Algebra: Gateway to a Technological Future
In early November of 2006, the Mathematical Association of America, in
a project funded by the National Science Foundation, brought together
representatives from the mathematics and mathematics education
communities across the entire K-16 spectrum to survey what has been
learned about the teaching of algebra and to identify common principles
that can serve as models for improvement. The approximately 50
participants were divided into five groups, corresponding to five
different levels of algebra instruction. The groups were (1)
Early Algebra, (2) Introductory Algebra, (3) Intermediate Algebra, (4)
College Algebra, and (5) Algebra for Prospective Teachers. Each
group reviewed research on what was known about the teaching of algebra
on that level and made suggestions for future directions that would
improve both the knowledge base and the actual teaching and learning of
final report from this
conference is available in pdf format.
The following is a summary of the findings and recommendations of each
the five groups.
Algebra: Gateway to a Technological
Future was funded by the National Science Foundation Division of
Elementary, Secondary and Informal Education Grant ESI-0636175.
It is now widely understood that preparing elementary students for the
increasingly complex mathematics of this century requires an approach
different from the traditional methods of teaching arithmetic in the
early grades, specifically, an approach that cultivates habits of mind
that attend to the deeper, underlying structure of mathematics and that
embeds this way of thinking longitudinally in students' school
experiences, beginning with the elementary grades. This approach
to elementary grades mathematics has come to be known as early
algebra. There is general agreement that early algebra comprises
two central features: (1) generalizing, or identifying, expressing and
justifying mathematical structure, properties, and relationships and
(2) reasoning and actions based on the forms of generalizations.
When early algebra is treated as an organizing principle of elementary
grades mathematics, the potential payoffs are tremendous: (1) It
addresses the five competencies needed for children's mathematical
proficiency: conceptual understanding, procedural fluency,
strategic competence, adaptive reasoning, and productive disposition.
(2) It creates children who understand more advanced mathematics in
preparation for concepts taught in secondary grades. (3) It
democratizes access to mathematical ideas so that more students
understand more mathematics and, thus, have increased opportunity for
There is already much research telling us what mathematics young
children are capable of learning in the early grades. Yet there
is still much to learn. Thus, we recommend three critical areas
for future research in early algebra:
- The development of "Early Algebra Schools"
schools that integrate a connected approach to early algebra across all
grades K-5 and provide all teachers with the essential forms of
professional development for implementing early algebra. The
systemic change implied by these schools should involve not only
elementary teachers, but also middle school teachers, principals,
administrators, education officials, math coaches, parents, and even
- Developing a coherent, connected early algebra content.
Although much is known about children's learning of basic ideas at
certain grade levels, there is a need to develop a more coherent
picture of early algebra throughout grades K-5 and connect this with
what follows in the higher grades.
- Understanding the pervasive nature of children's algebraic
thinking. Research has provided us with "existence
proofs" of the
kinds of algebraic thinking of which children are capable. But we
still need an understanding of how pervasive this knowledge can become.
One central problem in the teaching of introductory algebra is that
there are too many topics. Thus, the attempt to cover them all
impedes student learning of core concepts in depth. This problem
is exacerbated by the lack of logical connections between core concepts
and procedures. In addition, it is apparent that the transition
from using numbers to using symbols is much more difficult for many
students than has been assumed.
Basing our recommendations on these major problems and several others
as well, we suggest six major research directions for the future.
- Identify core concepts and procedures that should form the
content of introductory algebra. Through a number of conferences,
NSF should build a consensus on the content of the core of
algebra. Organizing introductory algebra around a core should
make the study more coherent. Of course, teachers themselves must
recognize the centrality of the important ideas, and must keep asking
students "why" so that the students become aware that
most ideas arise
in a number of contexts.
- Investigate the transition to symbolization and how teachers can
effectively facilitate it. Relatively little work has been done
investigating what occurs in the transition from work with numbers to
work with symbols. But this understanding is important in
designing ways that teachers could effectively facilitate students'
transitions. In addition, we need to know whether the process of
transition to symbolization differs in adults and in children.
- Investigate models that promote learning for students with
different needs, preparation, and backgrounds in the same
classroom. New pedagogical methods including community building,
group work, and inquiry learning can help all students, but we do not
know the best balance between such methods and more traditional ones
such as direct instruction or individual work. Projects should
study the use of these methods in different ways and pay particular
attention to their use in diverse classrooms.
- Prepare and sustain teachers in implementing good instructional
practices and curricular materials. There is a dearth of
curricular materials for professional development of new and practicing
teachers, especially materials which enable teachers to transfer their
own learning into new teaching practices. Investigation of the
particular content that would best support algebra learning is
- Identify systemic changes needed to support teacher growth.
Teachers need more structured time during the school day for
collaboration and growth. For school districts to fund such
expensive time, they need strong evidence that it will pay off and that
there are no cheaper alternatives. We therefore need to
investigate a variety of models that try alternative approaches to
providing such structured time and document the changes they produce.
- Determine what use of technology is appropriate in the
introductory algebra classroom. Research has show that graphing
calculators can enhance learning and computers can provide useful
practice. We need to compile evidence of what actually happens
when these are used, including what students learn with calculators
that they do not learn without them and what they fail to learn when
they use calculators that they learn without their use.
