Polygon 2012

Editorial Note: We, the editorial committee of Polygon, are pleased to publish the sixth issue of 2012 Spring Polygon includes six regular papers. Again we are pleased to present work from a diverse array of fields written by faculty from across the college. The editorial board of Polygon is thankful to the administration, staff and faculty of Hialeah Campus and Miami Dade College, in general, for their continued support and cooperation for the publication of Polygon.

The editorial committee would also like to cordially invite the MDC community to submit their articles for consideration for the 2013 Spring Issue of Polygon.

Sincerely, The Editorial Committee of Polygon Dr. M. Shakil (Mathematics), Dr. Jaime Bestard (Mathematics), and Professor Victor Calderin ( English) Patrons: Dr. Ana Maria Bradley-Hess, Dean of Academic and Student Services Prof. Djuradj Babich, Chair of Computer and Management Sciences Dr. Caridad Castro, Chair of LAS Prof. Maria Jofre, Chair of EAP and Foreign Languages Mission of Miami Dade College The mission of the College is to provide accessible, affordable, high--‐quality education that keeps the learner’s needs at the center of the decision--‐making process.

Miami Dade College District Board of Trustees Helen Aguirre Ferré, Chair Armando J. Bucelo Jr. Marielena A. Villamil Benjamin León III Marili Cancio Jose K. Fuentes Armando J. Olivera Eduardo J. Padrón, College President

Editorial Notes

i

Guidelines for Submission

ii-iii

Conceptual and Procedural Knowledge in Mathematics Education in the Case of Law of Exponents

1-23

D. Tsung

Teaching What to Think vs. Teaching How to Think in College

24-35

J. Guntin

Towards Development of Critical Thinking

36-44

J. Guntin

Factorial Experiment Design to Analyze Fuel Consumption of a Vehicle

45-57

J. Bestard

The Cost of Exclusion

58-73

M. Benitesl

Critical Pedagogy and English Language Acquisition

74-84

M. Benitesl

Comments about Polygon

1

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 Committee: Dr. Mohammad Shakil (Mathematics) Editor-in-Chief Dr. Jaime Bestard (Mathematics)

Editor

Prof. Victor Calderin (English)

Editor

Manuscript Submission Guidelines: Welcome from the New POLYGON Editorial Team: The Department of Liberal Arts and Sciences at the Miami Dade College–Hialeah Campus and the new 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.

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3 Instructions for the Preparation of Manuscripts for the Polygon (THE TITLE IS HERE) (12 pt, bold, 32 pt above) NAME IS HERE (11 pt16 pt above, 32 pt below)

ABSTRACT. Abstract is here, not exceeding 160 words. It must contain main facts of the work. (11 pt) Key words and phrases: (11 pt)

1. Introduction (11 pt, bold, 24 pt above, 12 pt below) Main Body

REFERENCES (11 pt, 30 pt above, 12 pt below)

[1] [2] [3] [4]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1970.

J. Galambos and I. Simonelli, Products of Random Variables – Applications to Problems of Physics and to Arithmetical Functions, CRC Press, Boca Raton / Atlanta, 2005.

S. Momani, Non-perturbative analytical solutions of the space- and time-fractional Burgers equations. Chaos, Solitons & Fractals, 28(4) (2006), 930-937.

Z. Odibat, S. Momani, A Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlin. Sci. Numer. Simulat. 1(7) (2006), 15-27. (11 pt)

XXXX YYYYY. Received his Master’s/Ph. D. Degree in Physics from the University of ZZZZZ (Country) in 1987 under the direction of Dr. M. N. OPQR. Since 1989, he has been at CCCC College in Hawaii, USA. His research interests focus on the Fractals, Solitons, Undergraduate Teaching of Physics, and Curriculum Development. (11 pt) Department of Liberal Arts & Sciences (Physics Program), CCCC College, P. O. Box 7777, Honolulu, Hawaii, USA. e-mail: xxyy@ccc (11 pt)

.

Polygon Spring 2012 1

Conceptual and procedural knowledge in Mathematics educationin the case of law of exponents

David H. Tseng Contact: (305)-237-0562 Email: dtseng@mdc.edu

Polygon Spring 2012 2 Abstract In this paper, a detailed research into students’ understanding of conceptual knowledge vs. procedural knowledge over the law of exponents has been conducted. An examination from students’ perspective will help teachers to teach law of exponents more effectively. This understanding can potentially impact the curriculum design to allow students to know the law for a longer time and not depending on their memorization on the law of exponents for a short while (usually before a test).

Polygon Spring 2012 3 According to an article entitled “procedures over concepts: the acquisition of decimal number knowledge” by Hiebert and Wearne of University of Delaware, conceptual knowledge is defined as knowledge of those facts and properties of mathematics that are recognized as being related in some way. In contrast, procedural knowledge is characterized by the absence of embedding relationships. This procedural knowledge contains the following two parts: 1. Part 1: the knowledge of written symbols in the synthetic system.

2. Part 2: the set of rules and algorithms that are used to solve mathematics problems. Therefore, conceptual knowledge is about relationship while procedural knowledge is about rules and strategies to solve problems.

Students need to use both knowledge, but more important, they need to connect these two knowledges together. Lacking either knowledge, a complicated problem cannot be solved thoroughly. However, some easy basic exponential problems, such as

, can be solved as

without much conceptual knowledge involved. Students can simply apply the law of

to get the correct answer. However, the

conceptual knowledge would be helpful to recognize that the exponential part of the answer should be a+b, instead of ab.

According to Hiebert’s article (1980) entitled “children’s mathematics learning: the struggle to link form and understanding”, there are 3 sites where links between conceptual and procedural knowledge are specified. The first site is the initial point in the problem-solving about interpretation of the problem. In the example of

, the

base of x and exponents of a and b must be interpreted as exponential symbols. The

Polygon Spring 2012 4 multiplication of “X” is the operational symbol. This symbol refers to the algorithm on relationship that exponent a should add to exponent b, instead of ab. This connection between exponential symbol (procedural knowledge) and operational symbol (conceptual knowledge) at site 1 are essential for establishing connections for the remaining sites.

The site 2 is about execution of procedure. In the example of

, students execute the rule by adding a to b. However, students

may not have conceptual rationale as to why a + b, instead of ab. Even if they got the right answer, they have not obtained mathematical proficiency to explain what is wrong with the answer of exponent ab.

The site 3 is the solution evaluation. In this site, it provides connection between conceptual and procedural knowledge. If connection is fully understood, the solution will be evaluated properly. Actually, students will be more confident about their answers. It is not just procedures (what) that have been executed, but the conceptual knowledge (why) has been applied. For example, if students know the concept of means

then

means

and

would be incorrect because 3 ’s add 4 ’s means 7 ’s. So,

at site 3, students will have a feeling about whether their answers make sense or not. This feeling came from conceptual knowledge. On the other hand, if students have no conceptual knowledge, they would not have the confidence that their answers are right, because they don’t have conceptual rationale to evaluate the result.

So far, it has been emphasized that conceptual knowledge is important to know why the procedural knowledge provides whatever it has provided. However, during a test, procedure knowledge has advantage of obtaining answers quickly by shortening the thinking process on why (conceptual knowledge). The conceptual knowledge can still be

Polygon Spring 2012 5 applied to evaluate the answer for correctness. However, this evaluation process should be reserved as the last step to double-check the answers after all test problems have been answered. This is particularly true when taking SAT or other tests for which there is almost no time to evaluate every answer by applying the conceptual knowledge.

In an article entitled “the strands of mathematical proficiency�, it describes mathematical proficiency should contain the following 5 components: 1. Conceptual understanding Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. It is a comprehension of mathematical concepts, operations and relations.

2. Procedural fluency Procedural fluency refers to skills and knowledge in performing a procedure appropriately. This includes when, how and what to do with method(s) of calculation.

3. Strategic competence Strategic competence refers to the ability to formulate, represent, and solve mathematical problems.

4. Adaptive reasoning Adaptive reasoning refers to capacity for logical thought, reflection, explanation and justification. It is a logical capacity in identifying the relationships among concepts and situations. It includes not only informal explanation and justification, but also intuitive and inductive reasoning based on pattern observed, analogy and metaphor.

Polygon Spring 2012 6 5. Productive disposition Productive disposition is a habitual inclination to see mathematics as a sensible, useful and worthwhile tool to problem-solving. To develop a productive disposition requires frequent opportunities to make sense of mathematics, to recognize the benefits of perseverance and to experience the rewards of sense making in mathematics.

Since these 5 components are interrelated to each other, the solving of mathematical problems relies on all 5 components. In the area of law of exponents, it is easy for teachers to jot down 10 laws of exponents. It may be easy for students to memorize these 10 laws for an immediate quiz or test. But it is not easy for students to remember these formulas after a longer period of time. The difference comes from the mathematical proficiency. If students have mathematical proficiency, these 10 laws of exponents will make sense to them. As such, they can remember laws easier and the memory will last much longer. On the other hand, if students just bluntly remember the formulas without any common sense built into the memory for mathematical proficiency, their memory about these laws can not last long. Furthermore, they will subtly develop an attitude of disliking Mathematics due to boring memorization. This is contrary to the fifth component of productive disposition.

In discussing the third component of strategic competence, these 10 laws of exponents can be strategically classified into 4 categories for easy memorization. From my personal teaching experience, I have found that it will be easier for students to remember the 10 laws by classifying them into the following 4 cases:

Polygon Spring 2012 7 1. Basic operations case 2. Separation case 3. Distribution case 4. Miscellaneous case Refers to attachment 1, we will find that in the basic operations case, 4 out of 10 laws are introduced. Their exponents to the base number of x are associated with the following basic operations: 1)

2)

3)

4)

In the separation case of the formulas, the radical sign will be separated by half. In the distribution case, the power of

is distributed to different bases, say

and

.In the final

miscellaneous case, we introduce 2 unique formulas: one with 0 as exponent, the other one uses a negative sign as exponent. With the above mentioned strategy, the 10 laws of exponents become easier to remember. Without these classified cases, students will have a hard time remembering 10 different, unrelated formulas. In the first component of mathematical proficiency--- conceptual understanding, students need to see examples first before each law of exponents is introduced. For example, in the first law of , students need to see an example that

. It means

. Therefore, they will realize that exponents should be added to become 3, instead of being multiplied to produce 1x2=2. This simple example has the following benefits for student who is struggling to memorize the formulas: a) Memorization will become easier and will last longer, because students know the relationship between exponent 1 and exponent 2.

Polygon Spring 2012 8 b) Even if students forget the formulas during a test or quiz, they can always conduct a simple mathematical experiment to recapture the formulas from their memory.

They simply ask themselves which one is more reasonable? ? or

?

c) With common sense built into memorizing the formulas, students can recall the formulas without much trouble. Hence, their confidence and interest in the area of law of exponents will be increased as they practice more exercise problems.

In the fourth component of mathematical proficiency- adaptive reasoning, teachers should point out the following two formulas as shown below: a)

and

b)

and

are horizontal type formulas. are vertical type formulas.

It is this kind of reflection or explanation which will help students to remember the category of distribution or separation. There will be basically two formulas. These two formulas are horizontal type versus vertical type.

With the above mentioned 4 components, interwoven into one another, the second component of procedure fluency becomes achievable because students remember these laws well due to their understanding of each law of exponent. Therefore, solving an exponential problem is nothing but applying these laws as procedures. Actually, each law is like a procedure. A complicated exponential problem means more procedures should be applied.

Next, we are going to analyze learning interference. This interference was identified in chapter 12 of Hiebert and Wearne’s book entitled “procedures over

Polygon Spring 2012 9 concepts- the acquisition of decimal number of knowledge”. When students are given two kinds of learning- relational learning for meaning and instrumental learning for procedure-skill development without reasons, there exists a possible interference.

Actually, in Hiebert and Wearne’s book, they pointed out 3 kinds of interferences as followed: 1) Cognitive interference: This interference occurs when previous understanding is powerful and affects subsequent learning. Teaching rational learning with reasons becomes important.

2) Attitudinal interference The second interference is called “attitudinal interference” as mentioned in chapter 12 entitled “interference of instrumental instruction in subsequent relational learning by Dolores D. Pesek and David Kirshner. Example in the law of exponents, students thought that all laws refer to the same base (refer to attachment), therefore,

and

may be treated mistakenly as x + y.

However, these two terms are not totally unrelated as they appear. Students did not examine closely into the relation of 4 and 2. If they do, they may discover that = 2 log 2. With this connection identified,

and

become

additive by finding an LCD (least common denominator) between log 4 and log 2.

1. Metacognitive interference This is the interference that derives from both cognitive and attitudinal varieties. When example of

is given to students,

students may be busy trying to simplify variables and their powers within the parenthesis, students fail to recognize that there is a zero outside

Polygon Spring 2012 10 parenthesis which can give them a quick answer of one as indicated by the formula in attachment. Another possible metacognitive interference is when students find that this problem is so complicated they my simply give up. If they do not have such an attitude, they can examine all powers and realize the outside power of zero would produce one as a quick and easy answer.

Although lack of experimental data on the law of exponents, the following two different types of treatments about relational learning and instrumental instruction offers some interesting findings. These findings are quoted from an article entitled “Interference of instrumental instruction in subsequent relational learning” by Dolores D. Pesek and David Kirshner. Their findings are based on two groups of study: 1. The I-R treatment group- teach students with instrumental instruction first then followed by relational instruction.

2. The R-O treatment group – teach students with relational instruction only.

Their findings are summarized below: 1. The R-O interviewees were able to make more sense of formula despite their lack of instructional guidelines.

2. The I-R interviewees relied heavily on formulas without much understanding of its meaning. This implies a metacognitive interference may have occurred.

Polygon Spring 2012 11 3. Instrumental instruction should not be preceded by relational instruction. Hence, interference of preoccupied knowledge can be avoided. This improvement can be considered as a potential classroom reform.

Based on experience on the findings by Pesek and Kirshner, I can borrow their experience in the area of exponential laws in the following manner: a. Always explain the meaning of law of exponents by examples first.

Otherwise, students will be confused by the relationship between base- tobase or power- to-power when coupled with basic operations (addition, subtraction, multiplication, and division). b. In the example of exponential laws, the procedural learning of formulas is still important because a complicated problem requires several laws to get to the final answer. If a student only has a fundamental understanding about the exponents without memorizing the formula, he or she will be slow in solving problems or maybe even blocked in the thinking due to the lacking of the adequate formulas.

