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Demos
and Projects that Make a Difference
INTRODUCTION
A
variety of sources acknowledge that there is currently a shift from an
instructor centered classroom toward one in which students are more
active participants in the teaching and learning process [1]-[4]. This
situation indicates that instructors need to establish an environment
that encourages active participation by students and find additional
learning tools that work well in this new environment.
An instructor may need to modify content delivery by using
techniques that address a variety of learning styles.
The modifications can be in the form of the instructor’s
actions or role in the classroom and in the form of independent
assignments that foster experiences in the scientific method. In
this article we look at two aspects of mathematics instruction.
First, we examine the instructor’s role in the classroom and
illustrate some effective instructional demonstrations.
Demos are intended to engage the learner’s attention and
encourage understanding of mathematical concepts beyond the
instructor’s oral presentation. Second,
we present some ideas for student-centered projects that encourage the
student to interact with the real world through projects.
These projects are designed to develop an appreciation of the
importance of mathematics as a tool for problem solving. INSTRUCTOR
AS FACILITATOR: EFFECTIVE
CLASSROOM DEMOS
As
the traditional learning scenario of an instructor centered approach
gives way to a more student centered atmosphere for learning, the role
of the instructor can be greatly improved by carefully planning what and
how we present in our mathematics classrooms. We must maintain a careful
balance on the type of communications we use in the classroom.
Communication via oral expression complemented with student
involvement and visual tools requires close coordination in order to
achieve a high level of effectiveness for the variety of learners we
teach. The verbal component
of instruction is crucial to expose students to the aspects of verbal
training requiring the ability to absorb, filter, and process
information involving words, numbers, symbols, and concepts.
These lecture oriented learner skills require that students
concentrate and remain focused as we perform our delivery functions.
As experienced instructors know, the student can much too easily
veer off the track of the ideas we are trying to present. To
keep students “with us” and help to maintain their focus, teachers
use a multitude of devices. As
technology has changed so have the teachers’ bag of tricks.
The tricks-of-the-trade now include physical manipulatives,
calculators, data banks, and computer driven tools, which are often
visually oriented. All of
these instructional devices are demonstrations designed to involve
students on learning levels complementing our oral presentations.
Since our time with our students is so precious and seemingly
quite short, the choice of demonstrations we use in class is extremely
important. The
use of demos in class can provide students with a situational or visual
model of a concept. Hence demos often set a mental reminder for a
concept that students recall later and provide teachers with an
opportunity to build connections between topics. Demos can be used for a
variety of purposes, including, introduction of a topic, clarification
of a point, emphasis of an idea, to add variety, to change focus, and to
highlight important points. Experienced
mathematics teachers have over time, developed, refined, and sifted
through collections of instructional demos, creating “teaching kits”
containing demos they have found to be effective. Communication of these
personal toolkits to others in the profession has the potential to
greatly improve mathematics instruction and benefit our students.
The NSF-funded project, Demos with Positive Impact, is directed toward establishing a
web-based collection of such effective instructional demos. The project Demos with Positive Impact has as its goals:
The
URL for Demos with Positive Impact is Have
your students ever asked, ‘Where do parabolas come from?’ We could
answer this by cutting a cone with an appropriate plane, but the demo Parabola!
takes a different approach and provides a dramatic example of how
parabolas arise as graphs of a natural phenomenon. A ball is given a
push up an inclined plane, and its progress up and then down is recorded
by a motion sensor. The time vs. distance graph appears on a computer
monitor. The graph is a parabola. This
demo has been used as a quickie with students studying quadratic
functions. It makes quite an impression on some of them:
one student said that this demo shows that parabolas are real. When
we introduce infinite series, it is often a painful experience for
instructors and students alike. As mathematics teachers we may recall
that, a "lets imagine that … " scenario and formal
definitions worked just fine for us when we were students, but, as we
know, this doesn't apply to all students. Students that are in
engineering and other applied fields often work better physically and
visually than they do abstractly. The
demo A "Sweet" Introduction
to Infinite Series apportions a donut to a group of students in
such a way that it holds the attention of a class and accurately
introduces the abstract ideas of the terms of a series, the partial sums
of a series, and convergence or divergence of a series. In fact, in one
50-minute lecture/demonstration/feast we get all that plus alternating
series and the divergence of the Harmonic series. By
the time our calculus courses get to the topic of partial derivatives,
students are able to differentiate many functions of a single variable
with few major difficulties. Then
we spring functions of several variables on them and effectively tell
them to use the differentiation techniques they already know
“pretending” that all the unknowns,
except the one with respect to
which they differentiate,
play the role of constants. Many
students successfully buy into this algebraic mechanism but very few
have a clue as to what is going on geometrically.
The demo Partial
Derivatives Geometrically provides a visual foundation for
partial derivatives of functions of two variables, z
= f(x,y).
The demo sketches a surface, cuts it with a plane perpendicular
to an axis to regard one unknown as a constant, generates the curve of
intersection, and animates a tangent line moving along the curve while
displaying the values of its slope. This demo provides a lecture tool
that has been used successfully in large and small classes for both
math/science and non-science majors. It is simple to use and lets the
instructor narrate the actions as they evolve on the screen.
It provides a visual enhancement to the usual algebraic
statements about partial derivatives. STUDENT
AS ACTIVE LEARNER: STUDENT-CENTERED
INVESTIGATIONS
In
general, students do not understand the purpose of many activities they
are asked to perform and make no connection between these activities and
the instruction that they have received [5].
