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Joseph A. Montagna
A second major portion of this unit is the presentation of material related to statistics. This portion of the narrative is my own assessment of what is appropriate for students to learn in the seventh and eighth grades. The math objectives for these levels include many of the subskills that are necessary to perform the mathematics related to statistics, i.e. finding percentages, calculating the arithmetic mean, graphing. Many of these objectives are also included in the curricula for other grades, making this unit easily adaptable for older or younger students. Further, many of the topics discussed in this unit are appropriate to the teaching of disciplines other than math, i.e. social sciences, career education.
These two goals of teaching students about themselves and about statistics will converge in an attempt to have students focus on the single question, “How do teenagers spend their time?” This question was the focus of a study conducted by a group of social scientists at the University of Chicago. The book about this project, Being Adolescent: Conflict and Growth in the Teenage Years, and Laurence Steinberg’s book, Adolescence, are the two major sources for this unit. I would strongly recommend them both.
The family is a system which, like all systems, must change as the circumstances change. As the needs of family members change, the family system changes to maintain equilibrium. As the child grows and reaches new levels, the expectations and roles for that family member will, or should, also change. The family responds to these changes in the child in such a way as to recognize these changes and adapt to them.
Our society makes it necessary to teach young people in large groups, in order to have reasonable assurance that the same messages are getting across to them. All individuals in our society are expected to learn the same set of norms. Modern society has created a system in which adolescents play an increasingly valuable role in teaching one another. According to the noted anthropologist, Margaret Mead, ours is a configurative culture. In a configurative culture changes occur so rapidly that much of what parents teach their young is outdated by the time they become adults. For adolescents in modern America, what peers teach them may be as important, if not more important, than what their parents teach them. The computer is an excellent means of illustrating this. Living with computers is easy for today’s youth. Computers have been a part of the world in their lifetimes. Very little anxiety is created within a youngster who is sitting in front of a computer and much of what they learn about computers is derived from their peers. On the other hand, many adults who are confronted by the green screen and blinking cursor experience anxiety often.
Adolescent peer groups are usually structured around cliques, groups of two to twelve individuals that structure the adolescent’s social activities. At first they are same sex groups, them change to mixed groups, then to opposite sex dyads as the adolescent moves from early to middle and late adolescence. The study of cliques is not as simple as the above may suggest. Occasionally, several cliques come together to form “crowds”, loosely organized around a particular activity or function, i.e. a school dance, a fundraising project, a party. Crowd formation seems to be a transitional stage that precedes dating. The crowd offers the sanctuary of the same sex peer group while affording the adolescent an opportunity to closely check out the opposite sex. Dating usually follows, generally led by the leaders of the cliques. Others follow as they resolve the dating issue within themselves. Middle adolescence is characterized by heterosexual cliques and crowds, although they are not quite ready to split off in opposite sex pairs. Eventually, as they move into late adolescence couples split off from the crowd, maintaining a loose tie With the larger group. This association with the larger group is through contact with other couples. The concept of the couple as the focus of social activity remains through adulthood. Early adolescence provides the seeds of an adolescent’s capacity for close relationships through friendships with same sex peers.
Cliques serve as a source of identity by acting as a reference group, a standard against which the adolescent evaluates his experiences, learns about himself, and forms judgments about his abilities. Cliques also serve as a provider of identity in the way one appears to other adolescents. Adolescents judge one another by the company they keep. Initially, cliques are formed on the bases of: same age, same sex, same social class and same race. However, the above list changes, eventually, to include two basic types of orientation, school orientation and youth culture orientation. These two determinants of clique formation provide a basis for friendships outside of one’s sexual, age and status groups.
If there is one thing that we can say about adolescence with complete predictability it is that it is unpredictable. The Chicago study found that adolescent experiences are marked by frequent and drastic mood changes, some of which occur within very short periods of time. These mood swings illustrate that the process of socialization of our youth is a never-ending conflict between the goals and rules of society and the instincts, values and habits of the adolescent. In this period of choice the adolescent is pulled and stretched in a number of directions by a variety of forces. How the individual copes with these conflicting forces is what determines what the person is likely to become in later life. Conflict is a necessary and inevitable part of our lives. If it is channeled into a constructive mode of thinking and behavior, the individual grows. If it is not, then the individual languishes, beginning a downward spiral that mitigates against growth.
