Volume 4, Issue B ::: September 2000
Making Peace in the Math Wars
Safford explores the theories that underlie different approaches to math instruction and envisions a math classroom that capture the positive aspects of them all
by Kathy Safford
A simplistic summary of Newton's Third Law of Motion states that for every action there is an equal and opposite reaction. An observer of mathematics education in the United States over the last 50 years could use the same words to describe the reform movements within that community. This historical roller coaster of theories and practices has shaped the mathematics histories of the adult students we meet in year 2000 classes. As I write this article there is, at every level of institutional education, disagreement about what should be taught, how the learner learns, and, therefore, how the teacher should teach. Like so many debates in the public forum, the participants often speak in terms of all or nothing and the controversy is sometimes bitter and downright nasty. Issues of rigor, exclusivity, and achievement lead to quarrels among even the best-intentioned participants. Perhaps nothing else could be expected of a marriage of theoretical mathematicians and educational psychologists. At the heart of the dispute is the question "What is mathematics?"
The repercussions of this controversy will sculpt the mathematics education we offer to our adult students. I hope to offer here a vision of an adult education math classroom that captures the positive aspects from all sides of the argument, based on my experiences as a math educator and the various educational psychology theories supporting them.
A Personal Journey
I began my mathematics education career in 1984, as the instructor of a community college basic mathematics course that met in the evenings. Because of the time slot, the majority of the students were adults. My teaching style reflected my own experiences in the classrooms of the 1950s and 1960s and was quite traditional. All students taking the basic mathematics course were tested -- at a central campus testing center -- five times a semester. The tests were multiple choice. My job boiled down to preparing the students to take the tests. In all probability I behaved like the teachers the students had met in their elementary school careers. Except, of course, that I had 45 hours to train them to perform arithmetic (often the word used in the text titles for courses at this level) tasks they had somehow failed to master in nine years of elementary education.
There were success stories. One semester all the students in my elementary algebra course passed the exit exam. Those familiar with developmental classes will realize the achievement that last sentence signifies. Certain doubts, however, continued to plague me. Students were frank in their evaluations of the utility of the skills they had acquired. The college did not allow them to use calculators, and they were expected to memorize the addition facts and multiplication tables during the first week of the class. Responses to that demand ranged from jocularity to open rebellion. Later in the course, when we were attempting to master the division of multiple-place decimal numbers into other multiple-place decimal numbers, one student announced that if she ever did that by hand at work she would be fired for wasting time and risking error. Her boss supplied a calculator that she was expected to use efficiently. Most damning was the inevitable last day of class when students thanked me politely and said that they had enjoyed the course, but had no idea when they would use the material in their "real" lives.
In 1990, I decided to expand my professional horizons and go back to school for a doctorate in either mathematics or mathematics education, settling after much thought upon the latter. The National Council of Teachers of Mathematics' Standards were fresh off the presses and Rutgers, the State University of New Jersey, was at the forefront of research on the applicability of constructivism to mathematics education. Constructivist theorists believe that "learners do not just absorb information at face value . . . they actively try to organize and make sense of it, often in unique, idiosyncratic ways" (Ormrod, 1999, p. 171). Like a convert to a new religion, I decided that a constructivist classroom was the way to go. Never again would I train students to perform tasks without understanding them, nor would I teach them rules that could be learned quickly and as quickly forgotten. Fortune provided two opportunities for me to implement these new beliefs. The first chance came via a position as the basic mathematics instructor at a manufacturing plant. The second opportunity came two years later at Rutgers, when the adult college within the university sanctioned a basic algebra course limited to their students who had been unsuccessful in classes with a predominance of traditional college-aged students. Reflections on these two experiences, as well as consequent immersion in research literature, form the basis of the adult class prototype I suggest here.
The Current Debate
The roots of the recent debate over what content and methods should compose the mathematics education of children, termed the "math wars" by the media, reach back two decades. A pamphlet published by the National Commission on Excellence in Education, entitled A Nation at Risk (1983), called for the strengthening of high school graduation requirements in the United States in five subject areas, including three years of mathematics study. Recommended content included the traditional topics of algebra and geometry and additional topics such as elementary probability and statistics. The Commission emphasized that their recommendations were not exclusive to the college bound but extended to mathematics classes for students who would not be continuing their formal education immediately (National Commission on Excellence in Education, 1983). The gauntlet for math education reform was picked up by the National Research Council, which published a trilogy, Everybody Counts, A Challenge of Numbers, and Moving Beyond the Myths, which addressed different aspects of the perceived problems with US mathematics education and recommended further action to address them.
