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1780 West 49th Street, Hialeah, Florida 33012, USA Editorial Note: Polygon is Miami Dade College, Hialeah Campus’ 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 2018 Spring Issue, which is the eleventh consecutive issue of Polygon. It includes thirteen regular research articles, including a Foreword from the President of our Hialeah Campus, Dr. Joaquín 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, the Miami Dade College District Board of Trustees, the Hialeah Campus President, Dr. Joaquín G. Martinez, Chairperson of the Hialeah Campus Liberal Arts and Sciences, Dr. Caridad Castro, Chairperson of the Hialeah Campus World Languages and Communication, Professor Liliana Cobas, Director of the Hialeah Campus Administrative Services, Ms. Andrea M. Forero, Director of the Hialeah Campus Network & Media Services, Mr. Juan Villegas, all the staff and faculty of the 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 and Dr. Jaime Bestard (Mathematics), and Professor Victor Calderin (English) Patrons: Dr. Joaquín G. Martinez, President, Hialeah Campus Dr. Caridad Castro, Chair of Liberal Arts and Sciences Professor Liliana Cobas, Chair of World Languages and Communication Ms. Andrea M. Forero, Director of Hialeah Campus Administrative Services Mr. Juan Villegas, Director of Hialeah Campus Network & Media Services Miami Dade College District Board of Trustees: Chair Bernie Navarro Vice Chair Marili Cancio Trustee Dr. Susan Amat Trustee José K. Fuentes Trustee Benjamin León III Trustee Dr. Rolando Montoya Trustee Juan Carlos Zapata Dr. Eduardo J. Padrón, College President


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Foreword By Dr. Joaquín G. Martínez Welcome to the eleventh 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

Teaching Hypothesis Testing to an Introductory Statistics Class with Statdisk Software Implementation - A Lesson Plan - Dr. Jack Alexander

1 - 10

A Note on a Classroom Lesson Plan for Teaching a Simplified Linear Regression Analysis, using Statdisk Software - Dr. Jack Alexander

11 - 17

The Impact of Learning Assistants in the Teaching and Learning Process for MAC1105 College Algebra at MDC, Hialeah Campus – Dr. Jaime Bestard, Ms. Laura Iglesias, Ms. Anette Sanz, Joel Gonzalez, Samara Rabassa, Danisley Muro, and Dayron Sanchez

18- 27

Effective Learning in Quantitative Reasoning and Problem Solving - Prof. Loretta Blanchette

28 - 66

The cubic and quartic equations - Dr. Mario Duran Camejo

67 - 70

Elliptic integrals, geometric analogies - Dr. Mario Duran Camejo

71 - 83

The one-seventh area triangle problem - Dr. Mario Duran Camejo

84 - 88

Polyconcs and Polyconvs - Dr. Mario Duran Camejo

89 – 126

Math Student Motivation Research Project – SWOOP – Prof. Lourdes España and Prof. Maria Alvarez

127 – 133

Considering the Student´s Mother Tongue in the Teaching-Learning Process of English - Dr. Raydel Hernandez Garcia

134 – 142

A Proposal for Scaffolding the Teaching-Learning Process of Speaking Skills in English - Dr. Raydel Hernandez Garcia

143 – 161

Statistical Analysis of Nutrition Facts of Some Common Foods - A Computer Project Assignment for teaching STA 2023 Course using Minitab and Statdisk Software - Dr. M. Shakil

162 - 178

Statistical Analysis of Some Greenhouse Gases and Air Quality Index Data - A Computer Project-Based Assignment for Teaching STA2023 Course - Dr. M. Shakil

179 - 201

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. 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.


iv Solicitation of Articles for the 2019 Issue (12th Issue) of Polygon: The editorial committee would also like to cordially invite the MDC community to submit their articles for consideration for the 2019 Issue (12th 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 efficiently 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. You may also please refer to the attached template for the preparation of the manuscripts.


v 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 coauthor’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 pt. 16 pt. above, 32 pt. below) ABSTRACT 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 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.


vi [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.) Author’s Biographical Sketch (Optional): Dr. Y. Abu received his Master’s and Ph. D. Degrees in Mathematics from the University of Small Town, USA, in 1987, under the direction of Dr. M. Opor. Since 1989, he has been teaching at the Community College of Small Town, USA. His research interests lie in the Fractals, Solitons, Undergraduate Teaching of Mathematics, and Curriculum Development. (11 pt.) Address: Department of Liberal Arts & Sciences (Mathematics Program), Community College of Small Town, P. O. Box 7777, Small Town, USA. E-mail: yabu@ccst. (11 pt.)


Teaching Hypothesis Testing to an Introductory Statistics Class with Statdisk Software Implementation - A Lesson Plan Dr. Jack Alexander Department of Mathematics Miami Dade College, North Campus Miami, FL 33167 E-mail: jalexan2@mdc.edu

ABSTRACT

In statistics a hypothesis is a claim or statement about a property of a population. A Hypothesis Test (or test of significance) is a procedure for testing a claim about a property of a population. There are three (3) hypothesis testing methods; the Traditional Method (T.M.), the P – Value Method (Pv), and the Confidence Interval Method (C.I.). Furthermore, there are three (3) types of tests; a one-tail test on the upper end, indicated by “>” in the claim, a one-tail test on the lower end, indicated by “<” in the claim, and a two-tail test, indicated by “≠” in the claim. This paper deals with developing a lesson plan using some examples to illustrate the three methods and the three types relevant to hypothesis testing and describe how these may be used these in introductory classes. The statdisk software implementation to these examples is also provided. It is hoped that by implementing the techniques discussed in this paper in preparing lesson plans on statistical hypothesis testing will help us to develop in our students the skills to solve problems using critical and creative thinking and scientific reasoning, , which is one of the Gen Ed Outcomes of Miami Dade College.

KEYWORDS:

Confidence interval, Hypothesis test, P-value, Statistical significance, Teaching.

2010 Mathematics Subject Classifications: 97C40, 97C70, 97D40, 97D50.

INTRODUCTION: One of the major aspects of inferential statistics is hypothesis testing about a property of a population parameter. There are three (3) hypothesis testing methods; the Traditional Method (T.M.), the P – Value Method (Pv), and the Confidence Interval Method (C.I.). Furthermore, there are three (3) types of tests; a one-tail test on the upper end, indicated by “>” in the claim, a one-tail test on the lower end, indicated by “<” in the claim, and a two-tail test, indicated by “≠” in the claim. SOME CLASSROOM EXAMPLES AND STATDISK IMPLEMENTATION: The examples below illustrate the three methods and the three types as discussed in the introduction. The statdisk implementation of these example are also provided..

1


Example 1: Forty (40) students were asked if they were first year or second year students. Twenty-two (22) said that they were first year students. Test the hypothesis that that the proportion of first year students is greater than 50% at the .01 significance level. Answer: The hypothesis set up is:

Null Claim

H0: p ≤ .50 H1: p > .50

Since this is a one tail test on the upper end, the “Traditional Method” requires that we will reject H0 (the null hypothesis) if the test statistic z is greater than the critical value (CV). In this case, p^ = 22/40 = .55 and q^ = 1 – p^ = 1 – .55 = .45. The Critical Value (CV) for α = .01 is zα/2 = 2.33. The test statistic for proportions is given by: pᶺ -- p z = --------√(pq/n) .55 – .50 z = ------------- = .63 √(.5 x .5/40) Hence, we do not reject H0. (DNRH0)

The requirements for the p-value method are: If the p-value is less than the significance level α, we reject H0. To make this determination, look up .63 in the positive Z score table. It turns out that .63  .7357. Since this a one tail test on the upper end, the p-value = 1 – .7357 = .2643. Since .2643 is not less than α = .01, we do not reject H0. (DNRHo) The “Confidence Interval” method employs the typical structure used for proportions. That is: p^ - E < p < p^ + E, where E = zα/2√(p^q^/n). In this case, E = 2.33√(.55 x.45/40) = .183. Hence, The required confidence Interval is given by: .55 – .183 < p < .55 + .183  .367 < p < .733. Rounded to two decimal places we have: .37 < p < .73 Note that .50 is in this interval. Therefore, we are again not in a position to reject H0. The Statdisk Implementation of the above example is given below: Alternative Hypothesis: 2


p > p(hyp) Sample proportion: 0.55 Test Statistic, z: 0.6325 Critical z: 2.3264 P-Value: 0.2635 98% Confidence interval: 0.3670076 < p < 0.7329924

Example 2: The average weight of adult men is 183 pounds with a standard deviation of 41 pounds. Thirty (30) men were asked their weight and the mean was calculated to be 180, Test the hypothesis that the mean is not 183 pounds using α = .05. This is an example of a Two-Tail Test. μ = 183 Answer: The hypothesis set up is: H0: H1: μ ≠ 183 Since this is a two tail test, the “traditional method” requires that we reject H0 if the test statistic z is either greater than the critical value or less than the negative of the critical value. The critical values (CV) for a two tail test at α = .05 are zα/2 = 1.96 and -1.96. The test statistic is given by: 3


x---- µ z = ---------(σ/√n)

z=

180 – 183 --------------- = -.40 (41/√30)

Since -.40 is not less than -1.96, we do not reject H0 (DNRH0). To employ the p-value method, we look up -.40 in the negative Z score table. This value is -.40  .3446. Since this is a two tail test pv = 2 x .3446 = .6892. Again, we do not reject H0 because this value is not less than the significance level α = .05. The “Confidence Interval” method employs the same structure typically used for means. That is: x-- - E < µ < x-- + E, where E = zα/2σ/√n = (1.96 x 41)/√30 = 14.67. Therefore, the required interval is: 180 – 14.67 < µ < 180 + 14.67 or 165.33 < µ < 194.67. Note that 183 is in this interval. This also indicates that we do not reject H0. The Statdisk Implementation of the above example is given below: Alternative Hypothesis: µ not equal to µ(hyp) z Test Test Statistic, z: -0.4008 Critical z: ±1.9600 P-Value: 0.6886 95% Confidence interval: 165.3286 < µ < 194.6714

4


Example 3 The mean height of adult women is 63.6 inches with a standard deviation of 2.5 inches. Forty (40) women were asked their height and the mean was calculated to be 61.5 inches. Test the hypothesis that the mean is less than 63.6 at the α = .05 significance level. Answer:

The hypothesis set up is:

H0: H1:

μ ≥ 63.6 μ < 63.6

Since this is a one tail test on the lower end, the “Traditional Method” requires that the test statistic z must be less than the critical value CV = zα/2 = -1.645. x---- μ z = ---------(σ/√n} 61.5 – 63.6 z = ----------------- = -5.31 (2.5/√40)

5


Since this value is, in fact, less than -1.645, we reject H0. This means that we agree with the claim that µ < 63.6. To find the p-value, we look up -5.31 in the negative Z score table. Because -5.31 is below -3.50, we use .0001 as the p-value. This indicates that we must reject H0 since .0001 < .05 (our signifi1cance level). The “Confidence Interval” would be set up in the same manner as was done in example 2 above. That is:

x-- - E < µ < x-- + E, where E = zα/2σ/√n = (1.645 x 2.5)/√40 = .65

Therefore, the required interval is: 61.5 – .65 < µ < 61.5 + .65 or 60.85 < µ < 62.15. Note that 63.6 is not in the interval. So, we again reject H0. The Statdisk Implementation of the above example is given below: Alternative Hypothesis: µ < µ(hyp) z Test Test Statistic, z: -5.3126 Critical z: -1.6449 P-Value: 0.0000 90% Confidence interval: 60.84982 < µ < 62.15018

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Example 4: Eighteen (18) students at MDC were asked how much they spent on text books this semester. The calculated mean was $142 and the calculated standard deviation s was $95.20. Test the hypothesis that the mean amount is greater than $100 using α =.01 Answer:

The hypothesis set up is:

H0: H1:

μ ≤ 100 μ > 100

Since we have a calculated standard deviation s = 95.20, this is a one tail test on the upper end. However, we must use the T Distribution Table to determine the critical value at α =.01 for 17 degrees of freedom. The appropriate value for the test is 2.567. For the “Traditional Method”, we will reject H0 if the test statistic is greater than 2.567. x---- μ In this case, the test statistic is: t = -----------. (s/√n) 142 – 100 t = ---------------- = 1,87 (95.20/√18)

7

Hence,


Since this value is not greater than 2.567, we do not reject H0 (DNRH0). The “Confidence Interval” set up in this case would be: x-- - E < µ < x-- + E, where E = tα/2s/√n = (2.567 x 95.20)/√18 = 57.6. Therefore, the required interval is: 142 – 57.6 < µ < 142 + 57.6 or 84.4 < µ < 199.6. Note that 100 is in the interval. Hence, we again do not reject H0 (DNRH0). The Statdisk Implementation of the above example is given below: Alternative Hypothesis: µ > µ(hyp) t Test Test Statistic, t: 1.8718 Critical t: P-Value:

2.5669 0.0393

98% Confidence interval:

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84.40088 < µ < 199.5991

CONCLUSION: Throughout the entirety of our lives we are faced with “claims”. Some are legitimate, while others are not. If you go to the doctor and he tells you that the medication he is now prescribing for you is superior to the one given to you previously, it is likely that this is true. On the other hand, if an advertiser or a salesperson tells you that a new product is better than their old product, this may or may not be the case. As an educated person, we owe it to ourselves to be in a position to make a determination about the legitimacy of a claim. This article illustrates how hypothesis testing employed in statistics assist in making up our own minds about “claims”. Also, it is hoped that the statdisk implementation will reinforce the teaching the concept of hypothesis testing to our students.

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Acknowledgments The author is thankful to the editor and reviewer for their suggestions, which improved the quality of the paper.

REFENENCES Bluman, A. G. 2017. Elementary Statistics, 10th Edition, Chapter 9; McGraw-Hill, New York. Larose, D. T. 2010. Discovering Statistics, Frist Printing, Chapter 9; W. H. Freeman and Company, New York. Triola, M. F. 2017. Elementary Statistics, 13th Edition, Chapter 8; Addison â&#x20AC;&#x201C; Wesley, Boston. http://www.statdisk.org/.

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A Note on a Classroom Lesson Plan for Teaching a Simplified Linear Regression Analysis, using Statdisk Software Dr. Jack Alexander Department of Mathematics Miami Dade College, North Campus Miami, FL 33167 E-mail: jalexan2@mdc.edu

ABSTRACT

One of the most useful economic tools of statistics is Linear Regression Analysis. This analysis allows us to make predictions on the basis of paired data. As it turns out, some relationships are, by nature, directly correlated while others are indirectly correlated. For example, the taller a person is the larger their foot size is likely to be. On the other hand, in an automobile piston chamber increased pressure produces decrease volume. While the formulas required for Linear Regression are rooted in partial differential calculus, those formulas can be easily evaluated if the requisite sums are detailed in a simple table. This paper gives the required formulas and gives an example that illustrates how the formulas can be applied to a real world situation. This simplified approach allows Linear Regression to be presented to beginning statistic students without their having to be first instructed in calculus. To reinforce the Regression Analysis Ideas, the statdisk software implementation is also provided.

KEYWORDS:

Linear functions, Slope, Y-intercept, Regression, Correlation

2010 Mathematics Subject Classifications: 97C40, 97C70, 97D40, 97D50.

INTRODUCTION: Regression Analysis is an invaluable tool for Economics. It allows us to make predictions on the basis of paired data. For example, all business concerns must be in a position to predict customer levels, expenses, debits and profits if they are to remain viable. In what follows, we provide the required formulas and gives an example that illustrates how the formulas can be applied to a real world situation. This simplified approach allows Linear Regression to be presented to beginning statistic students without their having to be first instructed in calculus. The statdisk software implementation is also provided.

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REGRESSION FORMULAS AND CORRELATION FORMULAS: The Line of regression (Line of best fit or prediction line) is given by Ŷ = B0 + B1X, where B0 = ӯ-- B1Ẍ

B1 =

and

n∑xy -∑x∑y --------------------------n∑x2 -- (∑x)2 The Coefficient of Correlation formula is given by:

n∑xy -∑x∑y R = -----------------------------------------------√((n∑x2 -- (∑x)2 )(n∑y2 -- (∑y)2 )) AN EXAMPLE: The table below shows the respective heights x and y of a sample of 12 fathers and their oldest sons in inches. The formulas given above can be used to determine the Line of Regression as well as the Coefficient of Correlation to test the goodness of fit of the regression line. Table 1 Father(x) 65 63 Son (y) 68 66

67 68

64 65

68 69

62 66

70 68

66 65

68 71

67 67

69 68

The next step is to develop Table 2 by adding 3 more columns as shown below

12

71 70


Table 2 x

y

x2

xy

y2

------------------------------------------------------------------------------------------------65 68 4225 4420 4624 63 66 3969 4158 4356 67 68 4489 4556 4624 64 65 4096 4160 4225 68 69 4624 4692 4761 62 66 3844 4092 4356 70 68 4900 4760 4624 66 65 4356 4290 4225 68 71 4624 4828 5041 57 67 4489 4489 4489 69 68 4761 4692 4624 71 70 5041 4970 4900 ---------------------------------------------------------------------------------------------------∑x = 790 ∑y = 811 ∑x2 = 52178 ∑xy = 53437 ∑y2 = 54849 Since B1 is part of the formula for B0, we must calculate its value first. 12 (53437) - 790 (811) B1 = --------------------------------- = 0.272 12(52178) - 7902 Therefore, B0 = 790/12 - (811/12) . (0.272) = 49.7 Hence, the Line of Regression is Ŷ = 49.7 + 0.272x. With this line of “Best Fit” we can make predictions. For example, if a father is 72 inches tall, we could predict that his oldest son would be 49.7 + 0.272 (72) = 69.284 ≈ 69 inches. To determine the accuracy of the prediction, we calculate the Coefficient of Correlation Using the formula given above. That is:

12(53437) – 790 (811) R = ------------------------------------------------------ = 0.568 √((12(52178 - 7902 ) (12(54849) -- 8112 )) Though this appears to be a relatively good correlation since correlations, by definition, must be in the range (-1 ≤ R ≤ 1), we need to perform a hypothesis testing. 13


The Statdisk Implementation of the above example is given below:

Sample size, n:

12

Degrees of freedom: 10 Correlation Results: Correlation coeff, r: 0.5681492 Critical r:

Âą0.5759826

P-value (two-tailed): 0.05395 Regression Results: Y= b0 + b1x: Y Intercept, b0: Slope, b1:

49.66994 0.2721022

Total Variation:

38.91667

Explained Variation: 12.56205 Unexplained Variation: 26.35462 Standard Error: Coeff. of Det. R^2:

1.623411 0.3227936

14


15


CONCLUSION: As indicated in the Abstract, Regression Analysis is an invaluable tool for Economics. All business concerns must be in a position to predict customer levels, expenses, debits and profits if they are to remain viable. This simplified approach to Regression makes it feasible for practitioners to make better decisions and accurate forecasts. The statdisk software implementation to the given example is also provided. It is hoped that by implementing the techniques discussed in this paper in preparing lesson plans on regression analysis will help us to develop in our students the skills to solve problems using critical and creative thinking and scientific reasoning, , which is one of the Gen Ed Outcomes of Miami Dade College.

16


Acknowledgments The author is thankful to the editor and reviewer for their suggestions, which improved the quality of the paper.

REFENENCES Adler, H. L. & Roessier, E. B. Introduction to Probability and Statistics, 6th Edition, Chapter 12; W. H. Freeman and Company, San Francisco. Bluman, A. G. 2017. Elementary Statistics, 10th Edition, Chapter 9; McGraw-Hill, New York. Larose, D. T. 2010. Discovering Statistics, Frist Printing, Chapter 9; W. H. Freeman and Company, New York. Triola, M. F. 2017. Elementary Statistics, 13th Edition, Chapter 8; Addison â&#x20AC;&#x201C; Wesley, Boston. http://www.statdisk.org/.

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The Impact of Learning Assistants in the Teaching and Learning Process for MAC1105 College Algebra at MDC, Hialeah Campus Dr. Jaime Bestard, Ph. D; Professor Department of Liberal Arts and Sciences - Mathematics, E-mail: jbestard@mdc.edu Ms. Laura Iglesias, STEM Program Grant Director Ms. Anette Sanz, STEM Program Specialist Joel Gonzalez, Samara Rabassa, Danisley Muro, Dayron Sanchez; L.A. STEM Program Miami Dade College, Hialeah Campus

Abstract This paper shows the results of the introduction of embedded Learning Assistants into gate keeper courses as a pedagogical strategy, to improve retention and success in critical courses like MAC1105 College Algebra. The activity of such pedagogic assistants in class has demonstrated its effectiveness in the course delivery and supporting studentsâ&#x20AC;&#x2122; work in class, but also out of class. The paper illustrates the statistical significance that the presence and the activity of the Learning Assistants produced in class in a semester where the study was conducted in three experimental units and one control group for the same instructor and course. This experience is extended, expecting similar results, not only to MAC1105 but in Biological Sciences and Chemistry courses in campus.

KEYWORDS:

Anova, Hypothesis test, Learning assistant, Statistical significance.

2010 Mathematics Subject Classifications: 97C40, 97C70, 97D40, 97D50.

Introduction Facilitating the delivery of concepts in a gate keeper course is a necessary action to make the course material accessible to students. According the literature (Thomas & Brown, Winkelmes) the facilitation of the concepts delivery is an outstanding component for High Impact Pedagogical Practices, and its effectiveness is demonstrated in student populations that have consistent and large proportion of minorities (Bass, Kuh) This paper is a compilation of the results of the participation of Learning Assistants (LA) in several courses during last Fall 2017 semester, as part of a program in campus since Spring 2017. There are trend analysis and the comparative study of the differences in performance that the students demonstrated when the learning assistants are present in class (and outside the class) 18


versus when there is no presence of those important factors of the teaching and learning process: the Learning Assistants. The use of Learning Assistants (LA) in critical gateway courses like MAC1105 College Algebra has been conducted by the Campus STEM Program with the intention to improve the performance of the students in such courses. The LA are students who passed and excel in the course and receive a training in several instructional techniques working with the students during the classes and after classes either in one to one or small groups peer tutoring sessions.

Methodology The experiment was conducted in four sections of MAC1105 College Algebra taught by the same instructor (first author) reference numbers: 6840, 6847, 6848 and 6850 all at the MDCHialeah Campus. The sections were organized in the form that the first three mentioned above were considered experimental units while the last was the control unit. During the experiment a collection of the studentsâ&#x20AC;&#x2122; results, attendance to laboratory sessions, and assessments in class were recorded. The analysis of the pre and post assessments of the studentsâ&#x20AC;&#x2122; performance are shown, Pre in the week V (some delay due to hurricanes class suspension) and Post assessment conducted during the week XIII. The assessments target basic concepts in the course from pre-requisites to MAC1105 and initial concepts, to the fundamental ideas about the quadratic functions, using the problem solving approach.

Results The analysis started describing the information of the pre and post assessments and compiling the results in stem and leaf diagrams shown below. Observe that, concurrent with the histograms, it is observed how all the units tend to shift concentration of students to the upper grades in performance, showing improvement. There is a regularity with the distributions of scores, provided the shifting to the upper values in most of the experimental units (except for ref. 6847, which showed not much improvement) as observed in the histograms. The side by side box plots show the differences pre and post in each case, it is remarkable that the control unit still showed improvement, given the concentration of high performers dual enrollment students. In all cases the central tendency improves, by the location of the median in the box and the location of the central 50 % of the respective distributions. It is also remarkable the increase in consistency observed in the comparisons pre vs. post assessments, by the compression (reduction of the length) of the boxes, indicating lower IQR. 19


The Descriptive statistics were computed showing differences that explain the progress as well and then, T-Tests with homogeneous variances were conducted between the pre and the post assessment in each unit showing highly significant differences (p-value <0.01) in favor of the post test for the experimental units ref. 6840 and ref. 6848, while significant (p-value <0.05) for the unit ref. 6847. The control group, ref. 6850, besides it is a dual enrollment group, with high performers, did not show the level of differences of the experimental units. A further ANOVA showed significant differences among the post tests of all the sections which is interpreted as improvement in all cases, including the section ref. 6847 where evidence of low attendance to laboratory session was observed as cause of lack of consistency in the data as in the progress. Stem-and-Leaf Displays: Pre 6840E1, Post6840E1, Pre6847E2, Post6847E2, Pre6848E3, Post6848E3, Pre 6850 E4, and Post 6850 E4 Stem-and-leaf of Pre 6840E1 N = 33 Leaf Unit = 1.0

8 0 00000000 11 1 005 15 2 0000 16 3 5 16 4 16 5 16 6 (16) 7 0000000000005555 1 8 0

Stem-and-leaf of Post6840E1 N = 31 Leaf Unit = 1.0

4 0 0000 4 1 4 2 4 3 4 4 7 5 000 12 6 00000 15 7 000 (6) 8 000055 10 9 00000055 2 10 00

Stem-and-leaf of Pre6847E2 N = 29 Leaf Unit = 1.0

1 0 5 9 1 00000055 14 2 00005 14 3

20


14 4 (1) 5 0 14 6 14 7 0000000555555 1 8 0

Stem-and-leaf of Post6847E2 N = 28 Leaf Unit = 1.0

7 0 0000000 7 1 7 2 7 3 7 4 8 5 5 13 6 00055 (2) 7 05 13 8 0005555 6 9 0000 2 10 00

Stem-and-leaf of Pre6848E3 N = 33 Leaf Unit = 1.0

5 0 00000 11 1 000005 16 2 00055 (6) 3 000055 11 4 0055 7 5 00 5 6 55 3 7 05 1 8 5

Stem-and-leaf of Post6848E3 N = 32 Leaf Unit = 1.0

3 0 000 3 1 3 2 3 3 3 4 3 5 5 6 00 8 7 005 15 8 0000005 (7) 9 0000005 10 10 0000000000

Stem-and-leaf of Pre6850CG N = 40 Leaf Unit = 1.0

2 7

2 00 3 00000

21


11 4 0055 (10) 5 0000055555 19 6 00555555 11 7 00055555 3 8 55 1 9 0

Stem-and-leaf of Post6850CG N = 40 Leaf Unit = 1.0

2 5 05 2 6 7 7 00055 16 8 000000555 (18) 9 000000000000000055 6 10 000000

Histogram of Pre 6840E1, Post6840E1, Pre6847E2, Post6847E2, ... 0

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Post6850CG

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0

0

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0

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Boxplot of Pre 6840E1, Post6840E1, Pre6847E2, Post6847E2, ... 100

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0 Pre 6840E1 Post6840E1 Pre6847E2 Post6847E2 Pre6848E3 Post6848E3 Pre6850CG Post6850CG

Descriptive Statistics: Variable N Pre 6840E1 33 Post6840E1 31 Pre6847E2 29 Post6847E2 28 Pre6848E3 33 Post6848E3 32 Pre6850CG 40 Post6850CG 40

Mean SE Mean StDev Var 0 41.52 5.68 32.63 1064.82 0 67.10 5.42 30.19 911.29 0 43.79 5.51 29.66 879.74 0 58.93 6.91 36.55 1335.85 0 30.15 4.09 23.50 552.32 0 79.22 5.03 28.43 808.24 0 56.00 2.85 18.02 324.62 0 85.75 1.80 11.41 130.19

Variable Median Q3 Pre 6840E1 70.00 70.00 Post6840E1 80.00 90.00 Pre6847E2 50.00 72.50 Post6847E2 72.50 85.00 Pre6848E3 30.00 45.00 Post6848E3 90.00 100.00 Pre6850CG 55.00 70.00 Post6850CG 90.00 90.00

CVar S.Sq 78.60 90950.00 44.99 166900.00 67.73 80250.00 62.02 133300.00 77.94 47675.00 35.89 225875.00 32.17 138100.00 13.31 299200.00

Max Range IQR Skewness Kurtosis 80.00 80.00 65.00 -0.20 -1.90 100.00 100.00 30.00 -1.32 0.93 80.00 75.00 60.00 -0.06 -2.00 100.00 100.00 71.25 -0.87 -0.87 85.00 85.00 35.00 0.62 -0.33 100.00 100.00 23.75 -2.11 3.91 90.00 70.00 25.00 -0.27 -0.59 100.00 50.00 10.00 -1.27 2.07

23

Min Q1 0.00 5.00 0.00 60.00 5.00 12.50 0.00 13.75 0.00 10.00 0.00 76.25 20.00 45.00 50.00 80.00


Two-Sample T-Test and CI: Pre 6840E1, Post6840E1 Two-sample T for Pre 6840E1 vs Post6840E1 N Mean StDev SE Mean Pre 6840E1 33 41.5 32.6 5.7 Post6840E1 31 67.1 30.2 5.4 Difference = μ (Pre 6840E1) - μ (Post6840E1) Estimate for difference: -25.58 95% upper bound for difference: -12.44 T-Test of difference = 0 (vs <): T-Value = -3.25 P-Value = 0.001 DF = 62 Both use Pooled StDev = 31.4727

Two-Sample T-Test and CI: Pre6847E2, Post6847E2 Two-sample T for Pre6847E2 vs Post6847E2 N Mean StDev SE Mean Pre6847E2 29 43.8 29.7 5.5 Post6847E2 28 58.9 36.5 6.9 Difference = μ (Pre6847E2) - μ (Post6847E2) Estimate for difference: -15.14 95% upper bound for difference: -0.41 T-Test of difference = 0 (vs <): T-Value = -1.72 P-Value = 0.046 DF = 55 Both use Pooled StDev = 33.2212

Two-Sample T-Test and CI: Pre6848E3, Post6848E3 Two-sample T for Pre6848E3 vs Post6848E3 N Mean StDev SE Mean Pre6848E3 33 30.2 23.5 4.1 Post6848E3 32 79.2 28.4 5.0 Difference = μ (Pre6848E3) - μ (Post6848E3) Estimate for difference: -49.07 95% upper bound for difference: -38.28 T-Test of difference = 0 (vs <): T-Value = -7.59 P-Value = 0.000 DF = 63 Both use Pooled StDev = 26.0432

Two-Sample T-Test and CI: Pre6850CG, Post6850CG Two-sample T for Pre6850CG vs Post6850CG N Mean StDev SE Mean Pre6850CG 40 56.0 18.0 2.8 Post6850CG 40 85.8 11.4 1.8 Difference = μ (Pre6850CG) - μ (Post6850CG) Estimate for difference: -29.75

24


95% upper bound for difference: -24.14 T-Test of difference = 0 (vs <): T-Value = -8.82 P-Value = 0.000 DF = 78 Both use Pooled StDev = 15.0799

One-way ANOVA: Post6840E1, Post6847E2, Post6848E3, Post6850CG Method Null hypothesis All means are equal Alternative hypothesis At least one mean is different Significance level Îą = 0.05 Equal variances were assumed for the analysis.

Factor Information Factor Levels Values Factor 4 Post6840E1, Post6847E2, Post6848E3, Post6850CG

Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Factor 3 14231 4743.8 6.44 0.000 Error 127 93540 736.5 Total 130 107771

Model Summary S R-sq R-sq(adj) R-sq(pred) 27.1391 13.21% 11.16% 7.19%

Means Factor N Mean StDev 95% CI Post6840E1 31 67.10 30.19 (57.45, 76.74) Post6847E2 28 58.93 36.55 (48.78, 69.08) Post6848E3 32 79.22 28.43 (69.73, 88.71) Post6850CG 40 85.75 11.41 (77.26, 94.24) Pooled StDev = 27.1391

25


Interval Plot of Post6840E1, Post6847E2, ... 95% CI for the Mean

100

90

Data

80

70

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50 Post6840E1

Post6847E2

Post6848E3

Post6850CG

The pooled standard deviation is used to calculate the intervals.

Concluding Remarks Concluding about the experiments bring that Course Embedded Learning Assistants improve significantly studentsâ&#x20AC;&#x2122; academic performance, improvements in success and retention are derived, from the confidence students showed in the end of course surveys. The activity of the Learning Assistants in class as outside the class brings an exemplary component to the Campus Supportive Academic Services and it is recommended to the MDC Learning Resources division in order to facilitate understanding in gate keeper courses and improving retention, performance and cooperative/ collaborative work students- LA.

Acknowledgments The authors are thankful to the editor and reviewer for their suggestions, which improved the quality of the paper.

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References Bass, R. (2012). Disrupting ourselves: The problem of learning in higher education. Educause Review, 47(2). Retrieved from http//www.educause.edu/EDUCAUSE+Review/EDUCAUSEReviewMagazineVolume47/Disrupt ingOurselvesTheProblem/247690 Bass, R., & Elmendorf, H. (2012). Social pedagogies. Retrieved from https://blogs.commons.georgetown.edu/bassr/social-pedagogies Kuh, G. D. (2008). High-impact educational practices: What they are, who has access to them, and why they matter. Washington, D.C.: American Association of Colleges and Universities. Kuh, G. D., Oâ&#x20AC;&#x2122;Donnell, K., & Reed, S. (2013). Ensuring quality and taking high-impact practices to scale. Washington, DC: Association of American Colleges and Universities. Thomas, D., & Brown, J. S. (2011). A new culture of learning: Cultivating the imagination for a world of constant change. Lexington, KY: CreateSpace. Winkelmes, M.A. Transparency in learning and teaching project. Retrieved from: https://www.unlv.edu/provost/transparency.