Csikszentmihalyi and Larson, in their book about the Chicago study, borrow a term from physics and information theory that is helpful in understanding the above—entropy. In physics, entropy describes a loss of energy in a system that is due to some disorder in that system. In information theory, entropy refers to some disorder in transmitting or receiving patterns that result in a loss of meaning. Psychic entropy; therefore, is the individual’s reception of conflicting information in consciousness, causing unpleasant experiences and a diminished capacity to perform.
Psychic entropy in the short term manifests itself as guilt, anxiety, alienation, frustration,or boredom. Positive growth can be realized by an individual who has such entopic experiences, that is, if the individual perceives them as temporary setbacks, and uses this feedback to turn one’s attention inward to resolve them, e.g. a person who is guilt ridden over letting down a loved one by not fulfilling a commitment, who then analyzes the situation and resolves either to follow through on commitments, or not to make commitments that cannot be kept.
There are four basic forms of disorder in consciousness that are termed psychic entropy: bad moods, passivity, diminished or absent motivation, and the inability to focus one’s attention. These experiences are inevitable in growing up and living. One point of view of the authors of the Chicago study is that a consistent pattern of disorder in consciousness may be permanently damaging to a person’s growth and productivity in adult life.
The opposite term of psychic entropy is psychic negentropy, order in consciousness. It is a state in which the person feels “whole”, positive and enthusiastic. It is a person performing at one’s optimal level. There are four main characteristics of psychic negentropy: positive feelings toward self and others, a sense of competence, one’s identification with the goals of the activity, and effective concentration. Psychic negentropy is not static. It is dynamic. New order is created out of experience and the self is constantly growing. Psychic negentropy does not seek to return to a previous state of order, however, since it is out of experience that growth and new order evolve. An illustration of this is a boy who has been jilted by his girlfriend. He works this out in his mind by rationalizing that there are other girls out there, ones whom he can love and will love him in return. However, the boy does not return to a previous state of innocence which existed before this incident. New order is created from this experience, and the boy becomes cautious about with whom he shares his love.
Can the above be useful to teachers and students? I believe that we can analyze instances whereby students have had good and bad experiences, and search for the underlying causes. If a student, for example, has a bad experience in math class, was it due to his (the student) not being prepared for class, or were there other reasons beyond his control? If the student was, indeed, not prepared for class, then what were the causes of this? Was it because he cut yesterday’s class and missed the lesson, or that he wasn’t paying attention? Perhaps the student needs to become better organized concerning his use of time? If only we can get our students to view the larger picture:.
As parents and teachers we cannot help but ask ourselves whether we are doing the right thing. Often, perhaps always, it is too late when the answer is realized—the youngster is now an adult. We can, however, look closely at indicators that exist in their present lives which will help us to assess their progress (or lack of it) toward a productive adult life. How the adolescent spends his time, the quality of his experiences, his habits and activities are all good indicators of the kind of person he is likely to become.
Where do adolescents spend their time? What do they do with their time? With whom do they spend their time? Laurence Steinberg calls these areas the contexts of adolescence. Csikszentmihalyi and Larson refer to them as the external landscape. We will be looking at these areas to map the daily experiences and interactions of adolescents.
School is where socialization is supposed to occur. The teenager’s consciousness is subjected to the attempts by adults to modify it, direct it toward the goals of the school, so that it conforms to adult standards. The school is also a place where the adolescent is exposed to the socializing influences of his peers. Of the 31% of time spent in school, 2/3 of that time was spent in classes and the remaining 1/3 was spent in fringe areas (cafeteria, corridors, student center). In the school in the Chicago study, as in others like it, these fringe areas are claimed by cliques of every variety (jocks, druggies, eggheads, Blacks, Italians).