From the start, the National Council of Teachers of Mathematics (NCTM) was also involved in the reform investigations. NCTM is the principal professional organization for primary and secondary school mathematics teachers in the United States and Canada. Its membership is overwhelmingly, but not exclusively, composed of individuals concerned with pedagogy and the instruction of children and adolescents. In 1989, NCTM released The Curriculum and Evaluation Standards for School Mathematics. The intent of the Standards was to ensure quality, indicate goals, and promote change (NCTM, 1989). It contained specific suggestions for content and practices that should receive increased or decreased attention from kindergarten through twelfth grade.
While other mathematics organizations have subsequently published documents that address the issues of reform, the NCTM Standards are at the center of the current maelstrom. They were first on the scene, made the most specific suggestions, and affected the greatest number of students. Some members of the adult basic education community found the original NCTM Standards useful as the basis for standards for their organizations. The Adult Basic Education Math Standards Project in Massachusetts used them as a guideline when composing The Massachusetts Adult Basic Education Math Standards (Leonelli & Schwendeman, 1994). The Mathematics Committee Division of Adult and Career Education did the same when writing their Adult Mathematical Literacy for the 21st Century (Milner, 1995). I, too, have written both a basic computation course and an introductory algebra course based on the recommendations of the 1989 Standards. The original document has been revised, reflecting the experiences of the past ten years, and the new version was released in April, 2000. The revised Standards captures the spirit of the original book while reorganizing and realigning some of the topics. Five content areas are addressed: number and operation; patterns, functions, and algebra; geometry and spatial sense; measurement; and data analysis, statistics, and probability. Cross-content standards of emphasis and methodology include problem-solving, reasoning and proof, communication, connections, and representation (NCTM, 2000).
What, you might ask, is the controversy all about? Superficially, the recommendations for decreased versus increased attention separate the two warring factions. Far more fundamental, however, is the question of how students learn mathematics. The answer to that requires an examination of learning theories and their appropriateness for the study of mathematics. While the NCTM wisely steered clear of endorsing a specific school of thought, the Standards do reflect the influence of the theory of constructivism. Much of the theory termed as "constructivist" stems from work by the Swiss psychologist Jean Piaget. The keystone of constructivism is the notion that all knowledge is constructed by individuals who act upon external stimuli and assimilate new experiences by building a knowledge base or altering existing schemas. At its most radical, constructivist theory holds that each person discovers truth and constructs his or her own unique knowledge base (von Glaserfeld, 1991). Constructivist advocates believe that students should explore mathematical situations and induce the general rules of mathematics from those experiences.
hold the view that students may fail to recognize correct patterns or may
construct erroneous rules that will be difficult to deconstruct. Many, though
certainly not all, of these individuals learned mathematics in the 1940s and 1950s,
when behaviorism was the prevailing learning theory and drill and practice
constituted the teaching methodology. A central focus of behaviorism is the
observation of a stimulus presented to the research subject and the resultant
response. As a result, behaviorism
is sometimes called S-R psychology. A simple example from mathematics is the
presentation of a problem involving a number fact (the stimulus) and the student
reply (response). Learning, in an
S-R sense, is considered to have occurred when the correct solution is given
consistently. Behaviorists, and their emphasis on observable, measurable
phenomena, reflect a general movement at that time toward rigor in the natural
sciences, and towards data that could be counted and quantitatively analyzed.
B.F. Skinner is perhaps the psychologist who first comes to mind when
behaviorism is mentioned. He proposed the ideas of operant conditioning and the
use of reinforcement to strengthen a desired response. The methods of operant
conditioning can be applied in learning situations to encourage desired behavior
as well as to discourage the undesirable (Ormrod, 1999). They are useful in the
treatment of anxiety. It is somewhat ironic that our adult students
often developed math anxiety because they were unsuccessful in
S-R math drills, yet knowledge of operant conditioning can help math educators to plan strategies to decrease and even overcome that anxiety. Course materials that appear different from classes in their past and a supportive classroom atmosphere invite anxious students to lower defensive attitudes that have blocked their previous mathematics learning (Ramus, 1997).