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Effective Learning in Quantitative Reasoning and Problem Solving Professor Loretta Blanchette, Associate Professor Senior Department of Liberal Arts and Sciences - Mathematics, E-mail: lblanchette@mdc.edu Miami Dade College, Hialeah Campus

Abstract

This paper discusses a research study conducted on effective learning of quantitative reasoning and problem solving skills. Specific teaching and learning strategies were implemented over the period of one semester in multiple classrooms of a large institute of higher learning in southeast Florida with the goal of enhancing effective learning in quantitative reasoning and problem solving. Data was then collected to determine the effectiveness of the particular set of learning strategies in targeted college algebra classes. All learning strategies employed were based on best practices found in neuroscience, psychology, and education disciplines pertaining to effective learning. The hypothesis of this study was that implementation of a specific set of best practices would result in enhanced quantitative reasoning and problem solving abilities. Statistics included summary data pertaining to class success and completion rates, proportion of students who score 75% or higher on a comprehensive final exam, and average proficiency level of students on a math discipline general education learning outcomes assessment. Findings from this study are hereby shared with colleagues in the educational field in hopes of affecting positive change in student learning.

KEYWORDS:

Effective learning, Problem solving, Quantitative reasoning.

2010 Mathematics Subject Classifications: 97C40, 97C70, 97D40, 97D50.

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Introduction Research Project Objective: Implement a teaching and learning strategy (treatment) designed to enhance effective learning in quantitative reasoning and problem solving. Targeted population: Miami Dade College (MDC) MAC1105 college algebra students. Need for Treatment: Students struggle with quantitative reasoning and problem-solving skills. This difficulty negatively impacts student success and completion. Literature Review Findings: Research indicates that the way humans learn directly impacts how much is learned as well as how effectively that learning is processed, stored, retrieved, and transferred. Students who fail to learn effectively struggle to succeed. A review of current literature reveals certain common principles of effective learning: effective learning integrates new knowledge and skills meaningfully with prior knowledge and skills (Hardiman, 2010), requires attentiveness, active participation, repetition, and practice (Ritter et al, 2013), occurs over time, and results in long-term memory processing and transfer, which in turn promotes critical thinking and enhanced reasoning and problem-solving capabilities (NRC, 2000). Dr. Mariale Hardiman explains that “the brain filters new information through the lens of prior experience and prior knowledge in order to create new meaning” (Hardiman, 2010). This has implications for education in that “what and how much is learned in any situation depends heavily on prior knowledge and experience” (Halpern, 2009). Halpern asserts that “the best predictor of what is learned at the completion of any lesson, course, or program of study is what the learner thinks and knows at the start of the experience” (Halpern, 2009). New learning builds upon prior learning. A student’s knowledge going in to a learning environment can be referred to as domain knowledge. John Dunlosky et al specify that “in an educational context, domain knowledge refers to the valid, relevant knowledge a student brings to a lesson” (Dunlosky, 2013). Effective learning requires making meaningful connections between new and prior learning. “One of the hallmarks of the new science of learning is its emphasis on learning 29


with understanding” (NRC, 2000). It is asserted that “knowledgeable individuals are more likely to be able to use what they have learned to solve novel problems – to show evidence of transfer” because they understand the relationships which exist between the various components they are studying (NRC, 2000). Effective learning requires not only factual knowledge but an understanding of how things fit together into a larger “conceptual framework” (NRC, 2000). Hardiman explains that “lack of conceptual understanding typically results in loss of retention of the disjointed facts and details” (Hardiman, 2010). Knowledge which is conceptualized effectively transfers to be broadly applicable in a variety of situations leading to critical thinking, higher-order reasoning and problem-solving abilities. Effective learning requires time: time to process and assimilate additional information and skills, time to practice, time to gain accuracy and mastery, and time to conceptualize by understanding how new knowledge fits into the bigger picture. As stated in “How People Learn,” “learning cannot be rushed; the complex cognitive activity of information integration requires time” (NRC, 2000). Yet time alone is not enough. Although it is undeniably true that “it takes time to learn complex subject matter,” it is also abundantly clear that “spending a lot of time … in and of itself is not sufficient to ensure effective learning” (NRC, 2000). Equally critical to effective learning is how that time is spent. Ritter et al emphasize connections between learning, retention, performance accuracy, practice, and the regular use of the newly learned knowledge and skills (Ritter et al, 2013). Students need to spend time practicing with deliberate attention to understanding, not merely to memorization. This practice must occur both inside and outside the classroom. Students need to be active and engaged learners. Spacing is another key component to long-term memory and retention. Kelley and Whatson explain the impact of spaced learning on creating long-term memories. They point out “the value of spaced practice (many short sessions) over massed practice (a single long session) in LTM.” Morris claims that the “spacing effect is large…studying in two spaced sessions can produce twice as much recall as a single session of equal length” (Morris, 2016). “The Whole Student Learning Series- Study Tips for Improving Long-Term Memory Retention and Recall” affirms that “restudying information at spaced intervals staves off forgetting and improves long-term retention and retrieval.”

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Closely linked to spaced learning is the matter of repetition. Hardiman asserts that “the most important factor in determining how well we remember information is the degree to which we rehearse and repeat that information” (Hardiman, 2010). This repetition is most effective if stretched out over time and if linked to meaningful pattern recognition. As stated in “Learning and Memory: What’s the Connection,” “repeated exposure to a stimulus or piece of information transfers it into long-term memory, especially if your review is distributed over time” (“Learning and Memory,” n.d.). When new knowledge or skills are revisited in multiple contexts over a span of time, they are learned on conceptual levels which subsequently become usable and retrievable in a variety of applications. This results in enhanced critical thinking and problem-solving abilities. Active participation of students in their own learning increases the likelihood of learning transfer. Active learning includes independent learning. As students applied new learning to problem solving they reinforced that learning, forming stronger links for future reference. Immediate feedback serves to correct misunderstandings and suggest alternate strategies with which to apply knowledge and skills to new situations. Cognitive science tells us that “immediate feedback strengthens learning memory patterns” (Hardiman, 2010). With active learning and regular feedback, students gain confidence, skill, and accuracy. Research shows that “experts’ abilities to reason and solve problems depend on well-organized knowledge that affects what they notice and how they represent problems” (NRC, 2000). By practicing effective learning strategies, students increase their useable knowledge and skill base which subsequently increases their ability to reason critically and solve problems effectively. Students must be taught to purposefully monitor their own thinking as they study, practice, and problem solve, both inside and outside of the classroom. In “Metacognition: Study Strategies, Monitoring, and Motivation,” William Peirce avers, “The value of a strategy lies more in the cognitive and metacognitive processes used than the steps in the strategy itself” (Peirce, 2003) Summary of Key Principles of Effective Learning: Effective learning integrates new knowledge and skills meaningfully with prior knowledge and skills, requires attentiveness, active participation, repetition and practice, occurs over time, benefits from spaced learning, results in long-term memory processing, retrieval and transfer, and promotes critical thinking and enhanced reasoning and problem-solving capabilities. 31


Proposed Teaching and Learning Strategy (Treatment): •

Pre-lesson learning and note taking

Mandatory attendance

Problem solving taught and modeled

Active classroom engagement

Peer teaching and learning

Spaced study sessions

Evidence of learning assignments

Timely feedback on problem solving

Free response assessment problems on exams

Comprehensive final assessment to include higher order thinking problems

Research Study Details Participating faculty members came from the Hialeah and North campuses. Each professor taught two MAC1105 classes in the fall 2017 term, one of which was randomly assigned as the control group and the other as the treatment group. Care was taken to ensure the experimental group class and the control group class of each professor followed the same scheduling format to minimize confounding variables. Participating professors were randomly assigned letters A, B, C, or D to provide anonymity in reporting and safeguard the confidentiality of their data. The confidentiality of all participating students was strictly maintained. All research study findings were reported to the researcher in the form of aggregate data and/or summary statistics. Students from both control and experimental group classes were informed of the research project and signed student consent forms (see Appendix G) as per college policy. All participating professors also signed faculty consent forms. The treatment implemented in this research study was based on best practices and included the following key components: (a) increase domain of knowledge through independent pre-lesson learning, (b) employ in class active learning with a focus on understanding 32


connections and concepts, training in critical thinking, problems solving, and metacognition, (c) provide timely feedback on evidence of learning assignments, (d) encourage after class review, practice, spaced learning, repetition and reinforcement, and (e) foster intentional practice of higher-order thinking, quantitative reasoning and problem solving. The proposed teaching and learning strategy included pre-lesson learning, mandatory attendance, in-class active learning, after class study sessions, evidence of learning problems, feedback on problem solving, free response assessment problems, and a comprehensive assessment which included higher order thinking problems. Students were required to pre-read the chapter/material for the upcoming lesson and take notes in advance of class. By reading and taking notes on the upcoming lesson, students were expected to increase their domain of knowledge prior to the lesson, thus increasing their likelihood of understanding and learning new topics and strategies (Halpern, 2009). In this way, the class lesson served as a repetition, reinforcing the prior reading material as well as clarifying matters that remained challenging or unclear to the students in their independent review. Mandatory attendance and enforcement of producing notes from the reading served as a form of accountability for the students and ensured repetitive exposure to the lesson material. Research fully supports the benefits of active learning; to that end, each class session purposefully included peer teaching and learning activities. Each student in the treatment group was assigned a class study partner and time was allotted each class period except exam days to peer teaching and learning as students worked together to solve problems. Students in the treatment group were active learners both inside and outside the classroom. For some, the prospect of explaining a math concept to another was daunting. Yet students paired together as study partners soon learned to verbalize what they are thinking, analyze the validity of their reasoning when questioned by a peer, and actively sought answers to puzzling questions. Students were encouraged to continue this partner learning experience after class as they studied in teams or in small groups while practicing skills and reviewing the topics of the course. In this way, the instructors provided an environment conducive to effective learning in the classroom by a variety of means â&#x20AC;&#x201C; both from their lecture as well as from the help of fellow classmates. Study sessions spread throughout the week corresponded to the principles of spaced learning (Kelley and Whatson, 2013). Research indicates that learning is more effective when 33


spaced apart as compared to compacted into one longer session (Morris, 2016). For this reason, students in the treatment group were instructed to schedule 3-6 study/practice session per week, each 30- 120- minutes long, in which to conduct their out-of-class learning. Students were encouraged to keep study logs in the form of a journal to record their study times. These logs were then submitted on exam day and accounted for 1 percent of the exam score. Providing a percentage point was considered necessary to ensure compliance with the learning strategy, however in practice this did not appear sufficient incentive to keep the log for many students. Even so, since pre-lesson learning was required for each new lesson, and evidence of learning assignments were due each class period, students necessarily spread their study sessions out throughout the week. The study log requirement was intended to serve as an additional incentive to incremental and regular learning sessions. All students in the experimental group classes were provided a Problem-Solving Components handout (see Appendix E) as a guide for learning key elements of problems solving. This guide served to inform student learning both inside and outside the classroom by training students to write down the reasoning and analysis necessary for effective problem solving. The process of writing and fully supporting all conclusions is an example of practicing metacognition and aligns with Miami Dade College’s Quality Enhancement Plan (QEP). As explained by Peirce, “Asking students to describe their thinking processes … develops their metacognitive abilities—a very necessary skill to improve thinking” (Peirce, 2003). Evidence of learning problems were assigned each class period except exam days and were due at the beginning of class on the following period. These problems were intended to provide students with necessary practice in critical thinking and problem solving and were a means for training students in effective problems solving strategies throughout the entire semester. Students were expected to verbalize the goals and objectives of each problem, delineate the processes needed, use algebraic methods to arrive at valid solutions, and provide meaning to their findings by purposefully tying their answers back to the original problem. Throughout this process, students were encouraged to evaluate their own logic, recognize gaps in their understanding, seek appropriate answers, and thus strengthening their conceptual learning and transfer. The evidence of learning problems provided an opportunity for the professors to provide meaningful and timely feedback to students prior to class exams, giving students opportunity to adjust their practices as needed. Students who were accustomed to multiple choice, matching, and/or fill in the blank problems had ample 34


opportunity to practice and learn how to use higher order reasoning and critical thinking skills by means of completing the evidence of learning problems throughout the semester. Students were scored by a pre-established rubric (see appendix F) to ensure consistency in expectation and evaluation of student mastery of problem solving. Exams included free response problems which were similarly assessed. Students in both the control and experimental group classes took a math discipline learning outcomes assessment (see Appendix D) prior to the end of the term. The average level of proficiency in quantitative reasoning and problem solving for each group was compared for any increase in proficiency by the experimental group. A comprehensive final exam served as a summative assessment of quantitative reasoning and problem-solving skills. Students who scored 75% or higher on the comprehensive final exam were considered to have achieved a measure of success. Students who score 70% or higher in the course succeeded in passing the class. The stated goals of this research were to: (1) Increase the proportion of students in MAC1105 college algebra classes who effectively learn quantitative reasoning and problemsolving skills as measured by the proportion of targeted college algebra students who score 75% or higher on a comprehensive final exam; (2) Increase the success rates of targeted MAC1105 college algebra classes; and (3) Raise studentsâ&#x20AC;&#x2122; average quantitative reasoning and problems solving proficiency level as measured by a standardized math discipline learning outcomes assessment.

Quantitative Experimental Design Population: MAC1105 college algebra students at Miami Dade College Samples: Students in targeted MAC1105 classes

Independent variable: specific teaching and learning strategy Resultant, dependent variables: â&#x20AC;˘

Success rate of the students as measured by their scores on a comprehensive final exam 35


Success quotient of the students in the MAC1105 classes of the study

Average proficiency level of these students on a math discipline learning outcomes assessment

Hypothesis Tests Conducted Proportion of students who achieved a grade of 75% or higher on their comprehensive final exam: •

z-test was used to determine whether a significant difference exists between pairs of classes taught by the same professor

z-test was also used to test whether the overall difference between the control groups’ proportion and the experimental groups’ proportion is significant at the 5% level of significance

Success quotients of targeted classes •

z-test was used to determine whether a significant difference existed between pairs of classes taught by the same professor

z-test was also used to test whether the overall difference between the control groups’ proportion and the experimental groups’ proportion is significant at the 5% level of significance

Average proficiency level on LOA Assessment •

t-test was used to test for a significant difference in average proficiency level using an aggregate of the control group classes and the experimental classes

Findings, Analysis and Conclusions 1. Proportion Tests for Success on Final Exam (See Appendix A for Data Collected) Characteristic of interest: Success on comprehensive final exam (scored 75% or higher)

36


Claim: The proportion of students in the experimental MAC1105 group who score 75% or higher on their comprehensive final exam will be higher than that of the control MAC1105 group. Population 1: Experimental MAC1105 group Population 2: Control MAC1105 group Parameter: proportion p

Else: p1 â&#x2030;¤ p2

Claim: p1 > p2

A. Professor A Final Exam Data Analysis Null Hypothesis:

H0: p1 = p2

Alternative Hypothesis:

H1: p1 > p2

(claim)

STATDISK Analysis: Alternative Hypothesis: p1 > p2 Pooled proportion: 0.4222222 Test Statistic, z: -1.1293 Critical z: P-Value:

1.6449 0.8706

90% Confidence interval: -0.4050209 < p1-p2 < 0.0716875

37

Right-tailed test


Conclusion: Fail to reject the null hypothesis since the test statistic does not fall into the critical region. There is not enough evidence to support the claim that the proportion of students in the experimental MAC1105 group who score 75% or higher on their comprehensive final exam will be higher than that of the control MAC1105 group. Also note: Since the 90% Confidence Interval includes zero, we can be 90% confident there is no significant difference in the population proportions.

B. Professor B Final Exam Data Analysis Null Hypothesis:

H0: p1 = p2

Alternative Hypothesis:

H1: p1 > p2

(claim)

STATDISK Analysis: Alternative Hypothesis: p1 > p2 Pooled proportion: 0.4102564 Test Statistic, z: -1.0352

38

Right-tailed test


Critical z: P-Value:

1.6449 0.8497

90% Confidence interval: -0.4220452 < p1-p2 < 0.0905235

Conclusion: Fail to reject the null hypothesis since the test statistic does not fall into the critical region. There is not enough evidence to support the claim that the proportion of students in the experimental MAC1105 group who score 75% or higher on their comprehensive final exam will be higher than that of the control MAC1105 group. Also note: Since the 90% Confidence Interval includes zero, we can be 90% confident there is no significant difference in the population proportions.

C. Professor C Final Exam Data Analysis Null Hypothesis:

H0: p1 = p2

Alternative Hypothesis:

H1: p1 > p2

(claim) 39

Right-tailed test


STATDISK Analysis: Alternative Hypothesis: p1 > p2 Pooled proportion: 0.7307692 Test Statistic, z: -1.0028 Critical z: P-Value:

1.6449 0.8420

90% Confidence interval: -0.2641999 < p1-p2 < 0.062486

Conclusion: Fail to reject the null hypothesis since the test statistic does not fall into the critical region. There is not enough evidence to support the claim that the proportion of students in the experimental MAC1105 group who score 75% or higher on their comprehensive final exam will be higher than that of the control MAC1105 group. Also note: Since the 90% Confidence Interval includes zero, we can be 90% confident there is no significant difference in the population proportions. 40


D. Professor D Final Exam Data Analysis Null Hypothesis:

H0: p1 = p2

Alternative Hypothesis:

H1: p1 > p2

(claim)

Right-tailed test

STATDISK Analysis: Alternative Hypothesis: p1 > p2 Pooled proportion: 0.6140351 Test Statistic, z: 2.2820 Critical z: P-Value:

1.6449 0.0112

90% Confidence interval: 0.0915617 < p1-p2 < 0.4971082

Conclusion: Reject the null hypothesis since the test statistic falls into the critical region. There is enough evidence to support the claim that the proportion of students in the experimental 41


MAC1105 group who score 75% or higher on their comprehensive final exam will be higher than that of the control MAC1105 group. Also note: Since the 90% Confidence Interval does not include zero, we can be 90% confident there is a difference in the population proportions. E. Aggregated Final Exam Data Analysis Null Hypothesis:

H0: p1 = p2

Alternative Hypothesis:

H1: p1 > p2

(claim)

STATDISK Analysis: Alternative Hypothesis: p1 > p2 Pooled proportion: 0.5799087 Test Statistic, z: -0.0138 Critical z: P-Value:

1.6449 0.5055

90% Confidence interval: -0.1106674 < p1-p2 < 0.1088316

42

Right-tailed test


Conclusion: Fail to reject the null hypothesis since the test statistic does not fall into the critical region. There is not enough evidence to support the claim that the proportion of students in the experimental MAC1105 group who score 75% or higher on their comprehensive final exam will be higher than that of the control MAC1105 group. Also note: Since the 90% Confidence Interval includes zero, we can be 90% confident there is no significant difference in the population proportions.

43


Findings in Three out of Four Cases There is not enough evidence to support the claim that the proportion of students in the experimental MAC1105 group who score 75% or higher on their comprehensive final exam will be higher than that of the control MAC1105 group. The 90% Confidence Interval included zero, thus we can be 90% confident there is no significant difference in the population proportions. Finding in One out of Four Cases There is enough evidence to support the claim that the proportion of students in the experimental MAC1105 group who score 75% or higher on their comprehensive final exam will be higher than that of the control MAC1105 group. Since the 90% Confidence Interval did not include zero, we can be 90% confident there is a difference in the population proportions. Finding for Aggregate Data There is not enough evidence to support the claim that the proportion of students in the experimental MAC1105 group who score 75% or higher on their comprehensive final exam will be higher than that of the control MAC1105 group. Since the 90% Confidence Interval included zero, we can be 90% confident there is no significant difference in the population proportions.

2. Success Quotient Measure (See Appendix B for Data Collected) Claim: The success quotient of the experimental MAC1105 class group will be higher than that of the control MAC1105 class group. Characteristic of interest: Success quotient of the class. The success quotient is a proportion which is calculated as a ratio of the number of students who earned A, B or C scores over the number of students who earned A, B, C, D, F, or I score in a class. Population 1: Experimental MAC1105 class group Population 2: Control MAC1105 class group Parameter: proportion p 44


Else: p1 â&#x2030;¤ p2

Claim: p1 > p2

A. Professor A Success Quotient Data Analysis Null Hypothesis:

H0: p1 = p2

Alternative Hypothesis:

H1: p1 > p2

(claim)

STATDISK Analysis: Alternative Hypothesis: p1 > p2 Pooled proportion: 0.4222222 Test Statistic, z: -1.1293 Critical z: P-Value:

1.6449 0.8706

90% Confidence interval: -0.4050209 < p1-p2 < 0.0716875

45

Right-tailed test


Conclusion: Fail to reject the null hypothesis since the test statistic does not fall into the critical region. There is not enough evidence to support the claim that the success quotient experimental MAC1105 class group will be higher than that of the control MAC1105 class group. Also note: Since the 90% Confidence Interval includes zero, we can be 90% confident there is no significant difference in the population proportions.

B. Professor B Success Quotient Data Analysis Null Hypothesis:

H0: p1 = p2

Alternative Hypothesis:

H1: p1 > p2

(claim)

STATDISK Analysis: Alternative Hypothesis: p1 > p2 Pooled proportion: 0.5641026 46

Right-tailed test


Test Statistic, z: -1.3298 Critical z: P-Value:

1.6449 0.9082

90% Confidence interval: -0.4760119 < p1-p2 < 0.0466641

Conclusion: Fail to reject the null hypothesis since the test statistic does not fall into the critical region. There is not enough evidence to support the claim that the success quotient experimental MAC1105 class group will be higher than that of the control MAC1105 class group. Also note: Since the 90% Confidence Interval includes zero, we can be 90% confident there is no significant difference in the population proportions.

47


C. Professor C Success Quotient Data Analysis Null Hypothesis:

H0: p1 = p2

Alternative Hypothesis:

H1: p1 > p2

(claim)

Right-tailed test

STATDISK Analysis: Cannot be done with this test since the sample size was small and there were less than 5 failures in the control class group.

D. Professor D Success Quotient Data Analysis Null Hypothesis:

H0: p1 = p2

Alternative Hypothesis:

H1: p1 > p2

(claim)

Right-tailed test

STATDISK Analysis: Cannot be done with this test since the sample size was small and there were less than 5 failures in the experimental class group.

E. Aggregate Success Quotient Data Analysis Null Hypothesis:

H0: p1 = p2

Alternative Hypothesis:

H1: p1 > p2

(claim)

STATDISK Analysis: Alternative Hypothesis: p1 > p2 Pooled proportion: 0.7077626 Test Statistic, z: 0.0801 Critical z: P-Value:

1.6449 0.4681 48

Right-tailed test


90% Confidence interval: -0.0961891 < p1-p2 < 0.1060356

Conclusion: Fail to reject the null hypothesis since the test statistic does not fall into the critical region. There is not enough evidence to support the claim that the success quotient for the experimental MAC1105 class group will be higher than that of the control MAC1105 class group. Also note: Since the 90% Confidence Interval includes zero, we can be 90% confident there is no significant difference in the population proportions.

Findings in Two out of Four Cases There is not enough evidence to support the claim that the success quotient experimental MAC1105 class group will be higher than that of the control MAC1105 class group. Since the 90% Confidence Interval included zero, we can be 90% confident there is no significant difference in the population proportions. 49


Findings in Remaining Two out of Four Cases In one case, the difference in proportions z test could not be done since the sample size was small and there were less than 5 failures in the control class group. In the other case, this test could not be done since the sample size was small and there were less than 5 failures in the experimental class group. Finding for Aggregate Data There is not enough evidence to support the claim that the success quotient for the experimental MAC1105 class group will be higher than that of the control MAC1105 class group. Since the 90% Confidence Interval includes zero, we can be 90% confident there is no significant difference in the population proportions. 3. Mean Test for Average Proficiency Level on the Miami Dade College Math Discipline

Learning Outcomes Assessment (LOA) (See Appendix C for Data Collected) Descriptive Statistics Analysis of Aggregate LOA Scores using Minitab Descriptive Statistics: Control Group Variable

Total Count

Mean

SE Mean

StDev

Control

99

2.8586

0.0986

0.9808

Descriptive Statistics: Experimental Group Variable

Total Count

Mean

SE Mean

StDev

Experimental

92

2.633

0.126

1.204

50


Boxplot of Control, Experi 4

Data

3

2

1

0 Control

Experi

Hypothesis Test for Average Proficiency Level on the Miami Dade College Math Discipline Learning Outcomes Assessment (See Appendix D for the LOA)

Claim: The average proficiency level on the math discipline learning outcomes assessment of the aggregated students in the experimental MAC1105 groups will be higher than that of the aggregated control MAC1105 groups. Characteristic of interest: proficiency level on a math discipline learning outcomes assessment Population 1: Aggregated Experimental MAC1105 groups Population 2: Aggregated Control MAC1105 groups Else: µ1 ≤ µ2

Parameter: mean µ Claim: µ1 > µ2 Aggregated Data Analysis Null Hypothesis:

H0: µ1 = µ2

51


Alternative Hypothesis:

H1: µ1 > µ2

(claim)

Right-tailed test

STATDISK Analysis: Not eq. vars: No Pool Alternative Hypothesis: µ1> µ2 Test Statistic, t: -1.4410 Critical t:

1.653567

P-Value:

0.9243

90% Confidence interval: -0.4939253 < µ1-µ2 < 0.0339253

Conclusion: Fail to reject the null hypothesis since the test statistic does not fall into the critical region. There is not enough evidence to support the claim that the average proficiency level on the math discipline learning outcomes assessment of the aggregated students in the experimental MAC1105 groups will be higher than that of the aggregated control MAC1105 groups. Also note: Since the 90% Confidence Interval includes zero, we can be 90% confident there is no significant difference in the population means. 52


Additional Hypothesis Test Run for Statistically Significant Difference in Mean of the Two Populations Claim: The average proficiency level on the math discipline learning outcomes assessment of the aggregated students in the experimental MAC1105 groups is different from that of the aggregated control MAC1105 groups. Null Hypothesis:

H0: µ1 = µ2

Alternative Hypothesis:

H1: µ1 ≠ µ2

(claim)

STATDISK Analysis: Not eq. vars: No Pool Alternative Hypothesis: µ1 ≠ µ2 Test Statistic, t: -1.4410 Critical t:

±1.973551

P-Value:

0.1514

95% Confidence interval: -0.5449977 < µ1-µ2 < 0.0849977

53

Two-tailed test


Conclusion: Fail to reject the null hypothesis since the test statistic does not fall into the critical regions. There is not enough evidence to support a claim that the average proficiency level on the math discipline learning outcomes assessment of the aggregated students in the experimental MAC1105 groups is different from than that of the aggregated control MAC1105 groups. Also note: Since the 90% Confidence Interval includes zero, we can be 90% confident there is no significant difference in the population means.

Summary of Findings: On all three measures the hypothesis tests lead us to conclude that there is no statistically significant difference in the two groups. There is insufficient evidence that the treatment had a significant impact on studentsâ&#x20AC;&#x2122; effective learning in quantitative reasoning and problem solving as measured by their success on a cumulative final exam, by their success in the class and their level of proficiency on a math discipline learning outcomes assessment.

Conclusions and Limitations: The treatment was applied over one semester in four different classes by four different professors. Care was taken to ensure random sampling and to minimize possible effects of confounding variables. All treatment design components were based upon 54


extensive literature review and employed known best practices. Even so, research data did not reflect expected improvement in quantitative reasoning and problem-solving skills. It appears that one semester may be insufficient time to develop a significant increase in quantitative reasoning and problem-solving skills using these methods. In the one instance where significant improvement was noted, it can be postulated that the positive results were influenced by additional variables. Students in the higher preforming experimental class group were dual enrolled, highly motivate students who responded well to the treatment and showed elevated levels of success. It cannot be concluded that the treatment itself drove this success since these high performing students were likely to have been more successful than typical MAC1105 students regardless the specific teaching and learning methods employed. Although the researcher sought faculty volunteers from across Miami Dade College, only faculty from two campuses agreed to participate in the research project. Of the faculty who volunteered to participate, it is possible that these particular faculties already employed a considerable number of key elements found in the treatment design in their day to day teaching. Faculty participants were encouraged to continue to teach their classes in their traditional manner while supplementing with the additional teaching and learning strategies of the treatment design in their experimental group class alone. Upon further conversing with these volunteer faculty during the implementation period, it became clear that those who agreed to participate did so, in large part, because they agreed with the proposed treatment strategy and already practiced many of its aspects in their classes. Thus, the differences between experimental and control group teaching and learning strategies may well have been minimal, leading to no significant impact of the proposed treatment. Should faculty who do not now practice such teaching and learning strategies begin to implement them in one class but not another, it is likely that a significant difference would emerge in student effective learning in quantitative reasoning and problem solving. Reflections and Recommendations: It would be interesting to conduct this experiment with a more diverse group of faculty volunteers, having varied teaching styles, to better ascertain its effectiveness in enhancing quantitative reasoning and problem solving. It may be necessary to implement the treatment with the same studentsâ&#x20AC;&#x2122; groups over multiple semesters to effect meaningful change. Such a longitudinal study would shed further light on the efficacy of the teaching and learning strategies proposed in this research study. It would also be informative to study student success in subsequent classes. This follow-up could serve to evaluate studentsâ&#x20AC;&#x2122; ability to transfer their learning and gauge the long-term benefit derived from the teaching and learning strategies of this study.

55


Acknowledgments The author is thankful to the editor and reviewer for their suggestions, which improved the quality of the paper.

