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Abstract

Teaching mathematics to prepare effectively students toward the achievement of their potential requires an understanding of what helps them learn and the adoption of innovative tools that exploit the existing technology in the classroom. The need to account for students’ abilities and weaknesses in designing a lecture and inspire them is at the basis of a proper teaching strategy aimed at acquiring high competency in mathematics tailored to their needs and at recruiting students to scientific disciplines and in engineering. The adoption of mathematical software integrated with the lectures and the use of interactive teaching tools are important modifications needed to reach this objective.
Hints on how to improve
mathematics instruction
Journal of e-Learning and Knowledge Society - EN
Vol. 7, n. 1, January 2011 (pp. 7 - 20)
ISSN: 1826-6223 | eISSN: 1971-8829
Flavia Colonna1, Glenn Easley2
1 Department of Mathematical Sciences, George Mason
University (Fairfax, Virginia, U.S.A.) - fcolonna@gmu.edu
2 System Planning Corporation (Arlington, Virginia, U.S.A.) -
geasley@sysplan.com
Invited Papers
Abstract
Teaching mathematics to prepare effectively students toward the
achievement of their potential requires an understanding of what helps them
learn and the adoption of innovative tools that exploit the existing technology
in the classroom. The need to account for students’ abilities and weaknesses
in designing a lecture and inspire them is at the basis of a proper teaching
strategy aimed at acquiring high competency in mathematics tailored to their
needs and at recruiting students to scientic disciplines and in engineering.
The adoption of mathematical software integrated with the lectures and
the use of interactive teaching tools are important modications needed to
reach this objective.
|
for citations:
Colonna F., Easley G. (2011), Hints on how to improve mathematics instruction. In Journal of
e-Learning and Knowledge Society, v. 7, n.1, 2011, English Edition, pp. 7-20. ISSN: 1826-6223,
e-ISSN:1971-8829
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Invited Papers - Vol. 7, n. 1, January 2011
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1 Introduction
Mathematics is an academic subject whose knowledge to some degree
is required in most disciplines, and in particular, in the natural sciences and
engineering. Unlike many scientic disciplines that have evolved over time,
particularly as a consequence of scientic discoveries and technological innova-
tions, the type of mathematics taught at the primary, secondary, and university
education levels has not changed signicantly in the last century. Aside from
an increased early exposure to probability and statistics, or a taste of set theory
even in elementary school, subjects such as algebra, geometry, or calculus, have
remained virtually unchanged from one generation to the next. Yet, in recent
times, the teaching of mathematics has been greatly analyzed and debated. How
do we prepare students? How do we create the conditions to attract them to
this subject? What can be done to close the gap in mathematics achievement
between men and women? How can we help our students achieve the best
performance possible?
Virtually everyone agrees that mathematics is important for research and
development, for technological advances, and for preparing our students and
help them become competitive men and women in the workplace. Economic
growth depends on the creation of new technologies. After all, mathematics is
classically a problem-solving discipline. Yet, in the United States it is common
to admit ignorance in mathematics almost with a sense of resignation or pride.
The popular culture is decidedly at odds with the vision of a mathematical whiz
as ‘cool’. Thus, it is necessary to change this popular culture; but how? The
starting point is to thoroughly change the teaching methods that only work for
a limited number of students. Of course, this change of mentality has to begin
in elementary school and continue throughout secondary school. Indeed, by the
time a student reaches the university, it is often too late to change bad habits
and abolish preconceived notions about the subject.
In recent years the division of undergraduate education of the National
Science Foundation (NSF) has been actively promoting research and awarding
grants for course curriculum development or modication in science, technolo-
gy, engineering and mathematics through a program called STEM. For details
on this and related programs, see <http://www.nsf.gov/funding/>. The NSF
publishes and regularly updates statistics on education, research, enrollment,
and graduation rates for undergraduates and graduate students, also categorized
by gender and minority status. For details, see <http://www.nsf.gov/statistics/>.
These statistics document the increase in doctoral degrees awarded in the U.S.
to foreign students, often out-ranking the U.S. citizens. This has raised signi-
cant concern, as many foreign graduates will return to their countries of origin
and enrich them culturally and economically. Through powerful investment in
F. Colonna, G. Easley - Hints on how to improve mathematics instruction
9
education, it is hoped to reverse this trend. In other countries, such as Italy, there
is the opposite problem: the need to stop the ‘brain drain’ through legislation
and nancial incentives.
In this paper, we share some ideas on how to make changes in the curricu-
lum to take steps toward the creation of a better learning environment. The main
focus here will be on teaching mathematics at the university level, although
most of what we indicate is applicable to most grades of secondary school.
