;&#x2DC;đ?&#x2018;Ś đ?&#x2018;&#x2018;đ?&#x2018;Ą 2

from which we obtain the equation

đ?&#x2018;&#x161;

đ?&#x2018;&#x2018;2đ?&#x2018;Ś + đ?&#x2018;&#x2DC;đ?&#x2018;Ś = 0 đ?&#x2018;&#x2018;đ?&#x2018;Ą 2

đ?&#x2018;Ś(đ?&#x2018;Ą0 ) = đ?&#x2018;Ś0

đ?&#x2018;Śâ&#x20AC;˛(đ?&#x2018;Ą0 ) = đ?&#x2018;Ł0

(4)

where đ?&#x2018;Ś represents the displacement from equilibrium position. The equation is, of course, identical to the one obtained for the block on a horizontal plane. Free Damped Motion To illustrate the effects of frictional forces, we introduce a retarding force proportional to the speed with which the object moves: đ?&#x2018;Ť = â&#x2C6;&#x2019;đ?&#x2018;?đ?&#x2019;&#x2014;, đ?&#x2018;? > 0 . Then equation (3) becomes = đ?&#x2018;&#x161;đ?&#x2018;&#x201D;đ?&#x2019;&#x160;Ě&#x201A; + đ?&#x2018;­đ?&#x2019;&#x201D; + đ?&#x2018;Ť

đ?&#x2018;&#x161;đ?&#x2019;&#x201A;

(5)

The corresponding scalar equation in the variable đ?&#x2018;Ś is đ?&#x2018;&#x2018;đ?&#x2018;Ś đ?&#x2018;&#x2018;đ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;Ś = đ?&#x2018;&#x161;đ?&#x2018;&#x201D; â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;â&#x2C6;&#x2020;đ?&#x2018;&#x2122; â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;đ?&#x2018;Ś â&#x2C6;&#x2019; đ?&#x2018;? đ?&#x2018;&#x2018;đ?&#x2018;Ą

đ?&#x2018;&#x161;đ?&#x2018;Śâ&#x20AC;˛â&#x20AC;˛ = đ?&#x2018;&#x161;đ?&#x2018;&#x201D; â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;(â&#x2C6;&#x2020;đ?&#x2018;&#x2122; + đ?&#x2018;Ś) â&#x2C6;&#x2019; đ?&#x2018;?

But again đ?&#x2018;&#x161;đ?&#x2018;&#x201D; = đ?&#x2018;&#x2DC;â&#x2C6;&#x2020;đ?&#x2018;&#x2122;. Therefore,

đ?&#x2018;&#x161;

đ?&#x2018;&#x2018;2 đ?&#x2018;Ś đ?&#x2018;&#x2018;đ?&#x2018;Ś = â&#x2C6;&#x2019;đ?&#x2018;&#x2DC;đ?&#x2018;Ś â&#x2C6;&#x2019; đ?&#x2018;? 2 đ?&#x2018;&#x2018;đ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;Ą

Hence, when we include a damping term, the equation is đ?&#x2018;&#x161;đ?&#x2018;Śâ&#x20AC;˛â&#x20AC;˛ + đ?&#x2018;?đ?&#x2018;Śâ&#x20AC;˛ + đ?&#x2018;&#x2DC;đ?&#x2018;Ś = 0

đ?&#x2018;Ś(đ?&#x2018;Ą0 ) = đ?&#x2018;Ľ0 đ?&#x2018;Śâ&#x20AC;˛(đ?&#x2018;Ą0 ) = đ?&#x2018;Ł0

For the block set up on the horizontal plane, the equation is3 = đ?&#x2018;Ł0

3

đ?&#x2018;&#x161;đ?&#x2018;ĽĚ&#x2C6; + đ?&#x2018;?đ?&#x2018;ĽĚ&#x2021; + đ?&#x2018;&#x2DC;đ?&#x2018;Ľ = 0

đ?&#x2018;Ľ(0) = đ?&#x2018;Ľ0 , đ?&#x2018;Ł(0)

It is important for students to get exposed to the various notations used for the derivative: đ?&#x2018;ĽĚ&#x2021; = đ?&#x2018;&#x2018;đ?&#x2018;Ľ/đ?&#x2018;&#x2018;đ?&#x2018;Ą.

(6)


14 The discriminant đ?&#x2018;? 2 â&#x2C6;&#x2019; 4đ?&#x2018;&#x161;đ?&#x2018;&#x2DC; determines the type of solutions the initial value problem will have. We summarize the possible scenarios in the table below. Discriminant đ?&#x2018;Ť

Roots of đ?&#x2019;&#x2018;(đ??&#x20AC;)

Solutions

Profile

â&#x2C6;&#x161;â&#x2C6;&#x2019;đ?&#x153;&#x2030; đ?&#x2018;Ą) 2đ?&#x2018;&#x161; â&#x2C6;&#x161;â&#x2C6;&#x2019;đ?&#x153;&#x2030; + đ?&#x2018;?2 sin ( đ?&#x2018;Ą)] 2đ?&#x2018;&#x161;

đ?&#x2018;?

đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ?&#x2018;&#x2019; â&#x2C6;&#x2019;2đ?&#x2018;&#x161;đ?&#x2018;Ą [đ?&#x2018;?1 cos (

2

đ?&#x2018;? â&#x2C6;&#x2019; 4đ?&#x2018;&#x161;đ?&#x2018;&#x2DC; <0

complex

đ?&#x2018;? 2 â&#x2C6;&#x2019; 4đ?&#x2018;&#x161;đ?&#x2018;&#x2DC; =0

Double root

đ?&#x2018;? 2 â&#x2C6;&#x2019; 4đ?&#x2018;&#x161;đ?&#x2018;&#x2DC; >0

Real and distinct

đ?&#x2018;?

Critically damped

đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ?&#x2018;&#x2019; â&#x2C6;&#x2019;2đ?&#x2018;&#x161;đ?&#x2018;Ą (đ?&#x2018;?1 + đ?&#x2018;?2 đ?&#x2018;Ą) đ?&#x2018;?

đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ?&#x2018;&#x2019; â&#x2C6;&#x2019; 2đ?&#x2018;&#x161;đ?&#x2018;Ą (đ?&#x2018;?1 đ?&#x2018;&#x2019;

(

1 2đ?&#x2018;&#x161;â&#x2C6;&#x161;đ?&#x153;&#x2030;)đ?&#x2018;Ą

+ đ?&#x2018;?2 đ?&#x2018;&#x2019;

(â&#x2C6;&#x2019;

Underdamped

1 2đ?&#x2018;&#x161;â&#x2C6;&#x161;đ?&#x153;&#x2030;)đ?&#x2018;Ą )

Overdamped

The last column refers to the type of motion that will ensue when the system is disturbed. When đ??ˇ < 0, there will be oscillations but with a decreasing amplitude (characterized by the factor đ?&#x2018;?

đ?&#x2018;&#x2019; â&#x2C6;&#x2019;2đ?&#x2018;&#x161;đ?&#x2018;Ą ). When đ??ˇ = 0, the system does not oscillate and comes to rest fairly quickly. Similarly, if đ??ˇ > 0 the damping is so severe that the system never returns to equilibrium, but approaches it indefinitely. It is a good exercise for students to show that when đ??ˇ > 0, the block remains on the same side of equilibrium for all đ?&#x2018;Ą > 0 regardless of the initial conditions. As an example, suppose a 12 lb. block is free to move in a medium in which it experiences a drag force đ?&#x2018;Ť = â&#x2C6;&#x2019;đ?&#x2018;?đ?&#x2019;&#x2014; where đ?&#x2018;? = 3 and it is put into motion from equilibrium by a blow that gives it an initial velocity of 1.5 ft/sec in the downward direction. Then, 3 8

đ?&#x2018;&#x2DC; = 24 đ?&#x2018;&#x161; = and đ?&#x2018;? = 3. Using (6) we have 3 đ?&#x2018;Śâ&#x20AC;˛â&#x20AC;˛ + 3đ?&#x2018;Śâ&#x20AC;˛ + 24đ?&#x2018;Ś = 0; đ?&#x2018;Ś(0) = 0 đ?&#x2018;Śâ&#x20AC;˛(0) = 1.5 8 The auxiliary equation is 3 2 đ?&#x153;&#x2020; + 3đ?&#x153;&#x2020; + 24 = 0 8 Simplifying,

đ?&#x153;&#x2020;2 + 8đ?&#x153;&#x2020; + 64 = 0

The roots are đ?&#x153;&#x2020;1 = â&#x2C6;&#x2019;4 + 4â&#x2C6;&#x161;3đ?&#x2018;&#x2013; and đ?&#x153;&#x2020;2 = â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 4â&#x2C6;&#x161;3đ?&#x2018;&#x2013;. Accordingly, the system is underdamped and has a general solution đ?&#x2018;Ś(đ?&#x2018;Ą) = đ?&#x2018;&#x2019; â&#x2C6;&#x2019;4đ?&#x2018;Ą (đ??´ cos 4â&#x2C6;&#x161;3đ?&#x2018;Ą + đ??ľ sin 4â&#x2C6;&#x161;3đ?&#x2018;Ą) Applying the first initial condition: đ?&#x2018;Ś(0) = 0 â&#x;š 1 â&#x2C6;&#x2122; (đ??´ â&#x2C6;&#x2122; 1 + đ??ľ â&#x2C6;&#x2122; 0) = 0 Thus, đ?&#x2018;¨ = đ?&#x;&#x17D; and the solution takes the form đ?&#x2018;Ś(đ?&#x2018;Ą) = đ??ľđ?&#x2018;&#x2019; â&#x2C6;&#x2019;4đ?&#x2018;Ą â&#x2C6;&#x2122; sin 4â&#x2C6;&#x161;3đ?&#x2018;Ą


15 To apply the second condition we first compute the derivative of đ?&#x2018;Ś(đ?&#x2018;Ą): đ?&#x2018;Śâ&#x20AC;˛(đ?&#x2018;Ą) = đ??ľ(â&#x2C6;&#x2019;4đ?&#x2018;&#x2019; â&#x2C6;&#x2019;4đ?&#x2018;Ą sin 4â&#x2C6;&#x161;3đ?&#x2018;Ą +đ?&#x2018;&#x2019; â&#x2C6;&#x2019;4đ?&#x2018;Ą 4â&#x2C6;&#x161;3 cos 4â&#x2C6;&#x161;3đ?&#x2018;Ą) đ?&#x2018;Śâ&#x20AC;˛(0) = 1.5 â&#x;š đ??ľ(4â&#x2C6;&#x161;3) = đ?&#x;&#x2018;

Thus, đ?&#x2018;Š = đ?&#x;&#x2013;â&#x2C6;&#x161;đ?&#x;&#x2018; =

3 2

â&#x2C6;&#x161;đ?&#x;&#x2018; â&#x2C6;&#x2122; đ?&#x;&#x2013;

The position function is therefore: â&#x2C6;&#x161;3 â&#x2C6;&#x2019;4đ?&#x2018;Ą đ?&#x2018;&#x2019; â&#x2C6;&#x2122; sin 4â&#x2C6;&#x161;3đ?&#x2018;Ą 8 When does the block return to equilibrium? The zeros of this function are given by đ?&#x2018;Ś(đ?&#x2018;Ą) =

sin 4â&#x2C6;&#x161;3đ?&#x2018;Ą = 0 This equation is satisfied when 4â&#x2C6;&#x161;3đ?&#x2018;Ą = đ?&#x2018;&#x203A;đ?&#x153;&#x2039;, đ?&#x2018;&#x203A; = 0,1,2, â&#x20AC;Ś Solving for t, đ?&#x2019;&#x2022;=

đ?&#x2019;?đ??&#x2026; đ?&#x;&#x2019;â&#x2C6;&#x161;đ?&#x;&#x2018;

, đ?&#x2019;? = đ?&#x;&#x17D;, đ?&#x;?, đ?&#x;?, â&#x20AC;Ś

A full cycle occurs when the block goes to maximum extension, then maximum compression, and finally back to the starting point. The object starts at the origin (đ?&#x2018;Ą = 0), đ??&#x2026; moves down to its max extension and returns to the origin when đ?&#x2019;&#x2022; = â&#x2C6;&#x2122; Then it goes to maximum compression and returns to the origin again when đ?&#x2019;&#x2022; = that â&#x20AC;&#x153;the period â&#x20AC;? is4 đ??&#x2026; đ?&#x2018;&#x2021;= â&#x2030;&#x2C6; đ?&#x;&#x17D;. đ?&#x;&#x2014;đ?&#x;? đ?&#x2019;&#x201D;đ?&#x2019;&#x2020;đ?&#x2019;&#x201E; đ?&#x;?â&#x2C6;&#x161;đ?&#x;&#x2018; 0.10

đ?&#x;?đ??&#x2026; đ?&#x;&#x2019;â&#x2C6;&#x161;đ?&#x;&#x2018;

đ?&#x;&#x2019;â&#x2C6;&#x161;đ?&#x;&#x2018;

đ??&#x2026;

= đ?&#x;?â&#x2C6;&#x161;đ?&#x;&#x2018; . We can say

đ?&#x2018;Ľ(đ?&#x2018;Ą)

0.05

0.5

0.05

1.0

1.5

2.0

2.5

3.0

t

T

0.10

As another example, if a 16 lb. block is suspended from a spring with spring constant đ?&#x2018;&#x2DC; = 2 lb/ft and after it comes to rest it is given an initial downward speed of 1 ft/sec in a medium in which the resistive force produces a force of 6 lb. for every đ?&#x2018;&#x201C;đ?&#x2018;Ą/đ?&#x2018; đ?&#x2018;&#x2019;đ?&#x2018;?, then the position function đ?&#x2018;Ś(đ?&#x2018;Ą) obeys the equation 1 đ?&#x2018;Śâ&#x20AC;˛â&#x20AC;˛ + 6đ?&#x2018;Śâ&#x20AC;˛ + 2đ?&#x2018;Ľ = 0; 2

4

đ?&#x2018;Ś(0) = 0, đ?&#x2018;Śâ&#x20AC;˛(0) = 1

This is not periodic motion since the amplitude continuously decreases so that the motion never repeats itself.


16 The discriminant đ??ˇ = â&#x2C6;&#x161;đ?&#x2018;? 2 â&#x2C6;&#x2019; 4đ?&#x2018;&#x161;đ?&#x2018;&#x2DC; = â&#x2C6;&#x161;32 > 0. Hence, the motion is under-damped. Solving the auxiliary equation 1 2 đ?&#x153;&#x2020; + 6đ?&#x153;&#x2020; + 2 = 0 2 gives us the roots đ?&#x2018;&#x;1 = â&#x2C6;&#x2019;6 + 4â&#x2C6;&#x161;2 and đ?&#x2018;&#x;2 = â&#x2C6;&#x2019;6 â&#x2C6;&#x2019; 4â&#x2C6;&#x161;2. The position function is given by đ?&#x2018;Ś(đ?&#x2018;Ą) = đ?&#x2018;&#x2019; â&#x2C6;&#x2019; 6đ?&#x2018;Ą (đ??´đ?&#x2018;&#x2019; 4â&#x2C6;&#x161;2đ?&#x2018;Ą + đ??ľđ?&#x2018;&#x2019; â&#x2C6;&#x2019;4â&#x2C6;&#x161;2đ?&#x2018;Ą ) Applying the initial conditions, đ?&#x2018;Ś(0) = 0 â&#x;š đ??´ + đ??ľ = 0 Since đ?&#x2018;Śâ&#x20AC;˛(đ?&#x2018;Ą) = â&#x2C6;&#x2019;6đ?&#x2018;&#x2019; â&#x2C6;&#x2019; 6đ?&#x2018;Ą (đ??´đ?&#x2018;&#x2019; 4â&#x2C6;&#x161;2đ?&#x2018;Ą + đ??ľđ?&#x2018;&#x2019; â&#x2C6;&#x2019;4â&#x2C6;&#x161;2đ?&#x2018;Ą ) + đ?&#x2018;&#x2019; â&#x2C6;&#x2019; 6đ?&#x2018;Ą (4â&#x2C6;&#x161;2đ??´đ?&#x2018;&#x2019; 4â&#x2C6;&#x161;2đ?&#x2018;Ą â&#x2C6;&#x2019; 4â&#x2C6;&#x161;2đ??ľđ?&#x2018;&#x2019; â&#x2C6;&#x2019;4â&#x2C6;&#x161;2đ?&#x2018;Ą ), the second initial condition tells us that (4â&#x2C6;&#x161;2 â&#x2C6;&#x2019; 6)đ??´ â&#x2C6;&#x2019; (4â&#x2C6;&#x161;2 + 6)đ??ľ = 1 Solving the system đ??´+đ??ľ =0 (4â&#x2C6;&#x161;2 â&#x2C6;&#x2019; 6)đ??´ â&#x2C6;&#x2019; (4â&#x2C6;&#x161;2 + 6)đ??ľ = 1 give us đ??´ = 1/8â&#x2C6;&#x161;2 and đ??ľ = â&#x2C6;&#x2019;1/8â&#x2C6;&#x161;2. Hence, đ?&#x2018;Ś(đ?&#x2018;Ą) =

1 8â&#x2C6;&#x161;2

đ?&#x2018;&#x2019; â&#x2C6;&#x2019; 6đ?&#x2018;Ą (đ?&#x2018;&#x2019; 4â&#x2C6;&#x161;2đ?&#x2018;Ą â&#x2C6;&#x2019; đ?&#x2018;&#x2019; â&#x2C6;&#x2019;4â&#x2C6;&#x161;2đ?&#x2018;Ą )

The graph of this function is shown below:

0.06

0.04

0.02

1

2

3

4

5

6

This figure illustrates that the object will quickly move downward (severe steepness of the curve initially) and then slowly returns to equilibrium. Other Models Oscillatory motion is not confined to spring-mass systems. For example, objects partially immersed in a fluid can be set into oscillatory motion, bobbing up and down when disturbed. The next example illustrates the differential equation that describes this motion. Example A cylindrical object of mass đ?&#x2018;&#x20AC;, radius đ?&#x2018;&#x; and altitude đ??ť is placed in a fluid of density đ?&#x153;&#x152;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122; . After it comes to rest with its base submerged to a depth h, it is pushed down a distance of đ?&#x2018;Ś0 and released. Find a differential equation that describes the objectâ&#x20AC;&#x2122;s subsequent motion. Solution


17 The figure below illustrates the forces that act on the object when it is in equilibrium. The small red circle is used as a reference point which moves in exactly the same way as the cylinder.

đ??š1

+đ?&#x2018;Ś 0

Area = A

P

â&#x201E;&#x17D;

đ?&#x2018;&#x20AC;đ?&#x2018;&#x201D; đ??š1 is the force due to atmospheric pressure and đ??š2 is the force due to the pressure at depth â&#x201E;&#x17D;. With this choice of coordinate system,

đ??š2

Since â&#x201E;&#x17D; < 0.

đ?&#x2018;&#x20AC;đ?&#x2018;&#x201D; = â&#x2C6;&#x2019;đ?&#x153;&#x152;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122; đ??´â&#x201E;&#x17D;đ?&#x2018;&#x201D;

Applying Newtonâ&#x20AC;&#x2122;s Second Law, đ??š2 â&#x2C6;&#x2019; đ??š1 â&#x2C6;&#x2019; đ?&#x2018;&#x20AC;đ?&#x2018;&#x201D; = 0 đ??š2 â&#x2C6;&#x2019; đ??š1 is the buoyant force đ??ľ. By Archimedesâ&#x20AC;&#x2122; Principle, which states that đ??ľ is equal to the weight of fluid displaced, đ??ľđ?&#x2018;&#x2019;đ?&#x2018;&#x17E;đ?&#x2018;˘đ?&#x2018;&#x2013;đ?&#x2018;&#x2122;đ?&#x2018;&#x2013;đ?&#x2018;?đ?&#x2018;&#x;đ?&#x2018;&#x2013;đ?&#x2018;˘đ?&#x2018;&#x161; = đ?&#x153;&#x152;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122; đ?&#x2018;&#x2030;đ?&#x2018;&#x201D; = â&#x2C6;&#x2019;đ?&#x153;&#x152;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122; đ??´â&#x201E;&#x17D;đ?&#x2018;&#x201D; where đ?&#x2018;&#x2030; is the volume of fluid displaced and đ?&#x153;&#x152;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122; đ?&#x2018;&#x2030;đ?&#x2018;&#x201D; is its weight. We must put a negative sign in order to ensure that đ??ľđ?&#x2018;&#x2019;đ?&#x2018;&#x17E;đ?&#x2018;˘đ?&#x2018;&#x2013;đ?&#x2018;&#x2122;đ?&#x2018;&#x2013;đ?&#x2018;?đ?&#x2018;&#x;đ?&#x2018;&#x2013;đ?&#x2018;˘đ?&#x2018;&#x161; > 0 since â&#x201E;&#x17D; < 0. Now if the cylinder is pushed some more and released, the buoyant and gravitational forces become unbalanced and the cylinder starts bobbing up and down. This is the motion we wish to describe relative to a coordinate system set up such that đ?&#x2018;Ś = 0 corresponds to the equilibrium position (see previous figure). At an arbitrary time đ?&#x2018;Ą > 0, the object might look something like this: đ?&#x2018;Š

+đ?&#x2018;Ś

0

đ?&#x2018;Ś â&#x201E;&#x17D;

đ?&#x2018;&#x20AC;đ?&#x2018;&#x201D;

The figure shows an instant in which the cylinder is submerged below its equilibrium point. In this state below equilibrium, the buoyant force will exceed đ?&#x2018;&#x20AC;đ?&#x2018;&#x201D; and a net upward


18 force will be exerted on the cylinder causing to accelerate upwards. The opposite is true when it overshoots the equilibrium point. We have đ?&#x2018;Š = â&#x2C6;&#x2019;đ?&#x153;&#x152;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122; đ??´(â&#x201E;&#x17D; + đ?&#x2018;Ś)đ?&#x2018;&#x201D;đ??Ł where đ??Ł is the unit vector in the positive y direction5. Again, the negative is there to accommodate our choice of coordinate system in which â&#x201E;&#x17D; < 0. Therefore, when đ?&#x2018;Ś is also negative đ?&#x2018;Š points up. Applying Newtonâ&#x20AC;&#x2122;s Second Law, đ?&#x2018;&#x20AC;đ?&#x2019;&#x201A; = â&#x2C6;&#x2019;đ?&#x2018;&#x20AC;đ?&#x2018;&#x201D;đ??Ł + đ?&#x2018;Š đ?&#x2018;&#x2018;2 đ?&#x2018;Ś đ?&#x2018;&#x20AC; 2 đ??Ł = â&#x2C6;&#x2019;đ?&#x2018;&#x20AC;đ?&#x2018;&#x201D;đ??Ł â&#x2C6;&#x2019; đ?&#x153;&#x152;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122; đ??´(â&#x201E;&#x17D; + đ?&#x2018;Ś)đ?&#x2018;&#x201D;đ??Ł đ?&#x2018;&#x2018;đ?&#x2018;Ą đ?&#x2018;&#x2018;2 đ?&#x2018;Ś đ?&#x2018;&#x20AC; 2 đ??Ł = (â&#x2C6;&#x2019;đ?&#x2018;&#x20AC;đ?&#x2018;&#x201D; â&#x2C6;&#x2019; đ?&#x153;&#x152;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122; đ??´â&#x201E;&#x17D;đ?&#x2018;&#x201D; â&#x2C6;&#x2019; đ?&#x153;&#x152;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122; đ??´đ?&#x2018;Śđ?&#x2018;&#x201D;)đ??Ł đ?&#x2018;&#x2018;đ?&#x2018;Ą But â&#x2C6;&#x2019;đ?&#x2018;&#x20AC;đ?&#x2018;&#x201D; â&#x2C6;&#x2019; đ?&#x153;&#x152;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122; đ??´â&#x201E;&#x17D;đ?&#x2018;&#x201D; = 0. Therefore, đ?&#x2018;&#x2018;2 đ?&#x2018;Ś + đ?&#x153;&#x152;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122; đ??´đ?&#x2018;&#x201D;đ?&#x2018;Ś = 0; đ?&#x2018;Ś(0) = đ?&#x2018;Ś0 , đ?&#x2018;Ł(0) = 0 đ?&#x2018;&#x2018;đ?&#x2018;Ą 2 Since the cylinder was initially pushed down, đ?&#x2018;Ś0 < 0. The angular frequency is given by đ?&#x2018;&#x20AC;

đ?&#x153;&#x152;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122; đ??´đ?&#x2018;&#x201D; đ?&#x153;&#x201D;=â&#x2C6;&#x161; đ?&#x2018;&#x20AC; This last equation is exactly the same as that of the spring-mass and pendulum equations except for the physical parameters inside the radical. You should check the units of the radicand as an exercise. Energy Methods Energy methods capitalize on the principle of conservation of energy. Every mechanical system may be described by an energy function which specifies at each instant in time the total mechanical energy đ??¸ of the system. The conservation of energy principle states that if there are no dissipative forces such as friction, then the total mechanical energy is conserved, that is, đ?&#x2018;&#x2018;đ??¸/đ?&#x2018;&#x2018;đ?&#x2018;Ą = 0. The total mechanical energy is the sum of the potential energy and kinetic energy of the system: đ??¸ =đ?&#x2018;&#x2C6;+đ??ž Hence, in order to apply this principle, one must first identify the systemâ&#x20AC;&#x2122;s mechanical energy in terms of a single dynamic variable (e.g. position). The Simple Pendulum A simple pendulum consists of a mass m (the bob) attached to a string of length l (assumed to have negligible mass) and allowed to swing about a fixed point P. The motion exhibited by the bob is also periodic and will in general experience decay due to frictional losses just as is the case 5

The moral of this story is: choose your coordinates well. It would have made a lot more sense to pick the downward direction as the positive direction in this example


19

with all mechanical systems. We are interested in studying the idealized case where there are no energy losses. The figure below illustrates the situation:

P đ?&#x153;&#x192;

đ?&#x2018;&#x2122; đ?&#x2018;&#x2021;

đ?&#x2018;&#x161;đ?&#x2018;&#x201D;

đ?&#x2018;Ł

In order to describe the pendulumâ&#x20AC;&#x2122;s motion we need to apply Newtonâ&#x20AC;&#x2122;s Laws, which requires that we identify all the forces at work and set up the equation corresponding to đ?&#x2018;&#x161;đ?&#x2019;&#x201A; = đ?&#x2018;­. Alternatively, we may use the energy approach that dispenses with the need for vectors. This approach demonstrates a very important technique used to solve complex problems and you are urged to spend time understanding it. The pendulumâ&#x20AC;&#x2122;s mechanical energy is derived from the gravitational potential energy đ?&#x2018;&#x2C6; associated with its position near earthâ&#x20AC;&#x2122;s surface. The potential energy is given by đ?&#x2018;&#x2C6; = đ?&#x2018;&#x161;đ?&#x2018;&#x201D;đ?&#x2018;Ś where y is the height above the 0 potential energy level (which we are free to choose). This 1 potential energy becomes kinetic energy đ??ž = đ?&#x2018;&#x161;đ?&#x2018;Ł 2 when the object moves with speed đ?&#x2018;Ł along 2 its circular path. Thus, if there are no frictional forces, the total energy of the system remains constant, alternating between gravitational potential energy and kinetic energy ad infinitum. The pendulumâ&#x20AC;&#x2122;s mechanical energy is given by 1 đ??¸ = đ?&#x2018;&#x161;đ?&#x2018;Ł 2 + đ?&#x2018;&#x161;đ?&#x2018;&#x201D;đ?&#x2018;Ś = đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018; đ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;Ą 2 We need to express this equation in terms of a single dynamical variable, and we choose this variable to be the angle đ?&#x153;&#x192;. Let us calculate đ??¸ at an arbitrary point along the bobâ&#x20AC;&#x2122;s path, and agree to use the line at the bottom of the swing as the line of 0 potential energy. Assume that the bob is initially displaced by an angle đ?&#x153;&#x192;0 with the horizontal that corresponds to a height đ?&#x2018;Ś0 from the zero potential line. đ?&#x153;&#x192;0

đ?&#x153;&#x192; đ?&#x2018;&#x2122; cos đ?&#x153;&#x192;

đ?&#x2018;Ś

đ?&#x2018;&#x2122; đ?&#x2018;&#x2122; sin đ?&#x153;&#x192;

đ??¸ = đ?&#x2018;&#x161;đ?&#x2018;&#x201D;đ?&#x2018;Ś0

đ??¸=

đ?&#x2018;&#x161;đ?&#x2018;&#x201D;

1 đ?&#x2018;&#x161;đ?&#x2018;Ł 2 + đ?&#x2018;&#x161;đ?&#x2018;&#x201D;đ?&#x2018;Ś 2

đ?&#x2018;&#x2C6;=0


20

Applying đ?&#x2018;&#x2018;đ??¸/đ?&#x2018;&#x2018;đ?&#x2018;Ą = 0,

đ?&#x2018;&#x2018; 1 ( đ?&#x2018;&#x161;đ?&#x2018;Ł 2 + đ?&#x2018;&#x161;đ?&#x2018;&#x201D;đ?&#x2018;Ś) = 0 đ?&#x2018;&#x2018;đ?&#x2018;Ą 2 or đ?&#x2018;&#x2018;đ?&#x2018;Ł đ?&#x2018;&#x2018;đ?&#x2018;Ś đ?&#x2018;&#x161;đ?&#x2018;Ł + đ?&#x2018;&#x161;đ?&#x2018;&#x201D; =0 đ?&#x2018;&#x2018;đ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;Ą

Cancelling đ?&#x2018;&#x161;, đ?&#x2018;Ł

đ?&#x2018;&#x2018;đ?&#x2018;Ł đ?&#x2018;&#x2018;đ?&#x2018;Ś +đ?&#x2018;&#x201D; =0 đ?&#x2018;&#x2018;đ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;Ą

As stated before, it is easier to express đ?&#x2018;Ł and đ?&#x2018;Ś in terms of đ?&#x153;&#x192;. Observe that đ?&#x2018;Ś = đ?&#x2018;&#x2122; â&#x2C6;&#x2019; đ?&#x2018;&#x2122; cos đ?&#x153;&#x192;. Also, since the object moves along a circular path of radius đ?&#x2018;&#x2122;, we đ?&#x2018;&#x2018;đ?&#x153;&#x192; have đ?&#x2018;Ł = đ?&#x2018;&#x2122; đ?&#x2018;&#x2018;đ?&#x2018;Ą . Thus, đ?&#x2018;&#x2122;

đ?&#x2018;&#x2018;đ?&#x153;&#x192; đ?&#x2018;&#x2018;2 đ?&#x153;&#x192; đ?&#x2018;&#x2018; â&#x2C6;&#x2122;đ?&#x2018;&#x2122; + đ?&#x2018;&#x201D; (đ?&#x2018;&#x2122; â&#x2C6;&#x2019; đ?&#x2018;&#x2122; cos đ?&#x153;&#x192;) = 0 đ?&#x2018;&#x2018;đ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;Ą 2 đ?&#x2018;&#x2018;đ?&#x2018;Ą

Simplifying, đ?&#x2018;&#x2122;2

đ?&#x2018;&#x2018;đ?&#x153;&#x192; đ?&#x2018;&#x2018;2 đ?&#x153;&#x192; đ?&#x2018;&#x2018;đ?&#x153;&#x192; â&#x2C6;&#x2122; 2 + đ?&#x2018;&#x201D;đ?&#x2018;&#x2122; sin đ?&#x153;&#x192; = 0 đ?&#x2018;&#x2018;đ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;Ą or đ?&#x2018;&#x2122;

đ?&#x2018;&#x2018;2 đ?&#x153;&#x192; + đ?&#x2018;&#x201D; sin đ?&#x153;&#x192; = 0 đ?&#x2018;&#x2018;đ?&#x2018;Ą 2

This is a non-linear equation in đ?&#x153;&#x192; and not an easy one to solve. However, it can be linearized using the small angle approximation6 sin đ?&#x153;&#x192; â&#x2030;&#x2C6; đ?&#x153;&#x192;: Using this approximation gives us đ?&#x2018;&#x2122;

đ?&#x2018;&#x2018;2 đ?&#x153;&#x192; + đ?&#x2018;&#x201D;đ?&#x153;&#x192; = 0 đ?&#x2018;&#x2018;đ?&#x2018;Ą 2

whose general solution is đ?&#x153;˝(đ?&#x2019;&#x2022;) = đ?&#x2018;¨ đ??&#x153;đ??¨đ??Ź đ??&#x17D; đ?&#x2019;&#x2022; + đ?&#x2018;Š đ??Źđ??˘đ??§ đ??&#x17D; đ?&#x2019;&#x2022; where đ??&#x17D; = â&#x2C6;&#x161;đ?&#x2019;&#x2C6;/đ?&#x2019;?. Interestingly, the angular frequency đ?&#x153;&#x201D; is independent of both the amplitude of oscillation and the mass of the bob. This is of course not true in general and only applies when we are justified in using the small angle approximation. Notice that the form of this equation is mathematically identical to that of the spring-mass system with đ?&#x2018;Ľ replaced by đ?&#x153;&#x192;, đ?&#x2018;&#x161; by đ?&#x2018;&#x2122;, and đ?&#x2018;&#x2DC; by g: đ?&#x2018;&#x2018;2 đ?&#x2018;Ľ đ?&#x2018;&#x2018;2 đ?&#x153;&#x192; + đ?&#x2018;&#x2DC;đ?&#x2018;Ľ = 0 đ?&#x2018;Łđ?&#x2018; . đ?&#x2018;&#x2122; + đ?&#x2018;&#x201D;đ?&#x153;&#x192; = 0 đ?&#x2018;&#x2018;đ?&#x2018;Ą 2 đ?&#x2018;&#x2018;đ?&#x2018;Ą 2 In fact, all mechanical systems that are subject to a restoring force obey the same mathematical law. đ?&#x2018;&#x161;

Rotating Wheel 6

This approximation is fairly good for angles up to about 17° if we round to two decimal places.


