*Proceedings of the Department of Education Algebra Initiative
Colloquium*, December 9--12, 1993, C. B. Lacampagne, W. Blair,
J. Kaput (ed.), U. S. Department of Education, 1995, 159-160

For all types of students, the first exposure to algebra is the
beginning of a transition from numbers to letters, from concrete
to abstract language. The importance of learning algebra as a
language rather than as a set of memorized rituals will not be
apparent until students recognize its power as a problem solving
tool. Important questions of "Why?" and "What if?"
can be posed and answered in this new language, a key feature
too often concealed from students.

A core set of algebra skills can be defined by identifying the
kinds of problems the target audience needs to solve. Besides
defining the core, those problems must motivate the study of algebra.
Indeed, one could argue that no concept of algebra should be taught
unless it can be motivated by a problem that is likely to be part
of the students' experience in the near future.

Unfortunately, the mathematics itself may be hiding in a spread
sheet, for example. Or its real value may appear in the need for
careful quantitative reasoning rather than in explicit manipulation
of symbols. Ideally, students can be motivated both by the need
to acquire quantitative thinking skills and by the demands of
specific technical challenges. In any case, the problems of uncovering
mathematics at use in the work place and conveying those experiences
to students are formidable challenges that must be overcome. (The
beginnings of one attempt at the graduate level in mathematics
are described in [1].)

More simply, one can identify some specific skills students need
at the pre-college level through widely available licensing examinations
for many trades and professions. Students who aspire to become
plumbers, electricians, or health care professionals, for example,
will be asked to find areas of plane figures and volumes of common
solids on their licensing examinations. They will have to manipulate
relationships involving rates and slopes to pass those exams and
to cope with their daily work.

Identifying these explicit challenges and conveying them to students
can provide motivation of a very concrete sort. Of course, an
algebra curriculum needs to go far beyond the need for formulas
if it aims to develop flexible quantitative thinking skills.

Can algebra be presented as the key that unlocks those doors rather
than the bar that blocks them? Can algebra become the language
of the relationships among dimensions, areas, and volumes? Among
rise, run, slope, and rate? Can it become the language of success,
opportunity, and access?

Mastery of the language of algebra also lays a foundation for
mathematical maturity. Familiarity with mathematical ideas enables
an auto mechanic to manage comfortably the business affairs of
an independent garage, to make informed decisions about loans
and equipment depreciation. Good algebra skills set the stage
for a trained secretary to undertake basic accounting and advance
to a position as manager of a small business. Those same skills
enhance the computer literacy of an electronic technician with
the ability to implement spreadsheet calculations that speed and
record a new procedure for testing equipment.

Necessity builds ownership, and genuine applications are evidence of the necessity of algebra in the professional life of the technical workforce. Students will take ownership if their algebra courses incorporate authentic applications like those they will encounter in their professions. The challenge is identifying those applications and using them as instructional vehicles.

1. P. W. Davis, Some views of mathematics in industry through focus groups, SIAM Mathematics in Industry Project, Report 1, Society for Industrial and Applied Mathematics, Philadelphia, PA (1-215-382-9800) 1991

Available electronically as siamrpt.dvi by anonymous ftp from /pub/forum on ae.siam.org, from the SIAM Gopher server at gopher.siam.org, or by mail from the author. Alternate versions have appeared in:

SIAM News26(13): Mathematicians in Industry: Credentials and Skills (January 1993), page 16; Industrial Problems Sources and Solutions (March 1993), page 10; Industrial Mathematics, The Working Environment (May 1993), page 16

Notices of the American Mathematical Society,40(7) Some Glimpses of Mathematics in Industry, (September 1993), pages 800-802.