The development of conceptual knowledge is through constructing relationships between pieces of information. As to how the process of a relationship is developed, the literature of Psychology and Mathematics education provides a varieties of insights and theories. In this research paper, I will introduce two types of growth in conceptual knowledge of learning Mathematics.

1) Observation

Polygon Spring 2012 12 In the example of

, students observe that there are ’s and ’s of base ,

therefore, the total exponent should be

. This is a direct observation to develop their

meaningful learning as a conceptual knowledge about the additive feature of law of exponents. Once this basic formula is understood, any simple problem such as would fall into the category of procedural knowledge. This kind of observation is direct and visually achievable. But if the observation process takes place in an abstract, non-visual manner, it will be a bit harder to obtain the complete conceptual knowledge. That’s why the next type of process which develops a complete conceptual knowledge is called creation.

2) Creation This process starts with observation, but creation will be needed to connect existing knowledge to new information that is just entering the system. In the example of , the existing formula

is apparently insufficient. Because there is

an additional c introduced and the problem becomes challenging. In this example, students need to be conceptually creative about problem becomes a typical case of

, which is

, therefore, the original . Meanwhile,

, so

. The final answer is achieved by applying another law of exponent which says

. Therefore, original problem which seems complicated and

unsolvable has now become easy and solvable. It requires the creation of through the law of

to obtain the final answer. Finally, after the use of

creativity, the conceptual knowledge can grow almost infinitely.

, then

Polygon Spring 2012 13 If the student conducts a careful observation of the first two conclusions by listing his/ her findings as below:

If the student has a sharp observation capability and rich imagination, he/she may discover a new “formula” as this one:

or even as complicated as:

.

In views of the combined procedural and conceptual knowledge, there are 3 types of exponential problem solvers. In a rather complicated exponential function simulating exponential growth in a spiral shell, a research entitled “purposeful choice: building mathematics through inquiry”, found the following different thinking processes have been utilized by basically 3 kinds of problem solvers: a) Problem solving expert clarify goals, organize, represent and interpret with varying levels of flexibility, monitor self and group solution progress, and analyze/ evaluate solution strategies. For details, see DeFranco & Hilton’s article, 1999, “ Distinguishing features of mechanical and human problem solving” Journal of Mathematical Behavior, 18, 79-84 and also see Greaser/ Person/ Huber’s article, 1992, “Mechanisms that generate questions”. b) Novice problem solvers rush to use a memorized algorithm, focus on superficial features of a problem and interpretation of explicit material and lack evaluative monitoring of progress (see DeFranco & Hilton’s article 1999).

Polygon Spring 2012 14 c) Transitional problem solvers, in contrast to experts and novices, exhibit expertlike problem solving behaviors in complex domains but may not have extensive content knowledge, see Walter, J.G. 2004, Tracing mathematical inquiry: High school students mathematizing a shell. Dissertation Abstracts International, 64, (12), 364A (UMI no. 3117648). Also see Greaser/ Person/ Huber’s article, 1992, “Mechanisms that generate questions”. Solvers keep practicing exponential problems and gain more experience in both conceptual and procedure knowledge, they eventually will become expert problem solvers themselves.

The difference in the above 3 kinds of problem solvers comes primarily from problems solver’s conceptual understanding. The novice problem solvers know little about concept, therefore, he/she just rush to apply formula or algorism as a procedure approach. They may or may not get the correct answer. On the other hand, the expert and transitional problem solvers have better conceptual knowledge, so they are working toward the correct direction. The only difference between the expert and transitional problem solvers is that the former one has extensive content knowledge than the later one. But if the transitional problem continues to practice, they eventually become problem-solving experts.

Since developing relationships which connect pieces of information, data, formulas, or facts is the key for conceptual knowledge, I am going to take a closer look at the relationship itself.

In chapter one of the book entitled “conceptual and procedural knowledge: the case of mathematics”, two kinds of relationships were proposed as follows: 1) Primary level of relationship:

Polygon Spring 2012 15 This level is logically and visually apparent. It is like when solving a right triangle problem, it is visually and logically apparent that Pythagorean Theorem may probably be needed for solving the right triangle problem.

A typical example in the area of exponential law would be like:

Students need to construct a relationship between log in the numerator versus log in the denominator. These two relationships are essentially visually available and are actually given in the 10 laws of exponents as procedural knowledge. Therefore, this example demonstrates a primary level of relationship which is visually apparent.

2) Reflective relationship: This is a higher and more abstract level of relationship. It requires imagination and the creation of something different than the given pieces of information.

The higher the reflective level of relationships, the more creation will be required. Example in the area of exponential law of this reflective level of relationship can be seen by the following example: log3x3+log273x= The challenge presented here is how to connect - with the other side of the logarithmics. Creative thinking to find the “mysterious relationship” will be needed at a higher and abstract manner.

According to the “Growth in Mathematics” article, the creative thinking needed for the solving of this problem requires more psychological processes

Polygon Spring 2012 16 such as primitive knowing, image making, imaging having and property noticing. If the problem still cannot be solved, then the “folding back” process will be needed over and over again until the barrier(s) of the challenging problem can be broken through.

For Mathematics competition problems, usually more than one barrier is presented. These types of problem will be very challenging and the psychological reward is “potentially greater.” The students who finally solved the problem will have tremendous gain in “productive disposition” as mentioned in the article entitled “The Strands of Mathematical Proficiency.”

The following part deals with issues about mathematical education on exponential functions : 1) Issue about the teacher’s understanding for the applied exponential problems using covariational reasoning.

The National Council of Teachers of Mathematics Principles and Standard (NCTM, 2000) advocates for high school and college mathematics curriculum to include the topic of exponential functions and conceptually multiplicative behavior through application problems from the real world.

Few papers discussed the role of covariation in the learning of exponential functions. In Carlson Jacobs, Coe, Larsen and Hsu’s article (2002) entitled “Applying covariational reasoning while modeling dynamic events,” covariational reasoning was defined “to be the cognitive activities involved in coordinating two

Polygon Spring 2012 17 varying quantities while attending to the ways in which they change in relation to each other (pg354).”

In Strom’s study about high school science and Mathematics teachers’ understanding on applied problems of exponential functions, Strom had an interesting finding that even teachers have difficulty with interpreting exponentiation within functions as a dynamic process. Coordinating images of two quantities changing in tandem over time proves to be a weakness for many high school teachers in the study. As a result of this study and findings, covariational reasoning abilities and ideas of proportionality among high school math and science teachers should be strengthened. This is particularly true in the application problems of decay and half-life of radioactive material.

In Strom’s article, it also mentioned that building multidirectional covariation ability (e.g. interpret graph from right to left and left to right) will provide a powerful mechanism for increasing the ability to reason through exponential function’s behavior.

In another study done by Strom entitled “secondary mathematical growth and decay: the case of Ben,” Storm discovered how emphasizing exponentiation as a process can result in an increased ability to describe exponential behavior in more powerful ways. The study, conducted by Strom, belongs to conceptual knowledge and not procedural knowledge. With this study on understanding reasoning abilities, whether from teachers or students, a practical guidance was offered to develop curricula activities on exponential functions. Because findings from Weber’s study suggested that students benefit from a more conceptual

Polygon Spring 2012 18 understanding of exponential operation as opposed to rote memorization of formulas as procedural learning. As an interesting activity to stimulate students’ conceptual understanding on exponential functions and its behavior, the following activity can be provided to students as a group project to help them develop conceptual understandings in the initial learning of the concept of exponential growth : 2) Issue about misperception of exponential growth. a. In a paper conducted by Willem Wagenaar of the Netherlands, entitled “misperception of exponential growth and psychological magnitude of numbers” which was compiled in a book called “social attitudes and psychophysical measurement”, Wagenaar found people underestimate the increase of the series 3,7,20,55,148…….. for perceive the increase series 1,10,100,1000, …….. for

but correctly . This is

because people usually have a better understanding about the base of “10” rather than the base of “e”. Since people do not feel comfortable with growth rate of e, they tend to grasp the meaning of exponential increase by taking the ratio of 2 successive numbers, instead of the growth rate of overall picture. b. Wagenaar also found another misperception of exponential growth. People seem to underestimate the increase of the series 3,7,20,55,148……. But correctly perceive the decline of the series 22026,8103,2981,1097,403……… A possible explanation is the increase of the series has no bound and decline of the series eventually stop to 0.

Polygon Spring 2012 19 As such, insufficient adjustment was made in exponential growth case results in underestimation.

Based on the above 2 misperceptions, the misperception of exponential growth is a societal phenomenon. The continuous intuitive underestimation of growth is caused by psychophysical transformation applied to the base number and dimension of the exponents.

Conclusion Conceptual understanding is the key for the development of procedural understanding. It is the essence of learning with understanding and not just memory work. As a final conclusion of this research paper, the following items summarize the findings with suggestions for learning and teaching exponential functions, exponential laws and the associated application problems as below: 1) High school teachers may need to improve their fundamental understanding on exponential functions before their students can become confident in exponential functions.

2) In teaching exponential functions, particularly about application problems in science, teachers should consider the incorporation of interesting projects on exponential functions as activities in the design of curriculum.

3) More research should be conducted into the thinking mind of students to address the following areas as a minimum: a) What are the possible confusion on exponential functions and why is the confusion arises in students’ mind as it is?

Polygon Spring 2012 20 b) Was there any prior knowledge from previously learned Algebra presents barrier to learn new topics of exponential functions and exponential laws? c) Develop interesting projects or activities of real world problems so that each of the 10 laws of exponents can be best utilized for problem solving. d) Other than 10 laws of exponents which summarize basic conceptual understanding in formula form, is there any missing concepts or links for students’ conceptual understanding of exponential function to be further improved? e) For smart students who already know the 10 laws of exponents and are capable to solve average problems, the challenge on educating these students is to increase their productive disposition. Therefore, perhaps collecting exponential competition problems on exponential functions from various domestic and foreign sources will be beneficial to further stimulate smart students’ intelligence and make their conceptual understanding and procedural fluency closer to perfection.

Polygon Spring 2012 21 References 1. Hiebert, J. (1984): Children’s mathematics learning: the struggle to link form and understanding Elementary school Journal, 84, 497-513.

2. “Conceptual and procedural knowledge: the case of mathematics” edited by James Hiebert.

3. National Research Council (2001), “Helping children learn mathematics” Washington D.C: National Academy Press.

4. Chapter 4 “representation and translations among representations in mathematics learning and problem solving” by Richard Lesh, Tom Post & Merlyn Behr.

5. “Exponential functions, rates of change and the multiplicative unit”. By Jere Confrey and Erick Smith. Educational Studies in Mathematics 26:135-164, 1994. Copyright of 1994 Kluwer Academic Publishers.

6. Carlson, M.P. Jacobs, S, Coe E, Larsen S. and Hsu E (2002). Applying covariational reasoning while modeling dynamic events: A frame work and a study Journal of Research in Mathematics Education.

7. The role of covariational reasoning in learning and understanding exponential functions by April D. Strom of Arizona State University.

8. “A rich lesson in exponential growth”, article from Curriculum Review, Dec. 1997, volume 37, Issue 4.

9. Wegener, Brend (1982). West Germany. “Social Attitudes and Pyschophysical Measurement”. Published by Lawrence Erlbaum Associates.

Polygon Spring 2012 22

Attachment 1

10 laws of exponents:

Polygon Spring 2012 23

Attachment 2 Project of Windfall Scenario (from Curriculum Review, December 1997, Volume 37, issue 4) Imagine you are 4 years old. A rich aunt wants to provide for your future with two options as financial aid: Options 1- she would give you $1000 per year until you reach age 21 (in 17 years).

Option 2- she would give you $ 1 this year, $2 next year, $ 4 the third year, thus doubling the amount from previous year until you reach age 21. a. Which option is the best option you should choose? b. If you only received money for 10 years, which options would bring the most money? c. How many years would you have the same amount of money with both options? d. Why did the money in option 2 increase so rapidly after the 14th year?

Polygon Spring 2012 24

Teaching What to Think vs. Teaching How to Think in College Jose A. Guntin ABSTRACT

The purpose of this study is to see whether or not helping psychology students identify content issues, conclusions, reasons, and ambiguity using the SEEI model will have an impact on development of critical thinking. Although research indicates that significant gains in critical thinking are both perceived and experienced by college students (Tsui, 2002), competence in critical thinking is lower than it should be at every stage in schooling (Norris, 1985). Psychology students enrolled in CLP1006 will be encouraged to utilize the SEEI (State, Elaborate, Exemplify, and Illustrate) model to assist them in identification of content issues, conclusions, reasons, and ambiguity during the semester. At the end of the semester, they will rate their own level of thought in a self-­‐assessment format as measured by Paul and Elder’s (2009) stages of critical thinking development (i.e.: unreflective thinker, challenged thinker, beginning thinker, practicing thinker, advanced thinker, and accomplished thinker). As a result of the intervention, it is expected that practice at identifying content issues, conclusions, reasons, and ambiguity by using the SEEI model will have a positive effect in critical thinking development.

Research Question: Will helping introduction to psychology students identify content issues, conclusions, reasons, and ambiguity by using the SEEI model have a positive effect on critical thinking development as measured by a self-­‐assessment?

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Literature Review Critical thinking (CT) is considered to be one of the key outcomes related to obtaining a college education (Ash, Clayton, and Atkinson, 2005; Renaud and Murray, 2007; Schamber and Mahoney, 2006; Stupnisky, Renaud, Daniels, Haynes, and Perry, 2008; Tsui, 2002; Solon, 2007). Interest in CT can be explored at the level of learning institutions, at the level of educators and at the level of the students. In 2007, the National Leadership Council for Liberal Education and America’s Promise (LEAP) identified “essential learning outcomes” that will help students become successful in the twenty-­‐first century (Association of American Colleges and Universities, 2007). The identified essential learning outcomes were subdivided into three sections: (1) intellectual and practical skills, which include critical and creative thinking, (2) personal and social responsibility, and (3) integrative learning, which includes synthesis and advanced accomplishment across general and specialized studies. Also in 2007, in a spearhead initiative, Miami Dade College (MDC) announced the Learning Outcomes Covenant Signing Ceremony. MDC faculty and student leaders pledged to “the development of knowledge, skills and attitudes that foster effective citizenship and lifelong learning”. MDC included critical thinking as one of the 10 learning outcomes.

