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 algebra. 

Algebra: Gateway to a Technological Future The final report from this conference is available in pdf format.

The following is a summary of the findings and recommendations of each of 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.

Early Algebra

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 lifelong success.

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:

  1. The development of "Early Algebra Schools" These are 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 university personnel.  
  2. 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.
  3. 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.

Introductory Algebra

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.

  1. 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.
  2. 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.
  3. 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.
  4. 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 particularly needed.
  5. 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.
  6. 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

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 college mathematics

  1. 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 following:
  2. 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
  3. 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.

College Algebra

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.

  1. 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.
  2. 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.
  3. 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 learning.
  4. 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.

  1. 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.
  2. 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.  
  3. 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.
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 such collaboration.

Contact Information

Lead editor for the project report: Victor Katz, University of the District of Columbia (emeritus), vkatz@udc.edu

Project Director at MAA: Michael Pearson, Director of Programs and Services, pearson@maa.org

Facilitators of the five working groups:

Early Algebra:  Maria Blanton, University of Massachusetts, Dartmouth, mblanton@umassd.edu

Introductory Algebra:  Diane Resek, San Francisco State University, resek@sfsu.edu

Intermediate Algebra:  William McCallum, University of Arizona, wmc@math.arizona.edu

College Algebra:  William Haver, Virginia Commonwealth University, whaver@vcu.edu

Algebra for Teachers:  Jim Fey, University of Maryland, jimfey@umd.edu