21 A wheel radius đ?&#x2018;&#x17D; and moment of inertia is đ??ź and mass đ?&#x2018;&#x161; is attached to a spring whose spring constant is đ?&#x2018;&#x2DC;. One end of the spring is secured to a rigid wall and the other to the axel of a wheel that is free to rotate without slipping. If the wheelâ&#x20AC;&#x2122;s center is displaced to some initial position đ?&#x2018;Ľ0 units from its equilibrium position, find the differential equation that đ?&#x2018;Ľ(đ?&#x2018;Ą) satisfies and the systemâ&#x20AC;&#x2122;s natural frequency.

đ??ź, đ?&#x2018;&#x161;

đ?&#x2018;&#x2DC;

đ??ź, đ?&#x2018;&#x161;

đ?&#x2018;&#x2DC;

đ?&#x2018;&#x17D;

đ?&#x2018;&#x17D;

0

1 đ??¸ = đ?&#x2018;&#x2DC;đ?&#x2018;Ľ02 2

đ?&#x2018;Ľ0

0

The energy equation is given by đ?&#x2018;Ź = đ??Šđ??¨đ??­đ??&#x17E;đ??§đ??­đ??˘đ??&#x161;đ??Ľ đ??&#x17E;đ??§đ??&#x17E;đ??Ťđ?? đ??˛ đ??Źđ??­đ??¨đ??Ťđ??&#x17E;đ??? đ??˘đ??§ đ??Źđ??Šđ??Ťđ??˘đ??§đ??  + đ??¤đ??˘đ??§đ??&#x17E;đ??­đ??˘đ??&#x153; đ??&#x17E;đ??§đ??&#x17E;đ??Ťđ?? đ??˛ đ??¨đ??&#x; đ?&#x2018;Şđ?&#x2018;´ + đ??Ťđ??¨đ??­đ??&#x161;đ??­đ??˘đ??¨đ??§đ??&#x161;đ??Ľ đ??&#x17E;đ??§đ??&#x17E;đ??Ťđ?? đ??˛ đ??¨đ??&#x; đ???đ??˘đ??Źđ??¤ Here, đ??śđ?&#x2018;&#x20AC; stands for the wheelâ&#x20AC;&#x2122;s center of mass (i.e. its center). At some time đ?&#x2018;Ą > 0, the objectâ&#x20AC;&#x2122;s center of mass is moving with velocity đ?&#x2018;Ł, the wheel is rotating with angular velocity đ?&#x153;&#x201D; and the spring is stretched by an amount đ?&#x2018;Ľ(đ?&#x2018;Ą), which may be negative is the spring is compressed: đ?&#x153;&#x201D; đ?&#x2018;&#x2DC;

đ?&#x2018;Ł

đ??ź, đ?&#x2018;&#x161; đ?&#x2018;&#x17D;

1

Kinetic Energy = đ?&#x2018;&#x161;đ?&#x2018;Ł 2 2

1

Rotational energy= đ??źđ?&#x153;&#x201D;2 2

1

Potential energy = đ?&#x2018;&#x2DC;đ?&#x2018;Ľ 2 2

đ?&#x2018;Ľ=0

đ?&#x2018;Ľ(đ?&#x2018;Ą)

Thus, the total energy at time đ?&#x2018;Ą is given by 1 1 1 đ??¸ = đ?&#x2018;&#x161;đ?&#x2018;Ł 2 + đ??źđ?&#x153;&#x201D;2 + đ?&#x2018;&#x2DC;đ?&#x2018;Ľ 2 2 2 2 Kinetic energy Rotational energy

potential energy

The â&#x20AC;&#x153;non-slipâ&#x20AC;? conditions are đ?&#x2018;Ł = đ?&#x2018;&#x17D;đ?&#x153;&#x201D; and đ?&#x2018;Ł = đ?&#x2018;&#x2018;đ?&#x2018;Ľ/đ?&#x2018;&#x2018;đ?&#x2018;Ą. This equation is just like an â&#x20AC;&#x153;accounting equationâ&#x20AC;? â&#x20AC;&#x201C; it accounts for all the energy of the system which must remain constant as long as there are no frictional losses. Setting the derivative of đ??¸ equal to 0 gives us the desired differential equation:


22 đ?&#x2018;&#x2018; 1 1 1 ( đ?&#x2018;&#x161;đ?&#x2018;Ł 2 + đ??źđ?&#x153;&#x201D;2 + đ?&#x2018;&#x2DC;đ?&#x2018;Ľ 2 ) = 0 đ?&#x2018;&#x2018;đ?&#x2018;Ą 2 2 2 đ?&#x2018;&#x2018;đ?&#x2018;Ł đ?&#x2018;&#x2018;đ?&#x153;&#x201D; đ?&#x2018;&#x2018;đ?&#x2018;Ľ đ?&#x2018;&#x161;đ?&#x2018;Ł + đ??źđ?&#x153;&#x201D; + đ?&#x2018;&#x2DC;đ?&#x2018;Ľ =0 đ?&#x2018;&#x2018;đ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;Ą Using đ?&#x153;&#x201D; = đ?&#x2018;Ł/đ?&#x2018;&#x17D; and đ?&#x2018;&#x2018;đ?&#x2018;Ł/đ?&#x2018;&#x2018;đ?&#x2018;Ą = đ?&#x2018;&#x2018;2 đ?&#x2018;Ľ/đ?&#x2018;&#x2018;đ?&#x2018;Ą 2 we may write a differential equation in terms of the variable đ?&#x2018;Ľ: đ?&#x2018;Ł đ?&#x2018;&#x2018;đ?&#x2018;Ľ đ?&#x2018;&#x2018;( ) đ?&#x2018;&#x2018;đ?&#x2018;Ľ đ?&#x2018;&#x2018;2 đ?&#x2018;Ľ đ?&#x2018;&#x17D; + đ?&#x2018;&#x2DC;đ?&#x2018;Ľ đ?&#x2018;&#x2018;đ?&#x2018;Ľ = 0 đ?&#x2018;&#x2018;đ?&#x2018;Ą đ?&#x2018;&#x161; +đ??ź 2 đ?&#x2018;&#x2018;đ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;Ą đ?&#x2018;&#x17D; đ?&#x2018;&#x2018;đ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;Ą Cancelling đ?&#x2018;&#x2018;đ?&#x2018;Ľ/đ?&#x2018;&#x2018;đ?&#x2018;Ą and simplifying, đ?&#x2018;&#x161;

đ?&#x2018;&#x2018;2 đ?&#x2018;Ľ 1 đ?&#x2018;&#x2018;2 đ?&#x2018;Ľ + đ??ź + đ?&#x2018;&#x2DC;đ?&#x2018;Ľ = 0 đ?&#x2018;&#x2018;đ?&#x2018;Ą 2 đ?&#x2018;&#x17D;2 đ?&#x2018;&#x2018;đ?&#x2018;Ą 2 or

(đ?&#x2019;&#x17D; +

đ?&#x2018;° đ?&#x2019;&#x2026;đ?&#x;? đ?&#x2019;&#x2122; ) + đ?&#x2019;&#x152;đ?&#x2019;&#x2122; = đ?&#x;&#x17D; đ?&#x2019;&#x201A;đ?&#x;? đ?&#x2019;&#x2026;đ?&#x2019;&#x2022;đ?&#x;?

The systemâ&#x20AC;&#x2122;s natural frequency is given by đ?&#x153;&#x201D;=â&#x2C6;&#x161;

đ?&#x2018;&#x2DC; (đ?&#x2018;&#x161; +

đ??ź ) đ?&#x2018;&#x17D;2

Compare this angular frequency to the one obtained for the block on a frictionless surface: đ?&#x153;&#x201D; = â&#x2C6;&#x161;đ?&#x2018;&#x2DC;/đ?&#x2018;&#x161;. Clearly the angular frequency of the present example is lower, and this makes sense; more energy is needed to make the wheel spin. Hence, the energy stored in the spring cannot be used in its entirety to produce translational motion and as a consequence, the wheel oscillates at a lower frequency than the block on a frictionless surface. As a final example that stresses the arbitrariness of â&#x20AC;&#x153;the point of zero potential energyâ&#x20AC;?, an object of mass đ?&#x2018;&#x161; is attached to a string which wraps around a pulley of radius đ?&#x2018;&#x17D; and moment of inertia is đ??ź. The string is attached to a spring with spring constant k which is secured to a wall. The mass is displaced from its equilibrium position and released. Determine a differential equation that governs its motion.

đ?&#x2018;&#x2DC;

đ??ź

a

đ?&#x2018;&#x17D;

đ?&#x2018;&#x161;

Let us place the zero gravitational potential energy line at the level at which the object is in equilibrium7. Since the spring is stretched by some amount once the object is attached, there is 7

This means that when the object is below the equilibrium position, its potential energy is negative.


23 some potential energy stored in the spring. If â&#x201E;&#x201C; denotes the amount by which this spring is 1 stretched, then that initial â&#x20AC;&#x153;springâ&#x20AC;? potential energy is given by đ?&#x2018;&#x2DC;â&#x201E;&#x201C;2 . The equilibrium state is 2 shown in the figure on the left: â&#x201E;&#x201C; đ?&#x2018;&#x2DC;

đ?&#x153;&#x201D; đ??ź

đ??ź đ?&#x2018;&#x2DC;(â&#x201E;&#x201C; â&#x2C6;&#x2019; đ?&#x2018;Ś)

đ?&#x2018;&#x2DC;â&#x201E;&#x201C; đ?&#x2018;&#x161;

đ?&#x2018;Ś

đ?&#x2018;&#x2C6;=0 đ?&#x2018;Ą>0

đ?&#x2018;&#x161;đ?&#x2018;&#x201D;

đ?&#x2018;&#x161;

đ?&#x2018;&#x2C6; = đ?&#x2018;&#x161;đ?&#x2018;&#x201D;đ?&#x2018;Ś đ?&#x2018;&#x2C6;=0

đ?&#x2018;&#x161;đ?&#x2018;&#x201D;

If the object is disturbed from this equilibrium position it will begin to oscillate. Let us denote that displacement by đ?&#x2018;Ś(đ?&#x2018;Ą) with the convention that values above equilibrium are positive. Now consider an arbitrary point in time when the block is, say, đ?&#x2018;Ś units above equilibrium (figure to the right). Then the systemâ&#x20AC;&#x2122;s energy is stored as gravitational potential and kinetic energy of the block, potential energy of the spring (this energy is 0 when the spring is unstretched), and rotational energy of the pulley. The sum total of these constitutes the total mechanical energy of the system at time đ?&#x2018;Ą: 1 1 1 đ??¸(đ?&#x2018;Ą) = đ?&#x2018;&#x161;đ?&#x2018;&#x201D;đ?&#x2018;Ś + đ??źđ?&#x153;&#x201D;2 + đ?&#x2018;&#x2DC;(â&#x201E;&#x201C; â&#x2C6;&#x2019; đ?&#x2018;Ś)2 + đ?&#x2018;&#x161;đ?&#x2018;Ł 2 2 2 2 If there are no frictional losses, đ??¸(đ?&#x2018;Ą) is constant and

đ?&#x2018;&#x2018;đ??¸ đ?&#x2018;&#x2018;đ?&#x2018;Ą

= 0. With this assumption, we can write

đ?&#x2018;&#x2018; 1 1 1 [đ?&#x2018;&#x161;đ?&#x2018;&#x201D;đ?&#x2018;Ś + đ??źđ?&#x153;&#x201D;2 + đ?&#x2018;&#x2DC;(â&#x201E;&#x201C; â&#x2C6;&#x2019; đ?&#x2018;Ś)2 + đ?&#x2018;&#x161;đ?&#x2018;Ł 2 ] = 0 đ?&#x2018;&#x2018;đ?&#x2018;Ą 2 2 2 or equivalently, đ?&#x2018;&#x161;đ?&#x2018;&#x201D;đ?&#x2018;Śâ&#x20AC;˛ + đ??źđ?&#x153;&#x201D;đ?&#x153;&#x201D;â&#x20AC;˛ + đ?&#x2018;&#x2DC;(â&#x201E;&#x201C; â&#x2C6;&#x2019; đ?&#x2018;Ś)(â&#x2C6;&#x2019;đ?&#x2018;Śâ&#x20AC;˛) + đ?&#x2018;&#x161;đ?&#x2018;Łđ?&#x2018;Łâ&#x20AC;˛ = 0 The relation between đ?&#x2018;Ś and đ?&#x153;&#x201D; is đ?&#x2018;Śâ&#x20AC;˛ = đ?&#x2018;&#x17D;đ?&#x153;&#x201D; â&#x;š đ?&#x153;&#x201D; = Therefore,

1 đ?&#x2018;Śâ&#x20AC;˛ đ?&#x2018;&#x17D;

1 1 đ?&#x2018;&#x161;đ?&#x2018;&#x201D;đ?&#x2018;Śâ&#x20AC;˛ + đ??ź đ?&#x2018;Śâ&#x20AC;˛ â&#x2C6;&#x2122; đ?&#x2018;Śâ&#x20AC;˛â&#x20AC;˛ + đ?&#x2018;&#x2DC;(â&#x201E;&#x201C; â&#x2C6;&#x2019; đ?&#x2018;Ś)(â&#x2C6;&#x2019;đ?&#x2018;Ś â&#x20AC;˛ ) + đ?&#x2018;&#x161;đ?&#x2018;Śâ&#x20AC;˛ â&#x2C6;&#x2122; đ?&#x2018;Śâ&#x20AC;˛â&#x20AC;˛ = 0 đ?&#x2018;&#x17D; đ?&#x2018;&#x17D; Simplifying this expression, 1 đ?&#x2018;&#x161;đ?&#x2018;&#x201D; + đ??ź 2 đ?&#x2018;Śâ&#x20AC;˛â&#x20AC;˛ â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;(â&#x201E;&#x201C; â&#x2C6;&#x2019; đ?&#x2018;Ś) + đ?&#x2018;&#x161;đ?&#x2018;Śâ&#x20AC;˛â&#x20AC;˛ = 0 đ?&#x2018;&#x17D; or 1 đ?&#x2018;&#x161;đ?&#x2018;&#x201D; + đ??ź 2 đ?&#x2018;Śâ&#x20AC;˛â&#x20AC;˛ â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;â&#x201E;&#x201C; + đ?&#x2018;&#x2DC;đ?&#x2018;Ś + đ?&#x2018;&#x161;đ?&#x2018;Śâ&#x20AC;˛â&#x20AC;˛ = 0 đ?&#x2018;&#x17D; But đ?&#x2018;&#x161;đ?&#x2018;&#x201D; = đ?&#x2018;&#x2DC;â&#x201E;&#x201C;. Therefore, đ?&#x2018;° (đ?&#x2019;&#x17D; + đ?&#x;? ) đ?&#x2019;&#x161;â&#x20AC;˛â&#x20AC;˛ + đ?&#x2019;&#x152;đ?&#x2019;&#x161; = đ?&#x;&#x17D; đ?&#x2019;&#x201A; with đ?&#x2019;&#x161;(đ?&#x;&#x17D;) = đ?&#x2019;&#x161;đ?&#x;&#x17D; , đ?&#x2019;&#x2014;(đ?&#x;&#x17D;) = đ?&#x;&#x17D;.


24 As can be seen, the energy method is a very powerful tool. The previous two problems would have been extremely difficult to solve using the usual vectorial approach. As a last example, it is always good to show students how a familiar problem may be generalized, and where energy considerations make calculations easier. This example also illustrates the importance of ensuring that a model, regardless of how its equation is obtained, should reduce to the usual equation in the limiting cases. Consider a block of mass đ?&#x2018;&#x161; attached to a spring that is secured to a rigid wall, but we now place the block on an incline of gradient đ?&#x203A;ź, as shown below:

đ?&#x203A;ź

If the block is disturbed from its rest position, obtain a differential equation that governs its motion. Intuition tells us that, in the absence of non-conservative forces (friction and drag), the motion should be at least similar to that of the block on a horizontal plane (đ?&#x203A;ź = 0°) or a vertical plane (đ?&#x203A;ź = 90°). Letâ&#x20AC;&#x2122;s see. Introduce a coordinate system whose axis is parallel to the surface of the incline, as shown below, and let đ?&#x2018;Ľ = 0 represent the location of the block when it is at equilibrium: đ?&#x2018;&#x2122;0 đ?&#x2018; 0 â&#x201E;&#x17D; 1

0

đ?&#x203A;ź

1

đ?&#x2018;Ľ(đ?&#x2018;Ą)

â&#x201E;&#x17D;2

đ?&#x203A;ź

2

1

đ??¸(0) = 2 đ?&#x2018;&#x2DC;đ?&#x2018;&#x2122;02 + đ?&#x2018;&#x161;đ?&#x2018;&#x201D;â&#x201E;&#x17D;1

đ??¸(đ?&#x2018;Ą) = 2 đ?&#x2018;&#x2DC;(đ?&#x2018;&#x2122;0 + đ?&#x2018;Ľ(đ?&#x2018;Ą)) + đ?&#x2018;&#x161;đ?&#x2018;&#x201D;â&#x201E;&#x17D;2 +

1 đ?&#x2018;&#x161;đ?&#x2018;Ł 2 2

1

1

= 2 đ?&#x2018;&#x2DC;đ?&#x2018;&#x2122;02 + đ?&#x2018; sin đ?&#x203A;ź

2

= 2 đ?&#x2018;&#x2DC;(đ?&#x2018;&#x2122;0 + đ?&#x2018;Ľ(đ?&#x2018;Ą)) + đ?&#x2018;&#x161;đ?&#x2018;&#x201D;(đ?&#x2018; â&#x2C6;&#x2019;

1

đ?&#x2018;Ľ(đ?&#x2018;Ą)) sin đ?&#x203A;ź + 2 đ?&#x2018;&#x161;đ?&#x2018;Ł 2

Here, đ?&#x2018; represents the distance from the base of the incline to the location of the block when it is at rest, đ?&#x2018;&#x2122;0 is the length by which the spring is stretched when the block is initially attached, đ?&#x2018;Ľ(đ?&#x2018;Ą) the blockâ&#x20AC;&#x2122;s displacement from equilibrium relative to the chosen origin, and đ?&#x2018;Ł its velocity at some time đ?&#x2018;Ą. In the absence of friction,

đ?&#x2018;&#x2018;đ??¸ đ?&#x2018;&#x2018;đ?&#x2018;Ą

= 0 or:

đ?&#x2018;&#x2018; 1 1 2 2 { đ?&#x2018;&#x2DC;(đ?&#x2018;&#x2122;0 + đ?&#x2018;Ľ(đ?&#x2018;Ą)) + đ?&#x2018;&#x161;đ?&#x2018;&#x201D;(đ?&#x2018; â&#x2C6;&#x2019; đ?&#x2018;Ľ(đ?&#x2018;Ą)) sin đ?&#x203A;ź + đ?&#x2018;&#x161;(đ?&#x2018;Ľ â&#x20AC;˛ (đ?&#x2018;Ą)) } = 0 đ?&#x2018;&#x2018;đ?&#x2018;Ą 2 2


25 Differentiating, đ?&#x2018;&#x2DC;(đ?&#x2018;&#x2122;0 + đ?&#x2018;Ľ(đ?&#x2018;Ą))đ?&#x2018;Ľ â&#x20AC;˛ (đ?&#x2018;Ą) + đ?&#x2018;&#x161;đ?&#x2018;&#x201D;(â&#x2C6;&#x2019;đ?&#x2018;Ľ â&#x20AC;˛ (đ?&#x2018;Ą)) sin đ?&#x203A;ź + đ?&#x2018;&#x161;đ?&#x2018;Ľ â&#x20AC;˛ (đ?&#x2018;Ą)đ?&#x2018;Ľâ&#x20AC;˛â&#x20AC;˛(đ?&#x2018;Ą) = 0 Cancelling đ?&#x2018;Ľâ&#x20AC;˛(đ?&#x2018;Ą) and rearranging the terms on the left-hand side of this equation, đ?&#x2018;&#x161;đ?&#x2018;Ľ â&#x20AC;˛â&#x20AC;˛ (đ?&#x2018;Ą) + đ?&#x2018;&#x2DC;(đ?&#x2018;&#x2122;0 + đ?&#x2018;Ľ(đ?&#x2018;Ą)) â&#x2C6;&#x2019; đ?&#x2018;&#x161;đ?&#x2018;&#x201D; sin đ?&#x203A;ź = 0 This is an example of a non-homogeneous second order linear differential equation: đ?&#x2019;&#x17D;đ?&#x2019;&#x2122;â&#x20AC;˛â&#x20AC;˛ (đ?&#x2019;&#x2022;) + đ?&#x2019;&#x152;(đ?&#x2019;?đ?&#x;&#x17D; + đ?&#x2019;&#x2122;(đ?&#x2019;&#x2022;)) = đ?&#x2019;&#x17D;đ?&#x2019;&#x2C6; đ??Źđ??˘đ??§ đ?&#x153;ś At this point it is worthwhile asking the student to determine if this equation reduces to the familiar ones derived for the limiting cases đ?&#x203A;ź = 0° (block on a horizontal plane), and đ?&#x203A;ź = 90° (block hanging vertically).

To summarize the different systems studied in these notes, +đ?&#x2018;Ś đ?&#x2018;Š

0

spring

â&#x201E;&#x17D; đ?&#x2018;Ł

Equation: đ?&#x2018;&#x161; đ?&#x2018;&#x2018;2 đ?&#x2018;Ś

đ??ź ) đ?&#x2018;&#x17D; 2 đ?&#x2018;&#x2018;đ?&#x2018;Ą 2

đ?&#x2018;&#x2018;2 đ?&#x2018;Ľ đ?&#x2018;&#x2018;đ?&#x2018;Ą 2

+ đ?&#x2018;&#x2DC;đ?&#x2018;Ľ = 0

đ?&#x2018;&#x2DC; >0

đ?&#x2018;Ś

đ?&#x153;&#x192;

đ?&#x2019;&#x17D;

đ?&#x153;&#x201D;

đ?&#x2018;&#x2122;

đ?&#x2018;&#x20AC;đ?&#x2018;&#x201D;

đ?&#x2018;Ś

đ?&#x2018;&#x161;đ?&#x2018;&#x201D;

đ?&#x2018;&#x2018;2 đ?&#x153;&#x192; đ?&#x2018;&#x2018;đ?&#x2018;Ą 2

đ??ź

đ?&#x2018;&#x161; đ?&#x2018;&#x161;đ?&#x2018;&#x201D;

+ đ?&#x2018;&#x201D;đ?&#x153;&#x192; = 0

đ?&#x2018;&#x2018;2 đ?&#x2018;Ś đ?&#x2018;&#x2018;đ?&#x2018;Ą 2

+ đ?&#x153;&#x152;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122; đ??´đ?&#x2018;&#x201D;đ?&#x2018;Ś = 0

đ?&#x2018;&#x2C6;=0

(đ?&#x2018;&#x161; +

+ đ?&#x2018;&#x2DC;đ?&#x2018;Ś = 0 1

đ?&#x2018;&#x2DC;

Frequency: đ?&#x2018;&#x201C; = 2đ?&#x153;&#x2039; â&#x2C6;&#x161;đ?&#x2018;&#x161;

1

đ?&#x2018;&#x201D;

đ?&#x2018;&#x201C; = 2đ?&#x153;&#x2039; â&#x2C6;&#x161; đ?&#x2018;&#x2122;

1

đ?&#x153;&#x152;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122; đ??´đ?&#x2018;&#x201D;

đ?&#x2018;&#x201C; = 2đ?&#x153;&#x2039; â&#x2C6;&#x161;

đ?&#x2018;&#x20AC;

đ?&#x2018;&#x201C;=

1 đ?&#x2018;&#x2DC; 2đ?&#x153;&#x2039; â&#x2C6;&#x161;đ?&#x2018;&#x161;+ đ??ź2 đ?&#x2018;&#x17D;

There is something very pleasing about these equations and it is hoped that they will help the student appreciate the beauty of mathematics beyond its usefulness. Qualitative Study of the spring-mass system Let us consider some qualitative aspects of spring-mass equation đ?&#x2019;&#x17D;đ?&#x2019;&#x2122;Ě&#x2C6; + đ?&#x2019;&#x192;đ?&#x2019;&#x2122;Ě&#x2021; + đ?&#x2019;&#x152;đ?&#x2019;&#x2122; = đ?&#x;&#x17D;

đ?&#x2019;&#x2122;(đ?&#x;&#x17D;) = đ?&#x2019;&#x2122;đ?&#x;&#x17D; , đ?&#x2019;&#x2014;(đ?&#x;&#x17D;) = đ?&#x2019;&#x2014;đ?&#x;&#x17D;

This differential equationâ&#x20AC;&#x2122;s characteristic polynomial is đ?&#x2018;?(đ?&#x153;&#x2020;) = đ?&#x2018;&#x161;đ?&#x153;&#x2020;2 + đ?&#x2018;?đ?&#x153;&#x2020; + đ?&#x2018;&#x2DC; whose roots are


26

đ?&#x153;&#x2020;=

â&#x2C6;&#x2019;đ?&#x2018;? Âą â&#x2C6;&#x161;đ?&#x2018;? 2 â&#x2C6;&#x2019; 4đ?&#x2018;&#x161;đ?&#x2018;&#x2DC; 2đ?&#x2018;&#x161;

Therefore, depending on the sign of the discriminant đ??ˇ = â&#x2C6;&#x161;đ?&#x2018;? 2 â&#x2C6;&#x2019; 4đ?&#x2018;&#x161;đ?&#x2018;&#x2DC; we will have one of three possible situations: two distinct real roots (D > 0) , two coincident real roots (D = 0), or a complex conjugate pair (D < 0). Accordingly, the solutions will involve exponential functions with real arguments in the first two cases and exponential functions with complex arguments â&#x20AC;&#x201C; which will lead to trigonometric functions â&#x20AC;&#x201C; in the third case. To simplify the analysis, denote đ?&#x2018;? 2 â&#x2C6;&#x2019; 4đ?&#x2018;&#x161;đ?&#x2018;&#x2DC; by đ?&#x153;&#x2030; (the Greek letter xi). Let us consider each case in turn. Case I: đ?&#x2018;? 2 â&#x2C6;&#x2019; 4đ?&#x2018;&#x161;đ?&#x2018;&#x2DC; > 0 [two real distinct roots] In this case, the position function is given by đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ?&#x2018;?1 đ?&#x2018;&#x2019;

(â&#x2C6;&#x2019;

đ?&#x2018;? 1 + 2đ?&#x2018;&#x161; 2đ?&#x2018;&#x161;â&#x2C6;&#x161;đ?&#x153;&#x2030;)đ?&#x2018;Ą

đ?&#x2019;&#x192;

= đ?&#x2019;&#x2020;â&#x2C6;&#x2019; đ?&#x;?đ?&#x2019;&#x17D;đ?&#x2019;&#x2022; (đ?&#x2018;¨đ?&#x2019;&#x2020;

(

+ đ?&#x2018;?2 đ?&#x2018;&#x2019;

đ?&#x;? đ?&#x;?đ?&#x2019;&#x17D;â&#x2C6;&#x161;đ??&#x192;)đ?&#x2019;&#x2022;

(â&#x2C6;&#x2019;

đ?&#x2018;? 1 â&#x2C6;&#x2019; 2đ?&#x2018;&#x161; 2đ?&#x2018;&#x161;â&#x2C6;&#x161;đ?&#x153;&#x2030;)đ?&#x2018;Ą

(â&#x2C6;&#x2019;

đ?&#x;? đ?&#x;?đ?&#x2019;&#x17D;â&#x2C6;&#x161;đ??&#x192;)đ?&#x2019;&#x2022; )

+ đ?&#x2018;Šđ?&#x2019;&#x2020;

Using Lâ&#x20AC;&#x2122;Hopitalâ&#x20AC;&#x2122;s Rule one can show that this function has a finite limit as đ?&#x2018;Ą â&#x2020;&#x2019; â&#x2C6;&#x17E;. Systems for which đ??ˇ > 0 are called over-damped systems because the damping forces are very large in magnitude and impede motion severely. CASE II: đ?&#x2018;? 2 â&#x2C6;&#x2019; 4đ?&#x2018;&#x161;đ?&#x2018;&#x2DC; = 0 [two real and equal roots] In this case, we have repeated roots and the position function is given by đ?&#x2019;&#x192;

đ?&#x2019;&#x2122;(đ?&#x2019;&#x2022;) = đ?&#x2019;&#x2020;â&#x2C6;&#x2019;đ?&#x;?đ?&#x2019;&#x17D;đ?&#x2019;&#x2022; (đ?&#x2018;¨ + đ?&#x2018;Šđ?&#x2019;&#x2022;) It is easy to see that this function also has limit 0 as t â&#x2020;&#x2019; â&#x2C6;&#x17E;. This is expected since the object is being subjected to retarding forces. Systems for which đ??ˇ = 0 are called critically damped systems. CASE III: đ?&#x2018;? 2 â&#x2C6;&#x2019; 4đ?&#x2018;&#x161;đ?&#x2018;&#x2DC; < 0 [complex roots] The roots are complex conjugates given by8 đ?&#x2018;&#x;=

â&#x2C6;&#x2019;đ?&#x2018;? Âą â&#x2C6;&#x161;đ?&#x2018;? 2 â&#x2C6;&#x2019; 4đ?&#x2018;&#x161;đ?&#x2018;&#x2DC; â&#x2C6;&#x2019;đ?&#x2018;? Âą đ?&#x2018;&#x2013;â&#x2C6;&#x161;|đ?&#x153;&#x2030;| = 2đ?&#x2018;&#x161; 2đ?&#x2018;&#x161;

The position function is given by đ?&#x2018;?

đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ?&#x2018;&#x2019; â&#x2C6;&#x2019;2đ?&#x2018;&#x161;đ?&#x2018;Ą (đ??´đ?&#x2018;&#x2019;

(

1 2đ?&#x2018;&#x161;â&#x2C6;&#x161;â&#x2C6;&#x2019;đ?&#x153;&#x2030;đ?&#x2018;&#x2013;)đ?&#x2018;Ą

+ đ??ľđ?&#x2018;&#x2019;

(â&#x2C6;&#x2019;

1 2đ?&#x2018;&#x161;â&#x2C6;&#x161;â&#x2C6;&#x2019;đ?&#x153;&#x2030;đ?&#x2018;&#x2013;)đ?&#x2018;Ą )

Using Eulerâ&#x20AC;&#x2122;s equation, đ?&#x2019;&#x192;

đ?&#x2019;&#x2122;(đ?&#x2019;&#x2022;) = đ?&#x2019;&#x2020;â&#x2C6;&#x2019;đ?&#x;?đ?&#x2019;&#x17D;đ?&#x2019;&#x2022; [đ?&#x2018;¨đ??&#x153;đ??¨đ??Ź (

8

â&#x2C6;&#x161;â&#x2C6;&#x2019;đ??&#x192; â&#x2C6;&#x161;â&#x2C6;&#x2019;đ??&#x192; đ?&#x2019;&#x2022;) + đ?&#x2018;Šđ??Źđ??˘đ??§ ( đ?&#x2019;&#x2022;)] đ?&#x;?đ?&#x2019;&#x17D; đ?&#x;?đ?&#x2019;&#x17D;

Since đ?&#x2018;? 2 â&#x2C6;&#x2019; 4đ?&#x2018;&#x161;đ?&#x2018;&#x2DC; < 0, |đ?&#x153;&#x2030;| = â&#x2C6;&#x2019;đ?&#x153;&#x2030;. Therefore, â&#x2C6;&#x2019;đ?&#x153;&#x2030; is positive.


27 Once again, it is easy to show that limđ?&#x2018;Ąâ&#x2020;&#x2019;â&#x2C6;&#x17E; đ?&#x2018;Ľ(đ?&#x2018;Ą) = 0. Systems for which đ??ˇ < 0 are called under-damped because the damping forces are not strong enough to prevent oscillatory motion. Nevertheless, the motion will cease eventually, driven by the exponentially decaying term đ?&#x2018;&#x2019; â&#x2C6;&#x2019;đ?&#x203A;˝đ?&#x2018;Ą/2đ?&#x2018;&#x161; .

Under-damped profile

Over-damped profile

The equation 2đ?&#x2018;ĽĚ&#x2C6; + 4đ?&#x2018;ĽĚ&#x2021; + đ?&#x2018;Ľ = 0; đ?&#x2018;Ľ(0) = 1 đ?&#x2018;ĽĚ&#x2021; (0) = 0 corresponds to a spring that has been displaced 1 unit to the right of equilibrium and released (0 initial speed). Since đ?&#x2018;? 2 â&#x2C6;&#x2019; 4đ?&#x2018;&#x161;đ?&#x2018;&#x2DC; = 16 â&#x2C6;&#x2019; 8 = 8 > 0, we have an over-damped system whose position function is đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ?&#x2018;?1 đ?&#x2018;&#x2019; (â&#x2C6;&#x2019;2+â&#x2C6;&#x161;2)đ?&#x2018;Ą + đ?&#x2018;?2 đ?&#x2018;&#x2019; (â&#x2C6;&#x2019;2â&#x2C6;&#x2019;â&#x2C6;&#x161;2)đ?&#x2018;Ą = đ?&#x2018;&#x2019; â&#x2C6;&#x2019;2đ?&#x2018;Ą (đ?&#x2018;?1 đ?&#x2018;&#x2019; â&#x2C6;&#x161;2đ?&#x2018;Ą + đ?&#x2018;?2 đ?&#x2018;&#x2019; â&#x2C6;&#x2019;â&#x2C6;&#x161;2đ?&#x2018;Ą ) Applying the initial conditions, gives us đ?&#x2019;&#x2122;(đ?&#x2019;&#x2022;) = đ?&#x2019;&#x2020;â&#x2C6;&#x2019;đ?&#x;?đ?&#x2019;&#x2022; (

â&#x2C6;&#x161;đ?&#x;? + đ?&#x;? đ?&#x;?â&#x2C6;&#x161;đ?&#x;?

â&#x2C6;&#x2122; đ?&#x2019;&#x2020;â&#x2C6;&#x161;đ?&#x;?đ?&#x2019;&#x2022; +

â&#x2C6;&#x161;đ?&#x;? â&#x2C6;&#x2019; đ?&#x;? đ?&#x;?â&#x2C6;&#x161;đ?&#x;?

â&#x2C6;&#x2122; đ?&#x2019;&#x2020;â&#x2C6;&#x2019;â&#x2C6;&#x161;đ?&#x;?đ?&#x2019;&#x2022; )

The figure below illustrates this situation. 1.0

0.8

0.6

0.4

0.2

2

đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ?&#x2018;&#x2019; â&#x2C6;&#x2019;2đ?&#x2018;Ą (

4

â&#x2C6;&#x161;2 + 2 2â&#x2C6;&#x161;2

6

â&#x2C6;&#x2122; đ?&#x2018;&#x2019; â&#x2C6;&#x161;2đ?&#x2018;Ą +

8

â&#x2C6;&#x161;2 â&#x2C6;&#x2019; 2 2â&#x2C6;&#x161;2

10

â&#x2C6;&#x2122; đ?&#x2018;&#x2019; â&#x2C6;&#x2019;â&#x2C6;&#x161;2đ?&#x2018;Ą )

On the other hand, the equation 2đ?&#x2018;ĽĚ&#x2C6; + 4đ?&#x2018;ĽĚ&#x2021; + 2đ?&#x2018;Ľ = 0; đ?&#x2018;Ľ(0) = 0 đ?&#x2018;ĽĚ&#x2021; (0) = 1 corresponds to a spring that starts at equilibrium and is pushed in the positive direction with an initial speed of 1 unit/sec.


28 In this case is đ??ˇ = 0 and therefore the system is critically-damped. The characteristic polynomial has a root đ?&#x2018;&#x; = â&#x2C6;&#x2019;1 of multiplicity 2. Thus, the solution is given by đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ?&#x2018;?1 đ?&#x2018;&#x2019; â&#x2C6;&#x2019;đ?&#x2018;Ą + đ?&#x2018;?2 đ?&#x2018;Ąđ?&#x2018;&#x2019; â&#x2C6;&#x2019;đ?&#x2018;Ą = đ?&#x2018;&#x2019; â&#x2C6;&#x2019;đ?&#x2018;Ą (đ?&#x2018;?1 + đ?&#x2018;?2 đ?&#x2018;Ą) The initial conditions require that đ?&#x2018;?1 = 0 and đ?&#x2018;?2 = 1. Hence, đ?&#x2019;&#x2122;(đ?&#x2019;&#x2022;) = đ?&#x2019;&#x2022;đ?&#x2019;&#x2020;â&#x2C6;&#x2019;đ?&#x2019;&#x2022; The only time this object is at equilibrium is when đ?&#x2018;Ą = 0. After being pushed to the right with an initial speed of 1, the block takes forever to get back to its equilibrium state again. 0.35 0.30 0.25 0.20 0.15 0.10 0.05

2

4

6

8

đ??Ľđ??˘đ??Ś đ?&#x2019;&#x2122;(đ?&#x2019;&#x2022;) = đ?&#x;&#x17D;

đ?&#x2019;&#x2022;â&#x2020;&#x2019;â&#x2C6;&#x17E;

Finally, consider an over-damped spring-mass system with no forcing function which has the following initial conditions: đ?&#x2018;Ľ(0) = 0, đ?&#x2018;ĽĚ&#x2021; (0) = đ?&#x2018;Ł0 . In other words, the system starts at rest and is set in motion by a sudden push in one or the other direction. Let us show that the only time at which these systems are at the equilibrium position is đ?&#x2018;Ą = 0. That is, if đ?&#x2018;Ą > 0 then đ?&#x2018;Ľ(đ?&#x2018;Ą) â&#x2030; 0. Proof The solution for over-damped systems has the form đ?&#x2018;?

đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ?&#x2018;&#x2019; â&#x2C6;&#x2019; 2đ?&#x2018;&#x161;đ?&#x2018;Ą (đ?&#x2018;?1 đ?&#x2018;&#x2019;

(

1 2đ?&#x2018;&#x161;â&#x2C6;&#x161;đ?&#x153;&#x2030;)đ?&#x2018;Ą

+ đ?&#x2018;?2 đ?&#x2018;&#x2019;

(â&#x2C6;&#x2019;

1 2đ?&#x2018;&#x161;â&#x2C6;&#x161;đ?&#x153;&#x2030;)đ?&#x2018;Ą )

where đ?&#x153;&#x2030; = đ?&#x2018;? 2 â&#x2C6;&#x2019; 4đ?&#x2018;&#x161;đ?&#x2018;&#x2DC;. Applying the first condition: đ?&#x2018;Ľ(0) = 0 â&#x;š đ?&#x2018;?1 + đ?&#x2018;?2 = 0 Therefore, đ?&#x2018;?1 = â&#x2C6;&#x2019;đ?&#x2018;?2 and we may replace these constants with a single constant, which we denote by đ?&#x2018;?. Thus, đ?&#x2018;?1 = đ?&#x2018;? and đ?&#x2018;?2 = â&#x2C6;&#x2019;đ?&#x2018;?. Thus, đ?&#x2018;?

đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ?&#x2018;?đ?&#x2018;&#x2019; â&#x2C6;&#x2019; 2đ?&#x2018;&#x161;đ?&#x2018;Ą (đ?&#x2018;&#x2019;

(

1 2đ?&#x2018;&#x161;â&#x2C6;&#x161;đ?&#x153;&#x2030;)đ?&#x2018;Ą

â&#x2C6;&#x2019;đ?&#x2018;&#x2019;

(â&#x2C6;&#x2019;

1 2đ?&#x2018;&#x161;â&#x2C6;&#x161;đ?&#x153;&#x2030;)đ?&#x2018;Ą )

Clearly đ?&#x2018;Ľ(0) = 0. Will it ever be 0 again? Assume that there is a value of đ?&#x2018;Ą > 0 such that đ?&#x2018;?

đ?&#x2018;?đ?&#x2018;&#x2019; â&#x2C6;&#x2019; 2đ?&#x2018;&#x161;đ?&#x2018;Ą (đ?&#x2018;&#x2019;

(

1 2đ?&#x2018;&#x161;â&#x2C6;&#x161;đ?&#x153;&#x2030;)đ?&#x2018;Ą

â&#x2C6;&#x2019;đ?&#x2018;&#x2019;

(â&#x2C6;&#x2019;

1 2đ?&#x2018;&#x161;â&#x2C6;&#x161;đ?&#x153;&#x2030;)đ?&#x2018;Ą )

đ?&#x2018;?

Since đ?&#x2018;? â&#x2030; 0 and đ?&#x2018;&#x2019; â&#x2C6;&#x2019; 2đ?&#x2018;&#x161;đ?&#x2018;Ą â&#x2030;  0, đ?&#x2018;&#x2019; That is

(

1 2đ?&#x2018;&#x161;â&#x2C6;&#x161;đ?&#x153;&#x2030;)đ?&#x2018;Ą

â&#x2C6;&#x2019;đ?&#x2018;&#x2019;

(â&#x2C6;&#x2019;

1 2đ?&#x2018;&#x161;â&#x2C6;&#x161;đ?&#x153;&#x2030;)đ?&#x2018;Ą

=0

=0


29 đ?&#x2018;&#x2019;

(

1 2đ?&#x2018;&#x161;â&#x2C6;&#x161;đ?&#x153;&#x2030;)đ?&#x2018;Ą

=đ?&#x2018;&#x2019;

(â&#x2C6;&#x2019;

1 2đ?&#x2018;&#x161;â&#x2C6;&#x161;đ?&#x153;&#x2030;)đ?&#x2018;Ą

or đ?&#x2018;&#x2019;

1 ( â&#x2C6;&#x161;đ?&#x153;&#x2030;)đ?&#x2018;Ą đ?&#x2018;&#x161;

=1 1

But we assumed đ?&#x2018;Ą > 0. Therefore, this equation is satisfied only if đ?&#x2018;&#x161; â&#x2C6;&#x161;đ?&#x153;&#x2030; = 0 or đ?&#x153;&#x2030; = 0. This is a contradiction because the system is over-damped so that đ?&#x153;&#x2030; > 0. Therefore, the mass will never pass though equilibrium again and its position function will have one of the following profiles, depending on the initial velocity: 0.4

0.4 0.2

0.2

2

4

6

8

2

4

6

8

0.2

0.2 0.4

0.4

đ?&#x2018;ĽĚ&#x2021; (0) > 0

đ?&#x2018;ĽĚ&#x2021; (0) < 0

How Linear was the Spring Used in Class? The spring used in class was a typical coil as the ones found in a physics lab of any public institution. The data bellow indicate that the spring used for this classroom demonstration was pretty linear. DATA FOR THE SPRING EXPERIMENT PERFORMED IN CLASS Extension (cm)

MASS (gm)

0

0

1.4

50

2.0

70

2.8

100

4.5

150

We can find đ?&#x2018;&#x2DC; from the slope of the line. Since we are using mass, if the spring truly obeys Hookeâ&#x20AC;&#x2122;s Law, then we should find that đ?&#x2018;&#x20AC;đ?&#x2018;&#x201D; = đ?&#x2018;&#x2DC; đ?&#x2018;&#x2DC;đ?&#x2018;Ľ or đ?&#x2018;&#x20AC; = đ?&#x2018;Ľ. In other words, the variable đ?&#x2018;&#x20AC;

m a s s

160 140 120 100 80 60 40 20 0 0

2

4

6

extension

đ?&#x2018;&#x201D;

(mass) should be a linear function of đ?&#x2018;Ľ.

The slope of the line is đ?&#x2018;&#x2DC;/đ?&#x2018;&#x201D; so by calculating the slope, we can obtain the value of đ?&#x2018;&#x2DC;. The computed value of the slope is 0.03226 Kg/cm or 3.226 Kg/m. Thus,


30 đ?&#x2018;&#x2DC; = đ?&#x2018;&#x201D; Ă&#x2014; 3.226 â&#x2030;&#x2C6; đ?&#x;&#x2018;đ?&#x;?. đ?&#x;&#x201D;đ?&#x;? đ???đ??­/đ??Ś The real test now is to measure the natural frequency of vibration and compare it to the observed 1 frequency. Using the empirical values of đ?&#x2018;&#x2DC; with a mass of 0.50 kg in the formula đ?&#x2018;&#x201C; = 2đ?&#x153;&#x2039; â&#x2C6;&#x161;đ?&#x2018;&#x2DC;/đ?&#x2018;&#x20AC; we obtain the theoretical value of frequency: 1 đ?&#x2018;&#x201C;đ?&#x2018;Ąâ&#x201E;&#x17D;đ?&#x2018;&#x2019;đ?&#x2018;&#x153;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;?đ?&#x2018;&#x17D;đ?&#x2018;&#x2122; = â&#x2C6;&#x161;31.61/0.5 â&#x2030;&#x2C6; đ?&#x;?. đ?&#x;?đ?&#x;&#x2022; đ??&#x2021;đ??ł. 2đ?&#x153;&#x2039; The observed frequency9 is đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;?đ?&#x2018; đ?&#x2018;&#x2019;đ?&#x2018;&#x;đ?&#x2018;Łđ?&#x2018;&#x2019;đ?&#x2018;&#x2018; =

10 â&#x2030;&#x2C6; đ?&#x;?. đ?&#x;?đ?&#x;&#x2019; đ??&#x2021;đ??ł. 8.03

Acknowledgments: I wish to thank, first and foremost, to my students of summer 2010 who were the first to review the more extensive set of notes on which this article is based â&#x20AC;&#x201C; they provided me with valuable input throughout the term, as well as after. I wish to also thank all the other students who, throughout the years after I started writing these notes, provided me with the encouragement to continue. My gratitude also goes to Professor Loretta Blanchett who took the time to read the article and provide me with corrections to several typographical errors. Last but not least, I wish to thank the physics department at the Kendall Campus of Miami Dade College for allowing me to use its equipment.

Bibliography 1. Comenetz, Michael. Calculus, the Elements. World Scientific. 2. Edwards and Penney. Differential Equations and Boundary Value problems, 4th Ed. PearsonPrentice Hall. 3. French, A.P. Vibration and Waves, The MIT press. 4. Morin, David. Introduction to Classical Mechanics with Problems and Solutions, 4th printing. Cambridge University Press. 5. Ohanian, Hans and Market, John. Physics for Engineers and Scientists, 3rd Ed. Norton and Company.

9

In order to measure the frequency, we time 10 oscillations. Thus, dividing this time by 10 allows us to obtain the observed frequency. Several trials of this should be done in order to find an average that is more representative of the true frequency. Three measurements were made resulting in 8.1, 8.0, and 8.0 seconds Thus, â&#x152;Šđ?&#x2018;&#x2021;â&#x152;Ş â&#x2030;&#x2C6; 8.03 sec. My gratitude to the physics department at Miami Dade Collegeâ&#x20AC;&#x2122;s Kendall Campus for lending me the equipment, and to Ms. Isabella Orozco for assisting me with time measurements.


31 CLASSROOM NOTES: BABY STEPS TOWARD THE FUNDAMENTAL THEOREM OF CALCULUS AND FIRST ORDER EQUATIONS Prof. Rene Barrientos Department of Mathematics, Miami Dade College | Hialeah Campus FL 33012 rbarrien@mdc.edu Abstract: Introducing students to some basic theory in mathematics courses can be often a traumatic experience. The common position taken by them (unless they happen to be math majors) is that applications and how to use the formulas is more important than a fundamental understanding of the underlying mathematics. By the time they get to differential equations, the damage has often been done, theory is shunned at all costs, and their understanding of, for example, the Fundamental Theorem of Calculus is superficial at best â&#x20AC;&#x201C; just enough to do applications involving the Riemann integral. The aim of this brief paper to help students bridge that divide between wrote learning and a deeper understanding of mathematics, and to hopefully encourage them to want to learn more. Key Words: Initial value problems, Fundamental Theorem of Calculus, first order equation, antiderivatives and integrals, Riemann integral, differential equation, differential. AMS Subject Classification: 02 The most basic initial value problem (ivp) in differential equations is the following: đ?&#x2018;Ś â&#x20AC;˛ = đ?&#x2018;&#x201C;(đ?&#x2018;Ľ); đ?&#x2018;Ś(đ?&#x2018;Ľ0 ) = đ?&#x2018;Ś0

(1)

whose solution, students learn, is obtained by â&#x20AC;&#x153;integrating and applying the initial conditionâ&#x20AC;?. Thus, to solve the equation đ?&#x2018;Ś â&#x20AC;˛ = đ?&#x2018;Ľ 2 + đ?&#x2018;&#x2019; â&#x2C6;&#x2019;đ?&#x2018;Ľ ; đ?&#x2018;Ś(0) = 1 We integrate, đ?&#x2018;Ś(đ?&#x2018;Ľ) = â&#x2C6;Ť(đ?&#x2018;Ľ 2 + đ?&#x2018;&#x2019; â&#x2C6;&#x2019;đ?&#x2018;Ľ ) đ?&#x2018;&#x2018;đ?&#x2018;Ą =

đ?&#x2018;Ľ3 â&#x2C6;&#x2019; đ?&#x2018;&#x2019; â&#x2C6;&#x2019;đ?&#x2018;Ľ + đ?&#x2018;? 3

And apply the initial condition đ?&#x2018;Ś(0) = 1 which tells us that đ?&#x2018;? = 2. Therefore, đ?&#x2019;&#x161;(đ?&#x2019;&#x2122;) =

đ?&#x2019;&#x2122;đ?&#x;&#x2018; â&#x2C6;&#x2019; đ?&#x2019;&#x2020;â&#x2C6;&#x2019;đ?&#x2019;&#x2122; + đ?&#x;? đ?&#x;&#x2018;

With this in mind, let us follow the same recipe to solve the following problem: 2

đ?&#x2018;Ś â&#x20AC;˛ = đ?&#x2018;&#x2019; â&#x2C6;&#x2019;đ?&#x2018;Ľ ; đ?&#x2018;Ś(1) = 2 Integrating as we did before, 2

đ?&#x2018;Ś(đ?&#x2018;Ľ) = â&#x2C6;Ť đ?&#x2018;&#x2019; â&#x2C6;&#x2019;đ?&#x2018;Ľ đ?&#x2018;&#x2018;đ?&#x2018;Ą + đ?&#x2018;? But now we cannot evaluate the integral in order to apply the initial condition. How do we obtain the arbitrary constant đ?&#x2018;? to complete the problem? The answer is: we donâ&#x20AC;&#x2122;t need to â&#x20AC;&#x201C; we invoke the Fundamental Theorem of Calculus, which now lies dormant in the studentsâ&#x20AC;&#x2122; memory, and


32 which is often only poorly understood by them. Let us first review what this important theorem says. The Fundamental Theorem of Calculus (FTC) The symbol â&#x2C6;Ť đ?&#x2019;&#x2021;(đ?&#x2019;&#x2122;) đ?&#x2019;&#x2026;đ?&#x2019;&#x2122; represents any function đ??š(đ?&#x2018;Ľ) with the property that đ??šâ&#x20AC;˛(đ?&#x2018;Ľ) = đ?&#x2018;&#x201C;(đ?&#x2018;Ľ). Since the derivative of a constant is 0, if đ??ş(đ?&#x2018;Ľ) = đ??š(đ?&#x2018;Ľ) + đ?&#x2018;?, then đ??şâ&#x20AC;˛(đ?&#x2018;Ľ) = đ??šâ&#x20AC;˛(đ?&#x2018;Ľ) = đ?&#x2018;&#x201C;(đ?&#x2018;Ľ). Thus, if đ??š and đ??ş are any two antiderivatives10 of đ?&#x2018;&#x201C; on some interval (đ?&#x2018;&#x17D;, đ?&#x2018;?), then đ??š(đ?&#x2018;Ľ) â&#x2C6;&#x2019; đ??ş(đ?&#x2018;Ľ) = 0 and by definition đ?&#x2018;&#x2018; â&#x2C6;Ť đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) đ?&#x2018;&#x2018;đ?&#x2018;Ľ = đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) đ?&#x2018;&#x2018;đ?&#x2018;Ľ on that interval. But these operations cannot be reversed: â&#x2C6;Ť

đ?&#x2018;&#x2018; đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) đ?&#x2018;&#x2018;đ?&#x2018;Ľ = đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) đ?&#x2018;&#x2018;đ?&#x2018;Ľ

is not true in general because integrals of functions are not uniquely determined. However, this is true when applied to the definite integrals. The Fundamental Theorem tells us how the indefinite integral and the definite (Riemann) integral are related, and that relation has two versions: Version I: Let đ?&#x2018;&#x17D; and đ?&#x2018;? be in the interval on which đ?&#x2018;&#x201C; is integrable, and let đ??š(đ?&#x2018;Ľ) be an antiderivative of đ?&#x2018;&#x201C; on that interval. Then, đ?&#x2019;&#x192;

â&#x2C6;Ť đ?&#x2019;&#x2021;(đ?&#x2019;&#x2122;) đ?&#x2019;&#x2026;đ?&#x2019;&#x2122; = đ?&#x2018;­(đ?&#x2019;&#x192;) â&#x2C6;&#x2019; đ?&#x2018;­(đ?&#x2019;&#x201A;) đ?&#x2019;&#x201A;

This is the version that is used by first-semester calculus students in order to solve area problems and other applications. We can rephrase this statement as follows: since đ??šâ&#x20AC;˛(đ?&#x2018;Ľ) = đ?&#x2018;&#x201C;(đ?&#x2018;Ľ), đ?&#x2019;&#x192;

â&#x2C6;Ť đ?&#x2018;­â&#x20AC;˛(đ?&#x2019;&#x2122;) đ?&#x2019;&#x2026;đ?&#x2019;&#x2122; = đ?&#x2018;­(đ?&#x2019;&#x192;) â&#x2C6;&#x2019; đ?&#x2018;­(đ?&#x2019;&#x201A;) đ?&#x2019;&#x201A;

Version II: Let đ?&#x2018;&#x201C; be continuous on [đ?&#x2018;&#x17D;, đ?&#x2018;?] and đ?&#x2018;&#x17D; â&#x2030;¤ đ?&#x2018;Ľ â&#x2030;¤ đ?&#x2018;?. Then the function đ?&#x2019;&#x2122;

đ??&#x2039;(đ?&#x2019;&#x2122;) = â&#x2C6;Ť đ?&#x2019;&#x2021;(đ?&#x2018;Ą) đ?&#x2019;&#x2026;đ?&#x2018;Ą đ?&#x2019;&#x201A;

is continuous on [đ?&#x2018;&#x17D;, đ?&#x2018;?] and differentiable on (đ?&#x2018;&#x17D;, đ?&#x2018;?), and đ??&#x2039;â&#x20AC;˛(đ?&#x2019;&#x2122;) = đ?&#x2019;&#x2021;(đ?&#x2019;&#x2122;) 11

Equivalently , More precisely, đ??šâ&#x20AC;˛ = đ??şâ&#x20AC;˛ on (đ?&#x2018;&#x17D;, đ?&#x2018;?) except possibly on a subset of measure 0, but this notion is not introduced at this stage of the learning process. đ?&#x2018;&#x2018; đ?&#x2018;Ľ 11 The equation â&#x2C6;Ťđ?&#x2018;&#x17D; đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) đ?&#x2018;&#x2018;đ?&#x2018;Ľ can lead to confusion because đ?&#x2018;Ľ is being used for two different purposes. It is therefore customary đ?&#x2018;&#x2018;đ?&#x2018;Ľ to distinguish between the variable upper limit of integration and the dummy variable of integration by using two different letters. 10


33 đ?&#x2019;&#x2026; đ?&#x2019;&#x2122; â&#x2C6;Ť đ?&#x2019;&#x2021;(đ?&#x2018;Ą) đ?&#x2019;&#x2026;đ?&#x2018;Ą = đ?&#x2019;&#x2021;(đ?&#x2019;&#x2022;) đ?&#x2019;&#x2026;đ?&#x2019;&#x2122; đ?&#x2019;&#x201A; Version 1 provides us with a method to solve initial value problems in elementary differential equations, and version 2 is used to prove certain things about their solutions. Because this theorem is so important, students should at the very least see an argument in its favor, and one that is not too difficult to follow so that it may be remembered. The following â&#x20AC;&#x153;proofâ&#x20AC;? also underscores the way the Fundamental Theorem is used in the context of first order linear differential equations.

Let đ?&#x2018;&#x201C; be a continuous function on the interval [đ?&#x2018;Ľ0 , đ?&#x2018;Ľ], and consider its graph on this interval. Partition the interval with points đ?&#x2018;Ą0 = đ?&#x2018;Ľ0 < đ?&#x2018;Ą1 < đ?&#x2018;Ą2 < â&#x2039;Ż < đ?&#x2018;Ąđ?&#x2018;&#x203A; = đ?&#x2018;Ľ. Then using the differential approximation â&#x2C6;&#x2020;đ?&#x2018;Śđ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 â&#x2030;&#x2C6; đ?&#x2018;&#x2018;đ?&#x2018;Śđ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 on the đ?&#x2018;&#x2013; đ?&#x2018;Ąâ&#x201E;&#x17D; subinterval the figure suggest the following: đ?&#x2018;Ś = đ?&#x2018;&#x201C;(đ?&#x2018;Ľ)

đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) â&#x2C6;&#x2019; đ?&#x2018;&#x201C;(đ?&#x2018;Ľ0 ) )

đ?&#x2018;&#x161; = đ?&#x2018;&#x201C; â&#x20AC;˛ (đ?&#x2018;Ľ)

đ?&#x2018;&#x2018;đ?&#x2018;Ś

â&#x2C6;&#x2020;đ?&#x2018;Śđ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 â&#x2C6;&#x2020;đ?&#x2018;Ą

đ?&#x2018;&#x201C;(đ?&#x2018;Ľ0 ) đ?&#x2018;Ľ0

đ?&#x2018;Ąđ?&#x2018;&#x2013;â&#x2C6;&#x2019;1

â&#x2C6;&#x2020;đ?&#x2018;Ą â&#x2C6;&#x2020;đ?&#x2018;Śđ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 â&#x2030;&#x2C6; (đ?&#x2018;&#x2018;đ?&#x2018;Ś)đ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 = đ?&#x2018;&#x201C;â&#x20AC;˛(đ?&#x2018;Ąđ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 )â&#x2C6;&#x2020;đ?&#x2018;Ąđ?&#x2018;&#x2013; đ?&#x2018;Ąđ?&#x2018;&#x2013;

đ?&#x2018;Ľ

Figure 1 Hence, đ?&#x2018;&#x203A;â&#x2C6;&#x2019;1

đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) â&#x2C6;&#x2019; đ?&#x2018;&#x201C;(đ?&#x2018;Ľ0 ) = â&#x2C6;&#x2018; â&#x2C6;&#x2020;đ?&#x2018;Śđ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 But â&#x2C6;&#x2020;đ?&#x2018;Śđ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 â&#x2030;&#x2C6; đ?&#x2018;&#x201C; â&#x20AC;˛ (đ?&#x2018;Ąđ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 )â&#x2C6;&#x2020;đ?&#x2018;Ąđ?&#x2018;&#x2013; . Therefore,

đ?&#x2018;&#x2013;=0 đ?&#x2018;&#x203A;â&#x2C6;&#x2019;1

đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) â&#x2C6;&#x2019; đ?&#x2018;&#x201C;(đ?&#x2018;Ľ0 ) â&#x2030;&#x2C6; â&#x2C6;&#x2018; đ?&#x2018;&#x201C;â&#x20AC;˛(đ?&#x2018;Ąđ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 )â&#x2C6;&#x2020;đ?&#x2018;Ąđ?&#x2018;&#x2013; In the limit as đ?&#x2018;&#x203A; â&#x2020;&#x2019; â&#x2C6;&#x17E;, the sum approaches

đ?&#x2018;&#x2013;=0 đ?&#x2018;Ľ â&#x2C6;Ťđ?&#x2018;Ľ đ?&#x2018;&#x201C;â&#x20AC;˛(đ?&#x2018;Ą)đ?&#x2018;&#x2018;đ?&#x2018;Ą 0

and therefore

đ?&#x2019;&#x2122;

đ?&#x2019;&#x2021;(đ?&#x2019;&#x2122;) â&#x2C6;&#x2019; đ?&#x2019;&#x2021;(đ?&#x2019;&#x2122;đ?&#x;&#x17D; ) = â&#x2C6;Ť đ?&#x2019;&#x2021;â&#x20AC;˛(đ?&#x2019;&#x2022;)đ?&#x2019;&#x2026;đ?&#x2019;&#x2022; đ?&#x2019;&#x2122;đ?&#x;&#x17D;

which is the statement of Version 1.

Experience tells us that students in an elementary course will be even more confused by this practice and therefore we will use one variable for both purposes.


