Chapter 2

Chapter II

Review of Related Literature

When discussing anxiety in mathematics, the main reference used by math instructors today would have to be Overcoming Math Anxiety, by Sheila Tobias. Originally presented in 1978, and revised in 1993, this is the best known reference in the topic. Tobias (1993) says:

For most people, mathematics is more than a subject. It is a relationship between themselves and a discipline purported to be "hard" and reserved only for an elite and powerful few. Thus, all people endure some mathematics anxiety, but it disables the less powerful - that is, women and minorities - more. There are certainly cures for math anxiety. In the short run, these involve changing attitudes and exploding myths about who can do mathematics and how mathematics competence is measured; in the long run, they require changing popular perceptions about mathematics. (p.10)

Kitchens (1995) continues:

A basic problem of anyone who feels helpless in math class is the problem of false perceptions. These false perceptions are in two areas: first, in your understanding of math anxiety - where it comes from, and how you control it; and second, in your understanding of mathematics, the discipline - how you think about it and how you learn it.(p.2)

One of those false perceptions is that of the "mathematical mind" Crawford (1980) elaborates:

Until recently, the idea of a math mind was almost universally accepted by mathematicians and non-mathematicians alike. It was assumed that one is either gifted with talents for doing math or one is not. The mathematically gifted have minds that think clearly and logically, understand new ideas at once, and have an uncanny ability to solve difficult puzzles and problems with both speed and accuracy. They are always the ones playing chess at social gatherings. Traditionally, these gifted individuals have been a small minority of our population, usually made up of men who went on to become our nuclear physicists, chemists, engineers, and of course, mathematicians. For the have-nots, with respect to math minds, it is assumed that one's potential must lie in verbal abilities. Standardized tests seem to reinforce the division between verbal and mathematical talents.

However, even though it is recognized that some have stronger verbal abilities than others, it is assumed that everyone should be able to develop a certain degree of competency in verbal skills. Furthermore, everyone should be capable of applying these skills to daily situations requiring abilities in reading and in writing.

Yet, when it comes to math abilities, you either have them or you don't - or so people once believed. The concept that most basic math skills can be developed by most individuals had developed only recently with the increased research into math anxiety.(P.142)

This concept of the "mathematical mind" does have some basis in theoretical science, primarily in the interpretation of Piaget's "Developmental Theory of Cognition". Published in the early 1950's in a series of three books, The psychology of intelligence, The origins of intelligence in children, and The construction of reality in the child.

Piaget described the development of human cognition in a series of progressive stages and sub stages that characterize the behavior of average humans. The principal stages are:

1. Sensorimotor (from birth to age 2): Cognition is typified by development of "schemes" or developing organized sensorimotor action sequences. Basically this means that most of the "thinking" in early stages of development is confined to reflexes, or reacting to stimulation in the environment, and assimilating responses that are productive. This tends to put limitations on cognition, to use a common phrase, "out of sight, out of mind". If the child cannot directly sense an object, it doesn't exit.

2. Preoperational Stage (age 2-6 yrs.): It is in this stage that the ability to represent ideas is first formed. The ability to use language develops, and symbols are used to imitate behavior. Characteristically, one word or name exists for one object, and the ability to generalize is rare. One primary characteristic is the essence of the object, and there is no conservation of ideas. Lerner (1984) states:

Conservation refers to the ability to know that one aspect of a stimulus array has remained unchanged although others have changed. To understand this concept, let us imagine that we present a five-year-old with two dolls, a mommy doll and a daddy doll. We then take four marbles and place them in a row beside the mommy doll, and we place four more marbles beside the daddy doll in positions directly corresponding to the mommy doll's marbles. If we show the five-year-old these materials arranged in this way and ask, "Which doll has more marbles to play with-the mommy doll or the daddy doll?" the child will most probably say that both dolls have the same amount of marbles to play with. However, if we spread out the mommy doll's marbles in the full view of the child (but leave the daddy doll's marbles in their original position), and then ask which doll has more, the preoperational five-year-old will answer that the mommy doll has more!(p.253)

This is the inability to conserve number. The child does not realize that the number has remained the same, instead, the positioning of the marbles takes precedence, probably because of its immediacy. It seems that the child cannot appreciate both characteristics of the situation at the same time. This is a cognitive error based on the child's inability to judge the effects of position versus number. The ability to judge the "reversibility" of the position, versis the constant value of the numeration appears beyond the grasp of the pre-orperational child.

3. Concrete Operational Stage (6 to 11 or 12 yrs.): It is now possible for the child to think of physical operations or actions without actually seeing it happen. This ability is limited to real objects and real events. If a child at this stage is asked, "What if coal were white?", the response would typically be "But coal is black". Concrete operational thought is limited to thinking about concrete "real" things.