Building a Philosophy
If you are new to teaching mathematics, or have been teaching many years without a firm grounding in the theory of mathematics instruction, you may want to follow these suggestions to building a basis for your teaching choices:
· Read the National Council of Teachers of Mathematics Principles and Standards for School Mathematics.
· Learn more about the various theories of learning highlighted in this article as well as others that space restrictions precluded.
· Think about the mathematical tasks you and your students perform in your daily lives.
· Read a current mathematics methods text, which will help you to clarify the essence of mathematical operations and number. These will often suggest problems you can alter to reflect adult situations. I like Elementary and Middle School Mathematics, by John Van De Walle. Published in 1997 by Addison-Wesley, it gives concepts and methods at the same time.
· Recognize mathematical "moments" in your daily life and start off the class with a problem structured on that experience. A moment might be a sound bite from the news you hear on the way to work, something that happened to you in the supermarket, or a mathematical pattern you noticed.
· Bring in clippings from the newspaper or magazines with graphs or reports of surveys. This year, for example, we will be seeing a lot of census information in the press. Share it with your students and discuss the implications for their community.
The classroom methods suggested here take time and adjustment on the part of the students and instructor. Preconceptions can undermine the establishment of this new learning climate. Students may come to the classroom with a clear vision of the teacherŮstudent relationship. To them, the teacher may be an ultimate source of knowledge and wisdom (Tennant & Pogson, 1995). What Friere terms the "banking concept" of education may be the model they expect, and they may be reluctant to surrender without a struggle (Friere, 1993). Others may feel that the teacher must be a provider and comforter (Tennant & Pogson, 1995). The nature of remembered school experiences may conflict with goals in a class or program that is striving to encourage independent thinking and learning. Adults may need some nudging to become active builders of their own knowledge.
An Adult Class Prototype
While time and experience have moderated my initial zeal for a strict constructivist andragogy - the art and science of teaching adults - it is still the driving theory upon which I believe mathematics instruction should be based. Constructivism is based on the fundamental assumption that people create knowledge from the interaction between their existing knowledge or beliefs and the new ideas or situations they encounter (Airasian & Walsh, 1997). Malcolm Knowles (1978), a pivotal figure in adult learning theory, wrote that as individuals mature they accumulate an expanding reservoir of experience that causes them to become increasingly rich resources for learning, and at the same time provides them with a broadening base to which to relate new learning (p. 56). Adult students bring to the mathematics classroom knowledge of situations that require mathematics as well as methods, sometimes rather ingenious, they have devised to solve problems involving mathematics. In the literature of adult mathematics research these are termed "street math" as opposed to "school math." One role of the instructor is to mediate these two "maths," to aid students in clarifying knowledge they already own, and to alter and enhance it with new knowledge acquired in our classrooms.
I try to begin each class with a problem tied to the topic of the day, but one that can be tackled with street skills. For example, how would the student figure out the tip in a restaurant? Since those skills vary, each student, or group of students if the class is working cooperatively, describes their solution strategy and it is recorded on the chalkboard. The class then examines the responses for similarities of solutions and strategies as well as differences. We evaluate and discard incorrect solutions, which often turn out to be correct solutions to a different problem. I guide the learners so that they recognize patterns that are emerging and develop the "rules" of mathematics themselves, in their own language.
Social Learning Theory
Sometimes no one can solve the problem or everyone is totally off base. Another school of thought, social learning theory, provides me with insight in these instances. One principle that underlies social learning theory is that people can learn by observing the behaviors of others and the outcomes of those behaviors. The work of Lev Vygotsky on "scaffolding" is generally classified as social constructivism (Ormrod, 1999). The process it describes is closely linked to learning from observation. Scaffolding posits that the learner functions as an apprentice to a master, and that there are four stages of learning. The first involves observation of the skilled individual. At the second level, the learner shadows the model, and performs the task simultaneously with the teacher. By the third level, the apprentice is practicing the skill under the watchful eye of the master until the fourth level, at which the master steps aside and allows the apprentice to perform unassisted.
In my teaching, I use scaffolding to teach problem-solving. When I begin teaching a course, I approach word problems by mulling aloud about the situation and organizing the information in a table in a seemingly offhand way. After a few class sessions, I begin to focus on that strategy and involve the students in the process of building the table. If students are rambling in their attempt to solve the problem, I take a direct approach and suggest that a good way to start organizing the information would be a table. I share with them the utility I have found in this approach. When solutions to problems are solicited, a tabular approach is the focal point of resolving differences. I consider the fourth level is achieved when tables show up on assignments or examinations.
