polygon 2012
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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
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)
Prof. Victor Calderin (English)
Editor
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.
General articles and research manuscripts: Potential authors are invited to submit papers for the next issues of the POLYGON. All manuscripts must be submitted electronically (via e-mail) to one of the editors at mshakil@mdc.edu, or jbestard@mdc.edu, or vcalderi@mdc.edu. This system will permit the new editors to keep the submission and review process as efficient as possible.
Typing: Acceptable formats for electronic submission are MSWord, and PDF. All text, including title, headings, references, quotations, figure captions, and tables, must be typed, with 1 1/2 line spacing, and one-inch margins all around. Please employ a minimum font size of 11. Please see the attached template for the preparation of the manuscripts.
Length: A manuscript, including all references, tables, and figures, should not exceed 7,800 words (or at most 20 pages). Submissions grossly exceeding this limit may not be accepted for review. Authors should keep tables and figures to a minimum and include them at the end of the text.
Style: For writing and editorial style, authors must follow guidelines in the Publication Manual of the American Psychological Association (5th edition, 2001). The editors request that all text
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pages be numbered. You may also please refer to the attached template for the preparation of the manuscripts.
Abstract and keywords: All general and research manuscripts must include an abstract and afew keywords. Abstracts describing the essence of the manuscript must be 150 words or less. The keywords will be used by readers to search for your article after it is published.
Book reviews: POLYGON accepts unsolicited reviews of current scholarly books on topics related to research, policy, or practice in higher education, Innovations at Work, and discipline related knowledge for the campus, college and service community to improve and increase information dissemination. Book reviews may be submitted to either themed or open-topic issues of the journal. Book review essays should not exceed 1,900 words. Please include, at the beginning of the text, city, state, publisher, and the year of the book’s publication. An abstract of 150 words or less and keywords are required for book review essays.
Notice to Authors of Joint Works (articles with more than one author). This journal uses a transfer of copyright agreement that requires just one author (the Corresponding Author) to sign on behalf of all authors. Please identify the Corresponding Author for your work when submitting your manuscript for review. The Corresponding Author will be responsible for the following:
• ensuring that all authors are identified on the copyright agreement, and notifying the editorial office of any changes to the authorship.
• securing written permission (via email) from each co-author to sign the copyright agreement on the co-author’s behalf.
• warranting and indemnifying the journal owner and publisher on behalf of all co-authors.
Although such instances are very rare, you should be aware that in the event a co-author has included content in their portion of the article that infringes the copyright of another or is otherwise in violation of any other warranty listed in the agreement, you will be the sole author indemnifying the publisher and the editor of the journal against such violation.
Please contact the editorial office if you have any questions or if you prefer to use a copyright agreement for all coauthors to sign.
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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] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1970.
[2] J. Galambos and I. Simonelli, Products of Random Variables – Applications to Problems of Physics and to Arithmetical Functions, CRC Press, Boca Raton / Atlanta, 2005.
[3] S. Momani, Non-perturbative analytical solutions of the space- and time-fractional Burgers equations. Chaos, Solitons & Fractals, 28(4) (2006), 930-937.
[4] Z. Odibat, S. Momani, 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)
.
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Polygon Spring 2012 1
Conceptual and procedural knowledge
in Mathematics education-
in 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 and
means then 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 are horizontal type formulas.
b) and 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- to-
base 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 , therefore, the original
problem becomes a typical case of , which is . 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 , then
through the law of to obtain the final answer. Finally, after the use of
creativity, the conceptual knowledge can grow almost infinitely.
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 expert-
like 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 but correctly
perceive the increase series 1,10,100,1000, …….. for . 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?
Polygon Spring 2012 25
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
Polygon Spring 2012 26
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
Polygon Spring 2012 27
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).
Polygon Spring 2012 28
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).
Polygon Spring 2012 29
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
Polygon Spring 2012 30
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
Polygon Spring 2012 31
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
Polygon Spring 2012 32
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.
Polygon Spring 2012 33
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.
Polygon Spring 2012 34
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.
Polygon Spring 2012 35
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: 513-
530.
Tsui, L.(2002). Fostering critical thinking through effective pedagogy: evidence from four
institutional case studies. Journal of Higher Education 73(6): 740-763.
Polygon Spring 2012
36
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.
Polygon Spring 2012
37
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 higher-
order 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.
Polygon Spring 2012
43
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.
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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.
