## Abstract
Students often have
difficulty understanding basic engineering procedures because of the gulf of
mathematical analysis that lies between the definition of a problem and the
implementation of its solution. We
describe a new Web-based introduction to engineering problem solving intended
to address this difficulty. Using Mathcad as a computational engine, analytic
details in the course are selectively de-emphasized, directing student
attention to the essentials of problem formulation and interpretation. The
approach integrates Mathcad's many pedagogical advantages with the
accessibilty, flexibility, and richness of the Web. ## I.
Introduction: The MathSoft Learning Site
The ubiquity of the World-Wide Web has brought both
new opportunities and new concerns to the teaching of the undergraduate
engineering curriculum. Indeed, the Web has forced a re-examination of the
curriculum itself. Using the Web has some obvious advantages—in terms of
availability, inter-connectivity, and access to rich sources of data and
reference information. On the other hand, spurious and distracting content, and
the difficulties of integrating open-ended instructional options into
established classroom practice, make effective use of the Web a continuing
challenge. Teachers are looking for workable strategies. With this in mind, MathSoft, the makers of Mathcad
software, have launched a Web-based educational resource center called The
Learning Site. It is located at: The goal of The Learning Site is to provide
teachers of engineering, science, and mathematics with reliable, ready-made
solutions to the problems of teaching with technology. The site offers online
training in the use of Mathcad, as well as integrated collections of learning
materials that make use of Mathcad and the Web to deliver their content through
engaging, technologically-informed presentations.
Figure 1. A Web page from the
MathSoft Learning Site. Mathcad itself is well known to engineers as an
easy-to-manipulate tool for everyday problem solving. Mathcad’s user interface
allows regions of text, equations, numerics, and graphics, as well as regions
connected to external applications, to be arranged freely on the page. The
inter-connectivity of the various regions is also freely arranged, resulting in
a creative computing environment that can be shaped to reflect thought
processes, metaphors of knowledge construction, and organizational goals. Mathcad’s
ability to hide or bypass selected intermediate computations provides a natural
way to highlight important cause-and-effect relationships. These features make
Mathcad ideal for engineering design work, but practiced teachers will
recognize the pedagogical value of a computing environment that can be adapted
to individual learning styles and focus student attention on essential
information. Used in a Web-based setting, Mathcad can also be used to direct
external resources into rich, integrated learning experiences. ## II. Engineering Problem Solving
Almost all engineering problems involve, among
other steps: 1)
Applying a
theoretical model. 2)
Solving the resulting
equations for the unknown or unknowns. 3)
Interpreting the
results. The intent of most of the textbooks used in engineering courses is to
focus on items 1 and 3. (See, for
example, Fox & McDonald A surprising amount of any given textbook, however, is actually
devoted to item 2. Students reading a fluid mechanics book to learn, say, how
to solve for the flow in a system of pipes, are often forced to devote a large
part of their mental resources to, e.g., solving non-linear differential
equations. All too often, the detailed mathematical analysis obscures the
problem’s essential physical mechanics, interrupting important cognitive connections
between theory and results. Worse, difficult symbolic and numeric computations
discourage students from pursuing the “what-ifs”’ that are the basis of
independent thought experiments and engineering design. This imbalanced
emphasis is especially marked in many textbook exercises. Working with Mathcad can redress these tendencies. The course introduces Mathcad as a modeling environment, and students
are taken through four separate modeling scenarios: - Determining the flow from a water tower through a pipe system.
- Analyzing the design characteristics of a building's modern
"tent" roof.
- Modeling vibrations in a three-mass system with springs.
- Calculating the deflections in a loaded cantilever beam.
Each case study includes a careful formulation, a
Mathcad implementation, and a discussion of the engineering implications.