While these ideas are not novel, for the most part they remain conned to cer-
tain schools of thought. We hope this work will have the effect of informing
educators about ongoing methodologies as well as establishing connections for
discussion and exchange of materials on teaching mathematics.
2 Different models in higher education
There are two substantially different models for university systems throu-
ghout the world. One, such as the U.S. undergraduate education system, is
the so called liberal arts model, which is based on teaching a wide variety of
subjects with a specic set of requirements, known as general education re-
quirements. These requirements have to be fullled by all students, regardless
of the eld in which they wish to specialize. Depending of the type of degree
sought, B.A. (Bachelors of Arts) or B.S. (Bachelors of Science), for a student
who seeks a degree in a specic discipline, the percent of courses related to
that discipline could vary roughly between 40 and 70 (i.e. see requirements in
the (George Mason University, 2010-2011)). The aim is the development of
a cultural breadth. The other model, such as the Italian university system, is
based on studying almost entirely courses related to the discipline of choice.
In the latter model, the aim could be seen as an over-specialized professional
education and focus. For this reason, the issues regarding teaching a subject
such as mathematics in the two models vary signicantly. For more information
on the liberal arts model, we recommend to the reader the American Association
of Colleges and Universities website: <http://www.aacu.org/leap/>.
2.1 The liberal arts model
The principle behind the liberal arts model is that cultural breadth provides
a student with a more complete education at an age when he/she has already
acquired a higher level of maturity. The in-depth specialization in one discipline
takes place after graduation, if so desired, in the pursuit of a Masters or a docto-
ral degree. It is the belief of those who favor this form of education, as opposed
to the specialized degree model, that students who graduate from a liberal-arts
institution are more t and prepared to enter the white-collar workforce.
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The rst issue that arises is: What constitutes a reasonable set of general
education requirements in mathematics? And, most importantly: What skills
should a student develop through the relative courses? In the U.S. model, a
desirable skill is competency in analytical reasoning. Thus, a prospective gene-
ral education course in mathematics has to teach how to think mathematically
using methodologies based on logic. A course of this type does not requi-
re an extensive background in mathematics, does not involve memorization
of formulas, and is not based on learning mathematical theories. A course in
analytical reasoning, in fact, has a modest amount of mathematics. The aim
is to teach how to think by starting with a concrete problem, interpreting it
symbolically, and using tools such as the principle of mathematical induction
or logical reasoning to derive a solution through a series of steps starting from
known facts. The ability of an instructor is to disassemble the problem into
parts in order to obtain as much information as possible, embed in the process
hints of what strategies pay off and discuss the options with the students, see-
king their opinions and raising their interest, while at the same time guarding
against logical ‘traps’ or circular arguments. An effective teacher should also
take the opportunity to explain why certain strategies do not pay off and offer
alternative routes.
The development of analytical reasoning skills is enforced throughout the
course by evaluating the students by means of exams and quizzes that contain
problems they have not seen before. However, given the limited amount of
time students have to work on test and quiz problems, most of the higher-level
learning by a student takes place in the form of demanding take-home assign-
ments for which students are given at least a week to complete.
Mathematics classes for non-mathematics and science majors are typically
given in medium size classrooms to facilitate communication between student
and teacher in the classroom. Since the objective is to teach how to solve
problems through analytical reasoning, it is useful to supplement the standard
course materials with applications from the real world. In fact, it is often the
case that even in publications such as newspapers and magazines, statisti-
cal information is given erroneously or the article writers are not sufciently
knowledgeable in the topics they cover and end up making mistakes in their
reporting. It is particularly important for a student to realize that, just because
an article appeared in a reputable publication, one should not assume that its
content is accurate without a proper analysis. When reporting, for instance, on
a study done to determine the effectiveness of a drug or to establish a cause-
and-effect link, such as between a high-fat diet and heart disease, one should
not deem as trustworthy a study done on a small sample or choosing a non-
random sample. Furthermore, there is so much uncontrolled advertising we are
daily exposed to, claiming the benets of a particular product on one’s health
F. Colonna, G. Easley - Hints on how to improve mathematics instruction
11
or tness or purporting the superiority of a certain commodity. How do we
discern facts from unproven claims? Students should therefore learn what to
look for in their analysis of advertisements and reports, even those considered
scientically sound.