Although there seems to be a level of interconnection among all learning outcomes, intuition suggests that CT could be considered a key or primary learning outcome that will facilitate attainment of the rest of the learning outcomes. For instance, an integrative learning, as defined by LEAP, presupposes a measure of CT as it includes connecting, synthesizing and transforming in its rubric (see http://www.aacu.org/value/rubrics/pdf/CreativeThinking.pdf ). Moreover, informal surveys conducted in campus indicate that most students and at least some professors agree that CT could be considered a key or basic learning outcome in the sense that its development facilitates attainment of other learning outcomes. In contrast, development of other learning outcomes does not suggest that CT will

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necessarily improve. In a similar vein, Carroll (2000) proposes that, regardless of intelligence or knowledge, people in general will make unreasonable decisions and arrive to unreasonable beliefs or take unjustifiable actions when they do not engage in CT. Student Performance in Critical Thinking Students agree that development of CT is desirable. Tsui’s (2002) research indicates that significant gains in CT are both perceived and experienced by college students. However, several researchers conclude that CT may not be at an acceptable level when students leave college. For instance, Norris (1985) concluded that competence in CT is lower than it should be at every stage of schooling. In addition, Keeley, Browne, and Kreutzer (1982) found that although seniors outperform freshmen at analyzing articles, they show “major deficiencies” in their performance. Moreover, Keeley (1992) found that both freshmen and seniors show “poor performance” at basic tasks, such as identifying assumptions. There is clearly a need to determine effective methods to assist students in developing CT skills, but it is equally important to find a way to measure student’s progress in the development of CT. Becoming a Critical Thinker We will first explore methods to assist students and we will later address the issue of measuring gains in CT. We must exercise caution when choosing a method to assist student to develop CT skills as some authors present concepts related to CT that may be difficult to operationalize and measure. Some authors such as Ruscio (2006) focus in the developing CT skills by exploring areas including reconceptualizing issues in multiple ways, keeping in touch with reality, formulating multiple working hypothesis, and danger of being mislead by testimonials. Some of those techniques, although informative, may prove to be difficult to use when trying to measure improvements in CT. Fortunately, other authors offer other methods that employ construct that are less ambiguous and therefore easier to measure. For instance, Carroll (2000) suggests focusing on identification of arguments and evaluation

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of arguments as a way to develop CT skills. Similarly, Smith (2002) offers seven guidelines to CT: (1) critical thinkers are flexible – they can tolerate ambiguity, (2) critical thinkers identify inherent biases and assumptions, (3) critical thinkers maintain an air of skepticism, (4) critical thinkers separate facts from opinions, (5) critical thinkers don’t oversimplify, (6) critical thinkers use logical inference process, and (7) critical thinkers examine available evidence before drawing conclusions. Finally, Browne and Keeley (2007) opt to explore issues, conclusions, reasons, ambiguity, assumptions, and fallacies in the reasoning in their guide to critical thinking.

As we can see, there is a wealth of possibilities from which to choose from to determine a method or system to help in the description and development of CT. There is abundant theoretical and pedagogical literature on CT in higher education but relative scarcity of published empirical work on the subject to guide educators in planning classes to facilitate CT (Solon, 2007). In fact, Stupnisky, Renaud, Daniels, Haynes, and Perry (2008) literature review found limited research on student’s CT disposition.

The new awareness related to CT seems to be part of a paradigm shift in education. Peters (2007) suggests that although CT tends to be treated ahistorically focusing on universal processes of logic and reasoning, it is important to look at CT in the context present day and age and considering kinds of thinking and styles of reasoning. In this view, distinct kinds of thinking operate in stages. Thinking skills can be considered to be transferable skills that provide learners with the means by which they can learn.

It is only in recent years that knowledge is considered to be a skill only after it is combined with real-­‐ world experience. This emphasis in the application of information and meta-­‐cognitive skills (i.e.: learning how to learn) is what separates the traditional educational system from the new educational system. Learning institutions, as well as educators, are no longer interested in producing graduates who think in ways specific to their major. Learning how to think is nowadays more important than learning what to think. Fortunately, teaching students skills so they can examine their own learning (i.e.: critical questioning and inquiry process) does not lead to loss of content (Osborne, Kriese, and Tobey, 2008).

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Methods and Assessment If the new interest in CT is a paradigm shift, it is important to identify basic skills to guide the emerging critical thinker, the path that allows for meta-­‐cognition and a measure to evaluate progress.

Basic skills to guide the emerging critical thinker: In order to find how to assist MDC students develop CT skills, I had several discussions with some colleagues and some of my students. The consensus seems to be that students could benefit from practicing a “simple CT system.” Following that advice, I propose a system based on Browne and Keeley (2007) and consisting on helping students in the identification of content issues, reasons, conclusions, and ambiguity. The system is combined with the seven guidelines to CT (Smith, 2002) and weekly instructional strategies to allow student to practice and gain progressively higher levels of mastery in CT. I discussed the idea of utilizing an instructional strategy to assist students in the assimilation of the simple CT system with colleagues attending the Scholarship of Teaching and Learning (SOTL) (2011) and with Enoch Hale, presenter at the SOTL seminar. They agreed that the State, Elaborate, Exemplify, and Illustrate (SEEI) model presented during the seminar by Hale would be the best instructional strategy for the purpose at hand. A similar training method was used by Bensley, Crowe, Bernhardt. Buckner, and Allman (2010) with encouraging results. They found that the group receiving explicit CT skills instruction showed significantly greater gains in their argument analysis skills than the groups receiving no explicit CT instruction.

Path towards CT: It is clear that the literature offers multiple paths towards CT. I choose Paul and Elder (2009) hierarchical organization of CT development on the basis of simplicity of their model and clarity of their definitions. They present six stages in CT development (i.e.: unreflective thinker, challenged thinker, beginning thinker, practicing thinker, advanced thinker, and accomplished thinker).

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Measure of CT: We must exercise all possible caution to avoid teaching to the test. Authentic assessments whereby students are asked to show their knowledge and skills by performing real-­‐world tasks are necessary if we aim to develop skills that students can generalize to their life outside the College. In an attempt to reduce the transmission of fragmented knowledge in the classroom, Clark (2010) utilized experiential essays as a means to connect content information with student’s thoughts and experiences. He reports that his students were able to use course concepts and focus on reporting applied understanding (i.e.: application of concepts to understand specific life events or experiences) and transformed understanding (i.e.: understanding of a topic, such as abnormality or intelligence, with impact beyond the course) as tools for better understand events and experiences in their lives beyond the classroom. A self-­‐assessment measure was chosen as it will allow students to reflect on their own level of critical thinking.

Participants and Procedure Approximately 90 to 100 students registered to two introductory educational psychology courses will participate in the study. One of the classes will act as control group; the other as experimental group.

The classes will run from August to December. Although students will not be randomly assigned, it is reasonable to expect that the students will constitute a representative sample of students attending MDC’s Kendall Campus. The course syllabus will indicate that data will be obtained during the course to enhance teaching effectiveness. Enrollment in this course typically consists of students who are 19-­‐26 years old, approximately 68% Hispanic, 60% female, and 52% first generation in college. Participation in the study will not compromise course objectives or interfere with any aspect of the course.

All students will obtain a rubric containing information about the Stages of Critical Thinking Development as presented by Paul and Elder (2009) and will complete a three to four page pre-­‐self-­‐ assessment rating their level of critical thinking. Students in the experimental group will be introduced

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to the “simple CT system” adapted from Browne and Keeley (2007). The students will utilize the

SEEI model at least once per week to gain practice in the identification and analysis of content issues, conclusions, reasons, and ambiguity. At the end of the semester, all students will complete a three to four page post-­‐self-­‐assessment rating their level of critical thinking and students in the experimental group will participate in a focus group.

Significant Results An independent rater will read all self-­‐assessments. When pre-­‐self-­‐assessments are compared with post-­‐self-­‐assessments, the following results are expected. First, students in the experimental group will report an increase in level of CT in their post-­‐self-­‐assessment as compared with pre-­‐self-­‐assessment. Second, students in the experimental group will report a higher level of CT than students in the control group. Third, the reasoning presented by students in the experimental group in the post-­‐self-­‐assessment will show clarity and ability to keep the reasons and conclusions straight, while students in the control group may use reasons that do not support their own conclusions. No significant correlation is expected between level of CT and final grades. The focus group will lead invaluable information that could be useful for future research.

Applicability to STEM Although the SEEI model has merit, it is not STEM specific and it is not sensitive to the needs of STEM students. STEM students have to successfully overcome a series of difficulties that have to be taken in consideration when one of the learning goals is enhancing CT in STEM. Basic issues to consider are the following: (1) students may not understand the task at hand; (2) they

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may have difficulty with scientific language; (3) they may solve problems in a mechanistic manner without considering alternative procedures that may prove more effective; and (4) they may not be able to transfer learned skills to alternative applications.

With this in mind, the STEM2 model (State the problem/issue, Translate to scientific language, Execute, Market/explain what you did and why did you use a particular technique, Make it applicable/extensive) was created to address needs specific to STEM students. This moded provides the student with a solid structure on which scientific knowledge can be learned while employing CT skills. The model allows the student to tackle new information systematically and addresses the four basic issues indicated above.

STEM2 model consists of five steps. Each one of the steps acts as a building block to facilitate comprehension and promote critical thinking and scientific reasoning. During the first step, State the problem or issue, the student is encouraged to State the problem or issue in his/her own words. This step will (1) prevent students from learning definitions without truly understanding important concepts, (2) assure that the student understands the issue presented in the classroom, and (3) promote curiosity as the student comes up with questions that need to be explained and challenges that need to be resolved scientifically. During the second step, Translate to scientific language, the student converts initial impressions into precise scientific language that facilitate communication and learning of scientific principles. During the third step, Execute, the student provides a solution to the task at hand. During the fourth step, Market, the student explains why a specific formula was utilized and shares about possible implications stemming from the derived solution. During the fifth step, Make it

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applicable/extensive, the student identifies (1) similar issues or problems that can be resolved with line of thinking presented and (2) nuances when the derived solution would be inadequate, (3) nuances when the chosen technique or formula to derive a solution would be inadequate.

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References Ash, S. L., Clayton, P. H., and Atkinson, M. P. (2005). Integrating reflection and assessment to capture and improve student learning. Michigan Journal of Community Service Learning Spring 49-60.

Bensley, D. Alan; Crowe, Deborah S.; Bernhardt, Paul; Buckner, Camille; Allman, Amanda L. (2010). Teaching and Assessing Critical Thinking Skills for Argument Analysis in Psychology. Teaching of Psychology, v37 n2 p91-96 2010. 6 pp.

Browne, M. N., Keeley, S. M. (2007). Asking the Right Questions A Guide to Critical Thinking Eighth Ed. (New Jersey, Pearson Prentice Hall).

Carroll, R. T. (2000). Becoming a Critical Thinker A Guide for the New Millennium. (Boston, Pearson Custom Publishing).

Clark, Kevin M. (2010). Applied and Transformed Understanding in Introductory Psychology: Analysis of a Final Essay Assignment. Journal of the Scholarship of Teaching and Learning, v10 n3 p41-57 Nov 2010. 17 pp.

Keeley, S. M. (1992). Are college students learning the critical thinking skill of finding assumptions? College Student Journal, 26, 316-322.

Keeley, S. M.. Browne, M. N., & Kreutzer, J. S. (1982). A compariron of freshmen and seniors on general and specific essay tests of critical thinking. Research in Hgher Education, 17, 139-154.

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Norris, S.P. (1985). Synthesis of research on critical thinking. Educationnal Leadership, 42(8), 40-45.

Osborne, Randall E., Kriese, Paul, Tobey, Heather (2008). Reflections on a Decade of Using the Scholarship of Teaching and Learning. InSight: A Journal of Scholarly Teaching, v3 p37-46 2008. 10 pp.

Paul, Richards, Elder, Linda (2009). The miniature Guide to Critical Thinking Concepts and Tools (The Foundation for Critical Thinking).

Peters, Michael A. (2007). Kinds of Thinking, Styles of Reasoning. Educational Philosophy and Theory, v39 n4 p350-363 Aug 2007. 14 pp.

Renaud, R. D., Murray, H. G. (2007). The validity of higher-order questons as a process indicator of educational quality. Research in Higher Education 48(3): 319-351.

Ruscio, John (2006). Critical Thinking in Psychology (Wadsworth).

Schamber, J. F., Mahoney, S. L. (2006). Assessing and improving the quality of group critical thinking exhibited in the final projects of collaborative learning groups. Journal of General Education 55(2): 103-137.

Smith, Randolph A. (2002). Changing Your Preconceptions Thinking Critically about Psychology (Wadsworth).

Solon, T. (2007). Generic critical thinking infusion and course content learning in introductory psychology. Journal of Instructional Psychology 34(2): 95-109.

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Stupnisky, R. H., Renaud, R. D., Daniels, L. M., Haynes, T. L., Perry, R. P. (2008). The interrelation of first-year college students’ critical thinking disposition, perceived academic control, and academic achievement. Research in Higher Education 49: 513530.

Tsui, L.(2002). Fostering critical thinking through effective pedagogy: evidence from four institutional case studies. Journal of Higher Education 73(6): 740-763.

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Towards Development of Critical Thinking in College Jose A. Guntin

ABSTRACT

Studies indicate that critical thinking improves as a result of attending college. Miami Dade College’s pioneer work includes critical thinking as one of the learning outcomes. Clearly, there is a higher value in teaching students how to think rather than what to think. Perry’s model of intellectual and ethical development suggests that college students will benefit from moving in a progression of increasingly higher less restrictive levels of thinking (i.e.: dualism, multiplicity, relativism, commitment in relativism) utilizing improved critical thinking skills. At higher levels the student’s level of functioning becomes improved. However, while the development of critical thinking skills is a fundamental objective of the general education of college students, it is not clear what specific elements contribute to the development of critical thinking. A literature review indicates that there may be three performance indicators to evaluate critical thinking: (1) higher-order questions (Renaud and Murray, 2007), (2) deep reflection (Moon, 2009), and addressing spirituality and the “big questions” (Walvoord, 2008).

In this study, introductory psychology students’ critical thinking skills will be measured in a pretest and posttest (i.e. a presentation) utilizing the rubric developed by MDC. The students are expected to show gains in the posttest after being exposed to higher-order questions, deep reflection, and spirituality questions including relating information presented in class to meaning of life.

