1. Project in Advance
Math
Submitted to:
Engr. Ravenal De Jesus
Submitted by:
Ma. Lea Javier
Renz Martin Elipe
Ariel Gonzales

2. Introduction:
What is mathematics?
We must have some idea of what mathematics is in order to start our discussion.
Unfortunately, a serious misconception already occurs here. Some things simply cannot
be defined in ordinary language and mathematics is almost certainly one of them. This
doesn’t mean that we can’t describe the subject in general terms. We just can’t sharply
limit it with a definition.
Over the years, a number of people have tried to define mathematics as “the study of
patterns” or “the language of science,” but professional mathematicians have avoided
trying to define mathematics. As best I can recollect, the nearest that a research
mathematician came to attempting a definition in print was Roy. Adler in the mid-1960’s
who suggested the semi-serious “Mathematics is what mathematicians do.”
A few years back a serious attempt at a short description of mathematics was given
privately by Norman Gottlieb at Purdue. He suggested “Mathematics is the study of
precisely defined objects.” A number of people participating in this discussion said, in
effect, “Yes, that’s very close, but let’s not publicize it since it would tend to confirm the
widely held belief that mathematics is boring and useless.”
Realistically, in describing what mathematics is, the best we can do is to discuss the
most important characteristics of mathematics. I suggest that these are
1. Precision (precise definitions of all terms, operations, and the properties of these
operations)
2. Stating well-posed problems and solving them. (Well-posed problems are problems
where all the terms are precisely defined and refer to a single universe where
mathematics can be done.)
It would be fair to say that virtually all of mathematics is problem solving in precisely
defined environments, and professional mathematicians tend to think it strange that
some trends in mathematics education isolate mathematical reasoning and problem
solving as separate topics within mathematics instruction.
Mathematical proficiency is an important predictor of engineering success. However, the
change of the demographics of engineering students has led to larger percentage of students
with weak mathematics skills. Furthermore, engineering courses are perceived as abstract and
irrelevant. Students struggle with required mathematics and have been shown to have
motivational difficulties when studying the subject.
ENGINEERING is the profession in which a knowledge of the mathematical and natural
sciences gained by study, experience, and practice is applied with judgment to develop
ways to utilize economically the materials and forces of nature for the benefit of
mankind.

3. Engineering skills
The Subject Benchmark statement for engineering details skills, attributes and qualities that are
thought necessary to enable the engineer to practice effectively in a professional manner. It is
expected that an engineering degree programme will foster, develop and inculcate such
attributes, skills and qualities. These attributes, skills and qualities are listed in the Subject
Benchmark Statement under five headings:
• Knowledge and Understanding
• Intellectual Abilities
• Practical Skills
• General Transferable Skills
• Qualities.
Basic information:
Engineering skills
Heading Demonstration of Skills / Attributes / Qualities
Knowledge & A Graduating Engineer should be able to demonstrate:
Understanding • Specialist (Discipline) knowledge
• Understanding of external constraints
• Business and Management techniques
• Understanding of professional and ethical
responsibilities
• Understanding of the impact of engineering solutions on
society
• Awareness of relevant contemporary issues
A Graduating Engineer should be able to demonstrate:
Intellectual • The ability to solve engineering problems, design
Abilities systems etc. through creative and innovative thinking
• The ability to apply mathematical, scientific and
technological tools
• The ability to analyse and interpret data and, when
necessary, design experiments to gain new data
• The ability to maintain a sound theoretical approach in
enabling the introduction of new technology
• The ability to apply professional judgement, balancing

4. issues of costs, benefits, safety, quality etc.
• The ability to assess and manage risks
A Graduating Engineer should be able to:
• Use a wide range of tools, techniques, and equipment
(including software) appropriate to their specific
discipline
• Use laboratory and workshop equipment to generate
valuable data
• Develop, promote and apply safe systems of work
A Graduating Engineer should be able to:
Practical Skills
• Communicate effectively, using both written and oral
methods
• Use Information Technology effectively
• Manage resources and time
• Work in a multi-disciplinary team
• Undertake lifelong learning for continuing professional
Development
A Graduating Engineer should be:
General • Creative, particularly in the design process
Transferable • Analytical in the formulation and solutions of problems
Skills • Innovative, in the solution of engineering problems
• Self-motivated,
• Independent of mind, with intellectual integrity,
Qualities particularly in respect of ethical issues
• Enthusiastic, in the application of their knowledge,
understanding and skills in pursuit of the practice of
engineering

5. Data gathered:
A study is undertaken to lay out in a structured manner the mathematics skills required of
undergraduate students in the Department of Aeronautics and Astronautics at the
Massachusetts Institute of Technology. The key objective of the research is to identify barriers
to deep mathematical understanding among engineering undergraduates. Data from
engineering course syllabi and interviews with engineering and mathematics faculty are
combined to form an implicit mathematics curriculum, which lists the mathematical skills
relevant to core engineering classes along with the flow of learning and utilization. Several
problematic areas are identified, including the concept of a function, linearization, and vector
calculus. Interview results show that many engineering faculty have an inadequate knowledge
of mathematics class syllabi, and often do not know where or how the skills they require are
taught, while mathematics instructors often have a limited understanding of how mathematical
concepts are applied in downstream engineering classes. A number of recommendations are
made, including increased communication between mathematics and engineering faculty,
development of joint resources for problematic areas, and dissemination of a formal catalogue
of mathematical skills and resources to engineering students and faculty.
Undergraduate programs; however, specific, well-documented examples of student difficulties
are often lacking, and the exact nature of the difficulty is frequently uncertain. Moreover, there
is often little communication between engineering and mathematics faculty dedicated to or
addressing mathematics skills related issues. Engineering faculty assume that certain concepts
are taught in the mathematics courses, but they are often not familiar with the specifics of the
mathematics curriculum, or the methods utilized (for example: terminology and context of use).
The level of mathematics skills of sophomores and juniors at MIT has been identified as a
problem by a number of the faculty that teach core subjects in the Department of Aeronautics
and Astronautics. This issue manifests itself in a number of ways and, in particular, has a
negative impact on students’ ability to grasp engineering subject material. Specific problems
are observed during lectures, where questions often arise regarding basic mathematic
manipulations. These questions are also posed in the form of “muddy cards” – cards on which
students anonymously write down the muddiest part of the lecture. Continue reading….
Analysis:
According to the data and discussion above it shows that there is a relationship between math
profiency and engineering skills. All the indicated engineering skills above are related to math.
Learning mathematics is the basic step in gaining the skills needed by an engineering student. It
seems that every move you make has an explanation in math, the first step to bridging the gap

