9.3: Early, Middle, and Late Renaissance Music
Early Renaissance Music (1400–1467)
This group gradually dropped the late medieval period’s complex devices of isorhythm and extreme syncopation, resulting in a more limpid and flowing style. What their music lost in rhythmic complexity, however, it gained in rhythmic vitality, as a “drive to the cadence” became a prominent feature around mid-century.
Middle Renaissance Music (1467–1534)
In the early 1470s, music started to be printed using a printing press. Music printing had a major effect on how music spread for not only did a printed piece of music reach a larger audience than any manuscript ever could, it did it far more cheaply as well. Also during this century, a tradition of famous makers began for many instruments. These makers were masters of their craft. An example is Neuschel for his trumpets.
Towards the end of the fifteenth century, polyphonic sacred music (as exemplified in the masses of Johannes Ockeghem and Jacob Obrecht) had once again become more complex, in a manner that can perhaps be seen as correlating to the increased exploration of detail in painting at the time. Ockeghem, particularly, was fond of canon, both contrapuntal and mensural. He composed a mass, Missa prolationum , in which all the parts are derived canonically from one musical line.
It was in the opening decades of the next century that music felt in a tactus (think of the modern-time signature) of two semibreves-to-a-breve began to be as common as that with three semibreves-to-a-breve, as had prevailed prior to that time.
In the early sixteenth century, there is another trend towards simplification, as can be seen to some degree in the work of Josquin des Prez and his contemporaries in the Franco-Flemish School, then later in that of G. P. Palestrina, who was partially reacting to the strictures of the Council of Trent, which discouraged excessively complex polyphony as inhibiting understanding the text. Early sixteenth-century Franco-Flemings moved away from the complex systems of canonic and other mensural play of Ockeghem’s generation, tending toward points of imitation and duet or trio sections within an overall texture that grew to five and six voices. They also began, even before the Tridentine reforms, to insert ever-lengthening passages of homophony, to underline important text or points of articulation. Palestrina, on the other hand, came to cultivate a freely flowing style of counterpoint in a thick, rich texture within which consonance followed dissonance on a nearly beat-by-beat basis, and suspensions ruled the day. By now, tactus was generally two semibreves per breve with three per breve used for special effects and climactic sections; this was a nearly exact reversal of the prevailing technique a century before.
Late Renaissance Music (1534–1600)
In Venice, from about 1534 until around 1600, an impressive polychoral style developed, which gave Europe some of the grandest, most sonorous music composed up until that time, with multiple choirs of singers, brass and strings in different spatial locations in the Basilica San Marco di Venezia. These multiple revolutions spread over Europe in the next several decades, beginning in Germany and then moving to Spain, France and England somewhat later, demarcating the beginning of what we now know as the Baroque musical era.
The Roman School was a group of composers of predominantly church music in Rome, spanning the late Renaissance and early Baroque eras. Many of the composers had a direct connection to the Vatican and the papal chapel, though they worked at several churches; stylistically they are often contrasted with the Venetian School of composers, a concurrent movement which was much more progressive. By far the most famous composer of the Roman School is Giovanni Pierluigi da Palestrina. While best known as a prolific composer of masses and motets, he was also an important madrigalist. His ability to bring together the functional needs of the Catholic Church with the prevailing musical styles during the Counter-Reformation period gave him his enduring fame.
The brief but intense flowering of the musical madrigal in England, mostly from 1588 to 1627, along with the composers who produced them, is known as the English Madrigal School. The English madrigals were a cappella, predominantly light in style, and generally began as either copies or direct translations of Italian models. Most were for three to six voices.
Musica reservata is either a style or a performance practice in a cappella vocal music of the latter, mainly in Italy and southern Germany, involving refinement, exclusivity, and intense emotional expression of sung text.
The cultivation of European music in the Americas began in the sixteenth century soon after the arrival of the Spanish, and the conquest of Mexico. Although fashioned in European style, uniquely Mexican hybrid works based on native Mexican language and European musical practice, appeared very early. Musical practices in New Spain continually coincided with European tendencies throughout the subsequent Baroque and Classical music periods. Among these New World composers were Hernando Franco, Antonio de Salazar, and Manuel de Zumaya.
In addition, many composers observed a division in their own works between a prima pratica (music in the Renaissance polyphonic style) and a seconda pratica (music in the new style) during the first part of the seventeenth century.