Intermediate algebra is generally designed as part of the
college-intending tract, and thus includes topics thought necessary for
students' later success in freshman college courses, primarily
calculus. Since the content of this course varies widely, our
focus was on describing the mathematical ways of thinking that under
gird algebra. We also propose an action plan to identify and
implement the best approach to preparing students for further study of
- Identify mathematical ways of thinking that are central to
algebraic reasoning. Weak coherence of algebra curricula results
in part from the absence of a central core of the subject. We
therefore propose a research and development program involving
collaboration between education researchers, mathematicians, and
teachers that will focus on mathematical ways of thinking as a way of
providing a common set of principles to guide the development of
curricula. Included among "mathematical ways of
thinking" are the
Build capacity to focus algebra instruction on mathematical ways
of thinking. We envision an evolutionary approach seeded by
smart, strategic moves that, after an initial phase of testing and
refinement, become self-disseminating and self-replicating. Some
of these moves could be
- The relation between form and function.
- Solving equations as a process of reasoning.
- Imagining a quantity's value varying continuously and imagining
invariant relationships between co varying quantities.
- The habit of finding algebraic representations even in a
problem that does not necessarily look mathematical.
- Anticipating the results of a calculation without doing it.
- Abstracting regularity from repeated calculation.
- Connections between representations, such as tables, graphs,
Advise policy makers on barriers and avenues to successful
implementation of sound instructional practices. Because
capacity-building is wasted if new ways of teaching run into
institutional or systemic barriers, we need to do determine those
institutional structures and policies that prevent the flow of
innovation as well as those that channel it in the right
direction. Thus mathematical researchers and practitioners must
collaborate with policy makers, administrators, parents, textbook and
testing companies, and the wider business community, to learn from each
other their respective concerns, to create a sense of shared
responsibility, and to encourage creative informed decision making.
- Equipping instructors with a framework for putting to new uses
the materials they have, especially in supporting mathematical ways of
- Establishing collaborative communities of mathematicians,
mathematics educators, and teachers that share a focus on student
learning of significant mathematical ideas.
- Finding ways of opening discussions about pedagogical
techniques among two and four year college faculty
- Encouraging all college faculty to understand that their course
is a potential teacher preparation course
- Approaching teacher professional development programs with a
focus on deepening mathematical content knowledge, encouraging
mathematical ways of thinking, and examining why students are not
Extensive studies over the past several years by the MAA and AMATYC
have shown that, to a large extent, the College Algebra courses taken
annually by some 700,000 college students are not successful.
Thus, the curriculum committees of both AMATYC and MAA have called for
replacing the current college algebra course with one in which students
address problems represented as real world situations by creating and
interpreting mathematical models. We therefore recommend several
programs designed to change the nature of college algebra courses
throughout the country and greatly improve the student success rate.
- A large scale program to enable institutions to refocus college
algebra. This would help a large number of institutions implement
the new guidelines, beginning with faculty development.
- Research on impact of refocused college algebra on student
learning. In connection with large scale implementation, there
needs to be a few extended longitudinal studies of student learning in
the refocused college algebra courses.
- Electronic library of exemplary college algebra resources.
This would provide classroom activities, extended projects, and videos
of lessons that would help instructors implement new ideas for student
- Establishment of national resource database on college
algebra. This resource would include information on funded
projects, textbooks, research articles, etc. that could help widely
disseminate positive results from exemplary algebra programs.
Algebra for Prospective Teachers
To improve the content and pedagogical knowledge of algebra teachers in
middle and high school, and thus to improve the achievement of their
students, we need answers to some basic research questions.
Given that answers to all of the above questions require collaborative
efforts among mathematicians, mathematics educators, and classroom
teachers, we also need to identify strategies that successfully nurture
- What is the role of teachers' algebraic knowledge for teaching as
it shapes their instructional practice? This needs to come from
observational studies of algebra teachers in action in a wide variety
of settings. Among other questions, we need to understand how
teachers respond to students' algebraic thinking as it occurs.
- How does the content and design of the abstract algebra course
typically taken by future teachers of algebra affect their later
teaching of school algebra? New types of abstract algebra courses
for teachers have been developed in recent years, but there is a need
for solid research studies on their effect.
- How does professional development in algebra content and pedagogy
affect teachers' classroom practices? This research must
begin with a careful analysis of the important algebraic concepts that
should inform teachers' understanding of the mathematics they are
teaching. Furthermore, we need to learn how teachers'
understanding of algebra and its teaching develops from the use of
different kinds of instructional materials.
Lead editor for the project report: Victor
Katz, University of the District of Columbia (emeritus), email@example.com
Project Director at MAA:
Michael Pearson, Director of Programs and Services, firstname.lastname@example.org
Facilitators of the five working
Early Algebra: Maria
Blanton, University of Massachusetts, Dartmouth, email@example.com
Diane Resek, San Francisco State University, firstname.lastname@example.org
Intermediate Algebra: William
McCallum, University of Arizona, email@example.com
College Algebra: William
Haver, Virginia Commonwealth University, firstname.lastname@example.org
Algebra for Teachers:
Jim Fey, University of Maryland, email@example.com