References •

Dunlosky, John, et al. “Improving Students’ Learning with Effective Learning Techniques: Promising Directions from Cognitive and Educational Psychology.” Association for Psychological Science, 2013. Retrieved from http://www.psychologicalscience.org/index.php/publications/journals/pspi/learningtechniques.html

Halpern, D. F. “Teaching for Long-Term Retention and Transfer.” Change, July/August 2003. Retrieved from http://cals.ufl.edu/trc/docs/tt/2009/2009-04Teaching%20for%20Long%20Term%20Retention.pdf

Hardiman, Mariale. “New Horizons for Learning: The Brain Targeted Teaching Model.” Johns Hopkins University. New Horizons for Learning, May 2010. Retrieved from http://education.jhu.edu/PD/newhorizons/Journals/spring2010/thebraintargetedteachingm odel/

Kelley, Paul and Whatson, Terry. “Making long-term memories in minutes: a spaced learning pattern from memory research in education.” Frontiers in Human Neuroscience, 2013; 7: 589. Retrieved from http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3782739/

References

“Learning and Memory: What’s the Connection?” Academic Resource Center handout. Duke University. n.d. Retrieve from https://arc.duke.edu/documents/Learning%20and%20Memory%20handout.pdf

56


Morris, Peter. “Memory – Structures and Functions.” 2016. Retrieved from http://education.stateuniversity.com/pages/2222/Memory-STRUCTURESFUNCTIONS.html

National Research Council. 2000. “How People Learn: Brain, Mind, Experience, and School: Expanded Edition.” Washington, DC: The National Academies Press. doi: 10.17226/9853. Retrieved from https://www.nap.edu/read/9853/chapter/5

Peirce, William. “Metacognition: Study Strategies, Monitoring, and Motivation.” Prince George's Community College, 2003. Retrieved from http://academic.pg.cc.md.us/~wpeirce/MCCCTR/metacognition.htm

Tokuhama-Espinosa, T. (2010). “Mind, Brain, and Education Science: The new brainbased learning.” New York, NY: W.W: Norton. Retrieved from http://education.jhu.edu/PD/newhorizons/Journals/Winter2011/Tokuhama5

“Whole Student Learning Series Study Tips for Improving Long-Term Memory Retention and Recall.” Michigan State University College, 7/14/14gb. Retrieved from http://com.msu.edu/Students/Academic_Guidance/long_term_retention_recall.pdf

Appendix A Final Exam Data Collected from the Research Implementation Final Exam C = Control E = Experimental

Data

Professor A

Professor B

Professor C

Professor D

Data

Data

Data

Data

Score

Score

Score

Score

d 75%

d 75%

d 75%

d 75%

or

or

or

or

Aggregate Data

Scored

Highe

Sampl

Highe

Sampl

Highe

Sampl

Highe

Sampl

75% or

Sampl

r

e Size

r

e Size

r

e Size

r

e Size

Higher

e Size

57


C

12

24

11

23

29

37

13

28

65

112

E

7

21

5

16

28

41

22

29

62

107

46.43

0.58035

58.04

%

7

%

75.86

0.57943

57.94

%

9

%

Final Exam Success Proportions

50.00 C

0.50

47.83

%

0.48

33.33 E

0.33

78.38

%

0.78

31.25

%

0.31

%

0.46

68.29

%

0.68

%

0.76

Appendix B Success Quotient Data Collected from the Research Implementation C = Control

Success Quotient Data

E = Experimental

Professor A

Professor B

Professor C

Professor D

Data

Data

Data

Data

Count

Count

Count

Count

of

of

of

of

Count

A+B+

A+B+

A+B+

A+B+

of

C

Sampl

Grades e Size

C

Sampl

Grades e Size

C

Sampl

Grades e Size

C

Aggregate Data

Sampl

Grades e Size

A+B+C

Sampl

Grades

e Size

C

12

24

15

23

34

37

18

28

79

112

E

7

21

7

16

35

41

27

29

76

107

64.29

0.70535

70.54

%

7

%

Success Quotients

50.00 C

0.50

%

65.22 0.65

%

91.89 0.92 58

%

0.64


33.33 E

0.33

%

43.75 0.44

%

85.37 0.85

93.10

%

0.93

71.03

% 0.71028

Appendix C LOA Data Collected from the Research Implementation LOA Data will be aggregated: Control Group (C) vs. Experimental LOA Data

Group (E) Professor B

Professor C

Professor D

Professor A Data

Data

Data

Data

Control

Control

Experi

Experi Control

Experi Control

Experi

1

2.50

3.25

4.00

4.00

2.00

2.00

3.00

4.00

2

3.75

3.75

3.00

3.00

3.00

1.00

3.00

4.00

3

1.75

2.50

3.50

3.50

3.00

4.00

1.00

4.00

4

3.25

3.25

3.50

3.50

2.00

1.00

3.00

1.00

5

3.00

3.50

0.50

3.00

3.00

4.00

4.00

0.00

6

2.25

3.25

3.50

3.50

3.00

3.00

3.00

0.00

7

3.00

3.50

3.50

3.00

3.00

3.00

4.00

1.00

8

2.25

1.50

2.00

3.50

4.00

3.00

1.00

4.00

9

3.50

3.25

1.50

3.50

3.00

3.00

3.00

1.00

10

3.50

3.00

1.50

3.50

2.00

3.00

1.00

1.00

11

3.00

4.00

3.50

3.50

4.00

3.00

2.00

3.00

12

2.00

3.25

3.00

1.50

4.00

3.00

0.00

3.00

13

3.25

3.50

2.00

2.50

2.00

3.00

2.00

2.00

14

3.25

3.25

4.00

3.50

2.00

3.00

4.00

0.00

15

3.50

3.25

3.00

1.50

3.00

4.00

4.00

4.00

16

3.25

3.25

4.00

4.00

1.00

4.00

3.00

17

3.25

3.50

2.50

4.00

1.00

3.00

4.00

18

3.75

3.00

4.00

3.00

4.00

4.00

3.00

19

3.00

3.00

1.50

4.00

4.00

1.00

2.00

n=15

59

%


20

3.75

4.00

4.00

3.00

21

3.50

1.00

3.00

2.00

22

3.50

1.50

3.00

1.00

23

3.25

3.50

3.00

3.00

0.00

24

3.00

2.00

3.00

0.00

4.00

1.00

2.00

2.00

4.00

1.00

27

2.00

2.00

3.00

28

2.00

3.00

0.00

29

4.00

2.00

1.00

30

2.00

31

1.00

n=29

n=29

32

4.00

n=19

25 26

n=23 n=24

n=32

Control:

99

Experimental:

92

Ordered Aggregate LOA Data Control

Experi

1

0.00

0.00

2

0.50

0.00

3

1.00

0.00

4

1.00

0.00

5

1.00

0.00

6

1.00

0.00

7

1.00

0.00

8

1.00

1.00

60

1.00

3.00 4.00

n=20

0.00


9

1.00

1.00

10

1.50

1.00

11

1.50

1.00

12

1.50

1.00

13

1.50

1.00

14

1.75

1.00

15

2.00

1.00

16

2.00

1.00

17

2.00

1.00

18

2.00

1.00

19

2.00

1.00

20

2.00

1.50

21

2.00

1.50

22

2.00

1.50

23

2.00

2.00

24

2.00

2.00

25

2.00

2.00

26

2.00

2.00

27

2.00

2.00

28

2.00

2.00

29

2.00

2.00

30

2.25

2.50

31

2.25

2.50

32

2.50

3.00

33

2.50

3.00

34

3.00

3.00

35

3.00

3.00

36

3.00

3.00

37

3.00

3.00

38

3.00

3.00 61


39

3.00

3.00

40

3.00

3.00

41

3.00

3.00

42

3.00

3.00

43

3.00

3.00

44

3.00

3.00

45

3.00

3.00

46

3.00

3.00

47

3.00

3.00

48

3.00

3.00

49

3.00

3.00

50

3.00

3.00

51

3.00

3.00

52

3.00

3.00

53

3.00

3.00

54

3.00

3.00

55

3.00

3.00

56

3.00

3.00

57

3.00

3.25

58

3.00

3.25

59

3.25

3.25

60

3.25

3.25

61

3.25

3.25

62

3.25

3.25

63

3.25

3.25

64

3.25

3.25

65

3.50

3.50

66

3.50

3.50

67

3.50

3.50

68

3.50

3.50 62


69

3.50

3.50

70

3.50

3.50

71

3.50

3.50

72

3.50

3.50

73

3.50

3.50

74

3.50

3.50

75

3.50

3.50

76

3.75

3.50

77

3.75

3.75

78

3.75

4.00

79

4.00

4.00

80

4.00

4.00

81

4.00

4.00

82

4.00

4.00

83

4.00

4.00

84

4.00

4.00

85

4.00

4.00

86

4.00

4.00

87

4.00

4.00

88

4.00

4.00

89

4.00

4.00

90

4.00

4.00

91

4.00

4.00

92

4.00

4.00

93

4.00

94

4.00

95

4.00

96

4.00

97

4.00

98

4.00 63


99

4.00

n=99 MEAN:

2.86 Control

n=92 2.63 Experi

StDeva: 0.980786 1.204075

Appendix E Problem Solving Components Handout A.

Context •

Given: capture the provided scenario, relevant facts, given formulas, graphs, constraints and/or limitations, and conditions as presented.

• B.

Goal: state the question(s) to be answered and/or instructions provided. Support

Sketch the situation if appropriate and helpful.

Write down the Formulas, Theorems, Rules, Definitions and/or Resources used.

Model relevant relationships by creating tables, functions, graphs, equations, inequalities, and/or systems as appropriate. Define all variables.

Process: show steps using appropriate methods of the course; follow the specific method indicated. Show all necessary analysis and calculations. Use accurate and appropriate notation. Follow valid logical steps. Write neatly and legibly throughout.

Explanation: If no calculations or process is required for a particular problem, give satisfactory (at course level) explanation or reasoning used to arrive at your conclusion.

C.

Meaning

Connect the final answer to the original problem.

Provide appropriate units. 64


â&#x20AC;˘

Interpret the results in context, using complete sentences as appropriate.

â&#x20AC;˘

Notation: Use appropriate notation in the final answer (and throughout the entire

problem-solving process). Appendix F Evidence of Learning (EOL) Rubric Letter Grade

Percentage

A+

100

A

95

A-

90

B

85

B-

80

C

75

C-

70

D

65

Criterion

Excellent at course level work including Context, Support, and Meaning. Utilized algebraic methods indicated and correct notation and logic throughout. Arrived at valid conclusions.

Great at course level work utilizing algebraic methods of that section as indicated BUT missing goal statement and/or instructions and/or clear identification of givens OR has a few notation errors and/or logic missteps but arrives at valid conclusions and gives proper meaning to answer.

Good at course level work utilizing algebraic methods of the section/course BUT missing context altogether OR fails to tie findings to context OR provides context, support and meaning but has several errors/omissions in notation and/or logic.

Weak utilization of algebraic methods of the section/course but effort made to use correct methods. Significant number of errors/omissions in notation/calculation/logic. Shows support for answer BUT provides no context and/or fails to tie findings to context. Has correct answer as per answer key and answer follows as per submitted work OR has incorrect answer due to some error but given answer is supported by work shown.

65


D-

60

Many errors and/or omissions. Some apparent effort to utilize the algebraic methods of the section/course. Answers are correct, and work appears to support the answer on some level, albeit without course level use of notation and/or logic OR fails to arrive at correct answers but begins the problems with some indication of appropriate methods.

F

50

Context + correct answers for problems that reasonably could have been worked mentally and/or with a calculator BUT there are trivial, incorrect or no support provided; failed to utilized method specified.

OR utilized appropriate methods to arrive at answers but answers and/or middle steps show severe errors/omissions and lack of understanding. Do/Redo

0

No submission/blank paper OR problems copied down with no effort to work the problems OR a list of unsupported answers

66


The cubic and quartic equations Dr. Mario Duran Camejo, Ph.D. Department of Liberal Arts and Sciences - Mathematics Miami Dade College, Hialeah Campus E-mail: mduranca@mdc.edu

Abstract This paper discusses the analytic method of solution of the quartic and cubic equations in historical perspective.

Keywords: Cubic equation; History of mathematics and mathematicians; Quartic equation. 2010 Mathematics Subject Classification: 00A05, 00A06, 01A05.

1. Introduction The analytic method of solution of the quartic and cubic equations is known since the Renaissance although a geometrical solution of the cubic equation was developed by the Persian poet and mathematician Omar Khayyám during the Middle Ages. Nonetheless, because of the attractiveness of its implications it is a recurrent theme in mathematics. Although the quartic equation can be solved by factorization, it is not possible to factorize a cubic equation because the constraints imposed to the coefficients lead again to a cubic equation and the problem becomes circular. The reason is that a quartic equation may be decomposed into two polynomials but the cubic has to be decomposed in a binomial and a polynomial and the coefficient of the binomial is one of the cubic roots. An alternative decomposition of the cubic would lead to infinite series rather than polynomials. In fact, some solutions of a cubic equation may be expressed by means of hypergeometric functions.

2. Cubic Equation Two, already classical, algebraic methods were developed for solving the cubic equation. The first one in chronological order is due to Scipione del Ferro and Niccolò Fontana Tartaglia but named after Girolamo Cardano who, later on, published the method. The other one was devised by François Viète. Both methods require to transform the whole cubic into a depressed equation. 67


Cardano's solution assimilates the depressed cubic to a quadratic equation whereas Viète's assimilates it to a biquadratic. As the transformation into a biquadratic actually doubles the number of solutions, both solutions from Viète's algebraic method are identical. The algebraic methods do really provide only one solution, and the other two have to be found multiplying the first one by the two complex cubic roots of 1, namely, â&#x2C6;&#x2019; 1â &#x201E;2 Âą â&#x2C6;&#x161;3 đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; â &#x201E;2. Additionally, Viète also found a trigonometric solution assimilating the depressed cubic equation to the formula for the cosine of the triple angle.

2.1 KhayĂĄmâ&#x20AC;&#x2122;s geometric solution KhayyĂĄm uses the intersection of two curves, a parabola and a circle, to get the solution of a depressed cubic. The solution is based in transforming the depressed cubic đ?&#x2018;Ľđ?&#x2018;Ľ 3 + đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; + đ?&#x2018;&#x2122;đ?&#x2018;&#x2122; = 0 into the incomplete depressed quartic đ?&#x2018;Ľđ?&#x2018;Ľ 4 + đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;Ľđ?&#x2018;Ľ 2 + đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122; = 0, multiplying the cubic by đ?&#x2018;Ľđ?&#x2018;Ľ. On the other hand, the đ?&#x2018;Ľđ?&#x2018;Ľđ?&#x2018;&#x192;đ?&#x2018;&#x192; coordinate of the point of intersection of the parabola đ?&#x2018;Śđ?&#x2018;Ś = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;Ľđ?&#x2018;Ľ 2 and the circle (đ?&#x2018;Ľđ?&#x2018;Ľ + đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;)2 + đ?&#x2018;Śđ?&#x2018;Ś 2 = đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; 2, is given by (đ?&#x2018;Ľđ?&#x2018;Ľđ?&#x2018;&#x192;đ?&#x2018;&#x192; + đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;)2 + (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;Ľđ?&#x2018;Ľđ?&#x2018;&#x192;đ?&#x2018;&#x192; 2 )2 = đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; 2 đ?&#x2018;Ľđ?&#x2018;Ľđ?&#x2018;&#x192;đ?&#x2018;&#x192;

4

đ?&#x2018;Ľđ?&#x2018;Ľđ?&#x2018;&#x192;đ?&#x2018;&#x192; 2 2đ?&#x2018;Ľđ?&#x2018;Ľđ?&#x2018;&#x192;đ?&#x2018;&#x192; đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; + 2 + =0 đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;2

Now, the solution of the incomplete depressed quartic can be assimilated to đ?&#x2018;Ľđ?&#x2018;Ľđ?&#x2018;&#x192;đ?&#x2018;&#x192; making đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; = 1â &#x201E;â&#x2C6;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; and đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; = đ?&#x2018;&#x2122;đ?&#x2018;&#x2122; â &#x201E;(2đ?&#x2018;&#x161;đ?&#x2018;&#x161;). As the intersection point is unique (because the origin is a trivial solution), and following what we have seen in the introduction above, the other two roots have to be obtained multiplying respectively by â&#x2C6;&#x2019; 1â &#x201E;2 Âą â&#x2C6;&#x161;3 đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; â &#x201E;2.

2.2 Alternative geometric solution

An interesting geometric solution would also be given by the intersection of the parabola đ?&#x2018;Śđ?&#x2018;Ś 2 = đ?&#x2018;Ľđ?&#x2018;Ľ + đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; and the equilateral hyperbola đ?&#x2018;Ľđ?&#x2018;Ľđ?&#x2018;Ľđ?&#x2018;Ľ = â&#x2C6;&#x2019;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; (for đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; > 0 and đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; > 0), which lead to the equation đ?&#x2018;Śđ?&#x2018;Śđ?&#x2018;&#x192;đ?&#x2018;&#x192; 3 + đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;đ?&#x2018;Śđ?&#x2018;Śđ?&#x2018;&#x192;đ?&#x2018;&#x192; + đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; = 0. When compared to the depressed cubic, we have đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; = đ?&#x2018;&#x161;đ?&#x2018;&#x161;, and đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; = đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;. Unlike the solution due to KhayyĂĄm, this intersection would give us the three unequal real roots of the cubic (in case they exist).

2.3 Cardanoâ&#x20AC;&#x2122;s algebraic solution

The solution is obtained by making the substitution đ?&#x2018;Ľđ?&#x2018;Ľ = đ?&#x2018;˘đ?&#x2018;˘ + đ?&#x2018;Łđ?&#x2018;Ł into the depressed cubic đ?&#x2018;Ľđ?&#x2018;Ľ 3 + đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; + đ?&#x2018;&#x2122;đ?&#x2018;&#x2122; = 0, which leads to đ?&#x2018;˘đ?&#x2018;˘3 + đ?&#x2018;Łđ?&#x2018;Ł 3 + (3đ?&#x2018;˘đ?&#x2018;˘đ?&#x2018;˘đ?&#x2018;˘ + đ?&#x2018;&#x161;đ?&#x2018;&#x161;)(đ?&#x2018;˘đ?&#x2018;˘ + đ?&#x2018;Łđ?&#x2018;Ł) + đ?&#x2018;&#x2122;đ?&#x2018;&#x2122; = 0. Then, imposing the condition 3đ?&#x2018;˘đ?&#x2018;˘đ?&#x2018;˘đ?&#x2018;˘ + đ?&#x2018;&#x161;đ?&#x2018;&#x161; = 0, it is possible to arrive to the two equations đ?&#x2018;˘đ?&#x2018;˘3 đ?&#x2018;Łđ?&#x2018;Ł 3 = â&#x2C6;&#x2019; đ?&#x2018;&#x161;đ?&#x2018;&#x161;3 â &#x201E;27, and â&#x2C6;&#x2019;(đ?&#x2018;˘đ?&#x2018;˘3 + đ?&#x2018;Łđ?&#x2018;Ł 3 ) = đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;. This is equivalent to consider đ?&#x2018;˘đ?&#x2018;˘3 and đ?&#x2018;Łđ?&#x2018;Ł 3 the solutions of the quadratic equation đ?&#x2018;§đ?&#x2018;§ 2 + đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;&#x2122; â&#x2C6;&#x2019; đ?&#x2018;&#x161;đ?&#x2018;&#x161;3 â &#x201E;27 = (đ?&#x2018;§đ?&#x2018;§ â&#x2C6;&#x2019; đ?&#x2018;˘đ?&#x2018;˘3 )(đ?&#x2018;§đ?&#x2018;§ â&#x2C6;&#x2019; đ?&#x2018;Łđ?&#x2018;Ł 3 ) = 0, so đ?&#x2018;˘đ?&#x2018;˘ = 3â&#x2C6;&#x161;đ?&#x2018;§đ?&#x2018;§1 , đ?&#x2018;Łđ?&#x2018;Ł = 3â&#x2C6;&#x161;đ?&#x2018;§đ?&#x2018;§2, and x is obtained summing up đ?&#x2018;˘đ?&#x2018;˘ and đ?&#x2018;Łđ?&#x2018;Ł. It is interesting to have also a look to the new approach of Cardanoâ&#x20AC;&#x2122;s solution, given in [1].

2.4 Vièteâ&#x20AC;&#x2122;s algebraic solution 68


The solution is obtained by substituting đ?&#x2018;Ľđ?&#x2018;Ľ, also in the depressed cubic đ?&#x2018;Ľđ?&#x2018;Ľ 3 + đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; + đ?&#x2018;&#x2122;đ?&#x2018;&#x2122; = 0, by đ?&#x2018;§đ?&#x2018;§ â&#x2C6;&#x2019; đ?&#x2018;&#x161;đ?&#x2018;&#x161;â &#x201E;(3đ?&#x2018;§đ?&#x2018;§), which leads to the equation đ?&#x2018;§đ?&#x2018;§ 6 + đ?&#x2018;&#x2122;đ?&#x2018;&#x2122;đ?&#x2018;§đ?&#x2018;§ 3 â&#x2C6;&#x2019; đ?&#x2018;&#x161;đ?&#x2018;&#x161;3 â &#x201E;27 = 0, biquadratic in đ?&#x2018;§đ?&#x2018;§ 3 , and, then, making the new substitution đ?&#x2018;§đ?&#x2018;§ 3 = đ?&#x2018;Łđ?&#x2018;Ł, the two values đ?&#x2018;Łđ?&#x2018;Ł1 and đ?&#x2018;Łđ?&#x2018;Ł2 may be obtained. Thus, đ?&#x2018;Ľđ?&#x2018;Ľ1 = 3 3 â&#x2C6;&#x161;đ?&#x2018;Łđ?&#x2018;Ł1 â&#x2C6;&#x2019; đ?&#x2018;&#x161;đ?&#x2018;&#x161;â &#x201E;(3 â&#x2C6;&#x161;đ?&#x2018;Łđ?&#x2018;Ł1 ), which has to be multiplied by â&#x2C6;&#x2019; 1â &#x201E;2 Âą â&#x2C6;&#x161;3 đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; â &#x201E;2, respectively, to obtain the whole set of three solutions. The other value, đ?&#x2018;Łđ?&#x2018;Ł2 , will yield the same set as đ?&#x2018;Łđ?&#x2018;Ł1 .

2.5 Vièteâ&#x20AC;&#x2122;s trigonometric solution

Starting with the depressed cubic, đ?&#x2018;Ľđ?&#x2018;Ľ 3 + đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; + đ?&#x2018;&#x2122;đ?&#x2018;&#x2122; = 0, let us set đ?&#x2018;Ľđ?&#x2018;Ľ = đ?&#x2018;˘đ?&#x2018;˘ cos đ?&#x153;&#x192;đ?&#x153;&#x192;, to make the equation coincide with the identity 4(cos đ?&#x153;&#x192;đ?&#x153;&#x192;)3 â&#x2C6;&#x2019; 3 cos đ?&#x153;&#x192;đ?&#x153;&#x192; â&#x2C6;&#x2019; cos(3đ?&#x153;&#x192;đ?&#x153;&#x192;) = 0. Then, we get đ?&#x2018;˘đ?&#x2018;˘3 (cos đ?&#x153;&#x192;đ?&#x153;&#x192;)3 + đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161;đ?&#x2018;&#x161; cos đ?&#x153;&#x192;đ?&#x153;&#x192; + đ?&#x2018;&#x2122;đ?&#x2018;&#x2122; = 0

Now, we choose đ?&#x2018;˘đ?&#x2018;˘3 = 4 and divide the equation by đ?&#x2018;˘đ?&#x2018;˘3 â &#x201E;4, which gives 4(cos đ?&#x153;&#x192;đ?&#x153;&#x192;)3 +

4đ?&#x2018;&#x161;đ?&#x2018;&#x161; 4đ?&#x2018;&#x2122;đ?&#x2018;&#x2122; cos đ?&#x153;&#x192;đ?&#x153;&#x192; + 3 = 0 2 đ?&#x2018;˘đ?&#x2018;˘ đ?&#x2018;˘đ?&#x2018;˘

Next, we make 4đ?&#x2018;&#x161;đ?&#x2018;&#x161;â &#x201E;đ?&#x2018;˘đ?&#x2018;˘2 = â&#x2C6;&#x2019;3 and 4đ?&#x2018;&#x2122;đ?&#x2018;&#x2122; â &#x201E;đ?&#x2018;˘đ?&#x2018;˘3 = cos(3đ?&#x153;&#x192;đ?&#x153;&#x192;). These two conditions lead to đ?&#x2018;˘đ?&#x2018;˘ = 2ďż˝â&#x2C6;&#x2019;

and the solutions are

đ?&#x2018;Ľđ?&#x2018;Ľđ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; = 2ďż˝â&#x2C6;&#x2019;

cos(3đ?&#x153;&#x192;đ?&#x153;&#x192;) =

đ?&#x2018;&#x161;đ?&#x2018;&#x161; 3

3 3đ?&#x2018;&#x2122;đ?&#x2018;&#x2122; ďż˝â&#x2C6;&#x2019; đ?&#x2018;&#x161;đ?&#x2018;&#x161; 2đ?&#x2018;&#x161;đ?&#x2018;&#x161;

đ?&#x2018;&#x161;đ?&#x2018;&#x161; 1 3đ?&#x2018;&#x2122;đ?&#x2018;&#x2122; 3 2đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039;đ?&#x153;&#x2039; ďż˝â&#x2C6;&#x2019; ďż˝ + cos ďż˝ cos â&#x2C6;&#x2019;1 ďż˝ ďż˝, 3 3 2đ?&#x2018;&#x161;đ?&#x2018;&#x161; đ?&#x2018;&#x161;đ?&#x2018;&#x161; 3

đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; = 0,1,2

Many years later, RenĂŠ Descartes made a geometric approach to Vièteâ&#x20AC;&#x2122;s trigonometric solution [2].

3. Quartic equation I'm bringing here a slight variation of the method due to RenĂŠ Descartes and improved later by Leonhard Euler. The point is that the solution can be reached in a simpler way if we transform the depressed quartic into a difference of squares instead of a product of two quadratic polynomials đ?&#x2018;Śđ?&#x2018;Ś 4 + đ?&#x2018;?đ?&#x2018;?đ?&#x2018;Śđ?&#x2018;Ś 2 + đ?&#x2018;&#x17E;đ?&#x2018;&#x17E;đ?&#x2018;&#x17E;đ?&#x2018;&#x17E; + đ?&#x2018;&#x;đ?&#x2018;&#x; = 0

(đ?&#x2018;Śđ?&#x2018;Ś 2 + đ??´đ??´)2 â&#x2C6;&#x2019; (đ??ľđ??ľđ??ľđ??ľ + đ??śđ??ś)2 = 0, 69


which gives the two quadratic equations đ?&#x2018;Śđ?&#x2018;Ś1 2 + đ??ľđ??ľđ?&#x2018;Śđ?&#x2018;Ś1 + (đ??´đ??´ + đ??śđ??ś) = 0, where đ?&#x2018;Śđ?&#x2018;Ś11 + đ?&#x2018;Śđ?&#x2018;Ś12 = đ??ľđ??ľ

đ?&#x2018;Śđ?&#x2018;Ś2 2 â&#x2C6;&#x2019; đ??ľđ??ľđ?&#x2018;Śđ?&#x2018;Ś2 + (đ??´đ??´ â&#x2C6;&#x2019; đ??śđ??ś) = 0, where đ?&#x2018;Śđ?&#x2018;Ś21 + đ?&#x2018;Śđ?&#x2018;Ś22 = đ??ľđ??ľ

Thus, (đ?&#x2018;Śđ?&#x2018;Ś11 + đ?&#x2018;Śđ?&#x2018;Ś12 )(đ?&#x2018;Śđ?&#x2018;Ś21 + đ?&#x2018;Śđ?&#x2018;Ś22 ) = â&#x2C6;&#x2019;đ??ľđ??ľ 2

The resolvent cubic is

and the other coefficients are

đ??´đ??´3 â&#x2C6;&#x2019;

đ??ľđ??ľ =

đ?&#x2018;?đ?&#x2018;? 2 1 đ??´đ??´ â&#x2C6;&#x2019; đ?&#x2018;&#x;đ?&#x2018;&#x;đ?&#x2018;&#x;đ?&#x2018;&#x; â&#x2C6;&#x2019; (đ?&#x2018;&#x17E;đ?&#x2018;&#x17E; 2 â&#x2C6;&#x2019; 4đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?) = 0, 2 8 ďż˝2đ??´đ??´ â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;? ; 2

đ??śđ??ś =

The rest of the formulation coincides with Euler's.

đ?&#x2018;&#x17E;đ?&#x2018;&#x17E;

2ďż˝2đ??´đ??´ â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?

4. Concluding Remarks and Directions for Future Research The aim of this work has been to bring together in a common frame two close related topics of algebra hoping that by doing this both will gain in perspective and clarity. Additionally, I have also provided a curious alternative to an ancient and, possibly, already forgotten method due to a great pre-classical mathematician. Although it is a quite hard task to add new insights on an old subject like the solution of algebraic equations, the reward may be expected on the side of making it more accessible and suggestive to the young students of mathematics.

Acknowledgements I would like to acknowledge in the first place the Editorial Committee of Polygon for accepting this work for publication. I also thank very sincerely Professor Rene Barrientos and Dr. M. Shakil for their warm and open support. Finally, a mention is deserved by the Department Mathematics, Miami Dade College, Hialeah Campus, which, giving me the opportunity to teach, has made all this work possible.

References [1] R. W. D. Nickalls, â&#x20AC;&#x153;A new approach to solving the cubic: Cardanâ&#x20AC;&#x2122;s solution revealedâ&#x20AC;?, The Mathematical Gazette (1993), 77 (November), 354â&#x20AC;&#x201C;359. 70


[2] R. W. D. Nickalls, â&#x20AC;&#x153;Viète, Descartes, and the cubic equationâ&#x20AC;?, The Mathematical Gazette (2006), 90 (July, No. 518), 203â&#x20AC;&#x201C;208.

. Elliptic integrals, geometric analogies Dr. Mario Duran Camejo, Ph.D. Department of Liberal Arts and Sciences - Mathematics Miami Dade College, Hialeah Campus E-mail: mduranca@mdc.edu Abstract This paper proposes an approach to the elliptic integrals of the first and second kinds from an alternative geometric point of view. This approach also suggests some transformations to be used in solving other different integrals.

Keywords: Elliptic integral; Geometry; Integration. 2010 Mathematics Subject Classification: 33E05.

1. Introduction "In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler (c.1750). Modern mathematics defines an elliptic integral as any function đ?&#x2018;&#x201C;đ?&#x2018;&#x201C; which can be expressed in the form đ?&#x2018;Ľđ?&#x2018;Ľ

đ?&#x2018;&#x201C;đ?&#x2018;&#x201C;(đ?&#x2018;Ľđ?&#x2018;Ľ) = ďż˝ đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; ďż˝đ?&#x2018;Ąđ?&#x2018;Ą, ďż˝đ?&#x2018;&#x192;đ?&#x2018;&#x192;(đ?&#x2018;Ąđ?&#x2018;Ą)ďż˝ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;, đ?&#x2018;?đ?&#x2018;?

where đ?&#x2018;&#x2026;đ?&#x2018;&#x2026; is a rational function of its two arguments, đ?&#x2018;&#x192;đ?&#x2018;&#x192; is a polynomial of degree 3 or 4 with no repeated roots, and đ?&#x2018;?đ?&#x2018;? is a constant. In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when đ?&#x2018;&#x192;đ?&#x2018;&#x192; has repeated roots, or when đ?&#x2018;&#x2026;đ?&#x2018;&#x2026;(đ?&#x2018;Ľđ?&#x2018;Ľ, đ?&#x2018;Śđ?&#x2018;Ś) contains no odd powers of đ?&#x2018;Śđ?&#x2018;Ś." I have borrowed the previous introduction from Wikipedia. Some subtle additions could be made, however, as including the name of John Wallis who, according to [1], would have 71


encountered the elliptic integrals in 1655-59, before G. Fagnano were born. It is important as well to emphasize the valuable contribution of A. M. Legendre in finding a better form to deal with the integration of the rational functions. In other order of things, the aforementioned book also introduces đ?&#x2018;&#x192;đ?&#x2018;&#x192;(đ?&#x2018;Ąđ?&#x2018;Ą), in a more general way, as an đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;th-degree polynomial. After some explanations, the degree is specified depending on which of the two forms is used to write down the integrals, either Weiertrass's form, where grade is 3, or Legendre's form, where it is 4. In addition to these two forms, the book makes a brief reference to a third form, suggested by A. R. Low in 1950. In more recent times, it has appeared a fourth form of expressing the elliptic integrals, proposed by B. C. Carlson (1993) [2], which is equivalent to Legendre's, and is called the Carlson symmetric form. This last form is rather similar to Low's. Here, I'm proposing some alternative approaches to the elliptic integral of the first and second kinds in Legendre's form, based on geometrical analogies, and using them to solve other integrals involving and not involving radicals.

2. A first geometrical approach Although in the elliptic integrals of the first and third kinds in Legendre's form the fourth-degree polynomial appears very clearly in the denominator, in the expression for the integral of the second kind this fact is not so obvious. However, since the integral of the second kind in Legendre's form, called đ??¸đ??¸, is defined by: đ??¸đ??¸ = đ??źđ??ź1 â&#x2C6;&#x2019; 2đ??źđ??ź2 , đ??˝đ??˝2 đ??źđ??ź2 = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;0 , đ??źđ??ź1 = đ??˝đ??˝0 , 2 đ?&#x2018;Ľđ?&#x2018;Ľ đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ??˝đ??˝đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; = ďż˝ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; , đ?&#x2018;Śđ?&#x2018;Ś

where đ?&#x2018;Śđ?&#x2018;Ś is the fourth-degree polynomial given by

đ?&#x2018;Śđ?&#x2018;Ś 2 = đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; 2 đ?&#x2018;Ľđ?&#x2018;Ľ 4 â&#x2C6;&#x2019; (1 + đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; 2 )đ?&#x2018;Ľđ?&#x2018;Ľ 2 + 1 ,

we can understand that, because both đ??źđ??ź1 and đ??źđ??ź2 contain đ?&#x2018;Śđ?&#x2018;Ś, đ??¸đ??¸ must contain it as well, although not explicitly, đ??¸đ??¸ = ďż˝ đ?&#x2018;Ąđ?&#x2018;Ą

đ?&#x2018;Ľđ?&#x2018;Ľ

= đ??˝đ??˝0 â&#x2C6;&#x2019; đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;0 đ??˝đ??˝2 ,

â&#x2C6;&#x161;1 â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; 2 đ?&#x2018;Ąđ?&#x2018;Ą 2 â&#x2C6;&#x161;1 â&#x2C6;&#x2019; đ?&#x2018;Ąđ?&#x2018;Ą 2

đ?&#x2018;Ľđ?&#x2018;Ľ

đ?&#x2018;Ľđ?&#x2018;Ľ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;Ąđ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; 2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; = ďż˝ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; = ďż˝ â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; ďż˝ 2 2 2 đ?&#x2018;Ąđ?&#x2018;Ą â&#x2C6;&#x161;1 â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; đ?&#x2018;Ąđ?&#x2018;Ą â&#x2C6;&#x161;1 â&#x2C6;&#x2019; đ?&#x2018;Ąđ?&#x2018;Ą đ?&#x2018;Ąđ?&#x2018;Ą đ?&#x2018;Śđ?&#x2018;Ś(đ?&#x2018;Ąđ?&#x2018;Ą) đ?&#x2018;Ąđ?&#x2018;Ą đ?&#x2018;Śđ?&#x2018;Ś(đ?&#x2018;Ąđ?&#x2018;Ą)

1 â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; 2 đ?&#x2018;Ąđ?&#x2018;Ą 2

đ?&#x2018;Ľđ?&#x2018;Ľ

because đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;0 = đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; 2 is the coefficient of đ?&#x2018;Ľđ?&#x2018;Ľ 4 . Equivalently, in trigoometric form, đ?&#x153;&#x2018;đ?&#x153;&#x2018;

đ??¸đ??¸ = ďż˝ ďż˝1 â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; 2 (sin đ?&#x203A;źđ?&#x203A;ź)2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;. 0

72

(1)


Now, making the substitutions đ?&#x203A;źđ?&#x203A;ź = 2đ?&#x153;&#x192;đ?&#x153;&#x192;; (sin đ?&#x203A;źđ?&#x203A;ź)2 = (1 â&#x2C6;&#x2019; cos đ?&#x153;&#x192;đ?&#x153;&#x192;)â &#x201E;2 ; â&#x2C6;&#x2026; = 2đ?&#x153;&#x2018;đ?&#x153;&#x2018;; đ?&#x2018;?đ?&#x2018;?2 = đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; 2 â &#x201E;2 , the last expression becomes â&#x2C6;&#x2026;

đ??¸đ??¸ = 2 ďż˝ ďż˝1 â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?2 + đ?&#x2018;?đ?&#x2018;?2 cos đ?&#x153;&#x192;đ?&#x153;&#x192; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; 0

If we assume that 1 â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?2 + đ?&#x2018;?đ?&#x2018;?2 cos đ?&#x153;&#x192;đ?&#x153;&#x192; (where 1 â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?2 > 0) represents the square of the distance đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; in the đ?&#x2018;Ľđ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x2018;Śđ?&#x2018;Ś plane, it may be depicted as,

whence đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; = ďż˝1 â&#x2C6;&#x2019; â&#x2C6;&#x161;1 â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; 2 ďż˝â &#x201E;2 , đ?&#x2018;?đ?&#x2018;? = ďż˝1 + â&#x2C6;&#x161;1 â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; 2 ďż˝â &#x201E;2 (or viceversa), and the integral becomes â&#x2C6;&#x2026;

đ??¸đ??¸ = 2 ďż˝ ďż˝đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;2 + đ?&#x2018;?đ?&#x2018;? 2 + 2đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; cos đ?&#x153;&#x192;đ?&#x153;&#x192; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;, 0

but đ?&#x2018;?đ?&#x2018;? cos đ?&#x153;&#x192;đ?&#x153;&#x192; = đ?&#x2018;Ľđ?&#x2018;Ľ and đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; = â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;â &#x201E;đ?&#x2018;Śđ?&#x2018;Ś = â&#x2C6;&#x2019; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;Ľđ?&#x2018;Ľâ &#x201E;â&#x2C6;&#x161;đ?&#x2018;?đ?&#x2018;? 2 â&#x2C6;&#x2019; đ?&#x2018;Ľđ?&#x2018;Ľ 2 . Then, đ??¸đ??¸ = â&#x2C6;&#x2019;2 ďż˝

đ?&#x2018;?đ?&#x2018;? cos â&#x2C6;&#x2026;

đ?&#x2018;?đ?&#x2018;?

