The following are generally referred to as the "laws" or "rules" of
exponents.
x^a x^b = x^(a+b)
(1)
x^a / x^b = x^(a-b) = 1/x^(b-a)
(2)
As with any formula, the most important thing is to be able to use them--that is, to understand what they mean. But it is also important to know where these formulae come from. And finally, in this case, the three should be memorized.
So...what do they mean? They are, of course, algebraic generalizations--statements that are true for any
x
,
a
, and
b
values. For instance, the first rule tells us that
7^12 cdot 7^4 = 7^16
which you can confirm on your calculator.
712 means twelve 7s multiplied;
74 means four 7s multiplied. Multiply them together, and you get sixteen 7s multiplied.
You can also think of the second rule as the inevitable consequence of the first rule. If
193 x 195 = 198, then
19^8/19^5
(which asks the question "195 times what equals 198?") must be 193.
Similarly, the third rule promises us that
(7^12)^4 = 7^48
These rules can be used to combine and simplify expressions.
In order to mark up the previous expression in ASCIIMath, you would have to write it as: (7^12)^4 = 7^48. The ^ symbol represents exponents in ASCIIMath.
Functionalists view education as one of the more important social institutions in a society. They contend that education contributes two kinds of functions: manifest (or primary)
functions, which are the intended and visible functions of education; and latent (or secondary) functions, which are the hidden and unintended functions.
There are several major manifest functions associated with education. The first is socialization. Beginning in preschool and kindergarten, students are taught to practice various
societal roles. The French sociologist Émile Durkheim (1858-1917), who established the academic discipline of sociology, characterized schools as "socialization agencies that
teach children how to get along with others and prepare them for adult economic roles. All education consists in a continuous effort to impose on the child ways of seeing, feeling,
and acting to which it would not spontaneously come." (Durkheim 1898)
Education also fulfills latent functions (shown in the table) . As you well know, much goes on in a school that has little to do with formal education. For example, you might notice
an attractive fellow student when he gives a particularly interesting answer in class—catching up with him and making a date speaks to the latent function of courtship fulfilled
by exposure to a peer group in the educational setting.
Manifest and Latent Functions of Education
Table 1:
According to functionalist theory, education contributes both manifest and latent functions.
| Manifest Functions: Openly stated functions with intended goals |
Latent Functions: Hidden, unstated functions with sometimes unintended consequences |
| Socialization |
Courtship |
| Transmission of culture |
Social networks |
| Social control |
Working in groups |
| Social placement |
Creation of generation gap |
| Cultural innovation |
Political and social integration |
Consider a large-city newspaper publisher. Ten years ago, when culling résumés for an entry-level copywriter, they were well assured that if they selected a grad with a GPA
of 3.7 or
higher, they'd have someone with the writing skills to contribute to the workplace on day one. But over the last few years, they've noticed that A-level students don't have the
competency evident in the past. More and more, they find themselves in the position of educating new hires in abilities that, in the past, had been mastered during their education.
This story illustrates a growing concern referred to as grade inflation—a term used to describe the observation that the correspondence between letter
grades
and the achievements they reflect has been changing (in a downward direction) over time. Put simply, what used to be considered C-level, or average, now often earns a student a B, or
even an A.
Which of the following is not a manifest function of education? Hint: The answer can be found in the "Manifest Functions" section.
- Cultural innovation
- Courtship
- Social placement
- Socialization
[ Show Solution ]
Because she plans on achieving success in marketing, Tammie is taking courses on managing social media. This is an example of ________.
- cultural innovation
- social control
- social placement
- socialization
[ Show Solution ]
The National Center for Fair & Open Testing (FairTest) is committed to ending the bias and other flaws seen in standardized testing. Their mission is to ensure that students,
teachers, and schools are evaluated fairly. You can learn more about their mission, as well as the latest in news on test bias and fairness, at their website:
http://fairtest.org/
Functionalists view education as one of the more important social institutions in a society. They contend that education contributes two kinds of functions: manifest (or primary)
functions, which are the intended and visible functions of education; and latent (or secondary) functions, which are the hidden and unintended functions.
There are several major manifest functions associated with education. The first is socialization. Beginning in preschool and kindergarten, students are taught to practice various
societal roles. The French sociologist Émile Durkheim (1858-1917), who established the academic discipline of sociology, characterized schools as "socialization agencies that
teach children how to get along with others and prepare them for adult economic roles" (Durkheim 1898).
Education also fulfills latent functions (shown in the table) . As you well know, much goes on in a school that has little to do with formal education. For example, you might notice
an attractive fellow student when he gives a particularly interesting answer in class—catching up with him and making a date speaks to the latent function of courtship fulfilled
by exposure to a peer group in the educational setting.
Conflict theorists do not believe that public schools reduce social inequality. Rather, they believe that the educational system reinforces and perpetuates social inequalities arising
from differences in class, gender, race, and ethnicity. Where functionalists see education as serving a beneficial role, conflict theorists view it more negatively. To them,
educational systems preserve the status quo and push people of lower status into obedience.
Feminist theory aims to understand the mechanisms and roots of gender inequality in education, as well as their societal repercussions. Like many other institutions of society,
educational systems are characterized by unequal treatment and opportunity for women. Almost two-thirds of the world's 862 million illiterate people are women, and the
illiteracy rate among women is expected to increase in many regions, especially in several African and Asian countries (UNESCO 2005; 'World Bank 2007).
Symbolic interactionism sees education as one way labeling theory is seen in action. A symbolic interactionist might say that this labeling has a direct correlation to those who are
in power and those who are being labeled. For example, low standardized test scores or poor performance in a particular class often lead to a student being labeled as a low achiever.
Such labels are difficult to "shake off," which can create a self-fulfilling prophecy (Merton 1968).
The major sociological theories offer insight into how we understand education. Functionalists view education as an important social institution that contributes both manifest and
latent functions. Functionalists see education as serving the needs of society by preparing students for later roles, or functions, in society. Conflict theorists see schools as a
means for perpetuating class, racial-ethnic, and gender inequalities. In the same vein, feminist theory focuses specifically on the mechanisms and roots of gender inequality in
education. The theory of symbolic interactionism focuses on education as a means for labeling individuals.
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