They do not link course topics, concepts, methods, or problems
they analyze with other disciplines or with their previous knowledge.
Projects can help students
make these associations as they apply course material in more
realistic settings. These
applications are usually more difficult and require more analysis than
standard homework problems. Higher
order thinking is required, especially when problems are open-ended in
nature. Exploring how ideas fit together and can be applied in a
variety of situations helps students to relate to the material they
learn and helps them to “take ownership” of course concepts.
Group or individual projects enable students to put course
material into context. This
provides an opportunity to integrate current concepts with
previous experiences from other courses, disciplines, or real life. Projects
provide students with an opportunity to clarify thinking through
discussions of and writing about course concepts and methods, to test
their ideas against those of other students, to appreciate new
perspectives, and to practice communication skills [6].
Discussions arising from project
analysis provide students with important
feedback regarding their understanding of course material.
Students get an opportunity to question, explain, express
opinions, admit confusion, and reveal misconceptions.
Simultaneously, they listen, respond to questions, and share
information to clear up confusion [4].
These activities help to emphasize to students that they are
central to the learning process. Another
valuable lesson is responsibility:
the group cannot succeed unless each member of the group
contributes. Writing a report or paper as
part of a project provides an opportunity for students to test their
understanding of concepts, symbols, and techniques through explanations
given in their own words. They
must clarify the problem and define all variables, notations, and
terminology used. In addition, they must communicate their approach and
elucidate the significance of solutions. These activities require
students to actively organize their own understanding of course concepts
and ideas in order to explain and solve the project “problem”. The
writing component of the project can be serious or creative in nature.
Rather than a workplace style report, a project paper might be in
the form of a story or letter in which the problem explanation,
analysis, and solution are presented. For example, presenting students with a narrative containing
a veiled optimization problem can be useful in helping students to
decode and solve problems presented within more realistic scenarios.
A written response such as a reply to the letter or a
continuation of the story can use characters, fictitious or real, to
carry out the discourse that provides the development of the project and
leads to the culmination of the assignment. While
providing a valuable learning experience for students, projects can keep
a course fresh and provide a challenge to the instructor.
Projects help to remove routine aspects from a course, energize
discussions, and motivate students to assume a more active role in learning.
The activities within a project can
enable the instructor to share interests or relevant developments in
the discipline. Projects
give an excellent opportunity to use and to teach students about
technology, such as the
graphing calculator, World Wide Web, or various computer programs like
Microsoft Excel, MATLAB, Derive, Maple, or Mathematica.
These technologies are useful for function and graph exploration
projects. Precalculus
students can explore and compare the graphs and equations for constant,
linear, and quadratic functions that result from taking different values
of coefficients for the function
While
a graph analysis project is an example of a project designed to help
students to make connections between concepts and associated graphical
and symbolic forms, a
Find-Your-Own-Data Project is a project designed to provide students
with an opportunity to apply techniques and analysis that they have
learned in a simulated workplace. One
scenario for a Find-Your-Own-Data
Project is an activity where students select a real data set and
analyze the data as if they were working for a company or for the
government. Students explore the World Wide Web as a source of
information and real data: they
can obtain current data for companies and government as well as access
newspapers, magazines, and almanacs.
In addition to performing any necessary background research
relevant to their data set, students analyze their data to determine
appropriate model type(s) and equation(s), provide realistic analysis of
the significance of trends displayed in the data, and discuss the
relevance of interpolation and extrapolation of the data. Students
can perform the data analysis and determine the model equation(s) using
both the graphing calculator and a spreadsheet program like Microsoft
Excel; this technology can enable students to generate graphs that
illustrate their analysis of the data for the project report. SUMMARY Demos and projects are
instructional tools that are designed to reach out to students and
involve them in a variety of ways.
Demos provide an immediate form of involvement by presenting a
path for “buying into” a concept or idea.
Students are provided with a type of physical link to the verbal
or symbolic expressions involved. Projects,
requiring more time, are designed so that students must backtrack,
assess, and solidify their perceptions about material.
Both demos and projects are opportunities for students to acquire
a better grasp of salient features of a course.
They are also opportunities for instructors to have a positive
impact on the learning process. Acknowledgement:
Partial support for this work was
provided by the National Science Foundation's Course, Curriculum and
Laboratory Improvement Program under grant DUE-9952306. REFERENCES
1.
Michael Yerushalmy and Juday Schwartz, ‘A procedural approach
to explorations in calculus’, Int.
J. Math. Educ. Sci. Technol.(30), No. 6, 1999, 903-914. 2.
Michael P. Lambert, ‘Context versus Content: fresh look at the
ongoing debate over place-based vs. distance education,’ Syllabus (13), No. 10, June 2000, 24. 3.
William H. Graves, ‘The Dot.xxx Challenge to Higher
Education,’ Syllabus
(13), No. 10, June 2000, 30-36. 4.
William J. McKeachie, Teaching
Tips: Strategies, Research, and Theory for College and University
Teachers, Ninth Edition, D.C. Heath, Lexington, Massachusetts, 1994. 5. Jon Saphier and Robert Gower, The Skillful Teacher: Building Your Teaching Skills, Research for Better Teaching, Acton, Massachusetts, 1997. 6.
Chet Meyers and Thomas B. Jones,
Promoting Active Learning: Strategies
for the College Classroom, Jossey-Bass, San Francisco, California,
1993.
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