Modeling and Applications in the Mathematics Classroom for Smooth Transition from High School to Engineering Education Riadh W. Y. Habash, PhD, P.Eng School of Information Technology and Engineering University of Ottawa, Ottawa,

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Modeling and Applications in the Mathematics
Classroomfor Smooth Transition from High School
to Engineering Education Riadh W. Y. Habash,
PhD, P.EngSchool of Information Technology and
EngineeringUniversity of Ottawa, Ottawa,
Canada.rhabash_at_site.uottawa.ca
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Transition from High School to University

• A large percentage of the young people in this
  country are going to university to study science
  and engineering.
• Students moving from high schools to
  mathematics-related programs at universities,
  such as engineering, often have difficulty
  applying their mathematical knowledge to new
  situations.
• They find that there are gaps in the knowledge
  and skills expected of them in a university
  program.
• Usually, these university programs depend
  critically on students experiences of learning
  mathematics and on their ability to make
  connections between the mathematics they learn in
  high school and the practical situations
  presented later in the university.

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• Challenges!
• Why do we have to learn this? When am I going to
  use this? These questions are asked with
  increasing frequency as students progress through
  study.
• While students enter school with large doses of
  curiosity, that curiosity may be overtaken by
  skepticism if we do not show them how their
  studies are relevant.
• This presents a challenge for all educators, but
  particularly those of us who teach math. That is
  because math is sometimes perceived by students
  as a collection of isolated topics that have
  little relationship to the real world.

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Role of the Teacher

• The role of the teacher at all levels is
  critical. Mathematics is difficult because we, as
  teachers, have made it difficult! Possibly we
  have been teaching too much mathematics and
  sometimes the wrong one! Much of the mathematics
  taught is rarely seen in their future careers. We
  have to make mathematics easier and more
  attractive.
• We have to remove the fear factor often
  associated with mathematics. We need to focus our
  attention on the typical student who does not
  have a deep interest in the subject but need to
  access the subject at a superficial level.

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• Often students arrive at university not having
  ever seen or realized differentiation, or can
  hardly differentiate. Many find even the simplest
  questions in integration almost impossible. Some
  will have never met complex numbers, vectors, or
  matrices.
• Sometimes engineering students complain that they
  physically cannot perceive mathematics concepts.
  In such case students can be given this
  relationship to realize.
• This relationship try to say that increasing the
  amount of work can easily compensate the limited
  ability of a student.

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Mathematics for Engineers

• It is recognized by engineering faculty that
  undergraduates in engineering programs should be
  better prepared in mathematics to successfully
  complete courses in their professional
  disciplines.
• Success in science and engineering depends
  heavily on the application of mathematical
  techniques to real world problems so increased
  use of engineering examples in mathematics
  courses can enhance the familiarity of students
  with mathematical concepts.
• However, developing custom courses in mathematics
  may not be economically viable since students in
  many different fields are taught by teachers from
  mathematics department.

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• High school students need to have it explained to
  them why knowledge of mathematics is essential
  for their future practical work.
• An understanding of key mathematical concepts
  together with a skill to apply them effectively
  to solve real world problems is an essential
  ability that every student must acquire.
• Mathematics should be regarded as a language for
  expressing physical and engineering laws.

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• Formal Lectures
• Although this method of teaching may meet the
  needs of students with high competence in
  mathematics, formal lectures do not appear to be
  the most effective method for teaching
  mathematics to engineering students for several
  reasons.
• Many students learn to solve theoretical problems
  without being able to apply that knowledge and
  further, are exposed to pure rather than applied
  mathematics.
• As well, a pure mathematicians perception of
  mathematics may be different from an engineers
  and the teachers perception of mathematics
  clearly affects the manner in which it is
  presented.

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• Bruner suggests We teach (mathematics) not to
  produce little living libraries on the subject
  but rather to get a student to think
  mathematically for himself. Current thought on
  mathematics teaching and learning suggests that
  the goal is to develop students who are able
  think mathematically rather than learners who
  simply memorize and apply procedures.
• J. S. Bruner, The Process of Education, Harvard
  University Press, 1960.