Masses
The fifteenth- and sixteenth-century masses had two kinds of sources that were used, monophonic and polyphonic, with two main forms of elaboration, based on cantus firmus practice or, beginning some time around 1500, the new style of pervasive imitation. Four types of masses resulted:
- Cantus firmus mass (tenor mass)
- The cantus firmus/imitation mass
- The paraphrase mass
- The imitation mass (parody mass)
Masses were normally titled by the source from which they borrowed. Cantus firmus mass uses the same monophonic melody, usually drawn from chant and usually in the tenor and most often in longer note values than the other voices.
Mannerism
In the late sixteenth century, as the Renaissance era closed, an extremely manneristic style developed. In secular music, especially in the madrigal, there was a trend towards complexity and even extreme chromaticism (as exemplified in madrigals of Luzzaschi, Marenzio, and Gesualdo). The term “mannerism” derives from art history.
Transition to the Baroque
Beginning in Florence, there was an attempt to revive the dramatic and musical forms of Ancient Greece, through the means of monody, a form of declaimed music over a simple accompaniment; a more extreme contrast with the preceding polyphonic style would be hard to find; this was also, at least at the outset, a secular trend. These musicians were known as the Florentine Camerata.
Contributors and Attributions
- Renaissance music. Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/Renaissance_music . License : CC BY-SA: Attribution-ShareAlike
- DESPREZ . Authored by : Clattuc . Provided by : Wikimedia Commons. Located at : https://commons.wikimedia.org/wiki/File:DESPREZ.jpg#/media/File:DESPREZ.jpg . License : Public Domain: No Known Copyright
- The Musicians. Provided by : Metropolitan Museum of Art. Located at : https://commons.wikimedia.org/wiki/File:Caravaggio_-_I_Musici.jpg . License : Public Domain: No Known Copyright


10.7: RLC Series Circuits
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Learning Objectives
By the end of this section, you will be able to:
- Determine the angular frequency of oscillation for a resistor, inductor, capacitor ( RLC ) series circuit
- Relate the RLC circuit to a damped spring oscillation
When the switch is closed in the RLC circuit of Figure \(\PageIndex{1a}\), the capacitor begins to discharge and electromagnetic energy is dissipated by the resistor at a rate \(i^2 R\). With U given by Equation 14.4.2 , we have
\[\frac{dU}{dt} = \frac{q}{C} \frac{dq}{dt} + Li \frac{di}{dt} = -i^2 R \nonumber \]
where i and q are time-dependent functions. This reduces to
\[L\frac{d^2q}{dt^2} + R\frac{dq}{dt} + \frac{1}{C}q = 0. \label{14.44} \]
This equation is analogous to
\[m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0, \nonumber \]
which is the equation of motion for a damped mass-spring system (you first encountered this equation in Oscillations ). As we saw in that chapter, it can be shown that the solution to this differential equation takes three forms, depending on whether the angular frequency of the undamped spring is greater than, equal to, or less than b /2 m . Therefore, the result can be underdamped \((\sqrt{k/m} > b/2m)\), critically damped \((\sqrt{k/m} = b/2m)\), or overdamped \((\sqrt{k/m} < b/2m)\). By analogy, the solution q ( t ) to the RLC differential equation has the same feature. Here we look only at the case of under-damping. By replacing m by L , b by R , k by 1/ C , and x by q in Equation \ref{14.44}, and assuming \(\sqrt{1/LC} > R/2L\), we obtain
Note
\[q(t) = q_0 e^{-Rt/2L} cos (\omega't + \phi) \label{14.45} \]
where the angular frequency of the oscillations is given by
Note
\[\omega' = \sqrt{\frac{1}{LC} - \left(\frac{R}{2L}\right)^2} \label{14.46} \]
This underdamped solution is shown in Figure \(\PageIndex{1b}\). Notice that the amplitude of the oscillations decreases as energy is dissipated in the resistor. Equation \ref{14.45} can be confirmed experimentally by measuring the voltage across the capacitor as a function of time. This voltage, multiplied by the capacitance of the capacitor, then gives q ( t ).
Note
Try an interactive circuit construction kit that allows you to graph current and voltage as a function of time. You can add inductors and capacitors to work with any combination of R , L , and C circuits with both dc and ac sources.
Note
Try out a circuit-based java applet website that has many problems with both dc and ac sources that will help you practice circuit problems.
