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  1. 1. Project in Advance MathSubmitted to:Engr. Ravenal De JesusSubmitted by:Ma. Lea JavierRenz Martin ElipeAriel Gonzales
  2. 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 cannotbe defined in ordinary language and mathematics is almost certainly one of them. Thisdoesn’t mean that we can’t describe the subject in general terms. We just can’t sharplylimit it with a definition.Over the years, a number of people have tried to define mathematics as “the study ofpatterns” or “the language of science,” but professional mathematicians have avoidedtrying to define mathematics. As best I can recollect, the nearest that a researchmathematician came to attempting a definition in print was Roy. Adler in the mid-1960’swho suggested the semi-serious “Mathematics is what mathematicians do.”A few years back a serious attempt at a short description of mathematics was givenprivately by Norman Gottlieb at Purdue. He suggested “Mathematics is the study ofprecisely defined objects.” A number of people participating in this discussion said, ineffect, “Yes, that’s very close, but let’s not publicize it since it would tend to confirm thewidely held belief that mathematics is boring and useless.”Realistically, in describing what mathematics is, the best we can do is to discuss themost important characteristics of mathematics. I suggest that these are1. Precision (precise definitions of all terms, operations, and the properties of theseoperations)2. Stating well-posed problems and solving them. (Well-posed problems are problemswhere all the terms are precisely defined and refer to a single universe wheremathematics can be done.)It would be fair to say that virtually all of mathematics is problem solving in preciselydefined environments, and professional mathematicians tend to think it strange thatsome trends in mathematics education isolate mathematical reasoning and problemsolving as separate topics within mathematics instruction.Mathematical proficiency is an important predictor of engineering success. However, thechange of the demographics of engineering students has led to larger percentage of studentswith weak mathematics skills. Furthermore, engineering courses are perceived as abstract andirrelevant. Students struggle with required mathematics and have been shown to havemotivational difficulties when studying the subject.ENGINEERING is the profession in which a knowledge of the mathematical and naturalsciences gained by study, experience, and practice is applied with judgment to developways to utilize economically the materials and forces of nature for the benefit ofmankind.
  3. 3. Engineering skillsThe Subject Benchmark statement for engineering details skills, attributes and qualities that arethought necessary to enable the engineer to practice effectively in a professional manner. It isexpected that an engineering degree programme will foster, develop and inculcate suchattributes, skills and qualities. These attributes, skills and qualities are listed in the SubjectBenchmark Statement under five headings:• Knowledge and Understanding• Intellectual Abilities• Practical Skills• General Transferable Skills• Qualities.Basic information:Engineering skillsHeading Demonstration of Skills / Attributes / QualitiesKnowledge & 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, designAbilities 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. 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 processTransferable • Analytical in the formulation and solutions of problemsSkills • 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. 5. Data gathered:A study is undertaken to lay out in a structured manner the mathematics skills required ofundergraduate students in the Department of Aeronautics and Astronautics at theMassachusetts Institute of Technology. The key objective of the research is to identify barriersto deep mathematical understanding among engineering undergraduates. Data fromengineering course syllabi and interviews with engineering and mathematics faculty arecombined to form an implicit mathematics curriculum, which lists the mathematical skillsrelevant to core engineering classes along with the flow of learning and utilization. Severalproblematic areas are identified, including the concept of a function, linearization, and vectorcalculus. Interview results show that many engineering faculty have an inadequate knowledgeof mathematics class syllabi, and often do not know where or how the skills they require aretaught, while mathematics instructors often have a limited understanding of how mathematicalconcepts are applied in downstream engineering classes. A number of recommendations aremade, including increased communication between mathematics and engineering faculty,development of joint resources for problematic areas, and dissemination of a formal catalogueof mathematical skills and resources to engineering students and faculty.Undergraduate programs; however, specific, well-documented examples of student difficultiesare often lacking, and the exact nature of the difficulty is