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• Similarly, Ward says In many ways we have, in
  the last 10 years, been teaching too much
  mathematics and been teaching the wrong
  mathematics. Much of the mathematics taught to
  engineers and scientists is rarely seen or used
  again in their future careers. This state of
  affairs has been driven, to a large extent, by
  the requirements laid down by the engineering
  institutions in order that engineering programs
  are suitably accredited. Dare to say that those
  mathematicians who advise the engineering
  institutions do not have their finger of the
  pulse of modern developments in this area? What
  we wish to focus on is what mathematics we should
  teach to engineers and scientists, and what
  electronic aids we should be using to teach that
  mathematics.
• J. P. Ward, Modern mathematics for engineers and
  scientists, Teaching Mathematics and its
  Applications, vol. 22, pp. 37-44, 2003.

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Make Connections

• Making connections in math is an important
  goal for all students because it underlies all
  other mathematical skills (problem solving,
  communicating, and reasoning). We should help
  students make the following math connections
• With real life.
• Within math (mathematical operations are
  logically connected).
• Between math and other subjects.
• Between conceptual (algorithms and formulas) and
  procedural knowledge (reasons why these formulas
  work). One without the other leads to senseless
  memorizing.
• Between concepts and physical quantities.

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Applications and Modeling

• The term applications and modeling has been
  increasingly used to denote many relationships
  between the real world and mathematics.
• Using mathematics to solve real world problems is
  often called applying mathematics, and a real
  world situation which can be tackled by means of
  mathematics is called an application of
  mathematics.
• The term modeling, on the other hand, is the
  process of representing the behaviour of a real
  system with a mathematical model, or collection
  of mathematical equations.

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• Engaging students in modeling reinforces
  mathematical concepts through their connection to
  real-world applications.
• Mathematical models can help in the understanding
  of practical systems, which is why they are so
  important to engineering.
• Simulation tools are very appropriate for
  expanding the range of options for approaches to
  teaching modeling and applications. They enhance
  the students experience of mathematising
  situations, designing and conducting simulations,
  and engaging in applied problem solving.

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Computer Packages

• Computers are extensions of human minds. They add
  two improvements speed and memory.
• The use of computer packages is now widespread
  and they have become more user-friendly. In some
  of them there is no need to write lines of code
  to sort out mathematical problems!
• For students who use these packages, they can
  carry sophisticated mathematical calculations
  without proper understanding of mathematics. Does
  this matter? For the majority of engineers and
  scientists it does not matter!
• For teachers, how does these packages impact
  teaching of mathematics? We may say that we
  should balance these two matters teach
  mathematical methods as well as using these
  packages effectively.

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Professional Development WorkshopFoundations of
Engineering Concepts

• We suggest a model of teachers teaching each
  other in a professional 2-4 day summer workshop
  that provides high school teachers with a
  foundation of engineering concepts and a means of
  applying mathematics and science.
• The content of the course focuses on the concepts
  and background of four units of study
• Engineering mechanics
• Fluids
• Heat
• Electricity

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Objectives of the Summer PD Workshop

• Provide teachers with a content-rich opportunity
  to learn, practice, and use engineering as a
  vehicle for the integration of math and science.
• Introducing more engineering ideas in the
  classroom so the students will acquire more
  options, accordingly, career doors will open and
  students can make informed choices rather than
  relying on high school guidance councilors or
  teachers.
• Combining engineering concepts with teachers
  understanding of teaching pedagogy to make a
  significant impact on their content knowledge and
  to take engineering back into their classrooms.

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What Mathematics do we Expect in Engineering?

• Rules of algebra proper notation, use of
  brackets, hierarchy of calculation.
• Persuade students to understand the structure of
  equations rather than to memorize the notation.
• Meaning of simple inequalities.
• Functions of a single variable.
• Limits, continuity, gradients, roots, etc.
• Polynomials. Complex roots. No emphasis on curve
  sketching.
• Standard functions exponential, sine, cosine,
  etc.
• Rates of change use the concept of modeling
  motion as a natural derivative.
• Sequences and series.

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• Complex numbers. No emphasis on algebra of
  complex numbers.
• Relationship between derivatives and integrals
  with applications. No emphasis on finding large
  numbers of integrals.
• Differential equations order, linear, nonlinear.
  No emphasis on methods of solving.
• Introduce vectors and their applications to the
  real world.
• Introduce linear equations no emphasis on
  solutions.
• Simple matrix algebra.
• Recognize when problem can be solved analytically
  and when numerically.