đ??¸đ??¸ = â&#x2C6;&#x2019;2â&#x2C6;&#x161;2đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; ďż˝

đ?&#x2018;?đ?&#x2018;? đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2026;

đ?&#x2018;?đ?&#x2018;?

â&#x2C6;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;2 + đ?&#x2018;?đ?&#x2018;? 2 + 2đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x161;đ?&#x2018;?đ?&#x2018;? 2 â&#x2C6;&#x2019; đ?&#x2018;Ľđ?&#x2018;Ľ 2

â&#x2C6;&#x161;đ??žđ??ž + đ?&#x2018;Ľđ?&#x2018;Ľ

â&#x2C6;&#x161;đ?&#x2018;?đ?&#x2018;? 2 â&#x2C6;&#x2019; đ?&#x2018;Ľđ?&#x2018;Ľ 2

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; ,

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; (2)

where đ??žđ??ž = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;2 + đ?&#x2018;?đ?&#x2018;? 2 )â &#x201E;(2đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;). With the change of variable đ?&#x2018;Ľđ?&#x2018;Ľ = 1â &#x201E;đ?&#x2018;Łđ?&#x2018;Ł â&#x2C6;&#x2019; đ??žđ??ž , the integral transforms into

73


đ??¸đ??¸ = 2â&#x2C6;&#x161;2 đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;

đ??¸đ??¸ = 2â&#x2C6;&#x161;2 đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;

1 đ?&#x2018;?đ?&#x2018;? cos â&#x2C6;&#x2026;+đ??žđ??ž

1 đ?&#x2018;?đ?&#x2018;? cos â&#x2C6;&#x2026;+đ??žđ??ž

ďż˝

1 đ?&#x2018;?đ?&#x2018;?+đ??žđ??ž

ďż˝

1 đ?&#x2018;?đ?&#x2018;?+đ??žđ??ž

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

đ?&#x2018;Łđ?&#x2018;Ł 2 ďż˝(đ?&#x2018;?đ?&#x2018;? 2 â&#x2C6;&#x2019; đ??žđ??ž 2 ) đ?&#x2018;Łđ?&#x2018;Ł + 2đ??žđ??ž â&#x2C6;&#x2019; 1â &#x201E;đ?&#x2018;Łđ?&#x2018;Ł đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

ďż˝(đ?&#x2018;?đ?&#x2018;? 2 â&#x2C6;&#x2019; đ??žđ??ž 2 ) đ?&#x2018;Łđ?&#x2018;Ł 5 + 2đ??žđ??žđ?&#x2018;Łđ?&#x2018;Ł 4 â&#x2C6;&#x2019; đ?&#x2018;Łđ?&#x2018;Ł 3

.

(3)

According to the expressions for a and b above, it can be proven that đ?&#x2018;?đ?&#x2018;? 2 â&#x2C6;&#x2019; đ??žđ??ž 2 < 0 regardless of how đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; and đ?&#x2018;?đ?&#x2018;? are chosen. Hence, in this case, the terms of third and fifth order will be always negative. Therefore, solutions will only exist if 2đ??žđ??žđ??žđ??ž < (đ??žđ??ž 2 â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;? 2 )đ?&#x2018;Łđ?&#x2018;Ł 2 + 1 (*), which implies 1â &#x201E;(đ??žđ??ž + đ?&#x2018;?đ?&#x2018;?) < đ?&#x2018;Łđ?&#x2018;Ł < 1â &#x201E;(đ??žđ??ž â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?). This condition is always attainable because đ?&#x2018;?đ?&#x2018;? â&#x2030;Ľ đ?&#x2018;Ľđ?&#x2018;Ľ â&#x2030;Ľ 0 and, consequently, 1â &#x201E;(đ??žđ??ž + đ?&#x2018;?đ?&#x2018;?) < đ?&#x2018;Łđ?&#x2018;Ł < 1â &#x201E;đ??žđ??ž . So the elliptic integral of the second kind in Legendre's form is a solution of đ?&#x2018;Ľđ?&#x2018;Ľ

ďż˝

đ?&#x2018;Ľđ?&#x2018;Ľ0

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

â&#x2C6;&#x161;đ??´đ??´đ?&#x2018;§đ?&#x2018;§ 5 + đ??ľđ??ľđ?&#x2018;§đ?&#x2018;§ 4 â&#x2C6;&#x2019; đ?&#x2018;§đ?&#x2018;§ 3

Notice that đ?&#x2018;Ľđ?&#x2018;Ľ0 cannot be arbitrary, rather it is related to đ??´đ??´ and đ??ľđ??ľ.

74


----------------------------------------------------------------------------------------------------------(*) In fact, the condition would be 2đ??žđ??žđ?&#x2018;Łđ?&#x2018;Ł 4 > |đ?&#x2018;?đ?&#x2018;? 2 â&#x2C6;&#x2019; đ??žđ??ž 2 |đ?&#x2018;Łđ?&#x2018;Ł 5 + đ?&#x2018;Łđ?&#x2018;Ł 3 but, since đ?&#x2018;Ľđ?&#x2018;Ľ > 0, đ?&#x2018;Łđ?&#x2018;Ł > 0, we divide the whole expression by đ?&#x2018;Łđ?&#x2018;Ł 3 . Also notice that đ??žđ??ž 2 â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;? 2 = |đ?&#x2018;?đ?&#x2018;? 2 â&#x2C6;&#x2019; đ??žđ??ž 2 |.

3. A second geometrical approach: The Jacobi's form

The scheme utilized before also leads to the Jacobi's form of the elliptic integral of the second kind. It is enough to realize that any distance đ?&#x2018;&#x201A;đ?&#x2018;&#x201A;đ?&#x2018;&#x201A;đ?&#x2018;&#x201A;â&#x20AC;˛ can also be seen as a projection đ?&#x2018;?đ?&#x2018;? cos đ?&#x153;&#x192;đ?&#x153;&#x192;, đ?&#x2018;?đ?&#x2018;? corresponding to the angle đ?&#x153;&#x192;đ?&#x153;&#x192; = 0. The coordinate đ?&#x2018;Ľđ?&#x2018;Ľâ&#x20AC;˛ of đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;â&#x20AC;˛ is constant and equal to đ?&#x2018;?đ?&#x2018;? cos đ?&#x153;&#x192;đ?&#x153;&#x192;. This is depicted in the figure below.

Now, we'll simply use (sin đ?&#x203A;źđ?&#x203A;ź)2 = 1 â&#x2C6;&#x2019; (cos đ?&#x203A;źđ?&#x203A;ź)2 in (1), obtaining đ?&#x153;&#x2018;đ?&#x153;&#x2018;

On the other hand,

đ??¸đ??¸ = ďż˝ ďż˝1 â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; 2 + đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; 2 (cos đ?&#x203A;źđ?&#x203A;ź)2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; 0

đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; â&#x20AC;˛ = ďż˝đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;2 + đ?&#x2018;?đ?&#x2018;? 2 (cos đ?&#x153;&#x192;đ?&#x153;&#x192;)2 â&#x2C6;&#x2019; 2đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; cos đ?&#x153;&#x192;đ?&#x153;&#x192; cos(180 â&#x2C6;&#x2019; đ?&#x153;&#x192;đ?&#x153;&#x192;â&#x20AC;˛) = ďż˝đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;2 + đ?&#x2018;?đ?&#x2018;? 2 (cos đ?&#x153;&#x192;đ?&#x153;&#x192;)2 + 2đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; cos đ?&#x153;&#x192;đ?&#x153;&#x192; cos đ?&#x153;&#x192;đ?&#x153;&#x192;â&#x20AC;˛

= ďż˝đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;2 + 2đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;?đ?&#x2018;? đ?&#x153;&#x2018;đ?&#x153;&#x2018; + đ?&#x2018;?đ?&#x2018;? 2 (cos đ?&#x153;&#x192;đ?&#x153;&#x192;)2 ,

whence, đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; = â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; cos đ?&#x153;&#x2018;đ?&#x153;&#x2018; + ďż˝1 â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; 2 (sin đ?&#x153;&#x2018;đ?&#x153;&#x2018;)2 and đ?&#x2018;?đ?&#x2018;? = đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;. It may be shown that đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; > 0 for 0 < đ?&#x153;&#x2018;đ?&#x153;&#x2018; < đ?&#x153;&#x2039;đ?&#x153;&#x2039;â &#x201E;2 and 0 < đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; < 1. Then, 75


đ?&#x153;&#x2018;đ?&#x153;&#x2018;

đ??¸đ??¸ = ďż˝ ďż˝đ??žđ??ž + đ?&#x2018;?đ?&#x2018;? 2 (cos đ?&#x153;&#x192;đ?&#x153;&#x192;)2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; 0

= â&#x2C6;&#x2019;ďż˝

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

0

â&#x2C6;&#x161;đ??žđ??ž + đ?&#x2018;Ľđ?&#x2018;Ľ 2

â&#x2C6;&#x161;đ?&#x2018;?đ?&#x2018;? 2 â&#x2C6;&#x2019; đ?&#x2018;Ľđ?&#x2018;Ľ 2

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;,

(4)

which is equivalent to the Jacobi's form of the elliptic integral of the second kind, where đ??žđ??ž = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;2 + 2đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; cos đ?&#x153;&#x2018;đ?&#x153;&#x2018;. Making the substitutions đ??žđ??ž + đ?&#x2018;Ľđ?&#x2018;Ľ 2 = 1â &#x201E;đ?&#x2018;Łđ?&#x2018;Ł 2 , đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; = â&#x2C6;&#x2019;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;â &#x201E;ďż˝đ?&#x2018;Łđ?&#x2018;Ł 2 â&#x2C6;&#x161;1 â&#x2C6;&#x2019; đ??žđ??žđ?&#x2018;Łđ?&#x2018;Ł 2 ďż˝, we arrive to đ??¸đ??¸ =

1 ďż˝ 2 (cos 2 đ?&#x153;&#x2018;đ?&#x153;&#x2018;) +đ??žđ??ž đ?&#x2018;?đ?&#x2018;?

ďż˝

1 ďż˝ 2 đ?&#x2018;?đ?&#x2018;? +đ??žđ??ž

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

đ?&#x2018;Łđ?&#x2018;Ł 2 â&#x2C6;&#x161;đ??´đ??´đ?&#x2018;Łđ?&#x2018;Ł 2 â&#x2C6;&#x2019; 1â&#x2C6;&#x161;1 â&#x2C6;&#x2019; đ??ľđ??ľđ?&#x2018;Łđ?&#x2018;Ł 2

,

(5)

where đ??´đ??´ = đ?&#x2018;?đ?&#x2018;? 2 + đ??žđ??ž and đ??ľđ??ľ = đ??žđ??ž. Then, the elliptic integral of the second kind in Legendre's form is also a solution of đ?&#x2018;Ľđ?&#x2018;Ľ

ďż˝

đ?&#x2018;Ľđ?&#x2018;Ľ0

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

đ?&#x2018;§đ?&#x2018;§ 2 â&#x2C6;&#x161;đ??´đ??´đ?&#x2018;§đ?&#x2018;§ 2 â&#x2C6;&#x2019; 1â&#x2C6;&#x161;1 â&#x2C6;&#x2019; đ??ľđ??ľđ?&#x2018;§đ?&#x2018;§ 2

Here it applies the same as in section 1 regarding the value of đ?&#x2018;Ľđ?&#x2018;Ľ0 . This integral greatly resembles the elliptic integral of the third kind in Jacobi's form.

4. Applying other substitutions and integrating by parts The approach given by (2) can be extended to the solution of other integrals by applying different substitutions and, then, integrating by parts the resulting expressions. In general, we can have the substitution đ?&#x2018;Łđ?&#x2018;Ł = (đ??žđ??ž + đ?&#x2018;Ľđ?&#x2018;Ľ đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; )đ?&#x2018;&#x17E;đ?&#x2018;&#x17E; , where đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; = [1, 2] and đ?&#x2018;&#x17E;đ?&#x2018;&#x17E; can be either an integer or a fraction. Let us see some examples. 4.1 Example 1 Using the substitution with đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; = 1 and đ?&#x2018;&#x17E;đ?&#x2018;&#x17E; = 1, we get đ?&#x2018;Ľđ?&#x2018;Ľ

đ??¸đ??¸ďż˝ = ďż˝ đ?&#x2018;Ľđ?&#x2018;Ľ0

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

â&#x2C6;&#x161;â&#x2C6;&#x2019;đ?&#x2018;§đ?&#x2018;§ 3 + đ??ľđ??ľđ?&#x2018;§đ?&#x2018;§ 2 + đ??śđ??śđ??śđ??ś

where đ??¸đ??¸ďż˝ indicates the elliptic integral affected by some numerical factor. 76


Now, suppose we are interested in solving the integral in the RHS. Integrating by parts, we obtain đ?&#x2018;Ľđ?&#x2018;Ľ

đ??¸đ??¸ďż˝ = đ?&#x2018;Ľđ?&#x2018;Ľ ďż˝ đ?&#x2018;Ľđ?&#x2018;Ľ0

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

â&#x2C6;&#x161;â&#x2C6;&#x2019;đ?&#x2018;§đ?&#x2018;§ 3 + đ??ľđ??ľđ?&#x2018;§đ?&#x2018;§ 2 + đ??śđ??śđ??śđ??ś

đ?&#x2018;Ľđ?&#x2018;Ľ

đ?&#x2018;§đ?&#x2018;§

đ?&#x2018;Ľđ?&#x2018;Ľ0

đ?&#x2018;Ľđ?&#x2018;Ľ0

â&#x2C6;&#x2019; ���

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

â&#x2C6;&#x161;â&#x2C6;&#x2019;đ?&#x2018;¤đ?&#x2018;¤ 3 + đ??ľđ??ľđ?&#x2018;¤đ?&#x2018;¤ 2 + đ??śđ??śđ??śđ??ś

ďż˝ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;,

which may be written as a linear differential equation đ??¸đ??¸ďż˝ = đ?&#x2018;Ľđ?&#x2018;Ľ

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; â&#x2C6;&#x2019; đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; , đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

whose exact solution (for đ?&#x2018;Ľđ?&#x2018;Ľ â&#x2030; 0) is the integral we want to solve.

4.2 Example 2

Using the substitution with đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; = 1 and đ?&#x2018;&#x17E;đ?&#x2018;&#x17E; = 1â &#x201E;2, we get đ?&#x2018;Ľđ?&#x2018;Ľ

đ??¸đ??¸ďż˝ = ďż˝ đ?&#x2018;Ľđ?&#x2018;Ľ0

Integrating by parts, we obtain đ?&#x2018;Ľđ?&#x2018;Ľ

đ??¸đ??¸ďż˝ = đ?&#x2018;Ľđ?&#x2018;Ľ ďż˝ but we know

đ?&#x2018;Ľđ?&#x2018;Ľ0

â&#x2C6;&#x161;â&#x2C6;&#x2019;đ?&#x2018;§đ?&#x2018;§ 4 + đ??ľđ??ľđ?&#x2018;§đ?&#x2018;§ 2 + đ??śđ??ś

đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§

â&#x2C6;&#x161;â&#x2C6;&#x2019;đ?&#x2018;§đ?&#x2018;§ 4 + đ??ľđ??ľđ?&#x2018;§đ?&#x2018;§ 2 + đ??śđ??ś đ?&#x2018;Ľđ?&#x2018;Ľ

ďż˝

đ?&#x2018;Ľđ?&#x2018;Ľ0

đ?&#x2018;§đ?&#x2018;§ 2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

đ?&#x2018;Ľđ?&#x2018;Ľ

đ?&#x2018;§đ?&#x2018;§

đ?&#x2018;Ľđ?&#x2018;Ľ0

đ?&#x2018;Ľđ?&#x2018;Ľ0

â&#x2C6;&#x2019; ���

đ?&#x2018;§đ?&#x2018;§ 2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

â&#x2C6;&#x161;â&#x2C6;&#x2019;đ?&#x2018;§đ?&#x2018;§ 4 + đ??ľđ??ľđ?&#x2018;§đ?&#x2018;§ 2 + đ??śđ??ś

=

where đ?&#x2018;?đ?&#x2018;? = â&#x2C6;&#x2019; đ??ľđ??ľâ &#x201E;2, and đ?&#x2018;&#x17E;đ?&#x2018;&#x17E; = đ??śđ??ś 2 + đ?&#x2018;?đ?&#x2018;?2 . So đ?&#x2018;Ľđ?&#x2018;Ľ

ďż˝ sin

đ?&#x2018;Ľđ?&#x2018;Ľ0

â&#x2C6;&#x2019;1

đ?&#x2018;¤đ?&#x2018;¤đ?&#x2018;¤đ?&#x2018;¤đ?&#x2018;¤đ?&#x2018;¤

â&#x2C6;&#x161;â&#x2C6;&#x2019;đ?&#x2018;¤đ?&#x2018;¤ 4 + đ??ľđ??ľđ?&#x2018;¤đ?&#x2018;¤ 2 + đ??śđ??ś

ďż˝ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;,

1 â&#x2C6;&#x2019;1 đ?&#x2018;Ľđ?&#x2018;Ľ 2 + đ?&#x2018;?đ?&#x2018;? sin ďż˝ ďż˝, 2 đ?&#x2018;&#x17E;đ?&#x2018;&#x17E;

đ?&#x2018;§đ?&#x2018;§ 2 + đ?&#x2018;?đ?&#x2018;? đ?&#x2018;Ľđ?&#x2018;Ľ â&#x2C6;&#x2019;1 đ?&#x2018;Ľđ?&#x2018;Ľ 2 + đ?&#x2018;?đ?&#x2018;? ďż˝ ďż˝ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; = sin ďż˝ ďż˝ â&#x2C6;&#x2019; đ??¸đ??¸ďż˝ . đ?&#x2018;&#x17E;đ?&#x2018;&#x17E; 2 đ?&#x2018;&#x17E;đ?&#x2018;&#x17E;

We have obtained the integral of an inverse trigonometric function with a quadratic argument. 4.3 Example 3 In this case we will keep (2) as it is and proceed as follows 77


đ??¸đ??¸ďż˝ = ďż˝

đ??¸đ??¸ďż˝ = ďż˝

đ?&#x2018;Ľđ?&#x2018;Ľ0 â&#x2C6;&#x161;đ?&#x2018;?đ?&#x2018;?

2

đ??žđ??žđ??žđ??žđ??žđ??ž

â&#x2C6;&#x161;đ??žđ??ž + đ?&#x2018;§đ?&#x2018;§

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

đ?&#x2018;Ľđ?&#x2018;Ľ0 â&#x2C6;&#x161;đ?&#x2018;?đ?&#x2018;?

2

â&#x2C6;&#x2019; đ?&#x2018;§đ?&#x2018;§ 2

đ?&#x2018;Ľđ?&#x2018;Ľ0 â&#x2C6;&#x161;đ?&#x2018;?đ?&#x2018;?

2

â&#x2C6;&#x2019; đ?&#x2018;§đ?&#x2018;§ 2 (đ??žđ??ž + đ?&#x2018;§đ?&#x2018;§)

= ďż˝

đ?&#x2018;Ľđ?&#x2018;Ľ

đ?&#x2018;Ľđ?&#x2018;Ľ

đ?&#x2018;Ľđ?&#x2018;Ľ

â&#x2C6;&#x2019; đ?&#x2018;§đ?&#x2018;§ 2 (đ??žđ??ž + đ?&#x2018;§đ?&#x2018;§)

đ??žđ??ž + đ?&#x2018;§đ?&#x2018;§

+ ďż˝

đ?&#x2018;Ľđ?&#x2018;Ľ

đ?&#x2018;Ľđ?&#x2018;Ľ0 â&#x2C6;&#x161;đ?&#x2018;?đ?&#x2018;?

Integrating by parts the second term in the RHS gives đ??¸đ??¸ďż˝ = ďż˝

đ?&#x2018;Ľđ?&#x2018;Ľ

đ?&#x2018;Ľđ?&#x2018;Ľ0 â&#x2C6;&#x161;đ?&#x2018;?đ?&#x2018;?

đ??žđ??žđ??žđ??žđ??žđ??ž

â&#x2C6;&#x2019; đ?&#x2018;§đ?&#x2018;§ 2 (đ??žđ??ž + đ?&#x2018;§đ?&#x2018;§)

2

đ?&#x2018;Ľđ?&#x2018;Ľ0

ďż˝đ?&#x2018;?đ?&#x2018;? 2 â&#x2C6;&#x2019; (đ?&#x2018;Ľđ?&#x2018;Ľ0 )2 (đ??žđ??ž + đ?&#x2018;Ľđ?&#x2018;Ľ0 )

where đ??žđ??ž0 = â&#x2C6;&#x2019;

đ??¸đ??¸ďż˝ = ďż˝ đ?&#x2018;Ľđ?&#x2018;Ľ0

đ?&#x2018;Ľđ?&#x2018;Ľ

đ?&#x2018;Ľđ?&#x2018;Ľ0 â&#x2C6;&#x161;đ?&#x2018;?đ?&#x2018;?

(đ??žđ??ž â&#x2C6;&#x2019; 1)đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

2

ďż˝đ?&#x2018;?đ?&#x2018;? 2 â&#x2C6;&#x2019; (đ?&#x2018;Ľđ?&#x2018;Ľ0 )2 (đ??žđ??ž+đ?&#x2018;Ľđ?&#x2018;Ľ0 )

â&#x2C6;&#x2019; đ?&#x2018;§đ?&#x2018;§ 2 (đ??žđ??ž + đ?&#x2018;§đ?&#x2018;§)

+

đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§đ?&#x2018;§

â&#x2C6;&#x2019; đ?&#x2018;§đ?&#x2018;§ 2 (đ??žđ??ž + đ?&#x2018;§đ?&#x2018;§) đ?&#x2018;Ľđ?&#x2018;Ľ

â&#x2C6;&#x161;đ?&#x2018;?đ?&#x2018;? 2 â&#x2C6;&#x2019; đ?&#x2018;Ľđ?&#x2018;Ľ 2 (đ??žđ??ž + đ?&#x2018;Ľđ?&#x2018;Ľ)

â&#x2C6;&#x2019; ďż˝ +

2

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

đ?&#x2018;Ľđ?&#x2018;Ľ

đ?&#x2018;Ľđ?&#x2018;Ľ0 â&#x2C6;&#x161;đ?&#x2018;?đ?&#x2018;?

2

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

â&#x2C6;&#x2019; đ?&#x2018;§đ?&#x2018;§ 2 (đ??žđ??ž + đ?&#x2018;§đ?&#x2018;§) đ?&#x2018;Ľđ?&#x2018;Ľ

â&#x2C6;&#x161;đ?&#x2018;?đ?&#x2018;? 2 â&#x2C6;&#x2019; đ?&#x2018;Ľđ?&#x2018;Ľ 2 (đ??žđ??ž + đ?&#x2018;Ľđ?&#x2018;Ľ)

â&#x2C6;&#x2019; đ??žđ??ž0

. The last equation may be written as a differential equation đ??¸đ??¸ďż˝ = (đ??žđ??ž â&#x2C6;&#x2019; 1)đ?&#x2018;&#x201E;đ?&#x2018;&#x201E; + đ?&#x2018;Ľđ?&#x2018;Ľ

which allows us to find the integral

ďż˝

đ?&#x2018;Ľđ?&#x2018;Ľ

đ?&#x2018;Ľđ?&#x2018;Ľ0 â&#x2C6;&#x161;đ?&#x2018;?đ?&#x2018;?

2

đ??žđ??ž + đ?&#x2018;§đ?&#x2018;§

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; + đ??žđ??ž0 , đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

â&#x2C6;&#x2019; đ?&#x2018;§đ?&#x2018;§ 2 (đ??žđ??ž + đ?&#x2018;§đ?&#x2018;§)

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

5. A third geometrical approach: The crank-rod model The crank-rod model is one of the oldest, still in use, mechanisms. It is depicted below, where it was assumed that the length of the crank is đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; and the length of the rod is đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192;đ?&#x2018;&#x192; = đ?&#x2018;&#x192;đ?&#x2018;&#x192;â&#x20AC;˛đ?&#x2018;&#x201E;đ?&#x2018;&#x201E;â&#x20AC;˛ = 1.

78


Now, this model is modified in the same way we did beforehand for transforming the first geometrical approach into the second one.

Applying geometrical relations, we get, đ?&#x2019;&#x152;đ?&#x2019;&#x152; cos đ?&#x153;&#x192;đ?&#x153;&#x192; sin đ?&#x153;&#x192;đ?&#x153;&#x192;â&#x20AC;˛ = sin đ?&#x203A;˝đ?&#x203A;˝ â&#x2C6;&#x2019; đ?&#x2019;&#x152;đ?&#x2019;&#x152; sin đ?&#x153;&#x192;đ?&#x153;&#x192; sin đ?&#x153;&#x192;đ?&#x153;&#x192;â&#x20AC;˛ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; + đ?&#x2019;&#x152;đ?&#x2019;&#x152; cos đ?&#x153;&#x192;đ?&#x153;&#x192; cos đ?&#x153;&#x192;đ?&#x153;&#x192;â&#x20AC;˛ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;â&#x20AC;˛ = cos đ?&#x203A;˝đ?&#x203A;˝ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; â&#x2C6;&#x2019; đ?&#x2019;&#x152;đ?&#x2019;&#x152; sin đ?&#x153;&#x192;đ?&#x153;&#x192; sin đ?&#x153;&#x192;đ?&#x153;&#x192;â&#x20AC;˛ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; + đ?&#x2019;&#x152;đ?&#x2019;&#x152; cos đ?&#x153;&#x2018;đ?&#x153;&#x2018; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x153;&#x192;đ?&#x153;&#x192; â&#x20AC;˛ = cos đ?&#x203A;˝đ?&#x203A;˝ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; cos đ?&#x153;&#x2018;đ?&#x153;&#x2018; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;â&#x20AC;˛ đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; = â&#x2C6;&#x2019; cos đ?&#x203A;˝đ?&#x203A;˝ cos đ?&#x203A;˝đ?&#x203A;˝ sin đ?&#x153;&#x192;đ?&#x153;&#x192; sin đ?&#x153;&#x192;đ?&#x153;&#x192;â&#x20AC;˛ đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; sin đ?&#x153;&#x192;đ?&#x153;&#x192; sin đ?&#x153;&#x192;đ?&#x153;&#x192; â&#x20AC;˛

(6)

Doing some algebra, the LHS of (6) becomes the 'imaginary' integrand of the elliptic integral of the first kind. Effectively, 79


cos 𝛽𝛽 = �1 − (𝒌𝒌 cos 𝜃𝜃 sin 𝜃𝜃′)2

= �1 − (𝒌𝒌 cos 𝜃𝜃)2 [1 − (cos 𝜃𝜃′)2 ] = �1 + (𝒌𝒌 cos 𝜑𝜑)2 − (𝒌𝒌 cos 𝜃𝜃)2

= �[1 − (𝒌𝒌 sin 𝜑𝜑)2 ] + (𝒌𝒌 sin 𝜃𝜃)2

= �1 − (𝒌𝒌 sin 𝜑𝜑)2 �1 −

(𝒌𝒌 sin 𝜃𝜃)2 (𝒌𝒌 sin 𝜑𝜑)2 − 1

(7)

= 𝑲𝑲𝑖𝑖�1 − (𝑲𝑲𝑲𝑲 sin 𝜃𝜃 )2 for 𝑘𝑘 > 1⁄sin 𝜑𝜑

On the other hand, we have

sin 𝜃𝜃 ′ = = = =

but, from eq. (7),

𝑦𝑦′ 𝒌𝒌 cos 𝜃𝜃

�(𝒌𝒌 cos 𝜃𝜃)2 − 𝑥𝑥′2 𝒌𝒌 cos 𝜃𝜃

�(𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃)2 − (cos 𝜑𝜑)2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃

�(sin 𝜑𝜑)2 − (sin 𝜃𝜃)2 , 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃

𝒌𝒌 sin 𝜃𝜃 cos 𝜃𝜃 𝑑𝑑𝑑𝑑

�[1 + (𝒌𝒌 sin 𝜑𝜑)2 ] + (sin 𝜃𝜃)2

,

and, because sin 𝛽𝛽 = �1 − (cos 𝛽𝛽)2 = 𝒌𝒌�(sin 𝜑𝜑)2 − (sin 𝜃𝜃)2 , we obtain

𝑑𝑑𝑑𝑑 =

𝒌𝒌 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 𝑑𝑑𝑑𝑑

�(𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑)2 − (𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃)2 �[1 + (𝒌𝒌 𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑)2 ] + (𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃)2

80


With all that, the RHS of (6) becomes =

cos 𝜑𝜑 𝑑𝑑𝑑𝑑′

�(𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑)2 − (𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃)2 sin 𝜃𝜃 �[1 + (𝒌𝒌 𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑)2 ] + (𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃)2 cos 𝜃𝜃 𝒌𝒌 sin 𝜃𝜃 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 𝑑𝑑𝑑𝑑

�[1 + (𝒌𝒌 𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑)2 ] + (𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃)2 �(𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑)2 − (𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃)2 =

𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑 cot 𝜃𝜃 𝑑𝑑𝑑𝑑′

�(𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑)2 − (𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃)2 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃

�(𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑)2 − (𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃)2 �[1 + (𝒌𝒌 𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑)2 ] + (𝑠𝑠𝑠𝑠𝑛𝑛 𝜃𝜃)2 (cos 𝜃𝜃)2 𝑑𝑑𝑑𝑑

− (8)

[(𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑)2 − (𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃)2 ] �[1 + (𝒌𝒌 𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑)2 ] + (𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃)2

Now, we have to express 𝑑𝑑𝑑𝑑′ as a function of 𝑑𝑑𝑑𝑑. For this, we know that cos 𝜃𝜃 ′ = 𝑥𝑥′⁄(𝒌𝒌 cos 𝜃𝜃) = cos 𝜑𝜑 sec 𝜃𝜃 . Then, − sin 𝜃𝜃 ′ 𝑑𝑑𝜃𝜃 ′ = cos 𝜑𝜑 sec 𝜃𝜃 tan 𝜃𝜃 𝑑𝑑𝑑𝑑 𝑑𝑑𝜃𝜃 ′ =

Substituting in eq. (8),

=

𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑡𝑡𝑡𝑡𝑡𝑡 𝜃𝜃 𝑑𝑑𝑑𝑑 sin 𝜃𝜃′ cos 𝜑𝜑 tan 𝜃𝜃 𝑑𝑑𝑑𝑑

�(sin 𝜑𝜑)2 − (sin 𝜃𝜃)2

(cos 𝜑𝜑)2 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = − − cos 𝛽𝛽 [(𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑)2 − (𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃)2 ] �[1 + (𝒌𝒌 𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑)2 ] + (𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃)2 (𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃)2 𝑑𝑑𝑑𝑑

[(𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑)2 − (𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃)2 ] �[1 + (𝒌𝒌 𝑠𝑠𝑠𝑠𝑠𝑠 𝜑𝜑)2 ] + (𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃)2

,

whence, since 𝑦𝑦 = sin 𝜃𝜃, and 𝑑𝑑𝑑𝑑 = cos 𝜃𝜃 𝑑𝑑𝑑𝑑, and making 𝐴𝐴 = sin 𝜑𝜑, and 𝐵𝐵 = �1 − (𝒌𝒌 sin 𝜑𝜑)2 ⁄𝒌𝒌, we find 81


(đ??´đ??´2 â&#x2C6;&#x2019; 1)đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; ďż˝1 â&#x2C6;&#x2019; đ?&#x2018;Śđ?&#x2018;Ś 2 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; = â&#x2C6;&#x2019; , cos đ?&#x203A;˝đ?&#x203A;˝ đ?&#x2019;&#x152;đ?&#x2019;&#x152;ďż˝1 â&#x2C6;&#x2019; đ?&#x2018;Śđ?&#x2018;Ś 2 ďż˝đ??´đ??´2 â&#x2C6;&#x2019; đ?&#x2018;Śđ?&#x2018;Ś 2 ďż˝đ??ľđ??ľ 2 + đ?&#x2018;Śđ?&#x2018;Ś 2 đ?&#x2019;&#x152;đ?&#x2019;&#x152;ďż˝đ??´đ??´2 â&#x2C6;&#x2019; đ?&#x2018;Śđ?&#x2018;Ś 2 ďż˝đ??ľđ??ľ 2 + đ?&#x2018;Śđ?&#x2018;Ś 2 which is an equivalent expression for the 'imaginary' integrand of the elliptic integral of the first kind. The complete integral could be found making đ?&#x153;&#x2018;đ?&#x153;&#x2018; = đ?&#x153;&#x2039;đ?&#x153;&#x2039;â &#x201E;2. In this case, đ??´đ??´ = 1, đ??ľđ??ľ = â&#x2C6;&#x161;1 â&#x2C6;&#x2019; đ?&#x2019;&#x152;đ?&#x2019;&#x152;2 â &#x201E;đ?&#x2019;&#x152;đ?&#x2019;&#x152;, and the expression simplifies to đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; = â&#x2C6;&#x2019; cos đ?&#x203A;˝đ?&#x203A;˝ đ?&#x2019;&#x152;đ?&#x2019;&#x152;ďż˝đ??ľđ??ľ 2 + đ?&#x2018;Śđ?&#x2018;Ś 2

Now, using đ?&#x2018;˛đ?&#x2018;˛ = đ??ľđ??ľđ?&#x2019;&#x152;đ?&#x2019;&#x152; and making đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; = đ?&#x2019;&#x152;đ?&#x2019;&#x152;đ?&#x2019;&#x152;đ?&#x2019;&#x152; (the elliptic modulus), we can write đ?&#x153;&#x2039;đ?&#x153;&#x2039; 2

đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;) = đ?&#x2018;&#x2013;đ?&#x2018;&#x2013; ďż˝ 0

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

ďż˝1 â&#x2C6;&#x2019; (đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; sin đ?&#x153;&#x192;đ?&#x153;&#x192;)2

1

= đ??ľđ??ľ ďż˝ 0

đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;

ďż˝đ??ľđ??ľ 2 + đ?&#x2018;Śđ?&#x2018;Ś 2

Applying the first of the identities from the Legendre's relation [3], the former can also reads as 1

1 1 đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018;đ?&#x2018;&#x2018; đ??žđ??ž ďż˝ ďż˝ â&#x2C6;&#x2019; đ??žđ??ž(đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; â&#x20AC;˛ ) = đ??ľđ??ľ ďż˝ . 2 + đ?&#x2018;Śđ?&#x2018;Ś 2 đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;â&#x20AC;˛ đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC;â&#x20AC;˛ ďż˝đ??ľđ??ľ 0

where đ??žđ??ž is the complete elliptic integral of the first kind, and đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; â&#x20AC;˛ = â&#x2C6;&#x161;1 â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;đ?&#x2018;&#x2DC; 2 is the complementary modulus. Notice that the integral in the RHS is explicit.