Table 1: 23 Factorial Design Table
A/C
Level
Traffic
Pattern
Tire Pressure
(psi) Miles per Gallon
A B C Replicate I Replicate II Replicate III
Low Light 28 18.11 21.30 20.81
Low Light 32 29.93 29.23 29.24
Low Heavy 28 17.92 18.36 20.56
Low Heavy 32 23.98 25.00 24.51
High Light 28 19.43 20.05 20.34
High Light 32 27.94 26.90 27.71
High Heavy 28 13.92 14.26 12.96
High Heavy 32 16.94 20.18 18.70
Table 2: Factor Code table
Factor (i) xi = -1 xi = +1
A/C Level (A) Low High
Traffic Pattern (B) Light Heavy
Tire Pressure (C) 28 psi 32 psi
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 21.595
βA -1.651
βB -2.653
βC 3.426
βAC -1.130
βBC -0.817
The regression model has an R2 value of 97.61% and Radj
2 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 Effect Coef SE Coef T P
Constant 21.595 0.2039 105.90 0.000
Block 1 -0.574 0.2884 -1.99 0.066
Block 2 0.314 0.2884 1.09 0.294
A -3.302 -1.651 0.2039 -8.10 0.000
B -5.306 -2.653 0.2039 -13.01 0.000
C 6.852 3.426 0.2039 16.80 0.000
A*B -2.260 -1.130 0.2039 -5.54 0.000
A*C -0.618 -0.309 0.2039 -1.52 0.152
B*C -1.633 -0.817 0.2039 -4.00 0.001
A*B*C 0.289 0.145 0.2039 0.71 0.490
S = 0.999004 PRESS = 41.0610
R-Sq = 97.61% R-Sq(pred) = 92.96% R-Sq(adj) = 96.07%
Analysis of Variance for MPG (coded units)
Source DF Seq SS Adj SS Adj MS F P
Blocks 2 3.964 3.964 1.982 1.99 0.174
Main Effects 3 516.098 516.098 172.033 172.38 0.000
A 1 65.427 65.427 65.427 65.56 0.000
B 1 168.942 168.942 168.942 169.28 0.000
C 1 281.729 281.729 281.729 282.29 0.000
2-Way Interactions 3 48.943 48.943 16.314 16.35 0.000
A*B 1 30.640 30.640 30.640 30.70 0.000
A*C 1 2.295 2.295 2.295 2.30 0.152
B*C 1 16.008 16.008 16.008 16.04 0.001
3-Way Interactions 1 0.502 0.502 0.502 0.50 0.490
A*B*C 1 0.502 0.502 0.502 0.50 0.490
Residual Error 14 13.972 13.972 0.998
Total 23 583.480
Obs StdOrder MPG Fit SE Fit Residual St Resid
1 1 18.1070 19.4987 0.6449 -1.3917 -1.82
2 2 19.4260 19.3642 0.6449 0.0619 0.08
3 3 17.9226 18.3750 0.6449 -0.4524 -0.59
4 4 13.9241 13.1421 0.6449 0.7820 1.02
5 5 29.9320 28.8923 0.6449 1.0397 1.36
6 6 27.9365 26.9421 0.6449 0.9944 1.30
7 7 23.9782 23.9230 0.6449 0.0552 0.07
8 8 16.9429 18.0320 0.6449 -1.0890 -1.43
9 9 21.2963 20.3870 0.6449 0.9093 1.19
10 10 20.0456 20.2525 0.6449 -0.2069 -0.27
11 11 18.3633 19.2633 0.6449 -0.9000 -1.18
12 12 14.2636 14.0303 0.6449 0.2332 0.31
13 13 29.2308 29.7806 0.6449 -0.5498 -0.72
14 14 26.9006 27.8304 0.6449 -0.9298 -1.22
15 15 25.0000 24.8113 0.6449 0.1887 0.25
16 16 20.1754 18.9202 0.6449 1.2552 1.65
17 17 20.8145 20.3321 0.6449 0.4824 0.63
18 18 20.3426 20.1976 0.6449 0.1450 0.19
19 19 20.5607 19.2084 0.6449 1.3524 1.77
20 20 12.9602 13.9755 0.6449 -1.0152 -1.33
21 21 29.2359 29.7257 0.6449 -0.4898 -0.64
22 22 27.7108 27.7755 0.6449 -0.0646 -0.08
23 23 24.5125 24.7564 0.6449 -0.2439 -0.32
24 24 18.6992 18.8654 0.6449 -0.1662 -0.22
Predicted Response for New Design Points Using Model for MPG
Point Fit SE Fit 95% CI 95% PI
1 19.4987 0.6449 (18.1156, 20.8818) (16.9484, 22.0490)
2 19.3642 0.6449 (17.9811, 20.7473) (16.8139, 21.9144)
3 18.3750 0.6449 (16.9919, 19.7581) (15.8247, 20.9252)
4 13.1421 0.6449 (11.7590, 14.5251) (10.5918, 15.6923)
5 28.8923 0.6449 (27.5092, 30.2754) (26.3420, 31.4426)
6 26.9421 0.6449 (25.5590, 28.3252) (24.3918, 29.4924)
7 23.9230 0.6449 (22.5399, 25.3061) (21.3728, 26.4733)
8 18.0320 0.6449 (16.6489, 19.4150) (15.4817, 20.5822)
9 20.3870 0.6449 (19.0039, 21.7700) (17.8367, 22.9372)
10 20.2525 0.6449 (18.8694, 21.6355) (17.7022, 22.8027)
11 19.2633 0.6449 (17.8802, 20.6463) (16.7130, 21.8135)
12 14.0303 0.6449 (12.6473, 15.4134) (11.4801, 16.5806)
13 29.7806 0.6449 (28.3975, 31.1637) (27.2303, 32.3309)
14 27.8304 0.6449 (26.4473, 29.2134) (25.2801, 30.3806)
15 24.8113 0.6449 (23.4282, 26.1944) (22.2610, 27.3616)
16 18.9202 0.6449 (17.5372, 20.3033) (16.3700, 21.4705)
17 20.3321 0.6449 (18.9490, 21.7152) (17.7818, 22.8824)
18 20.1976 0.6449 (18.8145, 21.5807) (17.6473, 22.7478)
19 19.2084 0.6449 (17.8253, 20.5915) (16.6581, 21.7587)
20 13.9755 0.6449 (12.5924, 15.3585) (11.4252, 16.5257)
21 29.7257 0.6449 (28.3426, 31.1088) (27.1755, 32.2760)
22 27.7755 0.6449 (26.3924, 29.1586) (25.2252, 30.3258)
23 24.7564 0.6449 (23.3733, 26.1395) (22.2062, 27.3067)
24 18.8654 0.6449 (17.4823, 20.2484) (16.3151, 21.4156)
B. MINITAB GRAPHS
Figure 1: Residual Plot (Fitted Value vs. Standardized Residual)
