Depository of Repetitive Internet-based probLems and Lessons (DRILL)

Vadim Ponomarenko
Department of Mathematics
Trinity University

{Abstract:} 
Modern calculus and pre-calculus education attempts to teach students three
major skills: mechanical computation, modeling, and mathematical maturity.
DRILL is proposed to supplement or substitute the traditional model for the
first of these skills, mechanical computation.  It is a web-based system for
automatically
generating problems, accepting student solutions, grading, and
reporting scores to the instructor.  Its widespread adoption would
revolutionize mathematics education and give unprecedented freedom
to those teaching it.

{Significance:}
Calculus is by far the most important subject taught in mathematics
departments at colleges and universities.  A vast proportion of all college
students take calculus or pre-calculus at some point before
graduation.  This cannot be said of any other subject in the
mathematics curriculum.  For example, the Trinity University
mathematics department in Fall 2000 offers twelve sections of
calculus, and three sections of pre-calculus. All other courses
together comprise twelve sections.  This is typical.  Furthermore,
calculus is a prerequisite to the other courses, so an improvement
to calculus and pre-calculus would be of benefit to all students
taking mathematics during their undergraduate careers.

Calculus and pre-calculus education in the modern era attempts to
train students in a wide variety of skills.  These include three
major goals: achieving competence in mechanical computation and
calculation of formal mathematical exercises, understanding and
practicing the methods by which real-world problems can be modeled
with formal mathematics, and learning the patterns of thought that
allow the construction and comprehension of a mathematical
argument.  Each of these has value in its own right, as well as
value to support understanding of subsequent material learned in
other courses (mathematics courses or otherwise).

The most basic and easily tested goal for students to achieve is
routine mathematical computation.  It involves manipulating
symbols in a systematic fashion to achieve a different, desired,
set of symbols.  Each successive course adds different symbols,
and different methods of manipulation.  For example, the
collection of symbols $3x=6$ can be manipulated by dividing both
sides by $3$ to get $x=2$.

Mathematics as a discipline is useful only when these mechanical
techniques can be applied to solve problems.  Forging that
connection is the second major goal.  Modeling allows students to
solve real problems by recognizing the applicability of
mathematical techniques and applying manipulative skills to
appropriate symbols. For example, the real world difficulty of
evenly dividing six dollars among three people can be resolved by
applying the manipulation in the above example.

Symbolic manipulation is one application of mathematical thinking.
More generally, mathematical thinking involves deductive
reasoning, careful assimilation of evidence, disregard for
irrelevancies, appreciation for existential and universal
qualifiers, and methodical treatment of all cases.  It can be
viewed as a major step toward maturation of naive thinking.  This
notion, sometimes called "mathematical maturity", is the one most
desirable to instill in all college graduates, yet is the most
elusive and difficult to convey.

{Current State of Relevant Research:}
The traditional model for learning calculus and pre-calculus involves some
combination of two forms of problems: take-home and in-class.  The first
includes brief weekly collections such as homework as well as larger
projects.  The second ranges from extremely brief quizzes, through chapter
tests, all the way to midterms and comprehensive finals.  Any of these
problems can address any of the three main goals.

Take-home problems are worked by students asynchronously, with
unknown levels of collaboration.  Solutions can be elaborate, and
may have involved intermediate, unsuccessful efforts not subject
to review.

In-class problems (this includes evening exams) are worked
synchronously. Generally, they are monitored and the level of
collaboration strictly controlled.  Due to time constraints,
solutions are typically somewhat short and generally all efforts
are subject to review.

Review generally takes the form of a numeric or letter grade,
together with written comments.  The primary purpose is to inform
students as to their level of understanding, with the hope that
they will address any deficiencies to achieve mastery.  A
secondary purpose, of course, is to help assign a course grade.

The traditional model provides little incentive for students to
use this feedback mechanism.  This is due entirely to
administrative reasons.  It takes time to generate problems, wait
for students to solve them, then grade and return them.  Even if
the instructor has unlimited resources for this process, one such
cycle can be no faster than several days, and is often a week or
more. By then there is new material to be studied, and the burden
on students to perform concurrent tasks becomes too great.  For
this reason summary exams are given, allowing a second or third
cycle through the same material during the semester.