6. Concluding Remarks and Directions for Future Research The elliptic integrals have been the subject of an enormous theoretical research. These special integrals are also linked with other functions and may expressed in many different ways. The goal of this paper is showing a different context whence the elliptic integrals make also their appearance. In fact, the idea for this work came from a physical problem concerning the growth of a spherical droplet. It will be interesting to investigate closer connections between this geometric approach and the original problem of finding the arc length of an ellipse.

Acknowledgements I would like to acknowledge in the first place the Editorial Committee of Polygon for accepting this work for publication. I also thank very sincerely Professor Rene Barrientos and Dr. M. Shakil for their warm and open support. Finally, a mention is deserved by the Department Mathematics, Miami Dade College, Hialeah Campus, which, giving me the opportunity to teach, has made all this work possible. 82


References [1] “Higher Trascendental Functions”, A. Erdérlyi, editor, 1953, reedited 1981, chapter XIII, prepared under the direction of F. G. Tricomi. [2] Carlson, B. C. (2010), "Elliptic integral", in Frank W. J. Olver, Daniel M. Lozier, Ronald F. Boisvert, Charles W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press. [3] Digital Library of Mathematical Functions, DLMF: 19.7 Connection formulas, https://dlmf.nist.gov/19.7#E3

83


The one-seventh area triangle problem Dr. Mario Duran Camejo, Ph.D. Department of Liberal Arts and Sciences - Mathematics Miami Dade College, Hialeah Campus E-mail: mduranca@mdc.edu

Abstract This paper presents a very simple proof of a relatively famous geometrical problem, only based in the properties of the triangles. As part of the proof a curious intermediate result is shown. The problem itself is a wonderful introductory problem to plane geometry.

Keywords: Triangle; Area; Euclidean geometry; Planar arrangements of lines. 2010 Mathematics Subject Classification: 51M04, 51M25, 52C30.

1. Introduction â&#x20AC;&#x153;For a triangle in the plane, if each vertex is joined to the point one-third along the opposite side (measured say anti-clockwise), prove that the area of the inner triangle formed by these lines is exactly one-seventh of the area of the initial triangle.â&#x20AC;? It is said that this problem was posed by Kai Li Chung, from the Stanford University, to the physicist Richard Feynman during a dinner conversation after a colloquium at the University of Cornell, and puzzled him for most of the evening. Because of that, it is also known as the Feynman's triangle. The story can be found in [1] but the authors don't say where the problem came from. In 2005, Michael de Villiers provided a generalization of this result, as a theorem, [2].

2. Proof of Theorem The theorem, as stated above, may be shown in different ways, a graphic one, very beautiful indeed, is due to Martin Gardner [3]. Here, I'm offering a proof of the theorem in three steps, that I haven't found elsewhere. The result of the first step might be considered as a small and beautiful theorem in itself. The intermediate step is, sometimes, assumed as evident although it must be demonstrated. Finally, the third step, making use of the previous ones, rounds the proof.

84


1. First step

From the figure above, we can write đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ = đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ + đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ + đ??´đ??´đ??ľđ??ľđ??ľđ??ľđ??ľđ??ľ + đ??´đ??´đ??ˇđ??ˇđ??ˇđ??ˇđ??ˇđ??ˇ ,

but

=

so In the same manner

= =

đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ = đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ + đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´

1 1

ďż˝ đ?&#x2018;?đ?&#x2018;?ďż˝ â&#x201E;&#x17D;đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ .

2 3

1 1

ďż˝ đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;ďż˝ â&#x201E;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; â&#x2C6;&#x2019; đ??´đ??´đ??ľđ??ľđ??ľđ??ľđ??ľđ??ľ ,

2 3 1 1

ďż˝ đ?&#x2018;?đ?&#x2018;?ďż˝ â&#x201E;&#x17D;đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; đ??´đ??´đ??śđ??śđ??śđ??śđ??śđ??ś ,

2 3

where â&#x201E;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; and â&#x201E;&#x17D;đ?&#x2018;?đ?&#x2018;? are the heights traced from the vertices A and B, respectively. Now, đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ = and

1 1

ďż˝ đ?&#x2018;?đ?&#x2018;?ďż˝ â&#x201E;&#x17D;đ?&#x2018;?đ?&#x2018;? +

2 3

1 1

1 1

ďż˝ đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;ďż˝ â&#x201E;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + 2 ďż˝3 đ?&#x2018;?đ?&#x2018;?ďż˝ â&#x201E;&#x17D;đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; (đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ + đ??´đ??´đ??ľđ??ľđ??ľđ??ľđ??ľđ??ľ + đ??´đ??´đ??śđ??śđ??śđ??śđ??śđ??ś ) + đ??´đ??´đ??ˇđ??ˇđ??ˇđ??ˇđ??ˇđ??ˇ ,

2 3

1 1 1 1 1 ďż˝ đ?&#x2018;?đ?&#x2018;?â&#x201E;&#x17D;đ?&#x2018;?đ?&#x2018;? + đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;â&#x201E;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;?â&#x201E;&#x17D;đ?&#x2018;?đ?&#x2018;? ďż˝ = (3đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ ) = đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ , 3 2 2 2 3

because each of the addends are equal to the area of the whole triangle. Therefore, đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ cancels out in each side and we get đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ + đ??´đ??´đ??ľđ??ľđ??ľđ??ľđ??ľđ??ľ + đ??´đ??´đ??śđ??śđ??śđ??śđ??śđ??ś = đ??´đ??´đ??ˇđ??ˇđ??ˇđ??ˇđ??ˇđ??ˇ ,

(1)

which is a curious result: The area of the central triangle equals the sum of the three small triangles in its corners.

85


2. Second step

In the previous figure, parallel lines to EF, FD and DE have been drawn through the vertices D, E and F, respectively. By consequence, we have đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(2 â&#x2C6;&#x2019; đ?&#x2018;&#x17E;đ?&#x2018;&#x17E;) đ?&#x2018;?đ?&#x2018;? đ??śđ??śđ??śđ??ś 3 3 = = 2đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?(đ?&#x2018;&#x;đ?&#x2018;&#x; + 1) đ??śđ??śđ??śđ??ś 3 3 đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?(2 â&#x2C6;&#x2019; đ?&#x2018;&#x;đ?&#x2018;&#x;) đ??śđ??śđ??śđ??ś 3 3 = = 2đ?&#x2018;?đ?&#x2018;? đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;?đ?&#x2018;? + 1) đ??śđ??śđ??śđ??ś 3 3 đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;?(2 â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;?) đ??śđ??śđ??śđ??ś 3 3 = = 2đ?&#x2018;?đ?&#x2018;? đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;(đ?&#x2018;&#x17E;đ?&#x2018;&#x17E; + 1) đ??śđ??śđ??śđ??ś 3 3 86


Solving the system for p, q and r, we obtain p = q = r = 1. Therefore đ??śđ??śđ??šđ??š = đ??šđ??šđ??šđ??š đ??´đ??´đ??´đ??´ = đ??ˇđ??ˇđ??ˇđ??ˇ đ??ľđ??ľđ??ľđ??ľ = đ??¸đ??¸đ??¸đ??¸

3. Third step Next, we will prove that the triangle ABE is six times the triangle ADG (shaded). In the same way it can be shown that BCF and ACD are equal to six times BEH and CFI, respectively.

Notice that the height of triangle ABE equal to 2h follows AD = DE. đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ =

đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ =

Therefore, đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ = 6đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ . Then, we have

1 đ?&#x2018;?đ?&#x2018;? â&#x201E;&#x17D;đ?&#x2018;?đ?&#x2018;? ďż˝ ďż˝â&#x201E;&#x17D; = 2 3 6

1 (2â&#x201E;&#x17D;)đ?&#x2018;?đ?&#x2018;? = â&#x201E;&#x17D;đ?&#x2018;?đ?&#x2018;? 2

đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ + đ??´đ??´đ??ľđ??ľđ??ľđ??ľđ??ľđ??ľ + đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ = 6(đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ + đ??´đ??´đ??ľđ??ľđ??ľđ??ľđ??ľđ??ľ + đ??´đ??´đ??śđ??śđ??śđ??śđ??śđ??ś ) = 6đ??´đ??´đ??ˇđ??ˇđ??ˇđ??ˇđ??ˇđ??ˇ đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ = 6đ??´đ??´đ??ˇđ??ˇđ??ˇđ??ˇđ??ˇđ??ˇ + đ??´đ??´đ??ˇđ??ˇđ??ˇđ??ˇđ??ˇđ??ˇ = 7đ??´đ??´đ??ˇđ??ˇđ??ˇđ??ˇđ??ˇđ??ˇ

đ??´đ??´đ??ˇđ??ˇđ??ˇđ??ˇđ??ˇđ??ˇ =

87

đ??´đ??´đ??´đ??´đ??´đ??´đ??´đ??´ 7


Concluding Remarks and Directions for Future Research In principle, this work seems to be a closed one mainly because of its specificity. Nevertheless, the implications of the quasi-theorem that stemmed from it is, perhaps, a source of new discoveries in the fascinating world of geometry, either Euclidean or not. Would it be something relatively similar in other geometries?

Acknowledgements I would like to acknowledge in the first place the Editorial Committee of Polygon for accepting this work for publication. I also thank very sincerely Professor Rene Barrientos and Dr. M. Shakil for their warm and open support. Finally, a mention is deserved by the Department Mathematics, Miami Dade College, Hialeah Campus, which, giving me the opportunity to teach, has made all this work possible.

References [1] R. J. Cook & G. V. Wood (2004), "Feynman's Triangle", Mathematical Gazette 88, p.299-302. [2] Michael de Villiers (2005), "Feynman's Triangle: Some Feedback and More", Mathematical Gazette 89, p.107. [3] James Randi (2001), â&#x20AC;&#x153;That Dratted Triangleâ&#x20AC;?, proof by Martin Gardner.

88


Polyconcs and Polyconvs Dr. Mario Duran Camejo, Ph.D. Department of Liberal Arts and Sciences - Mathematics Miami Dade College, Hialeah Campus E-mail: mduranca@mdc.edu

Abstract In this paper a non-conventional set of geometrical figures derived from the regular polygons subjected to some operations is displayed, and different tessellations obtained by using them are also explored. Additionally, interesting connections between these figures and the non-regular convex polygons are introduced.

Keywords: Polygon; Tessellation and tiling problems; Visual arts. 2010 Mathematics Subject Classification: 00A08, 00A66, 05B45, 52C20.

1. Introduction "A tile that tessellates obviously can have an infinite variety of shapes, but by imposing severe restrictions on the shape, the task of classifying and enumerating tessellations is reduced to something manageable. Geometers have been particularly interested in polygonal tiles, of which even the simplest present formidable problems. In this chapter we are concerned only with the task of finding all convex polygons that tile the plane. It is a task that was not completed until 1967, when Richard Brandon Kershner, assistant director of the Applied Physics Laboratory of Johns Hopkins University, found three pentagonal tilers that had been missed by all predecessors who had worked on the problem." [1] Referring to the previous quotation it looks peculiar that, although tessellations by means of non-regular convex polygons, has drawn the attention of many mathematicians, designers and math lovers, nobody has, apparently, felt attracted by the concave polygons. I think this rejection is unexplainable and unfair for two reasons: On one side, non-regular convex polygons that tessellate the plane are confined to 3, 4, 5 and 6 sides, while concave polygons are not, and, on the other, tessellations using concave polygons are very beautiful indeed. An additional argument in favor is that concave polygons are not just the addition of non-regular convex ones. I must say that not only the convex tiles have been considered for the tessellation of the plane. There are a lot of concave shapes among the forms called polyominoes, polyamants, and polyhexes, but they do not derive from the regular (convex) polygons, rather from the addition of congruent squares, triangles and hexagons, respectively. 89


A concave polygon cannot be regular because its angles are not equal, but its sides may be, so it is equilateral. Additionally, the angles that are not alike may be restricted to a minimum. The best way to construct an equilateral concave polygon is to reflect to the inside one or more vertices of the regular convex polygons. I will coin the word polyconc for such objects. It is not difficult to see that the smallest polyconc has five sides. In this work I will describe the first six polyconcs, with five, six, seven, eight, nine and ten sides. I would like to share with you the beauty of the patterns that can be obtained with them. Whereas the smaller polyconcs, the pentaconc and the hexaconc, have only one species each, the rest of polyconcs with the same number of sides constitutes a family and, as the number of sides increases, the number of its members increases as well. Sometimes the tessellation can be reached with only one of the species in the family, sometimes it needs more than one. There are also some species more exible than others as we will see later. It is quite intuitive to conjecture that all the polyconcs (because of their concavity) have to tessellate the plane. I do not know whether a formal proof of that conjecture, in case it is true, does exist. The six cases I will show you here do tessellate the plane, but in manners that are not trivial, and with quite different patterns. They share, however, a feature, to provide always, at least, one tessellation with rotational symmetry. One pertinent question is if a relation between the polyconcs and their kindred, the nonregular convex polygons (that I will call, henceforward, polyconvs) does exist. Of course, it does. This is particularly true regarding the polyconc of five sides, as we shall see afterwards.

2. The pentaconc Below, I'm showing the first of the polyconcs, the pentaconc. It is a single specimen but has a large hall of competitors because there are fifteen polyconvs that tessellate the plane. The pentaconc, named as equilateral pentagon, was mentioned by David Simonds in a paper published in 1977 [2].

The pentaconc has the property that it can be divided into three isosceles triangles, and this division can be recurrently applied to each triangle. This property is depicted in the next figure. 90


The pentaconc tessellates the plane in three different ways (This capacity seems to be typical of all the polyconcs). One way is periodically, the second, aperiodically (even, chaotically), and the third, with rotational symmetry. These three kinds of tessellation are shown below.

91


In an ordered tessellation of pentaconcs it is possible to devise some pentaconvs that, in turn, tessellates the plane. Most of these pentaconvs belongs to the aforementioned list of fifteen. In the next figures, the two tessellations are superimposed and two levels of relation are indicated, the red circles mark where the vertices of the two tessellations coincide, and the yellow circles mark where the vertices of either tessellation lie on a line pertaining to the other.

92


93


94


95


96


97


Next, we can see some other pentaconvs that also fit in the tessellation of pentaconcs but whose relation with it is much more distant, which becomes evident by the paucity of the red circles and the absence of the yellow ones. The pentaconv8 is practically an alien because it is based in a hexagonal grid. I have brought all them here, anyway, to show a spectrum of alternatives as much varied as possible.

98


99


3. The hexaconc The hexaconc is depicted in the first of the figures below. It is close related to the triangle in terms of its tessellation properties. In fact, the hexaconc is a triamant of fourth order, that is, it results from the aggregation of four congruent equilateral triangles. Other triamants may be created as well by the addition of hexaconcs, as the T-shirt, corresponding to a triamant of twelfth order. These two objects tessellate periodically the plane as can be seen in the second and third figures.

The hexaconc, however, in spite of its apparently orthodox nature, is able to produce a strongly aperiodic tessellation. Besides, it also tessellates the plane with rotational symmetry. See those configurations in the next three figures.

100


101


Additionally, the hexaconc and the T-shirt produces the two following pentaconvs, that I will call pentexconv1 and pentexconv2.

Finally, the following tesselation shows the regular hexagon as the result of the aggregation of six non-equilateral pentaconcs.

102


4. The non-single families The next non-single families of polyconcs have two and three members. The first two families, the heptaconc and the octaconc, are able to produce pentaconvs which tessellate the plane in different ways. The second two families, however, give rise to both pentaconvs (that only comes from the decaconc) and heptaconvs (coming from both the nonaconc and the decaconc), which can also tessellate the plane.

4.1 The heptaconc The heptaconc is the first polyconc with a family of more than one member. The two heptaconcs have a spatial symbiotic relation. The seed of the heptaconc, namely, the heptagon, is a quite mysterious object. It is the first polygon that breaks the ascendant line of those that can be constructed with compass and straightedge. It has been proved that a regular đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;-gon is constructible if and only if cos(2đ?&#x153;&#x2039;đ?&#x153;&#x2039;â &#x201E;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;) is a constructible number, that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots. For đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; being a prime number, the next constructible đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;-gon after the pentagon (đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; = 5) is the 17-gon (The method to construct the 17-gon using compass and straightedge is due to Carl Friedrich Gauss). So the heptagon cannot be constructed graphically. It's a pity but it is a reasonable limitation associated with the theorem of Pythagoras. The two heptaconcs are the hepatconc2 and the heptaconc3 and are shown next.

A line (dashed) from E, perpendicular to AB, divide the hepataconc2 into two congruent pentaconvs, named pentepconv1. Same, a line (dashed) from A, perpendicular to DE, divide the hepataconc3 into two congruent pentaconcs, the pentepconc. The denomination 2 and 3 indicates the number of sides forming the concavity. The heptaconc3 is autonomous in terms of tessellation but the heptaconc2 is not, which means that the heptaconc3 can tessellate the plane by itself but the heptacon2 has to be combined with the heptaconc3 in order to do that. The next figures show several tessellations, periodical and aperiodical, based on the heptaconc2 and the heptaconc3.

103


104


Notice that the first row in the previous tessellation is a border, that is, the heptaconcs cannot tessellate above it. The first tessellation in the next page makes clear that the addition of the pentepconv1 and the pentepconc produces a new pentepconv, the pentepconv2. The pentepconv2, obviously, also tessellates the plane. The second tessellation is another way to do it. The third tessellation, however, is not based in that simplification, rather makes use of the two heptaconcs indistinctly.

105


Finally, several tessellations with rotational symmetry.

106


4.2 The octaconc The octaconc is a more tractable family than the heptaconc. It has two members.

The pentocconv results from the partition of the octaconc2, which creates a beautiful tessellation with rotational symmetry. A tessellation with rotational symmetry can also be obtained with the octaconc3.

107


Some other examples of tesselations using octaconc2 and octaconc3 are shown below.

108


4.3 The nonaconc The nonaconc is the first family with three members. One of them, the nonacomb2, is a sort of deadbeat of the other two. Nevertheless, it generates, when divided, an equilateral pentaconv very known by its versatility [3]. The two other members, working together, can tessellate the plane, giving rise to the appearance of the heptanonc, which means that it is a heptaconc stemming from the nonaconc.

Next, the heptaconv derived from the nonaconc is shown.

109


The equilateral pentaconv or penteqconv, was found simultaneously in 1976 by a study group conducted by M. Hirschhorn and G. Szekeres, and by M. Rice [3].

110


The tessellations with rotational symmetry obtained with the nonaconcs are of two different types, the solid ones and the ones with a central hole. All of these tessellations have triangular and hexagonal symmetries.

111


112


113


4.4 The decaconc The decaconc, like the nonaconc, has three members. Although the first member, the decaconc2, is a lazybones in terms of tessellation, it creates the two pentadeconvs, namely, the pentadeconv1 and the pentadeconv2 which tessellates the plane separately by themselves but are also interchangeable. The decaconc3 is the only member of the family that can tessellate alone although the other two members are able to produce some kind of intrusions in its fabric.

114


115


The decaconc produces the same two types of tessellations with rotational symmetry as the nonaconc.

116


117


5. Appendix1: The relatiles This is, probably, a sort of digression. I was hesitating whether to include it or not because it has to do entirely with one family of polyconvs but, finally, here it is. It deals with the curious kinship between some known types of pentaconvs when small genetic changes are made to them. Because these tiles seem to be relatives I call them relatiles. There is a beautiful page in Wikipedia, named Pentagonal tiling. I will focus on the types described there as 11, 12, 14, and 15. They are represented below (where I'm keeping the aforementioned denomination).

As it can be seen, the four types are linked by two features: the presence of one right angle, and some special relation between their sides and some conditions imposed to their angles (Notice that types 11, 12, and 14 impose the same). However, the rest of their angles do not manifest any likeness. Whereas one type has angles that are all multiple of 15Ë&#x161;, the angles in the other types are rather disconnected or remain undetermined. Nevertheless, their tile patterns have a very close resemblance so we can feel there might be a mysterious kinship behind their identities. Let's explore this feeling a little bit more and, even, we will see a new 'type' emerge.

118


Now, let's particularize tile 11 making đ?&#x2018;?đ?&#x2018;? â&#x2C6;&#x2019; 2đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;, and tile 12 making đ?&#x2018;?đ?&#x2018;? = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; = đ?&#x2018;?đ?&#x2018;? , and đ?&#x2018;?đ?&#x2018;? = 2đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; = 2đ?&#x2018;?đ?&#x2018;?. This operation gives rise to tiles 11_1, 12_1, and 12_2. We can see how, suddenly, tile type 11 (converted in type 11_1) and tile type 14 become very similar. On the other hand, we see that tiles types 12_1 and 12_2 have the same angles although their sides b differ, and these angles are multiples of 15Ë&#x161; like in tile type 15 (They are also multiples of 30Ë&#x161;). Even more, we can obtain new tessellations with these two types.

119


120


Notice that, in tessellation 12_12, the yellow and green fields can extend indefinitely to the left and to the right, respectively. It is a curious tessellation because has two different uniform fields and a line of fracture in between.

121


6. Appendix2: The pairygons This is a sort of amusing pairs of polygons. Pairing is based in shear transforming a pair of regular polygons to make them tessellating the plane. Here, I will show a trio of pairygons, namely, the hepent, the hepsquare, and the nonarhomb. The first one transform both the regular heptagon and pentagon, the second one only transforms the regular heptagon while keeps the square, and the last one transforms the regular nonagon and keeps the rhombus.

122


123


A very interesting pseudo-pairygon can be obtained from the decagon. It is the decatrap. In fact, the decagon is decomposed into two isosceles trapezoids but the smallest of them reappears within the largest, generating, in turn, two reduced, large trapezoids.

124


7. Appendix3: Typo in reference [1] Figure 85 in p. 175 has an error in the angles. The figure represents the Marjorie Rice's tenth pentagonal tiling. The right approximate angles must be, starting from the bottom left corner and going clockwise, 77˚, 143˚, 63˚, 149˚, and 108˚.

8. Concluding Remarks and Directions for Future Research This work results from the confluence of two parallel disciplines Mathematics and Architecture with which I have been engaged in different times of my professional life. The exploration of this sort of geometrical structures is surprising because it feeds from quite different mental attitudes and provides a very rich field for experimentation. The link between polyconcs and polyconvs (much more studied and classified) is still a promising land of investigation.

Acknowledgements I would like to acknowledge in the first place the Editorial Committee of Polygon for accepting this work for publication. I also thank very sincerely Professor Rene Barrientos and Dr. M. Shakil for their warm and open support. Finally, a mention is deserved by the Department Mathematics, Miami Dade College, Hialeah Campus, which, giving me the opportunity to teach, has made all this work possible.

References [1] Martin Gardner “Time travel and other mathematical bewilderments”, 1988, W. H. Freeman and Co. [2] David Simmons, “Central tesselations with an equilateral pentagon”, Mathematics Teaching MT 081, December 1977. [3] Marjorie Rice & Doris Schattschneider, “The incredible pentagonal versatile”, Mathematics Teaching MT 093, December 1980.

Math Student Motivation Research Project – SWOOP 125


Professor Lourdes España Associate Professor, Sr. of Mathematics Mathematics Department Miami Dade College, North Campus, Fl. 33167, USA E-mail: lespana@mdc.edu and Professor Maria Alvarez Associate Professor, Sr. of Mathematics Mathematics Department Miami Dade College, Kendall Campus, 33176, USA Email: malvare1@mdc.edu

Abstract

A team of Math and Psychology faculty joined forces in an experimental project, WOSOP – Wish, Outcome, Success, Obstacle, Plan, to try to improve student motivation in their classes. The study included students who are enrolled in gateway mathematics and psychology courses at Miami Dade College. We are attempting to test if strategies toward developing self-motivation and self-volition will enhance student learning in mathematics and psychology courses. Students that participated in the study were assigned to study conditions and were the recipient of faculty instructional methods designed to enhance learning, motivation, and volition. The authors of this study are Drs. Juan Abascal, Dominic Brucato, and Kimberly Coffman, faculty in Psychology at MDC’s Kendall Campus. As participants of the research project, we would like to share our observations and results.

Keywords:

SWOOP, WOSOP, Motivation, Volition, Success, Teaching, Math

2010 Mathematics Subject Classifications: 97C40, 97C70, 97D40, 97D50.

1. Introduction 126


During the summer of 2016, several psychology and mathematics professors joined forces in the discussion to implement an experimental study called “WOSOP Group Work and Student Success Project” to try to improve student motivation in their classes. The study was designed by several Psychology Faculty from the Social Science Department - Dr. Juan Abascal, Dr. Dominic Brucato, and Dr. Kimberly Coffman. They submitted a research proposal addressing the proposed methodology to be implemented during fall of 2016. The research proposal was entitled, “Effects of WOSOP – A Mental Contrasting with Implementation Intentions (MCII) Derivative – and Base Group Involvement on Students’ Success in Entry Level Mathematics and Psychology Courses.” The acronym WOSOP stands for Wish, Outcome, Success, Obstacle and Plan. After implementation of the project during the Fall of 2016, data was collected and it is being currently evaluated. Upon reflection and discussion of our implementation experiences during fall of 2016, some team members decided to continue the use of WOSOP in their classes, but in a slightly different order which became known as SWOOP – Success, Wish, Outcome, Obstacle, Plan. The study’s purpose was to determine if strategies toward developing self-motivation and self-volition will enhance student learning in mathematics and psychology courses. Students that participated in the study were assigned to study conditions and were the recipient of faculty instructional methods designed to enhance learning, motivation, and volition.

As professors “we are confronted with people from all walks of life who come with high expectations of success, but lack the necessary cognitive and non-cognitive skills to make that success real” (Abascal & Brucato, 2015, p.1). How many of you agree that the great majority, if not all of our students, are quite certain that they will succeed by the end of the semester and then end up failing short of what they had envisioned (Abascal & Brucato, 2015, p. 1)?

“Achievement falls into two separate dimensions. One is motivation, which is the desire for a goal. The second is volition, which is the ability to do what is necessary to achieve the goal. Each of these is essential to accomplish long term objectives, but neither is sufficient alone” (Abascal & Brucato, 2015, p. 1). “…Our students come very motivated to succeed in our classes, but unless they have the volition, the self-control and frustration tolerance to go through necessary

127


periods of discomfort, their goals remain nice dreams and no more” (Abascal & Brucato, 2015, p. 1).

So, how does Motivation and Volition result in Success? Motivation is important because it is the person’s decision to accept responsibility for a task and to pursue a goal (Keller and Deimann, as cited in Reiser & Dempsey, 2018, p. 79). Different from motivation is volition which is the learner’s ability “on overcoming obstacles and hindrances during the learning process” (Keller and Deimann, as cited in Reiser & Dempsey, 2018, p. 78). In other words, “motivation to learn is protected and maintained when learners employ volitional (self-regulatory) strategies to protect their intentions” (Keller and Deimann, as cited in Reiser & Dempsey, 2018, p.83).

In convincing our students of the importance of the study that they were about to participate, two concepts were presented to our students: Neural Plasticity and Neurogenesis. Students were taught that Neural Plasticity is the increment “of connections and activities among our billions of neurons which continues throughout our life time depending on our experiences” (Abascal, Brucato & Coffman, 2016, WOSOP Rationale). We mentioned that we know that genetics account for about “no more than 40% of what we are able to learn” (Abascal, Brucato & Coffman, 2016 p. WOSOP Rationale). Thus, “our belief that our learning abilities are caused by genetics affects our ability to learn” (Abascal, Brucato & Coffman, 2016, WOSOP Rationale). In the past, we thought that our brain developed until about age 18 and deterioration began. But, advances in neuroscience has shown that the brain is affected by experiences (Abascal, Brucato & Coffman, 2016, WOSOP Rationale). The brain “is plastic in that it molds itself in response to our behavior, our thoughts, in short, our experiences” (Abascal, Brucato, & Coffman, 2016, WOSOP Rationale). Not only is the brain neuroplastic, but it also creates “new neurons based on our experiences” (Abascal, Brucato & Coffman, 2016, WOSOP Rationale). This is known as Neurogenesis. So, we emphasized to our students that “when we engage in new learning, our brains literally grow” (Abascal, Brucato & Coffman, 2016, WOSOP Rationale).