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Review of the Literature The development of critical thinking is considered to be an essential component of the general education of college students (Ash, Clayton, and Atkinson, 2005; Renaud and Murray, 2007; Schamber and Mahoney, 2006; Stupnisky, Renaud, Daniels, Haynes, and Perry, 2008; Tsui, 2002; Solon, 2007). At the level of higher learning institutions, Miami Dade College (MDC) in a spearhead initiative included critical thinking as one of the 10 learning outcomes as a result of obtaining a college education. However, it is surprising that research on student’s critical thinking dispositions is so limited (Stupnisky, Renaud, Daniels, Haynes, and Perry, 2008). Little is found in terms of actual approaches to promote critical thinking in classrooms that have gone through research scrutiny. For instance, Solon’s (2007) literature review found abundant theoretical and pedagogical literature on critical thinking in higher education, but relative scarcity of published empirical work on the subject to be able to guide educators in planning their classes to facilitate the development of critical thinking.

One of the reasons research in critical thinking is scarce could be that increase in critical thinking skills may not necessarily be related to significant improvement in overall test scores.

Perceived academic control, for instance, was found to have a stronger impact on student’s GPA than critical thinking disposition (Stupnisky, Renaud, Daniels, Haynes, and Perry, 2008).

Even though there is indication that critical thinking may not lead to an increase in GPA, Tsui’s (2002)research states that significant gains in critical thinking are both perceived and experienced by college students. The question is whether or not these desired gains are at favorable levels. Several researchers conclude that critical thinking may not be at an acceptable level when students leave college. Even more distressing is Norris’ (1985) conclusion that competence in critical thinking is lower than it should be at every stage of schooling. Also,

Polygon Spring 2012 38 Keeley, Browne, and Kreutzer (1982) found that although seniors outperform freshmen at analyzing articles, they show “major deficiencies” in their performance. Moreover, Keeley (1992) found that both freshmen and seniors show “poor performance” at identifying assumptions.

If seems clear that students could benefit from a curriculum that emphasizes the importance of improving critical thinking as learning institutions, educators, and students are in agreement that improvements in critical thinking is desirable. A possible answer to the problem would be to offer students specific courses to improve critical thinking skills. In fact, there is no scientific basis to support the notion that one particular course, other than a critical thinking type of course, can make any positive measurable difference to increase critical thinking (Solon, 2007).

The above alternative may not be highly effective as some students will be unable to generalize the learned skills outside the classroom. A second alternative would be to promote faculty initiatives to enhance critical thinking. Here some may be concerned that making emphasis on critical thinking may compromise course content learning. But there is evidence to suggest that this is not the case. For example, no significant differences in psychology learning were found as reflected by test scores between a group of students who have been exposed to a “moderate infusion of generic critical thinking material” and a control group (Solon, 2007).

In an attempt to improve critical thinking without compromising course content learning, three methods were found in the literature: frequency of higher-order questions, working with diverse groups, and reflection.

Frequency of Higher-Order Questions Renaud and Murray (2007) found that frequency of higher-order questions is related to student’s critical thinking skills. They presented three studies in their research. One compared the amount with higher-order questions on tests and assignments in actual classes to pre-test-post-

Polygon Spring 2012 39 test gains in critical thinking. a second experimental study compared groups of students given lower- vs. higher –order questions in actual classes. The third was a true experiment done in a laboratory that related level of review questions to pre-test-post-test gains while controlling for possible confounding variables. Overall, they found that students are more likely to improve their critical thinking skills when they have answered higher-order questions in their coursework.

Also in relation to the kind of questions asked, Walvoord (2008) explored the possibility of using spirituality and the “big questions” to promote critical thinking. He gathered survey data from 533 classes and more qualitative data from 66 classes whose teachers had been recommended by their department chairs as “highly effective” and found that it is possible to manage issues, such as spirituality by giving students tools to deal with life’s questions.

Working in Diverse Groups and Reflection Laird (2005) found that students with positive interactions with diverse peers are more likely score higher in critical thinking disposition. Interestingly, he also found that experience with diverse groups leads to gains in self-confidence in one’s academic and intellectual abilities.

Self-confidence may be a component in perceived academic control which was found by Stupnisky, Renaud, Daniels, Haynes, and Perry (2008) to have a greater effect on GPA than critical thinking disposition. Thus, working with diverse peers becomes a desirable aspect to be studied in combination with critical thinking to see if it leads to significant effects on exam scores. In addition, Schamber and Mahoney (2006) assessed critical thinking in collaborative learning groups through authentic assessment using two interventions: revision in writing and a critical thinking rubric and found gains in student’s group critical thinking skills. Moreover, Moon (2009) suggests that the graduated scenario method could be used to improve critical thinking.

Briefly, the technique was developed to facilitate reflective learning. It consists on developing alternatives of the same account; each alternative scenario represents a distinct level of depth of reflection. The accounts are made available to the group whose task is to identify the strands

Polygon Spring 2012 40 that change between the accounts and make one more deeply reflective than the first. Perry’s (1999) developmental scheme (i.e.: dualism, multiplicity, relativism, and commitment) could be adopted to devise alternative accounts to be used in the graduated scenario method. Reflection has been found to be a key component in student’s learning (Ash, Clayton, and Atkinson, 2005).

Reflective writing results in better quality learning (Moon, 2009). Reflection can be adopted to promote development in critical thinking.

Summary Although critical thinking is both a desired and expected outcome of college education for learning institutions, educators, and students, some researches found that competence in critical thinking is lower than it should beat every stage of schooling (Keeley, 1992; Keeley, Brown, and Kreutzer, 1982; Norris, 1985). The pedagogical literature about critical thinking may be rich but literature about empirical work is needed. Although some researches, such as Stupnisky, Renaud, Daniels, Haynes, and Perry (2008) promote that perceived academic control has a stronger impact on students’ GPA than critical thinking disposition, it is my view that it is likely that more can be done to promote adequate gains in critical thinking. Once adequate levels of critical thinking disposition are obtained, improvements in elaborative learning resulting from (1) make use of of higher-order questions and (2) moving from dualistic absolutism towards generalized relativism and commitment (Perry, 1999) has to be expected with both gains in GPA and perceived academic control. Since research to address the above issues would be too much of an undertaking, the proposed study will be focused on promoting critical thinking and comparing grades undergoing treatment with other students taking the same class with the same course.

Methods and Assessment

Polygon Spring 2012 41 The purpose of this study is to explore increase in critical thinking skills for a group of students taking an introduction to psychology class by means of pre-test-post-test. Students will be divided into collaborative learning groups accounting for diversity. Each group will conduct two presentations. The first presentation will take place during the second week of the course and the last second presentation will take place one week before final examinations. Two rubrics will be utilized by independent raters as pre-test-post-test. The MDC rubric will be compared with a rubric developed based on Perry’s cognitive development model (1999). MDC rubric yields four levels of competency (emerging, developing, proficient, exemplary) created by faculty to assess critical thinking, while Perry’s model describes the steps that move students from simplistic categorical view of knowledge to a more complex view (i.e.: dualism, multiplicity, relativism, commitment) based on sound theory. While MDC rubric allows for both qualitative and quantitative data collection, the rubric based on Perry’s model allows for qualitative data collection and an explanation based on a theoretical model.

Participants and Procedure A total of approximately 45 to 50 students registered to an introductory educational psychology course. The classes will run from August to December. Although students will not be randomly assigned, it is reasonable to expect that the students will constitute a representative sample of students attending MDC’s Kendall Campus. Enrollment in this course typically consists of students who are 19-26 years old, approximately 68% Hispanic, 60% female, and 52% first generation in college. Participation in the study will not interfere with any aspect of the course and course objectives will not be compromised. Students will be tested in groups ranging in size from 5 to 7. Before the experiment begins, each participant will sign a consent form.

Performance in group presentation will be measured before and after treatment. The treatment will consist on (1) having students work in collaborative learning groups to find answers to higher-order questions about information presented in class, (2) application of Moon’s graduate

Polygon Spring 2012 42 scenario technique (2009) by having the groups of students discuss and evaluate four alternative accounts varying in depth of critical thinking and cognitive development as suggested by Perry, and (3) having students write four reflection papers in response to higherorder questions.

Significant Results It is expected that comparison of means obtained in pre-test-post-test by utilizing the MDC rubric will yield significant results suggesting that the treatment was successful in bringing gains in critical thinking. Also, it is expected that students will show improvement in their cognitive development and will be able to process information moving away from simple dualism into multiplicity and relativism. The implication of having significant results would be that introduction to psychology courses can be design to promote cognitive development and increase critical thinking for students. Also, it would be interesting to know if any relationship exists between MDC’s critical thinking rubric and Perry’s model. Finally, final grades of students participating in this study could be compared to final grades obtained by students who did not participate in this study but took the same course. If significant results would be obtained, a higher scale study could be developed comparing test scores of students who are subject to (1) collaborative learning using higher-order questions, (2) application of the graduate scenario method, and (3) reflection with students taking the same tests but attending traditional courses.

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References Ash, S. L., Clayton, P. H., and Atkinson, M. P. (2005). Integrating reflection and assessment to capture and improve student learning. Michigan Journal of Community Service Learning Spring 49-60.

Hittner, J. B. (1999). Fostering critical thinking in personality: the trait paper assignment. Journal of Instructional Psychology 26(2): 92-97.

Holliway, D. (2009). Towards a sense-making pedagogy: writing activities in an undergraduate learning theories course. International Journal of Teaching and Learning in Higher Education 20(3): 447-461.

Keeley, S. M. (1992). Are college students learning the critical thinking skill of finding assumptions? College Student Journal, 26, 316-322.

Keeley, S. M.. Browne, M. N., & Kreutzer, J. S. (1982). A compariron of freshmen and seniors on general and specific essay tests of critical thinking. Research in Hgher Education, 17, 139-154.

Moon, J. (2009). The use of graduate scenarios to facilitate the learning of complex and difficultto-describe concepts. Art, Design & Communication in Higher Education 8(1) 57-70.

Nelson Laird, T. F. (2005). College students’ experiences with diversity and their effects on

Polygon Spring 2012 44 academic self-confidence, social agency, and disposition towards critical thinking.

Research in Higher Education 46(4): 365-387.

Norris, S.P. (1985). Synthesis of research on critical thinking. Educationnal Leadership, 42(8), 40-45.

Renaud, R. D., Murray, H. G. (2007). The validity of higher-order questons as a process indicator of educational quality. Research in Higher Education 48(3): 319-351.

Schamber, J. F., Mahoney, S. L. (2006). Assessing and improving the quality of group critical thinking exhibited in the final projects of collaborative learning groups. Journal of General Education 55(2): 103-137.

Solon, T. (2007). Generic critical thinking infusion and course content learning in introductory psychology. Journal of Instructional Psychology 34(2): 95-109.

Stupnisky, R. H., Renaud, R. D., Daniels, L. M., Haynes, T. L., Perry, R. P. (2008). The interrelation of first-year college students’ critical thinking disposition, perceived academic control, and academic achievement. Research in Higher Education 49: 513530.

Tsui, L.(2002). Fostering critical thinking through effective pedagogy: evidence from four institutional case studies. Journal of Higher Education 73(6): 740-763.

Walvoord, B. E. (2008). How to help students confront life’s “big questions”? Chonicle of Higher Education 54(49).

Factorial Experiment Design to Analyze Fuel Consumption of a Vehicle

Eng. Int. Joaquin A. Bestard Adjunct Professor Department of Mathematics Liberal Arts and Sciences Miami Dade College, Hialeah Campus Hialeah, Florida 33012 E-mail: jbestar1@mdc.edu

ABSTRACT The focus of this experiment is to discover which factors affect the gas mileage of a vehicle by adjusting three driving settings. The experiment is performed on a 2009 Honda Civic, but the targeted population is vehicles of any make and model since the results can be extended to all car models since all vehicles use the same basic functioning principles.

It is hoped that the presented study can be useful to the applied researchers in various fields.

1. INTRODUCTION With increasing gas prices and the lack of alternate fuels the mileage per gallon of a vehicle has become a concern to many drivers and consumers. The focus of this experiment is to discover which factors affect the gas mileage of a vehicle by adjusting three driving settings. The experiment will be performed on a 2009 Honda Civic, but the targeted population is vehicles of any make and model since the results can be extended to all car models since all vehicles use the same basic functioning principles. The experiment tests three driving settings. The car’s A/C temperature and tire pressure are two settings which can be readily adjusted by the driver at any moment before driving.

The tire pressure cannot be controlled during the trip, but it will stay steady for a short trip. The third setting cannot be controlled at the flip of a switch like the A/C, but one can avoid high traffic zones knowing during what time intervals they occur, of course it is not always possible to drive through the ideal light traffic zone. Heavy traffic is avoidable to an extent, but it could be very random and also during a trip there are parts in which the traffic is heavy and parts in which it is light that is why this setting is left to the discretion

of the driver. The driver during the experiment considered it heavy traffic when it took more than 30 minutes to drive through a 10 mile zone, which is equivalent to driving at an average of 20 mph. During the series of tests, the A/C level, tire pressure and traffic pattern each had two adjustable settings. The different combinations of these settings were replicated three times to assure a high volume of data instead of single observations which would have led to only eight observations of mileage per gallon. The goal of these observations is to increase the miles per gallon consumption ratio of the car.

From the analysis of the experiment it was concluded that the highest mileage per gallon could be obtained and replicated by lowering the A/C level, keeping the tire pressure at 32 psi (recommended by Honda), and by driving during low traffic hours.

2. PROBLEM STATEMENT Increasing gas prices and the lack of alternate fuels have made the mileage per gallon of a vehicle a primary concern to many consumers. The goal of this experiment is to determine which factors affect the amount of miles a vehicle travels per gallon by adjusting different driving settings.

3. EXPERIMENT DESIGN Description of Design Factors and Response Variable: 1) Traffic Pattern is a random experimental variable. Traffic cannot be controlled but if a route is observed there usually two patterns. Heavy traffic patterns are usually during rush hour traffic, it takes a lot of breaking and accelerating to get from point A to point B. Light traffic patterns usually happen late at night or early in the morning of weekend days, characterized by smooth driving at constant velocity.

The time of the day a route is used matters because traffic patterns change according to time of day. For this experiment the driver will be held fixed to eliminate the variation due to the driver’s manner of driving and the route used will be held constant. The route is a 10 mile-long section of the Palmetto Expressway from 8th Street to 103rd St. We will be doing test runs during Monday through Friday on regular work days. The schedule for a heavy traffic pattern will be any time from 7:30 AM to 10:30 AM or 3:00 PM to 5:00 PM, for a light traffic pattern 8:00 PM to 11:00 PM. We will be checking the effect of light and heavy traffic on the gas mileage by driving from the Chevron Gas Station (located at 950 Southwest 87th Avenue, Miami, FL 33174) to a BP Lakeside Station (located at 8314 Northwest 103rd Street, Hialeah, FL 33016)

2) Air conditioning is a controllable experimental variable. The air conditioning system of a 2009 Honda Civic contains two knobs one to control temperature. The other to control the intensity of the A/C. The intensity knob (located to the left) will be held constant at its maximum level while the temperature knob (located on the right) will be changed to two different temperatures very cold (High) and regular (Low). The high setting of the knob is located at the point when the temperature knob cannot be turned anymore in the counter-clockwise direction and the low when the pointer of the knob points to the narrowest of the blue marks.