The public places within which these adolescents exist are varied and, as mentioned earlier, governed by varying rules. IN some cases, such as work, the adolescent’s time is structured in terms of adult goals. Churches, restaurants and stores demand adult-like behavior. Behavior at concerts, movies and dances are more influenced by the values of the youth cultures. Other locations are a mix of adult goals and youth values, such as the automobile. While driving a car the teenager must conform to the rules of the road, if he wants to continue to drive, while the automobile is also a means to power (acting out one’s wishes) and freedom (withdrawal from adult scrutiny).
The school and the home are environs in which one can easily identify the seat of control by either adults or peers. The public sphere, however, is one in which adult behavior is required and opportunities exist for unsupervised activities. Teen life is split between adult values domination and peer values domination.
With whom do adolescents spend their time? The reports of the students in the study indicate that they are seldom in the company of adults. Approximately 20% of their time is spent with family and a very small portion of this time is spent with parents only. Other adult company adds a mere 2% to this time. A bit more than one quarter of their waking hours is spent in solitude, and more than half of their time is spent with peers (23% with classmates, 29% with friends). These figures clearly illustrate the influence that peers have on adolescents, or, at least, the potential for having a great influence.
What does the teenager’s life feel like? What is the internal landscape? We have taken a look at where and with whom a teenager spends his time. This gives us a fairly clear picture of the investment of time. What about the investment of psychic energy? The question of when psychic entropy and negentropy occur will give us a clearer picture of how life is experienced. For teens to develop into productive individuals, who contribute to society, it is necessary for them to experience enjoyment and order in life. If they derive enjoyment from school we can be reasonably assured that they will turn out to be motivated adults. If they receive enjoyment only from idle interactions with peers or in disruptive activities, then we have cause for concern about what they will become in adult life.
The authors of this study consider intrinsic motivation to be the single best indicator of harmony in teenagers’ consciousness. To measure this dimension they asked students to rate how much they “wished” to be doing the activity in which they were engaged when signaled. This measure was the student’s own rating of whether they were doing something because they had to, or because they were truly invested in it. They were also asked to measure their emotional state and cognitive efficiency.
The results of this aspect of the study are astonishing. Twenty-five percent of the time they reported a wish to be doing what they were doing. They reported varying degrees of wishing to be doing something else the rest of the time. However, their ratings of emotional harmony (favorable moods) are positive 71% of the time. Approximately 6% of the time they reported being as happy and cheerful as the rating scale allowed. As one might expect these teenagers were more likely to report positive feelings and high concentration and activation in activities and locations that are less structured by adults (sport, hobbies, games, etc.).
One might be inclined to think that an adolescent’s life is structured to such a point as to be bland and routine. The data reveal that this is not so. The teenager is subject to numerous mood swings in a single day. The students in the study were randomly chosen to provide greater detail about their daily lives for a full week. The reports clearly show that teenagers’ moods fluctuate dramatically. A student could be ecstatic one minute, and in a depressive mood the next. Two case studies are presented in the book. The first one is about a 16 year old “druggie”, Greg. Through his dress and behavior Greg makes it clear that he rejects the school’s goals and the values of the so-called straight community. School is boring to him, and his only excitement is derived from hanging out with his friends, male and female, and in getting high. Greg has mood changes, as does any other adolescent, yet his moods generally stay on the negative side.
Kathy is seventeen years old. She is a good student with clear life goals. She has a serious commitment to playing orchestral music. Her moods swing from utter boredom to exhilaration just as quickly as Greg’s, however, Kathy appears to have more negentropic experiences in a variety of settings. Kathy experiences frustration, yet has learned to channel this frustration into something constructive. For example, when she has a poor practice session with her violin, Kathy uses this feedback to her advantage, making her next one better than ever. Psychic entropy for Kathy seems to improve her performance because she has learned how to use it.
Each of these two students has mood swings. When one places these moods on charts it is easy to see that Greg’s moods are consistently on the negative side. His only negentropic experiences come from getting high or hanging out with his friends. Kathy has moods that go as far on the negative side as Greg’s, yet she has negentropic experiences from a variety of settings, including school. The major difference regarding these moods is how each student handles the challenges and frustrations of their daily lives. For Kathy, she has learned the meaning of the adage “when life deals you lemons, make lemonade.”