So, research on brain science has made it clear that regardless of the contribution of genetics, how well your students do in class is mostly affected by whether or not they do the steps necessary to learn the subject matter. Believing they can succeed is important, but not enough. They must 128


also believe that they will take the steps necessary to succeed and then follow through. (Abascal, Brucato & Coffman, 2016, WOSOP Rationale)

2. Implementation

We took several steps to implement SWOOP in our classes. We administered the GRIT Survey at the start and the end of the semester. Grit predicts â&#x20AC;&#x153;who will succeed in college, in the work place, and in life overall. It has absolutely nothing to do with IQ, or math skills, or talentâ&#x20AC;? (Abascal & Brucato, 2016, p. 2). Grit represents the passion for a goal and the perseverance in doing everything necessary to reach that goal whatever the obstacles (Abascal & Brucato, 2016, p. 2). See pre- and post- Grit Scale below (Abascal, Brucato & Coffman, 2016, Appendix K).

Then every week, for 6 weeks (if it is an 8-wk. semester) or for 14 weeks (if it is a 16-wk. semester), the instructor reads a meditation script which involves the five steps to success. During meditation students reflect on how SWOOP applies to them. Then they answer the SWOOP questions either in paper and pencil format or through a classroom response system. We used a classroom response system called Learning Catalytics. Samples of the SWOOP questions has been attached below. 129


3. Methodology

Dr. Kimberly Coffman collected results from the implementation of WOSOP, in the math and psychology courses, during the Fall of 2016 and is in the process of evaluating the data. Betweensubjects SPSS data base will be constructed with primary demographic information about each student to match student pre-post survey results. Also, a three-way ANOVA analyses will be conducted. Frequency data and main effects and interactions will be analyzed and compiled. Finally, GRIT pre- and post- data will be entered, and correlation data will be generated. (Abascal, Brucato & Coffman, 2016)

4. Discussion of Preliminary Results

Prof. Maria Alvarez implemented WOSOP in two College Algebra Courses (MAC1105) during the fall of 2016. The experimental group had a 7% higher passing rate than the control group. In the Fall of 2017, Prof. EspaĂąa continued the use of the strategy, SWOOP, in her MAT 0029 and MGF1107 classes. In MAT0029, her experimental group had a 92% passing rate versus 130


the control group with a passing rate of 71%. In MGF 1106, her experimental group had an 82% passing rate versus the control group with a 77% passing rate. Then in the Spring of 2017, Professor España’s MGF 1107 experimental group had a 90% passing rate versus the control group with an 83% passing rate. Consequently, all four courses MAC1105, MAT 0029, MGF 1106 and MGF 1107 students did much better in the experimental groups compared to the control groups.

The majority of the student’s feedbacks were positive. Prof. España’s students stated that SWOOP helps them set goals and realize what they need to do in order to accomplish them. One student stated that, “SWOOP is like a layout plan, a guide, kind of guiding them through until they reach the goal they wrote down for the week with the obstacles they may encounter. Knowing what those problems are could really help them better their current situation. Also, that this strategy should be continued being used.” Another student stated that SWOOP allows them to figure out the problem they may have in class and also figure out habits they need to change to be successful in passing this class. Finally, when students were asked on a scale from 1 to 5 (5 being you really like it & 1 don’t like it at all) rate SWOOP the majority gave a score of 4 and 5.

5. Concluding Remarks

Although official results from the original implementation of WOSOP from the fall 2016 is pending, our observations and experiences of WOSOP and SWOOP are strongly positive. We recommend continued implementation and data collection to determine statistical significance.

Acknowledgements Special thanks to Dr. Juan Abascal, Dr. Dominic Brucato, and Dr. Kimberly Coffman to introducing us to WOSOP and allowing us to participate in their study. We would also like to thank Dr. M. Shakil and Dr. Jaime Bestard for encouraging us to write this paper on this research and our experiences.

References

131


Abascal, J. R., & Brucato, D. (2015). Teaching2 Series: Psychology and Neuroscience in the Classroom - Motivation + Volition = Success. Miami, FL: Mindworks International Inc.

Abascal, J.R., Brucato, D. & Coffman, K. (2016). Effects of WOSOP – A Mental Contrasting with Implementation Intentions (MCII) Derivative – and Base Group Involvement on Students’ Success in Entry – Level Mathematics and Psychology Courses. Unpublished research proposal, Miami Dade College, Miami, FL.

Keller, J.M. & Deimann, M. (2018). Motivation, Volition, and Performance. In R.A. Reiser & J.V. Dempsey (Eds.), Trends and Issues in Instructional Design and Technology (pp. 78-86). United States of America: Pearson Education, Inc.

Reiser, R. A. & Dempsey, J. V. (2018). Trends and Issues in Instructional Design and Technology. United States of America: Pearson Education, Inc.

Considering the Student´s Mother Tongue in the 132


Teaching-Learning Process of English Dr. Raydel Hernandez Garcia, PhD Department of World Languages and Communication Miami Dade College, Hialeah Campus email: rherna24@mdc.edu

Abstract The process of learning English as a Second Language or English for Academic Purposes is hindered occasionally because of a diversity of contradictions, which arise due to contrasts between studentsâ&#x20AC;&#x2122; mother tongue and the foreign language they are studying. In addition, there are wrong generalizations, which lead to some difficulties too. Teachers should be equipped to detect these incongruities in order to improve their methodological procedures, built upon linguistic foundations. This article intends to explain how instructors can use the didactic principle of considering the studentâ&#x20AC;&#x2122;s mother tongue, based on the linguistic comparative method. With the results obtained from the comparison, professors will have at their disposal tools to develop habits in the teaching-learning process. A user-friendly model of linguistic comparison is provided in this paper. The examples of contradictions come from Hispanic learners who are studying EAP or ESOL.

Keywords:

Comparative

linguistics,

Habits,

Contradictions,

Synchronic,

Diachronic,

Semasiological, Onomasiological.

Introduction The investigation of the similarities and differences between language families is the concern of comparative linguistics. As a major branch of general linguistics, this discipline first developed rigorous methods of linguistic analysis. The study of linguistic change was possible at the beginning of the nineteenth century by means of a diachronic analysis, whose main goal was to establish the genetic relationships between 133


languages and language families. However, there were limitations because linguists realized that languages were not static, they also went through a process of linguistic change. The synchronic comparison made it possible to analyze different historical stages of the language. In the decades that have followed, the two approaches - the diachronic and the synchronic â&#x20AC;&#x201C; have combined usefully to yield abundant results. Today, some principles have been made clear, which were not apparent to the great comparatist scholars of the nineteenth century. It is now an accepted fact that all language comparison is based on descriptive data resulting from synchronic analysis. The comparison will be only as accurate as these data will allow it to be. On the other hand, in the evolution of comparative linguistics there have been many approaches and methods, each of them has had its own impact. In the relation content-form, there are two approaches: the Semasiological Approach and the Onomasiological Approach. The Semasiological Approach starts from the analysis of form/s to arrive at the content expressed. In other words, it starts from structures to get to the meaning conveyed. The semasiological approach has been, for quite a long time, the major way to deal with language comparison because of the influence of structural descriptivism. The Onomasiological Approach 1 begins from a given content to determine the forms that can be used to express it. Thus, it moves from meaning to form/s. In this paper, we follow the Onomasiological Approach for the comparison because we are in tune with the notion of communicative language teaching 2; the point of departure is communicative functions and later structural components (phonology, lexicology, syntax, etc.) are analyzed. In other linguistic comparative studies, scientists begin by describing grammatical structures and later their forms of expression (semasiological approach). So, how important is the linguistic comparison for foreign language instructors in current times?

1

For more information about the onomasiological and semasiological approaches refer to Bermello and Vega (2007), Vega (2011), Hernandez (2014). 2 The communicative approach has been widely spread in the last twenty to thirty years. Communicative language teaching has attempted to change the focus of foreign language teaching for many years, which goes beyond the mastery of linguistic rules, and takes into consideration contextual, semantic and strategic elements. (Savignon,1976).

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Nowadays, communication in a foreign language, especially in English, opens many doors to people in a variety of contexts. English is the official language in the United States, and it is considered a Lingua Franca around the world since it allows people who do not have the same language to communicate in events, conferences, among other exchanges. In our specific milieu, Miami Dade College, English for Academic Purposes is a program that has been designed for foreign language students to make the transition to the American academic life. Our institution plays a key role in this regard, thanks to our educators and administrators who are constant and effective. However, we need to continue improving the quality of the teaching-learning process of EAP courses.

A set of categories, principles and laws govern foreign language instruction (Antich, 1987), (Brown, 1994), (Font, 2006), (González, 2011). One the most important principles in this field is considering the student´s mother tongue. This principle is built upon the results of comparative linguistics, which makes it possible to establish the similarities and differences between students’ mother tongue and the target language.

No instructor in areas such as EAP and ESOL can overlook this principle because the results of the comparison will help pre-establish possible inter-linguistic transfer (positive or negative), design activities and use the comparison to explain any contradictions that is obstructing the learning process. The results of comparative linguistics lead to the improvement of the teachinglearning process. The data obtained from the comparison can be used to develop new linguistic habits. There are three different types of habits according to A. A. Leóntiev (1981): •

The first type is the simple transference of the habits existing in the mother tongue to the foreign language (transferable habits). For example, the pronunciation of letter /m/ in initial and middle position in English and Spanish. In addition, some words of international use such as television, radio.

The second type is the transference which requires corrections and clarification (modifiable habits). For instance, the articulation of phoneme /m/ in final position in Spanish is not bilabial like in English. This new habit should be modified as a result.

135


The third type is where the operation must be formed from scratch (new habit). For example, asking questions in English depends on the use of auxiliary verbs, word order, and intonation, whereas in Spanish the most prominent feature is intonation.

The three types of habits abovementioned should be taken into account in the teaching-learning process of EAP or ESOL. Close attention should be paid to the second and third types, because if they are not partially or completely modified, communication in the foreign language will be hindered. In addition, we need to raise awareness of words that look similar yet do not transfer to the target language correctly. These words are known as false cognates. Some examples of false cognates for Hispanic learners of English are carpet (“carpeta”), actually (“actualmente”), among many others.

Not only should habits be developed in the linguistic component (sounds, lexical items, syntax) of the language being taught, teachers should also be aware that there are also contradictions due to new sociolinguistic and sociocultural contents and realities which students face on regular basis. For instance, in the Cuban culture it is very easy to strike up a conversation in a variety of contexts, while in the American setting this process can be harder. In Cuba, at a bus stop or a doctor’s office, if a person sits down next to you, it is common to start talking to him or her, whereas this type of sociolinguistic behavior in the USA is not always welcomed and may be considered invasive or even rude. Students need to learn this cultural habit.

The following chart will help teachers analyze just a few contradictions students come across when learning the foreign language. It will provide the teacher with an easy model to apply the principle of considering the student’s mother tongue. The fruits of the linguistic results will be useful to enhance our teachers’ methodological procedures. 3

This model provided is practical, unlike a pure linguistic paper or dissertation, in which the description of both languages needs to be thorough, and after that, an exhaustive comparison is carried out based on the detailed description of each language involved in the study. 3

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Contradiction

*Potential Error

Spanish

English

Description of the contradiction

Asking

A) You studied

¿Estudiaste

Did you study

The main cause in example A is

questions in

English yesterday?

inglés ayer?

English

inter-linguistic due to the subject-

yesterday?

verb inversion of periphrastic -do

both languages is expressed

(yes-no question)

in English, which does not occur

with different

in Spanish. That is the reason why

forms of

students omit very often the

expressions.

auxiliary verb in the simple form for the present (do, does) or the

(linguistic

past(did). This habit should be

contradiction)

taught from scratch (third type). Also, students will be likely to B) Do you know

¿Sabes a

where did she go?

donde fue

(embedded/included

Do you know

write two question marks in

where she went? questions in English as they do in Spanish. This habit should be

ella?

modified (second type). Another

question)

element to mention is intonation in yes-no questions. In Spanish intonation for this type of questions is rising-falling, whereas in English it is rising. We must modify this habit in our students (second type).

Example B is the result of an intralinguistic error. These types of errors are the result of overgeneralizing a rule already

137


learned and extending it to new structures where it should not be applied. After the students fix the habit of structuring the question “where did she go?”, they are likely to maintain the same structure of this question when it is embedded in expressions such as: Do you know…? Can you tell me ...? I wonder... This is a habit that needs to be modified. Expressing

Tengo 10

I am 10 years

This is a negative transfer in

años.

old.

English that comes from Spanish.

Tengo miedo.

I am afraid.

temperatures,

Tengo

I am cold/hot.

(third type).

etc.

frío/calor.

In the Cuban

In the American

This is an example of sociocultural

arrived from Cuba

culture people

culture people

difference that leads to culture

was introduced to

hug and kiss

do not kiss and

an American one

each other.

hug so

oneself about age, fear, heat or cold,

I have 10 years old. I have afraid. I have hot/cold.

Students use the verb “have” and not “be”. This is new habit to form

(linguistic contradiction) Physical contact A person who just (sociocultural contradiction)

day. The next day

effusively.

when they saw each

138


other, the Cuban

shock. This is a habit to be

hugged him. The

modified (second type). 4

American felt awkward.

Talking about

I am teacher.

Soy maestro.

I am a teacher.

jobs.

This a negative transfer in the use of articles. In English it is obligatory to use the indefinite article “a/an” when talking about a

Making generalizations

The life is beautiful. La vida es

Life is beautiful.

bella.

person’s job, not in Spanish. The use of the definite article “the” is incorrect in English when generalizing, whereas in Spanish

(linguistic

its use is obligatory. This is a new

contradiction)

habit to modify (third type). Comparing two

She is more tall

Ella es más

She is the taller

This is a negative transfer from

or more

than her sister.

alta que su

than her sister.

Spanish. In English, the tendency

elements. (linguistic contradiction)

She is the most tall in her class.

hermana. Ella es la más alta de su

She is the tallest in her class

is to add -er to monosyllabic (onesyllable) adjectives in the comparative form and -est in the superlative form. The error can

aula.

have intra-linguistic causes. This is a new habit to create from scratch (third type).

4

This problem is related to one of the categories of nonverbal communication, specifically the notion of proxemics. It is vital that students be aware of the appropriate physical space between speakers according to the culture in which they currently reside.

139


Conclusions The application of the didactic principle of considering the studentâ&#x20AC;&#x2122;s mother tongue will help instructors be aware of a great deal of contradictions in the teaching-learning process of a language. These difficulties can be the result of inter-linguistic transfer because the codes of verbal and extraverbal communication are not partially or totally shared between the target language and the studentâ&#x20AC;&#x2122;s mother tongue. Besides, there can be intra-linguistic problems due to wrong overgeneralizations in the learning process. In our current setting, it is important to raise awareness of contrasts in the linguistic, semantic and pragmatic components of communicative competence in English and Spanish, especially because of the widespread presence of the Hispanic population in our institution. In this paper, the author strongly encourages instructors to carry out similar theoretical or applied comparative studies with other contradictions that interfere the teaching-learning process. It is recommended that the comparison follow an onomasiological approach. It is hoped that this work can be discussed by EAP instructors in Miami Dade College as well as other institutions.

Acknowledgements The author of this article would like to thank Dr. Gladys Bermello and Dr. Juan Carlos Vega, for their endless unconditional support in their supervision in the application of comparative analysis with an onomasiological approach and the application of the principle of considering the studentâ&#x20AC;&#x2122;s mother tongue. Moreover, the author is grateful for the support of colleagues in the revision of the article; they are Liliana Cobas (Chairperson of World Languages and Communication), Orestes Vegas and Nora Hentschel. Finally, special thanks to all my students for the chance to share the teaching-learning process with them, our Institution Miami Dade College for the opportunity to teach and do research here, and to the Editorial Committee of Polygon for their suggestions to improve the design of this paper.

References 140


Antich, R. (1987). Metodología de la enseñanza de lenguas extranjeras. La Habana: Editorial Pueblo y Educación.

Bermello Lastra, G. & Vega, J. C. (2007). An English Grammar for Spanish speaking teachers-to-be of English. La Habana: Editorial Pueblo y Educación.

Brown, D. (1994). Teaching by Principles. New Jersey: Prentice Hall.

Font, S. (2006). Metodología para la asignatura Inglés en la secundaria básica desde una concepción problémica del enfoque comunicativo (Tesis en opción al grado científico de Doctor en Ciencias Pedagógicas). La Habana: Instituto Superior Pedagógico Enrique Jose Varona.

González Cancio, R (2011). La clase de lengua extranjera II. Teoría y práctica. La Habana: Editorial Pueblo y Educación.

Hernández García, R. (2014). Estrategia linguo-didáctica para el proceso de enseñanzaaprendizaje de los enunciados interrogativos en la lengua inglesa (Tesis en opción al grado científico de Doctor en Lingüística). La Habana: Universidad de La Habana.

Leontiev, A. (1981). Actividad, conciencia y personalidad. La Habana: Editorial Pueblo y Educación.

Savignon, S. (1976). Communicative Competence: Theory and Classroom Practice. Conference on the Teaching of Foreign Languages. Paper presented at the Central States Conference on the Teaching of Foreign Languages. Detroit, Michigan.

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A Proposal for Scaffolding the Teaching-Learning Process of Speaking Skills in English Dr. Raydel Hernandez Garcia, PhD Department of World Languages and Communication Miami Dade College, Hialeah Campus email: rherna24@mdc.edu

Abstract This proposal intends to organize systematically and scaffold explicitly a variety of activities, techniques, and resources we come across in the literature about the teaching learning process of speaking skills, also known as speech. This paper also raises awareness of the roles of learners and teachers in each stage of the learning cycle to understand better the gradual release of responsibility, toward a point where students can independently and confidently develop speaking skills creatively. The learning-cycle designed here will not only include the stages in the classroom, but a new stage that will take place out of the classroom. The cycle is characterized by problem solving strategies to make the teacher-learning process student-centered.

Keywords: Scaffolding, Learning cycle, Speaking skills, Problem solving

Introduction

There is an increase of interesting techniques and activities that can be found in foreign language teaching literature. However, sometimes these classroom practices are not organized following a systemic and an interrelated learning cycle. Providing the proper scaffolding in pedagogy will ensure that students make a smooth and gradual transit in every stage of the learning process. Scaffolding is a term that comes from the field of construction, in reference to a temporary structure for supporting workers and materials. According to Sherin, B., Reiser, B. J., & Edelson, 142


D. (2004), â&#x20AC;&#x153;The scaffolding metaphor was originally developed to describe the support given by a more expert individual in a one-on-one interaction.â&#x20AC;? The notion of scaffolding, also known as learning cycle (Font, 2006) or the gradual release model (Fisher and Frey, 2008), can also be applied to the didactics of foreign language. In this article, scaffolding in the field of language teaching is defined as the interrelated, systemic and paced learning cycle that is composed of stages language learners go through with the ultimate goal of becoming functional in communication.

The teaching-learning cycle in languages has been scaffolded in different ways by prominent authors. Many proponents of learning cycles depart from a first stage that is teacher-centered and signalled by a controlled practice of language items. 1. Keith Johnson (1982): 1. Mechanical exercises, 2. Mechanical-communicative exercises, 3. Communicative practice. 2. Hector Hammerly (1982): 1. Mechanical exercises with primary attention to form., 2. Meaningful exercises with equal attention to form and meaning, 3. Communicative activities with primary attention to meaning. 3. Donn Byrne (1989): Practice in the reproduction of the fixed elements of language, 2. Opportunities for the expression of personal meaning. 4. Crookes and Chaudron (1991): 1. Controlled techniques, 2. Semi-controlled techniques, 3. Free techniques.

If we analyse one of the principles of foreign language teaching 5: the determining role of the need for communication in a concrete social situation, we understand that we cannot depart from something that is not communication itself. These cycles are not in tune with the principle mentioned because their main focus in the first stage is structural components. Communicative language teaching follows the logical aspects of real communication, first we activate a plane of 5

The communicative approach has been widely spread in the last twenty to thirty years. Communicative language teaching has attempted to change the focus of foreign language teaching for many years, which goes beyond the mastery of linguistic rules, and takes into consideration contextual, semantic and strategic elements. (Hymes, 1972), (Savignon,1976), (Richards, 2006).

143


content (intentions, context, speakers, etc.), and later we move toward the plane of expression (grammar, vocabulary, prosodic elements, etc.). On the other hand, a very common learning cycle has been the presentation, practice and production model (PPP). According to Harmer (2001) it is â&#x20AC;&#x153;used as a main default for the new teaching of new language formsâ&#x20AC;?. It is split into three phases that moves from a strict teacherâ&#x20AC;&#x2122;s control in the presentation stage, towards a freer learner in the practice stage. Very similar to this model is the gradual responsibility model also known as I do, you do and you do (Pearson and Gallagher, 1983) (Fisher and Frey, 2008). The PPP model has been highly criticized because it is teacher-centered (Harmer, 2001). Likewise, the second model faces the same problem due to its similarities with the first one. They both share a teacher who starts modelling stage one. Thus, the role of the student is null in this moment, depriving him/her of any participation. These two models do not let the student be an active participant from the first moment of the learning cycle. The current situation brings us to a point of interest in relation to the way we can organize a teaching-learning process that better responds to the educational needs of our society, one that is tailored to the specific needs of our didactic process, depending on our school, our classroom, the subject we teach. For this article, the focus will be on how to create a learning cycle for the development of the teaching-learning process of speaking skills.

What characteristics should this learning cycle have in the didactic process of speaking skills? What will make the difference from other existing models? In this article, some questions will be answered considering linguistic, didactic and psycholinguistic factors that affect the learning process. What should be the departing point in a learning cycle? What is the role of the teacher and the learners in each stage of the learning cycle? What needs to be taught first in the language learning cycle, grammar or communicative functions? Is it possible to follow this scaffolding model other language skills (listening, reading and writing)? The first element to mention is that unlike other cycles, this one will begin with a communicative practice, with an interaction among the students and the teacher. By using the problem-solving method, the teacher will communicate knowledge in a dynamic way. The teacher will present students with tasks that interest them and leads to find ways and means to solve them, which favors 144


not only the acquisition of new knowledge, but also methods of action and research 6. Problem solving methods use the heuristic conversation as a technique. The teacher interacts with students asking questions that spark their interest. Singer et al (2013) explain that if we go back in time, Socrates established an efficient method of learning through a continuous dialog based on posing and answering questions to stimulate critical thinking and illuminate ideas. The proposal of a learning cycle presented now is based on a problem-solving conception of the communicative approach in a context where English is taught as a second language. It is divided into six stages: 1. Initial problemic communicative practice: In this initial communicative activity the students use the communicative resources that they already have to interact about a topic that gradually leads them to feel the need to use the new elements of the language to express their ideas. The elicitation of the new elements, provoked by the teacher, creates an additional motivational element that allows better learning because the new elements become necessary and meaningful, and the students face a problemic-communicative situation which they cannot cope with just with the knowledge and abilities they already have. The practice is communicative because the teacher allows the students to use the linguistic resources freely. He does not impose anything, nor does he demand the use of one structure or another. Instead, he becomes another participant in the process, guiding the students towards the need of the new knowledge. During this initial practice, the student gets deeply involved in a communicative activity from which contradictions arise in one or more of the components of communicative competence. The problemic situation occurs in the studentsâ&#x20AC;&#x2122; mind as a result of the contradiction. For example, in the initial problemic communicative activity, the teacher presents a communicative situation in which the students talk about their daily activities. The teacher gets the students to use the simple present. Then, the teacher asks about the day before. The students do not know the simple past in English from the morphological point of view. They are only familiarized with time expressions such as yesterday, last night, among others.

6

For more information about the problem-solving method see MartĂ­nez (1998). For problem-solving in foreign language teaching refer to Font (2006).

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Then the student says: ‘I watch TV yesterday’. This utterance is the most logical thing to say, assuming that the student only knows the present tense. Then, the teacher indicates that the answer is incorrect. The student faces a contradiction: between the knowledge he has acquired up to this moment, and the fact that the teacher is telling him that his answer is incorrect. Why is it incorrect? What is the correct way? Other students participate in the conversation providing utterances such as: “I play baseball with my brother.” “I listen to music at home.”, to refer to actions that happened the day before. The problemic situation is created: using what I know, I cannot communicate what I am trying to say. How do I figure this out? 2. Definition, analysis and solution to the problem: This moment of the process is favorable to the student’s immersion in a situation in which he has to solve a problem. The new communicativelinguistic element can bring about contradictions with what was previously learned. The contradiction is assimilated by the student, who turns it into a problemic situation in his mind. The problemic situation is what is unknown to him, and he cannot solve just from his experience. When the student understands what he should look for, then he has defined his problem. The teacher isolates the example “I watch TV yesterday.” on the board and includes some others that are similar, always trying to bring examples not coming from the book, but from the problemic-communicative practice situation described in the first stage of the cycle. The student discovers the regularities, the rule, departing from different problemic methods, mainly partial search (guided discovery) and heuristic conversation. This is a stage where consciousness plays a governing role. In the analysis of a communicative linguistic phenomenon, its use must be contextualized in specific social situations, that is, the analysis should not cover only form, but meaning and use as well 7. 3. Controlled practice aimed at form: This stage turns out to be vital for habit formation after the phenomenon has been understood. Stage No. 2 does not in itself make it possible for the students to talk, nor does its combination with this third stage achieve it by itself either. The most important focus is form: accurate reproduction of grammatical patterns and pronunciation. It is important that

7

In this stage the contradiction may be due to negative transfer from the student’s mother tongue. The teacher can resort to the linguistic comparative method in order to explain the cause of the error. See Hernandez (2014).

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the student knows what he is saying in every moment, and that the teacher be able to listen to one of them among the others, to make the necessary corrections. In this stage the teacher has to correct, in order to guarantee a minimum of success in later stages in which the content of what is being said is as important as its form of expression. As Wilga Rivers (1987) sees it teaching grammar “should be done through providing activities that enable people to perform rules so that they are actually becoming familiar with the structures and accumulating a performance memory and integrating the material into their semantic networks”. It is useful to remember that it is about teaching English for Speakers of Other Languages (ESL/ESOL) and therefore this familiarization process with structures that Wilga Rivers refers to entails major significance.

4. Guided practice: Unlike controlled practice, in this fourth stage the teacher gives rise to the use of the linguistic element without making explicit his intention that the student should use it. The student faces exercises demanding the use of the new linguistic element, not in a mechanical way, but “almost” communicatively. This type of practice has also been called pseudo-communicative exercises, and it constitutes the so-called missing link that existed when the student was expected to communicate fluently only by doing mechanical repetition exercises. This practice is carried out in a gradual way, that is, the previous exercise is more guided than the next one, until the student is able to get close to using the linguistic element freely and automatically. A well-known example is the exercise named “Find someone who”. The students receive instructions to look for unknown information from one or more persons in the classroom. To accomplish his task, the student has to ask all over the classroom until he finds the person(s) he is looking for. The questionnaire can include several questions, involving one or more verbal tenses, and one or more intonation patterns, according to the content that is being practiced. Guided practice is very favourable for the development of problemic situations resulting from contradictions in the sociolinguistic and sociocultural components. It is in this stage of the cycle that the student is gradually aware of the social implication of the linguistic element he uses, and how the social and cultural contexts influence the selection of certain linguistic structure by the speaker or writer. A very useful guided practice exercise is role play. Role play is an eminently communicative task and therefore it is based on information, opinion or judgment gaps. However, this type of exercise

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demands preparation, mainly from those students whose communicative skills are not highly developed, and they have a poor command of the vocabulary. For example, there is a role play exercise with two characters: A and B. There are several couples in the classroom that use the same role play exercise, which means that there is more than one student A and one student B. The final objective of this exercise is that A and B interact without previous knowledge of what they will tell each other. This objective is not achieved many times, because either A or B cannot say anything, because their development does not allow them to. Role-play rehearsal allows two or more students A, and two or more students B to practice together. This exercise permits A1 to solve a communicative problem assigned in his task, and if he has any difficulties, A2 can help him get to a higher level, which might not be as high as the desirable one, but it is at least close to it. In this example, A2 is “the other” (as defined by Vygotsky 8), that is pulling up A1, who can slowly climb up the steps of his own zone of proximal development, that is not the same as that of A2. 5. Integrated free practice: Unlike in the initial practice, it is in this stage that the student is ready to get involved in a free-expression communicative activity, in which he puts into practice the skills developed in the new unit together with the other skills that were previously acquired. 6. Creative application: This stage takes place mainly outside of the classroom. In this stage the student applies the acquired knowledge and abilities to solve communicative problems by himself, in the social context he is preparing to face. Learning does not end in the classroom. If the five previous stages are satisfactorily carried out, in a way that the student can learn how to solve problems without the teacher’s permanent guidance, then he will be able to go through the sixth stage. How can we guide the student’s activity in this stage? Guidance has to be done in a way that the student puts into practice creatively, the cultural and linguistic contents, communication skills, learning strategies, and the values and modes of action acquired in the classroom.

8

This concept is built upon the Zone of Proximal Development, one of the notions created by Vygotsky (1981). He

is also known for his sociocultural theory in learning through social interactions.

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It is important to reassert the fact that this stage implies creativity. But what does being creative mean in foreign language learning? According to Font (2006) creativity in foreign language learning can be defined as the student’s capacity to implement the linguistic-discursive, cultural and sociolinguistic knowledge and abilities, and his communicative strategies to solve communication problems on his own. It should be carried out in an independent and not reproductive way so that the process is creative. They should be able to interact correctly and appropriately in situations that can be distant from those used in the classroom. The student’s performance will be more creative inasmuch as the communicative situation moves away from what was exercised, reproduced and produced in the classroom. The creative application tasks will include: objective, content to put into practice, context where they will take place (temporal, spatial, relational and material), activities to be done by the students, and the way the professors will get feedback. The creative application tasks may not be the same for all the students. It is necessary to consider the students’ abilities, their individualities in terms of their likes, material conditions, and their will to carry out the tasks. It can be possible that some students in the group do not go through the sixth stage. Creativity in foreign language learning is not accomplished at once; it is a process that takes a lot of time and effort by all the professors and students. Each of these stages is assessed by the teacher along their development. The process either moves ahead or goes back to a previous step, depending on this continuous feedback. This constant feedback is very important so that the process does not develop on weak foundations. It does not mean that each stage should be limited clearly. It is possible and logical that all the stages interweave, because the foreign language teaching-learning process is characterized by the presence of transfers, moving from one stage to another is not merely a quantitative change, but rather an eminently qualitative progress. Besides, a language lesson, however thoroughly planned, is subject to the appearance of unexpected situations as a result of the student’s experiences that can lead the teacher to integrate stage 1 into stage 2, or just speed up the third stage because it is not necessary to make it longer in the presence of quick assimilation. The whole process is accompanied by feedback, correction (with different procedures according to the stage), and 149


assessment (as the qualitative evaluation of individual learning and of the process in general, rather than a measurement in itself). The picture below is a visual representation of the six-stage learning cycle.

The author of this paper considers this learning cycle to be the most suitable among all the models analysed for several reasons. First, it starts with communicative practice and interaction between the teacher and the students, unlike the other more restrictive models where the teacher begins modelling and the students are passive receivers. Second, stages three and four are shown separately with different objectives and procedures. This distinction is very important from a methodological point of view, because it makes the professors aware of the need for guided practice with very specific tasks. Finally, it includes a new stage of creative application, which takes place mainly out of the classroom. In this stage the students will put into practice their creativity in solving problems in communication. Now it is time to illustrate how the scaffolding process can occur in the learning process of speaking skills of a given unit. The tasks in the unit presented are aligned, graded and systematically connected. Every stage of the cycle can have one or more tasks. Each task will have

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6 components: goal, input, activity, teacher role, learner role and setting. The components of the tasks were adapted from Nunan (1989).

STAGE 1: Initial Communicative Practice Task No. 1 Goal: The students should be able to: •

Interact in oral communication using the communicative functions learned in previous units.

Feel the need to use new communicative contents related with the topic of places they have been to.

Identify and reproduce new language items related with the new communicative functions.

Input: The initial input will consist in a key expression written on the board: Places you have been to

Activities: The teacher will ask the students about places they have been to, whether in the country or abroad. The teacher will write a few questions on the board, linking them together with the initial key expression he had written. The students will mention “PLACES THEY HAVE BEEN TO”. The teacher will then elicit from the students as much information as possible on the issue of places in the country or abroad. Learner role: The learners will get involved in a communicative activity in which they will play a very active role, contributing ideas and interacting with their classmates and teacher. Teacher role: The teacher will basically aim at eliciting information from the students so that they can put into practice the knowledge and skills previously acquired. He will take part in the conversation and will ask questions to encourage the students to participate. Setting: Whole class and pair work.

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STAGE 2: Definition, analysis and solution to the problem

Task No. 2 Goal: The students should be able to: â&#x20AC;˘

Infer the Present perfect form in yes or no questions and short answers. Some past participles of irregular verbs. Just with recent past.

â&#x20AC;˘

Identify and solve the contradictions in the use of the new grammatical structure, involving not only morphology and syntax but pronunciation and lexical items as well.

â&#x20AC;˘

Discover the rule and formulate it in such a way that it is useful for them to generate new utterances.