2.2 The specialized professional model
A model based on studying courses related to the eld of interest should in
theory have a more integrated picture of the role of any required discipline for
learning that eld. However, in practice there is often a disconnect between the
teaching of the discipline and how it could be used effectively in the specialized
eld. The reasons for this dissociation are often due to poor communication
between different departments and the belief that the subject at hand is needed
in its ‘pure form’, without much thought on applications to make it more rele-
vant to the students. In addition, instructors of the so called ‘service courses’
(such as many mathematics courses) often delegate the coverage of applications
to the specialized eld. However, the context in which an application is shown
is different and, by the time the application is shown in the specialized eld,
the mathematical aspect are often long forgotten or no longer fully understood
by the student. This delay has detrimental effects on retaining and seeing in
action the mathematical techniques learned in the course.
It is our experience that even for students of disciplines that have a heavy
mathematical content, such as engineering, there is a need for a more integrated
course structure with plenty of examples from engineering where a particular
mathematical tool is needed. An effective teacher shows how to deal with a
particular application before and after learning a certain technique. Only then
can a student fully appreciate the power of the method.
Another important component on learning in the pursuit of a specialized
degree is to assign projects with a creative component by offering precise
guidelines, but also a certain freedom to reach a certain practical objective. In
fact, these assignments are often the best opportunities for meaningful lear-
ning to take place. Better yet is to assign open-handed problems, in which a
student is asked to accomplish a certain task, possibly too difcult to be done
independently, without much direction or initial guidance. However, to offset
this push for autonomy it is important for the instructor to remain available for
questions, clarications and suggestions. In Section 6, we shall discuss more in
detail the issues regarding the teaching of mathematics courses with an applied
component to be implemented at the computer that we have taught.
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3 Teaching philosophy
What constitutes the perfect teaching style? There cannot be denitive an-
swers to this question since students have different style preferences based
on their individual strengths. However, most of us would agree on certain
parameters in evaluating teaching effectiveness. The lecture should be clear,
stimulating, whenever possible even thought provoking, and rich in interac-
tion between the teacher and the audience. The amount of detail presented in
class should not be excessive and students should be asked to try lling in the
holes of proofs that follow arguments students have already been exposed to
or complete calculations on their own. The best way to learn a new subject is
to make connections with other related topics already known, emphasize the
key ideas of the new theory or technique and at the end of class summarize the
main points raised in the course of the lecture. The repetition of some aspects
covered at length previously helps a student sort out the important information
from the rest. Not all students are capable to be uent at synthesizing crucial
elements of a concept or method. Ideally, each student should be able to write
a short paragraph on what was learned that day.
A very important quality every teacher should possess is to inspire. A stu-
dent cannot become excited about learning by being a passive spectator in a
classroom governed by an instructor embedded in silence. For a subject such as
mathematics, this is a particularly challenging issue. Aside from the logistics on
how to create an environment amenable to a two-way communication between
students and teacher, there is the issue of class size, the need for moving for-
ward with the curriculum, and to involve in the discussion those students who
prefer to stay by the sidelines. Thus, while this interactive learning environment
is not practical in a class with over 35 students, there are other strategies that
could be implemented as a way to increase the communication at least outside
the classroom.
In the mid 1990s one of us participated in a year-long series of workshops
on the pedagogy of math and science. In the course of the workshop, there was
a debate on how to best increase students’ participation, test their mastery of the
subject and give them the tools to learn at a deeper level. A comparison among
different lecture styles for different cultures was shown. Videos showing a
typical class were analyzed. The videos from a non-English speaking class had
subtitles to help the viewer understand the dialogs. The workshop’s participants
then discussed the positives and the negatives of the different styles.
One of the problems in a class of geometry in a high school in the United
States was the excessive repetition of facts that did not go beyond learning the
name of a geometric entity. There was no discussion on the purpose of the con-
F. Colonna, G. Easley - Hints on how to improve mathematics instruction
13
cept being taught, or how this concept could be implemented. While the class
took place in a lively classroom with an informal and relaxed atmosphere, the
students were distracted, bored, and did not pay attention to the teacher.
By contrast, in a Japanese classroom, in which there was a much more
formal atmosphere, the teacher started the class with a practical problem on
how to create a fence dividing the lands of two neighbors according to certain
criteria. He sought the students’ opinions and the discussion lasted for a long
time. Everyone participated in the discussion and only at the end the teacher
developed the concept in full. By then the students had already learned the key
ideas and this made the lecture very easy to understand. At the end the students
were divided into groups to explore new problems related to the topic develo-
ped. There was nothing repetitive about the approach adopted by this teacher
and the students seemed genuinely interested.
This workshop made an impression on the participants who then wonde-
red why no university faculty is required to be trained in teaching methods.