The daily lives of adolescents are fraught with things that can, and often do, go wrong. It is not realistic for adolescents to try to avoid conflict and entropy in their experiences. Growing up in our modern society presents a predicament for adolescents to meet standards set up by adults, or reject them. Many adolescents are rejecting these standards, refusing to enlist in the established social order. Each generation has had its own fashion of displaying this rejection; the 50s had beatniks, the 60s had hippies, the 70s had punks, and the 80s have the “what’s in it for me’ kids. Suicide, the third highest cause of teenage death, and drug abuse are just two examples of adolescent rejection of the norms of society. The most important question for today’s adolescents is not “how do I avoid conflict? it should be “How can I learn to live with conflict and use it to promote my personal growth?”
Why is it that many people have flow experiences outside of leisure activities, i.e. work setting, school,etc? The answer lies in the skills that the individual possesses and the challenges presented by the activity. If an activity is not challenging enough to meet the skills of the individual, boredom results. When an activity presents challenges that far exceed the skill one possesses, the result is frustration and anxiety. Only a match between skill and challenge can produce the flow experience. The challenge of the activity must exceed the skill by a margin that is attainable by the individual. A good example is that of a youngster who plays the piano. If he is left to practice the basics that were mastered long ago, he will become bored. No one enjoys having to practice something that has clearly been shown to be mastered. On the other hand, if this youngster is presented with a classical piece that is far beyond his skill level, then he will become frustrated. As new skills are mastered by the youngster new challenges must be presented that promote further growth in his ability to play the piano. Growth occurs as a result of mastering new skills, paving the way the introduction of skills that are of a greater complexity, and so on.
The implications of the flow experience for teaching should be clear. The teacher should be aware of the skills levels of each of his pupils, and at the same time continue to challenge each one of them. This is no easy task for a teacher of 125 students each day, yet it is not impossible. If we can provide enough flow experiences early in their school careers, our students may derive enough enjoyment from learning to help them through the later years.
Statistical knowledge has applications to subjects other than mathematics. While it is true that some mathematical training is necessary to deal with statistical matter in a sophisticated manner, meaningful experiences with statistics can be gleaned even at the primary school levels. Nearly every subject in the curriculum can incorporate even a small amount of statistics.
Statistics can be found in areas that require very little coaxing by the teacher to get students motivated to learn, such as sports. Nearly every student has a favorite sport or sports figure. The motivation in these cases is provided free of charge. All one needs to do is to help students discover the application of statistics to these areas of high interest. Such a strategy is recommended if the teacher deems it necessary for his class. This will enable students to build up their confidence before moving on to other applications.
The relevance of statistics to the real world can be readily shown to students. Statistics is one way that we can represent what occurs in the real world. We must present students with the basics to be able to deal with the mass of data that is thrown at us every day, for as the world becomes increasingly complex even the smallest, most personal decisions will depend on how well we understand these data. A prime illustration of this is the projections for job availability in the year 2000. Positive growth is projected in all areas of engineering and health related fields, while negative growth is projected for college professors. It would be wise to prepare for a career that has a promising future.
What follows is one way of teaching statistics. Some of the lessons are rather elementary. I am assuming nothing regarding students’ prior knowledge and experience. Only the teacher is able to judge which classes can handle the later material without taking on the earlier lessons. The lessons move from having nothing to do with adolescence, building toward the necessary skills to accomplish the work in the culminating study.
N = the number of scores
· = the mathematical verb which tells us to sum the scores
Give students a set of scores to find the mean of:.
14, 24, 29, 19, 12, 21, 17 The sum is 136. Dividing 136 by 7 gives us 19.4285
We can leave the number as is, or we can round it to any of several places.
Give students enough practice in class, and assign a suitable homework.
Take the set of scores that were used as an example in 1.1, and add an extreme score to it Below is that set with 100 added to it.
14, 24, 29, 19, 12, 21, 17, 100 Now have students find the mean of this set. 236 = 29.5
This second mean is approximately 50% larger than the first by adding the extreme value to the set. The mean is very sensitive to extreme scores. Try this again with the first set adding a low extreme score to it.
Give practice in class in calculating the means of sets which have extreme scores in them. One suggested way to do this is to use some of the previous lesson’s work, adding extreme scores to them.
Ordering the scores from 1.1 gives us: 12,14,17,19,21,24,29
The median is 19. Three scores lie on either side of 19.
12,14,17,19,21,24,29,100 Because this set of numbers has an even number of elements (8), we have to find the mean of the two numbers on either side of the middle, 19 and 21. The median now is 20. In this case the median is effected only slightly.
Give students an opportunity to check this out in a few other cases.
Give students practice in finding the mode of sets of values.
Time allotted: one class period
Measure each student’s height in centimeters.
Put the individual heights in order, from tallest to shortest.
2.1 Find the mean height of the class (one class period) 2.2 Find the median height of the class 2.3 Find the mode (if one exists) 2.4 Discussion of findings
Time allotted: one class period Refer to the findings in the lessons involving the class project mentioned above. Draw students into a discussion of how these three measures of central tendency relate to one another. How do they differ?
Have students order themselves by height in front of the class.
Which student is in the exact middle? Is this person the median or the mean?
Refer to the mean value obtained. Which person(s) are that height? Which persons represent the extremes in this class?
Build a list on the blackboard that includes students’ suggested topics. Each student should have one of his/her own. Ex: “What is your favorite automobile?”
Have students construct a data collection sheet similar to the one shown. They should begin data collection as soon as they have finished constructing the data collection sheet. They will certainly have to finish outside of class. Collection is facilitated for this lesson by students verbally asking the question.
Upon completion of collecting the raw data, discuss some display methods one may use (histogram. pictograph, “pie” graph).
For the purposes of this lesson we will employ the pie graph method. This method is a personal favorite of mine because it requires the use of a number of math subskills.
Before actually constructing the pie graph students will have to properly fill in the next three columns on the data collection sheet.
Construct the pie graph using the information they compiled on their data collection sheets.
- -Fraction column The count becomes the numerator of the fraction. The denominator is the total number who were surveyed.
- -Decimal column Dividing the denominator into the numerator yields a decimal fraction. If This decimal is taken to the thousandths place.
- -Percent column Students can round the decimal column to the nearest hundredth. Dropping the decimal point and adding the percent sign produces the percent figures.
- -Degrees column Multiplying the decimal column figures by 360 (number of degrees in a circle) yields the size of each “slice” of the pie graph. Students should round to the nearest whole degree. Note that the total number of degrees may add up to more than 360 due to rounding.
Present results of individual surveys to entire class.
It is recognized that this lesson will require more time than the normal class period. While some of this may be completed outside of class, students will need close supervision by the teacher. Two class periods with homework assigned may be enough time.
Collect data in class. Have students copy down the priority numbers that you read to them off of the students’ questionnaires. Their homework assignment is to find the mean value for each activity.
(an alternative method is to use small groups calculate the means)
WHAT DOES IT TAKE TO GET INTO THE LEADING CROWD
IN YOUR SCHOOL?
__ be an athlete
__ good grades
__ good looks
__ have money
__ good clothes
__ come from right neighborhood
This is the book about the study of adolescents in a Chicago suburb. The narrative of this unit refers to this book on a number of occasions. It is recommended for reading by any teacher who wishes to understand more about adolescents in modern America. The book is easily read and contains an enormous mass of data from the study.
Runyon, Richard P., & Haber, Audrey, Fundamentals of Behavioral Statistics, 5th ed., Reading, MA, Addison-Wesley, 1984
This is a basic textbook on statistics. The first five chapters is all the teacher would need to read in order to carry out the spirit of the lessons in this unit. The book would certainly come in handy for anyone who plans to take a statistics course in graduate school.
Steinberg, Laurence, Adolescence, New York: Alfred A. Knopf, 1985
This book is also cited extensively in the narrative of this unit. The book provides an excellent framework for studying and understanding adolescence as a part of the total life. It is an excellent book that would be a fine addition to any teacher’s library.
Contents of 1985 Volume VIII | Directory of Volumes | Index | Yale-New Haven Teachers Institute