A Role for S/R
listening to student voices, I have come to realize that S-R theory still has a
place in adult mathematics instruction, but that it should be the cart rather
than the horse. A major complaint from students about my constructivist stance
was the lack of adequate practice of skills and rules
constructed during investigations of problems (Ramus, 1997). Repetition
and practice hone skills that become automatic, a process termed
"automaticity" in cognitive science. This frees working memory to deal with
more challenging, nonroutine tasks (Ormrod, 1999). Imagine yourself having to
think about the meanings of the colors in traffic lights every time you approach
one. Not having to do so frees you to think about pedestrians, the bus ahead,
and the myriad other distractions you encounter while driving. If students have
constructed and own the rules they are practicing, they can reconstruct them at
some later time. Practice diminishes
the need for such activity and releases their brains to engage
in more constructive work.
The field of adult basic education is broad and that education is delivered in a variety of institutions. In some situations attendance is erratic and individualized instruction more practical than whole-class or small-group work. For others, a multitude of languages and literacy levels make mathematics word problems a challenge. In whole-class settings there may be a great discrepancy of mathematical knowledge among the participants, offering the challenge of meeting everyone's needs without holding some back or leaving others behind. In the box on the previous page, I offer suggestions for building your mathematics program grounded in the theories discussed and based on my own experiences at the task.
While the math wars rage outside our classrooms, we soldier on inside. The last 20 years have brought changes in the content we emphasize in mathematics classes, the tools we use to teach that content, and the methods we use to deliver instruction. This article has attempted to offer a compromise plan to ABE teachers by sharing some strategies for effective math teaching and the learning theories that support good practice. Instructors who are informed of their choices are equipped to design instructional experiences that will assist students towards both personal and credentialing goals.
Airasian, P., & Walsh, M. (1997) "Constructivist cautions." Kappan, 78, 6, 444-449.
Friere, P. (1993). Pedagogy of the Oppressed. New York, NY: Continuum Press.
Knowles, M. (1978). The Adult Learner: A Neglected Species. (2nd ed.) Houston: Gulf Publishing.
Knowles, M. (1990). The Adult Learner: A Neglected Species. (4th ed.) Houston, TX: Gulf Publishing.
Leonelli, E. & Schwendeman, R. (eds.) (1994). The Massachusetts Adult Basic Education Math Standards. Malden, MA: Massachusetts ABE Math Team.
Milner, M. (ed.) (1995). Adult Mathematical Literacy for the 21st Century. Los Angeles, CA: Division of Adult and Career Education, LA Unified School District.
National Commission on Excellence in Education (1983). A Nation at Risk: The Imperative for Educational Reform. Washington, DC: US Department of Education.
National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: NCTM.
National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.
National Research Council (1989). Everybody Counts. Washington, DC: National Academy Press.
National Research Council (1990), A Challenge of Numbers. Washington, DC: National Academy Press.
National Research Council (1991). Moving Beyond the Myths. Washington, DC: National Academy Press.
Ormrod, J. (1999). Human Learning (3rd ed.). Upper Saddle River, NJ: Prentice Hall.
Ramus, K. (1997). "How the mathematics education reforms pertain to undergraduate curriculum: An introductory study of an experimental developmental algebra course for adults." Dissertation Abstracts International, 9717243.
Steen, L. (1999). "Theories that gyre and gimble in the wabe - a review of mathematics education as a research domain: A search for identity." Journal for Research in Mathematics Education, 30, 235-241.
Tennant, M., & Pogson, P. (1995). Learning and Change in the Adult Years: A Developmental Perspective. San Francisco, CA: Jossey-Bass Publishers.
von Glaserfeld, E. (1991). "An exposition of constructivism: Why some like it radical." In Davis, R., Maher, C., & Noddings, N. (eds.) Constructivist Views on the Teaching and Learning of Mathematics. Reston, VA: NCTM.
Vygotsky, L. (1978). Mind in Society. Cambridge, MA: Harvard University Press.
About the Author
Kathy Safford received her doctorate in mathematics education from Rutgers University. Her research concentrates on adults learning or relearning mathematics. She is an assistant professor of mathematics at St. Peter's College, Jersey City, NJ, where she teaches both developmental and undergraduate mathematics courses.