Instructors are certainly affected by their teaching experiences in the course
of their academic careers, yet, often they tend to emulate their own professors
without giving much thought on whether there are better teaching alternatives,
thus perpetuating a model that has the potential to be greatly improved. This is
precisely why an exchange program that allows faculty to experience teaching
in different settings can bring the breath of fresh air needed to make classroom
improvements.
4 The gender gap
Statistics in the United States indicate that the number of women who cho-
ose to specialize in elds such as mathematics, computer science and engi-
neering is quite small and this gender gap persists in the their career choices
after graduation. The motivation for this gap has been attributed to women’s
lower achievement in mathematics and science in secondary school and at the
university than that of men. This realization has been a key factor in trying to
promote more awareness of the necessity to increase interest in mathematics
among girls in the early grades, since girls’ negative attitude toward mathe-
matics has the effect of turning them away from many scientic disciplines.
Initiatives at giving priority to women for admission at selective colleges and
universities in disciplines in which they are underrepresented have been hotly
debated. Aside from the deserving male applicant who complains of having
been unfairly turned down to make room for a less qualied female applicant,
there is a signicant concern that the quality of education may suffer as a con-
sequence of this policy.
In (Else-Quest et al., 2010), Else-Quest, Hyde and Linn analyzed cross-na-
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tional patterns of gender differences in mathematics. This work was motivated
by the continuing concern on the impact that gender differences in mathematics
achievement and attitudes has on the underrepresentation of women at the hi-
ghest levels of science, technology, mathematics, and engineering, and by the
persisting stereotypes according to which girls and women lack mathematical
ability, despite growing evidence of similarities in math achievement between
genders, as documented in several recent studies such as (Hapern et al., 2007).
In their study, in which two large international data sets were analyzed, in
(Else-Quest et al., 2010) the authors showed that a gender gap in mathematics
achievement persists in some nations but not in others, and that on the avera-
ge, males and females differ very little in math achievement, despite the more
positive math attitudes among males. The differences are more pronounced
at the top levels of performance. However, their ndings were not consistent
with the theory supported by several researchers (e.g. (Baker & Jones, 1993))
suggesting that cross-national patterns of gender differences in mathematics
achievement are related to gender inequities in educational and economic op-
portunities present in a culture. They argue that factors that have a more direct
effect on learning, such as the curriculum and the quality of instruction, may
be more useful to lessen the gender inequity on mathematics achievement.
Another theory on a factor inuencing the gender gap is the stereotype threat
experienced by members of a gender, race, class or status when they believe
that they might be treated negatively simply because of their social identity.
According to this hypothesis, girls perform less well in mathematics because
they are stigmatized as a result of their gender identication. Consequently,
women who are stigmatized in elds with a heavy mathematical component
will tend to avoid them. For more information on this theory, see (Murphy et
al., 2007) and (Rydell et al., 2010).
One of us has been involved for several years in the outreach program
‘Blueprints to the future’ organized in the 1990s by the American Associa-
tion of University Women. This program recruited participants among several
middle school students in Fairfax County (Virginia, U.S.A.) to participate to
a math awareness day in which many well-known speakers in different scien-
tic elds were invited to talk about the nature of their work and the students
were invited to take part in hands-on activities. The emphasis of this program
was on sending out the message on the importance of mathematics in all elds
presented in the program.
Other initiatives aimed at reducing the gender gap are individual mentoring
of girls and the active recruitment of female students through advertisements
and public speeches by academics and researchers from the private sector
through school visits.
An interesting realization conrming the inuences of the environment
F. Colonna, G. Easley - Hints on how to improve mathematics instruction
15
on math achievement can be seen by the effect of video game playing. Girls
can improve performance in mathematics and science by playing specialized
video games designed to improve visual processing. Indeed, there is ample
literature supporting the hypothesis that playing video games helps the player
strengthen the hand-eye coordination as well as visual and spatial abilities. In
2009, U.S. president Barack Obama launched a campaign called Educate to
Innovate whose scope is to improve the mathematical, scientic, technologi-
cal, and engineering abilities of American students by harnessing the power
of interactive games.
5 Different learning styles
An effective teacher must have the ability to gear his/her lectures to a di-
verse audience. Thus, the style that is best suited to have a strong impact on
learning is one that relies on multiple techniques: visual, for those students
who have difculty with verbal instruction, auditory or through the adoption
of manipulatives, for those who do not interpret geometric features properly
or have less developed spatial abilities, and illustrative examples for those
students who have poor abstract reasoning skills. In addition, instructors must
keep in mind that an increasing number of students has limited prociency in
the language spoken in the classroom or learned differently how to perform
arithmetic operations, which may have a negative impact on his/her ability to
follow a calculation or the mathematical steps of a proof.
