Chapter1

Chapter I

Introduction

A common topic in mathematics education, especially with respect to the difficulty experienced by female students, is known as "Math Anxiety". Tobias, (1978) said:

The first thing people remember about failing at math is that it felt like sudden death. Whether it happened while learning word problems in sixth grade, coping with equations in high school, or first confronting calculus and statistics in college, failure was sudden and very frightening. An idea or a new operation was not just difficult, it was impossible! And instead of asking questions or taking the lesson slowly, assuming that in a month or so they would be able to digest it, people remember the feeling, as certain as it was sudden, that they would never go any further in mathematics. (p.44)

Many classroom teachers' experience has taught there is an additional problem specifically with respect to algebra; call it "Algebra Anxiety". A number of students who experienced no particular "Math Anxiety" experienced considerable "Algebra Anxiety" when faced with algebra for the first time.

The comments by some students, and also interviews with people who are no longer students, point to a frustration with the letters used as variables. Ironically, this is not addressed in the current mathematical literature. It is assumed that the ability to learn mathematics has to do with the innate ability of the student to grasp abstraction. There appears to be no currently available study which addresses problems which may be unique to the learning of algebra.

A common approach is like that of Schwebel and Ralph, (1973) who said about proceeding from the "concrete" to the "abstract":

Educators have progressed to the point of realizing that children do not learn by simply being told or having things explained to them verbally. We now recognize the necessity of having concrete experiences first and buying lots of manipulatives from toy companies.

Few people have asked, however, what concrete experiences are, and what abstraction means. Concrete experience usually refers to any direct contact with real objects and events, and abstract thinking generally refers to the use of representation and the so-called higher order concepts. Piaget's conceptualization of experience and abstraction gives guidelines to the teacher as to how to make moment-to-moment decisions in the classroom to develop children's intelligence. (p.207)

Algebra is traditionally taught at an age at which human beings are expected to be able to think abstractly.

Jacob, (1984), says:

What is it about the action of the adolescent that warrants our speaking of a new equilibrium, a new form of thought? This question can best be answered in terms of: (a) the functions that adolescents can perform that were previously not in their repertoire, and (b) the structures whose presence must be inferred to be operative given that adolescents function in new ways. An obvious method for exploring these questions is to analyze the thinking processes of children and adolescents as they attempt to solve certain tasks, tasks which are specifically designed to tease apart the thought processes of the two sets of subjects. Such tasks have been worked out in great detail by Inhelder and Piaget (1958) in The Growth of Logical Thinking from Childhood to Adolescence, which is truly a landmark in this area. (p.37)

However, there are many factors which may affect the ability of the student to develop the skills to manipulate equations, to solve for variables, and to interpret word problems with equations. Aptitude, maturity, motivation, and personal environment are important and for the most part beyond the ability of the classroom teacher to affect.

Students learn more than what teachers actually teach. elementary teachers usually follow a prescribed sequence of skills, and usually reading and writing are taught together. Math commonly stands alone as the learning about numbers, although in some systems, math and science are associated at an early age. The underlying message here, or subliminal message is that letters are used for the construction of words for reading and writing, and numbers are used for mathematics. Indeed that is what is true for the first seven or eight years of elementary school. This elementary, untaught (at least not directly) message is commonly incorporated into the student's body of practical experience that is retained as part of the student's neural network of brain cells. The subliminal message (untaught) becomes perceived reality. This perceived reality makes the use of letters in algebra an additional problem for students with a pre-existing difficulty for abstraction. Because these subliminal messages are consistent, they are addressable by specific teaching strategies.

Why should subliminal messages have such an effect, when the direct efforts of dedicated professionals seem to have no effect? The answer lies in the research of the human brain that has progressed dramatically in recent years. With the advent of computer imaging, and the unlikely source of computer researchers who have been attempting to design computers that are based on a neural network (as opposed to binary computers), only recently have there been significant connections between the way brain cells actually work, and the process of learning.

In the Review of Related Literature, Chapter II, this thesis investigates the specifics of current mathematical references, Piaget's conceptualization theories, and the probable cause of students' perceived reality. Often this perception conflicts with the required need to deal with the abstract for beginning Algebra students.

Chapter III is an analysis of student writings prepared for a remedial college math course. The students commonly do not recognize their problem directly, yet indirectly when asked to write about their previous experiences in mathematics, an anxiety separate and distinct from general "Math Anxiety" begins to emerge.

Chapter IV describes the perceived reality of students. "Connectionist" theory is explained, and then used to describe how perceived reality becomes the student's reality.

Chapter V draws conclusions and makes recommendations for further study. The first Appendix contains a useable lesson for students beginning algebra entitled, "Algebra without the Letters". A copy of the original student essays is also provided in Appendix 2.

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