34 In an introductory differential equations course, this version is used in the following way: consider again the initial value problem (1): đ?&#x2018;&#x2018;đ?&#x2018;Ś = đ?&#x2018;&#x201C;(đ?&#x2018;Ľ); đ?&#x2018;Ś(đ?&#x2018;Ľ0 ) = đ?&#x2018;Ś0 đ?&#x2018;&#x2018;đ?&#x2018;Ľ Let us integrate đ?&#x2018;Śâ&#x20AC;˛(đ?&#x2018;Ľ) on the interval [đ?&#x2018;Ľ0 , đ?&#x2018;Ľ]. Then, đ?&#x2018;Ľ

đ?&#x2018;Ľ

â&#x2C6;Ť đ?&#x2018;Śâ&#x20AC;˛(đ?&#x2018;Ľ) đ?&#x2018;&#x2018;đ?&#x2018;Ľ = â&#x2C6;Ť đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) đ?&#x2018;&#x2018;đ?&#x2018;Ľ đ?&#x2018;Ľ0

đ?&#x2018;Ľ0

Then by the Fundamental Theorem, đ?&#x2018;Ľ

đ?&#x2018;Ś(đ?&#x2018;Ľ) â&#x2C6;&#x2019; đ?&#x2018;Ś(đ?&#x2018;Ľ0 ) = â&#x2C6;Ť đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) đ?&#x2018;&#x2018;đ?&#x2018;Ľ đ?&#x2018;&#x17D;

So that đ?&#x2019;&#x2122;

đ?&#x2019;&#x161;(đ?&#x2019;&#x2122;) = đ?&#x2019;&#x161;đ?&#x;&#x17D; + â&#x2C6;Ť đ?&#x2019;&#x2021;(đ?&#x2019;&#x2122;) đ?&#x2019;&#x2026;đ?&#x2019;&#x2122; đ?&#x2019;&#x201A;

More formally, Theorem 1 Let đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) be a continuous function on an open interval đ??ź containing đ?&#x2018;Ľ0 . Then (1) has a solution given by đ?&#x2018;Ś(đ?&#x2018;Ľ)

đ?&#x2018;Ľ

= đ?&#x2018;Ś0 + â&#x2C6;Ť đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) đ?&#x2018;&#x2018;đ?&#x2018;Ľ on the interval đ??ź.

(2)

đ?&#x2018;Ľ0

Proof We first show that this solution satisfies the initial condition: đ?&#x2018;Ľ0

đ?&#x2018;Ś(đ?&#x2018;Ľ0 ) = đ?&#x2018;Ś0 + â&#x2C6;Ť đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) đ?&#x2018;&#x2018;đ?&#x2018;Ľ = đ?&#x2018;Ś0 đ?&#x2018;Ľ0

Therefore, the initial condition is satisfied. Next we show that it solves the differential equation. By the Fundamental Theorem (Version 2), đ?&#x2018;Ľ đ?&#x2018;&#x2018;đ?&#x2018;Ś đ?&#x2018;&#x2018; = (đ?&#x2018;Ś0 + â&#x2C6;Ť đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) đ?&#x2018;&#x2018;đ?&#x2018;Ľ) đ?&#x2018;&#x2018;đ?&#x2018;Ľ đ?&#x2018;&#x2018;đ?&#x2018;Ľ đ?&#x2018;Ľ0

=

đ?&#x2018;&#x2018; đ?&#x2018;Ľ â&#x2C6;Ť đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) đ?&#x2018;&#x2018;đ?&#x2018;Ľ đ?&#x2018;&#x2018;đ?&#x2018;Ľ đ?&#x2018;Ľ0

= đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) Hence, (2) is a solution of (1). Q.E.D A Uniqueness Theorem


35 The next question asks whether (2) is the only solution of (1). The following theorem says it is. Theorem 2 The ivp (1) has as unique solution đ?&#x2018;Ľ

đ?&#x2018;Ś(đ?&#x2018;Ľ) = đ?&#x2018;Ś0 + â&#x2C6;Ť đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) đ?&#x2018;&#x2018;đ?&#x2018;Ľ đ?&#x2018;Ľ0

Proof: Let đ?&#x2018;Ś1 (đ?&#x2018;Ľ) and đ?&#x2018;Ś2 (đ?&#x2018;Ľ) be two solutions in the interval đ??ź. Then đ?&#x2018;Śâ&#x20AC;˛1 = đ?&#x2018;&#x201C;(đ?&#x2018;Ľ); đ?&#x2018;Ś1 (đ?&#x2018;Ľ0 ) = đ?&#x2018;Ś0 and đ?&#x2018;Śâ&#x20AC;˛2 = đ?&#x2018;&#x201C;(đ?&#x2018;Ľ); đ?&#x2018;Ś2 (đ?&#x2018;Ľ0 ) = đ?&#x2018;Ś0 Hence

(đ?&#x2018;Ś1 â&#x2C6;&#x2019; đ?&#x2018;Ś2 )â&#x20AC;˛ = đ?&#x2018;Śâ&#x20AC;˛1 â&#x2C6;&#x2019; đ?&#x2018;Śâ&#x20AC;˛2 = đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) â&#x2C6;&#x2019; đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) = 0

Therefore, đ?&#x2018;Ś1 (đ?&#x2018;Ľ) â&#x2C6;&#x2019; đ?&#x2018;Ś2 (đ?&#x2018;Ľ) is a constant, call it đ?&#x2018;?: đ?&#x2018;Ś1 â&#x2C6;&#x2019; đ?&#x2018;Ś2 = đ?&#x2018;? What could the value of the constant đ?&#x2018;? be? Exercise: use the fact that both solutions must satisfy the same initial condition to show that đ?&#x2018;? = 0. Having established that đ?&#x2018;? = 0, we conclude that đ?&#x2018;Ś1 (đ?&#x2018;Ľ) = đ?&#x2018;Ś2 (đ?&#x2018;Ľ) and therefore all solutions of the IVP are given by (2). With these results in mind, we can return to the problem 2

đ?&#x2018;Ś â&#x20AC;˛ = đ?&#x2018;&#x2019; â&#x2C6;&#x2019;đ?&#x2018;Ľ ; đ?&#x2018;Ś(1) = 2 Invoking (2), the (unique) solution of this equation is given by đ?&#x2018;Ľ

2

đ?&#x2018;Ś(đ?&#x2018;Ľ) = 2 + â&#x2C6;Ť đ?&#x2018;&#x2019; â&#x2C6;&#x2019;đ?&#x2018;Ľ đ?&#x2018;&#x2018;đ?&#x2018;Ľ 1

This is perhaps a good place to stress the need for two variables in equation (2). What is đ?&#x2018;Ś(1.5)? It is very easy for an inexperienced student to write 1.5

2

đ?&#x2018;Ś(1.5) = 2 + â&#x2C6;Ť đ?&#x2018;&#x2019; â&#x2C6;&#x2019;1.5 đ?&#x2018;&#x2018;đ?&#x2018;Ľ 1

which is, of course, incorrect. This ambiguity can be eliminated by introducing a different letter for the dummy variable of integration, say đ?&#x2018;Ą, and write instead đ?&#x2018;Ľ

2

đ?&#x2018;Ś(đ?&#x2018;Ľ) = 2 + â&#x2C6;Ť đ?&#x2018;&#x2019; â&#x2C6;&#x2019;đ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;Ą 1

It is now clear what needs to be done in order to approximate đ?&#x2018;Ś(1.5):


36 1.5

2

đ?&#x2018;Ś(1.5) = 2 + â&#x2C6;Ť đ?&#x2018;&#x2019; â&#x2C6;&#x2019;đ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;Ą 1

The values of this function may be approximated using any of several methods of numerical integration. Concluding remarks: Although mathematical rigor is important, it sometime hinders students from engaging in a deeper study of the subject. By relaxing it and providing intuitive arguments such as the ones presented in these notes, one can help students to appreciate the growing pains through which an idea goes before becoming a full-fledged mathematical theorem, deprived of all the scaffoldings that supported it while it matured. Acknowledgements I wish to express my gratitude to Blanca, my soulmate if there be a soul, whose encouragement and patience prompted me to write-to-publish. Otherwise, these and other notes that I have written over the years would not be seeing the light of day except for the times that my students make use them (I hope). I would also like to thank Professor Matthew Page who took the time to read this short article and provide me with valuable suggestions and corrections. References 1. Hirsch, Morris, et. al., Differential Equations, Dynamical Systems and an Introduction to Chaos, 2nd Ed. 2. Stewart, James, Calculus, Early Transcendentals, 7th Ed.


37 Becoming Human: Building Bridges from Li to Logos A comparative study of the Way toward the Good according to Confucius and Plato

Professor Sarah Jacob Department of Philosophy Arts & Letters Miami Dade College, West Campus 3800 NW 115th Avenue Doral, FL 33178 email: sjacob@mdc.edu

ABSTRACT The period known as the â&#x20AC;&#x153;Axial Age,â&#x20AC;? a coin termed by Karl Jaspers to describe a huge shift in thought that took place roughly from 800-200 BCE, is a time that scholars return to again and again to rediscover the earliest development of human ethical systems. In the ancient Chinese world, one of the earliest (and definitely the most renowned) recorded Chinese philosophers who sought to rethink humanity in terms of ethical development is Confucius (551-479 BCE.) In Ancient Greece, the first systematic philosopher is Plato (circa 428-348BCE). These two thinkers, disparate in their analysis and method, reinterpreted the human quest as that which is ethical. Understanding what it is to become human according to the ancient thinkers is as important today as it was 2,500 years ago. By doing so we may establish implicit connections that are relevant for us as we too struggle with becoming human in a world that too often pushes us towards the inhumane. This paper offers a brief overview of some of these connections.

Keywords: Confucius, Plato, virtue, ethics, axial age

Becoming Human: Building Bridges from Li to Logos


38 Introduction In the ancient Chinese world, one of the earliest (and definitely the most renowned) recorded Chinese philosophers who sought to rethink humanity in terms of ethical development is Confucius (551-479 BCE.) In Ancient Greece, the first systematic philosopher is Plato12 (circa 428-348BCE). These two thinkers, disparate in their analysis and method, reinterpreted the human quest as that which is ethical. Ren, the Chinese term roughly translated as benevolence, humanity and goodness, is the ultimate goal of the individual; and according to Confucius it is only achievable through Li (ritual), the practice of which rectifies relationships and prescribes behavior accordingly. For Plato, we move toward the Good when the human soul is reminded of it through earthly transactions that bring to mind a perfection that lies beyond this world. The Confucian Sage-King and Platonic Philosopher-King are described respectively by Confucius and Plato as paragons of virtue, providing a Utopian vision of what humans ought to become,13 emphasizing that living virtuously is the only way to accomplish a life that is meaningful, authentic and good. It is indisputable that Confucius and Plato helped to reshape human desires, values and purpose for future generations in their respective Eastern and Western cultures and even today maintain an unwavering influence on thought. Although there are irreconcilable differences between the two philosophies of life, purpose and meaning proposed by both thinkers, there are also areas where profound bridges can be made and stunning moments of harmony become apparent. Both shared an emphasis on self-cultivation and education, and prescribed virtuous thought and behavior as integral to the development of the ethical human and the harmonious society. Understanding what it is to become human according to the ancient thinkers is as important today as it was 2,500 years ago. By doing so we may establish implicit connections that are relevant for us as we too struggle with becoming human in a world that too often pushes us towards the inhumane. Li and Logos14 For Confucius, Li15 plays a vital role in the process of becoming human; and in this way, Li is one of the primary Confucian virtues. Li acts as a container for human potential and it is only through the purposeful practice of Li that Ren (goodness/humane-ness), or at least the movement toward Ren, can be achieved. Li can be likened to repetition of thought and action and helps to mold one’s intentions and concept of “right” action as “natural” action. It is both a restrictive and liberating force as it supports relationships and actions according to one’s social place, but at the same time creates an environment of security where human flourishing can take place. Li is paradigmatic. It is the boundary that allows the expression of belief and the safe application of desire. We participate in Li as it helps us develop our relationship with the community of which we are part. As Chad Hansen notes, “The goal was not simply to master the Li (ritual), but to cultivate the more abstract and intuitive Ren (humanity).”16 Confucius, who witnessed the Socrates in some ways serves as a more compatible partner for Confucius. However, most of Socrates insights are documented by, and then integrated into, Plato’s elaborate metaphysics, epistemology, political philosophy and ethics. By comparing Confucius with Plato, I am also to some degree comparing Confucius with Socrates. Certainly some main ideas were first discussed by Socrates. The concept that the soul is the seat of the human virtues is one believed to be first offered by Socrates. 13 Plato explicitly points out that to become a Philosopher-King requires a virtuous, enlightened soul, hence it is for most an improbability. 14 Logos is used here to mean “the reason through which order follows”. 15 How we interpret Li is dependent upon the context. It can refer to tradition, religious rites/ceremony, socialization, the art of governing, personal spiritual practices, self-cultivation, symbolic communication/gestures and ritual as it pertains to the term De (virtue) Ren (good human) and Dao (the way). Li is integral in the way of the good human. (interpretations taken from Dr.Zhang class hand-out) 16 Hansen, Chad, Daoist Theory of Chinese Thought, “Confucius: The Baseline” (New York: Oxford University Press., 1992), p. 59. 12


39 fragmentation of the once cohesive Zhou empire into warring states, interpreted that it is the breakdown of ritual propriety that was key to the disharmony that ensued. The rituals, which were more or less still implemented had lost their meaning, or were being altered without thought. Confucius determined that participating in Li is not a simple formation of habit as they had turned into, instead it is the cultivation of what has been translated as “harmonious ease.” In a discussion on Li, Confucius notes, When it comes to the practice of ritual, it is harmonious ease that is to be valued. It is precisely such harmony that makes the Way of the Former Kings so beautiful. If you merely stick rigidly to ritual in all matters, great and small, there will remain that which you cannot accomplish. Yet if you know enough to value harmonious ease but try to attain it without being regulated by the rites, this will not work either.17 This passage suggests that the empty practice of Li as mechanical does not fulfill the role that Li plays in helping to establish a peaceful, harmonious community. Accordingly, we are not merely “vessels of ritual” (thoughtless habitual action), but instead the goal is that human “nature” is cultivated (not oppressed) through “Li.” As noted by Halls and Ames, Ritual action is a pervasive condition of Confucius’ vision of social harmony because, by definition, it not only permits, but actually requires personalization…people are unique, and…they must be orchestrated into relationships that permit expression of this uniqueness. A formal ceremony without this commitment is hollow, meaningless, and even antisocial parody; on the other hand, a ceremony that coordinates and expresses the genuineness of its participants is a source of social cohesion and enjoyment.18 We are purposeful beings, and for Confucius Li is “The Way” to achieve that purpose. It is through Li that humans “broaden” the Way. A deeper interpretation of the role of Li can infer that not only does it force the individual to internalize certain normative behaviors, but it also inhibits and directs desire, thereby cultivating the objects of desire. Li is the blueprint upon which the correct “Way” of living can emerge as natural. It is indeed the essence of Confucian philosophy that presents Confucius as both moral exemplar and an astute sociologist: humans are ritualistic beings. Whether guided or not, humans create rituals and imbue these rituals with meaning.19 Hence, the type of rituals developed are of the utmost importance in the development of Ren as it pertains to both the individual and society. For Confucius, it is lack of the right rituals as established by the Zhou Empire that is at the root of a corrupt society rife with greed and selfinterest. It is Li that paves the Way of Tian (heaven) and through ritual that, as the Confucian Xunzi elaborates, that we can create order from our desires. Desire is part of the human condition, but the object of desire is not. Ritual helps determine the objective. Xunzi asserts, From what did ritual arise? I say: Humans are born having desires. When they have desires but to not get the objects of their desires, then they cannot but see some means of satisfaction. If there is no limit to their seeking, then they cannot help but struggle with each other. If they struggle with each other then there will be chaos, and if there is chaos then they will be impoverished. The former kings hated such chaos, and so they “The Analects,” Readings in Classical Chinese Philosophy, ed. Philip J. Ivanhoe and Bryan W. Van Norden. (Indianapolis: Hackett Publishing Inc.,1992) ch.1.2 18 Hall, David and Ames, Roger, “The Chinese Community without Transcendence” Thinking from the Han. (Albany: State University of New York Press) p.272 19 A great example of this tendency can be seen in the formation of gangs, the leader of which often requires pernicious initiation tests before membership is allowed. While this does not meet Confucian ritual requirements, it does imply that as a community we have a tendency toward ritualistic behavior. 17


40 established rituals and the standards of righteousness in order to allot things to people, to nurture their desires, and to satisfy their seeking. They caused their desires never to exhaust material goods, and material good never to be depleted by desires, so that the two support each other and prosper. This is how ritual arose.20 Li is the foundation of the establishment and in this case, it is class conscious. The rituals engaged in by the king, from the type of robes he will wear to the sort of funeral he will have differs considerably from the ritual attire worn by a farmer or the sort of ceremony he will expect or even want at his funeral. The type of food the farmer desires, has access to, and way it is prepared and served does little to resemble the kind of food the King desires, has access to, and the way it is prepared and served. Therefore, rituals play a significant role in socialization (the rectification of relationships), naturalizing the concept of social classes, and is a means to manage the surplus and scarcity inherent in the natural world. Of that which is abundant, the common farmer can take freely; of that which is scarce is kept for the rare class of Kings and Princes. There is a correlation between one’s station in life and the products that are rightfully his. Xunzi offers the most prescriptive and civilizing explanation of the role of ritual. He notes, Thus, ritual is a means to nurture… the mouth…the nose…the eyes…the ears…the body. The gentleman not only obtains its nurturing but also loves its differentiations. What is meant by “differentiations”? I say: It is for noble and lowly to have their proper ranking, for elder and youth to have their proper distance, and for poor and rich, humble and eminent each to have their proper weights. And so, in the Grand Chariot of the Emperor there are cushions, as a means to nurture his body…21 Li brings out the natural distinctions between things and people; and it can be said that it likens every gesture to an event that is ordered. The sun is different from the stars that surround it. These value distinctions are what, according to both Confucius and Xunzi, make us human. Karl Jaspers in his brief review of Confucius emphasizes the totality of Li: Ritual is life. He notes, Confucius set forth the li as a whole; he observed them, collected the, formulated and arranged them. His vision embraced the whole world of Chinese customs: the right way of walking, greeting, behaving in company, always in accordance with the particular situation; the ways of conducting sacrifices and observing holidays; the rites of marriage, birth, death, and burial; the rules of administration; the customs governing work, war, the family, the priesthood, and court; the order of the days and seasons, the stages in life.22 Hall and Ames note that the Chinese character shen, implicit in the character li, means both spirituality and divinity. However, its root meaning is “extension” and Hall and Ames note, “As persons become increasingly spiritual through the effective performance of communal life forms (li), and becomes increasingly inspirational for their communities, they are exalted and move in the direction of divinity.”23 According to this interpretation, the individual extends beyond his or her physical boundaries by participating in Li which not only establishes behavior for the future, but derives it from cultural heroes from the past, so they may still bear influence on us today. For Confucius, the Duke of Zhou was a constant source of inspiration because of his apparent commitment to Ren. For post-Confucius Confucians, it is Confucius himself who is that source. Ibid. Hall and Ames, “Xunxi” “Discourse on Ritual” p. 274 Ibid. pp. 274-275 22 Popper, Karl, Socrates, Buddha, Confucius, Jesus, ed. Hannah Arendt, trans. Ralph Manheim, (New York: Harcourt, Brace & World, Inc., 1957) p. 45 23 Ibid. Hall and Ames, “The Chinese Community without Transcendence” p. 265 20 21


41 The concept of individual extension also pierces through the boundaries of life and death. Through memory and ideation of these profoundly influential people, they are kept alive and remain influential and are still metaphorically alive in the community. This is indicated in The Analects, when Confucius says “How seriously I have declined! It has been so long since I dreamed of meeting the Duke of Zhou.”24 The Duke of Zhou, though deceased, was a role model and one might say, an existent friend of Confucius. Notwithstanding the metaphysical and dualistic overtone of Plato’s philosophy as well as the “lack” of ritual prescribed by the Platonic philosophical system, Plato too implies that human mastery comes from directing and cultivating human nature. In Phaedrus, the character Socrates describes the human condition in the Allegory of the Charioteer,25 Let us then liken the soul to the natural union of a team of winged horses and their charioteer…To begin with our driver is in charge of a pair of horses; second, one of his horses is beautiful and good and from stock of the same sort, while the other is the opposite and has the opposite sort of bloodline. This means that chariot-driving in our case is inevitably a painfully difficult business. (246a)26 The Allegory implies this dialectic between immanence and transcendence. The soul has wings to fly, but is pulled toward the Earth through material desires. He notes, By their nature wings have the power to lift up heavy things and raise them aloft where the gods all dwell, and so, more than anything that pertains to the body, they are akin to the divine, which has beauty, wisdom, goodness, and everything of that sort. These nourish the soul’s wings, which grow best in their presence, but foulness and ugliness make the wings shrink and disappear. (246e)27 Plato identifies “goodness” as a transcendent form. But this passage can also be read as indicating the importance of overcoming immanent desires; of controlling the waves of emotions that surge and recede like a sometimes calm, sometimes chaotic sea. The idea that we must cultivate desire is key for both Confucius and Plato. While the Confucian concept of Li is not created out of an elaborate metaphysical account, both Li and Logos are employed as two different ways toward self-management and self-cultivation. In both philosophers, it is vital to internalize virtue28: Li serves this purpose for Confucius, Logos for Plato. Confucius notes, “…If, however, you guide

Ibid. “The Analects,” 7.5 Both Plato and Aristotle believed we have a tripartite soul: appetitive (vegetative), ambitious (animal) and intellectual (human). The first two souls are shared by other species, but the intellectual/human soul is uniquely human. Therefore, it is this (rational) aspect of each individual that holds the key to becoming human. 26 “Phaedrus”, 246a, The Complete Works of Plato, ed. John M. Cooper, (Indianapolis: Hackett Publishing Inc., 1997) p. 524 27 Ibid. 246e. p.525 • 28 There are differences between ancient Greek and Confucian virtues. The four main Greek virtues include Wisdom, Courage, Temperance and Justice. Love (the varying kinds of love as expressed by Eros (sexual desire), Philia (deep friendship), Agape (Universal love), Pragma (long-standing/pragmatic love), Ludus (playful love) and Philautia (love of self, not be confused with Narcissism) is not explicated as a virtue, although Platonic love brings the lover closer to the Form of Good. Therefore, for Plato it is somewhat instrumental as we move toward an understanding of true reality. Confucian virtues emphasize Ren (humanity/goodness/benevolence/compassion) Yi (righteousness, Li (rites/rituals), Zhi (knowledge) and Xin (integrity). Undoubtedly Plato (and the ancient Greek philosophers) and Confucians emphasized different virtues, but it is also apt to note that the plausibility of justice, without compassion or righteousness without courage or integrity without balance are not feasible options. It is also relevant that predominant virtues are contextual and indicate those qualities that are required given the existent circumstances and culture. 24 25


42 them (the people)29 with Virtue, and keep them in line by means of ritual the people will have a sense of shame and will rectify themselves.”30 Both ancient traditions move away from virtue/ethical behavior as being enforced from external powers. No longer is good behavior merely a method for placating the authorities (both human or divine), but it is a method of becoming fully human; of cultivating that which is “good.” For Confucius, this goodness is in relationship with the community. For Plato, this goodness is a personal journey toward an abstract idea. However, for both, being virtuous and practicing virtue is something sought after by those who are on the way to becoming human. For both, it is key for enduring happiness, or Eudaimonia as expressed by the ancient Greeks. However, it cannot be missed that the internalized belief that true human flourishing cannot be achieved void of virtuous thought and action is, besides anything else, also a brilliantly manipulative concept. Li (for Confucius) and Logos (for Plato) are both prescriptive mechanisms that help to quell the individual potential for relativism, which potentially leads to chaos and anarchy. Vital for social harmony is the sacrifice of self-interest for the belief in a greater reward. Through Li or Logos, we determine what is the right thing to do and can live freely within a world that has been cultivated accordingly. It is Li or Logos that establishes our boundaries, but it is also apparent that these are fictitious boundaries created for the good of community. In other words, both Confucius and Plato are redesigning cultural norms even though both argue that they are mere transmitters of the right way to conduct life. Confucius, creatively transmits Li as prescribed by the Golden Era (which also understood itself as transmitting the wisdom of the three Sage-Kings) and Plato too is transmitting the already existent Greek virtues of Wisdom, Courage, Temperance and Justice, reformulated as essences of the soul that must be cultivated by the newly designed, responsible individual. Both Confucius and Plato are creative interpreters of traditional morality. Confucius, who has no concept of a transcendent reality, uses Li to “cultivate” the good. Plato, whose true reality is transcendent uses Logos for human subjects to “know” the good.31 However, both result in educating people to act in the world in a more enlightened manner for social and political harmony. Wuwei, Wuzhi, Wuyu Much speculation and comparative work has been done between Confucius and Laotzi, and while this paper does not have the room to explore the relationship between Taoism and Confucianism, the Taoist concepts of Wuwei (effortless action), Wuzhi (unprincipled knowing) and Wuyu (objectless desire) offer us interesting insight into the human condition dealt with by both Chinese philosophers and also in this exploration, by Plato. To provide a brief account, the following will highlight some differences between Laotzi and Confucius: Laotzi criticized Confucius for his emphasis on Li and instead prescribed quietude and passive-action. He notes, “When the great Way is abandoned, there are benevolence and righteousness…Cut off benevolence, abandon righteousness, and the people will return to being filial and kind.”32 For Laotzi, Confucianism forces roles and abandons The Way through dogmatic doctrines that emphasize control. The Confucian prescribed Rectification of Names, so vital for social order, is a concept that Daoist philosophy whole-heartedly opposes: it forces meaning into language that can My clarification. Ibid. Hall and Ames, “The Analects” 2.3 31 The Good as a concept differs considerably for both Confucius and Plato. For the former, the Good as it pertains to humanity (Ren) is about honoring the Way. However, it is not a fixed, “pure” goodness, but rather a contextualized event created through ritual. For Plato, the Form of Good is generally read as being a fixed, unchanging idea that then descends upon various things or situations. Hence the Good for Plato is the ultimate pure, objectless object and for Confucius it is a harmonious human creation that is always relative to its opposite. 32 “Daodejing” Readings in Classical Chinese Philosophy, ed. Philip Ivanhoe and Bryan Van Norden. (Indianapolis: Hackett Publishing Inc., 1992) Chapter 18-19, page 171. 29 30


43 only ever be a distortion of reality. Whereas Confucius supports the hierarchical feudal system, Laotzi advocates small communities that require little to no leadership. However, both Laotzi and Confucius are social and political critics that agree the world in which they are part is rife with greed and corruption emanating from the highest ranks of the social sphere. For Confucius, Li is the way to restore harmony. For Laotzi, it is only through withdrawal that we can manage the desires for the “10,000 things.” In the quiet, we can contemplate “the Great Way” and we can see, hear, smell, taste and feel more clearly, without objects of desire that only serve to distort reality. It is here that we see the wholly opposite methodology between Confucius and Laotzi, but not a wholly opposite goal. Both are ethical and political philosophers who interpret the Dao (The Way) very differently. For Confucius the concepts of Wuwei, Wuzhi and Wuyu must be directed through Li. Effortless action is directed through prescribed action. Unprincipled knowledge is directed through the learning of the six arts (archery, calligraphy, computation, music, chariot driving and ritual.) Objectless desire is directed through prescribed desires. At first glance, these negations of the human propensity to alter reality (act) to become an expert (to know something) and to yearn for something (desire) bear little in relation to Platonic metaphysics, but I would like to propose that the Chinese concept of Wu (as objectless, unprincipled and effortless) can serve to illuminate the Platonic Realm of Forms - most specifically the ultimate forms for humanity: Beauty and the Good. For Plato, through rational thinking we move closer to seeing the world as it really is. That is, when we see a beautiful sunset or person or listen to beautiful music, we are mesmerized. And this “yearning” for beauty (such as falling in love) is for Plato, actually the desire for something which is not in the thing, but instead is “objectless”. The object we find beautiful or that we love is merely a reminder or reflection of the objectless forms of true Beauty and Goodness. For Plato these forms are what we all on this pitiful Earthly realm are searching for when we desire some-thing. We long for the Good because it is our soul’s purpose to know Goodness; because we have a memory of the Good. But Good is no-thing. Beauty is no-thing. Love is no-thing. Knowledge and Truth are no-things. Beauty, love and truth as objectless is implicated by Plato’s alternate realm, but impossible for humans in their immanence to perceive. We always need to attach beauty or target our desire to some-thing, and when we do, it helps us to recall The Good and The Beautiful in their purest objectless form. The memory of a more perfect realm is mirrored by Confucius, only his memory is of a more perfect time. Plato’s memory is abstract and objectless. There is no one or nothing to attach it to. It is instead, the memory of objectless perfection. Albeit, rife with many differences, the concept that desire for Plato is not really of the object, but of that which is objectless, offers a connection to the Daoist concept of desire which is potentially harmful to the well-being of the social-sphere if desires attach themselves to objects33. Plato, in his occasionally expressed disdain for the material world can in some ways be compared to Laozi who would prefer a life of quiet contemplation away from worldly desires. However, according to The Republic, Plato’s Utopian polis, he aligns himself with Confucius. He agrees desire, knowledge and action must be given objectives in the Earthly realm, and by doing so, the harmonious, Utopian polis can be created. Laozi, arguably skeptical of human exceptionalism as advocated by both Confucius and Plato instead pursues the path of the monk, limiting the activation of objective desire, ego-induced action and principled knowledge. The Noble Lie and Confucian “As If” Philosophy: The Making of Meaning It is implied in The Analects that we should engage in the ritual practices not because they are ordained by the gods, but because through them, we can create the human world that is good. To

The problem is that when desire attaches itself to objects, then the potential for suffering is increased and social disharmony probable. We cannot always get the objects of desire. 33


44 do this, Confucius seems to advocate the use of an “As If” philosophy.34 We practice the rites As If they do have meaning; and when we do this, we actually imbue the rites with meaning ourselves. As Confucius himself recommends, ““Sacrifice As If (they were) present” means that, when sacrificing to the spirits, you should comport yourself As If the spirits were present.”35 Hence, when rituals are ignored or broken, the behavior is not merely an oversight, but it too is imbued with meaning. This is apparent when he notes of the Ji Family, “They have eight rows of dancers performing in the courtyard. If they condone this what are they not capable of?”36 This seemingly benign act is a defiant and disrespectful message by the family implying that they are as powerful as the Zhou Kings. Hence, the Ji family’s use of eight dancers is more than an empty practice, it symbolically disrupts the social order. At the same time, if you do not know ritual propriety, then eight dancers versus six means absolutely nothing. Hence, the “As If” philosophy is contingent upon communal participation and agreement. However, it mustn’t be assumed that Confucius is a passive follower of rules and ritual. Rather, he is an interpreter of rituals as defined by the Zhou dynasty and he creatively applies them to the contemporary context of the time. Some ritual alterations are justified. Confucius notes, “A ceremonial cap made of linen is prescribed by the rites, but these days people use silk. This is frugal and I follow the majority.”37 Similarly, he discerns whether or not social norms are justified and if not he does not uphold them. Of Gongye Chang, he notes “He is marriageable. Although he was once imprisoned as a criminal, he was in fact innocent of any crime.”38 Further, he gave his daughter’s hand to marry this individual who was probably scorned by most. This offers a picture of Confucius as a discerning Master, using ritual to achieve social harmony and eliminating or altering ritual and even “meaning” based upon the context. It expounds an “As If” philosophy that is not rigid and requires constant interpretation of context and events. It is relevant for this paper to show how the “As if” philosophy fits with Plato’s political as well as metaphysical ideas that ultimately also have pragmatic ends. An obvious “As if” moment occurs with Plato’s thought experiment, that justifies the use of a Noble Lie in order to create The Republic, a city of virtue that would mirror the beautiful soul in its cultivation of wisdom, courage, temperance and, perhaps most importantly, justice. The Republic uses the Platonic theory of the tripartite soul as its blueprint, and establishes three classes, each correlating to a different part of the soul. The merchant class represents the appetitive soul, the warriors/military class represents the ambitious soul and finally the smallest social group, the Philosopher-Kings, represent the intellectual soul. The Noble Lie propagates the myth that each individual has a blood type that corresponds with one of these classes. The Merchant class is predominantly bronze; the Military class is predominantly silver; and the Philosopher-King class is predominantly gold. The practice of forming these classes begins soon after birth, when the children are observed at play. Dependent upon the behavior they exude, each one is allocated to a class. Once allocated, each child receives education pertaining to the activities and desires “natural” to which tier of society he/she belongs. The merchants are the wealthiest as they are the most pleased by possessions. Their desires need “objects” and so The Republic is organized so these material desires are The Philosophy of As If is the title of a book by German philosopher Hans Vaihinger (1852-1933). In a short autobiography on the genesis of this idea, he recalls a young professor who opened up the world of Plato to him. He recalls, “…he read to us in Greek the myth in Phaedrus of the nature of the soul, and the description of the cave from the Republic. This opened up a new world to me, the world of “Ideas,” and as he also spoke of Plato’s myths the seed was sown then of that conception which later I myself named the “World of As if…I called this work The Philosophy of “As if” because it seemed to me to express more convincingly than any other possible title that I wanted to say, namely that “As if,” i.e., appearance, the consciously-false, plays an enormous part in science, in world-philosophies and in life. 35 Ibid. “The Analects”3.12 36 Ibid. “The Analects” 3.1 37 Ibid. “The Analects” 9.3 38 Ibid. “The Analects” 5.1 34