Special-needs students, such as dyslexics, students affected by attention-
decit disorder or other forms of impairment deserve a particular care by the
instructors. It is therefore important to establish contacts outside the classroom
to understand what learning strategies may help them and create lesson plans
accordingly, or develop supplemental materials to hand out to the students
who need them.
A helpful teaching tool is to provide students several days after the lecture
with a copy of the instructor’s own lecture notes, preferably formally typed. A
common problem that arises is that a student who is engaged in taking notes
may lose touch with the instructor’s verbal explanations, which prevents the
lecture from being processed in a coherent and complete manner. The preem-
ptive distribution of lecture notes may have the undesirable effect of discoura-
ging students’ participation since they could feel not motivated to attend. We
believe that attendance and class participation are important factors for optimal
learning. It is important for the students to write down as much of the lecture as
possible in order to retain most of the information to be recalled afterwards. It
would lessen the students’ anxiety the knowledge that they can have access to a
written copy or an electronic version of the lecture at a later time provided that
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they continue attending regularly. Of course, this strategy is time-consuming
and may not be feasible or practical for many lower level courses.
For an excellent resource on teaching strategies for students with different
abilities, we recommend to the reader (Armstrong, 1994; Gardner, 1993).
6 Technology in the classroom
Many classes at universities across the U.S. are being held in ‘smart clas-
srooms,’ that is classrooms equipped with a multimedia console hooked to a
computer and connected to a DVD player, a projector, and a lit writing board
that can be projected onto a large screen. An instructor can then have instan-
taneous access to lectures online as well as demonstrations of math software
with plotting and computing capabilities. There are many advantages in the use
of these technological tools. The screen is better lit and more visible from the
back of the room than the traditional blackboard. Instructors can easily access
old materials and more steps can be shown using color coding and other visual
devices aimed at emphasizing certain aspects of the lecture. These tools can
also be used to project onto the screen pictures, formulas or problems from a
textbook or another source. Aside from the occasional computer glitch or equi-
pment failure, these classrooms have revolutionized the ways many subjects are
taught and have opened up many new possibilities on how to upgrade classroom
instruction, both in terms of content and visually.
An optimal environment for learning math courses such as calculus or diffe-
rential equations is based on the adoption of software whose aim is to obtain a
quick answer to a computation with the push of a button or graphing instantly
a function for demonstration purposes. The types of math software we are most
familiar with are Maple and Matlab.
Maple is suitable for a wide variety of uses, ranging from solving an equa-
tion numerically, to differentiating and nding limits of a function, as well as
plotting the graph of a curve in polar or parametric form. This software is also
necessary in order to compute denite integrals of functions whose antideriva-
tives cannot be determined analytically and sums of convergent series. There
are commands that allow for an increased resolution to yield a more precise
plot, by far superior to what is feasible with a hand graphing calculator.
Matlab, which is heavily used in engineering applications, can also be uti-
lized for graphing, but its main purpose is to carry out matrix operations or to
implement codes based on the use of linear algebra.
These tools are very useful also to explain the meaning of a linear appro-
ximation of a function. By zooming in on a suitably small interval or grid and
plotting simultaneously the function and its linear approximation, it is possible
to demonstrate the accuracy of the approximation and even determine the size
F. Colonna, G. Easley - Hints on how to improve mathematics instruction
17
of the error before a mathematical estimate of the error could be determined
analytically or through a formula. A follow-up problem that can be tackled
next is to see how a quadratic approximation, or more generally, a polynomial
approximation can be implemented for better results and how the degree of the
polynomial should be chosen to obtain the desired level of accuracy with the
smallest number of operations.
The intent of changing the calculus curriculum to one aimed at making de-
cisions on modeling, or approaching a problem through the use of technology
is to make students more aware of the methodologies taught by rst giving
them a hands-on experience. The aim is also to teach students how to apply
these tools by focusing less on programming and more on how to choose the
technology that best ts their needs.
A problem that arises in teaching specialized mathematics courses that are
applied in nature, such as wavelet theory and Fourier analysis, is that these
courses are typically structured by exposing the students mostly to the theore-
tical background and leaving very little room to the applications. Such courses
offer unique educational challenges, not only for the difculty integrating the
pure and the applied side of the subject, but also because the type of mathema-
tics needed in the application is often different, leaving a gap between the two
to be lled. Indeed, going back to the examples mentioned above, both wavelet
theory and Fourier analysis have a rich theoretical background in analysis and
can be taught by just focusing on their classical theory. Teaching these courses
by following such an approach is what is often done because this is how the
instructors themselves have learned these topics and is the way in which the
material is presented in most textbooks. Yet, such instructions can miss an
important connection with applications. An important value of these subjects
is indeed in their applications. Ironically, a student may successfully complete
such courses without being able to do any real world applications.
An important issue is that the most useful applications of these subjects are
done by using computers. When the focus of these courses is too theoretical,
no basic instruction is given on how to implement the concepts covered on a
computer. Not only does this create an obstacle for helping students develop
new technology using such theories, but it can also be misleading. Indeed, often
the applications in the digital domain become nite and discrete, and hence
associated to linear algebra rather than analysis. This implies that many of
the theoretical results should also be formulated according to the language of
linear algebra. Yet, doing this adaptation, in addition to presenting the analysis
results, takes away much of the available time in the lectures, leaving little
time left to present the related algorithms. Thus, a useful technique is to design
assignments that gear the students toward developing some of this knowledge
by themselves. However, in order for this to work, the students still need to see
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many basic demonstrations in the classroom.
It is our experience that these demonstrations seem to work best when pre-
sented as combinations of improvising and following prepared scripts, which
should be provided to the students at a later time. It should be noted that im-
provising with a computer software package such as Matlab is most helpful
at creating a friendly student-teacher interaction. Part of this happens because
the instructor will often need to sit and type some commands creating a less
imposing posture and an atmosphere more conducive to having an open dialog.
In addition, the improvisation can be enjoyable, exposes the students to what
is like to do code development, and give them an impression of what active
researchers do. This is important because the classical theory, while funda-
mental, can often leave the students wondering what could possibly be done
that is new to the eld.
7 Technology and the publishing industry
The impact that technology can have on teaching has been understood for
some time and the publishing industry has been quick at taking advantage of
the emerging technologies and at marketing new textbooks with a large num-
ber of calculator or computer applications. Most well established and popular
textbooks in lower level mathematics courses have been replaced by updated
versions by the same authors with increasing frequency, due especially to how
quickly technologies are evolving. It is not uncommon to have a new edition
in the market only two or three years after the preceding one has appeared.
In addition, a new generation of technology-rich textbooks and user-friendly
tools has emerged.
New learning tools that are gaining in popularity are complete online courses
afliated with textbooks by a specic publisher. The instructor can create online
courses for as many textbooks as desired and can choose whether or not to make
the course available to the students. The instructor can also set aside practice
problems with solutions, multiple choice quizzes and tests for credit and allow
students to access other material. One of us is familiar with the interactive
website called MyMathLab, which, working jointly with Pearson’s Education
Publishing Company, provides self-test and works through practice exercises
with step-by-step help. MyMathLab includes multimedia learning aids, videos,
animation, as well as live tutorial help. For more information, see <http://www.
mymathlab.com> and <http://www.pearsonhighered.com>. Another interactive
website that offers course development services to help create engaging courses
to t the instructor’s needs is Cengage Learning (for information, see <http://
www.cengage.com/custom/>. This type of online course set up, however, is
not sufcient for most students to be a satisfactory substitute of the standard
F. Colonna, G. Easley - Hints on how to improve mathematics instruction
19
in-classroom learning environment. Indeed, several of our students who chose
to use this online material requested help from one of us because they could
not understand the solutions provided or why a certain answer they gave was
incorrect. In order to provide this service to the students, the instructor has to
monitor the choice of materials and testing and be available for online support
when requested.
We believe that, as it is already happening in other elds, ever for mathe-
matics a combination of in-class (according to the criteria described above)
and interactive online course set up is ideal for providing clarity and support
and for assessing effectively what the student has learned. Moreover, it has
the potential to become the prototype of a self-paced course for students who
cannot come to class regularly.
Conclusions
In this article, the most important aspects on teaching mathematics have
been outlined. The main objective of this work was to promote discussion and
expand the study on how to foster a better learning environment for mathema-
tics at the university. It is our hope that this article will stimulate an exchange
of ideas on pedagogy and the creation of new tools for learning both inside and
outside the classroom, as well as emphasize the importance of paying particular
attention to girls’ learning and of adapting the teaching style to students with
different learning abilities.
In the United States there are several federal agencies and academic ma-
thematical organizations (most notably, the National Science Foundation, the
American Mathematical Society, the Mathematical Association of America,
and the Association of Women in Mathematics) whose mission is to facilitate
the exchange of information among academics and researchers. These organi-
zations have been instrumental in the cultural and social awareness raised in
their newsletters and publications, and through conferences. There are many
opportunities to expand this exchange by establishing new networks through
blogs and other means of communication. A rst step is to establish cross-
cultural connections and assess periodically the progress being made socially
as well as in the teaching and research arenas.