3) Tire pressure will also be random experimental variable since one can measure tire pressure, but due to friction the temperature of the tires increases and tire pressure increases along with it. The tire pressure will be measured when the tires have been sitting for more than 3 hours and we will test the effect of tire pressure on the gas consumption at two different levels 28 psi and 32 psi.

Other factors that affect fuel consumption are acceleration, braking, traffic and weather conditions. These factors cannot be controlled easily therefore they will be accounted for by the error term of our model. Weather is impossible to control therefore we will take this variable as part of the error. The driver of the vehicle and vehicle brand may affect mileage per gallon but to make this experiment practical they will be will be held fixed. Route determines road condition and this factor will be held fixed as well as the distance traveled.

The response variable is the mileage per gallon of the vehicle. Miles per gallons will be the dependent variable of the experiment. This variable will be calculated by dividing the distance traveled by the gallons consumed.

Data Collection Procedure: To collect data the plan is to use the same route for all the experimental trials in order to reduce variation due to route. Trials will be held during different random weekdays and the time of the day will be chosen randomly between the times of day mentioned in the Traffic Pattern variable. In order to measure the amount of fuel consumed, the following steps will be taken: 1) Fill the fuel tank of the car to the maximum on a gas station located at the beginning of the route.

2) Take note of the initial mileage of the car.

3) Drive through the route until the other gas station located at the end of the route is reached.

4) Take note of the final mileage of the car.

5) Fill the tank and make a note of the gallons that were pumped into the tank.

6) This process will be repeated while varying A/C, tire pressure settings, and the traffic density. These settings will be adjusted according to the type of test run.

Experiment Design and Analysis of Data: Since all of the factors were tested at two levels, the 23 Factorial Design (Design and Analysis of Experiments 7th Edition; Douglas Montgomery; Section 6.3; pg. 215) will be used to analyze the results gathered. During the experiment the replicates will be blocked on. Table 1 shows the summary of raw data and Table 2 shows the codes assigned to the factors in order to perform the experiment.

Three replicates of each treatment were measured for a total of 24 test runs. The data was analyzed using an ANOVA table that compares the main effect of each factor as well as the two-factor and three factor interactions.

A/C Level A Low Low Low Low High High High High

Traffic Pattern B Light Light Heavy Heavy Light Light Heavy Heavy

Table 1: 23 Factorial Design Table Tire Pressure Miles per Gallon (psi) C Replicate I Replicate II Replicate III 28 18.11 21.30 20.81 32 29.93 29.23 29.24 28 17.92 18.36 20.56 32 23.98 25.00 24.51 28 19.43 20.05 20.34 32 27.94 26.90 27.71 28 13.92 14.26 12.96 32 16.94 20.18 18.70 Table 2: Factor Code table Factor (i) xi = -1 xi = +1 Low High A/C Level (A) Light Heavy Traffic Pattern (B) 28 psi 32 psi Tire Pressure (C)

4. EXPERIMENT ANALYSIS ANOVA Results: The null and alternate hypothesis being tested by in the ANOVA is: H0: β0 = βA = βB = βC = βAB = βAC = βBC = βABC = 0

None of the factors or interactions have a significant effect on the response variable.

H0: βi ≠ 0 At least one main effect/interaction have a significant effect on the response variable.

According to the ANOVA results (page 7) generated by Minitab 16, Air conditioning level (Factor A), Traffic Pattern (Factor B), and Tire Pressure (Factor C) all have a significant effect on the Mileage per Gallon (MPG) of the vehicle. Air conditioning and traffic pattern interact to cause a significant effect on the mileage per gallons, and so do the traffic pattern and tire pressure.

Residual Analysis: The residual plot in Figure 1 (page 9) shows no pattern. In addition, the residuals are distributed equally above and below the y = 0 line. This suggests that the model generated by the 23 Factorial Design Experiment is unbiased and the predictions generated by the model do not tend to underestimate or overestimate the actual data values. Figure 2 shows that if the data values are organized by replication number and by factor combination it is very random.

The normal plot for the 23 Factorial Design on Figure 3 shows that the data is normally distributed, because all of the points on the plot follow a linear pattern. Those points that do not fall on the line can pass the normality test using the fat pencil test for normality. In Figure 4 the results of the Kolmogorov-Smirnov test for normality appear on the label, KS = 0.149 and the corresponding p-value is 0.150 which is greater than the significance level of 0.05 and this corroborates the fat pencil test results by saying that the data is indeed normally distributed. The normal plot of standardized effects in Figure 5 supports the ANOVA results by showing how the main effects of A, B, and C and the interactions AB and BC are have a significant standardized effect and fall far from the normal line.

This figure also shows the sign of the effect coefficients; if the points lie on the left the effects will have a negative effect coefficient on the regression model, while the points on the right will have a positive effect coefficient.

Regression Model, Contour Plots and Surface Plots Analysis: The significant regression model coefficients obtained from Minitab (page 7) are listed on the following table and the equation of the model is shown below the table. β0 βA βB βC βAC

21.595 -1.651 -2.653 3.426 -1.130

βBC

-0.817

̂

The regression model has an R2 value of 97.61% and Radj2 of 96.07% which assures us that the regression model closely resembles the actual data.

The contour plots of this experiment are all shown in Figures 6 through 11 (starting in page 10). In order to optimize the gas economy of the vehicle, we are interested in the dark green regions on these contour plots which represent a predicted mileage per gallon above 28 mpg. Contour plots in Figures 7, 8, 10, and 11 all show the dark green region of interest. Three of the contour regions in the analysis show that a favorable combination to increase the gas mileage is low setting of factor A and B and the high setting of C, which means driving the vehicle with the A/C at a low level at a low traffic zones and with 32 psi tire pressure (which is suggested in the manual of the vehicle). Figure 11 shows that the low level of factor B and high level of A and C will create a desirable effect, but this is questionable because only one contour plot shows this effect and if we compare the same observations to the other graphs the color is actually a lighter shade of green which corresponds to a lower level of gas mileage. Response Surface Graphs are the 3D models of the contour plots and they show the same results analyzed in the contour plots.

5. CONCLUSION Using the regression model generated by the 23 Factorial Design: ̂

The study suggests that the best factor combination to increase the number of miles per gallon of a 2009 Honda Civic is low A/C, light traffic zones, and to use the suggested 32 psi tire pressure setting suggested by the car’s manufacturer.

This experiment can be extended to other vehicle make and models and further tests concentrating in more factors can be performed to obtain more accurate results. During this experiment many source of error existed due to the lack of measuring equipment and software. It is suggested to lessen variation to keep the gasoline supplier constant; this source of variability was ignored for the simplicity of this experiment. Another improvement to the design of the experiment is to use tools to measure tire pressure variations during the experiment, in this experiment the equipment to instantaneously

measure this variable factor was not present. If tire pressure increases with the increase in temperature due to friction, then the gas mileage shown in the results is improved due to friction encountered by the road conditions.

ACKNOWLEDGEMENT The author would like to thank his professor, Dr. Sneh Gulati, Department of Mathematics and Statistics, Florida International University, for her guidance in the completion of this project.

Also, the author would like to thank the editors for their useful comments and suggestions which considerably improved the presentation of the paper.

REFERENCES [1] Douglas Montgomery; Design and Analysis of Experiments 7th Edition: Wiley, 2008; Section 6.3; pp 215.

[2] "Factors That Affect Fuel Economy." U.S. Department of Energy. 27 Nov 2011 < http://www.fueleconomy.gov/feg/ratings2008.shtml >.

APPENDIX A. MINITAB RESULTS Factorial Fit: MPG versus Block, A, B, C Estimated Effects and Coefficients for MPG (coded units) Term Constant Block 1 Block 2 A B C A*B A*C B*C A*B*C

Effect

-3.302 -5.306 6.852 -2.260 -0.618 -1.633 0.289

S = 0.999004 R-Sq = 97.61%

Coef 21.595 -0.574 0.314 -1.651 -2.653 3.426 -1.130 -0.309 -0.817 0.145

SE Coef 0.2039 0.2884 0.2884 0.2039 0.2039 0.2039 0.2039 0.2039 0.2039 0.2039

PRESS = 41.0610 R-Sq(pred) = 92.96%

T 105.90 -1.99 1.09 -8.10 -13.01 16.80 -5.54 -1.52 -4.00 0.71

P 0.000 0.066 0.294 0.000 0.000 0.000 0.000 0.152 0.001 0.490

R-Sq(adj) = 96.07%

Analysis of Variance for MPG (coded units) Source Blocks Main Effects A B C 2-Way Interactions

DF 2 3 1 1 1 3

Seq SS 3.964 516.098 65.427 168.942 281.729 48.943

Adj SS 3.964 516.098 65.427 168.942 281.729 48.943

Adj MS 1.982 172.033 65.427 168.942 281.729 16.314

F 1.99 172.38 65.56 169.28 282.29 16.35

P 0.174 0.000 0.000 0.000 0.000 0.000

A*B A*C B*C 3-Way Interactions A*B*C Residual Error Total Obs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

StdOrder 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

1 1 1 1 1 14 23

30.640 2.295 16.008 0.502 0.502 13.972 583.480

MPG 18.1070 19.4260 17.9226 13.9241 29.9320 27.9365 23.9782 16.9429 21.2963 20.0456 18.3633 14.2636 29.2308 26.9006 25.0000 20.1754 20.8145 20.3426 20.5607 12.9602 29.2359 27.7108 24.5125 18.6992

Fit 19.4987 19.3642 18.3750 13.1421 28.8923 26.9421 23.9230 18.0320 20.3870 20.2525 19.2633 14.0303 29.7806 27.8304 24.8113 18.9202 20.3321 20.1976 19.2084 13.9755 29.7257 27.7755 24.7564 18.8654

30.640 2.295 16.008 0.502 0.502 13.972

SE Fit 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449

30.640 2.295 16.008 0.502 0.502 0.998

Residual -1.3917 0.0619 -0.4524 0.7820 1.0397 0.9944 0.0552 -1.0890 0.9093 -0.2069 -0.9000 0.2332 -0.5498 -0.9298 0.1887 1.2552 0.4824 0.1450 1.3524 -1.0152 -0.4898 -0.0646 -0.2439 -0.1662

30.70 2.30 16.04 0.50 0.50

0.000 0.152 0.001 0.490 0.490

St Resid -1.82 0.08 -0.59 1.02 1.36 1.30 0.07 -1.43 1.19 -0.27 -1.18 0.31 -0.72 -1.22 0.25 1.65 0.63 0.19 1.77 -1.33 -0.64 -0.08 -0.32 -0.22

Predicted Response for New Design Points Using Model for MPG Point 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Fit 19.4987 19.3642 18.3750 13.1421 28.8923 26.9421 23.9230 18.0320 20.3870 20.2525 19.2633 14.0303 29.7806 27.8304 24.8113 18.9202 20.3321 20.1976 19.2084 13.9755 29.7257 27.7755 24.7564 18.8654

SE Fit 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449 0.6449

95% (18.1156, (17.9811, (16.9919, (11.7590, (27.5092, (25.5590, (22.5399, (16.6489, (19.0039, (18.8694, (17.8802, (12.6473, (28.3975, (26.4473, (23.4282, (17.5372, (18.9490, (18.8145, (17.8253, (12.5924, (28.3426, (26.3924, (23.3733, (17.4823,

CI 20.8818) 20.7473) 19.7581) 14.5251) 30.2754) 28.3252) 25.3061) 19.4150) 21.7700) 21.6355) 20.6463) 15.4134) 31.1637) 29.2134) 26.1944) 20.3033) 21.7152) 21.5807) 20.5915) 15.3585) 31.1088) 29.1586) 26.1395) 20.2484)

95% (16.9484, (16.8139, (15.8247, (10.5918, (26.3420, (24.3918, (21.3728, (15.4817, (17.8367, (17.7022, (16.7130, (11.4801, (27.2303, (25.2801, (22.2610, (16.3700, (17.7818, (17.6473, (16.6581, (11.4252, (27.1755, (25.2252, (22.2062, (16.3151,

PI 22.0490) 21.9144) 20.9252) 15.6923) 31.4426) 29.4924) 26.4733) 20.5822) 22.9372) 22.8027) 21.8135) 16.5806) 32.3309) 30.3806) 27.3616) 21.4705) 22.8824) 22.7478) 21.7587) 16.5257) 32.2760) 30.3258) 27.3067) 21.4156)

B. MINITAB GRAPHS Figure 1: Residual Plot (Fitted Value vs. Standardized Residual)

Figure 2: Residual Plot (Standardized Residual vs. Observation Order) Versus Order

Versus Fits

(response is MPG)

2

2

1

1

Standardized Residual

Standardized Residual

(response is MPG)

0

-1

-2

0

-1

-2 15

20 Fitted Value

25

2

30

4

6

18

20

22

Figure 4: Normal Probability Plot (Response Variable vs. Percent)

Normal Probability Plot

Probability Plot of MPG 99

95

95

90

90

80

80

70

70

60 50 40 30

20

10

10

5

5

-2

-1

0 1 Standardized Residual

2

3

Mean StDev N KS P-Value

60 50 40 30

20

-3

24

Normal

99

Percent

Percent

10 12 14 16 Observation Order

Figure 3: Normal Probability Plot (Standardized Residual vs. Percent) (response is MPG)

1

8

1

10

15

20

25 MPG

30

35

21.60 5.037 24 0.149 >0.150

MINITAB GRAPHS Figure 5: Normal Probability Plot (Standardized Effect vs. Percent) Normal Plot of the Standardized Effects (response is MPG, Alpha = 0.05)

99

Effect Ty pe Not Significant Significant

95 C

90

F actor A B C

Percent

80 70 60 50 40 30

BC AB A

20 10

B

5

1

-10

-5

0 5 10 Standardized Effect

15

20

N ame A B C

MINITAB GRAPHS Figure 6: Contour Plot of A/C Level vs. Traffic Pattern (Tire Pressure at 28 psi)

Figure 7: Contour Plot of A/C Level vs. Traffic Pattern (Tire Pressure at 32 psi)