Exercise \(\PageIndex{1}\)
In an RLC circuit, \(L = 5.0 \, mH\), \(C = 6.0 \, \mu F\), and \(R = 200 \, \Omega\). (a) Is the circuit underdamped, critically damped, or overdamped? (b) If the circuit starts oscillating with a charge of \(3.0 \times 10^{-3}C\) on the capacitor, how much energy has been dissipated in the resistor by the time the oscillations cease?
- Answer
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a. overdamped; b. 0.75 J


13.3: Factors Affecting Capacitance
There are three basic factors of capacitor construction determining the amount of capacitance created. These factors all dictate capacitance by affecting how much electric field flux (relative difference of electrons between plates) will develop for a given amount of electric field force (voltage between the two plates):
PLATE AREA: All other factors being equal, greater plate area gives greater capacitance; less plate area gives less capacitance.
Explanation: Larger plate area results in more field flux (charge collected on the plates) for a given field force (voltage across the plates).
PLATE SPACING: All other factors being equal, further plate spacing gives less capacitance; closer plate spacing gives greater capacitance.
Explanation: Closer spacing results in a greater field force (voltage across the capacitor divided by the distance between the plates), which results in a greater field flux (charge collected on the plates) for any given voltage applied across the plates.
DIELECTRIC MATERIAL: All other factors being equal, greater permittivity of the dielectric gives greater capacitance; less permittivity of the dielectric gives less capacitance.
Explanation: Although its complicated to explain, some materials offer less opposition to field flux for a given amount of field force. Materials with a greater permittivity allow for more field flux (offer less opposition), and thus a greater collected charge, for any given amount of field force (applied voltage).
“Relative” permittivity means the permittivity of a material, relative to that of a pure vacuum. The greater the number, the greater the permittivity of the material. Glass, for instance, with a relative permittivity of 7, has seven times the permittivity of a pure vacuum, and consequently will allow for the establishment of an electric field flux seven times stronger than that of a vacuum, all other factors being equal.
The following is a table listing the relative permittivities (also known as the “dielectric constant”) of various common substances:
An approximation of capacitance for any pair of separated conductors can be found with this formula:
A capacitor can be made variable rather than fixed in value by varying any of the physical factors determining capacitance. One relatively easy factor to vary in capacitor construction is that of plate area, or more properly, the amount of plate overlap.
The following photograph shows an example of a variable capacitor using a set of interleaved metal plates and an air gap as the dielectric material:
As the shaft is rotated, the degree to which the sets of plates overlap each other will vary, changing the effective area of the plates between which a concentrated electric field can be established. This particular capacitor has a capacitance in the picofarad range, and finds use in radio circuitry.


8.2: Concept
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Imagination is the ability to form images, perceptions, and concepts. In fact, images, perceptions and concepts are not perceived through sight, hearing or other senses. Imagine the work of the mind and help create fantasy. Imagination helps to provide meaning and provide an understanding of knowledge; imagination is the basic ability of people to create meaning for the world; imagination also plays a key role in the learning process. The basic training method for cultivating imagination is to listen to stories; when listening to stories, the accuracy of wording is the basic factor of “generating the world”. Imagine any power we face. We combine what we touch, see and hear into a "picture" by imagination. It is widely believed that as an intrinsic ability, as a factor in perceiving the public world from the senses, in the process of inventing a partial or complete personal domain in the mind, the term is used in the professional use of psychology, meaning the mind. In the recovery process, restores the sensory object previously presented to the perception. Because the use of this term contradicts the use of everyday language, some psychologists tend to describe this process as an "image process" or "image", or as a "regeneration" imaginary process, using "generating or constructive" The imaginary process "images of imagination" are seen with the "eye of the soul." Imagination can also be expressed through fairy tales or imaginary situations. The most famous invention or entertainment product is created by one's imagination. One hypothesis about the evolutionary process of human imagination is that imagination allows conscious species to solve problems by using psychological simulations. Children are often considered to be imaginative groups. Because their way of thinking has not yet formed, there are fewer ideological restrictions and rules than adults. Therefore, it is often imaginative. Children often use stories or pretend games to practice their imagination. When children have fantasies, they play a role on two levels: on the first level, they use role-play to achieve what they create with imagination; at the second level, they pretend to believe that the situation is to play games.