45 satisfied. The military, who derive their pleasure and sense of worth from their role as protectors of the community and the honor it begets, have less material wealth. Finally, the PhilosopherKing who is the paragon of virtue, derives his pleasure from realizing the Realm of Forms, owns little to nothing. He is a reluctant leader, but does it because he is obligated to bring about justice and harmony in The Republic. This Utopian society is of course much more intricate than needs to be developed in this paper and Confucius would necessarily be at odds with this kind of Utopia vision for many reasons that cannot be taken up. However, what is apparent here is the connection between Confucian Li and the Platonic Noble Lie that is both operate “As if” a fully virtuous society is possible. Implicit in Plato’s Republic is the concept that desires and individual capacities can be nurtured and channeled. Due to the “proper” education, the merchant class would not even consider becoming warriors, because their entire education and development is limited to merchant appropriate knowledge and the development of merchant appropriate/necessary capabilities and desires. “Correct” behavior and desire is class specific. Confucius himself hints at a similar notion, The Prime Minister asked Zigong, Your Master is a sage, is he not? How is it, then, that he is skilled at so many menial tasks?...When the Master heard of this, he remarked, “How well the Prime Minister knows me! In my youth I was of humble status, so I became proficient in many menial tasks. Is the gentleman broadly skilled in trivial matters? No he is not.39 A gentleman, much less a Sage does not have to know how to do anything useful. A Sage is the embodiment of Ren. He is a model for humanity who has overcome his inhumane/uncultivated tendencies. He does not live for himself, but for the greater whole of the community; and his benefit for the community is eternal, passing down through the ages, continuing to shape society. The Sage-King and Philosopher-King Although, Confucius admitted he knew of no actual Sages, the Chinese dynasties had kept alive the legend of the three pre-Axial age Sage Kings, Yao, Shun and Yu (circa, 2,250-2,205BCE) who were pivotal in the development of China. He was, however, hopeful that in the future there would be more Sage-Kings. This offers us a Utopian vision of the goal of becoming human. For all intents and purposes, Confucius imagines the Sage-King existed once before in the early Zhou Empire and imagines that he will exist again. Therefore, Confucianism offers us a philosophy of hope. Once again, we can apply Vaihinger’s “As If” philosophy to the concept of the Sage-King. We live as if this most “human” state of being is possible. Even though Confucius does not look above immanent life on Earth for answers, he does look beyond the here and now. He looks back and then looks forward, undermining the implications of reality: in the present, the desire to be good is often overshadowed by the desire to be clever, right, rich or lazy. However, his faith in the Sage-King is without reservation, “If a true king were to arise, we would certainly see a return to Goodness after a single generation.”40 A true King would rectify the names, ensuring only those ministers whose actions corresponded to their privileged title would remain in power. A true King would implement the rituals, ensuring that the well-ordered family relationship will be the blueprint upon which communities and nations are built. Confucius’s belief in the Sage-King is a desire to recover virtue as it allegedly used to be - as it should be. Whereas Confucius is more readily comparable with Aristotle for his empirical method and tangible practices, by this account the Sage-King is a Utopian vision for humanity and in this way is more compatible with Platonic idealism. It must be noted that the Confucian Sage is not the mystical Daoist Sage who

39 40

Ibid. “The Analects” 9.6 Ibid. “The Analects” 13.12


46 separates himself from the world of “10,000 things,” but instead he is the moral exemplar standing at the helm of a well-ordered society. For the purposes of this paper, it is interesting to look at how the Confucian Sage-King and Platonic Philosopher-King compare to each other. The Allegory of the Cave, and perhaps also the idealized character of Socrates in all of Plato’s writings, also offers an illustration of the Philosopher-King as a Utopian human-being. While his wisdom comes from his transcendence into the rational world of ideas, we see a similar fictional idea emerging albeit differently. The Philosopher-King operates on a higher-sphere seeing beyond the illusion of the 10,000 things to the truer reality beyond/behind it. He has, to some degree, mastered the art of objectless desire because he is never swayed by material wealth or physical indulgences. He has had a glimpse of the Good and understands that all yearning is yearning for this perfect Formless Form of Good. In some ways he aligns himself more with the Daoist monk, except that the Philosopher King is not merely living for his own enlightenment. He too has a duty and obligation to uplift the polis and to lead the people toward the Good. However, when comparing both the Sage and the Philosopher, we can see that both are “beyond” the struggles of the mundane world. They have overcome the petty desires and have become paragons, indeed, super-humans. These two figures are offered as destinations of humanity and are void of the imperfections that most of us experience. Hence, we will be destined to always be becoming human, without ever actually achieving the unachievable goal. Ren and The Good There are numerous differences implied by Confucian Ren and the Platonic Form of the Good, the most apparent being that for Confucius, Ren is cultivated, therefore created, through Li, whereas for Plato, The Good is akin to the Sun. It nourishes us from above. It exists as an eternal reality whether or not we have a rational understanding of it. Understanding the implications of Ren is a difficult task as noted by Hall and Ames, Given the central importance of ren in Confucian thought, the decidedly underdetermined status of this notion as it is presented in the Analects has been a source of concern for commentators over the centuries. On six different occasions, Confucius is called upon to define ren, and six times he gives a different answer. The best that Confucius can do in stipulating the content of ren is to point to historical exemplars.41 Ren is translated as benevolence or humanity or goodness. But perhaps all fall short. Hall and Ames continue, The problem with benevolence is that it psychologizes ren, reducing a holistic and resolutely social conception of person to someone’s particular moral disposition. “Humanity,” a broader and hence more adequate term, still fails to do justice to the profoundly religious dimension of ren, and vitiates the uniqueness inherent in becoming ren.42 Ren as an individual essence is only established within community. That is, to be Ren, is to be Ren within a context. Ren is a learned phenomenon. We learn from others that are closer to Ren than we. Confucius notes, “…To live in the neighborhood of the Good is fine. If one does not choose to dwell among those who are Good, how will one obtain wisdom?”43 Ren is promoted as Ibid. Halls and Ames “Transcendence and Cultural Clues” p. 262 Ibid. p.262 43 Ibid. “The Analects” 4.1 41 42


47 key to enduring happiness and is able to “truly love others or truly despise others,”44 offering a description of Ren that knows right from wrong, good from bad. Ren loves that which is Ren. The individual that has embodied Ren is no longer for himself, but is for the community. He has become the embodiment of the community and the community the embodiment of him. Hall and Ames explain this as an intrinsic motif in Yin Yang philosophy. They note “Yin is always “becoming yang,” and yang is always “becoming yin,” just as day is a “becoming night” and night is a “becoming day.” For the ji/ren distinction, “oneself” is always “becoming other,” and another is always “becoming oneself.””45 Therefore, Ren as a desired state of being is always connected with community. I cannot become good for myself. Alas, despite the promise of happiness, the cultivation of Ren is a path less followed. Confucius notes, I have yet to meet a person who truly loved Goodness or hated a lack of Goodness…Is there a person who can, for the space of a single day, simply devote his efforts to Goodness? I have never met anyone whose strength was insufficient for this task. Perhaps such a person exists, but I have yet to meet him.46 To be Ren requires disciplined cultivation through the rituals and learning. The undertaking requires both internal and external discipline as explicated in the ancient text, The Highest Order of Cultivation, What is meant by the words: ‘achieving a state of integral wholeness within one’s innermost consciousness is that one must avoid all manner of self-deception, just as spontaneously as one feels distaste for a distasteful smell, or takes delight in a delightful sight. This is what is meant by the expression ‘being at ease with oneself’. For this reason, it is imperative that the man of noble character pay great heed to the core of his own individuality.47 The misconception of Li as an externally applied practice that leads to Ren, misses the point if the internal practices are not also undertaken with the same duty bound commitment. It is the individual’s responsibility to nurture a straight heart and mind, …when one’s personal relations are governed by animosity and resentment, then one incapable of achieving this straightness of mind; when one’s consciousness is occupied by feelings of fondness and delight, one is incapable of achieving straightness of mind…In cases such as these…one looks but does not see, listens but does not hear…This is what is meant by the statement: ‘the cultivation of one’s moral character as an individual is predicated upon setting straight the seat of one’s emotive and cognitive faculties.48 Ren, as both journey and destination, requires constant internal and external discipline. Popper notes, “Jen is all-embracing, not a virtue among others, but the soul of all virtues…It is through Jen that the particular virtue becomes truth.”49 Ren implies an intuitive understanding of what is good and for all intents and purposes, remains ambiguous. We cannot determine the Ren. The right behavior and ritual propriety does not necessarily imply that the intention and desire is Ren. It is something that is beyond obligation but it is realized in the world through obligation. Ren is Ibid. 4.3 Ibid. Hall and Ames, “The Focus-Field Self in Classical Confucianism” p.27 46 Ibid. 4.6 47 Confucius, The Highest Order of Cultivation, trans. Andrew Plaks, (London: Penguin Group, 2003) p.11 48 Ibid. p. 12 49 Ibid. Karl Popper, p.50 44 45


48 difficult to reconcile with the Platonic Form of Good except if we remember that similarly to Ren, the Good, like Ren, is also the source of all ancient Greek virtues. As these are cultivated in the beautiful soul, we can also see how the form of Good, like Ren has a practical and communal value. Like Ren, Greek Virtue reveals itself in situations. It cannot be observed from the outside because the right action is not merely an abstract duty, but it is intrinsically a situational “the right action”. To bring balance to an unbalanced situation is virtuous. To bring justice where there is none is just. To apply wisdom as needed and to be courageous when courage is called for is to apply virtue correctly. The Good as expressed through virtue, like Ren, is ambiguous, offering a perspective of ancient Greek philosophy that betrays its reputation of having a focus on absolute certainty. Conclusion Over 2,500 years ago, a shift in human consciousness became apparent in both Ancient China and Ancient Greece. It was a shift in the understanding of self and society that fostered an inquiry into what being human implicated, an inquiry that is as relevant today as it was then. The two philosophers focused upon in this paper, Confucius and Plato, lived in vastly different regions, were part of vastly different cultures and spoke two vastly different languages. Confucius’ philosophy provided a pragmatic blueprint for humanity to achieve Ren, whereas Plato’s philosophy offered a sophisticated metaphysical account of a true reality that lay beyond the immanent Earthly realm. And yet, both thinkers provide us with similar prescriptions for a good, flourishing life: we must cultivate virtue and lead rational lives. Then and only then, can we move toward becoming human.50

The task of this paper has not been to study the negative consequences of either Confucian or Platonic philosophy however, it seems appropriate to note that Confucianism can become (did become) dogmatic, oppressive and regimented, binding people to roles that are subservient and even abusive. Plato’s Republic also has pernicious implications with its fixed social classes and an almost divine King (who is unquestionably virtuous), not to mention the disembodied Realm of Forms and its influence on the formation of Christianity that often undermined the body and life on Earth. 50


49 Bibliography Confucius, “The Analects,” Readings in Classical Chinese Philosophy, ed. Philip Ivanhoe and Bryan, Van Norden. (Indianapolis: Hackett Publishing Inc., 1992) Confucius, “The Highest Order of Cultivation,” Ta Hsueh and Chung Yung (The Highest Order of Cultivation and On the Practice of the Mean) trans. Andrew Plaks, (London: Penguin Group, 2003) Hansen, Chad, Daoist Theory of Chinese Thought, “Confucius: The Baseline” (New York: Oxford University Press., 1992) Hall, David and Ames, Roger, Thinking from the Han. (Albany: State University of New York Press, 1998) Lao Tzi, “Daodejing” Readings in Classical Chinese Philosophy, ed. Philip Ivanhoe and Bryan Van Norden, (Indianapolis: Hackett Publishing Inc., 1992) Plato, “Phaedrus”, 246a, The Complete Works of Plato, ed. John M. Cooper, (Indianapolis: Hackett Publishing Inc., 1997) Popper, Karl, Socrates, Buddha, Confucius, Jesus, ed. Hannah Arendt, trans. Ralph Manheim, (New York: Harcourt, Brace & World, Inc., 1957) Vaihinger, Hans, The Philosophy of As If, 6th edition. ed. David Payne, trans. C.K. Ogden, (New York: Harcourt, Brace and Company, 1925) Reprinted by Random Shack, 2015.


50 Students’ Statistics Research Projects in STA2023 using STATDISK - An Introduction M. Shakil, Ph.D. Professor of Mathematics Liberal Arts & Sciences Department Miami Dade College, Hialeah Campus, FL 33012, USA E-mail: mshakil@mdc.edu and J. Bestard, Ph.D. Professor of Mathematics Liberal Arts & Sciences Department Miami Dade College, Hialeah Campus, FL 33012, USA E-mail: jbestard@mdc.edu Abstract Due to the tremendous development of computers and other technological resources, the use of Statistical Software for data analysis, in a Statistical Methods (STA 2023) course, cannot be underemphasized and ignored. In this paper, we have presented the use of STATDISK to conduct students’ statistics projects in Statistical Methods (STA 2023) courses. It is hoped that the findings of the paper will be quite useful for educators and researchers in various disciplines.

2010 Mathematics Subject Classifications: 97C40, 97C70, 97D40, 97D50. Keywords: STATDISK, Projects, Statistics, STA 2023, Teaching.

1. Introduction Statistics is one of the important sciences at present. Due to the tremendous development of computers and other technological resources, the use of Statistical Software for data analysis, in a Statistical Methods (STA 2023) course, cannot be underemphasized and ignored. According to Professor Mario F. Triola, the author of "Elementary Statistics" textbooks, “Statistical Software is now a common technology choice used in introductory statistics courses. An important reason many educators choose Statistical Software is its extensive use in corporate America. The world of business and industry has embraced the spreadsheet as an efficient and effective tool for the analysis of data, and many Statistical Software such as SPSS, SAS, Excel, MINITAB, STATDISK, among others, have become the premier spreadsheet programs. Motivated by a desire to better serve their students by better preparing them for


51 professional careers, many instructors now include a Statistical Software as the technology tool throughout the statistics course. This marriage of statistics concepts and spreadsheet applications is giving birth to a generation of students who can enter professional careers armed with knowledge and skills that were once desired, but are now demanded” (http://www.statdisk.org/). In this paper, we have presented the use of STATDISK to conduct students’ projects in Statistical Methods (STA 2023) courses. For details on STATDISK, see Triola (2010), http://www.statdisk.org/, and https://www.youtube.com/playlist?list=PLiuxxNbKiuJ5YgOqcJmowweivKgTIJzda, among others. The organization of this paper is as follows. In Section 2, some special features of STATDISK are presented. Section 3 contains the uses of STATDISK to conduct students’ projects in Statistical Methods (STA 2023) courses, along with some applications to real world data. In section 4, some concluding remarks are given. 2. Some Special Features of STATDISK STATDISK is a full featured statistical analysis package. It includes over 70 functions and tests, dozens of built-in datasets, and graphing. Statdisk is free to users of any of Pearson Education Triola Statistics Series textbooks. Some special features of STATDISK are described below. Help Overview, Sample Editor / Data Window  Many individual modules include their own Help comments. Here we provide some comments about the STATDISK main menu bar at the top. 

File: Click on File to open an existing file or to save a file that has been created in the STATDISK Data Window. The "Open" and "Save As" features require that you select the location of the file to open or saved. If the default that is displayed is not what you want, click on the small box to the right of the default location and proceed to select the desired location.

Edit: Click on Edit to Copy a STATDISK file to another application or to Paste a file from another application.

Analysis: Click on Analysis to access many of the STATDISK modules, including those related to such features as confidence intervals and hypothesis testing.

Data: Click on Data to access STATDISK features such as those related to descriptive statistics, histograms, and boxplots.

Data Sets: Click on Datasets to access the list of data sets in Appendix B of the textbook.


52 

Window: The Window menu lists all of the windows currently open in your STATDISK session. You can click a window (or use its hotkey combination) to bring it to the front.

Help: The help menu will open help from this site for the various STATDISK windows. There are also links to additional resources such as the STATDISK Workbook and the Triola Statistics website.

The STATDISK Sample Editor serves as a basic starting point for using STATDISK. Many of the modules in STATDISK require raw data in order to perform a statistical analysis. The STATDISK Data Window allows you to manually enter lists of raw data.

3. Students’ Projects in STA2023 using STATDISK: In the following examples, the uses of STATDISK are demonstrated. 3.1. Descriptive Statistics: In this subsection, using STATDISK, we provide an example of a “Descriptive Statistics” of “the ATMOSPHERIC CARBON DIOXIDE CONCENTRATIONS IN PARTS PER MILLION (PPM) DATA AT MAUNA LOA AND CAPE KUMUKAHI, HAWII”, (see, http://cdiac.esd.ornl.gov/trends/trends.htm, 13 June 2002). These are shown in Figure 1 below.


53

FIGURE 1: DESCRIPTIVE STATISTICS OF THE ATMOSPHERIC CARBON DIOXIDE CONCENTRATIONS IN PARTS PER MILLION (PPM) DATA AT MAUNA LOA AND CAPE KUMUKAHI, HAWAII (http://cdiac.esd.ornl.gov/trends/trends.htm, 13 June 2002)


54 3.2. Histogram and Frequency Distribution: See Figure 2 below.

Figure 2: THE ATMOSPHERIC CARBON DIOXIDE CONCENTRATIONS IN PARTS PER MILLION (PPM) DATA AT MAUNA LOA AND CAPE KUMUKAHI, HAWII (http://cdiac.esd.ornl.gov/trends/trends.htm, 13 June 2002) Figure 2: THE ATMOSPHERIC CARBON DIOXIDE CONCENTRATIONS IN PARTS PER MILLION (PPM) DATA AT MAUNA LOA AND CAPE KUMUKAHI, HAWAII (http://cdiac.esd.ornl.gov/trends/trends.htm, 13 June 2002)


55 (c) Boxplots: See Figure 3 below.

Figure 3: THE ATMOSPHERIC CARBON DIOXIDE CONCENTRATIONS IN PARTS PER MILLION (PPM) DATA AT MAUNA LOA AND CAPE KUMUKAHI, HAWAII (http://cdiac.esd.ornl.gov/trends/trends.htm, 13 June 2002) (d) Normal Quantile Plots: See Figure 4 below. (d) Normal Quantile Plots: See Figure 4 below.

Figure 4: THE ATMOSPHERIC CARBON DIOXIDE CONCENTRATIONS IN PARTS PER MILLION (PPM) DATA AT MAUNA LOA AND CAPE KUMUKAHI, HAWAII (http://cdiac.esd.ornl.gov/trends/trends.htm, 13 June 2002)


56

(e) Confidence Interval Estimates and Hypothesis Test Mean â&#x20AC;&#x201C; Two Independent Samples: See Figure 5 below.

Figure 5: THE ATMOSPHERIC CARBON DIOXIDE CONCENTRATIONS IN PARTS PER MILLION (PPM) DATA AT MAUNA LOA AND CAPE KUMUKAHI, HAWII (http://cdiac.esd.ornl.gov/trends/trends.htm, 13 June 2002) (f) Pie Charts: See Figure 6 below.

Frequency Distribution

Relative Frequency Distribution (%)

Figure 6: Blood Types of the 25 Army Inductees (SOURCE: BLUMAN) Frequency Distribution and Relative Frequency Distribution (%)


57 (f) Regression: See Figure 7 below.

Figure 7: Number of Absences and Final Grades (%) (SOURCE: BLUMAN) Scatterplot, Correlation and Regression Results 4. Concluding Remarks: In this paper, the uses of STATDISK are demonstrated through some examples. It is hoped that this paper will be helpful in teaching any introductory course in statistics such as courses in Statistical Methods (STA 2023) using STATDISK and conduct students’ statistics research projects. Further, as there is a great emphasis on statistical literacy and critical thinking in education these days, it is hoped that, with the help of STATDISK, the students will be able to conduct statistical research projects in their STA2023 courses, and will be able to achieve the following: I. Search or web-search any real world data. II.

Analyze the data statistically using STATDISK, that is, 

Compute descriptive statistics for any real world data;

Draw histograms and other statistical graphs for data sets;

Discuss the distributions of data sets;

Other Statistical Analysis.

III. Write a statistical research project or report by incorporating the above findings. IV. Present the research project.


58 Acknowledgment

Parts of this paper were presented by the authors during the Math Retreat 2015, at the Miami Dade College, Hialeah Campus, on March 19, 2015. Further, the authors would like to thank the Editorial Committee of Polygon for accepting this paper for publication in Polygon. Also, the authors are thankful to their institution, Miami Dade College, for providing them the opportunity to serve it. Further, the authors would like to thank their wives for their patience and perseverance for the period during which this paper was prepared. Lastly, the authors would like to dedicate this paper to their late parents.

References 1. Triola, M. F. (2010). Elementary Statistics, 11th Edition. Addison-Wesley, New York. 2. http://www.statdisk.org/ 3. https://www.youtube.com/playlist?list=PLiuxxNbKiuJ5YgOqcJmowweivKgTIJzda 4. http://www.tec.org/ 5. http://www.texascenter.org/almanac/Air/AIRCH6P3.HTML 6. http://data.giss.nasa.gov/, http://eosweb.larc.nasa.gov/PRODOCS/narsto/table_narsto.html 7. http://cdiac.esd.ornl.gov/trends/trends.htm


59

Using Sustainability Data to Teach MAC 2233 and MAC 2311 Courses via Maple and Mintab Software vis-a-vis Developing a Lesson Plan Dr. M. Shakil, Ph.D. Professor of Mathematics Liberal Arts & Sciences Department Miami Dade College, Hialeah Campus FL 33012, USA E-mail: mshakil@mdc.edu Abstract This paper deals with developing a lesson plan to show how the teaching of basic mathematical concepts in MAC 2233 and MAC 2311 Course via Maple and Mintab Software can help the students to gain the knowledge and insights from various aspects of sustainability data, such as examining the Global Average Temperature data, making a graph, finding some mathematical models, and making predictions. Such studies are important in view of the facts that there is a great emphasis on students’ analytical reasoning, logical and critical thinking, and to enable the students to develop mathematical models to solve different real world problems in mathematics courses these days. Due to the tremendous development of computers and other technological resources, the use of mathematical software for data analysis, in a Business Calculus (MAC2233) or Calculus 1 (MAC2311) course, cannot be underemphasized and ignored. Therefore, in this paper, we present how to use sustainability data to teach MAC 2233 and MAC 2311 Courses via Maple and Mintab software. It is hoped that by implementing the techniques discussed in this paper in preparing lesson plans in MAC 2233 or MAC 2311 Course via Maple and Mintab Software will help us to develop in our students the skills of quantitative reasoning and analysis of real world data, which is one of the Gen Ed Outcomes of Miami Dade College. It is also expected that this approach will enable the students to develop the competency of analyzing the real world problem as well as to enable them to develop mathematical models to solve different real world problems.

2010 Mathematics Subject Classifications: 97C40, 97C70, 97D40, 97D50. Keywords: Calculus, Global Temperature, Lesson Plan, Maple, Mathematical Analysis, Minitab, Quantitative reasoning, Sustainability, Teaching.

1. Introduction: Sustainability of our environment is an issue of emerging international importance and significance. There are a number of potential impacts on the sustainability of ecosystems including warmer temperatures and rising sea levels, changes in rainfall patterns, and increased storm and cyclone intensity due to climate change. Consequently, these impacts are presenting many challenges for our world’s environment, community and economy. Since mid-1980s, considerable research into various aspects of issues related to environmental sustainability has been conducted. For example, the concentration of ‘greenhouse’ gases (namely, carbon dioxide, methane, nitrous oxide, ozone, and chlorofluorocarbon) in the Earth’s atmosphere, resulting in a gradual increase in temperatures at the Earth’s surface, is an important


60 issue and area of research. There is a rising trend of average global temperatures over recent years. Ecologists around the world are concerned about global warming. There are many causes and effects of such a global warming. One possible effect is the melting of the icecaps. According to the research, the scientists believe that the â&#x20AC;&#x153;greenhouse effectâ&#x20AC;? may also be one of the causes of global warming. As greenhouse and climate change are fundamental issues of environmental sustainability, the purpose of this project is to study and analyze past trends in Global Average Temperature by applying some mathematical concepts from a Business Calculus (MAC2233) or Calculus 1 (MAC2311) course and using appropriate technology. There is a great emphasis on mathematical literacy and critical thinking in education these days. An introductory course Business Calculus (MAC2233) or Calculus 1 (MAC2311) at Miami Dade College can easily provide such avenues to our students. In any Business Calculus (MAC2233) or Calculus 1 (MAC2311) course, the students have already learned about the following mathematical concepts, some in their Business Calculus (MAC2233) or Calculus 1 (MAC2311) course, and some in their previous mathematics courses (namely, College Algebra, Intermediate Algebra, etc.), which are important tools and techniques for describing, exploring, summarizing, and comparing data sets. (i) Graph linear, quadratic and cubic functions. (ii) The slope of a line given two points that lie on the line. (iii) The Average Rate of Change of a function y = f(x), over a closed interval [a, b] , with respect to x. (iv)

The concept of derivative

dy of a function y = f(x) with respect to x, and its dx

applications in finding the slope of the tangent to the

graph of y = f(x) at a

certain point x on it, in determining whether the function y = f(x) is increasing or decreasing, and finding the critical numbers and relative extrema of the function, as well as finding the absolute extrema on a specified interval. Due to the tremendous development of computers and other technological resources, the use of Mathematical Software for data analysis, in a Business Calculus (MAC2233) or Calculus 1 (MAC2311) course, cannot be underemphasized and ignored. Mathematical Software is now a common technology choice used in Business Calculus (MAC2233) and Calculus 1 (MAC2311) courses. An important reason many educators choose Mathematical Software is its extensive use in science, industry and corporate America. The world of science, business and industry has embraced many Mathematical Software such as Maple, Mathematica, Matlab, Excel, Minitab, among others, as an efficient and effective tools for the analysis of data. Motivated by a desire to better serve their students by better preparing them for professional careers, many instructors now include a Mathematical Software as the technology tool throughout the mathematics course. This marriage of mathematical concepts and applications of mathematical software is giving birth to a generation of students who can enter professional careers armed with knowledge and skills that were once desired, but are now demanded. This paper develops a lesson plan how to use sustainability data to teach a Business Calculus (MAC2233) or Calculus 1 (MAC2311) Course via Maple and Mintab Software. As greenhouse and climate change are fundamental issues of environmental sustainability, this project aims at studying and conducting mathematical analysis of some global average temperature data. The organization of this paper is as follows. In Section 2, mathematical concepts from algebra and calculus courses required for mathematical analysis of some global average temperature data are presented. Section 3 contains the methodology and the study of the past trends in average global temperature by applying some mathematical concepts from calculus, and using appropriate technology. Analysis of data and graphs are also presented. Concluding remarks are given in Section 4.


61

2. Some Mathematical Concepts required for the Study: Some mathematical concepts, which the students have already learned in their algebra and calculus courses required for mathematical analysis of the global average temperature data, are given below. 

Definition 1. Average Rate of Change (or Difference Quotient) of a function y = f(x), over a closed interval [a, b] , is

f (b)  f (a) , which is also the Slope of Secant ba

Line passing though the points (a, f(a)) and (b, f(b)) of the graph of y = f(x). 

Definition 2. The derivative a function y = f(x) is given by

dy f ( x  h)  f ( x ) .  f ( x)  lim h  0 dx h 

Definition 3. For a function y = f(x), the derivative

dy also defines the slope of dx

the tangent to the graph of y = f(x) at x. 

Definition 4. Critical Number and Critical Point: A number x = c in the domain of the function y = f(x) is said to be a critical number of f(x) if f (c)  0 or f (c) is undefined. The corresponding point (c, f(c)) on the graph of y = f(x) is called critical point for y = f(x). Note that critical points are important because these are the points where the function changes from increasing to decreasing or from decreasing to increasing.

Definition 5. Relative Extrema: Relative extrema occur when a function changes from increasing to decreasing, or from decreasing to increasing. The two types of extrema are relative minimum points and relative maximum points. If the function has relative extrema they will occur at the critical points.

First-Derivative Test: Let x= c be a critical number for y = f(x) and let f(c) be defined. Then, f(c) is a relative maximum if f ( x)  0 to the left of c and f ( x)  0 to the right of c, i.e., the function is increasing to the left of c and the function is decreasing to the right of c. On the other hand, f(c) is a relative minimum if f ( x)  0 to the left of c and f ( x)  0 to the right of c, i.e., the function is decreasing to the left of c and the function is increasing to the right of c. Note that f(c) is neither a relative maximum nor a relative minimum if f (x) has the same sign on both sides of c.

3. Methodology and and the Study of the Past Trends in Average Global Temperature: In what follows, we will discuss the methodology, and study and analyze past trends in Global Average Temperature (in degrees Celsius) from 1950 to 2001, as provided in Table 1 below, by


62 applying the mathematical concepts discussed above, and using the appropriate technology, namely, MAPLE and MINITAB software. Table 1 Global Average Temperature, 1950-2001 Year (x) 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982

Temperature (degrees Celsius) (y) 13.83 13.97 14.04 14.11 13.91 13.91 13.83 14.09 14.09 14.04 13.98 14.10 14.05 14.04 13.76 13.86 13.93 13.99 13.90 14.00 14.05 13.90 13.95 14.18 13.94 13.98 13.78 14.15 14.07 14.12 14.27 14.40 14.09


63 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

14.34 14.16 14.13 14.18 14.34 14.42 14.28 14.49 14.44 14.16 14.18 14.31 14.45 14.34 14.38 14.69 14.41 14.38 14.53

(Source: Goddard Institute for Space Studies, NASA Goddard Space Flight Center Earth Sciences Directorate, "Global Temperature Anomalies in .01 C," http://www.giss.nasa.gov/data, updated December 2002; C.D. Keeling, T.P. Whorf, and the Carbon Dioxide Research Group, "Atmospheric Carbon Dioxide Record from Mauna Loa," Scripps Institution of Oceanography, University of California, http://cdiac.esd.ornl.gov/trends/trends.htm, 13 June 2002.) 3.1. Methodology: This is a Computer Project using MAPLE and MINITAB. Using the above data, write a research project on the following topic, by following the instructions given below.  “A STUDY OF PAST TRENDS IN AVERAGE GLOBAL TEMPERATURE FROM 1950 – 2001 BY APPLYING SOME MATHEMATICAL CONCEPTS FROM CALCULUS AND USING APPROPRIATE TECHNOLOGY” Instructions: Please note the data in Table 1 for the Global Average Temperature (in degrees Celsius) from 1950 to 2001. Now answer the following questions. Questions: Examine the data (in Table 1), make a graph, find some mathematical models, and make predictions. Use the following steps to write your research project. Step I: Using Minitab, draw the following graphs. Take “Year as x” along X-axis and “Temperature as y” along Y-axis. Use a (Year, Temperature), i.e., (x, y) Coordinate System. For convenience, let x = 0 in 1950. (i) Scatterplot of Temperature (degrees Celsius) vs Year “With Connect Line”; (ii) Scatterplot of Temperature (degrees Celsius) vs Year With “Regression”.


64 (iii)

Describe the trends in the Global Average Temperature based on your graphs in (i) and (ii) above.

Step II: Note the following: The annual change in the Global Average Temperature for year x is y(x 1) - y(x) degrees Celsius. The total change in the Global Average Temperature between times x = a and x = b is y(b) - y(a) degrees Celsius. The average annual change the Global Average Temperature between times x = a and x = b is

y(b) - y(a) degrees Celsius. ba

Following the above concepts, compute the following (using any Scientific Calculator): a) Consider the data on the Global Average Temperature. Using Minitab, fit a linear polynomial function to the data. b) Estimate the Global Average Temperature in the years 2002, 2003, 2004, 2005 and 2050. c) The annual change in the Global Average Temperature for 2002. d) The change in Global Average Temperature between the years 1950 (x = 0) and 2001 (x = 51). e) The average annual change in Global Average Temperature between x = 0 and x = 51. Step III: Consider the data on the Global Average Temperature. Using Minitab, fit a cubic polynomial function to the data and make predictions of the Global Average Temperature in the year 2050. Step IV: Ecologists are forewarning that the Global Average Temperature will reach between 16 degrees Celsius (i.e., 60.8 degrees Fahrenheit) and 19 degrees Celsius (i.e., 66.2 degrees Fahrenheit) by the year 2050. How well does each of your regression models in Steps II and III confirm that prediction? Step V: Using your cubic polynomial function fitted to the Global Average Temperature data, the concepts of Relative Extrema, as described above, and using the software “MAPLE,” find the Relative Extrema of the Global Average Temperature and the corresponding years, in which these occurred. Step VI: By incorporating the above findings, write your research project. Step VII: Submit your findings, graphs, tables, charts, and report on Monday, 20th November, 2017, positively, in the class, with your group name, student I. D., class, semester, course, and reference number, typed and put in a folder. Late submission of the project will not be accepted or graded. 3.2. Analysis of Data and Graphs: These are described below. It is observed that the a cubic polynomial function fits to the data better than a linear polynomial function, and thus can make better predictions of the Global Average Temperature in the future.

RESULTS USING MAPLE AND MINITAB


65 (A) RELATIVE EXTREMA OF GLOBAL AVERAGE TEMPERATURE USING MAPLE > with(Student[Calculus1]): CriticalPoints( -0.000006*x^3+0.000765*x^2-0.01290*x+14.01 );

> extrema( -0.000006*x^3+0.000765*x^2-0.01290*x+14.01,{},x );

"SCATTERPLOT WITH CONNECT LINE"

"GLOBAL AVERAGE TEMPERATURE (Y) (IN DEGREE CELSIUS) VS. YEAR (X)"

Average Temp. in Deg. Cels. (Y)

14.7 14.6 14.5 14.4 14.3 14.2 14.1 14.0 13.9 13.8 0

10

20 30 YEAR (X)_1950_To_2001

For convenience, we have let X = 0 in 1950.

FIGURE 5: Scatterplot

40

50


66

"SCATTERPLOT WITH REGRESSION"

"GLOBAL AVERAGE TEMPERATURE (Y) (IN DEGREE CELSIUS) VS. YEAR (X)"

Average Temp. in Deg. Cels. (Y)

14.7 14.6 14.5 14.4 14.3 14.2 14.1 14.0 13.9 13.8 0

10

20 30 YEAR (X)_1950_To_2001

40

50

For convenience, we have let X = 0 in 1950.

FIGURE 6: Scatterplot with Regression Fitted Line Plot

Y = 13.85 + 0.01112 X 14.7

S R-Sq R-Sq(adj)

14.6 14.5 14.4

Y

14.3 14.2 14.1 14.0 13.9 13.8 0

10

20

30

40

X

FIGURE 7: Fitting of a Line

50

0.134611 61.5% 60.8%


67

Fitted Line Plot

Y = 14.01 - 0.01290 X + 0.000765 X**2 - 0.000006 X**3 14.7

S R-Sq R-Sq(adj)

14.6

0.123163 69.1% 67.2%

14.5 14.4

Y

14.3 14.2 14.1 14.0 13.9 13.8 0

10

20

30

40

50

X

FIGURE 8: Fitting of a Cubic Polynomial 4. Concluding Remarks and Suggestion: In this paper, we have discussed about developing a lesson plan to show how the teaching of basic mathematical concepts in a Business Calculus (MAC2233) or Calculus 1 (MAC2311) Course via Maple and Mintab Software can help the students to gain the knowledge and insights from various aspects of sustainability data, such as examining and studying the Global Average Temperature d, and conducting its mathematical analysis. It is hoped that this paper will be helpful in teaching Business Calculus (MAC2233) and Calculus 1 (MAC2311) Courses via Maple and Mintab Software at Miami Dade College. Further, as there is a great emphasis on mathematical literacy and critical thinking in education these days, it is hoped that, with the help of Maple and Mintab Software, the students will be able to conduct mathematical research projects in their MAC2233 and MAC2311 courses, and will be able to achieve the following: III. Search or web-search any real world data. IV. V. VI.

Analyze the data mathematically using Maple and Mintab, as discussed above. Write a mathematical research project or report by incorporating the above findings. Present the research project.

Finally, it is hoped that by implementing the techniques discussed in this paper in preparing lesson plans will help us to develop in our students the quantitative analytic skills to evaluate and process numerical data, which is one of the Gen Ed Outcomes of Miami Dade College. The students would be able to successfully investigate the above projects. It should be noted that there are many aspects of “SUSTAINABILITY” and “OUR ENVIRONMENT” to which Undergraduate Mathematics can be easily applied. It is necessary that our mathematics students should be exposed to this important subject. As we always talk about the students’ lack of interest


68 in mathematics, student success, QEP Mathematics, etc., it will be a great idea to introduce this important subject into our curriculum.

Acknowledgments The author would like to thank the Earth Ethics Institute of Miami Dade College for providing us an opportunity to attend a workshop on "Seeing Systems Peace, Equity, Regeneration (part 2)" (EEI1001-3), Spring Term, 2017, at Hialeah Campus, during which this paper was prepared. Also, the author is thankful to the Editorial Committee of Polygon for accepting this paper for publication in the 10th Issue of Polygon (2017). I would also like to acknowledge my sincere indebtedness to the works of various authors and resources on the subject which I have consulted during the preparation of this research project. Last but not least, the author is grateful to Miami Dade College for giving him the opportunity to be of service to this institution, without which it would have been impossible to conduct his research. References 1. Hoffman, L. D., and Bradley, G. L. (2004), Calculus for Business, Economics, and the Social and Life Sciences, 8th Edition, McGraw-Hill, New York. 2. Introduction â&#x20AC;&#x201C; Minitab, http://support.minitab.com/en-us/minitab/17/gettingstarted/introduction/. 3. Teaching Calculus with Maple: A Complete Kit, http://www.maplesoft.com/contact/webforms/CalculusKit.aspx.


69

Survey of Students’ Opinion about “Service Learning - Math Mentorship Program” - A Statistical Analysis

M. Shakil, Ph.D. Professor of Mathematics Department of Liberal Arts and Sciences Miami Dade College, Hialeah Campus FL 33012, USA E-mail: mshakil@mdc.edu Abstract In this paper, Students’ Opinion about “Service Learning - Math Mentorship Program” has been studied from a statistical point of view. By administering a survey on it in some math courses, the data have been analysed statistically which shows some interesting results. It is hoped that the findings of the paper will be quite useful for educators and researchers in various disciplines. 2010 Mathematics Subject Classifications: 97C40, 97C70, 97D40, 97D50. Keywords: ANOVA, Cooperative Learning, Excel, Hypothesis Testing, Math Mentorship, Minitab, Service Learning, Student-Centered Learning.

1. INTRODUCTION In recent years, many educators and researchers have pointed on the importance of student-centered and cooperative learning. For example, among various cooperative group learning approaches of instructions in various disciplines, there has been a great emphasis on the implementation of “Math Mentorship Program” in all level mathematics courses, and whether this program is helpful to the students to build a strong foundation of skills in these courses, and also, at the same time, is helpful in improving their performance in their math courses. The importance of “Math Mentorship Program” in the present day instruction of mathematics at various levels cannot be ignored. As observed by Angelo and Cross (1993), “the goals of college teachers differ, depending on their disciplines, the specific content of their courses, their students, and their own personal philosophies about the purposes of higher education. All faculty, however, are interested in promoting the cognitive growth and academic skills of their students”. According to


70 Greive (2003, p. 48), “classroom assessment is an ongoing sophisticated feedback mechanism that carries with it specific implications in terms of learning and teaching.” Grieve (2003) further observes, “The classroom assessment techniques emphasize the principles of active learning as well as student-centered learning.” According to Dr. De Gallow of the Problem-Based Learning Faculty Institute, University of California, Irvine (see http://www.pbl.uci.edu/whatispbl.html), “Student-centered refers to learning opportunities that are relevant to the students, the goals of which are at least partly determined by the students themselves”. This does not mean that the teacher abdicates her authority for making judgments regarding what might be important for students to learn; rather, this feature places partial and explicit responsibility on the students’ shoulders for their own learning. Creating assignments and activities that require student input presumably also increases the likelihood of students being motivated to learn”. For details on student-centered and cooperative learning approaches of instructions, the interested readers are also referred to Slavin (1995), Oon-Seng (2003), Timpson and Bendel-Simso (2003), Savin-Baden and Major (2004), Johnson et al. (2008), and Eggen and Kauchak (2010), among others. The effects of problem-based and project-based learning approaches of instructions on students’ learning have been studied and analyzed by many researchers. See, for example, Kaw and Yalcin (2008), Shakil (2008, 2015), Gok and Silay (2010), and Khairiree and Kurusatian (2009), among others. Motivated by the importance of “Math Mentorship Program” in the present day instruction of mathematics at various levels, during the fall term 2016, I had implemented the “Service Learning - Math Mentorship Program” in my six teaching courses as one of the means of grading the students, besides other assignments. Then, after the completion of these projects, the students were asked to complete the online survey questions, through the Blackboard Course Management, as stated in Section 2 above, about their opinions towards the “Service Learning - Math Mentorship. It appears from the literature that not much work has been done on the Students’ Opinion about “Math Mentorship Program”. In this paper, in view of the importance of “Math Mentorship Program” at various levels of mathematics instructions, we have statistically investigated and analyzed the Students’ Opinion about “Service Learning - Math Mentorship Program”, as implemented in six courses of my fall term 2016. The organization of this paper is as follows. Section 2 discusses the methodology. The results are given in section 3. The discussion and conclusion are provided in Section 4. 2. METHODOLOGY A survey consisting of 10 multiple choice questions (see Appendix I) were constructed for the survey on “Students’ Opinion about Service Learning - Math Mentorship Program”. It was administered to six different math courses in the Fall Term of 2016. The courses were MAC 1147 (two sections), MAC 2233 (one section), MAC 2311 (one section), and STA 2023 (two sections), which will be referred as MAC and STA2023. The survey was administered online in Blackboard by the instructor in each of these courses. A total of 148 students (out of 196 enrolled students) participated in the survey, the details of which are provided in the following Table 1 below.


71

Table 1: Surveyed Courses Discipline STA MAC

Total

Courses STA2023 (Two Sections) MAC1147 (Two Sections), MAC 2233 (One Section), MAC 2311 (One Section), 6

Respondents 55 93

148

After the completion of the “Service Learning - Math Mentorship Program”, the students of the above six math courses were asked to answer the following survey questions about their opinions towards the “Math Mentorship Program”. These survey questions were same for each class, and were posted online in the Blackboard Course Management. Survey of Students’ Opinion about “Service Learning (iCED) - Math Mentorship Program”, Fall Term, 2016 Question 1: Did the “Service Learning (iCED) - Math Mentorship Program” help you to build a strong foundation of skills in your math course? a) Yes b) No Question 2: Did you find the above Learning Assignments Approach of Teaching, “Service Learning (iCED) - Math Mentorship Program”, helpful in your field of studies? a) Yes b) No Question 3: Do you think that the “Service Learning (iCED) - Math Mentorship Program” will be helpful in improving your performance in your current math course (Fall Term, 2016)? a) Yes b) No Question 4: Did you find the Completion of 20 Hours Math Mentorship Program - Service Learning (iCED) on time difficult? a) Yes b) No Question 5: Did you find the Completion of 20 Hours Math Mentorship Program - Service Learning (iCED) on time easy? a) Yes b) No


72

Question 6: Did you find the MDC Hialeah Campus Math Lab, Learning Resource and Service Learning (iCED) Staff helpful in the Completion of your “20 Hours Math Mentorship Program”, for your math course?

a) Yes b) No Question 7: Would you recommend the implementation of “Service Learning (iCED) - Math Mentorship Program” for your future math courses?

a) Yes b) No Question 8: Would you recommend that the other math professors should also implement the “Service Learning (iCED) - Math Mentorship Program” in their courses?

a) Yes b) No Question 9: Out of the following Learning Assignments Approach of Teaching, which approach you would prefer most in future math courses? a) Service Learning (iCED) - Math Mentorship Program b) Problem Based Learning Assignments b) Project (Computer) Based Learning Assignments d) Project (Writing) Based Learning Assignments d) All of the above e) None of the above Question 10: Choose one of the following rating scales for the above Learning Assignments Approach of Teaching Mathematics Courses, that is, “Service Learning (iCED) - Math Mentorship Program”. a) Excellent b) Good c) Average e) None of the above


73 3. DISCUSSIONS OF RESULTS As pointed above, during the fall term 2016, I had implemented the “Service Learning - Math Mentorship Program” in my six teaching courses as one of the means of grading the students, besides other assignments. After the completion of these projects, the students were asked to complete the online survey questions, through the Blackboard Course Management, as stated in Section 2 above, about their opinions towards the “Service Learning - Math Mentorship”. A total of 148 students (out of 196 enrolled students) participated in the survey, and completed the survey online within the specified time without any difficulty. The students’ responses on the above ten survey questions and data analysis are provided, using Minitab and Excel, in Figures 1 - 10 and Tables 1 6 below. The “TWO-WAY ANOVA: STUDENTS’ RESPONSE versus QUESTION (# 9), and QUESTION (# 10), including INTERACTION PLOTS (DATA MEANS)”, are also provided, using Minitab; see Tables 5 and 6 respectively, and Figures 7 and 10 respectively. These tables and figures are self-explanatory. One can easily draw inferences from the statistical analysis of students’ opinions towards the “Service Learning - Math Mentorship” learning approach of instructions implemented in some mathematics courses during Fall 2016 term. From the interaction plots as provided below (Figures 7 and 10), one can also observe some interaction between different factors. TABLE 1 STUDENTS’ RESPONSES versus QUESTIONS (# 1, 2, 3) (Average %) QUESTIONS 1. Did the “Service Learning (iCED) - Math Mentorship Program” help you to build a strong foundation of skills in your math course? 2. Did you find the above Learning Assignments Approach of Teaching, “Service Learning (iCED) - Math Mentorship Program”, helpful in your field of studies? 3. Do you think that the “Service Learning (iCED) - Math Mentorship Program” will be helpful in improving your performance in your current math course (Fall Term, 2016)?

Yes (%)

No (%)

96.45

3.55

92.63

7.37

96.17

3.83


74

STUDENTS’ RESPONSES

STUDENTS’ RESPONSES versus QUESTIONS (# 1, 2, 3) (Average %) 120.00% 100.00% 80.00% 60.00% 40.00% 20.00% 0.00%

1

2

3

Yes (%)

96.45%

92.63%

96.17%

No (%)

3.55%

7.37%

3.83%

FIG. 1: Percentage of Students’ Responses towards Questions # (1, 2, 3)

TABLE 2 STUDENTS’ RESPONSES versus QUESTIONS (# 4, 5, 6, 7, 8) (Average %) QUESTIONS 4. Did you find the Completion of 20 Hours Math Mentorship Program - Service Learning (iCED) on time difficult? 5. Did you find the Completion of 20 Hours Math Mentorship Program - Service Learning (iCED) on time easy? 6. Did you find the MDC Hialeah Campus Math Lab, Learning Resource and Service Learning (iCED) Staff helpful in the Completion of your “20 Hours Math Mentorship Program”, for your math course? 7. Would you recommend the implementation of “Service Learning (iCED) - Math Mentorship Program” for your future math courses? 8. Would you recommend that the other math professors should also implement the “Service Learning (iCED) - Math Mentorship Program” in their courses?

Yes (%)

No (%)

21.01

78.99

Unanswered (%) 0

0 81.45

18.55

0.5 96.5

3.0

83.55

15.69

0.76

85.65

13.73

0.62


75

STUDENTS’ RESPONSES

STUDENTS’ RESPONSES versus QUESTIONS (# 4, 5, 6, 7, 8) (Average %) 120.00% 100.00% 80.00% 60.00% 40.00% 20.00% 0.00%

Q. 4

Q. 5

Q. 6

Q.7

Q. 8

Yes (%)

21.01%

81.45%

96.50%

83.55%

85.65%

No (%)

78.99%

18.55%

3.00%

15.69%

13.73%

0%

0%

0.50%

0.76%

0.62%

Unanswered (%)

FIG. 2: Percentage of Students’ Responses towards Questions # (4, 5, 6, 7, 8)

TABLE 3 Q. 9: Out of the following Learning Assignments Approach of Teaching, which approach you would prefer most in future math courses? COURSE

MAC2311 MAC1147-A MAC1147-B MAC2233 STA2023-A STA2023-B AVERAGE

Service Learning (iCED) Math Mentorship Program (%) 63.64 62.963 38.10 52.17 70.83 54.84

Problem Based Learning Assignments (%)

Project (Writing) Based Learning Assignments (%)

All of the above (%)

None of the above (%)

TOTAL

9.09 3.704 19.05 8.70 4.17 0

Project (Computer) Based Learning Assignments (%) 9.09 0 4.76 8.70 0 6.45

0 0 0 0 0 0

18.18 22.222 33.33 26.08 20.83 29.03

0 11.111 4.76 4.35 4.17 9.68

100 100 100 100 100 100

57.0905

7.452333333

4.833333333

0

24.9453

5.6785

100


76

Q. 9: Out of the following Learning Assignments Approach of Teaching, which approach you would prefer most in future math courses?

80 70 60 50 40 30 20 10 0

MAC2311 MAC1147-A MAC1147-B MAC2233 STA2023-A STA2023-B

FIG. 3: Percentage of Students’ Responses towards Question # (9)

TABLE 4 Q. 10: Choose one of the following rating scales for the above Learning Assignments Approach of Teaching Mathematics Courses, that is, “Service Learning (iCED) - Math Mentorship Program”. COURSE Excellent (%) Good (%) Average (%) None of the TOTAL (%) above (%) MAC2311 72.73 22.73 4.54 0 100 MAC1147-A 55.56 37.04 3.70 3.70 100 MAC1147-B 80.95 14.29 4.76 0 100 MAC2233 39.13 43.48 4.35 13.04 100 STA2023-A 70.83 29.17 0 0 100 STA2023-B 83.87 16.13 0 0 100 AVERAGE 67.17833 27.14 2.89166667 2.79 100


77

Q. 10: Choose one of the following rating scales for the above Learning Assignments Approach of Teaching Mathematics Courses, that is, “Service Learning (iCED) - Math Mentorship Program”.

90 80 70 60 50 40 30 20 10 0

Excellent (%) Good (%) Average (%) None of the above (%)

FIG. 4: Percentage of Students’ Responses towards Question # (10)

TABLE 5 Two-way ANOVA: RESPONSE (%)-Q. 9 versus COURSE, Teaching Approach Source COURSE Teaching Approac Error Total S = 7.369

COURSE MAC1147-A MAC1147-B MAC2233 MAC2311 STA2023-A STA2023-B

DF 5 5 25 35

SS 108.9 13463.4 1357.7 14930.0

R-Sq = 90.91%

F 0.40 49.58

P 0.843 0.000

R-Sq(adj) = 87.27%

Individual 95% CIs For Mean Based on Pooled StDev ---------+---------+---------+---------+ (------------*-----------) (-----------*------------) (-----------*------------) (-----------*------------) (-----------*------------) (-----------*------------) ---------+---------+---------+---------+ 15.0 20.0 25.0 30.0

Mean 21.3333 16.6667 16.6667 16.6667 16.6667 16.6667

Teaching Approach All of the a Math Mentors None of the Problem Base Project (Com Project (Wri

MS 21.78 2692.68 54.31

Mean 25.7417 57.0905 3.9933 7.4523 8.5370 1.8518

Individual 95% CIs For Mean Based on Pooled StDev --+---------+---------+---------+------(--*--) (---*--) (--*--) (--*--) (--*--) (--*--) --+---------+---------+---------+------0 20 40 60


78

Normal Probability Plot of the Residuals

Histogram of the Residuals (response is RESPONSE (%)-Q. 9)

(response is RESPONSE (%)-Q. 9)

99

12

95

10

80

90

Percent

Frequency

14

8 6

70 60 50 40 30 20

4

10

2

5

0

-16

-8

0 Residual

8

16

1

-20

-15

-10

-5

0 Residual

5

10

15

FIG. 5: Residual Histogram for RESPONSE (%)-Q. 9 FIG. 6: Normplot of Residuals for RESPONSE (%)-Q. 9

Interaction Plot (data means) for RESPONSE 30

C O URSE MA C 1105 MA C 1147 MA C 2233 MA C 2311 STA 2023-A STA 2023-B

25

Mean

20 15 10 5 0 NO

YES OPINION

FIG. 7: Interaction Plot (data means) for RESPONSE - QUESTION (# 9)

(A) Hypothesis Test: As a consequence of the results in Table 5 and Figure 7, using the P-value approach, we can draw the following conclusion from the hypothesis test among the means for the main effect due to the Survey Questions (COURSE) as follows:


79

Since the P-value (0.843) > 0.05, we fail to reject the null hypothesis. That is, at the 5% level of significance, we cannot conclude that the average of the students’ responses for the survey question # 9 (COURSE) is significantly different from each other.

Conclusion:

(B) Hypothesis Test: As a consequence of the results in Table 5 and Figure 7, using the P-value approach, we can draw the following conclusion from the hypothesis test among the means for the main effect due to the Students’ Opinions (Teaching Approach) as follows: 

Since the P-value (0.000) < 0.05, we reject the null hypothesis. That is, at the 5% level of significance, we can conclude that the average of the students’ responses for the survey question # 9 on the Students’ Opinions (Teaching Approach) is significantly different from each other.

Conclusion:

TABLE 6 Two-way ANOVA: RESPONSE(%)-Q. 10 versus COURSE, RATING-Math Mentorship Program Source COURSE RATING-Math Ment Error Total S = 12.31

DF 5 3 15 23

SS 0.0 16593.9 2271.8 18865.7

COURSE MAC1147-A MAC1147-B MAC2233 MAC2311 STA2023-A STA2023-B

R-Sq = 87.96%

RATING-Math Mentorship Program Average Excellent Good None of the

Mean 25 25 25 25 25 25

MS 0.00 5531.31 151.45

F 0.00 36.52

P 1.000 0.000

R-Sq(adj) = 81.54%

Individual 95% CIs For Mean Based on Pooled StDev ---+---------+---------+---------+-----(------------------*-----------------) (------------------*-----------------) (------------------*-----------------) (------------------*-----------------) (------------------*-----------------) (------------------*-----------------) ---+---------+---------+---------+-----14.0 21.0 28.0 35.0

Mean 2.8917 67.1783 27.1400 2.7900

Individual 95% CIs For Mean Based on Pooled StDev ---+---------+---------+---------+-----(---*---) (---*---) (---*---) (---*---) ---+---------+---------+---------+-----0 25 50 75


80

Normal Probability Plot of the Residuals

Histogram of the Residuals (response is RESPONSE(%)-Q. 10)

(response is RESPONSE(%)-Q. 10)

99

6

95

5

80

90

Percent

Frequency

7

4 3

70 60 50 40 30 20

2

10

1 0

5

-30

-20

-10 Residual

0

10

1

-30

-20

-10

0 Residual

10

20

30

FIG. 8: Residual Histogram for RESPONSE (%)-Q.10 FIG. 9: Normplot of Residuals for RESPONSE (%)-Q.10

Interaction Plot (data means) for RESPONSE(%)-Q. 10 90

C O URSE MA C 1147-A MA C 1147-B MA C 2233 MA C 2311 STA 2023-A STA 2023-B

80 70

Mean

60 50 40 30 20 10 0 Average

Excellent Good None of the above RATING-Math Mentorship Program

FIG. 10: Interaction Plot (data means) for RESPONSE - QUESTION (# 10)

(A) Hypothesis Test: As a consequence of the results in Table 6 and Figure 10, using the P-value approach, we can draw the following conclusion from the hypothesis test among the means for the main effect due to the Survey Questions (COURSE) as follows:


81

Conclusion:

Since the P-value (1.000) > 0.05, we fail to reject the null hypothesis. That is, at the 5% level of significance, we cannot conclude that the average of the students’ responses for the survey question # 10 (COURSE) is significantly different from each other.

(B) Hypothesis Test: As a consequence of the results in Table 5 and Figure 7, using the P-value approach, we can draw the following conclusion from the hypothesis test among the means for the main effect due to the Students’ Opinions (RATING-Math Mentorship Program) as follows: 

Conclusion:

Since the P-value (0.000) < 0.05, we reject the null hypothesis. That is, at the 5% level of significance, we can conclude that the average of the students’ responses for the survey question # 10 on the Students’ Opinions (RATING-Math Mentorship Program) is significantly different from each other.

4. CONCLUDING REMARKS Based on our observations and analysis, it is clear that, among various cooperative group learning approaches of instructions in various disciplines, “Math Mentorship Program” learning approach of instructions is very important, and would be helpful to the students to build a strong foundation of skills in these courses, and also, at the same time, would be helpful in improving their performance in their math courses. Incorporating this approach of instruction in classes can help teachers to measure the effectiveness of their teaching by finding out what students are learning in the classroom and how well they are learning. In addition, , “Math Mentorship Program” learning approach of instructions can provide an efficient avenue of input and a high information return to the instructor without spending much time and energy. It is recommended that, in future, more , “Math Mentorship Program” learning approach of instructions be developed and implemented in other mathematics classes for better learning and more effective teaching. It is hoped that the findings of the paper will be quite useful for researchers in various disciplines. Acknowledgment The author would like to express his sincere gratitude and acknowledge his indebtedness to his students of the courses, MAC 2311, MAC 1147 (two sections), MAC 2233, and STA 2023 (two sections), during the Fall Term of 2016, for their cooperation in participating in the survey. Further, the author would like to thank the Editorial Committee of Polygon for accepting this paper for publication in Polygon. I would also like to acknowledge my sincere indebtedness to the works of various authors and resources on the subject which I have consulted during the preparation of this research project. The author is thankful to his wife for her patience and perseverance for the period


82 during which this paper was prepared. The author would like to dedicate this paper to his late parents, brothers and sisters. Last but not least, the author is thankful to the Miami Dade College for providing him an opportunity to serve this college, and allowing him to take a Graduate Course in “Analysis of Teaching”, without which it was impossible to conduct his research. References Angelo, T. A., and Cross, K. P. (1993). Classroom Assessment Techniques – A Handbook for College Teachers. Jossey-Bass, San Francisco. Eggen, P., and Kauchak, D. (2010). Educational Psychology: Windows on Classrooms, 8th Edition. Pearson Education, Inc., Upper Saddle River, NJ. Gallow, D. http://www.pbl.uci.edu/whatispbl.html. Greive, D. (2003). A Handbook for Adjunct/Part-Time Faculty and Teachers of Adults, 5th Edition. The Adjunct Advocate, Ann Arbor. Johnson, D. W., Johnson, R. T., and Holubec, E. J. (2008). Cooperation in the classroom, 8th Edition. Interaction Book Co., Edina, MN. Kaw, A. K., and Yalcin, A. (2008). Problem-centered approach in a numerical methods course. Journal of Professional Issues in Engineering Education and Practice, 134(4), 359-364. Khairiree, K., and Kurusatian, P. (2009). Enhancing Students' Understanding Statistics with TinkerPlots: Problem-Based Learning Approach, Electronic Proceedings of the Fourteenth Asian Technology Conference in Mathematics, Beijing, China. Oon-Seng, T. (2003). Problem-based learning Innovation: Using problems to power Learning in the 21st Century. Thomson Learning Asia, Singapore, Singapore. Savin-Baden, M., and Major, C. H. (2004). Foundations of Problem-Based Learning. 1st Edition. Open University Press, Houston, Texas. Shakil, M. (2008). Classroom Assessment Techniques and their Implementation in a Mathematics Class. Polygon, 2008, 1 - 21. Shakil, M. (2015). A Statistical Analysis of Students’ Opinions towards Project-Based and Problem-Based Learning Approaches of Instructions in Some Mathematics Courses. Polygon, 2015, 30 - 44.


83 Slavin, R. E. (1995). Cooperative learning theory, research and practice. Allyn & Bacon, Boston. Timpson, W., and Bendel-Simso, P. (2003). Concepts and Choices FOR TEACHING. Atwood Publishing, Madison, WI.


84

Survey of Students’ Attitudes towards Mathematics and Statistics A Statistical Analysis M. Shakil, Ph.D. Professor of Mathematics Department of Liberal Arts and Sciences Miami Dade College, Hialeah Campus FL 33012, USA E-mail: mshakil@mdc.edu ABSTRACT In this paper, “Students’ Attitudes towards Mathematics and Statistics” have been studied from a statistical point of view. By administering a survey on it in some math and stat courses, the data have been analysed statistically which shows some interesting results. It is hoped that the findings of the paper will be quite useful for educators and researchers in various disciplines. Keywords: Attitude, Learning Outcomes, Learning Process, Mathematics, Statistics, Survey. MSC: 97C20, 97C30, 97C40, 97C70.

1. INTRODUCTION Recently, I attended the 2017 Math Retreat which was held on April 6th, 2017, at the Kendall Campus. The theme of the Math Retreat was: “MIND OVER MATH: PSYCHOLOGY OF STUDENTS IN A MATH CLASS”. Related to this theme (directly or indirectly, in one way or the other), there were various presentations during the Math Retreat. All these presentations were related to the theme of Math Retreat: “MIND OVER MATH: PSYCHOLOGY OF STUDENTS IN A MATH CLASS”, directly or indirectly, in one way or the other, which one cannot ignore in a mathematics classroom in order to improve student motivation, engagement and success. For example, a group of presenters dealt with “Learning Catalytics”, emphasizing their benefits in teaching a math class, for example, student engagement, critical thinking skills, cooperative learning, and a flipped class, among others. The other groups of presenters discussed about “Peer Led Team Learning” and “Teaching with a Tablet. As the importance of the students’ mindset in a math class cannot be denied or underemphasized, in my opinion, all the above strategies of teaching a math class are very important and, if implemented, would be very useful in helping students build mindset, develop a positive attitude towards mathematics, and, at the same time, have a great impact on the students’ mindset needed for success in a mathematics class. I must also mention about a topic, “Psychology of Students in the Mathematics Classroom”, discussed by a group of presenters, which mostly dealt with the


85 techniques that could be used to improve the perception of mathematics in the classroom and help struggling students develop a positive attitude towards mathematics needed for success. Classroom assessment is one of the most significant teaching strategies. It is a major component of classroom research at present. Classroom Assessment Techniques are designed to help teachers measure the effectiveness of their teaching by finding out what students are learning in the classroom and how well they are learning it. As observed by Angelo and Cross (1993), “the goals of college teachers differ, depending on their disciplines, the specific content of their courses, their students, and their own personal philosophies about the purposes of higher education. All faculty, however, are interested in promoting the cognitive growth and academic skills of their students”. Assessing accomplishments in the cognitive domain has occupied educational psychologists for long, (see, for example, Angelo and Cross (1993), and references therein). Many researchers have worked and developed useful theories and taxonomies on the assessment of academic skills, intellectual development and cognitive abilities, both from the analytical and quantitative point of view. The development of the general theory of measuring the cognitive abilities began with the work of Bloom et al. (1956), known as “Bloom Taxonomy.” Further developments continued with the contributions of Ausubel (1968), Bloom et al. (1971), McKeachie et al. (1986), and Angelo and Cross (1993), among others. “Active engagement in higher learning implies and requires self-awareness and self-direction,” which is defined as “metacognition” by cognitive psychologists. The discipline of mathematics taught at Miami Dade College has its own identity, importance and educational values. Besides Utilitarian, Disciplinary, and Cultural values and aims of the teaching of mathematics, another important objective is to develop in our students the quantitative analytic, critical thinking, communication and technological skills to evaluate and process numerical data, which belong to the following General Education Learning Outcomes as defined by the Miami Dade College:     

Communicate effectively using listening, speaking, reading, and writing skills. Use quantitative analytical skills to evaluate and process numerical data. Solve problems using critical and creative thinking and scientific reasoning. Formulate strategies to locate, evaluate, and apply information. Use computer and emerging technologies effectively.

The objective of developing the quantitative analytic, critical thinking, communication and technological skills varies for different levels of mathematics classes. However, for any level of mathematics class, some of the common objectives of the above General Education Learning Outcomes can be specified as follows:  To develop the power of analytical reasoning and logical thinking.  To develop the speed and accuracy in computing.  To develop the skills of quantitative reasoning and analysis of real world data.  To develop the competency of analyzing the real world problem.  To enable the students to develop mathematical models to solve different real world problems. Besides student success, our concern is also “how to incorporate the general education


86 (learning) outcomes into a course.” In order to achieve these goals, it is understood that all the math instructors at Miami Dade College have been incorporating the various general education learning outcomes into their math classes, in one way or another. As pointed out by Leder (1985), “Discussions about the nature of mathematics learning typically include both cognitive and affective variables such as attitude”. According to the Encyclopedia Britannica, “Attitude, in social psychology, is defined as a cognition, often with some degree of aversion or attraction (emotional valence), that reflects the classification and evaluation of objects and events. While attitudes logically are hypothetical constructs (i.e., they are inferred but not objectively observable), they are manifested in conscious experience, verbal reports, overt behaviour, and physiological indicators” (https://www.britannica.com/topic/attitude-psychology). As noted by McLeod (2014), “An attitude is a relatively enduring organization of beliefs, feelings, and behavioral tendencies towards socially significant objects, groups, events or symbols" (Hogg and Vaughan 2005, p. 150). Further, as pointed out by Judi et al. (2011), “Student's attitude towards a course is important because it affects the entire learning process. A positive attitude enables students to develop statistical thinking skills, to apply knowledge acquired in everyday life, and to have an enjoyable experience throughout the course.” The interested readers are also referred to Ashaari eta al. (2011), Di Martino and Zan (2011), Pepin (2011), and Mata et al. (2012), among others. In view of the importance of the students’ mindset needed for success in a mathematics class, to improve the perception of mathematics in the classroom and help struggling students develop a positive attitude towards mathematics needed for success, I conducted a Survey on Students’ Attitudes towards Mathematics and Statistics in some of my math and stat classes. For the interest of researchers and educators, the results of these surveys are described below (Section 2). 2. METHODOLOGY Two different surveys on “Students’ Attitudes towards Mathematics and Statistics”, consisting of 20 multiple choice questions, for each of Mathematics and Statistics Courses, were constructed using different resources. The survey on “Students’ Attitudes towards Mathematics” was administered to the students of three different math courses (MAT0020, MAC 1105, MAC 2233) in one semester), which will be referred as MAC. The number of students who took this survey was 74. The survey on “Students’ Attitudes towards Statistics” was administered to the students of two different sections of statistics courses (STA2023) in another semester), which will be referred as STA. The number of students who took this survey was 49. The survey was administered online by the instructor in each of these courses by using an online tool provided by the CT & D of Miami Dade College. A total of 123 students participated in the survey the details of which are provided in the following Table 1 below. Table 1: Surveyed Courses Discipline Courses Respondents MAC MAT0020, MAC1105, MAC 2233 74 STA STA2023 (Two Sections) 49 Total 123


87 The students of the above courses were asked to respond to the following survey questions about their attitude towards mathematics and statistics respectively. Instructor: Dr. M. Shakil Courses: MAT0020, MAC 1105, MAC 2233 Title Survey: Survey of Student Attitudes toward Mathematics The number of students who took this survey was: 74 Strongly Strongly agree Agree Disagree disagree 1. I am confident that I can get good grades in mathematics.

[27]

[35]

[6]

[2]

2. The skills I learn in mathematics will help me in other subjects.

[39]

[25]

[6]

[0]

3. I am always stressed in a mathematics class.

[13]

[18]

[29]

[10]

4. Doing mathematics lets me think creatively.

[20]

[37]

[9]

[4]

[46]

[21]

[3]

[0]

6. I would like people to think that I am smart in mathematics.

[27]

[30]

[12]

[1]

7. Mathematics is one of the most important subjects for people to study.

[33]

[27]

[9]

[1]

8. Mathematics courses will be very helpful no matter what my major is.

[36]

[27]

[5]

[2]

[17]

[33]

[12]

[8]

10. Mathematics helps develop my mind and teaches me to think.

[33]

[30]

[4]

[2]

11. An understanding of mathematics is needed by artists and writers as well as scientists.

[24]

[38]

[5]

[1]

[9]

[39]

[17]

[5]

[2]

[11]

[32]

[24]

[17]

[9]

5. It is important to do well in mathematics.

9. I usually enjoy studying mathematics.

12. I am good at mathematics. 13. Mathematics is not important for most jobs. 14. Solving mathematics problems is fun.

[9]


88 [35] 15. I avoid mathematics whenever I can. [3]

[18]

[34]

[13]

[5]

[15]

[35]

[15]

17. I get more nervous before a mathematics test than before tests in other subjects.

[19]

[26]

[16]

[9]

18. Most people should study some mathematics.

[27]

[40]

[3]

[0]

19. There is usually more than one way to solve a math problem.

[34]

[36]

[0]

[0]

20. Mathematical thinking helps me make intelligent decisions.

[32]

[33]

[4]

[1]

16. I am bad at mathematics.

Instructor: Dr. M. Shakil Courses: STA 2023 (Two Sections) Title Survey: Survey of STA2023 Student Attitudes toward Statistics The number of students who took this survey was: 49 Strongly Strongly agree Agree Disagree disagree 1. I am confident that I can get good grades in statistics.

[13]

[23]

[12]

[1]

2. The skills I learn in statistics will help me in other subjects.

[18]

[27]

[4]

[0]

3. To succeed in life you need to be able to do statistics.

[7]

[30]

[12]

[0]

[15]

[23]

[10]

[1]

[19]

[28]

[1]

[0]

6. I would like people to think that I am smart in statistics.

[17]

[23]

[8]

[1]

7. Statistics is one of the most important subjects for people to study.

[8]

[26]

[11]

[4]

8. Statistics courses will be very helpful no matter what my major is.

[15]

[23]

[10]

[1]

4. Doing statistics lets me think creatively. 5. It is important to do well in statistics.


89 9. I usually enjoy studying statistics. [9]

[17]

[19]

[4]

10. Statistics helps develop my mind and teaches me to think.

[17]

[27]

[5]

[0]

11. An understanding of statistics is needed by artists and writers as well as scientists.

[14]

[23]

[12]

[0]

[6]

[20]

[20]

[3]

[2]

[12]

[24]

[11]

[4]

[23]

[18]

[4]

[9]

[10]

[24]

[6]

[4]

[17]

[21]

[6]

[15]

[13]

[16]

[5]

[10]

[33]

[6]

[0]

[23]

[25]

[3]

[0

12. I am good at statistics. 13. Statistics is not important for most jobs. 14. Solving statistics problems is fun. 15. It scares me to have to take statistics. 16. I am bad at statistics. 17. I get more nervous before a statistics test than before tests in other subjects. 18. Most people should study some statistics. 19. Taking statistics is a waste of time. 20. Statistical thinking helps me make intelligent decisions.

[0] [12]

[1] [34]

3. DISCUSSIONS OF RESULTS As pointed out above, during the two different semesters, I had administered the online surveys on “Students’ Attitudes towards Mathematics and Statistics” to the students of some my math and stat classes respectively. A total of 123 students participated in the two surveys within the specified time without any difficulty. 3.1. Students’ Attitudes towards Mathematics: In order to analyze the mathematics students’ responses to their attitude towards mathematics on the above survey, ten following typical survey questions were selected. 

Question 2: The skills I learn in mathematics will help me in other subjects.


90 

Question 7: Mathematics is one of the most important subjects for people to study.

Question 8 Mathematics courses will be very helpful no matter what my major is.

Question 9: I usually enjoy studying mathematics.

Question 10: Mathematics helps develop my mind and teaches me to think.

Question 11: An understanding of mathematics is needed by artists and writers as well as scientists.

Question 12: I am good at mathematics.

Question 15: I avoid mathematics whenever I can.

Question 16: I am bad at mathematics.

Question 20: Mathematical thinking helps me make intelligent decisions.

The data analysis of the above ten selected survey questions is provided in Figures 1 – 5 below. Survey of Student Attitudes towards Mathematics The number of students that took this survey was: 74

Q.20 Q.10 Q. 15 Q.16 Q. 12 Q. 11 Q. 9 Q. 8 Q. 7 Q. 2

Strongly disagree Disagree Agree Strongly agree

0

10

20

30

40

FIGURE 1

50


91

Strongly agree Q.20 14% Q.10 14%

Q. 15 1% Q.16 2%

Q. 2 17%

Q. 12 4%

Q. 7 14%

Q. 11 11% Q. 9 7%

Q. 8 16%

FIGURE 2

Agree Q.20 12% Q.10 11%

Q. 2 9% Q. 7 9% Q. 8 9%

Q. 15 6% Q.16 5%

Q. 12 14%

Q. 9 12% Q. 11 13%

FIGURE 3


92

Q.10 3%

Q.20 3%

Disagree

Q. 2 Q. 7 4% 7% Q. 8 4%

Q. 15 26%

Q. 9 9%

Q.16 27%

Q. 11 4%

Q. 12 13%

FIGURE 4

Q.10 4%

Q.20 2%

Strongly disagree Q. 2 0%

Q. 7 2%

Q. 8 4%

Q. 9 17%

Q. 15 27%

Q. 12 11%

Q. 11 2%

Q.16 31%

FIGURE 5 These figures are self-explanatory. From the statistical analysis of the above selected survey questions based on these figures, one can easily draw some inferences on mathematics students’ attitude towards mathematics, for example, as illustrated below for the sake of completion. 

17 % of the Participant “Strongly Agreed” on Question 2: The skills I learn in mathematics will help me in other subjects.


93 

14 % of the Participant “Strongly Agreed” on Question 7: Mathematics is one of the most important subjects for people to study.

16 % of the Participant “Strongly Agreed” on Question 8: Mathematics courses will be very helpful no matter what my major is.

7 % of the Participant “Strongly Agreed” on Question 9: I usually enjoy studying mathematics.

14 % of the Participant “Strongly Agreed” on Question 10: Mathematics helps develop my mind and teaches me to think.

11 % of the Participant “Strongly Agreed” on Question 11: An understanding of mathematics is needed by artists and writers as well as scientists.

4 % of the Participant “Strongly Agreed” on Question 12: I am good at mathematics.

1 % of the Participant “Strongly Agreed” on Question 15: I avoid mathematics whenever I can.

2 % of the Participant “Strongly Agreed” on Question 16: I am bad at mathematics.

14 % of the Participant “Strongly Agreed” on Question 20: Mathematical thinking helps me make intelligent decisions.

3.2. Students’ Attitudes towards Statistics: In order to examine the statistics students’ responses to their attitude towards statistics on the above survey, five following typical survey questions were selected. 

Question 2: The skills I learn in statistics will help me in other subjects.

Question 7: Statistics is one of the most important subjects for people to study.

Question 8: Statistics courses will be very helpful no matter what my major is.

Question 9: I usually enjoy studying statistics.

Question 11: An understanding of statistics is needed by artists and writers as well as scientists.

The data analysis of the above five selected survey questions is provided in Figures 6 – 10 below.


94

Survey of STA2023 Student Attitudes toward Statistics The number of students who took this survey was: 49

Q. 11 Q. 9

Strongly disagree Disagree

Q. 8

Agree Q. 7

Strongly agree

Q. 2 0

5

10

15

20

25

30

FIGURE 6

Strongly agree Q. 2

Q. 7

Q. 8

22%

Q. 9

28%

14% 13% 23%

FIGURE 7

Q. 11


95

Agree Q. 2

Q. 7

Q. 8

20%

Q. 9

Q. 11

23%

15% 22% 20%

FIGURE 8

Disagree Q. 2

Q. 7

Q. 8

Q. 9

7% 21% 20%

34%

18%

FIGURE 9

Q. 11


96

Strongly disagree Q. 2

Q. 7

Q. 8

Q. 9

Q. 11

0% 0%

44%

45%

11%

FIGURE 10 These figures are self-explanatory. From the statistical analysis of the above selected survey questions based on these figures, one can easily draw some inferences on statistics students’ attitude towards statistics, for example, as illustrated below for the sake of completion. 

28 % of the Participant “Strongly Agreed” on Question 2: The skills I learn in statistics will help me in other subjects.

23 % of the Participant “Strongly Agreed” on Question 8: Statistics courses will be very helpful no matter what my major is.

22 % of the Participant “Strongly Agreed” on Question 11: An understanding of statistics is needed by artists and writers as well as scientists.

14 % of the Participant “Strongly Agreed” on Question 9: I usually enjoy studying statistics.

13 % of the Participant “Strongly Agreed” on Question 7: Statistics is one of the most important subjects for people to study.

4. CONCLUDING REMARKS Based on our observations and data analysis of the survey questions on “Students’ Attitudes towards Mathematics and Statistics” in some of my math and stat classes, it is


97 clear that the importance of the students’ mind-set needed for success in a mathematics or a statistics class cannot be denied or ignored. It is very important to improve the perception of mathematics and statistics in the classroom and help struggling students to develop a positive attitude towards mathematics and statistics needed for success. For example, incorporating the various approaches of instructions in classes, as discussed in Section 1, including the problem-based and project-based learning approaches of instructions, besides others, would be very useful in helping students build mind-set, develop a positive attitude towards mathematics and statistics, and, at the same time, have a great impact on the students’ mind-set needed for success in a mathematics or a statistics class. These approaches can help teachers to measure the effectiveness of their teaching by finding out what students are learning in the classroom and how well they are learning. In addition, these techniques can provide an efficient avenue of input and a high information return to the instructor without spending much time and energy. I must also mention about, “Psychology of Students in the Mathematics Classroom”, as discussed in Section 1, could be used to improve the perception of mathematics in the classroom and help struggling students develop a positive attitude towards mathematics and statistics needed for success. It is recommended that, in future, more “Learning Catalytics” of instructions be developed and implemented in all of our mathematics and statistics classes for better learning and more effective teaching. It is hoped that the findings of the paper will be quite useful for researchers and educators in various disciplines.

ACKNOWLEDGMENTS The author would like to express his sincere gratitude and acknowledge his indebtedness to his students of the courses, MAT 0020, MAC 1105, MAC 2233 and STA 2023, for their cooperation in participating in the survey. Further, the author would like to thank the CT & D for providing him with the tool to administer his online survey on “Students’ Attitudes towards Mathematics and Statistics” in his above courses. Also, the author is thankful to the Editorial Committee of Polygon for accepting this paper for publication in the 10th Issue of Polygon (2017). I would also like to acknowledge my sincere indebtedness to the works of various authors and resources on the subject which I have consulted during the preparation of this research project. Last but not least, the author is grateful to Miami Dade College for giving him the opportunity to be of service to this institution, without which it would have been impossible to conduct his research.

REFERENCES Angelo, T. A., and Cross, K. P. (1993). Classroom Assessment Techniques – A Handbook for College Teachers. Jossey-Bass, San Francisco. Ashaari, N. S., Judi, H. M., Mohamed, H., & Wook, M. T. (2011). Student's Attitude towards Statistics Course. Procedia-Social and Behavioral Sciences, 18, 287-294.


98 Ausubel, D. P. (1968), Educational Psychology: A Cognitive View. Holt, Reinhart & Winston, Troy, Mo. Bloom, B. S., Hastings, J. T., and Madaus, G. F. (1971). Handbook on Formative and Summative Evaluation of Student Learning. McGraw-Hill, New York. Bloom, B. S., Engelhart, M. D., Furst, E. J., Hill, W. H., and Krathwohl, D. R. (1956). Taxonomy of educational objectives: The classification of educational goals. Handbook I: Cognitive domain. David McKay Company, New York. Di Martino, P., & Zan, R. (2011). Attitude towards mathematics: A bridge between beliefs and emotions. Zdm, 43(4), 471-482. Hogg, M., and Vaughan, G. (2005). Social Psychology (4th edition). Prentice-Hall, London. Judi, H. M., Ashaari, N. S., Mohamed, H., and Wook, T. M. T. (2011). Students profile based on attitude towards statistics. Procedia-Social and Behavioral Sciences, 18, 266272. Leder, G. C. (1985). Measurement of attitude to mathematics. For the learning of Mathematics, 5(3), 18-34. Mata, M. D. L., Monteiro, V., and Peixoto, F. (2012). Attitudes towards mathematics: Effects of individual, motivational, and social support factors. Child development research, 2012. McKeachie, W. J., Pintrich, P. R., Lin, Yi-Guang, and Smith, D. A. F. (1986). Teaching and Learning in the College Classroom: A Review of the Research Literature. National Center for Research to Improve Postsecondary Teaching and Learning. University of Michigan, Ann Arbor. McLeod, S. A. (2014). Attitudes and Behavior. Retrieved from www.simplypsychology.org/attitudes.html Pepin, B. (2011). Pupilsâ&#x20AC;&#x2122; attitudes towards mathematics: a comparative study of Norwegian and English secondary students. ZDM, 43(4), 535-546. https://www.britannica.com/topic/attitude-psychology.


99

THE DEMARCATION PROBLEM Dr. Melissa Lammey Associate Professor, Sr., Philosophy Miami Dade College Hialeah Campus E-mail: mlammey@mdc.edu

ABSTRACT The legal battle over science education in the United States that began with Scopes and has seen its most recent conclusion with Dover underscores the significance of key debates in the philosophy of science. The purpose of this paper is to investigate the debate over what distinguishes science from nonscience and the debate about the relevance of philosophical assumptions in settling that question. In the context of science education in the US, a recent question is whether intelligent design theory is more properly considered science or religion – or perhaps nonscience of some other variety. My aim here will be to show that when due consideration is given to the history of science, perhaps the most useful contribution intelligent design theory may be understood to make is one that Phillip Kitcher introduces in his book, Living with Darwin. Here, Kitcher suggests that intelligent design theory might simply intend to identify the limits of natural science. He neither believes that this truly is its intention, nor that it succeeds in doing so. Still, I think this element of intelligent design is worth exploring further. In its attempts and given the particular nature of its attempts, intelligent design might serve as an unintended impetus to advance scientific research in the same way as did Newton’s and Mendel’s ‘theistic inspirations.’ Toward achieving this interpretation, I will consider arguments that intelligent design theory is not properly considered science. However, I will also suggest that this does not necessarily mean that it is of no value to science. It could be that important challenges to philosophical naturalism are a worthwhile consequence of intelligent design theory and that these might have further and farreaching effects.

Keywords: philosophical naturalism, methodological naturalism, demarcation problem


100 DISTINGUISHING SCIENCE FROM NONSCIENCE The question of what distinguishes science from nonscience is not a new question in philosophy, but it has held a new and special significance since McLean. As Judge Overton relied on the criteria set forth by Michael Ruse in determining that creationscience does not qualify as science and the same criteria were relevant to testimony in the Dover trial, they have been brought under attack in the years since. A full treatment of what has come to be known as ‘the demarcation problem’ is offered by Larry Laudan. According to Laudan, the question of what makes a belief scientific is uninteresting. In his view, we should instead be concerned with what makes a belief well-founded. He goes further to question the criteria offered by Ruse, and contends that the “victory in the Arkansas case was hollow, for it was achieved only at the expense of perpetuating and canonizing a false stereotype of what science is and how it works.”51 In The Demise of the Demarcation Problem, Laudan addresses the issue of whether there are epistemic features of science that are unique to it and distinguish it from other systems of belief.52 Ultimately, Laudan thinks philosophers have failed to answer this question. As he explains, this question has been before the attention of philosophers since Aristotle offered two criteria of demarcation between knowledge and opinion. According to Aristotle, knowledge of a phenomenon requires: 1) certainty, or the infallibilism of its foundations, and 2) comprehension of its first causes. Laudan explains that through the seventeenth and eighteenth centuries, Aristotle’s first criterion, ‘infallibilism’, was embraced, but that the second was rejected. Galileo and Newton, for instance, thought that knowing how natural systems functioned was properly considered scientific even without knowing why the systems functioned as such. They did, however, regard the certainty of their conclusions as key in qualifying them as fact and distinct from opinion. The preferential status of a functional explanation of a phenomenon even if an ontological explanation is lacking persists in science today, but infallibilism gave way completely to fallibilism in the nineteenth century. Ruse’s falsifiability criterion seems to be a testament to this fact. Laudan represents fallibilism as marking the end of identifying the distinction between science and nonscience as the distinction between knowledge and opinion. He says that “the unambiguous implication of fallibilism is that there is no difference between knowledge and opinion: within a fallibilism framework, scientific belief turns out to be just a species of the genus opinion.”53 Given the distinction between knowledge and opinion as proposed by Aristotle, Laudan is certainly correct about this. I think, though, that a separate issue is involved here. The term ‘opinion’ seems to more properly denote claims that are addressed to matters of opinion, which are by definition not subject to independent verification and are not candidates for scientific status for that reason. That a claim is falsifiable does not make it an opinion. The opposite seems to be true. It seems that opinions are unfalsifiable in virtue of the subject matter to which they are addressed. I suppose this is a question of semantics, but it seems more appropriate to understand ‘opinion’ in this way. It does 51

Laudan, Larry. Science at the Bar – Causes for Concern, in But is it Science? 2nd ed., Robert T. Pennock and Michael Ruse, eds. New York: Prometheus Books. 2009. 52 Laudan, Larry. The Demise of the Demarcation Problem, in But is it Science? 2nd ed., Robert T. Pennock and Michael Ruse, eds. New York: Prometheus Books. 2009. 53 Ibid,. p. 316.


101 seem that the word ‘opinion’ functions at times to mean one’s take on a matter of opinion and at other times to mean one’s best guess about a matter of fact. I think the former is the better usage and can help to avoid certain misunderstanding about the status of scientific claims. Scientific claims are certainly falsifiable, but opinions, understood as I am proposing, are not those sorts of claims. If we did qualify the definition of opinion in this way, we could perhaps avoid the sort of mistake made when one says that my belief, for instance, that the earth is more than 4 billion years old is simply my opinion. With infallibilism abandoned, Laudan explains that the next significant attempt at demarcation was to focus on the methodology of science as distinct from other types of methodologies. This idea, beginning in the nineteenth century, took the scientific method to be the best method available to produce knowledge claims. Even if the claims produced are fallible, the method itself is self-corrective and can eventually correct mistakes. The problem, according to Laudan, is that there was no agreement in the nineteenth century on what the scientific method actually was. Philosophers of science disagreed on exactly what constituted science and worse still, what they offered as criteria for science often did not describe the methods that scientists were actually using. For this reason, Laudan claims that philosophers failed again to distinguish science from nonscience. He goes further to claim that no one had managed to show that the methods proposed as scientific were “epistemically superior to their rivals.”54 After considering the history of the demarcation effort up to the twentieth century, Laudan raises some questions that he believes should be answered in moving forward. Specifically, he asks: 1) What conditions of adequacy should a proposed demarcation criterion satisfy? 2) Is the criterion under consideration offering necessary or sufficient conditions? 3) What actions or judgments are implied by the claim that a certain belief or activity is “scientific” or “unscientific”? 55 In answering these questions, Laudan contends that 1) a demarcation criterion must adequately explain how we normally distinguish between science and nonscience and it must precisely differentiate between scientific and nonscientific enterprises. In other words, it must account for the epistemic status of scientific claims over nonscientific claims. He also explains that 2) it must provide the necessary and sufficient conditions of scientific status. Otherwise, it can only tell us what might be science in the event that it only offers necessary conditions; or it is precluded from telling us what is certainly nonscience in the event that it only offers sufficient conditions. Both necessary and sufficient conditions are required to tell us what all and only activities or statements count as scientific. Finally, he argues that 3) adequate criteria of demarcation must have ‘judgmental implications,’ insofar as what is called ‘scientific’ is better, or more reliable, knowledge than what is ‘unscientific.’56 Two twentieth century approaches to demarcation that Laudan rejects are verificationism and falsificationism. ‘Verificationism’ is a theory of meaning proposed by the logical positivists which suggests that meaningful claims are those claims that can be exhaustively verified. Laudan rejects verificationism as demarcation because, as he explains, many scientific statements are not open to exhaustive verification. He cites all 54

Ibid., pp. 318-319. Ibid., p. 319. 56 Ibid., pp. 319-322. 55


102 universal laws as a case in point. Also, he argues that a statement is verifiable to the extent that we can “specify a class of possible observations which would verify it.”57 Because every belief that has been proposed and rejected as part of science was ‘falsifiable’, it was at least partially verifiable. In order to falsify a claim, we must know conditions under which it would be verified. For this reason, Laudan claims that verificationism has failed as a criterion of demarcation. ‘Falsificationism’, on the other hand, was proposed by Karl Popper and is the view that a claim is scientific to the extent that it can be falsified. Laudan rejects falsificationism as a criterion of demarcation because it could render any statement about what exists as scientific. What is required by falsificationism is that the proponent of a claim can offer some conditions under which that claim might be false. It seems that such conditions could be offered for any claim and so, according to Laudan, Popper has brought us no closer to an acceptable criterion. Laudan finally considers two other criteria that have been proposed in the twentieth century, the criterion of well-testedness and the criterion of cognitive progress. He explains that the attempt to qualify claims as scientific to the extent that they are welltested fails because there are other fields that are not properly considered scientific that advance well-tested claims. He offers, as an example, the claim from football strategy that “offside kicks are not usually fumbled.” He contends that such claims are often better tested than many claims that are properly considered scientific. Also, he argues that claims that have yet to be tested within science would not count as scientific on this criterion. He rejects the criterion of cognitive progress for similar reasons. There are many disciplines that are not properly considered scientific that have shown considerable cognitive progress and a number of fields in science that, at least in certain periods, have exhibited none. Laudan concludes this article with the contention that the attempt to distinguish science from nonscience is in vain and that there is nothing interesting to be said toward this effort. He goes as far as to say that we should “drop terms like ‘pseudoscience’ and ‘unscientific’ from our vocabulary; they are just hollow phrases that only do emotive work for us.”58 On his view, we should instead be concerned with which claims are well substantiated by evidence and which are not. For this reason, he does believe that some claims have greater epistemic value than others. However, this is not a matter of the types of claims they are but is only a matter of the quality of the evidence that supports them. I think Laudan is correct that scientists are properly focused on investigating evidence for claims that are advanced within their enterprise, but I disagree that philosophers of science have nothing interesting to say about what distinguishes science from nonscience. While a clear line of demarcation has yet to be drawn, there are some claims and methods that seem clearly scientific and some that seem clearly unscientific. What accounts for this? Attempting to determine what accounts for this is an effort that philosophers of science rightly pursue, particularly within the context of the debate about science education. Given the legal context that is relevant to that debate, there is certainly something interesting to say toward distinguishing science from nonscience. Also, I disagree that the failure of philosophers to agree upon a demarcation criterion implies that the effort should be abandoned. There are a number of philosophical debates that have not been resolved to satisfaction but that we still find worthwhile to pursue. And 57 58

Ibid., p. 323. Ibid., p. 328.


103 again, the legal context of the debate over science education precludes abandoning the effort altogether. Even if the legal issues at hand are the only reason to pursue the philosophical debate further, we cannot afford to be academic purists when the future of one of academia’s most important disciplines is at stake. Laudan has also taken aim at the demarcation criteria advanced in McLean, where the significance of the legal context clearly cannot be denied. As explained in chapter 1, the criteria adopted by Judge Overton were offered in Michael Ruse’s testimony and are that a theory may properly be counted as scientific if 1) it is guided by natural law, 2) it is necessarily explanatory by reference to natural law, 3) it is testable against the empirical world, 4) its conclusions are tentative, and 5) it is falsifiable. In Science at the Bar – Cause for Concern, Laudan admits that the ruling in McLean is “probably to be commended.” But he goes on to say that “it was reached for all the wrong reasons and by a chain of argument that is hopelessly suspect.”59 By applying the arguments discussed above, Laudan contends that creation-scientists make claims that are both testable and falsifiable. For instance, they claim that the Earth is very young and has experience a great flood. In denying that these claims are testable and falsifiable, Laudan thinks that science is deprived of its strongest arguments against creationism. For Laudan, these would be arguments that such claims are not substantiated by evidence and so should be rejected as false. He goes further to argue that the tenants of creation-science only fail to be tentative in the minds of those who advance it and that it is the epistemic status of creationism that is in question, not the particular beliefs of its proponents. Finally, Laudan contends that creation-science cannot be shown to lack the requisite conditions that it is guided by and explanatory by reference to natural law. On his view, the problem might simply be that we have yet to discover the laws that would account for these occurrences. He argues that this is not enough to show that they cannot be explained in terms of natural law. In Pro Judice, Ruse responds to Laudan’s critique of Judge Overton’s ruling in McLean.60 Given the legal context, Ruse argues that Laudan’s view that creation-science is better understood as weak science than as nonscience is insufficient. As he explains, the US Constitution does not prohibit the teaching of weak science – it only prohibits the teaching of religion. The charge in the case was that Act 590 set out to advance religion in science education. In response, the defendants claimed that creation-science was science, not religion. The plaintiffs, then, set out to show both that creation-science was not properly considered science and that it was properly considered religion. They rejected the dual-model approach, and in doing so advanced independent arguments for both claims. Ruse grants that a clear line of demarcation has not been drawn, but he argues that his criteria do distinguish what he calls the ‘black and white’ cases. Perhaps his criteria cannot distinguish science from nonscience in the ‘gray areas’, but creation-science does not fall there. As natural law is concerned here, Ruse rejects Laudan’s suggestion that creation-science might depend upon laws that are simply not yet known. Rather, the 59

Laudan, Larry. Science at the Bar – Causes for Concern, in But is it Science? 2nd ed., Robert T. Pennock and Michael Ruse, eds. New York: Prometheus Books. 2009. p. 331. 60

Ruse, Michael. Pro Judice, in But is it Science? 2nd ed., Robert T. Pennock and Michael Ruse, eds. New York: Prometheus Books. 2009.