Contour Plot of MPG vs B, A

Contour Plot of MPG vs B, A MPG < 14 14 – 16 16 – 18 18 – 20 20 – 22 22 – 24 24 – 26 26 – 28 > 28

B

0.5

0.0

1.0

MPG < 14 14 – 16 16 – 18 18 – 20 20 – 22 22 – 24 24 – 26 26 – 28 > 28

0.5

B

1.0

0.0

Hold Values C -1

Hold Values C 1

-0.5

-1.0 -1.0

-0.5

-0.5

0.0 A

0.5

-1.0 -1.0

1.0

-0.5

0.0 A

0.5

1.0

Figure 8: Contour Plot of A/C Level vs. Tire Pressure (Traffic Pattern is Light)

Figure 9: Contour Plot of A/C Level vs. Tire Pressure (Traffic Pattern is Heavy)

Contour Plot of MPG vs C, A

Contour Plot of MPG vs C, A

1.0

0.0

Hold Values B -1

-0.5

MPG < 14 14 – 16 16 – 18 18 – 20 20 – 22 22 – 24 24 – 26 26 – 28 > 28

0.5

C

C

0.5

-1.0 -1.0

1.0

MPG < 14 14 – 16 16 – 18 18 – 20 20 – 22 22 – 24 24 – 26 26 – 28 > 28

0.0

Hold Values B 1

-0.5

-0.5

0.0 A

0.5

1.0

-1.0 -1.0

-0.5

0.0 A

0.5

1.0

MINITAB GRAPHS Figure 10: Contour Plot of Traffic Pattern vs. Tire Pressure (A/C Level at Low)

Figure 11: Contour Plot of Traffic Pattern vs. Tire Pressure (A/C Level at High)

Contour Plot of MPG vs C, B

Contour Plot of MPG vs C, B MPG < 14 14 – 16 16 – 18 18 – 20 20 – 22 22 – 24 24 – 26 26 – 28 > 28

C

0.5

0.0

1.0

MPG < 14 14 – 16 16 – 18 18 – 20 20 – 22 22 – 24 24 – 26 26 – 28 > 28

0.5

C

1.0

0.0

Hold Values A -1

-0.5

Hold Values A 1

-0.5

-1.0 -1.0

-0.5

0.0 B

0.5

-1.0 -1.0

1.0

-0.5

0.0 B

0.5

1.0

Figure 12: Response Surface Plot A/C Level vs. Traffic Pattern (Tire Pressure at 28 psi)

Figure 13: Response Surface Plot A/C Level vs. Traffic Pattern (Tire Pressure at 32 psi)

Surface Plot of MPG vs B, A

Surface Plot of MPG vs B, A Hold Values C -1

30

20 MPG

Hold Values C 1

1

18

1 MPG

16

0

14 -1 0 A

-1 1

B

25 0

20

-1

-1

0 A

1

B

MINITAB GRAPHS Figure 14: Response Surface Plot A/C Level vs. Tire Pressure (Traffic Pattern is Light)

Figure 15: Response Surface Plot A/C Level vs. Tire Pressure (Traffic Pattern is Heavy)

Surface Plot of MPG vs C, A

Surface Plot of MPG vs C, A Hold Values B -1

Hold Values B 1

30

MPG

25

MPG

25

20

1

1 15

20

0

0

C

-1

C

-1 0 A

1

0

-1 A

1

-1

Figure 16: Response Surface Plot Traffic Pattern vs. Tire Pressure (A/C Level at Low)

Figure 17: Response Surface Plot Traffic Pattern vs. Tire Pressure (A/C Level at High)

Surface Plot of MPG vs C, B

Surface Plot of MPG vs C, B Hold Values A 1

Hold Values A -1

30 25 MPG

25

MPG

20 1

1 15

20 0

0

C -1

-1 0 B

1

0

-1 B

1

-1

C

THE COST OF EXCLUSION

1 The Cost of Exclusion

Mary Ann Benites Associate Professor Department of English for Academic Purposes Miami Dade College, Hialeah Campus E-mail: mbenite2@mdc.edu

Abstract This paper will address the importance of in-state tuition benefits for undocumented college bound students. In the United States, each and every year, approximately 65,000 undocumented students graduate from American high schools and are faced with an educational paradox. This quandary is due to the fact that, in 1982, the U.S Supreme ruled in favor of providing equal access to public K-12 education regardless of a child's immigration status. While this ruling should be considered a noble act, undocumented college bound students face serious legal restrictions to higher education upon graduation. Clearly, without sound federal educational policy that includes the ability to obtain in-state tuition for higher education, thousands of American raised high school graduates are doomed for a life of poverty. The literature will interpret current immigration policy and prospects for undocumented college bound youth.

Consideration will be given to the reality that if undocumented students are provided opportunities to pursue higher education, these students could greatly benefit the U.S. tax base and help to promote the national economy.

Key words: Dream Act, Illegal Immigration Reform and Immigrant Responsibility Act, Phyler v Doe 1982, Undocumented Students.

THE COST OF EXCLUSION

2 Introduction

The Plight of Undocumented Students In a recent 2011 study compiled by Educators for Fair Consideration, it was estimated that 11.2 million undocumented immigrants of all ages live in the United States of America. Of the millions of undocumented residents, 1.1 million are children under the age of eighteen. In addition, there are approximately 65,000 undocumented students who graduate from high school each year. Through no fault of their own, undocumented college bound students are not allowed to participate in higher education and as a tax payer society because they were brought to the United States illegally as children.

As a nation, one should consider a college education is a personâ&#x20AC;&#x2122;s strongest barrier against a life of poverty and dependence on governmental funds. Of the millions mentioned, there are only 7,000 to 13,000 undocumented students enrolled in colleges throughout the United States. However, more would attempt to participate if the federal government would mandate that all fifty states provide students the ability to pay in-state college tuition. As it stands, only thirteen states have passed laws that allow undocumented students to qualify for in state tuition (Educators for Fair Consideration, 2011).

In order to comprehend this debatable state of affairs, one must investigate the characteristics of college bound undocumented students. Most of these students have lived in the U.S. almost all of their lives, and they have learned to view themselves as Americans. These students have become acculturated to pursue the American dream. In like manner, many have excelled academically in high school and desire to pursue a college education. Still, a costly social problem exists due to the fact that there is no federal pathway to supporting these students in becoming legal residents.

Moreover, one must consider that undocumented students who attend our nationâ&#x20AC;&#x2122;s public K-12 classrooms have been acculturated to believe in themselves, and follow American core values of competition and hard work equals material success in the United States. However, on graduation day, undocumented students face extreme financial uncertainty due to their legal status and inability to pursue the American dream. As a result, most undocumented high school graduates continue to exist within a deficient economic cycle. Currently, nearly 40% of undocumented children live below the poverty level as opposed to 17% of native born children (Gonzales & Orozco, 2009). These statistics are alarming, and one must keep in mind that due to the Phyler v. Doe 1982 ruling, the United States educates undocumented children in public schools. Yet, there is no easy transition to higher education after the fact.

In order to comprehend the argument behind instate tuition and or aid for undocumented college bound students, one must realize it was the Supreme Court of the United States that ruled in favor of allowing undocumented K-12 students to attend our nations classrooms in the first place. In brief, the Court cited the Equal Protection Clause in the Fourteenth Amendment The Court also ruled that regardless of status under immigration laws, an alien is a "person." As such, illegal resident children are also protected by the Due Process Clauses of the Fifth and Fourteenth Amendments. In this landmark case, the Court struck down a Texas state statute that denied educational funding to illegal aliens. The Court also ruled against school districts which attempted to charge illegal immigrant children a $1,000 annual tuition fee per student. Prior to this class action court case, there were no financial provisions to educate undocumented school children (Phyler v Doe, 1982, 457, U.S. pp. 210-216).

Moreover, in a 5 to 4 ruling, the Supreme Court cited that denying illegal immigrant students the right to a proper education would contribute to the creation of a subclass of illiterates within the nation. The Court cited, "Public education has a pivotal role in maintaining the fabric of our society and in sustaining our political and cultural heritage." They added by ruling, "Depriving a person of an education takes an inestimable toll on the social, economic, intellectual, and psychological wellbeing of the individual, and it poses an obstacle to individual achievement." (Phyler v Doe, 1982, 457, U.S. pp. 216-224).

Just as in the 1954 Brown vs The Board of Education ruling, the Court did not mention provisions regarding higher education. However, in today's challenging economy, one requires more than a high school diploma to remain financially solvent. What's more, appropriate educational policies must be taken to reduce this sub group's dependence on public assistance.

Camarota (2004) discusses the fact that on average, illegal households i.e. undocumented parents with children born in the United States pay approximately $4,200 annually in forms of federal taxes, yet these families impose higher costs on the national economy of approximately $6,950 per household. Camarota adds that, "The median income of a college graduate with a bachelor's degree is $49,900 of which an estimated $11,800 is paid in annual taxes." Clearly, solutions must be found to bridge higher education for college bound undocumented students as opposed to the maintenance of the status quo.

Still, it is the mismatch of federal policies that enable the status quo. The enactment of the Illegal Immigration Reform and Immigrant Responsibility Act of 1996 ("IIRIRA"), on September 30, 1996, resulted in significant changes to U.S. immigration laws. With reference to higher education, IIRIRA Section 505 in particular addresses the following criterion: An alien who is not lawfully present in the United States shall

not be eligible on the basis of residence within a State ... for any postsecondary education benefit unless a citizen or national of the United States is eligible for such a benefit (in no less an amount, duration, and scope) without regard to whether the citizen or national is such a resident.

Harmon, Carne, Lizardy-Hajbi, and Wilkerson (2010) argue that while most states interpret IIRIRA to mean that undocumented immigrants cannot receive in-state tuition rates, there are states which have opted to disagree. It appears that the use of the word â&#x20AC;&#x153;unless,â&#x20AC;? has allowed states to interpret this verbage as a loophole. As such, thirteen states have passed laws that allow undocumented students to qualify for "in-state tuition" at public colleges or universities within their state of residence.

Moreover, California, Illinois, New Mexico and Texas are the only four states which allow students access to state funded financial aid. These laws require that undocumented students attend high school for a specified time frame in order to qualify for in-state tuition benefits. This aid includes grants, work study programs, or publically funded loan programs (Educators for Fair Consideration, 2011).

However, there are difficult financial barriers solidly in place that block the pathway between high school and higher education for the undocumented. It is estimated that the annual cost of full-time "instate tuition" at a two year college is $2,713.00; while, the cost of instate tuition at a public four year university averages at $7,605.00. To add, in academic year 2009-10, more than $154 billion in federal and state aid was awarded to undergraduate students. Also, much of these funds do not require repayment by the student. In a recent report publish by College Board, the average amount of financial aid dispersed to undergraduate students is

$11,500; this includes approximately $6,000 in grants which do not require repayment (What It Costs to Go to College, 2011). Clearly, undocumented college bound students must view higher education as an unattainable American dream.

What has been done to address this plight? The most notable effort to addressing the predicament of college bound undocumented students the Development, Relief, and Education for Alien Minors (DREAM Act). Multiple DREAM Act bills have been introduced and debated to address the undocumented college bound student population. The DREAM Act was initially introduced by Orrin Hatch (R) as a legislative proposal on August 1, 2001 with the latest version recently reintroduced by Senator Harry Reid (D) on May 11, 2011 (S.952. Library of Congress, 2011). Yet, to date, every attempt to pass this bi-partisan legislative proposal has failed.

The primary issue in the DREAM Act legislation is the language stated in the bill. It appears in the 1996 bill, there is a two prong approach which enabled some unauthorized alien students to become U.S. legal permanent residents during their college tenure or after military service. In the 111th Congress, the House attempted to approve revising the DREAM Act language as part of an unrelated bill. However, the Senate failed, on a 55-41 vote, to invoke revision of the language on the DREAM Act amendment and the bill died. Yet, once again another attempt to pass the DREAM Act failed in December 2010 when the 112 Congress failed to enact the bill into law (Bruno, 2011).

It appears another glaring drawback to the passage of the DREAM Act is the fact that undocumented college students would be eligible for federal work-study funds. Those against the passage of the DREAM Act argue that by spreading financial aid dollars, in a bad economy, resources would be limited for American or legal resident students. Moreover, throughout the decade of the DREAM Act, opponents pointed out that in-state tuition benefit laws would result

in added costs to taxpayers (Zota, 2009). In addition to the financial argument against the DREAM Act, there is the belief that by making financial aid benefits more available to the undocumented, it would encourage more unauthorized immigration into the nation. As such, opponents strongly objected to subsidizing educational benefits for what they believe are people who are in clear violation of our nationâ&#x20AC;&#x2122;s immigration laws.

Yet, what are the alternatives? First, the status quo, only continues to approach undocumented students from differing perspectives with few benefiting and most not. This nonsensical approach fails to provide college bound undocumented students a route to continued education. Clearly, our current immigration policies most likely appeal to conservative constituencies; however, from the standpoint of what is morally right for the nation, these policies greatly fail to improve upward mobility in the United States of America. It makes no sense to provide undocumented students a free K-12 education and then welcome them to a life on welfare benefits after high school. These educational legal barriers must be addressed to help foster our nation's economic growth as well (Jewell, 2009).

As such, the literature will address the lack of sound federal policies which must be changed and implemented to guarantee equal access for college bound undocumented students who have met specific criterion for in-state tuition benefits.

Review of the Literature Unclear Federal Policies Abrego and Gonzales (2010) cite this problem as one caused because there are no clear federal policies enacted which support undocumented college bound students. As there are no clear federal policies which indicate what to do with undocumented students after high school graduation, states have increasingly attempted to create their own legislative solutions. As

previously mentioned, only thirteen states have attempted to bypass language and allow undocumented students to pay in-state tuition fees. The following states which allow undocumented students to pay reduced costs are: California, Connecticut, Illinois, Kansas, Maryland, Nebraska, New Mexico, New York, Rhode Island, Texas, Utah, Washington, and Wisconsin.

(Olivias, 2009) adds that in said states, the student is generally required to have resided for a specific time (usually 5 years), attend and graduate from a high school in that state as well as make a legitimate attempt to obtain lawful residency status. While this is a noble attempt to aid these college bound students, there are numerous students, who currently reside in other states, that do not meet in-state tuition criterion.

