1. 1 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, Canada. rhabash@site.uottawa.ca

2. 2 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.

3. 3 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.

4. 4 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.

5. 5 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.

6. 6 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.

7. 7 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.

8. 8 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 mathematician’s perception of mathematics may be different from an engineer’s and the teachers’ perception of mathematics clearly affects the manner in which it is presented.

9. 9 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.

10. 10 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.

11. 11 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.

12. 12 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.

13. 13 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.

14. 14 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.

15. 15 Professional Development Workshop Foundations 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

16. 16 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 teacher’s understanding of teaching pedagogy to make a significant impact on their content knowledge and to take engineering back into their classrooms.

17. 17 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.

18. 18 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.