In terms of behavior, what they create seems to be a real reality, and this already exists in the myth of the story. Artists often need a lot of imagination because they are engaged in creative work, and imagination can help them break existing rules and bring aesthetics into a new model. Imagine the universal use of this term, which means the process of forming an image that has not been experienced before in the mind, or the process of forming an image that is at least partially experienced or formed by different combinations. Some typical examples are as follows: fairy tale Fiction The illusion of forms inspired by fantasy novels and science fiction spurs readers to pretend that these stories are real, and by resorting to fantasy objects such as books or fantasy, these objects are not in the fantasy world. Imagination in this sense (not limited to the precise knowledge gained from actual needs) is somewhat exempt from objective limitations. Imagine being in the position of another person. This ability is very important for social relationships and social understanding. Einstein said: "Imagination... is more important than knowledge. Knowledge is limited. Imagination encompasses the world." [8] However, in all fields, even imagination is limited: therefore, one Human imagination does violate the basic law, or the inevitable principle of violating the actual possibility, or the possibility of rationality in a certain situation, is considered a mental disorder. The same limitations place imagination in the realm of scientific assumptions. The advancement of scientific research is largely due to some temporary explanations; these explanations are based on imagination. However, these assumptions must be made up of previously identified facts and must be coordinated with the principles of a particular science. Imagination is an experimental part of thinking that creates functional-based theories and ideas. Separate objects from real perceptions and use complex "features" to envision creating new ideas or changing ideas. This part of the thinking is crucial to improving and improving the way new and old tasks are completed. These experimental ideas can be implemented steadily in the field of simulation; then, if the concept is very likely and its function is real, then the concept can be implemented in reality. Imagination is the key to the new development of the soul. It shares with others and progresses together. Imagination can be divided into: Automatic imagination (from dreams and daydreams) Don't automatically imagine (renewable imagination, creative imagination, future dreams)


1.6: Consequences of Light Travel Time
There is another reason the speed of light is such a natural unit of distance for astronomers. Information about the universe comes to us almost exclusively through various forms of light, and all such light travels at the speed of light—that is, 1 light-year every year. This sets a limit on how quickly we can learn about events in the universe. If a star is 100 light-years away, the light we see from it tonight left that star 100 years ago and is just now arriving in our neighborhood. The soonest we can learn about any changes in that star is 100 years after the fact. For a star 500 light-years away, the light we detect tonight left 500 years ago and is carrying 500-year-old news.
Because many of us are accustomed to instant news from the Internet, some might find this frustrating.
“You mean, when I see that star up there,” you ask, “I won’t know what’s actually happening there for another 500 years?”
But this isn’t the most helpful way to think about the situation. For astronomers, now is when the light reaches us here on Earth. There is no way for us to know anything about that star (or other object) until its light reaches us.
But what at first may seem a great frustration is actually a tremendous benefit in disguise. If astronomers really want to piece together what has happened in the universe since its beginning, they must find evidence about each epoch (or period of time) of the past. Where can we find evidence today about cosmic events that occurred billions of years ago?
The delay in the arrival of light provides an answer to this question. The farther out in space we look, the longer the light has taken to get here, and the longer ago it left its place of origin. By looking billions of light-years out into space, astronomers are actually seeing billions of years into the past. In this way, we can reconstruct the history of the cosmos and get a sense of how it has evolved over time.
This is one reason why astronomers strive to build telescopes that can collect more and more of the faint light in the universe. The more light we collect, the fainter the objects we can observe. On average, fainter objects are farther away and can, therefore, tell us about periods of time even deeper in the past. Instruments such as the Hubble Space Telescope (Figure 1.5) and the Very Large Telescope in Chile (which you will learn about in the chapter on Astronomical Instruments), are giving astronomers views of deep space and deep time better than any we have had before.


5.15: Skills Checklist- Nail Care
NOTE: Nail care for clients with diabetes should be performed by a Registered Nurse (RN).
- Gather Supplies: Basin, warm water, soap, lotion, two washcloths, one towel, barrier, gloves, manicure stick, emery board, nail clipper, and linen bag or hamper
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Routine Pre-Procedure Steps:
- Knock on the resident’s door.
- Perform hand hygiene.
- Maintain respectful, courteous, and professional communication at all times.
- Introduce yourself and identify the resident.
- Provide for privacy.
- Explain the procedure to the resident.
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Procedure Steps:
- Put on gloves.
- Fill the basin with warm water and place it on a flat surface with a barrier underneath. Have the resident check the water temperature by placing their hand in the basin or putting a wet washcloth on the back of their hand.