Nonetheless, Abrego and Gonzales (2010) postulate that the DREAM Act which was aimed to provide immigration relief to all undocumented students would have helped to remove unclear state policies and open the legal barriers to higher education. This legislature could have increased the likelihood of upward social mobility. The authors cite that the DREAM Act, also provides the best pathway to legal residency. Moreover, it should be considered the sole solution to the economic predicament of the undocumented college bound student. To add, in a federal policy evaluation which attempts to study the effects of passing the DREAM Act, Flores (2010), theorizes the educational completion rates of undocumented high school students are particularly important in assessing if they would likely benefit from a federal mandate which favors in-state resident tuition assistance. However, there are no valid baselines of measurement to date.

Nevertheless, Flores (2010) attempts to draw a hypothesis and indicates facts. As it appears, under the current educational system, one-sixth of the undocumented student population is under age 18. In like manner, the author cites a recent 2010 study conducted by the Urban

Institute which estimates 49% of â&#x20AC;&#x153;unauthorized youthâ&#x20AC;? of all races and ethnicities do not complete high school. This is in comparison to 21% of their legal immigrant counterparts and 11% of native-born students in the United States. In her own study, Flores adjusts her sampling to address the metropolitan Latino population in states with an in-state tuition policy versus states which lack one.

The results of Flores' policy change initiative indicate - The sample students were 1.52 times more likely to enroll in college directly after graduation if allowed to pay in-state tuition fees. Flores (2010) concludes that the availability of an in-state tuition policy significantly affects higher education decisions of undocumented students, yet the author adds we know less about how they perform in postsecondary institutions.

Indeed, one may conclude it is obvious undocumented students would attempt to gain access to higher education if it were available to them, but this issue is not black and white. As previously mentioned, in 1996, Congress sought to clarify and block the status of undocumented immigrant students within higher education settings by implementing Section 505 of the Illegal Immigration Reform and Immigration Responsibility Act (IIRIRA). Perry (2006) elaborates IIRIRA operates under the assumption that the law does not completely ban states from offering in-state tuition to undocumented students due to interpretation of the language by individual states.

Arbrego and Gonzalez (2010) cite Glassi (2003) and add that the federal government has never issued actual regulations specifying how the IIRIRA provision should be interpreted and enforced. Therefore, due to the lack of clarity at the federal level, this law continues to be interpreted differently by each state. As such, it appears the states which provide in-state tuition for undocumented immigrants cite their educational policies do not violate IIRIRA because

undocumented students have attended high schools and meet more stringent residency requirements than their U.S. counterparts.

Nonetheless, there are legislators and voters who strongly believe that students who do not carry legal residency are not entitled to higher education. North (2009) points out that state legislatures opt to bar undocumented immigrant students from in-state tuition benefits to appease their constituents. North (2009) discusses Olivias (2008) research on the effect of Arizona Proposition 300 as a clear example of appeasing the voters. In 2006, Arizona voters approved Proposition 300 which mandated that university students who were not U.S. citizens or permanent residents, or who do not have lawful immigration status, would not be eligible for instate tuition status or state subsidized by financial aid. The Arizona initiative resulted in the removal of almost 5,000 students from in-state tuition status.

Without federal policies enacted, undocumented students can attend college classes one day, and be ousted the next. In a similar manner, a waiver system in Georgia had once allowed each public college to award in-state status to undocumented students for up to two percent of the collegeâ&#x20AC;&#x2122;s headcount. However, in 2007, voters approved SB529, the Georgia Security and Immigration Compliance Act, and by 2008, undocumented students were unable to establish instate residency (North, 2009).

Clearly, this educational scenario is a paradox in action. As a general rule, undocumented students are ineligible for state and federal financial aid packages despite the fact that their parents file tax returns and pay into the system. Olivas (2009) postulates studies indicate many undocumented parents file their returns with an individual taxpayer identification number (ITIN).

In essence, this practice is the federal governmentâ&#x20AC;&#x2122;s method to collect funds from all citizens regardless of their immigration status. Ring and Svensson (2007) add that these unsympathetic

paradoxical policies, at the federal level, allow states to run renegade with the future of undocumented college bound students.

Furthermore, Ring and Svensson (2007) add if large groups of undocumented residents are simply ignored the ramifications could be quite severe in terms of social ills in the United States. There is a correlation between social status, crime, and frustration and fueled by a lack of resources. North (2009) agrees that a growing portion of undocumented high school graduates who are not able to obtain a higher education due to finances or opportunities, may generate feelings of anger toward those who have the chance to better themselves. As such, the problems of exploitation, crime, and social class intensify. Also, due to archaic federal policy, Jewell (2009) adds these students will most likely not take their K-12 education seriously as they can't see themselves with a future in this nation.

Shiu, Kettler & Johnson (2009) concur with Jewell (2009) and postulate, with the implementation of sound fiscal immigration policy, the potential numbers of crime, health, social related ills could go down significantly as undocumented young adults would be given the chance to obtain meaningful employment. Furthermore, the enactment of sound fiscal immigration policy which favors all of our nation's college bound youth would have numerous social benefits in particular, those who encompass a marginalized population group.

Possible Solutions and Potential Outcomes If the United States is prepared to pay for added crime and fund added police officials and welfare agencies, then the do nothing approach works best. Furthermore, due to the Plyler v Doe 1982 ruling, illegal immigrant students will continue to fill our nation's already overcrowded classrooms. As such, the do nothing approach will continue to result in burdening the United States welfare system as the growing number of undocumented students reach

adulthood. Edwards (2010) cites that undocumented students have trouble seeking ways to make ends meet with merely a high school diploma and no legal status. Edwards (2010) continues his point by stating taxpayers pay $4.3 billion dollars every year toward undocumented residents living in the United States.

In order to illustrate the quandary associated with maintaining the status quo versus finding a better solution for the good of the nation, one must explore viable solutions. In particular, a viable return on higher education investment for all of our nation's children.

Fraum (2011) cites two Rand Studies published in 2001 and in 1996 respectively to elaborate on the known fact that higher levels of education translate into greater public fiscal returns. The Rand Studies cites the potential economic benefits of doubling the rate at which U.S. born Latinos receive college degrees. U.S. born Latinos were tracked due to the current laws which discriminate against undocumented college students.

As it stands, both studies estimated a cost of $6.5 billion to double the rate of Latinos earning a bachelorâ&#x20AC;&#x2122;s degree. The studies found that doing resulted in an increase of $13 billion in public revenues in the form of increased funds from taxes and contributions to Medicare and Social Security. Furthermore, Fraum (2011) postulates that the RAND researchers found the return on investment would only take 13 to 15 years for the public to recoup the costs of their investment.

Noorani & Belanger, (2009) cite the most viable alternative to producing an answer that makes sense for the American people is education and immigration enforcement combined with a pathway to legal residency. There must be a significant coupling of ideological streams that unite in one direction and not a back and forth in a continued discourse of problems. Moreover, Noorani and Belanger continue that a resolution is often difficult to reach, yet key stakeholders

and interests groups must see how beneficial it is to give qualified students a gateway to education and legal immigration status as opposed to added financial ills to our society.

Frum (2011) brings a fresh perspective to this debate and adds 30 states will see changes in the number of students graduating from high school in the next ten years. The ranges are noted as increases of less than 10 percent to more than 100 percent. Also, the remaining 20 states will see a decline in high school graduates. Frum continues by indicating this scenario will have an adverse effect on the number of college enrollments.

Therefore, the author elaborates that a balance should be implemented which allows undocumented students to attend colleges which suffer from low enrollments. Frum (2011) summarizes her policy idea and elaborates that by admitting undocumented students from neighboring states, institutions would be able to fill previously vacant higher education seats and in the process colleges can receive new tuition dollars.

Indeed, the step to establishing sound educational and fiscal policy is the ability to step back and review its long term value. Barach (2009) adds that sound educational and fiscal policy should not be created for the purpose of putting an end to all of the illegal immigration problems of the United States, but rather to provide a chance at hope for this narrowly defined population group who have lived in the United States, for more than five years, and have graduated from American high schools. Clearly, the long-term ramifications of doing nothing are apparent and have already been documented. The alternative to maintaining the status quo has already proven to be extremely difficult to police, and it is even more costly to the American taxpayer.

Conclusion To conclude, the fact still remains that a large number of undocumented college bound students, in the United States, are faced with a multitude of obstacles as they struggle to gain

access to higher education. Many have the academic preparation to pursue a postsecondary education, but their economic and social mobility is severely restricted by their undocumented status (Gonzalez & Orozco, 2009). Indeed, the implementation of sound fiscal policy toward illegal immigration could be a singular move toward improving access to college as well as defining a structured path from which to obtain legal status.

Moreover, sound policy would play a major role in rejuvenating the economy, as more students will be eligible to matriculate into college. With this matriculation, there will be an increase in the number of qualified workers who can commit themselves as contributing members of the marketplace. Thus, not only could these long-term policy effects provide a much-needed jolt to economic revitalization, it could also refresh the national landscape for undocumented high school graduates, their families and future generations of Americans.

Consequently, the positive impact of sound fiscal educational policy in favor of undocumented college bound students could resonate positively for years to come.

References Abrego L., & Gonzales, R. (2010), Blocked paths, uncertain futures: The postsecondary education and labor market prospects of undocumented Latino youth. The Post Journal of Education for Students Placed at Risk, pp. 144â&#x20AC;&#x201C;157.

Bardach, E. (2009). A practical guide for policy analysis: The eightfold path to more effective problem solving. Washington, DC: Sage Publications.

Bruno, A. (2011). Unauthorized Alien Students: Issues and DREAM Act Legislation March 22, 2011 Report, Congressional Research Office.

Camarota, S. (2004), the high cost of cheap labor: Illegal immigration and the federal government. Center for Immigration Studies, Washington, D.C. Educators for Fair Consideration (2011) Retrieved from www. e4fc.org Edwards, J. (2010). The medicaid costs of legalizing illegal aliens. Retrieved from http://cis.org/medicaid-costs Flores, S. (2010). State dream acts: The effect of in-state resident tuition policies and undocumented Latino students. The Review of Higher Education Frum, J. (2011). Post secondary educational access for undocumented students: Opportunities and constraints. American Academic (3).

Gonzales, R. G. & Orozco, M. (2009), Lives on hold: The college dreams of undocumented students. College Board Advocacy Report April 2009.

Harmon, C., Carne, G., Lizardy-Hajbi, G, & Wilkerson, E. (2010). Access to higher education for undocumented students: outlaws of social justice, equity, and equality. Journal of Praxis in Multicultural Education 5(1), 67-82.

Jewell, M. (2009), Undocumented-with College Dreams: What happens when an undocumented

student decides to go to college? Educational Leadership, 66 (7), 48-53. Retrieved from http://0-ehis.ebscohost.com.

North, D. (2009). The immigrant paradox: The stalled progress of recent immigrantsâ&#x20AC;&#x2122; children.

Retrieved from http://cis.org/ImmigrantParadox.

Noorani, A., & Belanger, M. (2009). The need for progressive immigration reform. Social Policy, Spring, pp. 13-14.

Olivas, M. (2009). Undocumented college students, taxation, and financial aid: A technical n note. The Review of Higher Education, (32) 3 Spring 2009, pp. 407-416.

Perry, A. (2006).Toward a theoretical framework for membership: The case of undocumented immigrants and financial aid for postsecondary education. The Review of Higher Education, (30) 1. pp. 21-40.

Pyler v. Doe No. 80-1538 (1982), Cornell University Law School. Legal Information Institution.

Retrieved from www.lawcornell.edu.

Ring, J. & Svensson, R. (2007). Social class and criminality among young people: A study considering the effects of school achievement as a mediating factor on the basis of Swedish register and self-report data. Journal of Scandinavian Studies in Criminology and Crime Prevention, 8, 210-233.

Shiu, A., Kettler, T., & Johnsen, S. (2009). Social effects of Hispanic students enrolled in an AP class in middle school. Journal of Advanced Academics, 21(1), 58-82.

Zota S. (2009). Unauthorized immigrants access to higher education: Fifty states, different directions. Spring/Summer Popular Government.

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Critical Pedagogy and English Language Acquisition Mary Ann Benites Associate Professor

Department of English for Academic Purposes Miami Dade College, Hialeah Campus E-mail: mbenite2@mdc.edu Abstract This paper aims to address the extent to how Critical theory should function in teaching and learning English. The English language is frequently utilized to communicate across borders, and it plays a significant factor in the current era of globalization. As such, immigrants, students and business professionals partake in learning English as a Second or Foreign Language. Yet, one must take into account the influences of Critical Pedagogy when teaching English to second and foreign language learners. The purpose of this paper is to comprehend the impact of critical theory on English language teaching and learning in EFL/ESL classrooms. The literature reveals insight as to the power of language as well as how to implement culturally sensitive pedagogy in the classroom and curriculum.

Keywords: Culture, Critical Pedagogy, Critical Awareness, English as a Foreign Language, English as a Second Language.

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Introduction The requirement for mastering English as a means for written and spoken communication and study has grown due to our interconnected global environment. This linguistic demand requires English as a Second or Foreign Language educators to comprehend not only a variety of methodologies for instruction but also apply a culturally sensitive framework in the classroom.

Therefore, teacher education is needed that supports critical reflection and pedagogy. Educators must reflect upon content which attempts to erase miscellany in the name of a democratic society or first world mind-set. Teaching strategies should be adapted to maintain a student's cultural identity as well as foster resistance to oppressive policies dictated within English language curricula. This requires the need to take into account the liaison between language and culture.

One should commence with a clear understanding of critical pedagogy and apply its principles in the classroom to avoid implementing ethnocentric dominance on English language learners. Critical pedagogy has its origin from the Frankfurt School, yet the North American School viewpoint can be traced back to the traditions of progressivism as exemplified in the writings of John Dewey (Biesta, 1998). Yet, it is through the work of Paulo Freire (1993), which infuses critical pedagogy into present-day education.

Haque (2007) pinpoints that Freireâ&#x20AC;&#x2122;s pedagogy follows a critical model by supplying a framework that provides an explanation of the world particularly in the manner that it sanctions and fosters inequalities and injustice. It also provides the tool for transformational change further along within analytical processes. This means advocating for this process through a critical lens within co-intentional education. Thus, education should be a place where teachers and students

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are both subjects, not only in the task of unveiling a reality real world experiences, but also in the task of reinforcing knowledge (Freire, 1993).

In essence, within Freire’s critical framework is an aspiration for emancipation and freedom from oppression. Thus, the teaching and learning environment must be dialogic, provide empowerment and incorporate the concept of voice (Haque, 2007). Crookes and Lehner (1998) specify that critical pedagogy should be carefully considered as goals in the teaching of ESL and EFL. As such, three goals must be considered: (1) Simultaneous development of English communicative abilities (2) Application of knowledge to develop a critical awareness of the world (3) Ability to act on knowledge and awareness to improve matters. In like manner, an understanding and respect of the student's culture without the execution of ethnocentric dominance is paramount to classroom success.