A second flaw with the traditional model is the balance between
difficulty and collaboration.  Asynchronous work can be made
difficult, but the level of collaboration is impossible to
monitor.  In particular, it is extremely difficult to enforce
expectations on allowed levels of collaboration.  Consequently,
synchronous work is also generally given.  However, due to time
constraints, this work cannot be made very difficult -- it is
unreasonable to administer frequent exams that last four hours.

{Objectives and Methodology:}
The abstract primary objective of DRILL is to provide a new model for
problems, that includes a third source which is neither at-home nor
in-class.  That source will be a computer program, written in the commonly
available language Java and accessible through the World Wide Web. It will
generate an endless supply of problems of any type, addressing any
combination of computational skills.

The problems generated by DRILL will address the goal of
mechanical computation, and only that goal.  This is by design;
objective, computerized grading is only possible of problems of
this type.  Human, professional, evaluation will be necessary to
determine progress toward the other goals.

The major benefit of this new source is that of fast turnaround.
If a student performs poorly, that student is told instantly and
that very minute can study further so as to improve.  DRILL will
always generate new problems.  Therefore, memorizing the correct
pattern of solutions will not help -- the student really must
learn the appropriate manipulative concept in order to succeed.
Further, DRILL will display context-sensitive help, specific to
the error the student made.  This will further assist the student
in improving the deficiency.

The concrete primary objective of DRILL is to teach and assess the goal of
mechanical symbolic manipulation.  The instructor can then focus on
assigning problems that are longer and more in-depth, without the need for
separate testing of routine calculation.  This will allow synchronous
problems of greater difficulty, relieving some of the strain between
difficulty and collaboration.

Another objective of DRILL is to give the instructor more
flexibility in structuring the course.  It will be easy to
configure to provide a variety of paradigms, including:
* Tiny sets of problems, covering a single day's worth of material
* Small sets of problems, covering a week's worth of material
* Moderately sized collections of problems, covering a chapter
* Moderately sized collections of problems, covering all material
cumulatively up to any given point
* Long, exhaustive collections of problems on a single topic
* Long, exhaustive collections of problems on the entire course

A secondary objective of DRILL is to give the instructor more
information about student progress.  All attempts made by all
students will be recorded.  The instructor can easily determine
which problems are causing the most difficulty and which students are
struggling more than others, thereby tailoring lectures
appropriately.

The methodology of DRILL will involve the principal investigator
developing software on machines available at Trinity University.
The work will be completed over the course of summer 2001.
Computer code in the Java programming language will be written to
expand the small existing core of DRILL to a general-purpose
program of widespread utility. This software will be made freely
available via the Trinity University web site.

{Qualifications:}
The principal investigator has written in 1998 a preliminary
version of the core Student Package, and has maintained it since.
It can be presently found at
http://www.trinity.edu/vadim/algebratrig.html
This package has been used successfully in several courses taught
by the principal investigator at the University of Wisconsin and
Trinity University.

{Anticipated Output:} 
The expected software will include several packages.  These packages will be
freely available on a Trinity University World Wide Web page.  They will be
accompanied with simple instructions for their use, together with a detailed
manual to answer technical questions.

* Student Package - this generates problems, displays them,
accepts a solution, and provides an immediate grade
* Instructor Package - this monitors a collection of students,
their attempts and successes, and allows configuration of the
Student Package interface
* Submitter Package - this interface allows instructors to
submit additional questions which will be reviewed and added to the
depository

Once written and successful, the software will be advertised
through a variety of sources. Some electronic sources of
dissemination will include Usenet and advertising email to be sent
chairs of departments of mathematics of the ACS.  Other sources of
dissemination will include talks given by the principal
investigator at professional mathematics conferences, and articles
to be published in journals related to mathematics pedagogy, such
as The American Mathematics Monthly, The Journal for Research in
Mathematics Education, and Mathematics Teacher.

{Evaluation:} 
Once implemented, the software will immediately be subject to both an
internal and external evaluation process.  The internal evaluation will
consist of using the software as a tool in teaching calculus.  Student and
faculty comments will be collected and compiled.  The external evaluation
process will consist of polling those that visit the web page to use the
software.  Again, both student and faculty comments will be solicited and
compiled.

There are therefore three measures of success: whether the
software was useful and appropriate when implemented by an expert
(the author), whether the software was useful and appropriate when
implemented by a nonexpert (outside faculty), and whether the
advertising of the software was successful in persuading others to
use it.