4. Formal Operational Stage (11-12 years till death): Basically cognition begins to correspond to formal, logical thinking. Lerner (1984) explains:

In the formal operational stage, thought becomes hypothetical in emphasis. Now discriminating between thoughts about reality and actual reality, the child comes to recognize that his or her thoughts about reality have an element of arbitrariness about them, that they may not actually be real representations of the true nature of experience. Thus the child's thoughts about reality take on a hypothetical "if . . . then" characteristic: "If something were the case, then something else would follow." In forming such hypotheses about the world, the child's thought can be seen to correspond to formal, scientific, logical thinking. This emergence accounts for the label applied to this stage- the formal operational stage.(p256)

Adolescent thought patterns are changing significantly in this period. E.D. Neimark (1975) notes:

Although the properties and relations at issue during the concrete operational stage are abstract in the sense of being derived from objects and events, they are still dependent upon specifics of the objects and events from which they derive; that is, they are empirically based abstractions rather than pure abstractions. In this sense the elements of concrete operation thought are "concrete" rather than "abstract" or "formal." On the other hand, propositions, the elements of formal operational thought are abstract in the sense that the truth value of a statement can be freed from a dependence upon the evidence of experience and instead, determined logically from the truth values of other propositions to which it bears a formal, logical relationship. This type of reasoning, deriving from the form of propositions rather than their content, is new in the development of the child: deductive rather than inductive thought. (p.547-548)

In mathematics education, this is the age that we usually attempt to teach algebra. As the formal operations stage according to Piaget starts at age 11-12, and most eighth graders are at least 12, it would be logical to assume that algebra should be able to be grasped by all eighth graders, and for sure all ninth graders.

However, not all individuals develop cognitive skills at the same rate. Jackson (1965) found that only 7 of 16 persons thirteen to fifteen years old reached a conventional stage of formal operational thought. Dale (1970) found less than 75% of fifteen year olds studied had reached the formal stage, and in two studies of college students, Elkind(1962) and Tower and Wheatley (1971), gave evidence that only 60% of the college students sampled, showed an appropriate performance on tasks requiring the conservation of volume.

The assumption then would be that those individuals who develop formal operations cognition at an early age, are the same individuals that gain the reputation in the classroom as having a "math mind." Those who do not reach the formal operations stage at a young age, are commonly left behind in mathematics, and develop a self fulfilling prophecy where they think they cannot do math, and as a result they cannot.

Mathematics educators have several resources which attempt to deal with the problem. Although emotion is a topic that is not normally associated with the teaching of mathematics, the strategies used to help students with math anxiety commonly work on the student's attitude, and inform the student that the emotions that they are experiencing are a large part of the problem. Tobias (1987) says:

The understanding and recall pathways have become cluttered by emotions. There is an inability to think, but not because the "hardware" is inadequate. The input, memory, and understanding and recall systems are just as good as they were before. But, because the pathways have been blocked, you cannot remember. You lose self-confidence because you don't seem able to analyze the problem. You may even doubt that you have the intelligence to do the job. But, in truth, the only reason you cannot work is that your feelings have created too much "static" in your brain. Soon, your pencil stops moving. Your brain stops functioning. You can't work, you assume, because you can't think. But in fact, it's just the reverse: you can't think because you have stopped working.(P.87)

Buxton (1991) says:

Mathematics is commonly seen as a study based on reason, with the emotions rarely engaged. There is a belief in its firmness, precision, and fixity; this is associated with a feeling in some that it is not created, is certainly not man-made, but is external and intractable.

A mathematician would not accept these views, but that such beliefs are widely held is a fact. They lead to an odd antithesis - that math is not connected with emotion and that in many people it demonstrably evokes decidedly strong emotional responses of a very negative sort. It raises in a pointed fashion the question, old but largely unresolved, of the relationship between reason and emotion.(p.272)

In Fear of Math, How to get over it and get on with Your Life, Claudia Zaslavsky (1994) states:

In the United States most people would be ashamed to admit that they never could learn to read, yet it is perfectly respectable to confess that one can't do math. Mathematics has such a bad reputation in this country that it can be used to induce emotional stress. Physicians measuring blood flow under various stress conditions give their patients "a barrage of mental arithmetic problems" as a surefire method of inducing stress. Many people recall math as punishment.(p.14)

Nowhere in the literature is there a distinguishment made between math anxiety in general, and algebra anxiety more specifically. Yet many individuals who experienced no particular math anxiety before, experienced considerable anxiety when faced with algebra.

Crawford(1980) states:

In "Math Without Fear" I often asked adults to try to pinpoint the time when their troubles with math began. The most common response I received was algebra. Somehow algebra seemed to be the point at which math made simple became math made difficult. (p.127)

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