6. between mathematics and engineering is to comprehend the barriers to deep mathematical
understanding among engineering undergraduates. In order to gain such understanding, it is
critical to identify specifically what mathematical skills are expected and where in the
engineering curriculum these skills are gained. While there were many suppositions regarding
this issue in the above discussions such identification had not been formally carried out or
documented. This paper describes an effort to formally identify and document the implicit
mathematics curriculum in the undergraduate degree program.
As the study tells during the data gathering, interviews were scheduled with engineering faculty
with the intention to incorporate their feedback on the curriculum structure and to collect their
opinion on mathematics-related problems they might have encountered in core undergraduate
teaching. The goal was to formally document and organize the implicit mathematics curriculum
and to trace the flow of skills learning and utilization. The first question to each faculty member
was whether they would modify the mathematics topics list in any way, reorganize or add
topics. This helped to form an exhaustive list as viewed by the engineering faculty. The second
part of the interview concentrated on the particular set of mathematics skills relevant to the
course taught by the interviewee. In particular, he or she had to specify in detail precisely how
a mathematics skill is relevant to the class, and whether it is taught anew, reviewed, and/or
utilized. If a skill was reviewed and/or utilized, the faculty member was asked to identify the
prior course in which the knowledge was assumed to be gained.
After this preliminary round of data was collected, feedback and input was sought from
mathematics faculty. The data was discussed in detail with faculty involved in teaching a
freshman calculus course and a freshman/sophomore differential equations course, which are
respectively pre-requisite and co-requisite for sophomore students in the Department of
Aeronautics and Astronautics. This communication with mathematics faculty not only enabled
precise identification of mathematics courses and topics where engineering mathematics
knowledge and skills are introduced, but also allowed some major gaps and misconceptions to
be identified.
All in all we conclude that understanding of the impact of engineering solutions on society ,
awareness of relevant contemporary issues, the ability to solve engineering problems, design
systems etc. through creative and innovative thinking, the ability to apply mathematical,
scientific and technological tools, ability to analyse and interpret data and, when necessary,
design experiments to gain new data, ability to maintain a sound theoretical approach in
enabling the introduction of new technology, ability to apply professional judgement, balancing
issues of costs, benefits, safety, quality etc. Finally the ability to assess and manage risks, use a
wide range of tools, techniques, and equipment (including software) appropriate to their
specific discipline, use laboratory and workshop equipment to generate valuable data. Develop,
promote and apply safe systems of work, these are all gain fom mathematical profiency.
Reference:

7. [1] Adamczyk, B., Reffeor, W. and Jack, H., 2002, “Math Literacy and Proficiency in Engineering
Students,” Proceedings of the 2002 American Society for Engineering Education Annual
Conference and Exposition.
*2+ d’Aimp Project, 2003, MIT Department of Mathematics, http://www-math.mit.edu/daimp/
*3+ Darmofal, D., “Concerns Regarding Undergraduate Mathematical Skills,” Internal Memo,
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 2002.
Proceedings of the 2004 American Society for Engineering Education Annual Conference &
Exposition Copyright © 2004, American Society for Engineering Education
[4] Diefes-Dux, H., 2002, “Does a Successful Mathematics Bridge Program Make for Successful
Students?” Proceedings of the 2002 American Society for Engineering Education Annual
Conference and Exposition.
*5+ Lingefjard, T., 2002, “To study mathematics in an engineering program,” The Third Swedish
Mathematics Education Research Seminar, Norrköping, Sweden, January 23-25, 2002,
http://www.mai.liu.se/SMDF/madeng.htm.
*6+ Mosteller, F., “The Muddiest Point in the Lecture as a Feedback Device in Teaching and
Learning,” Journal of the Harvard-Danford Center, Vol. 3, April 1989, pp. 10-21.
*7+ Pines, D., Nowak, M. and Alnajjar, H., 2002, “Integrating Science and Math into the
Freshman Engineering Design Course,” Proceedings of the 2002 American Society for
Engineering Education Annual Conference and Exposition.
[8] Project Links, Renssselaer Polytechnic Institute, http://links.math.rpi.edu/
[9] Unified Engineering Class Web Site, source: http://web.mit.edu/16.unified/www/, Date
Accessed: June 9, 2003
[10] 18.01 Single Variable Calculus, source:
http://wwwmath.mit.edu/18.01/syllabus/index.html, Date Accessed: August 1, 2003
[11] 18.02 Multivariable Calculus - Fall 2002, source: http://www-math.mit.edu/18.02/info.pdf,
Date Accessed: August 1, 2003
[12] 18.03 Differential Equations Fall 2002, source: http://web.mit.edu/18.03/, Date Accessed:
August 1, 2003