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... Student-centered learning requires the active involvement of students, which can be implemented based on constructivism theory (Bergsten, 2008;Dubinsky, 1991Dubinsky, , 2001. Studies have shown that the constructivism theory is effective in a computer-technology-integrated environment (Colonna & Easley, 2011;Ward, 2003). With technology, students are flexible in adjusting their learning strategy based on their learning style (Salleh & Zakaria, 2012b). ...
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The objective of this research is to investigate the effectiveness of a learning strategy using Maple in integral calculus. This research was conducted using a quasi-experimental nonequivalent control group design. One hundred engineering technology students at a technical university were chosen at random. The effectiveness of the learning strategy was examined through three variables on two groups of these students. Data were analyzed using Hotelling’s T2 and explained by interview data. The advantages offered in Maple enable students’ thinking to be amplified. Students benefit from the conceptual and procedural understanding of integral calculus. However, they need more time to improve their metacognitive awareness. The transformation of the integral calculus learning approach using Maple has the potential to overcome engineering technology students’ underpreparedness. As a result, the nation’s inadequacy in the related workforce may be overcome.
... This allows for an enhancement of teaching and the implementation of learning calculus using computers to improve their calculus performance. This is parallel to the needs in learning calculus in its optimal environment, which is a software adoption teaching and learning paradigm (Colonna & Easley, 2011). Nordin et al. (2010) emphasized that the usage of technological tools in mathematics classes will produce generations who are not only information communication technology (ICT) literate but also knowledgeable mathematically. ...
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The purpose of this study was to investigate students’ perceptions towards integral calculus difficulty and their readiness towards using technology in learning integral calculus. A total of 191 students were selected at random from two lecture groups of Technical Mathematics 1. The students were given a set of questionnaire with two parts. The first part was used to measure students’ perceptions on integral calculus difficulty. The second part was used to measure students’ computer readiness in learning. Three main contributing factors of students’ readiness towards computers were adapted from The Computer Aversion, Attitudes, and Familiarity Index (CAAFI).For measuring computer readiness using CAAFI, Pearson correlations and the mean values were determined. The inter-correlations between factors in this instrument were statistically significant. More than three quarters of the students with school calculus background perceived integral calculus as difficult or a very difficult topic. The students were found to have positive attitudes towards computers, a low computers aversion level and an average level of computers familiarity. These findings allow for an enhancement of teaching and implementation of learning calculus using computers. DOI: 10.5901/mjss.2015.v6n5s1p144
... The most advantageous environment for learning mathematics including calculus is based on the integration of mathematical software such as Maple and Matlab [10]. Previous researches have discussed about various ways that have been implemented in improving students' calculus learning through the use of educational technology as a tool [11][12][13][14] A well-designed instructional approach in using technology for the teaching and learning can produce positive impacts on students' mathematical understanding. ...
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Students entering engineering and engineering technology courses at the university are expected to be competent in mathematics. This is due to the fact that mathematics, including calculus is a primary gateway to an engineering and engineering technology careers. However, the issue of under preparedness of students with diversity in mathematics background enrolling these two courses is at an alarming rate. One factor which can be improved to minimize this gap is to enhance the teaching technique of the subject. With the evolution of various forms of technology, a new approach of teaching integral calculus was developed in this study. Accordingly, this paper discusses the process of integrating Maple software in the teaching of the first year integral calculus topic. In addition, this paper also discusses how this approach can help to deepen the engineering technology students’ understanding in integral calculus topic at the university. Two groups of Technical Mathematics 2 with various mathematics backgrounds were randomly chosen to undergo the treatment. As an initial measurement, a pre-test on integral calculus was administered to these students to check their integral calculus background. The highest mark for the test was far below the passing mark, indicates that students involved in this study have a minimum understanding in secondary school integral calculus. These students were exposed to the treatment for five weeks. At the end of the treatment a post-test was administered to see the effect of the treatment on students’ understanding in integral calculus. Students in the experimental group were found to outperform their peers in the control group in both constructs investigated in this study. The effectiveness of the use of the technology in learning this topic may be enhanced further if its prerequisite topics were also exposed to the same strategy.
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Mathematics is a gateway to engineering technology courses. However, students choosing to enrol in engineering technology courses were found to be underprepared in this subject including integral calculus. Non-proficiency in mathematics affects students' understanding in other subjects, which will eventually delay the completion of their studies. Therefore, in this study, a new approach of teaching and learning integral calculus was carried out to enhance students' understanding in this topic. Technology integration was used due to its capability in distributing knowledge to diverse types of students. In order to investigate its effectiveness, a quasi-experimental non-equivalent comparison group design was employed. An achievement test and a questionnaire were used to gather information. The results indicated that the integration of Maple software in the learning of integral calculus was able to enhance students' understanding but not their metacognitive awareness. Further research is required to explore new possibilities in enhancing not only students' understanding in integral calculus, but also to enhance their metacognitive awareness in learning. © Sila Science. All Rights Reserved.
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Stereotype threat (ST) refers to a situation in which a member of a group fears that her or his performance will validate an existing negative performance stereotype, causing a decrease in performance. For example, reminding women of the stereotype "women are bad at math" causes them to perform more poorly on math questions from the SAT and GRE. Performance deficits can be of several types and be produced by several mechanisms. We show that ST prevents perceptual learning, defined in our task as an increasing rate of search for a target Chinese character in a display of such characters. Displays contained two or four characters and half of these contained a target. Search rate increased across a session of training for a control group of women, but not women under ST. Speeding of search is typically explained in terms of learned "popout" (automatic attraction of attention to a target). Did women under ST learn popout but fail to express it? Following training, the women were shown two colored squares and asked to choose the one with the greater color saturation. Superimposed on the squares were task-irrelevant Chinese characters. For women not trained under ST, the presence of a trained target on one square slowed responding, indicating that training had caused the learning of an attention response to targets. Women trained under ST showed no slowing, indicating that they had not learned such an attention response.
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A gender gap in mathematics achievement persists in some nations but not in others. In light of the underrepresentation of women in careers in science, technology, mathematics, and engineering, increasing research attention is being devoted to understanding gender differences in mathematics achievement, attitudes, and affect. The gender stratification hypothesis maintains that such gender differences are closely related to cultural variations in opportunity structures for girls and women. We meta-analyzed 2 major international data sets, the 2003 Trends in International Mathematics and Science Study and the Programme for International Student Assessment, representing 493,495 students 14-16 years of age, to estimate the magnitude of gender differences in mathematics achievement, attitudes, and affect across 69 nations throughout the world. Consistent with the gender similarities hypothesis, all of the mean effect sizes in mathematics achievement were very small (d < 0.15); however, national effect sizes showed considerable variability (ds = -0.42 to 0.40). Despite gender similarities in achievement, boys reported more positive math attitudes and affect (ds = 0.10 to 0.33); national effect sizes ranged from d = -0.61 to 0.89. In contrast to those of previous tests of the gender stratification hypothesis, our results point to specific domains of gender equity responsible for gender gaps in math. Gender equity in school enrollment, women's share of research jobs, and women's parliamentary representation were the most powerful predictors of cross-national variability in gender gaps in math. Results are situated within the context of existing research demonstrating apparently paradoxical effects of societal gender equity and highlight the significance of increasing girls' and women's agency cross-nationally.
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This study uses data on sex differences in the eighth-grade mathematical performance of over 77,000 students in 19 countries, 1964 and 1982 data on such differences in 9 countries, and data on gender stratification of advanced educational and occupational opportunities to explore when and where gender will affect students' performance in mathematics. The analyses show that there is cross-national variation in the performance of mathematics and that it is related to variation in the gender stratification of educational and occupational opportunities in adulthood, that sex differences have declined over time, and that school and family factors leading to higher mathematical performance are less stratified by gender when women have more equal access to jobs and higher education.
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This study examined the cues hypothesis, which holds that situational cues, such as a setting's features and organization, can make potential targets vulnerable to social identity threat. Objective and subjective measures of identity threat were collected from male and female math, science, and engineering (MSE) majors who watched an MSE conference video depicting either an unbalanced ratio of men to women or a balanced ratio. Women who viewed the unbalanced video exhibited more cognitive and physiological vigilance, and reported a lower sense of belonging and less desire to participate in the conference, than did women who viewed the gender-balanced video. Men were unaffected by this situational cue. The implications for understanding vulnerability to social identity threat, particularly among women in MSE settings, are discussed.
Creating gender equality: cross-national gender
  • D P Baker
  • D P Jones
D.P. Baker, D.P. Jones (1993), Creating gender equality: cross-national gender
Multiple intelligences in the classroom, Association for Supervision and Curriculum Development
  • T Armstrong
T. Armstrong (1994), Multiple intelligences in the classroom, Association for Supervision and Curriculum Development, Alexandria, Virginia, U.S.A., 1994. D.P. Baker, D.P. Jones (1993), Creating gender equality: cross-national gender