104 doctrine of creation-science itself relies upon laws that stand outside of natural law and could never be known. There is no testability or tentativeness in creation-science because, as Ruse explains, one of its central tenants is that it never deviates from the Biblical texts. Finally, he rejects Laudan’s claim that the claims of creation-science are falsifiable or revisable because there are no conditions under which those who advance them would give them up. Laudan has argued that creation-science should be understood on its own terms and apart from the beliefs of those who advance it. However, Ruse rightly points out that it is not like other enterprises in science that are properly considered within the domain of ‘humanity’s cultural heritage.’ Rather, creation-science exists only in the minds of a small group of people. For this reason, the motivation for Laudan’s suggestion here is indeed unclear. Laudan has offered a response to Ruse in More on Creationism.61 His response, however, seems to miss the point of Ruse’s replies. Interestingly, he levies the same charge against Ruse, claiming that he has not properly addressed his arguments that some scientific theories fail to meet the criteria advanced in McLean and that some nonscientific theories do. He says, “Ruse fails to see the absolute irrelevance to my argument of his rehearsing examples that ‘fit’ Overton’s analysis.” 62 I should note, first, that in this article, Laudan fails to consistently employ the quotations he places around “scientific” and “nonscientific” as he does in The Demise of the Demarcation Problem and Science at the Bar – Cause for Concern that were clearly meant to denote that he was simply mentioning these terms rather than using them. This suggests to me that the ‘black and white’ distinction between the two, which is all that Ruse is arguing for, is necessary even to making his own central argument against it. In any case, what Laudan thinks Ruse fails to see is that if his criteria do not provide a clear line of demarcation then creation-science might indeed be counted as science or, at least, could make some small adjustments to meet the requirements. He seems to think that Ruse is demonstrating that they in fact do, and is only offering one example to illustrate this. A better interpretation of Ruse’s reply, though, is that he is not attempting to provide a clear line of demarcation. Rather, he is attempting to distinguish what is clearly scientific from what is clearly nonscientific – the ‘black and white’ cases – and he is arguing that creation-science is one of those black and white cases. Laudan has given no reason to think he has failed to do so. Laudan also seems to miss the point that Ruse makes in stating that creationscience does not exist in the domain of humanity’s cultural heritage, but rather exists in the mind of a small group of people. He stresses again that the beliefs themselves and not the mindset of those who advance them should be evaluated. However, Ruse does seem to be addressing the beliefs themselves. That they issue from a small group that is homogonous in its relevant philosophical assumptions and do not appeal broadly to humanity suggests that they are more than just weakly supported claim – it suggests that they are religious beliefs. This is relevant in their evaluation, especially in the context of a court that is attempting to determine if they are properly considered scientific claims or religious claims.

61

Laudan, Larry. More on Creationism , in But is it Science? 2nd ed., Robert T. Pennock and Michael Ruse, eds. New York: Prometheus Books. 2009. p. 331. 62 Ibid., p. 347.


105 I think then, that Ruse has gained his point. First, the legal context gave birth to the creation-science movement so it is relevant in evaluating it. This to not say that Laudan’s larger philosophical point cannot be made. However, it is only within the legal context that this is an important debate, especially since we are considering the beliefs of a small group of people. If not for McLean, we would still be debating demarcation criteria, but we would not be debating the scientific status of creation-science. Further, it is not necessary to solve the demarcation problem to show that creation-science is not science. There are clearly cases of science and nonscience, even if there are gray areas in between. Creation-science does not fall in the gray area – it is clearly not science. The same hold true, then, for intelligent design theory. As was demonstrated in the Dover decision, intelligent design theory has really gone no further than replacing ‘creation’ with ‘design.’ Intelligent design theory could avoid the charge of advancing religion, I suppose, if it were a stringent in its adherence to a natural designer, but this is only weakly suggested by Dembski and the most preliminary investigation of his true views and those of and other proponents of intelligent design betrays their religious commitments. Still, rejecting these commitments would only show that intelligent design might not be religion. It would not be enough to show that it was science. Perhaps it could count as science if it proposed a natural design and set about to illustrate the nature of that design and the mechanism by which it designs, but scientists have been engaged in that pursuit for more than a hundred years and the fruit of their efforts is evolutionary theory. PHILOSOPHICAL NATURALISM & METHODOLOGICAL NATURALISM Despite Laudan’s attempts to divorce the debate over demarcation from philosophical assumptions, the nature of such assumptions has become quite significant to the larger debate about evolutionary theory and intelligent design theory. Specifically, each side charges the other with holding the ‘wrong’ sorts of assumptions for what they advance to properly count as science and not religion. As discussed in chapter 1, the efforts of Phillip Johnson have shifted the debate into this context and provided the theoretical framework for the advancement of intelligent design to replace creationscience. The effort was to eliminate prima facie religious commitments in order to put a new face on creationism that would meet the requirements of the court for entering public education. The effort has failed in the court thus far, but the philosophical debate continues. While I think that Laudan’s sincere and charitable – albeit perhaps misguided – effort to evaluate the claims of creation-science in themselves is unproductive, I do think that Johnson’s effort to shift the debate into the realm of philosophy is interesting and might yield valuable results. I do not think, though, that the nature of those results would be satisfactory to the proponents of intelligent design. In his testimony during Dover, Robert Pennock argues that intelligent design theory is not merely unscientific, but is essentially religious.63 Pennock accepts that science can be understood as a set of methods that require appeal to natural explanations. As he explains, intelligent design theory violates this fundamental tenant of science from the beginning by appealing to explicitly supernatural explanations. He does not believe 63

Pennock, Robert T. Kitzmiller v. Dover Area School District Expert Report, in But is it Science? 2nd ed., Robert T. Pennock and Michael Ruse, eds. New York: Prometheus Books. 2009.


106 that science necessarily denies the existence of supernatural entities or causes, but the fact that such things are, by definition, outside of the realm of empirical evidence precludes them from being included in anything that is properly considered scientific. He believes that this alone is enough to count intelligent design theory as religion and not science. To illustrate his claim that intelligent design necessarily appeals to the supernatural, Pennock points to the work of Johnson, who he recognizes has affirmed that “God is objectively real as Creator.”64 The problem here, according to Pennock, is not that Johnson and other proponents of intelligent design hold an antecedent belief that God exists. The problem is that this belief functions in their explanations, making them fundamentally not naturalistic explanations. They simply reject the methodology of science.65 Johnson’s intellectual honesty, at least, is impressive insofar as he never denies that he has fundamentally different assumptions about the natural world, its causes, and how we should understand them than do those who claim to be doing ‘real science.’ In his book, Reason in the Balance, he contends that the demand of science that its explanations be wholly natural assumes that a deeper philosophical commitment, philosophical naturalism, is true. Most of what science tells us about the universe, according to Johnson, is tainted by naturalism and theists should work toward advancing their own theory of knowledge. He does not hide his view that this new theory should be grounded in religious doctrine. He says: “The most important statement in Scripture about creation is not contained in Genesis but in the opening verses of the Gospel of John: In the beginning was the Word, and the Word was with God, and the Word was God. He was in the beginning with God. All things came into being through him, and without him not one thing came into being, (John 1:1-3) This statement plainly says that creation was by a force that was (and is) intelligent and personal. The essential, bedrock position of scientific naturalism is the direct opposite of John 1:1-3. Naturalistic evolutionary theory, as part of the grand metaphysical story of science, says that creation was by impersonal and unintelligent forces. The opposition between the biblical and naturalistic stories is fundamental, and neither side can compromise over it. To compromise is to 66 surrender.”

He clearly sees science and theism as necessarily opposed to one another, though many, including Ruse, have argued that this is not the case. He continues: “Because in our universal experience unintelligent material processes do not create life, Christian theists know that Romans 1:20 is also true: ‘Ever since the creation of the world [God’s] eternal power and divine nature, invisible though they are, have been understood and seen through the things he has made.’ In other words, there is absolutely no mystery about why living organisms appear to be the products of intelligent creation, and why scientific naturalists have to work so hard to keep themselves from perceiving the obvious. The reason living things give that appearance is that they are actually what they appear to be, and this fact is evident to all who do not cloud their minds with naturalistic philosophy or some comparable 67 drug.”

Later, in The Wedge of Truth, he contends that science as it is currently understood is not a metaphysically innocent empirical enterprise. Rather, he argues that it depends upon 64

Ibid., p. 459. Ibid., p. 463. 66 Johnson, Phillip E. Reason in the Balance: The Case against Naturalism in Science, Law, and Education. Downers Grove: InterVarsity Press. 1995. pp. 107-108. 67 Ibid., p. 108. 65


107 philosophical naturalism insofar as it accepts only naturalistic explanations. The assumption of naturalism, Johnson appears to think, opens the door to the broader debate about what metaphysical claims are reasonable to accept. He says: “By any realistic definition, naturalism is a religion, and an extremely dogmatic one. It rests on a basic conviction about ultimate reality that is held by a kind of faith, and it incorporates its own definitions of ‘knowledge’ and ‘reason.’ It says that knowledge comes ultimately from our senses and that the more complex forms of knowledge come from scientific investigation. By naturalistic definitions, there can be no such thing as knowledge of the supernatural. Statements about God are either nonrational (if frankly presented as mere subjective belief) or irrational (if they purport to make objective factual claims). This system of categories allows the metaphysical naturalists to mollify the potentially troublesome religious people by assuring them that that 68 science does not rule out ‘religious belief’ (so long as it does not pretend to be knowledge).”

In these passages, it is clear that Johnson’s attempt is to move the problem of demarcation into the realm of philosophical assumptions and it appears that he believes that the nature of these assumptions is that they are either theistic or atheistic. This is a version of the dual-model approach, it seems, and whether or not he is correct in this approach remains to be seen. Still, he has introduced some philosophical issues of interest that deserve further attention. Namely, he has raised questions about whether science, insofar as it offers only naturalistic explanations, can truly be understood apart from any religious doctrine and what the answer to this question suggests for its fate in the legal context. Interestingly, there is little rejection of Johnson’s claim that philosophical naturalism is problematic. Thus far, responses have aimed to move the debate out of the realm of philosophy and back into the realm of practice, or methodology. It seems then, that there might be something significant to address in the larger philosophical context, regardless of whether Johnson’s position is correct. To his credit, Johnson appears to concede this point. He says: “If Christian theists can summon the courage to argue that preexisting intelligence really was an essential element in biological creation and to insist that the evidence be evaluated by standards that do not assume the point in dispute, then they will make a great contribution to the search for 69 truth, whatever the outcome.”

I will return to these questions and how they have been addressed in recent literature in Chapter 3. For the purposes of this discussion, though, it is important to consider how science was distinguished from intelligent design in the Dover case, and methodological naturalism was important to achieving this end. In his testimony, Pennock does address Johnson’s concern that science depends upon a type of religious dogma, but he does not argue against him on this point at any length here. Rather, he relies upon the methodological approach to defining science offered by Ruse during McLean and claims that “Science does not reject the supernatural dogmatically, but rather because it cannot be tested by empirical evidence.”70 Clearly this would not count as a satisfactory reply for Johnson because as quoted above, he believes that without a commitment to philosophical naturalism, we are free to see that things are ‘actually what they appear to be.’ This suggests that for Johnson, the nature of empirical observation itself is determined by one’s antecedent philosophical commitments and if 68

Johnson, Phillip E. The Wedge of Truth: Splitting the Foundations of Naturalism. Downers Grove: InterVarsity Press. 2000. p. 148. 69 Johnson, Phillip E. Reason in the Balance: The Case against Naturalism in Science, Law, and Education. Downers Grove: InterVarsity Press. 1995. P. 110 70 Pennock, p. 458.


108 this is true, then Pennock is simply begging the question. In any case, later in his testimony, Pennock draws the key distinction between philosophical naturalism and methodological naturalism that keeps the debate in the court centered on practice rather than ideology. He says: “As we have seen, the defining element of IDC is its essential reliance upon supernatural begins and powers – entities that are unconstrained by either lawful necessity or chance processes. The ID movement thus rejects a basic element of scientific evidence, namely, that explanations appeal only to natural causal processes. Scientific explanations need not cite a specific law of nature, but they are always understood to be restricted to the physical realm of law-bound cause and effect relations. In science, this is a principle of method, not a metaphysical dogma. This is typically spoken of as methodological naturalism in contrast to metaphysical naturalism (also sometimes 71 referred to as ontological or philosophical naturalism).”

Although Judge Jones accepted the criteria of methodological naturalism in his decision, many are unsatisfied with what it requires. Alvin Plantinga, for instance, advances a number of arguments against it. Plantinga attacks Ruse, in particular, for claiming in Darwinism Defended that science – by definition – deals only with the natural.72 According to Plantinga, Ruse’s explanation of methodological naturalism leaves something to be desired for three reasons. First, he references the demarcation problem and suggests that if Ruse were in fact appealing to a definition, then it should offer necessary and sufficient conditions for what counts as scientific. Second, he thinks the characteristics of science appealed to by Ruse are insufficient. These are that science deals with things that are 1) repeatable, 2) merely natural, and 3) governed by natural law. Plantinga addresses the first and the third of these. He thinks that the first condition fails to characterize theories like the Big Bang that are clearly scientific. He argues that the Big Bang was unique and possibly unrepeatable, but that this does not prevent us from considering investigation into the Big Bang scientific. In attacking the third, he argues that that there is some debate about whether or not natural law even exists. He appeals to the work of Bas van Fraassen who he claims argues for the conclusion that there are no natural laws. Plantinga argues that regularities do exist, but that something further is required of law. Law should explain regularities and should be necessary in some way. He contends that perhaps the requirements for a law to exist may not be met.73 Finally, Plantinga argues that regardless of the conditions offered by a definition of science, the debate over demarcation cannot be settled simply by appealing to a definition. He thinks what is at stake is not a question about whether the word ‘science’ properly denotes a hypothesis that makes reference to god, but rather, it is a question of whether a hypothesis that references god could be part of science.74 It is curious that Plantinga raises the same objection to Ruse that Laudan raised more than a decade earlier. This is just what his first concern amounts to. Without responding to Ruse’s explanation that he is only offering a characterization of the black and white cases of science and nonscience, Plantinga raises the demarcation problem as an argument against Ruse’s account of methodological naturalism. In Methodological

71

Ibid., p. 464. Plantinga, Alvin. Methodological Naturalism? in Intelligent Design Creationism and Its Critics: Philosophical, Theological, and Scientific Perspectives. Robert Pennock, ed., Cambridge: MIT Press. 2001. 73 Ibid., pp. 344-345. 74 Ibid., p. 345. 72


109 Naturalism Under Attack, Ruse replies to Plantinga’s concern.75 He emphasizes that he is not looking to solve the demarcation problem and that he is not attempting to offer an analytic definition of ‘science’ in offering criteria for what counts as scientific. For the reasons I discussed in the previous section, I think his replies are sufficient to dismiss this concern. Ruse also replies to Plantinga’s attacks on his characterization of methodological naturalism. According to Ruse, Plantinga’s concern over repeatability is misguided because there have been many events that are unique but are still explicable in terms of natural laws. He appeals to the extinction of dinosaurs, in particular, and argues that despite the fact that dinosaurs only existed once and will never reappear, their extinction by way of an asteroid or comet hitting the earth is completely explicable in terms of natural law. It does seem clear that the particular state of affairs upon which natural law operates will determine the uniqueness of the events that result. For this reason, some events are more likely to be repeated and others are less likely. It is not at all clear why Plantinga thinks that this is such a problem for methodological naturalism. It seems that many examples from natural history can be produced to illustrate this point. What is interesting, and what should be noticed by Plantinga, is that the same natural laws are operate in producing unique states of affairs. This suggests that it is indeed a feature of the state of affairs upon which laws operate – and not any problems concerning the repeatability criteria of methodological naturalism – that produce unique events like the Big Bang and the extinction of the dinosaurs. Plantinga’s worry that there might be no natural laws is also addressed by Ruse. He explains that it is legitimate to question the necessity of laws, but that Plantinga has misinterpreted van Fraassen’s claims. According to Ruse, laws just are the regularities that are presupposed in science and van Fraassen, nor any average scientists would deny this. Plantinga has gone further to claim that laws must be more than regularities – they must provide an explanation of the necessity of regularities. It is not clear what Plantinga seeks here, nor is it clear what would satisfy him. It seems that a regularity might be something like the claim that ‘an apple falls to the earth when it is dropped.’ The law that is invoked to explain this is the law of gravity. Is Plantinga characterizing gravity as regularity rather than a law? And if so, is he further demanding to know what necessitates that gravity exerts its force upon objects? It seems that the regularity of apples falling to the earth as a matter of necessity is explained by the law of gravity. It is not clear what further demand Plantinga is making. The only candidate that seems plausible is an explanation of the necessity of the law of gravity itself. But then, if we were to offer such an explanation, what would make that the law and not just another regularity to be explained? This suggests to me that if Plantinga is interpreting what most understand as law as mere regularity, then there can be no explanation that will satisfy him.

75

Ruse, Michael. Methodological Naturalism Under Attack in Intelligent Design Creationism and Its Critics: Philosophical, Theological, and Scientific Perspectives. Robert Pennock, ed., Cambridge: MIT Press. 2001.


110 SCIENCE AND SUPERNATURALISM Thus far, I have focused on the effort made in the demarcation debate to characterize science and to show that intelligent design is not science. This effort points to features of intelligent design that are specifically unscientific – most importantly, that it appeals to supernatural causes. In response, proponents of intelligent design – or at least critical of philosophical and methodological naturalism – have argued that what properly counts as science is not adequately characterized by the criteria that has been offered thus far. There is another strategy in the literature that deserves to be discussed. This is the strategy of showing what science, properly considered, has in common with intelligent design theory, as opposed to what intelligent design theory has in common with science. Proponents of this ‘supernatural strategy’ typically focus on supernatural assumptions that have been instrumental in grounding the development of certain theories in the past and contend that we consider those theories to be scientific. So, why should we begrudge intelligent design theory for introducing a supernatural cause? In response to this strategy, Ruse suggests that sometimes science is mingled with nonscience and this fact does not make the nonscience into science. To the extent that one deals with natural law, one is dealing in science. To the extent that one does not, one is dealing in nonscience. He goes further to argue that the scientific enterprise itself has evolved and what might have been considered acceptable in the past is no longer acceptable, nor should it be acceptable. 76 I think these are the right sort of replies to the supernatural strategy and that the involvement of supernatural causes precludes a theory from being properly considered scientific. I also think, though, that there might be a separate issue at hand and I think this issue is perhaps suggested by Steve Fuller in his article, A Step Toward the Legalization of Science Studies.77 Here, Fuller does invoke the supernatural strategy and he in fact served as a rebuttal witness for the defense in the Dover trial specifically to argue against the National Academy of Science’s declaration that science requires methodological naturalism. As he explains, the history of science is full of examples of supernatural hypotheses and explanation. He says: “Typically, these supernatural hypotheses – expressions of what is less prejudicially called ‘metaphysical realism’ – receive their initial grounding in a mathematically significant pattern that points to a deeper level of explanation. In the case of, say, Newton’s appeal to gravitational attraction or Mendel’s to hereditary factors, these hypotheses have had theistic origins that 78 survive in contemporary IDT –namely, ideas of a divine plan and special creation.”

For this reason, Fuller does not think it is appropriate to characterize science as the absence of any supernatural elements that function in explanation because science has, in fact, done this for centuries. Still, though, he recognizes that the decision in the Dover case was the right one and he says that the right precedent was set by Ruse’s testimony in the McLean case. Yet, he is not completely satisfied with the Dover ruling. He says: 76

Ruse, Michael. Pro Judice, in But is it Science? 2nd ed., Robert T. Pennock and Michael Ruse, eds. New York: Prometheus Books. 2009. pp. 339-340. 77

Fuller, Steve. A Step Toward the Legalization of Science Studies in But is it Science? 2nd ed., Robert T. Pennock and Michael Ruse, eds. New York: Prometheus Books. 2009. 78 Ibid., p. 488.


111 “My own view is that the defense did indeed have the weaker case, but equally that the judge did an injustice to the relevant philosophy, politics, and ultimately to science. IDT may be inept in its 79 self-understanding and self-presentation, but it did not deserve to be dismissed outright.”

These are the closing remarks of his article, so it is not completely clear what injustice has been done in Fuller’s view. If he thinks that Ruse’s criteria set the right precedent, then it seems that he is not too bothered by counting intelligent design theory as nonscience, though he thinks a historical interpretation of the development of science makes this less than a charitable characterization. Also, the ruling was about a proposed educational policy – and he thinks it was the correct ruling – so what is the injustice? Perhaps Fuller thinks that the ruling has ended debate in a significant sense and, in doing so, has deprived us of further and fruitful discussion. I do not think this follows from the ruling at all. Without question, the debates over demarcation, philosophical naturalism, and methodological naturalism continue and the courtroom can certainly be revisited if the proponents of intelligent design theory are able to get a significant foothold in them. And it is, to be clear, the context of the courtroom that is driving these debates for many involved. Otherwise, why would gaining the status of science be so significant to proponents of intelligent design? Why not just let others think what they may and continue to work on their own projects? The institutes that support them are quite wealthy and, as they like to emphasize, a clear majority of the American public already supports them. To be sure, it is to achieve the goals of the creationist movement in public education that intelligent design advocates continue in their efforts and so the debate will likely continue in a manner similar to the way it has thus far for some time to come. And of course, there will be good scientists and philosophers who will stand every ready to meet the challenge – as they should because the logic is on their side and because there is much at stake. I believe Ruse’s criteria demonstrate that intelligent design theory is not science and this has proven sufficient to satisfy the courts. As he explains, the courts do not prohibit the teaching of bad science so it is important that it is understood to be a black and white case of nonscience. However, as Fuller’s concerns suggest, the debate still exists and indeed continues in other contexts – namely, the academic and social contexts. In his books, Abusing Science: The Case Against Creationism and Living with Darwin: Evolution, Design, and the Future of Faith, Phillip Kitcher is concerned with these contexts, particularly the social context. As he explains, the debate over creation and evolution continues to persist in the social context no matter how successfully academics have defended evolution or refuted creationism. For this reason, his aims in these two books are similar. He hopes to provide a guide for people with little training in the subject matter so that they may appreciate the strength of Darwinian evolution, understand the flaws of creationism, and develop a ready response to the defenders of creationism that they encounter. The first of these books, Abusing Science, was published in 1982 and so focuses on creation science.80 The second book, Living with Darwin, was published in 2007 and

79 80

Ibid., p. 489. Kitcher, Phillip. Abusing Science: The Case Against Creationism. Cambridge: MIT Press. 1982


112 deals with the new challenge of intelligent design theory.81 While intelligent design theory is properly considered nonscience in the legal context, if it is still insisted to be science in continuing academic and social debates, Kitcher’s work is useful in showing that it is bad science. He thinks arguing that intelligent design in bad science rather than nonscience is a more successful approach for two reasons. First, he explains that proponents of intelligent design can simply alter their position so that it does meet the necessary criteria to be counted as science. For instance, he believes it is possible to conceive of intelligence without personifying it. One strategy he suggests is proposing an ‘intelligent operation’ in the place of an intelligent designer to avoid personification of intelligence.82 It seems that we might see such a move from the creationist camp and this is a move that would mirror the transition from creation-science to intelligent design theory. Second, Kitcher advocates the supernatural strategy as effective against attempts at demarcation insofar as he thinks religious hypotheses and theories have functioned throughout the history of scientific inquiry.83 Kitcher’s own view is that intelligent design theory should be viewed as ‘dead science,’ or bad science, precisely because of its entanglements with religious sentiments. Similar to Ruse, Kitcher argues that intelligent design might have once been considered science, but it is not properly considered so today. Also similar to Ruse, he recognizes that the legal context requires more than the claim that intelligent design is dead science to keep it out of the classroom. However, he does not believe it is necessary to argue that it is not science in order to satisfy the courts. Rather, his approach is to show that the only motivation for introducing it into the classroom is a religious motivation. For Kitcher, the fact that intelligent design remains silent on the mechanisms of intelligent operation – specifically that would explain when it operates and what it does when it operates – suggests that it is not primarily concerned with doing science.84 If it were, and if it were concerned with doing good science, then it would seek to offer explanations of its primary mechanisms. Given the failure of creation science and the subsequent rise of intelligent design theory, Kitcher rightly addresses the puzzling persistence of the creationist movement in Living with Darwin. Here, he not only addresses the academic arguments for intelligent design, but also the attitudes that have made its support by average citizens possible. According to Kitcher, a driving force that sends defenders of evolution and creationism to the courts time and again is public sentiment. As he explains, the average person is convinced by creationists that to accept Darwin is to reject god. He notes that this dilemma has been argued against since Darwin’s day and that even the eulogies offered at Darwin’s funeral in Westminster Abbey show that it is a mistake – one can understand Darwinian evolution as consistent with a divine plan.85 Still, the notion that accepting evolution requires atheism persists and is a motivating factor in anti-Darwin sentiments. In addition to addressing the mistaken assumption that evolution is atheism, though, Kitcher also believes it is important to understand how lives can matter in a way that is 81

Kitcher, Philip. Living with Darwin: Evolution, Design, and the Future of Faith. Oxford: Oxford University Press. 2007. 82

Ibid.. p. 100. Ibid., pp. 8-9. 84 Ibid., pp. 103-106. 85 Ibid., p. 120. 83


113 consistent with evolution. Clearly the average citizen is not convinced, and so he concludes this book with the suggestion that, “We should articulate, as clearly as can be done, the possible routes along which lives can find significance.”86 I think Kitcher is right that views of meaning in life consistent with evolution must be advanced in addition to arguments that evolution is not atheism. I also believe he is correct in his belief that until these issues are adequately addressed, the debate surrounding evolution and creationism will continue to arise. A clear example of this can be found in a paper published in 2008 by Thomas Nagel on intelligent design and public education and in his positive review of a book by Stephen Meyer that advances intelligent design theory. Nagel has extended an olive branch to proponents of intelligent design despite the fact that he does not agree with their religious sentiments. He does, however, agree with their view that accepting evolution requires accepting atheism insofar as he believes that evolutionary theory precludes the possibility of a divine intelligence that intervenes in the natural world. Nagel’s arguments have stunned philosophers and scientists, but perhaps may be understood to have been predicted by Kitcher. One interpretation of intelligent design theory offered by Kitcher suggests that its proponents might simply be attempting to explain the ways in which scientific investigation is limited. He says: “Perhaps, however, there is an alternative interpretation, one that recognizes the intelligent design-ers as intending only to identify the limits of natural science. On this reading, they would argue only that some evolutionary transitions cannot be understood in terms of the operation of natural selection – or indeed in terms of any other natural process – and they would modestly decline to advance any explanation of why such transitions have occurred.”87

Kitcher does not think that this is a proper interpretation of intelligent design theory, and even if it were, it would still have trouble showing the limits of science that its proponents advocate. However, it seems that an effort to explore the limits of science would be useful philosophically, and might provide some catalyst toward scientific inquiry. I am not sure what those limits might be or, more properly, where they might be, or even if such limits exist. It seems fair, though, to explore this question. I would not, however, start where intelligent design proponents start. The literature overwhelmingly illustrates how their effort to show the inadequacies of certain scientific explanations has failed miserably. However, exploring the limits of science and what may or may not be said beyond these limits, but also from the standpoint of reason, is useful philosophical inquiry that stands to move the academic debate in new directions. The legal debates concerning the limits of science have mostly stymied since Dover. I think, though, that we have not seen an end to the demarcation problem in the legal context. What science studies and by what methods science proceeds are issues that have been properly, though perhaps not adequately, discussed in the academic context. However, toward what end public science education aims is a subject ripe for a new discussion. Where it is found that public education aims at some profit, who’s profit is relevant to determining who’s science public funding facilitates. I suspect the demarcation problem is disjunctive and that an analysis of statutes in a given state will reveal not only the means by which, but also the extent to which economic gain drives public education. The methodological questions that follow such an analysis will (and must) address the extent 86 87

Ibid., p. 166. Ibid., p. 114.


114 to which the intended profit of science affects the methods by which it is pursued and ultimately, taught. Here, we have left the traditional religious context in which the demarcation problem historically developed. However, the economic context introduces new questions of justice that are unique to philosophical inquiry and should be explored.


115

Comments about Previous Issues Polygon Polygon Spring 2010 Vol. 4, 81-82 COMMENTS ABOUT POLYGON *********************************************************************** Dr. Norma Martin Goonen President, Hialeah Campus Miami Dade College Thank you, Dr. Shakil, for providing scholars a vehicle for sharing their research and scholarly work. Without opportunities for sharing, so many advances in professional endeavors may have been lost. NMG Dr. Norma Martin Goonen President, Hialeah Campus Miami Dade College *********************************************************************** Dr. Ana María Bradley-Hess Academic and Student Dean, Hialeah Campus Miami Dade College Welcome to the third edition of Polygon, a multidisciplinary peer-reviewed journal of the Arts & Sciences! In support of the Miami Dade College Learning Outcomes, one of the core values of Hialeah Campus is to provide “learning experiences to facilitate the acquisition of fundamental knowledge.” Polygon aims to share the knowledge and attitudes of the complete “scholar" in hopes of better understanding the culturally complex world in which we live. Professors Shakil, Bestard and Calderin are to be commended for their leadership, hard work and collegiality in producing such a valuable resource for the MDC community. Ana María Bradley-Hess, Ph.D. Academic and Student Dean Miami Dade College – Hialeah Campus 1800 West 49 Street, Hialeah, Florida 33012 Telephone: 305-237-8712 Fax: 305-237-8717 *********************************************************************** Dr. Caridad Castro, Chairperson English & Communications, Humanities, Mathematics, Philosophy, Social & Natural Sciences Hialeah Campus Miami Dade College POLYGON continues to grow and to feature our local MDC scholars. Thanks to you and your staff for providing them this opportunity. Cary Caridad Castro, J.D., Chairperson English & Communications, Humanities, Mathematics, Philosophy, Social & Natural Sciences Miami Dade College – Hialeah Campus 1776 W. 49 Street, Hialeah, FL 33012 Phone: 305-237-8804 Fax: 305-237-8820 E-mail: ccastro@mdc.edu


116 *********************************************************************** Dr. Arturo Rodriguez Associate Professor Chemistry/Physics/Earth Sciences/Department North Campus Miami Dade College I want to congratulate you and the rest of the colleagues who created the POLYGON that is occupying an increasingly important place in the scholarly life of our College. Now, the faculties from MDC have a place to publish their modest contributions. arturo Dr. Arturo Rodriguez Associate Professor Chemistry/Physics/Earth Sciences/Department North Campus Miami Dade College 11380 NW 27th Avenue Miami, Florida 33167-3418 phone: 305 237 8095 fax: 305 237 1445 e-mail: arodri10@mdc.edu *********************************************************************** Dr. Mattie Roig-Watnik, Ed.D. President, Hialeah Campus Miami Dade College 1780 W.49th St. Rm 301 Hialeah, Fl 33012

Polygon is a tribute to the scholarship and dedication of the faculty at Miami Dade College in interdisciplinary areas. Miami Dade Collegeâ&#x20AC;&#x2122;s esteemed faculties have contributed their scholarly works in this edition (2014) and previous issues of Polygon. The interdisciplinary articles and approaches to teaching and learning are a true tribute to and scholarly pursuits of faculty at Miami Dade College. The sharing of information, data and teaching/learning enhances the commitment to education every day. Thank you for continuing to enrich our lives. Mattie

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Polygon is a tribute to the scholarship and dedication of the faculty at Miami Dade College in interdisciplinary areas.

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