Therefore, to develop an awareness of unbiased English language instruction within the tenets of critical pedagogy, a discussion on critical language awareness, and the power of language must be addressed. The literature shall also support strategies to reduce cultural dominance when teaching English to second or foreign language learners.

Review of the Literature Critical Pedagogy Critical pedagogy makes it explicit of how education is constantly shaped by ideologies arising from “power, politics, history, and culture” (Huang, 2009). (Giroux, 1994) adds that “Critical pedagogy aims to illuminate the relationship among authority and power.” Within an educational framework, it questions the power relationships between teachers, students, institutions and society. Moreover, it pays noteworthy consideration to the relationship between

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knowledge and power. As such, it questions the role of institutional power within the process of knowledge creation. Huang (2009) indicates that critical pedagogy dictates the fact of how and why some realities are legitimated while others are silenced or made invisible, and whose knowledge is officially validated and which version of truth is actually left out. Wink (2005) sums up critical pedagogy as the concern which challenges unequal power relations in interactions between individuals and institutions. Furthermore, Wink (2005) draws attention to the cultural, political, social, and historical influences on schools and brings to light the issues of power and its relationship to classroom practices of teaching and learning. Wink continues that critical pedagogy is concerned with how methodology can be decisive, that is, how the method of delivery influences the process and content of knowledge construction.

Accordingly, educators whose practices are derived from critical philosophy deem that teaching and learning should also be connected to real life contexts that are associated with communities at large. These methods should also transform studentsâ&#x20AC;&#x2122; and teachersâ&#x20AC;&#x2122; lives to the degree that pupils are encouraged to perform as change agents in society (Huang, 2009).

Clearly, critical pedagogy can be distinguished from other educational philosophies in that its main concern lies with classroom practices with an emphasis to learning that extends beyond the classroom into the community at large. Educators may wonder just how critical theories translate into the English as a Second or Foreign Language classroom.

Norton and Toohey (2004) postulate that critical pedagogy correlates to English language learning with reference to social change in varying socio-economic levels of society.

Moreover, the authors claim that Critical pedagogy enables the view of language as a social practice that constructs the ways learners can better understand themselves when acquiring

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English. Thus, students become more aware their surroundings as well as their contribution to society at large if they can comprehend the culture representations and speak English.

Critical Language Awareness In order to comprehend the tenets of critical principles in English as second or foreign language learning, one must analyze the nature of the relationship between language and culture.

Shaul and Furbee (1998) state that language and culture are systematic to a large degree and are observable and describable. The authors indicate that systematic description of language is noted as linguistics while the description of cultures is called ethnography. Brown (2007) adds that both are tightly interwoven that the two cannot be separated without losing the significance of one another. Kuang (2007) cites that, "Language is the carrier of culture and culture is the content of language, and it may be hard to learn a language without knowing its culture" (pp.

75).

Restating principles derived by McLeod (1976), Kuang (2007) maintains that second and foreign language educators should teach their students about the cultural activities of the target language whether or not it is indicated in the curriculum. Moreover, Kuang (2007) adds that students will become empowered if they comprehend new cultural principles while learning the language. In essence, language teaching is actually culture teaching. More importantly, Cox and Assis-Peterson (1999) ascertain that a critical pedagogical curriculum requires English as Second or Foreign teachers to scrutinize the dominant discourse presented and consider if they are contributing colonial dominance in the teaching of language.

Cox and Assis-Peterson (1999) add that further reflection should be taken into account, by educators, with reference to the fabrication of discriminatory and unjust social structures which perpetuate the hegemonic power of English. As such, educators are directly responsible

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for guiding students in an attempt to use their new found knowledge of English for democratic purposes like social alteration and to help the less fortunate in society.

Fairclough (1992) also points out that English language education should serve as an avenue to resolve social inequalities that arise from power relationships. Fairclough (1992) proclaims that linguistics which contents itself with solely descriptive language practices and no attempt to relate the content to social power relations is missing an important point.

Furthermore, Fairclough (1992) also indicates the acquisition of mechanical skills of coding and decoding of linguistic structures without a critical element that discusses how the messages attain different meanings in social functions deceives the learner. This deception takes place in both the true nature of language as well as cultural practices. This dishonesty also deprives students of their full potential for effective citizenship in a new society.

The Power of Language Brandon, Baszile, & Berry (2009) postulate that in modern United States discourse, the division between “good” and “bad” language usage gained saliency with the connection between Standard American English and what is considered patriotism. At the turn of the 20th century, Crawford, (1992) cites that President Theodore Roosevelt’s restrictive language policy made this relationship very clear. Roosevelt stated, “We have room but for one language here, and that is the English language." Moreover, the Roosevelt claimed, "We as a nation intend to see that the crucible turns our people out as Americans, of American nationality and not dwellers in a polyglot boarding house” (Crawford, 1992).

It appears that Roosevelt's powerful messages generated an “either/or” paradigm which solidified understandings of who was and wasn't viewed as a patriotic American. As such, this nation became synonymous with the ability to speak appropriate Standard American English. In

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like manner, anthropologists through the Theory of Evolution also generated culturally fixed linguistic relationships between the civilized colonialists and their savage subjects (Brandon, Baszile, & Berry (2009). Both nuances created an “imperialist” versus “subjects” mentality in order to justify colonization practices in ESL/EFL curricula. Furthermore, Nieto (2004) argues that in U.S. classrooms Standard American English has become the instrument used to transmit and maintain Anglo-American culture and language on immigrant children. Indeed, these children are taught early on that if they want to succeed in American society and culture, they need to acquire the language of dominant discourse. As such, linguistic diversity in the U.S. has come to be viewed as a rather troublesome barrier, and many schools still attempt to rid immigrant students of this burden as quickly as possible (Nieto, 2004).

Moreover, Henry Trueba's (1993) Theory of Castification has been viewed as an aspect to consider in the teaching of English to non native speakers. Trubea (1993) states that castification of language minority families in the United States is caused by institutional oppression and by the dominant group. In the case of language minority students, castification oppresses ESL students by words utilized in public discourse. Trueba (1993) postulates that verbiage used in the English language promotes racism and a loss of dignity. Such common language includes words like resident aliens or illegal aliens which are commonly used in government documentation.

Cultural Considerations Citing from Peterson and Coltrane (2003), Thu (2010) points out that English and culture can be instructed without preconceptions. Moreover, Peterson & Coltrane (2003) and Thu (2010) cite that language and cultural considerations should be provided in a non biased manner that does not place judgment on the distinctions between the student's culture and the dominant one

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being taught. In like manner, it is of utmost importance to not only learn facts about a new language, but also values and behaviors that support the language in order to bridge cultural misunderstandings.

Cox and Assis-Peterson (1999) also indicate "Those who teach English must be critical of the dominant discourse that represents the internationalization of English as good and as a passport to the first world." This should be particularly considered in our era of globalization.

In like manner, the authors cite that, "Teachers must also consider the relationship of their work to the spread of the language as well as critically evaluate the implications of their practice in the production or reproduction of social inequalities." Thus, teachers should avoid maintaining that learning English will bring the student into a higher level of society. Furthermore, teachers must question whether they are contributing to the perpetuation of domination and find ways to avoid coming across in such a manner in their curricula (Cox & Assis-Peterson, 1999).

In like manner, Brandon, Baszile & Berry (2009) cite work by Cummins, (1996) and concur, educators must provide students space in the classroom to share their cultural expressions in ways which are both validated and celebrated. Thus, teachers must apply this knowledge as a premise for enabling students to succeed in society. Most importantly, the authors conclude that educators must be aware that English language acquisition is neither absolute nor linear, but it involves the creation of novel democratic discourse which empowers the student in the process of learning (Brandon, Baszile, Berry (2009) & Cummins (1999).

Embedded within the content of democratic discourse is critical consciousness which can be fostered through contextual patterns and cycles of texts. Barnawi (2010) analyzes contextual patterns and cycles as a means which includes multiple readings, discussing, analyzing, and open questioning of required tasks. This requires English language educators to promote critical

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thinking woven into teaching methods and learning activities. Barnawi (2010) asserts that the implementation of critical consciousness should include narratives, advocacy letters to policy makers and research based assignments. In essence, students are receiving critical transformative teaching as opposed to rote linguistic methodology which often imposes cultural dominance.

Younga, Sachdevb and Seedhouse (2009) concur that an approach to English language teaching and student learning which steers away from stereotyping will clearly be more successful, and the authors also promote an intercultural format to weave into curricula.

Fernstein (2008) also elucidates that allowing sociopolitical discourse empowers instructors and students to discuss language and cultural differences openly because it invites students to discuss any feelings of conflict. According to Fernstein (2008), by adapting a sociopolitical ideology educators can become open to comprehending the tensions which can arise when English as a Second Language (ESL) learners do not feel culturally included by institutions. Moreover, by maintaining openness in the classroom, ESL learners will not feel a sentiment of dominance or invisible privilege on behalf of their instructor.

Thu (2008) adds that beyond awareness of critical discourse in the socio political sense, educators should find the means to develop materials that integrate non bias cultural learning into language teaching. Textbooks and handouts indeed play a crucial role in integrating culture and the English language. As such, educators should take the time to critically assess the topics and provide guidelines in curricula.

Conclusion This paper has attempted to shed light on issues faced by English language educators and learners. Moreover, the literature supports the need to reevaluate ESL/EFL programs which have often been centered on colonial methodology. Indeed, educators are faced with methodological

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challenges when providing English instruction to international students. Thus, critical pedagogical considerations have been presented to help foster awareness. In like manner, assumptions behind pedagogical tasks presented in the literature have been discussed to provide teaching methods which are contextualized and socio-politically appropriate to the learner. By applying this critical lens, educators will become insightful to the fact there are alternatives to help ESL/EFL students participate in a global society without prompting feelings of submission.

With reference to my own practice as an English as a Second/Foreign language faculty member within higher education, I plan to implement critical pedagogy which is culturally sensitive and promotes social consciousness. Learning to teach English as a Second or Foreign Language within a critical framework can only help my students become successful and productive bi-lingual individuals within our global society.

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References Biesta, G. (1998) Say you want a revolution … suggestions for the impossible future of critical pedagogy. Educational Theory, 48(4), 499–510.

Brandon, L., Baszile, D. & T.Berry (2009) Linguistic moments: Language, teaching and teacher education in the U.S. Educational Foundations, Winter-Spring.

Brown, H. D. (2007). Principles of language learning and teaching. New York, NY: Pearson Cox, P.M & Assis-Peterson A, A. (1999) Critical approaches to TESOL. TESOL Quarterly, 33 (3) pp.433-452.

Crawford, J. (1992). Hold your tongue: Bilingualism and the politics of “English only.”

Reading, MA: Addision-Wesley.

Crookes, G. & Lehner, A. (1998) Aspects of process in an ESL critical pedagogy teacher education course, TESOL Quarterly, 32(2), 319–328.

Fairclough, N. (1992). Introduction. In N. Fairclough (Ed.). Critical language awareness (pp. 130). London: Longman.

Freire, P. (1993) Pedagogy of the oppressed. pp. 50-51. New York, NY: Continuum.

Haque, E. (2007). Critical pedagogy in English for academic purposes and the possibility for tactics of resistance. Pedagogy, Culture & Society. 15 (1) March pp. 83–106. DOI: 10.1080/14681360601162311.

Kuang, J. F. (2007). Developing student’s cultural awareness through foreign language teaching.

Sino US English Teaching, 4 (12), pp. 74-81.

McLeod, B. (1976). The relevance of anthropology to language teaching. TESOL Quarterly, 10.

(2), pp. 211-220 Nieto, S. (2004). Affirming diversity: The sociopolitical context of multicultural education, 4th ed. Boston, MA: Pearson Press.

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Polygon Spring 2010 Vol. 4, 81-82

COMMENTS ABOUT POLYGON

*********************************************************************** Dr. Norma Martin Goonen President, Hialeah Campus Miami Dade College Thank you, Dr. Shakil, for providing scholars a vehicle for sharing their research and scholarly work. Without opportunities for sharing, so many advances in professional endeavors may have been lost.

NMG Dr. Norma Martin Goonen President, Hialeah Campus Miami Dade College *********************************************************************** Dr. Ana María Bradley-Hess Academic and Student Dean, Hialeah Campus Miami Dade College Welcome to the third edition of Polygon, a multi disciplinary peer-reviewed journal of the Arts & Sciences! In support of the Miami Dade College Learning Outcomes, one of the core values of Hialeah Campus is to provide “learning experiences to facilitate the acquisition of fundamental knowledge.” Polygon aims to share the knowledge and attitudes of the complete “scholar" in hopes of better understanding the culturally complex world in which we live. Professors Shakil, Bestard and Calderin are to be commended for their leadership, hard work and collegiality in producing such a valuable resource for the MDC community.

Ana María Bradley-Hess, Ph.D. Academic and Student Dean Miami Dade College – Hialeah Campus 1800 West 49 Street, Hialeah, Florida 33012 Telephone: 305-237-8712 Fax: 305-237-8717

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Comments About â&#x20AC;Ś

*********************************************************************** Dr. Caridad Castro, Chairperson English & Communications, Humanities, Mathematics, Philosophy, Social & Natural Sciences Hialeah Campus Miami Dade College POLYGON continues to grow and to feature our local MDC scholars.

Thanks to you and your staff for providing them this opportunity.

Cary Caridad Castro, J.D., Chairperson English & Communications, Humanities, Mathematics, Philosophy, Social & Natural Sciences Miami Dade College â&#x20AC;&#x201C; Hialeah Campus 1776 W. 49 Street, Hialeah, FL 33012 Phone: 305-237-8804 Fax: 305-237-8820 E-mail: ccastro@mdc.edu *********************************************************************** Dr. Arturo Rodriguez Associate Professor Chemistry/Physics/Earth Sciences/Department North Campus Miami Dade College I want to congratulate you and the rest of the colleagues who created the POLYGON that is occupying an increasingly important place in the scholarly life of our College. Now, the faculties from MDC have a place to publish their modest contributions. arturo Dr. Arturo Rodriguez Associate Professor Chemistry/Physics/Earth Sciences/Department North Campus Miami Dade College 11380 NW 27th Avenue Miami, Florida 33167-3418 phone: 305 237 8095 fax: 305 237 1445 e-mail: arodri10@mdc.edu

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