Necessary Mathcad features are reviewed in context. Links to relevant Web
resources are provided as needed. The following example contrasts a traditional
problem solving approach with the pedagogy employed in ## Example
The traditional approach
to solving this problem is to first set up the energy equation:
Known quantities here are L
and _{elbow}K. Unknowns are the velocity _{ent}V and the friction
factor f, which depends non-linearly on V via the Reynolds number
Re = rVD/m:
Students often pause at this point because they
have to solve a coupled set of non-linear equations, (1) and (2), to solve for
the velocity In the traditional approach In
Figure 3. A Mathcad worksheet
focusing on formulation and interpretation. Once the model has
been formulated, Mathcad solves the mathematical problem quickly. Students may
then focus on obtaining real-time feedback related to design conjectures. After
completion of the course, students will have been exposed to several classic
engineering problems, and have had the opportunity to perform many “thought
experiments” to test their knowledge and understanding of engineering science.
In doing so, they may overcome their hesitancy about tackling real-world
problems that typically involve complicated calculations. Fundamentally, the course is designed to teach students how to use
Mathcad to solve problems by ü
Students learn how to
effectively incorporate technology into their problem solving. ü
Students are
encouraged to explore the “what-ifs” that make general principles real. ## III. Teaching
with Web-based Learning Materials
Within the Learning Site’s online environment, students read course
material, write and save notes, participate in lesson discussions, work through
activities and interactive exercises, and evaluate their progress with graded
on-line tests. Hints, questions, reviews, links to external information, and
the exploratory nature of the Mathcad worksheets themselves create an arena in which
students can freely experiment with their understanding of the material. Each
lesson offers guidance along a central thread, but there is sufficient
supplementary content to let every student find their level, their pace, and
points of reference that will relate to their own interests. They can do all of
this accordingly to their own schedule, location, and interest level. Many
learning styles are accommodated by the non-linear structure and the richness
of the integrated environment. Most importantly, however, Learning Site courses free teachers to do
what they do best: Teach! Learning Site materials are essentially modular. This
means that teachers and students are never forced into a program of instruction
that does not suit their needs. They may pick and choose among lessons within a
course, subtopics within a lesson, and individual worksheets within a subtopic
without breaking any pre-programmed notion of how to proceed. Teachers pick and
choose materials as they need them, and shape classroom time accordingly. While modular, Learning Site materials are
nevertheless fully integrated. This means that however a student or teacher
chooses to use individual pieces, they can count on different materials
speaking to one another in a unified voice. Students presented with material
from the course in a classroom setting, or those proceeding through the
materials on their own, learn to appreciate the connections within the subject
while looking at key concepts in many ways. Teachers give Learning Site materials appropriate context and meaning,
without having to create the basic building blocks on their own. The materials
free teachers to give students individual attention as they work through the
mechanical aspects of a subject and consider the feedback that they are
receiving. In this classroom model, students learn not only their subject, but
the confidence that comes with mastering the technical tools necessary for
future success. CALL FOR AUTHORS: Bibliography 1. Fox & McDonald, 2. Beer & Johnston, 3. White, WILLIAM MUELLER is Mathematics
& Education Specialist at MathSoft, Inc. in Cambridge, MA. He received his
undergraduate degree at MIT and his PhD at Duke, both in mathematics. He has
been involved in national efforts to develop technology-based instruction for
many years. Further information is available at
http://www.wmueller.com.
He can be contacted at
wmueller@alum.mit.edu
.
Write for a review account on any Learning Site course. PHILIP J. PRITCHARD is an Associate Professor in Mechanical Engineering at Manhattan College, where he is Director of Freshman Engineering. He received his B. Tech. at the University of Bradford (UK), an M.S. at SUNY Stony Brook, and an M. Phil. and Ph.D. at Columbia. His main interests are now in the area of engineering education. In recent years he has specialized in the use of computer software in teaching engineering analysis, in particular the use of Mathcad and Excel. Further information is available at http://www.engineering.manhattan.edu/mechanical/faculty/pritchar.html. He can be contacted at ppritcha@manhattan.edu. |