Figure 3: Normal Probability Plot (Standardized Residual vs. Percent)
Figure 2: Residual Plot (Standardized Residual vs. Observation Order)
Figure 4: Normal Probability Plot (Response Variable vs. Percent)
30252015
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-2
Fitted Value
Sta
nd
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Versus Fits(response is MPG)
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Standardized Residual
Pe
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Normal Probability Plot(response is MPG)
24222018161412108642
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Observation Order
Sta
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353025201510
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60
50
40
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MPG
Pe
rce
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Mean 21.60
StDev 5.037
N 24
KS 0.149
P-Value >0.150
Probability Plot of MPGNormal
MINITAB GRAPHS
Figure 5: Normal Probability Plot (Standardized Effect vs. Percent)
20151050-5-10
99
95
90
80
70
60
50
40
30
20
10
5
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Standardized Effect
Pe
rce
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A A
B B
C C
Factor Name
Not Significant
Significant
Effect Type
BC
AB
C
B
A
Normal Plot of the Standardized Effects(response is MPG, Alpha = 0.05)
MINITAB GRAPHS
Figure 6: Contour Plot of A/C Level vs. Traffic Pattern (Tire Pressure at 28 psi)
Figure 8: Contour Plot of A/C Level vs. Tire Pressure (Traffic Pattern is Light)
Figure 7: Contour Plot of A/C Level vs. Traffic Pattern (Tire Pressure at 32 psi)
Figure 9: Contour Plot of A/C Level vs. Tire Pressure (Traffic Pattern is Heavy)
A
B
1.00.50.0-0.5-1.0
1.0
0.5
0.0
-0.5
-1.0
C -1
Hold Values
>
–
–
–
–
–
–
–
< 14
14 16
16 18
18 20
20 22
22 24
24 26
26 28
28
MPG
Contour Plot of MPG vs B, A
A
C
1.00.50.0-0.5-1.0
1.0
0.5
0.0
-0.5
-1.0
B -1
Hold Values
>
–
–
–
–
–
–
–
< 14
14 16
16 18
18 20
20 22
22 24
24 26
26 28
28
MPG
Contour Plot of MPG vs C, A
A
B
1.00.50.0-0.5-1.0
1.0
0.5
0.0
-0.5
-1.0
C 1
Hold Values
>
–
–
–
–
–
–
–
< 14
14 16
16 18
18 20
20 22
22 24
24 26
26 28
28
MPG
Contour Plot of MPG vs B, A
A
C
1.00.50.0-0.5-1.0
1.0
0.5
0.0
-0.5
-1.0
B 1
Hold Values
>
–
–
–
–
–
–
–
< 14
14 16
16 18
18 20
20 22
22 24
24 26
26 28
28
MPG
Contour Plot of MPG vs C, A
MINITAB GRAPHS
Figure 10: Contour Plot of Traffic Pattern vs. Tire Pressure (A/C Level at Low)
Figure 12: Response Surface Plot A/C Level vs. Traffic Pattern (Tire Pressure at 28 psi)
Figure 11: Contour Plot of Traffic Pattern vs. Tire Pressure (A/C Level at High)
Figure 13: Response Surface Plot A/C Level vs. Traffic Pattern (Tire Pressure at 32 psi)
B
C
1.00.50.0-0.5-1.0
1.0
0.5
0.0
-0.5
-1.0
A -1
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>
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–
–
–
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< 14
14 16
16 18
18 20
20 22
22 24
24 26
26 28
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Contour Plot of MPG vs C, B
1
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016
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18
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B
A
C -1
Hold Values
Surface Plot of MPG vs B, A
B
C
1.00.50.0-0.5-1.0
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0.5
0.0
-0.5
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A 1
Hold Values
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< 14
14 16
16 18
18 20
20 22
22 24
24 26
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Contour Plot of MPG vs C, B
1
200
-1
25
0
30
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1
MPG
B
A
C 1
Hold Values
Surface Plot of MPG vs B, A
MINITAB GRAPHS
Figure 14: Response Surface Plot A/C Level vs. Tire Pressure (Traffic Pattern is Light)
Figure 16: Response Surface Plot Traffic Pattern vs. Tire Pressure (A/C Level at Low)
Figure 15: Response Surface Plot A/C Level vs. Tire Pressure (Traffic Pattern is Heavy)
Figure 17: Response Surface Plot Traffic Pattern vs. Tire Pressure (A/C Level at High)
1
200
25
-1
30
0 -11
MPG
C
A
B -1
Hold Values
Surface Plot of MPG vs C, A
1
20
0
25
-1
30
0 -11
MPG
C
B
A -1
Hold Values
Surface Plot of MPG vs C, B
1
15
0
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0 -11