Input: A set of examples coming from the discussions held in the previous stage of the cycle for this unit. These examples may have been provided by the students themselves in their interaction with each other, or by the teacher in the same process. Activities: The teacher or a student will write some of the examples on the board. These examples will include the linguistic structures that include places they have been to, present perfect forms, and some past participles to fulfil the communicative function and provide as much information as possible about places in the country or abroad. The teacher will guide the students towards discovering the rules that govern the use of these linguistic structures, through a process of analysis and synthesis. The students will state the rules, provide new examples of their own and will be made to repeat some of the examples to practice pronunciation. Learner role: The students will get involved in an active process of thinking, directed at discovering the rules underlying the linguistic structures presented. Teacher role: The teacher will play an important role in leading the students through a process of discovery. The teacher may provide some grammar books to help the students find the rule themselves and provide new examples. He will make any contradictions apparent to the students and will help them resolve them. Setting: Whole-class or group work.

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STAGE 3. Controlled practice aimed at form Task No. 3. Adapted from Spectrum 3, Unit 1 (Byrd,1987) Goal: The students should be able to: •

Paraphrase the following communicative functions: Introduce someone, talk about places you have been to, talk about recent past, offer help.

Drill the following grammatical structures: present tense, some past participles of irregular verbs, “just” with recent past.

Imitate the model provided of prosodic elements.

Input: a picture, a written dialogue from Spectrum 3, Unit 1 “Taking off” (Presentation dialogue) Activities: The system of activities should be done in every segment of the dialogue. 1. Warm up: The teacher will ask the students where they would like to go if they were to travel that day. Everybody will mention different places. 2. Then, the teacher will have the students work with the introductory speech, to try to anticipate what the dialogue will be about. 3. The teacher will start with the artwork. He will ask questions of the students about the picture in order to have them predict what might happen in the conversation. 4. The teacher will assign the first listening and the students will check if what they anticipated previously has something to do with the real listening. (All the conversations will be covered). The teacher plays the first listening and after that asks the students general comprehension questions. 5. The teacher will assign more specific questions about the first segment. Then, he plays the second listening. 6. Then the teacher will have the students decipher each utterance. He will ask the students to identify every utterance they hear in the conversation. 7. The teacher will make the students repeat what they hear. Repetition will be made in groups, in pairs or individually, always trying to make the students pronounce as closest to the native model as possible, for which effective techniques of phonetic correction will be used in order to achieve accuracy. 153


8. Then the teacher will ask the students to paraphrase by using simple substitution exercises. 9. The teacher will start the “Time to Ask” section 9. He will prepare questions about the conversation in indirect speech so that the students can build up the different question patterns in direct speech. This is a very important moment for the students since asking questions is one of the weaknesses they have. Ask X if: the woman is a friend of the man’s/the woman spent a few days with her husband/ Laura is going back to Chicago/it has been a long time they haven’t seen each other/the man started a new computer business/Laura was told about the man’s new job by his mother. 10. The students will follow the reading as they listen to the dialog in order to become aware of the phoneme-grapheme correspondence in speech. 11. The students will make a brief narration of the segment. The same steps will be followed for the other four segments, except for the “Time to Ask” section. Learner role: The students have a negotiating and active role in the learning process. Teacher role: The teacher will play the role of a guide in this activity. He will provide assistance to the students in carrying out their activities. He will make the students use specific language items. He will make corrections whenever it becomes necessary to fix problems that may occur in communication. The teacher will focus on accurate drilling of grammatical patterns and pronunciation they use as they speak. The teacher will encourage the students to speak English all the time, and will elicit from the learners other relevant questions and comments. Setting: Individual and pair work. STAGE 4. Guided Practice Task No. 4 Goal: The students should be able to:

9

This is a vital activity that must be done in speaking. Asking for information is macro-communicative function in conversations. Many students make mistakes when asking questions in the learning process of English due to new formal features of these types of utterances, which are different from the structures they use in their mother tongue. For example, asking questions is very difficult for students who speak Spanish or French as their mother tongue mainly due to the differences in structural features. (Hernandez, 2014).

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Use the vocabulary and grammatical structures learned in the unit in pseudocommunicative situations in which they interact with their classmates with some teacher guidance.

Input: The students will work with a guide to a find-someone-who exercise. This guide will include the most important items about which they will have to ask their partners. Find someone who: has visited Tallahassee, has been to Miami Beach, has seen a performance at the American Airlines Arena, has ever been abroad, has spent all his/her vacation out of his house. Activities: A find-someone-who activity in which the students will interact in a question-andanswer process on the topic of places and personal relationships. Learner role: A very active role, in which they will have to encode questions based on the items in the guide, and decode and jut down their partners’ answers. Teacher role: The teacher will act as one more participant in the information quest. He will monitor the students’ performance and will make all necessary corrections and explanations. Setting: Whole class. STAGE 5 Integrated free practice Task No. 5 Goal: The students should be able to: •

Use the communicative contents of the unit without much teacher guidance.

Interact orally, playing roles that include the communicative functions learned throughout the unit

Input: Role-play situations. A1: You are walking on a street in Miami Downtown and suddenly run into another friend of yours. Greet him / her, talk about the things you have done and places you have been to. You have been to the following countries lately: Mexico, Belize and Honduras.

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A2: You just came from Paris because you work at the American Embassy there. You stop at a place in Downtown Miami to buy something when suddenly you see a face that looks familiar to you. Talk about your lives and also about the places you have been to. Activities: The teacher will place strips of papers with the role-play situations on a table. The students will sit in isolation and they will have five minutes to prepare. In pairs, the students then act out the roles that they have been given for about 10 minutes. They will discuss what has happened in their lives and talk about places they have been to. During the conversation the students will focus on language form, including the syntax, morphology and pronunciation of the language items studied in previous lessons. Learner role: The student plays an active role in the process, interacting with his partner. Teacher role: The teacher will play a guiding role, making sure that the students comply with the task given, and supplying them with language items that may become necessary to fulfill their communicative task. He can get involved in the activity to encourage the students. Setting: Pair work. Stage 6. Creative application Task No. 6 Goal The students should be able to: â&#x20AC;˘

Interact in oral communication applying the knowledge and skills developed throughout the unit.

Input: There are two kinds of input for this task. The first input is directly connected with the first goal declared for this task. The teacher provides the students with a list of items they should use to interview an airline pilot. This list of items may, of course, be enriched with the studentsâ&#x20AC;&#x2122; own contributions. This list will not include the direct questions the students will ask. It will consist 156


of a number of items which are crucial to obtain the information necessary to fulfil the second goal of the task. The input to fulfil the second goal will come from the answers the pilots will provide. List of items for the interview: Full name/ Marital status/Children/Date of graduation/Special characteristics to be a pilot/Any good or bad experience he/she had in his/her professional life/ Training or studies abroad, if any (where, when, how long, language spoken and experiences with it, experiences as a pilot) Activity: The students will interview a pilot who speaks English. They will ask him/her as many questions as possible. They will write down all the information obtained in the interview, and will come to the classroom ready to talk about the pilotâ&#x20AC;&#x2122;s career. Learner role: The students have an active role in the process and they have control over the activity. They will have freedom of choice for linguistic forms. Teacher role: The teacher participates in the orientation of the task, but has little control over its implementation. Setting: The orientation to carry out the task will be done in the classroom with the teacher. The actual interview will take place out of the place out of the classroom. The results of the interviews will be checked in the classroom either in small groups or in a whole-class setting. Conclusions The learning cycle proposed in this article has been designed for speaking skills. It was possible to examine some teaching models found in foreign language teaching literature. These models have not shown full functionality because from the very first stage the departing point is the analysis of structural components and not communicative functions. In addition, the first stage in the learning cycles analysed here is often too teacher-centered, with an instructor who is modelling and telling what students need to do. Thus, problem-solving and creativity are hindered. The model conceived here does not end in the classroom, but outside of it, where the student needs to interact alone with other people using the linguistics and communicative resources previously learned in class. It is strongly recommended that new techniques and activities for speaking skills are integrated into this model or a similar one, so the student can go through a gradual and paced process in the 157


way they learn. This scaffolding process can be can be adapted to the rest of the other language skills considering their particularities. It might also be adapted to other languages and pedagogical areas.

Acknowledgements In this article I would like to thank Dr. Sergio Font for his supervision and leadership in the field of foreign language teaching. Special thanks to all my colleagues in Cuba, the USA, IATEFL, TESOL and MDTESOL for the learning opportunities as a professional. The author would like to thank Nora Hentschel (MA in TESOL) and Liliana Cobas (Chairperson of World Languages and Communication, MS in Applied Linguistics), for their comments and suggestions after revising this article. Special thanks to Miami Dade College and Florida International University that have given me the chance to enjoy the teaching-learning process of language teaching, as well as the possibility to apply the results of my research in my pedagogical practice. Finally, special thanks to all my students for the chance to share the teaching-learning process with them, our Institution Miami Dade College for the opportunity to teach and do research here, and to the Editorial Committee of Polygon for their suggestions to improve the design of this paper.

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Statistical Analysis of Nutrition Facts of Some Common Foods - A Computer Project Assignment for teaching STA 2023 Course using Minitab and Statdisk Software* Dr. M. Shakil, Ph.D. Professor of Mathematics Department of Liberal Arts and Sciences - Mathematics) 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 statistical methods using Minitab and Statdisk software can help the students to gain the knowledge and insights from various aspects of nutrition facts data, such as exploratory data analysis, studying and conducting statistical analysis of the sodium, carbohydrates and calories, among others, contained in some common foods. Such studies are very important in view of the facts that there is a great emphasis on health literacy and redefining health these days. An introductory course in statistics such as Statistical Methods (STA 2023) at Miami Dade College can easily provide such avenues to our students. In any Statistical Methods (STA 2023) Class, the students are first taught the frequency distributions, statistical graphs, basic descriptive statistics, that is, center, variation, distribution, and outliers, and exploratory data analysis (EDA), which are important tools and techniques for describing, exploring, summarizing, and comparing data sets. 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. Therefore, this paper shows how to use nutrition facts data to teach Statistical Methods (STA2023) course using Minitab and Statdisk software. 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.

Keywords: Health, Lesson Plan, Minitab, Nutrition Facts, Statistical Methods, Teaching. 2010 Mathematics Subject Classifications: 97C40, 97C70, 97D40, 97D50.

* Part of this article was presented during the EEI Workshop â&#x20AC;&#x153;Health: Connecting People, Places, and Planet Part I Series (EEI1011-3)â&#x20AC;?, held from September 25 to November 06, 2017, at the Hialeah Campus of Miami Dade College.

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1. Introduction: In recent years, there has been a great emphasis on redefining health and well-being of human beings, not in U.S.A. only but in all countries of the world. These are the issues of emerging international importance and significance. As pointed out by Bircher (2005), “Researchers, policymakers, and healthcare professionals of all political stripes agree that we are in the midst of a multifactorial and growing crisis of health care systems”. Further, according to Badash et al. (2017), “Since the earliest days of mankind, cultures around the world have sought the elusive understanding of what it means to be healthy. This definition has evolved many times, often reflecting the specific beliefs and the levels of scientific and medical understanding of that particular era. Understanding these changes provides a context for the new definition that is needed in the present age”. It appears from the above studies that considerable research into various aspects of issues related to assessing the quality of life and health has been conducted. Many researchers and authors have also developed a number of models to predict the risk of mortality, incapacity and health in children, young and adults, see, for example, McClintock et al. (2016). However, “Health is more than just the absence of disease. Health also includes physical, psychological, and social well-being. Many traditional models of health, however, focus mostly on conditions such as diabetes, cancer, and heart disease”; cf. https://www.nih.gov/news-events/nih-research-matters/redefining-health-well-being-older-adults (2016). There are many factors that affect the health and well-being of human-beings. For, example, it appears from literature that many people suffer from the lack of health awareness and health literacy, see, for example, Osborn et al. (2007), among others. Moreover, they do not have sufficient knowledge of “nutrition facts” of the foods they consume on daily basis. This is despite the fact that most of the food items we consume contain nutrition facts labels these days. As pointed out by Sharif et al. (2014), “The Nutrition Facts label was mandated by the Nutrition Labeling and Education Act (NLEA) of 1990 requiring nutrition labeling on most packaged foods. The label was modified to facilitate consumer use in 1993. In March 2014, the Food and Drug Administration (FDA) proposed significant changes to the Nutrition Facts label for the first time since it was created over 20 years ago. The proposed changes were motivated by findings in nutrition science that have advanced our understanding of how diet impacts health as well as by data documenting trends in dietary practices and chronic disease in the United States. The proposed changes, explained in detail elsewhere, are intended to improve both the content provided and the presentation of the information in order to aid consumers’ interpretation of the nutritional quality of the food item. The original purpose of the Nutrition Facts label, however, remains the same: to provide information about the nutritional characteristics of foods, in order to facilitate healthy dietary choices. Research has demonstrated that utilization of the label is associated with healthier dietary habits including reduction of fat and overall energy intake and an increase in fruits and vegetables. Yet, rates of utilization of the information on the label are low and limited comprehension of the label is the most commonly cited barrier to using it. Moreover, levels of utilization and comprehension are lower among vulnerable subgroups including ethnic minorities, low-income populations and people who have not completed high school”. For details, the interested readers are referred to Sharif et al. (2014). As stated above, these nutrition facts labels may enable us to evaluate the healthiness of the food products before we buy them or consume them. For example, the nutrition facts labels may contain the following food components: total fat, saturated fat, cholesterol, total carbohydrate, 162


dietary fiber, sodium, potassium, and protein, among others, the knowledge of which are necessary before their intake as severe consumption of some of these may be harmful to our bodies. For the sake of completeness, these food components are illustrated in the following Figure 1 of a typical “Nutrition Facts Label", which contains the “Amount Per Serving”, including the “% Daily Value”, among others of these food components; (cf. https://www.fda.gov/Food/GuidanceRegulation/GuidanceDocumentsRegulatoryInformation/ LabelingNutrition/ucm063113.htm).

Figure 1: The Nutrition Facts Label “Nutritional information is based on product as packaged 101.9(b)(9)” (Source: https://www.fda.gov/Food/GuidanceRegulation/GuidanceDocumentsRegulatoryInformation/ LabelingNutrition/ucm063113.htm)

Thus, motivated by the importance of “nutrition facts” of the foods we consume, this project aims at studying and conducting statistical analysis of some of the nutrition facts of some common foods which most people consume very frequently on daily basis. It should be pointed out that “Statistics is one of the important sciences at present”. In order to summarize any type of data, we cannot underestimate its role, uses and importance in the modern world. There is a great emphasis on statistical literacy and critical thinking in education these days. An introductory course in statistics such as Statistical Methods (STA 2023) at Miami Dade College can easily provide such avenues to our students. In any Statistical Methods (STA 2023) Class, the students are first taught the frequency distributions, statistical graphs, and basic descriptive statistics, that is, center, variation, distribution, and outliers, which are important tools and techniques for describing, exploring, summarizing, comparing data sets, and other aspects of exploratory data analysis (EDA). Due to the tremendous development of computers and other technological 163


resources, the use of Statistical Software for data analysis, in a Statistical Methods (STA 2023) class, 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 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/). This paper develops a lesson plan how to use nutrition facts data to teach Statistical Methods (STA2023) via Minitab software. For details on Minitab demonstration with examples, the interested readers referred, for example, to McKenzie and Goldman (2005), Bluman (2013), and Triola (2010, 2014), among others. The organization of this paper is as follows. In Section 2, the uses of Minitab and Statdisk in teaching Statistical Methods (STA 2023) courses are presented. Section 3 contains the applications of Minitab and Statdisk in the statistical analysis of nutrition facts data in teaching Statistical Methods (STA 2023) courses. The concluding remarks are given in Section 4.

2. Uses of Minitab and Statdisk: This is a computer project on the statistical analysis of some nutrition facts using Minitab and Statdisk. It is expected that the students have already learned about the following topics in the class: •

Exploratory Data Analysis (EDA): The five-number summary, namely, the minimum value, the 1st quartile, the median (i.e., the 2nd quartile or the 50th percentile), the 3rd quartile (i.e., the 75th percentile), and the maximum value are used to construct the boxplot.

Measures of central tendency (namely, mean, median, and mode) are used to indicate the “typical” value in a distribution. A comparison between the median and mean are used to determine the shape of distribution, while the mode measures the most frequently occurred data.

Measures of dispersion or variation (namely, range, standard deviation, and variance) are used to determine the “spread out” of the data.

Some statistical graphs, for example, histograms, can also be used to describe the shape of distribution

2.1. Minitab and Some of its Special Features: In what follows, we will discuss some special features of Minitab and its use for constructing frequency distributions, histograms, descriptive statistics and exploratory data analysis (EDA). Minitab is one of the most important statistical 164


analysis software. When Minitab is opened, there are two windows displayed first, as described below: • •

Session Window: It is an area which displays the statistical results of the data analysis and can also be used to enter commands. Worksheet Window: It is a grid of rows and columns (similar to a spreadsheet) and is used to enter and manipulate the data.

There are also other windows in Minitab as given below: • • •

Graph Window: When graphs are generated, each graph is opened in its own window. Report Window: There is also a report manager which is helpful in organizing the results in a report. Other Windows: Minitab has also other windows known as History and Project Manager. For details on these, the interested readers are referred to Minitab help.

2.2. Minitab Tools (or Menus): In general, various tools (called menus) of Minitab are described in the following Table 1, which, when Minitab is opened, are displayed in the menu bar on the top. For statistical data analysis, these menus can be used appropriately by following the instructions provided in Minitab help; also see McKenzie and Goldman (2005), Bluman (2013), and Triola (2010, 2014), among others. Table 1 File Menu

Edit Menu

Data Menu

Calc Menu

Stat Menu

Open and save files; Print files; etc.

Undo and redo actions; Cut, copy, and paste; etc.

Undo and redo actions; Cut, copy, and paste; etc.

Calculate statistics; Generate data from a distribution; etc.

Regression and ANOVA; Control charts and quality tools; etc.

Graph Menu

Editor Menu

Tools Menu

Help Menu

Scatterplots; Bar charts; etc.

Graph, Data, and Session window editing; Modify active window; etc.

Window Menu Arrange windows; Select active window; etc.

Change Searchable Minitab Help; defaults; StatGuide; etc. Create and modify toolbars and menus; etc. 2.3. 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. 165


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.

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. Applications of Minitab and Statdisk in the Statistical Analysis of Some Nutrition Facts Data: Redefining health and well-being of human beings are important issues of international significance. As pointed out by Badash et al. (2017), “Since the earliest days of mankind, cultures around the world have sought the elusive understanding of what it means to be healthy. This definition has evolved many times, often reflecting the specific beliefs and the levels of scientific and medical understanding of that particular era. Understanding these changes provides a context for the new definition that is needed in the present age”. It appears from literature that considerable research into various aspects of issues related to redefining health and well-being of human beings has been conducted. For example, the nutrition facts 166


(namely, total fat, saturated fat, cholesterol, total carbohydrate, dietary fiber, sodium, potassium, and protein, among others) of many commonly used foods items, is an important area of research, since the knowledge of these are necessary before their intake as severe consumption of some of these may be harmful to human bodies. Many researchers have examined and analyzed different aspects of nutrition facts, both statistically and experimentally. A majority of statistical studies, whether experimental or observational, are comparative in nature. The simplest type of a comparative study compares two populations based on samples drawn from them. As redefining health are fundamental issues of well-being of human beings, this project aims at studying and conducting statistical analysis of some of the nutrition facts of some common foods which most people consume very frequently on daily basis. These are described below. To access to these data, the interested readers are referred to Bluman (2013). Minitab is an efficient and effective tool for analyzing such data. As such, we shall use Minitab for constructing frequency distributions, generating histograms, finding basic descriptive statistics and exploratory data analysis (EDA), for the analysis of some nutrition facts data, which are presented in the following paragraphs. 3.1. Frequency Distributions and Graphs using Minitab: In this subsection, we explore traditional statistics, that is, data are organized by using a frequency distribution. Then, from this Distribution, the graph such as the histogram is constructed to determine the shape or nature of the distribution. In addition, various statistics such as the mean and standard deviation have been computed to summarize the data. We illustrate these through the following example. Example (Analyzing Calories of Some Food Items): Using Minitab, a dietitian is interested in analyzing calories of some food items by constructing a frequency distribution and presenting the data by constructing a histogram. The data is given in Table 2 below (Source: https://neosguide.org/content/diet-problem-solver). The frequency distribution is given in Table 3 and the histogram is provided in Figure 2. The descriptive statistics of calories is provided in Table 4.

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Table 2 (Calories Information for Each Food) Name

Calories

Name

Calories

Frozen Broccoli

73.8

Butter, Regular

35.8

Carrots, Raw

23.7

Cheddar Cheese

112.7

Celery, Raw

6.4

3.3% Fat, Whole Milk

149.9

Frozen Corn

72.2

2% Lowfat Milk

121.2

108.3

Bologna, Turkey

56.4

Potato Chips, BBQ

139.2

Frankfurter, Beef

141.8

Poached Eggs

74.5

Scrambled Eggs

Spaghetti W/ Sauce

358.2

Tomato, Red, Ripe, Raw

25.8

Apple, Raw, w/Skin

81.4

710.8

Popcorn, AirPopped

20

277.4

Pork

99.6

Peppers, Sweet, Raw

Roasted Chicken

188.5

115.6

85.5

88.2

Peanut Butter

White Tuna in Water

Skim Milk

Tofu

98.7

49.9

2.6

171.5

Macaroni, cooked

Calories 100.8

Sardines in Oil

Lettuce, Iceberg, Raw

Potatoes, Baked

Name Couscous

Pretzels

108 142

Ham, Sliced, Extralean

37.1

Tortilla Chips

Kielbasa, Pork

80.6

Chicken Noodle Soup

150.1

Cap'N Crunch

119.6

Splt Pea&Ham Soup

184.8

Veggie Beef Soup

158.1

Cheerios

111

Banana

104.9

Grapes

15.1

Corn Flakes, Kellogg's

110.5

New Eng Clam Chwd

175.7

Kiwifruit, Raw, Fresh

46.4

Raisin Bran, Kelloggâ&#x20AC;&#x2122;s

115.1

Tomato Soup

170.7

Oranges

61.6

Rice Krispies

112.2

New Eng Clam Chwd, w/Mlk

163.7

168


Table 2 (continued) (Calories Information for Each Food) Name

Calories

Name

Bagels

78

Wheat Bread

65

White Bread

65

Oatmeal Cookies

81

Apple Pie

67.2

Chocolate Chip Cookies

78.1

Calories

Special K

110.8

Oatmeal

145.1

Malt-O-Meal, Choc

607.2

Pizza w/Pepperoni

181

Hamburger w/Toppings

275

Hotdog, Plain

Name Crm Mshrm Soup, w/Mlk

Calories 203.4

Bean Bacon Soup, w/Watr

242.1

Source: https://neos-guide.org/content/diet-problem-solver. Table 3 (Minitab Output of Frequency Distribution of Calories) CalCodes

Count (Frequency)

CumCnt

Percent

CumPct

2:103 104:205 206:307 308:409 410:511 512:613 614:715

28 28 3 1 0 1 1

28 56 59 60 60 61 62

45.16 45.16 4.84 1.61 0 1.61 1.61

45.16 90.32 95.16 96.77 96.77 98.39 100

169

172


Histogram (with Normal Curve) of Calories 40

Mean StDev N

131.1 118.9 62

Frequency

30

20

10

0

0

200

Calories

400

600

Figure 2: Minitab Output of Histogram of Calories

Table 4 Minitab Output of Descriptive Statistics of Calories Variable N N* Mean StDev Variance Minimum Q1 Median Q3 Maximum Calories 62 0 131.1 118.9 14135.9 2.60 71.0 109.4 159.5 710.8 Variable Range Skewness Kurtosis Calories 708.2 3.10 12.20

3.2. Exploratory Data Analysis (EDA) using Minitab and Statdisk: The concept of exploratory data analysis (EDA) was developed by John Tukey; see, for example, Tukey (1977). The objective of exploratory data analysis is to analyze data in order to find out information about the data such as the center and the spread. In this subsection, we discuss exploratory data analysis (EDA) using Minitab. For example, we organize data using a stem and leaf plot. We compute the median which is the measure of central tendency, and also we compute the interquartile range which is the measure of variation. Further, in EDA, we represent the data graphically using a boxplot (sometimes called a box-and-whisker plot). These plots involve five specific values, called a â&#x20AC;&#x153;five-number summaryâ&#x20AC;? of the data set, as defined below:

170


i. ii. iii. iv. v.

The lowest value of the data set, that is, minimum Q1 , called the first quartile Q2 , called the second quartile, or the median Q3 , called the third quartile The highest value of the data set, that is, maximum

We illustrate the above exploratory data analysis (EDA) through the following example taken from Bluman (2013). Example (Comparing the Sodium Content of Real Cheese with the Sodium Content of a Cheese Substitute): Using Minitab and Statdisk, a dietitian is interested in comparing the sodium content of real cheese with the sodium content of a cheese substitute. The data for two random samples are given in Table 5 below. The stem and leaf plots are provided in Table 6. In order to compare the distributions of two samples, the descriptive statistics and the five-number summaries of two data sets are provided in Table 7. The boxplots of two data sets are provided in Figure 3 using Statdisk. Table 5 Sodium Contents of a Real Cheese and a Cheese Substitute Real Cheese Cheese Substitute 310, 420, 45, 40, 220, 240, 180, 90 270, 180, 250, 290, 130, 260, 340, 310 Source: Netzer (2011). The Complete Book of Food Counts. Table 6 Minitab Output of Stem-and-Leaf Display: Real Cheese, Cheese Substitute Stem-and-leaf of Real Cheese Stem-and-leaf of Cheese Substitute N =8 N =8 Leaf Unit = 10 Leaf Unit = 10

2 3 3 4 4 2 2 1 1

0 0 1 1 2 2 3 3 4

44 9

1 2 2 (4) 2

8 24 1 2

171

1 1 2 2 3

3 8 5679 14


Table 7 Descriptive Statistics: Real Cheese, Cheese Substitute Variable RealCheese

N N* Mean StDev Minimum Q1 Median Q3 Maximum 8 0 193.1 133.2 40.0 67.5 200.0 275.0 420.0

Variable RealCheese

Range IQR 380.0 207.5

Variable N N* Mean StDev Minimum Q1 Median Q3 Maximum CheeseSubstitute 8 0 253.8 68.6 130.0 215.0 265.0 300.0 340.0 Variable Range IQR CheeseSubstitute 210.0 85.0

Figure 3: Statdisk Outputs of Boxplots of Real Cheese and Cheese Substitute Here Col 1): Real Cheese; Col 2): Cheese Substitute Comparison of Two Data Sets: We observe from the five number summaries of two data sets (Table 7) and the boxplots (Figure 3) that the median for the distribution for the cheese substitute 172


data is higher median than the median for the distribution for the real cheese data. Also, we observe that, since the interquartile range (IQR) for the distribution for the real cheese data is 236.3, which is greater than the interquartile range (IQR) for the distribution for the cheese substitute data, 107.5. Thus there is more variation or spread for the distribution of the real cheese data than the variation for the distribution of the cheese substitute data. 3.3. Confidence Interval Estimates and Hypothesis Test Mean â&#x20AC;&#x201C; Two Independent Samples: We illustrate this through the following example taken from Bluman (2013). Example (Comparing the Carbohydrates Content of Chocolate and Nonchocolate Candy): Using Minitab and Statdisk, a dietitian is interested in comparing the carbohydrates content of chocolate candy with the carbohydrates content of nonchocolate candy. The number of grams of carbohydrates contained in 1-ounce servings of randomly selected chocolate and nonchocolate candy is provided in Table 8 below. The Minitab outputs of the descriptive statistics of two samples are provided in Table 9 below. We want to test if there is sufficient evidence to conclude that the difference in the means is significant using a significance level of Îą = 0.10 . We conducted the test of hypothesis using both Minitab and Statdisk. Table 8 Chocolate Nonchocolate 29 41 25 41 17 37 36 29 41 30 25 38 32 39 29 10 38 29 34 55 24 29 27 29 Source: Borushek (2004). The Doctorâ&#x20AC;&#x2122;s Pocket Calorie, Fat, and Carbohydrate Counter. Table 9 Descriptive Statistics: Chocolate, Nonchocolate Variable N N* Mean SE Mean StDev Minimum Q1 Median Chocolate 13 0 29.69 1.80 6.50 17.00 25.00 29.00 35.00 Nonchocolate 11 0 34.36 3.38 11.20 10.00 29.00 37.00 41.00 Variable Maximum Chocolate 41.00 Nonchocolate 55.00

173

Q3


The Statdisk outputs of two-sample hypothesis test (t-test) and 90% confidence interval estimate of the difference of two population means, µ1 − µ 2 , for chocolate vs nonchocolate candy, using the no pool method of not equal variances, are provided in Figure 4 and Table 10 below:

Figure 4: Statdisk Outputs of Hypothesis Test Mean – Two Independent Samples (Chocolate vs Nonchocolate Candy) Table 10 Two-Sample T-Test and CI: Chocolate, Nonchocolate Method of Analysis: Not Equal Variances: No Pool Null Hypothesis, H 0 : µ1 = µ 2 Alternative Hypothesis, H 1 : µ1 ≠ µ 2 (Claim) Test Statistic, t: -1.2200 Critical t: ±1.749591 P-Value: 0.2408 Degrees of freedom: 15.4654 90% Confidence interval: -11.36745 < µ1 − µ 2 < 2.027448 Decision: Fail to reject H 0 : µ1 = µ 2 . Conclusion: There is not enough evidence to support the claim that the means are not equal, that is, µ1 ≠ µ 2 .

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4. Concluding Remarks: In this paper, we have discussed about developing a lesson plan to show how the teaching of basic statistical methods via Minitab and Statdisk software can help the students to gain the knowledge and insights from various aspects of nutrition facts data, such as exploratory data analysis, studying and conducting statistical analysis of the sodium, carbohydrates and calories, among others, contained in some common foods. It is hoped that this paper will be helpful in teaching any introductory course in statistics such as courses in Statistical Methods (STA 2023) at Miami Dade College using Minitab and Statdisk. 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 Minitab and 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 Minitab and 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. 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.

Acknowledgments The author is thankful to the editors of the Polygon for their suggestions, which improved the quality of the paper. Further, the author would like to thank the Earth Ethics Institute (EEI) of Miami Dade College for providing us an opportunity to attend the EEI workshop on “Health: Connecting People, Places, and Planet Part I Series (EEI1011-3)”, at Hialeah Campus. Also, the author is thankful to Miami Dade College for providing him an opportunity to serve as a mathematics faculty in the college at its Hialeah Campus, without which it was impossible to conduct his research. The author would like to thank his wife for her patience and perseverance for the period during which this paper was prepared. Lastly, the author would like to dedicate this paper to his late parents.

175


References

1. Badash, I., Kleinman, N. P., Barr, S., Jang, J., Rahman, S., and Wu, B. W. (2017). Redefining Health: The Evolution of Health Ideas from Antiquity to the Era of ValueBased Care. Cureus, 9(2), 2 - 9. 2. Bircher, J. (2005). Towards a dynamic definition of health and disease. Medicine, Health Care and Philosophy, 8, 335 â&#x20AC;&#x201C; 341. 3. Bluman, A. G. (2013). Elementary Statistics, A Brief Version, 6th Edition. McGraw-Hill Co., New York. 4. Borushek, A. (2004). The Doctor's Pocket Calorie, Fat & Carbohydrate Counter. Borushek, Allan & Associates, Incorporated, New York. 5. McClintock, M. K., Dale, W., Laumann, E. O., and Waite, L. (2016). Empirical redefinition of comprehensive health and well-being in the older adults of the United States. Proceedings of the National Academy of Sciences, 113(22), E3071 - E3080. 6. McKenzie, J. and Goldman, R. (2005). The Student Guide to MINITAB Release 14, 14th Edition. Pearson Addison-Wesley, New York. 7. Netzer, C. T. (2011). The Complete Book of Food Counts, 9th Edition. Dell Book, New York. 8. Osborn, C. Y., Weiss, B. D., Davis, T. C., Skripkauskas, S., Rodrigue, C., Bass, P. F., and Wolf, M. S. (2007). Measuring adult literacy in health care: performance of the newest vital sign. American Journal of Health Behavior, 31(1), S36 - S46. 9. Sharif, M. Z., Rizzo, S., Prelip, M. L., Glik, D. C., Belin, T. R., Langellier, B. A., and Ortega, A. N. (2014). The association between nutrition facts label utilization and

176


comprehension among Latinos in two East Los Angeles neighborhoods. Journal of the Academy of Nutrition and Dietetics, 114(12), 1915 - 1922. 10. Triola, M. F. (2010). Elementary Statistics, 11th Edition. Addison-Wesley, New York. 11. Triola, M. F. (2014). Elementary Statistics Using Excel, 5th Edition. Addison-Wesley, New York. 12. Tukey, J. (1977). Exploratory Data Analysis, Addison-Wesley, New York. 13. http://www.statdisk.org/. 14. https://www.fda.gov/Food/GuidanceRegulation/GuidanceDocumentsRegulatoryInformati on/ LabelingNutrition/ucm063113.htm. 15. https://neos-guide.org/content/diet-problem-solver.