- Have the resident perform hand hygiene with sanitizer.
- Immerse the client’s hands in warm water for 5 to 20 minutes.
- Place their hand on a barrier.
- Using a manicure stick, clean underneath each nail, wiping any debris on the barrier after each nail.
- If necessary, trim nails using a clipper. Sanitize the clipper prior to and after use.
- Using an emery board, file each nail from the outside of the nail towards the middle of the nail.
- Check each nail for snags and file until smooth.
- Rinse the hand in water, return to the barrier, and dry.
- Repeat the procedure for the second hand.
- Offer lotion. If applying lotion, wear gloves.
- Rub the lotion gently into the skin if requested.
- Wipe off any excess lotion with a dry towel.
- While wearing gloves, empty the equipment.
- Rinse the equipment.
- Dry the basin.
- Return the equipment to storage.
- Dispose of soiled linen in a designated laundry hamper.
- Remove the gloves, turning them inside out.
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Post-Procedure Steps:
- Perform hand hygiene.
- Check for resident comfort and ask if anything else is needed.
- Ensure the bed is low and locked. Check the brakes.
- Place the call light or signaling device within reach of the resident.
- Open the door and privacy curtain.
- Perform hand hygiene
- Document and report any skin or nail issues or changes noted with the resident.
View a YouTube video [1] of an instructor demonstrating n ail care:
- Chippewa Valley Technical College. (2022, December 3). Nail Care. [Video]. YouTube. Video licensed under CC BY 4.0 . https://youtu.be/T32Csl2Rx0s ↵


10.7.3: Traditional and Contemporary Violence against Women
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Female Genital Mutilation
Female genital mutilation (FGM) is the traditional cutting, circumcision, and removal of most or all external genitalia from women with the end result of closing off some or part of the vagina until such time as the woman is married and cut open. Female genital mutilation is often performed in order to preserve the purity of females before marriage—a cultural ideal in some societies. In some traditions, there are religious underpinnings. In others, there are customs and rituals that have been passed down. In no way does the main body of any world religion condone or mandate this practice. Many countries where this takes place are predominantly Muslim; yet local traditions have corrupted the purer form of the religion and its beliefs. Female genital mutilation predates Islam. 47
There are no medical or therapeutic benefits from female genital mutilation. Quite the contrary, there are many adverse medical consequences that result from it, ranging from pain, difficulty in childbirth, illness, and even death. Many human rights groups, the United Nations, scientists, advocates, the United States, the World Health Organization, and others have made aggressive efforts to effect the cessation of this practice worldwide, but progress has come very slowly. Part of the problem is that women often perform the ritual and carry on the tradition as it was perpetrated upon them.
Foot Binding
A small foot in China, no different from a tiny waist in Victorian England, once represented the height of female refinement. For families with marriageable daughters, foot size translated into its own form of currency and a means of achieving upward mobility. Tiny feet were a symbol of wealth, as women whose feet were bound were unable to work in the fields or on their feet for a long period of time. This economic symbol eventually translated into a sexually desirable symbol for possible suitors. The most desirable bride possessed a three-inch foot, known as a “golden lotus.” It was respectable to have four- inch feet—a silver lotus—but feet five inches or longer were seen as too large and named “iron lotuses.” The marriage prospects for a girl with feet five inches or larger were slim.
Do you have an iPhone? If so, hold it up. Your iPhone (doesn’t matter the model) is close to five inches long. So feet the size of your iPhone were seen as unattractive, minimizing a women’s opportunity for marriage to someone with high social worth. In fact, women with feet one inch shorter than that iPhone were still not allotted the same worth as women whose feet were two inches shorter than that phone.
How does foot binding work?
First, beginning at the age of two or three, her feet were plunged into hot water and her toenails clipped short. Then the feet were massaged and oiled before all the toes, except the big toe, were broken and bound flat against the sole, making a triangle shape. Next, her arch was strained as the foot was bent double. Finally, the feet were bound in place using a silk strip measuring ten feet long and two inches wide. These wrappings were briefly removed every two days to prevent blood and pus from infecting the foot. Sometimes “excess” flesh was cut away or encouraged to rot. The girls were forced to walk long distances in order to hasten the breaking of their arches. Over time the wrappings became tighter and the shoes smaller as the heel and sole were crushed together. After two years the process was complete, creating a deep cleft that could hold a coin in place. Once a foot had been crushed and bound, the shape could not be reversed without a woman undergoing the same pain all over again. 48
The truth, no matter how unbelievable, is foot-binding was experienced and enforced by women. Though the practice is rejected in China today—the last shoe factory making lotus shoes did not close until 1999—it survived for a thousand years in part because of women’s social investment in the practice.