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C
A
B 1
Hold Values
Surface Plot of MPG vs C, A
1
15
0
20
25
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0 -11
MPG
C
B
A 1
Hold Values
Surface Plot of MPG vs C, B
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 person’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 nation’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 “unless,” 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 nation’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 “unauthorized youth” 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 in-
state 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 college’s headcount. However, in 2007, voters approved SB529, the Georgia Security and
Immigration Compliance Act, and by 2008, undocumented students were unable to establish in-
state 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 government’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 bachelor’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
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education and labor market prospects of undocumented Latino youth. The Post Journal of
Education for Students Placed at Risk, pp. 144–157.
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 immigrants’ children.
Retrieved from http://cis.org/ImmigrantParadox.
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Policy, Spring, pp. 13-14.
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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
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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
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Polygon Spring 2012 74
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.
Polygon Spring 2012 75
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 Freire’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
Polygon Spring 2012 76
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
Polygon Spring 2012 77
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 students’ and teachers’ 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
Polygon Spring 2012 78
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
Polygon Spring 2012 79
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
Polygon Spring 2012 80
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
Polygon Spring 2012 81
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
Polygon Spring 2012 82
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
Polygon Spring 2012 83
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.
Polygon Spring 2012 84
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. 1- 30). 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.
Norton, B., & Toohey, K. (2004). Critical pedagogies and language learning: An introduction. In B. Norton & K. Toohey. Critical pedagogies and language learning (pp. 1-17). UK: Cambridge University Press.
Shaul, D. L., & Furbee, N. L. (1998). Language and culture. Prospect Heights, IL: Waveland.
Trueba, H. (1993). Healing multicultural America: Mexican immigrants rise to power in rural California. London, UK. Falmer Press.
Wink, J. (2005). Critical pedagogy: Notes from the real world. (3rd ed.). New York: Pearson.
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Polygon
Spring 2010 Vol. 4, 81-82
COMMENTS ABOUT POLYGON
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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.
N
N
M
M
G
G
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
Comments About …
82
***********************************************************************
Dr. Caridad Castro, Chairperson
English & Communications, Humanities, Mathematics, Philosophy,
Social & Natural Sciences
Hialeah Campus
Miami Dade College
POLYGON continues to grow and to feature our local MDC scholars.
Thanks to you and your staff for providing them this opportunity.
Cary
Caridad Castro, J.D., Chairperson
English & Communications, Humanities, Mathematics, Philosophy,
Social & Natural Sciences
Miami Dade College – Hialeah Campus
1776 W. 49 Street, Hialeah, FL 33012
Phone: 305-237-8804
Fax: 305-237-8820
E-mail: ccastro@mdc.edu
***********************************************************************
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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