177


Statistical Analysis of Some Greenhouse Gases and Air Quality Index Data A Computer Project-Based Assignment for Teaching STA2023 Course* Dr. M. Shakil, Ph.D. Professor of Mathematics Department of Liberal Arts and Sciences - Mathematics) 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 statistical methods, such as exploratory data analysis, statistical hypothesis, among others, using statistical software (Minitab and Excel), can help the students to gain the knowledge and insights from some air pollution data, such as the greenhouse gases emissions and air quality index data, affecting the health, environment and well-being of our communities. Such studies are very important in view of the facts that there is a great emphasis on health literacy and building healthy communities. 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.

Keywords: Air Pollution, Excel, Greenhouse Gases, Health, Lesson Plan, Minitab, Statistical Methods, Teaching.

2010 Mathematics Subject Classifications: 97C40, 97C70, 97D40, 97D50.

*Part of this article was presented during the EEI Workshop â&#x20AC;&#x153;Health: Connecting People, Places, and Planet - Part II (EEI1015-2)â&#x20AC;?, Spring 2018, held from 01/29/2018 to o3/12/2018, at Miami Dade College, Hialeah Campus.

178


1. Introduction: In recent years, there has been a great emphasis on the health and environmental effects of air pollution, and well-being of our communities. These are the issues of emerging national and international importance and significance. It appears from literature that many people suffer from the lack of the knowledge and insights of various aspects of air pollution data affecting the health, environment and well-being of our communities. Such studies are very important in view of the facts that there is a great emphasis on health literacy and building healthy communities. Thus, motivated by the importance of “building healthy communities”, this project aims at studying and conducting statistical analysis of some aspects of air pollution data such as the greenhouse gases data emissions, affecting the health, environment and well-being of our communities. It should be pointed out that “Statistics is one of the important sciences at present”. In order to summarize any type of data, we cannot underestimate its role, uses and importance in the modern world. There is a great emphasis on statistical literacy and critical thinking in education these days. An introductory course in statistics such as Statistical Methods (STA 2023) at Miami Dade College can easily provide such avenues to our students. In any Statistical Methods (STA 2023) Class, the students are first taught the frequency distributions, statistical graphs, and basic descriptive statistics, that is, center, variation, distribution, and outliers, which are important tools and techniques for describing, exploring, summarizing, comparing data sets, and other aspects of exploratory data analysis (EDA). 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) class, cannot be underemphasized and ignored. This computer-based project develops a lesson plan how to use air pollution data to teach Statistical Methods (STA2023) via Minitab and Excel software. For details on Minitab and Excel, the interested readers referred, for example, to McKenzie and Goldman (2005), Bluman (2013), and Triola (2010, 2014), among others. The organization of this paper is as follows. In Section 2, an overview of some basic concepts of statistics needed for our project is presented. In Section 3, an overview of some basic facts of air pollution, and its effects on health and environment are presented. In Section 4, uses of some statistical software, such as Minitab and Excel, are presented. Section 5 contains the applications of Minitab and Excel in the statistical analysis of some greenhouse gases data. The concluding remarks are given in Section 6.

2. An Overview of Some Basic Concepts of Statistics: This is a computer project on the statistical analysis of some air pollution data using Minitab and Statdisk. It is expected that the students have already learned about the following topics in the class, which, for the sake of completeness, are stated here. For details on these, the interested readers referred, for example, to Bluman (2013), and Triola (2010, 2014), among others. •

Exploratory Data Analysis (EDA): The five-number summary, namely, the minimum value, the 1st quartile, the median (i.e., the 2nd quartile or the 50th percentile), the 3rd quartile (i.e., the 75th percentile), and the maximum value are used to construct the boxplot.

Measures of central tendency (namely, mean, median, and mode) are used to indicate the “typical” value in a distribution. A comparison between the median and mean are used to

179


determine the shape of distribution, while the mode measures the most frequently occurred data. •

Measures of dispersion or variation (namely, range, standard deviation, and variance) are used to determine the “spread out” of the data.

Some statistical graphs, for example, histograms, can also be used to describe the shape of distribution

3. Uses of Minitab and Excel: In what follows, we will discuss some special features of Minitab and Statdisk. 3.1. Minitab and Some of its Special Features: In what follows, we will discuss some special features of Minitab and its use for constructing frequency distributions, histograms, descriptive statistics and exploratory data analysis (EDA). Minitab is one of the most important statistical analysis software. When Minitab is opened, there are two windows displayed first, as described below: • •

Session Window: It is an area which displays the statistical results of the data analysis and can also be used to enter commands. Worksheet Window: It is a grid of rows and columns (similar to a spreadsheet) and is used to enter and manipulate the data.

There are also other windows in Minitab as given below: • • •

Graph Window: When graphs are generated, each graph is opened in its own window. Report Window: There is also a report manager which is helpful in organizing the results in a report. Other Windows: Minitab has also other windows known as History and Project Manager. For details on these, the interested readers are referred to Minitab help.

3.2. Minitab Tools (or Menus): In general, various tools (called menus) of Minitab are described in the following Table 3.2.1, which, when Minitab is opened, are displayed in the menu bar on the top. For statistical data analysis, these menus can be used appropriately by following the instructions provided in Minitab help; also see McKenzie and Goldman (2005), Bluman (2013), and Triola (2010, 2014), among others.

180


Table 3.2.1 File Menu

Edit Menu

Data Menu

Calc Menu

Stat Menu

Open and save files; Print files; etc.

Undo and redo actions; Cut, copy, and paste; etc.

Undo and redo actions; Cut, copy, and paste; etc.

Calculate statistics; Generate data from a distribution; etc.

Regression and ANOVA; Control charts and quality tools; etc.

Graph Menu

Editor Menu

Tools Menu

Help Menu

Scatterplots; Bar charts; etc.

Graph, Data, and Session window editing; Modify active window; etc.

Change Minitab defaults; Create and modify toolbars and menus; etc.

Window Menu Arrange windows; Select active window; etc.

Searchable Help; StatGuide; etc.

3.3. Some Special Features of Excel: Uses of Excel: This is an Excel project. It is expected that the students have already learned about the following topics in the class: •

Measures of central tendency (namely, mean, median, and mode) are used to indicate the “typical” value in a distribution. A comparison between the median and mean are used to determine the shape of distribution, while the mode measures the most frequently occurred data.

Measures of dispersion or variation (namely, range, standard deviation, and variance) are used to determine the “spread out” of the data.

Some statistical graphs, for example, histograms, can also be used to describe the shape of distribution

The five-number summary, namely, the minimum value, the 1st quartile, the median (i.e., the 2nd quartile or the 50th percentile), the 3rd quartile (i.e., the 75th percentile), and the maximum value are used to construct the box-plot.

In what follows, we will discuss some special features of Excel and its use for constructing frequency distributions, histograms, and descriptive statistics. Some Special Features of Excel: These are described below. Statistical Analysis Tools: Microsoft Excel provides a set of data analysis tools-called the 181


Analysis ToolPak-that you can use to save steps when you develop complex statistical or engineering analyses. You provide the data and parameters for each analysis; the tool uses the appropriate statistical or engineering macro functions and then displays the results in an output table. Some tools generate charts in addition to output tables. Related Worksheet Functions: Excel provides many other statistical, financial, and engineering worksheet functions. Some of the statistical functions are built-in and others become available when you install the Analysis ToolPak. Accessing the Data Analysis Tools: The Analysis ToolPak includes the tools described below. To access these tools, click Data Analysis on the Tools menu. If the Data Analysis command is not available, you need to load the Analysis ToolPak add-in program. Descriptive Statistics Analysis Tools: The Descriptive Statistics Analysis tool generates a report of univariate statistics for data in the input range, providing information about the central tendency and variability of your data. Histogram Analysis Tools: The Histogram analysis tool calculates individual and cumulative frequencies for a cell range of data and data bins. This tool generates data for the number of occurrences of a value in a data set. How to Perform a Statistical Analysis: These are described below.  On the Tools menu, click Data Analysis. If Data Analysis is not available, load the Analysis ToolPak as follows: 

On the Tools menu, click Add-Ins.

In the Add-Ins available list, select the Analysis ToolPak box, and then click OK.

If necessary, follow the instructions in the setup program.

 In the Data Analysis dialog box, click the name of the analysis tool you want to use, then click OK.  In the dialog box for the tool you selected, set the analysis options you want.

4. An Overview of Some Basic Facts of Air Pollution, and Its Effects on Health and Environment: These are presented below in subsections 4.1 and 4.2. 4.1. Types of Air Pollution: According to Agency for Toxic Substances and Disease Registry (ASTDR), the following are major types of air pollution; please see www.atsdr.cdc.gov for details: 182


(I) Gaseous pollutants: The most common gaseous pollutants are carbon dioxide, carbon monoxide, hydrocarbons, nitrogen oxides, sulfur oxides and ozone, produced by the burning of fossil fuel, cigarette smoking, the use of certain construction materials, cleaning products, home furnishings, volcanoes, fires, and industry. The most commonly recognized type of air pollution is smog. Smog generally refers to a condition caused by the action of sunlight on exhaust gases from motor vehicles and factories. (II) Greenhouse Effect: A greenhouse gas is a gas that both absorbs and emits thermal radiation or heat, and when present in the atmosphere, causes a warming process called the greenhouse effect. It includes carbon dioxide, carbon monoxide, methane, nitrous oxide, ozone, water vapor (H2O), and chlorofluorocarbon (CFC). Many scientists believe that this is causing global warming. (III) Acid Rain: It is formed when moisture in the air interacts with nitrogen oxide and sulfur dioxide released by factories, power plants, and motor vehicles that burn coal or oil. This interaction of gases with water vapor forms sulfuric acid and nitric acids. Eventually these chemicals fall to earth as precipitation, or acid rain. (IV) Ozone: It is a form of oxygen found in the earth's upper atmosphere. Itâ&#x20AC;&#x2122;s believed that the damage to the ozone layer is primarily caused by the use of chlorofluorocarbons (CFCs). The depletion of ozone is causing higher levels of sun's ultraviolet (UV) radiation on earth, endangering both plants and animals. (V) Particulate Matter or PM (PM10 and PM2.5): It is the general term used for a mixture of solid particles and liquid droplets found in the air. When PM is breathed in, it can irritate and damage the lungs causing breathing problems. (VI) Climatic Effects: These are caused by the wind patterns, clouds, rain, and temperature, because of which pollutants move away quickly from an area to another area and thus creating air pollution. 4.2. The Air Quality Index: The Air Quality Index (AQI) is a tool used by U.S. Environmental Protection Agency (EPA) and other agencies to provide the public with timely and easy-tounderstand information on local air quality and whether air pollution levels pose a health concern. The AQI is focused on health effects that can happen within a few hours or days after breathing polluted air, as described in the following Tables 4.2.1 and 4.2.2.

183


Table 4.2.1 (Air Quality Index) Air Quality Index (AQI)

Levels of Health Concern

Colors

When the AQI is in this range:

...air quality conditions are:

...as symbolized by this color.

0 to 50:

Good

Green

51 to 100:

Moderate

Yellow

101 to 150:

Unhealthy for Sensitive Groups Orange

151 to 200:

Unhealthy

Red

201 to 300:

Very unhealthy

Purple

301 to 500:

Hazardous

Maroon

Values

Source: ATSDRâ&#x20AC;&#x2122;s Web site at www.atsdr.cdc.gov.

184


Table 4.2.2 (AQI Values) AQI Level

Numerical

Ozone

PM2.5

Value

Carbon Monoxide

Good

0-50

0-59 ppb

0-15.4 µg/m3

0-4.4 ppm

Moderate

51-100

60-75 ppb

15.5-35.4 µg/m3

4.5-9.4 ppm

Unhealthy for

101-150

76-95 ppb

35.5-65.4 µg/m3

9.5-12.4 ppm

Unhealthy

151-200

96-115 ppb

65.5-150.5 µg/m3

12.5-15.4 ppm

Very Unhealthy

201-300

116-375 ppb

150.5-250.4 µg/m3

15.5-30.4 ppm

Hazardous

>300

>375 ppb

>250.5 µg/m3

>30.5 ppm

Sensitive Groups

Sources: http://airnow.gov; http://www.epa.gov/ttn/naaqs/

5. Statistical Analysis of Greenhouse Effect Data: In this section, we present the statistical analysis of some greenhouse gases data using the software Minitab and Excel. The concentration of ‘greenhouse’ gases in the earth’s atmosphere, resulting in a gradual increase in temperatures at the earth’s surface, is an important area of research. Emissions of greenhouse gases worldwide resulting from human activities are expected to contribute to future climate changes. As greenhouse and climate change are fundamental issues of environmental sustainability and building healthy communities, this project aims at studying and conducting some statistical analysis of the greenhouse gas emissions data (namely, carbon dioxide, methane, nitrous oxide, and fluorinated gases) from 2001 to 2015, as reported by the annual inventory of United States Environmental Protection Agency (EPA). For details on these, please visit EPA’s website at the link: https://www3.epa.gov/climatechange/ghgemissions/inventoryexplorer/#iallsectors/allgas/gas/all. These greenhouse gas emissions data are provided in the following Table 5.1.

185


Table 5.1 (U.S. Greenhouse Gas Emissions by Gas, 2001 â&#x20AC;&#x201C; 21015) Year / Gas

Carbon dioxide

Methane

Nitrous oxide

Fluorinated gases

2001

5902.71

695.77

363.85

132.83

2002

5943.95

684.23

362.21

140.62

2003

5990.73

684.41

365.98

130.39

2004

6105.43

675.81

385.93

137.46

2005

6131.83

680.94

361.64

138.86

2006

6051.5

682.08

371.13

141.76

2007

6130.63

685.96

378.84

153.63

2008

5932.98

695.09

361.6

155.64

2009

5495.69

690.39

362.27

151.75

2010

5699.93

692.12

370.5

162.92

2011

5569.52

672.1

364.04

171.09

2012

5362.10

666.07

340.73

169.39

2013

5514.02

658.77

335.53

171.74

2014

5565.50

659.14

335.48

179.57

2015

5411.41

655.72

334.81

184.71

Minitab and Excel are efficient and effective tools for analyzing data. As such, we shall describe the frequency histograms, basic descriptive statistics and exploratory data analysis (EDA), for statistical analysis of the above-sated greenhouse gas emissions data, using Minitab and Excel. 5.1. Descriptive Statistics and Histograms: In this subsection, we present the descriptive statistics and the above-sated greenhouse gas emissions data, namely, carbon dioxide, methane, nitrous oxide, and fluorinated gases. Table 5.2 Minitab Output of Descriptive Statistics of Greenhouse Gas Emissions Data , 2001 â&#x20AC;&#x201C; 2015 Total Variable Count N N* Mean StDev Minimum Q1 Median Q3 Carbon dioxide 15 15 0 5787.2 279.6 5362.1 5514.0 5902.7 6051.5 Methane 15 15 0 678.57 13.44 655.72 666.07 682.08 690.39 Nitrous oxide 15 15 0 359.64 15.92 334.81 340.73 362.27 370.50 Fluorinated gas 15 15 0 154.82 17.64 130.39 138.86 153.63 171.09

Variable Maximum Carbon dioxide 6131.8 Methane 695.77 Nitrous oxide 385.93

186


Gas Emissions (million metric tons of carnbon dioxide equivalents)

Excel Output of Histogramsof Greenhouse Gas Emissions Data (2001 - 2015) 7000 6000 5000 Carbon dioxide

4000

Methane

3000

Nitrous oxide

2000

Fluorinated gases

1000 0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Figure 5.1: Excel Output of Histograms of Greenhouse Gas Emissions Data, 2001 â&#x20AC;&#x201C; 2015 Histogram of Carbon dioxide, Methane, Nitrous oxide, Fluorinated gases Normal

4

Carbon dioxide

Methane

3

3

C arbon dioxide Mean 5787 StDev 279.6 N 15

2

Methane Mean 678.6 StDev 13.44 N 15

Frequency

2 1

1 0 6.0

5200 5400 5600 5800 6000 6200 6400

Nitrous oxide

0 4

4.5

3

3.0

2

1.5

1

0.0

330 340 350

360 370

650

380 390

0

660

670

680

690

Fluorinated gases

700

710

Nitrous oxide Mean 359.6 StDev 15.92 N 15 Fluorinated gases Mean 154.8 StDev 17.64 N 15

120 130 140 150 160 170 180 190

Figure 5.2: Minitab Output of Histograms of Greenhouse Gas Emissions Data, 2001 â&#x20AC;&#x201C; 2015 5.2. Exploratory Data Analysis (EDA): The concept of exploratory data analysis (EDA) was developed by John Tukey; see, for example, Tukey (1977). The objective of exploratory data analysis is to analyze data in order to find out information about the data such as the center and 187


the spread. In this subsection, we discuss exploratory data analysis (EDA) using Minitab. For example, we organize data using a stem and leaf plot. We compute the median which is the measure of central tendency, and also we compute the interquartile range which is the measure of variation. Further, in EDA, we represent the data graphically using a boxplot (sometimes called a box-and-whisker plot). These plots involve five specific values, called a “five-number summary” of the data set, as defined below: vi. vii. viii. ix. x.

The lowest value of the data set, that is, minimum Q1 , called the first quartile Q2 , called the second quartile, or the median Q3 , called the third quartile The highest value of the data set, that is, maximum

The exploratory data analysis (EDA) of the greenhouse gas emissions data (namely, carbon dioxide, methane, nitrous oxide, and fluorinated gases), from 2001 to 2015, is reported below, using Minitab, in Figure 5.3 (Boxplots) and Table in 5.3, below. Minitab Output of Boxplots of Greenhouse Gas Emissions Data, 2001 - 2015 6000 5000

Data

4000 3000 2000 1000 0 Carbon dioxide

Methane

Nitrous oxide

Fluorinated gases

Figure 5.3: Minitab Output of Boxplots of Greenhouse Gas Emissions Data, 2001 – 2015 Table 5.3 Minitab Output of Stem-and-Leaf Display: Greenhouse Gas Emissions Data, 2001 – 2015 Stem-and-leaf of Carbon dioxide N = 15 Leaf Unit = 10

Stem-and-leaf of Methane N = 15 Leaf Unit = 1.0

Stem-and-leaf of Nitrous oxide N = 15 Leaf Unit = 1.0

188

Stem-and-leaf of Fluorinated gases N = 15 Leaf Unit = 1.0


5

53 6 3 54 19 6 55 166 7 56 9 7 57 7 58 (4) 59 0349 4 60 5 3 61 03

3 3 4 5 6 (4) 5 4 2

65 66 66 67 67 68 68 69 69

589

5 3 4 4 4 4 (6) 5 4 2

6 2 5 0244 5 02 55

5 5

33 34 34 35 35 36 36 37 37

33 4 55 0

112234 5 01 8 38 38 5

2 4 6 6 (2) 7 6 5 4 2

5

13 13 14 14 15 15 16 16 17 17

02 78 01 13 5 2 9 11 9 18 4

Total Variable Count N N* Mean StDev Minimum Q1 Median Q3 Carbon dioxide 15 15 0 5787.2 279.6 5362.1 5514.0 5902.7 6051.5 Methane 15 15 0 678.57 13.44 655.72 666.07 682.08 690.39 Nitrous oxide 15 15 0 359.64 15.92 334.81 340.73 362.27 370.50 Fluorinated gas 15 15 0 154.82 17.64 130.39 138.86 153.63 171.09

Variable Maximum Carbon dioxide 6131.8 Methane 695.77 Nitrous oxide 385.93

From the above Figure 5.3 (Boxplots) and Table in 5.3, we can easily infer the â&#x20AC;&#x153;five-number summary, that is, minimum value, Q1 , Q2 , Q3 , and maximum valueâ&#x20AC;? of the greenhouse gas emissions data, that is, carbon dioxide, methane, nitrous oxide, and fluorinated gases respectively, as given above. We observe that the median for the distribution for carbon dioxide emissions data is higher than the median for the distributions for methane, nitrous oxide, and fluorinated gases emissions data. Also, we observe that the interquartile range (IQR) for the distribution for carbon dioxide emissions data is greater than the interquartile range (IQR) for the distributions for methane, nitrous oxide, and fluorinated gases emissions data. Thus there is more variation or spread for the distribution of carbon dioxide emissions data than the variation for the distribution of methane, nitrous oxide, and fluorinated gases emissions data.

5.3. Time Series Analysis: Using Excel, the time-series analysis of the greenhouse gas emissions data (namely, carbon dioxide, methane, nitrous oxide, and fluorinated gases), in the United States in the United States for the recent ten-year period, from 2001 to 2015, is reported in Figure 5.4, below.

189


7000 6000 5000 Carbon dioxide

4000

Methane

3000

Nitrous oxide

2000

Fluorinated gases

1000 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Figure 5.4: Excel Output of Time Series of Greenhouse Gas Emissions Data, 2001 â&#x20AC;&#x201C; 2015 It is apparent from the above time-series graph (Figure 5.4) that the amounts of the four greenhouse gas emissions, (namely, carbon dioxide, methane, nitrous oxide, and fluorinated gases), in the United States for the recent ten-year period, from 2001 to 2015, are decreasing. 5.4. Scatterplot Analysis: In order to explore the relationship of carbon dioxide emissions data with methane, nitrous oxide, and fluorinated gases emissions data respectively, we have provided the scatterplots, using Minitab, in the following Figures 5.5 â&#x20AC;&#x201C; 5.7. Scatterplot of Carbon dioxide vs Nitrous oxide 6200 6100

Carbon dioxide

6000 5900 5800 5700 5600 5500 5400 5300

330

340

350

360 Nitrous oxide

190

370

380

390


Figure 5.5: Minitab Output of Scatterplot of Carbon Dioxide vs Nitrous Oxide Emissions Scatterplot of Carbon dioxide vs Methane 6200 6100

Carbon dioxide

6000 5900 5800 5700 5600 5500 5400 5300

650

660

670

Methane

680

690

700

Figure 5.6: Minitab Output of Scatterplot of Carbon Dioxide vs Methane Emissions Scatterplot of Nitrous oxide vs Fluorinated gases 390

Nitrous oxide

380 370 360 350 340 330

130

140

150

160 Fluorinated gases

191

170

180

190


Figure 5.7: Minitab Output of Scatterplot of Carbon Dioxide vs Fluorinated Gas Emissions It is obvious from the above scatterplots (Figures 5.5 â&#x20AC;&#x201C; 5.7) that there is no relationship of carbon dioxide emissions data with methane, nitrous oxide, and fluorinated gases emissions data respectively.

6. Statistical Analysis of Air Quality Data: As air quality index values and concentration of pollutants are also fundamental issues of environmental sustainability and building healthy communities, this project aims at studying and conducting some statistical analysis of daily mean PM2.5 concentration and corresponding daily AQI value for the months of January, 2017 and December, 2017, collected at outdoor monitors across the Metropolitan Core Based Statistical Areas (CBSA), namely, Miami-Fort Lauderdale-West Palm Beach, FL, with CBSA CODE: 33100, and reported by the United States Environmental Protection Agency (EPA). For details on these, please visit EPAâ&#x20AC;&#x2122;s website at the link: https://www.epa.gov/outdoorair-quality-data/air-data-basic-information. These data are provided in the following Table 6.1. Table 6.1 Outdoor Air Quality Data (January and December 2017) (Criteria Pollutant: PM2.5; Daily Air Quality Index Value) Metropolitan Core Based Statistical Areas (CBSA) NAME: Miami-Fort Lauderdale-West Palm Beach, FL; CBSA CODE: 33100 JAN2017 Daily Mean PM2.5 Concentration 1/1/2017 - 1/31/2017

DEC2017 Daily Mean PM2.5 Concentration 12/1/2017 - 12/31/2017

JAN2017 DAILY AQI VALUE 1/1/2017 - 1/31/2017

DEC2017 DAILY AQI VALUE 12/1/2017 - 12/31/2017

24.1

76

5.6

23

7.2

30

7.3

30

5.7

24

8.1

34

5.9

25

7.8

33

8.2

34

5.9

25

8.5

35

6.3

26

8.9

37

6.1

25

7.6

32

8

33

3.3

14

5.7

24

4

17

4.6

19

4

17

7.2

30

5.4

23

5.8

24

5.3

22

12

50

5.4

23

8.2

34

5.7

24

9.5

40

6.4

27

9.7

40

6

25

10

42

7.6

32

8.8

37

192


7.2

30

8.2

34

8.4

35

8.5

35

11.2

47

10.3

43

8.7

36

11

46

9.7

40

8.2

34

10.1

42

8.8

37

8.7

36

8.6

36

12.2

51

6.7

28

11.9

50

6.5

27

17.1

62

5.6

23

4.4

18

8.7

36

9.2

38

10.2

43

6.8

28

10.6

44

In what follows, using the data in Table 6.1, we have provided various statistical analysis in Tables 6.2 â&#x20AC;&#x201C; 6.5, and Figures 6.1 â&#x20AC;&#x201C; 6.7, such as the descriptive statistics, histograms, hypothesis testing, confidence interval estimates, box plots, time series plot, among others, from which we can easily draw some inferences about the daily mean PM2.5 concentration and corresponding daily AQI value for the months of January, 2017 and December, 2017, collected at outdoor monitors across the Metropolitan Core Based Statistical Areas (CBSA), namely, Miami-Fort Lauderdale-West Palm Beach, FL, with CBSA CODE: 33100. Table 6.2 Minitab Output of Descriptive Statistics: JAN-2017-AQIValue, DEC-2017-AQI-Value Variable N N* Mean StDev Variance CoefVar Minimum Q1 JAN-2017-AQIValu 31 0 33.23 13.52 182.85 40.70 14.00 24.00 DEC-2017-AQI-Val 31 0 33.39 7.80 60.85 23.36 19.00 26.00 Variable Median Q3 Maximum JAN-2017-AQIValu 32.00 38.00 76.00 DEC-2017-AQI-Val 34.00 40.00 50.00

193


Histogram (with Normal Curve) of JAN-2017-AQIValue Mean StDev N

9 8

33.23 13.52 31

Frequency

7 6 5 4 3 2 1 0

10

20

30 40 50 60 JAN-2017-AQIValue

70

80

Figure 6.1: Histogram (with Normal Curve) of JAN-2017-AQIValue

Histogram (with Normal Curve) of DEC-2017-AQI-Value 12

Mean StDev N

10

Frequency

8 6 4 2 0

15

20

25

30 35 40 DEC-2017-AQI-Value

194

45

50

33.39 7.800 31


Figure 6.2: Histogram (with Normal Curve) of DEC-2017-AQI-Value

Table 6.3 Minitab Output of Two-Sample T-Test and CI: JAN-2017-AQIValue, DEC-2017-AQI-Value Two-sample T for JAN-2017-AQIValue vs DEC-2017-AQI-Value N Mean StDev SE Mean JAN-2017-AQIValu 31 33.2 13.5 2.4 DEC-2017-AQI-Val 31 33.39 7.80 1.4

Difference = mu (JAN-2017-AQIValue) - mu (DEC-2017-AQI-Value) Estimate for difference: -0.161290 95% CI for difference: (-5.769637, 5.447057) T-Test of difference = 0 (vs not =): T-Value = -0.06 P-Value = 0.954 DF = 60 Both use Pooled StDev = 11.0384

Boxplot of JAN-2017-AQIValue, DEC-2017-AQI-Value 80 70

Data

60 50 40 30 20 10 JAN-2017-AQIValue

DEC-2017-AQI-Value

Figure 6.3: Boxplot of JAN-2017-AQIValue, DEC-2017-AQI-Value

195


Table 6.4 Minitab Output of Descriptive Statistics: Jan2017-Daily-PM2.5, Dec2017-Daily-PM2.5 Variable N N* Mean StDev Variance CoefVar Minimum Q1 Jan2017-Daily-PM 31 0 8.219 4.111 16.902 50.02 3.300 5.700 Dec2017-Daily-PM 31 0 8.016 1.851 3.425 23.09 4.600 6.300 Variable Median Q3 Maximum Jan2017-Daily-PM 7.600 9.200 24.100 Dec2017-Daily-PM 8.200 9.500 12.000

Histogram (with Normal Curve) of Jan2017-Daily-PM2.5 Mean StDev N

10

Frequency

8

6

4

2

0

0

4

8 12 16 Jan2017-Daily-PM2.5

20

24

Figure 6.4: Histogram (with Normal Curve) of Jan2017-Daily-PM2.5

196

8.219 4.111 31


Histogram (with Normal Curve) of Dec2017-Daily-PM2.5 Mean StDev N

7

8.016 1.851 31

6

Frequency

5 4 3 2 1 0

4

6

8 10 Dec2017-Daily-PM2.5

12

Figure 6.5: Histogram (with Normal Curve) of Dec2017-Daily-PM2.5

Table 6.5 Minitab Output of Two-Sample T-Test and CI: Jan2017-Daily-PM2.5, Dec2017-Daily-PM2.5 Two-sample T for Jan2017-Daily-PM2.5 vs Dec2017-Daily-PM2.5 N Mean StDev SE Mean Jan2017-Daily-PM 31 8.22 4.11 0.74 Dec2017-Daily-PM 31 8.02 1.85 0.33

Difference = mu (Jan2017-Daily-PM2.5) - mu (Dec2017-Daily-PM2.5) Estimate for difference: 0.203226 95% CI for difference: (-1.416534, 1.822985) T-Test of difference = 0 (vs not =): T-Value = 0.25 P-Value = 0.803 DF = 60 Both use Pooled StDev = 3.1880

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Boxplot of Jan2017-Daily-PM2.5, Dec2017-Daily-PM2.5 25

Data

20

15

10

5

Jan2017-Daily-PM2.5

Dec2017-Daily-PM2.5

Figure 6.6: Boxplot of Jan2017-Daily-PM2.5, Dec2017-Daily-PM2.5

Daily Mean PM2.5 Concentration - January and December 2017 25

Variable Jan2017-Daily -PM2.5 Dec2017-Daily -PM2.5

Data

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15

10

5 3

6

9

12

15 18 Index

21

24

27

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Figure 6.7: Time Series Plot of Jan2017-Daily-PM2.5, Dec2017-Daily-PM2.5 198


6. Concluding Remarks: In this paper, we have discussed about developing a lesson plan to show how the teaching of basic statistical methods via Minitab and Excel software can help the students to gain the knowledge and insights from the greenhouse gases emissions and air quality index data, in the United States and the World, affecting the health, environment and well-being of our communities. It is hoped that this paper will be helpful in teaching any introductory course in statistics such as courses in Statistical Methods (STA 2023) at Miami Dade College using Minitab and Excel. 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 Minitab and Excel, the students will be able to conduct statistical research projects in their STA2023 courses, and will be able to achieve the following: III.

Search or web-search any real world data.

IV.

Analyze the data statistically using Minitab and Excel, 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. 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.

Acknowledgments The author is thankful to the editors of the Polygon for their suggestions, which improved the quality of the paper. Further, the author would like to thank the Earth Ethics Institute (EEI) of Miami Dade College for providing us an opportunity to attend the EEI workshop on “Health: Connecting People, Places, and Planet Part II Series (EEI101)”, at Hialeah Campus. The author is also thankful to the agencies, such as U.S. Environmental Protection Agency (EPA), Agency for Toxic Substances and Disease Registry (ASTDR), http://airnow.gov, http://www.epa.gov/ttn/naaqs/, and http://www.dep.state.fl.us/air/airquality.htm, among others, for providing valuable information resources and data on their websites, which were freely consulted during the preparation of this paper. Also, the author is thankful to Miami Dade 199


College for providing him an opportunity to serve as a mathematics faculty in the college at its Hialeah Campus, without which it was impossible to conduct his research. Further, the author would like to thank his wife for her patience and perseverance for the period during which this paper was prepared. Lastly, the author would like to dedicate this paper to his late parents.

References 1. Bluman, A. G. (2013). Elementary Statistics, A Brief Version, 6th Edition. McGraw-Hill Co., New York. 2. McKenzie, J. and Goldman, R. (2005). The Student Guide to MINITAB Release 14, 14th Edition. Pearson Addison-Wesley, New York. 3. Triola, M. F. (2010). Elementary Statistics, 11th Edition. Addison-Wesley, New York. 4. Triola, M. F. (2014). Elementary Statistics Using Excel, 5th Edition. Addison-Wesley, New York. 5. Tukey, J. (1977). Exploratory Data Analysis, Addison-Wesley, New York. 6. http://airnow.gov. 7. http://www.epa.gov/ttn/naaqs/. 8. www.atsdr.cdc.gov. 9. http://www.dep.state.fl.us/air/airquality.htm. 10. https://www3.epa.gov/climatechange/ghgemissions/inventoryexplorer/#iallsectors/allgas/ gas/all. 11. https://www.epa.gov/outdoor-air-quality-data/air-data-basic-information.

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Polygon 2018  

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

Polygon 2018  

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

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