Child Marriage
Child marriage, defined as a formal marriage or informal union before age 18, is a reality for both boys and girls, although girls are disproportionately affected. Today, about a third of women aged 20-24 years old in the developing world are married as children. Children who are married before the age of 18 are more at risk for domestic violence, rape from their husbands, and even murder. Some 10 million girls a year are married off before the age of 18 across the world, according to a UNICEF report released this year. While the majority of child marriages in any singular region are performed in Sub-Sarah Africa, India is responsible for a disproportionate amount of these underage unions, as well.
Figure \(4.3.1\) Chart of “The Highest Rates of Child Marriage are Found in sub-Saharan Africa” graph depicting the highest rates of child marriage are found in sub-Saharan Africa.
Child marriages are illegal in India, and are punishable with a fine and two years in prison for anyone who performs, conducts, or negligently fails to prevent a child marriage. But this tradition is so ingrained in Indian culture that, especially in remote villages, child marriage is usually fully supported by the entire community, and it is rare for someone to inform the police so these marriages can be stopped.
In many communities girls are seen as an economic burden, and marriage transfers the responsibility to a girl’s new husband. Poverty and marriage expenses such as the dowry may lead a family to marry off a daughter at a young age to reduce these expenses.
Patriarchy, class, and caste also influence the norms and expectations around the role of women and girls in India. In many communities restrictive norms limit girls to the roles of daughter, wife, and mother. Girls are seen as the property of their father and then of their husband. Poor educational opportunities for girls, especially in rural areas, also increase girls’ vulnerability to child marriage.
 Rape is not the same as sex. Rape is violence, motivated primarily by men and primarily for power. Rape is dangerous and destructive and more likely to happen in the United States than in most other countries of the world. There are 195 countries in the world today. The U.S. typically is among the top five percent in terms of rape. Consecutive studies performed by the United Nations Surveys on Crime Trends and the Operations of Criminal Justice Systems confirm that South Africa is the most dangerous, crime-ridden nation on the planet in all crimes including rape. 49
The United Nations reported, according to World Bank data, women aged 15 to 44 are more at risk from rape and domestic violence than from cancer, motor accidents, war, or malaria. 50
A 1997 study on the non-institutionalized, non-military population by the U.S. Bureau of Justice Statistics, which defines rape as forced penetration by the offender, 51 found that 91% of reported rape victims are female and 9% are male. 52
The majority of rapes in the United States go unreported. 53 According to the American Medical Association (1995), sexual violence, and rape in particular, is considered the most under- reported violent crime. 54 Some of the most common reasons given by victims for not reporting rape are fear of retaliation, shame, and blaming themselves for the occurrence of the act itself. Under-reporting affects the accuracy of this data.
Other Forms of Oppressions
Veiling
The mandatory covering of females’ bodies from head to toe has been opposed by some and applauded by others. Christians, Hindus, and many other religious groups have the practice of covering or veiling in their histories. Yet, over the last 30 years, fundamentalist Muslim nations and cultures have returned to their much more traditional way of life. Hijab is the Arabic word that means to cover or veil and has become more common in recent years (ħijāb or حجاب .(,Often Hijab means modest and private in the day-to-day interpretations of the practice. For some countries it is a personal choice, while for others it becomes a crime not to comply. The former Taliban punished such a crime with death (they also punished formal schooling of females and the use of makeup by death). Many women’s rights groups have brought public attention to this trend, not so much because the mandated covering of females is that oppressive, but because the veiling and covering is symbolic of the religious, traditional, and labor- forced patterns of oppression that have caused so many problems for women and continue to do so today.
Misogynistic Language
The public demeaning of women has been acceptable throughout various cultures because publicly demeaning members of society who are privately devalued and or considered flawed, fits the reality of most day-to-day interactions. Misogyny is the hatred of women often manifested as physical or verbal abuse and oppressive mistreatment of women. Verbal misogyny is unacceptable in public in most Western Nations today. With the ever-present technology found in cell phones, video cameras, and security devices, a person’s private and public misogynistic language can be easily recorded and shared. