Finding Common Ground, Indianapolis, March 2006
Algebra Group Report
Working Group
David Bressoud, Macalester College
Cindy Bryant, Missouri Math Academy
John Carter, Adlai E. Stevenson High School, IL
Susan Forman, Bronx Community College
Ira Papick, University of Missouri
Alan Tucker, SUNY Stony Brook
Hung-Hsi Wu, University of California-Berkeley
Preface
The universal value of algebra lies in its use as a tool for problem
solving, strengthening reasoning skills, and interpreting functional
relationships. A good foundation in algebra is a prerequisite to
success in the study of mathematics, science, engineering, and the
social sciences. Mathematics becomes more meaningful to students when
they see the connection to their lives, prior learning experiences, or
future learning.
Mathematics Learning Prior to Algebra
It is important that elementary and middle school students be
appropriately prepared to take Algebra I in Grade 8 or Grade 9.
Experience with symbols and generalizations should increase in
the years preceding Algebra I. (There was disagreement as to when
students should take Algebra I.)
Elementary and Middle School Mathematics
- Reinforce the understanding of whole number arithmetic through
mathematical explanations of algorithms, i.e., use of the commutative,
associative, and distributive properties.
- Gradually incorporate the use of generality and symbols. One
natural way to do this is in the development of rational numbers and
operations on them.
- Introduce linear relations, linear functions, and basic graphing
techniques.
- Provide students with problems that exemplify the power of
generalization. For example, 18 is divisible by 3 and 27 is divisible
by 3 is (18 + 27) divisible by 3 Evaluate (43 x 63) + (57 x 63).
- Use well-structured patterns to help build students’ capacity for
generalization. Students should be able to explain the process they
used to arrive at a generalization. (In process.)
Algebra I
Algebra begins as generalized arithmetic. In algebra classes, students
learn to handle the concept of generality and to use symbols to
translate verbal information into symbolic language. In our search for
common ground, there was agreement that Algebra I should contain:
solving linear equations, linear functions and their graphs, systems of
linear equations, quadratic equations and their graphs, solving general
quadratic equations, operations on polynomials, and an introduction to
exponential growth. Other topics include rate of change and linear
inequalities.
Algebra II
Students should complete the equivalent of Algebra I and Algebra II by
the end of 12th grade. There was agreement that Algebra II primarily
comprises an extension of topics in Algebra I, complex numbers,
in-depth study of functions including composition and inverses,
rational functions and the asymptotes of their graphs, radical
functions, logarithmic and exponential functions and their graphs, and
polynomial functions.
Alternate Algebra II
The group considered an alternative to the Algebra II course described
above. Suggested topics for inclusion in this alternative course are
in-depth study of functions, logarithmic and exponential functions,
linear programming, data analysis and modeling, and topics in discrete
mathematics such as recurrence relations, permutations and
combinations, and matrices.
Finally…
Due to the fact that algebra requires students to think at a higher
level of abstraction and generality, due attention should be paid to
precise definitions, logical reasoning, and mathematical closure.