frequently uncertain. Moreover, thereis often little communication between engineering and mathematics faculty dedicated to oraddressing mathematics skills related issues. Engineering faculty assume that certain conceptsare taught in the mathematics courses, but they are often not familiar with the specifics of themathematics 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 aproblem by a number of the faculty that teach core subjects in the Department of Aeronauticsand Astronautics. This issue manifests itself in a number of ways and, in particular, has anegative impact on students’ ability to grasp engineering subject material. Specific problemsare observed during lectures, where questions often arise regarding basic mathematicmanipulations. These questions are also posed in the form of “muddy cards” – cards on whichstudents 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 mathprofiency 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. Itseems that every move you make has an explanation in math, the first step to bridging the gap
  6. 6. between mathematics and engineering is to comprehend the barriers to deep mathematicalunderstanding among engineering undergraduates. In order to gain such understanding, it iscritical to identify specifically what mathematical skills are expected and where in theengineering curriculum these skills are gained. While there were many suppositions regardingthis issue in the above discussions such identification had not been formally carried out ordocumented. This paper describes an effort to formally identify and document the implicitmathematics curriculum in the undergraduate degree program.As the study tells during the data gathering, interviews were scheduled with engineering facultywith the intention to incorporate their feedback on the curriculum structure and to collect theiropinion on mathematics-related problems they might have encountered in core undergraduateteaching. The goal was to formally document and organize the implicit mathematics curriculumand to trace the flow of skills learning and utilization. The first question to each faculty memberwas whether they would modify the mathematics topics list in any way, reorganize or addtopics. This helped to form an exhaustive list as viewed by the engineering faculty. The secondpart of the interview concentrated on the particular set of mathematics skills relevant to thecourse taught by the interviewee. In particular, he or she had to specify in detail precisely howa mathematics skill is relevant to the class, and whether it is taught anew, reviewed, and/orutilized. If a skill was reviewed and/or utilized, the faculty member was asked to identify theprior course in which the knowledge was assumed to be gained.After this preliminary round of data was collected, feedback and input was sought frommathematics faculty. The data was discussed in detail with faculty involved in teaching afreshman calculus course and a freshman/sophomore differential equations course, which arerespectively pre-requisite and co-requisite for sophomore students in the Department ofAeronautics and Astronautics. This communication with mathematics faculty not only enabledprecise identification of mathematics courses and topics where engineering mathematicsknowledge and skills are introduced, but also allowed some major gaps and misconceptions tobe 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, designsystems 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 inenabling the introduction of new technology, ability to apply professional judgement, balancingissues of costs, benefits, safety, quality etc. Finally the ability to assess and manage risks, use awide range of tools, techniques, and equipment (including software) appropriate to theirspecific 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. 7. [1] Adamczyk, B., Reffeor, W. and Jack, H., 2002, “Math Literacy and Proficiency in EngineeringStudents,” Proceedings of the 2002 American Society for Engineering Education AnnualConference and Exposition.*2+ d’Aimp Project, 2003, MIT Department of Mathematics,*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 SuccessfulStudents?” Proceedings of the 2002 American Society for Engineering Education AnnualConference and Exposition.*5+ Lingefjard, T., 2002, “To study mathematics in an engineering program,” The Third SwedishMathematics Education Research Seminar, Norrköping, Sweden, January 23-25, 2002,*6+ Mosteller, F., “The Muddiest Point in the Lecture as a Feedback Device in Teaching andLearning,” 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 theFreshman Engineering Design Course,” Proceedings of the 2002 American Society forEngineering Education Annual Conference and Exposition.[8] Project Links, Renssselaer Polytechnic Institute,[9] Unified Engineering Class Web Site, source:, DateAccessed: June 9, 2003[10] 18.01 Single Variable Calculus, source:, Date Accessed: August 1, 2003[11] 18.02 Multivariable Calculus - Fall 2002, source:,Date Accessed: August 1, 2003[12] 18.03 Differential Equations Fall 2002, source:, Date Accessed:August 1, 2003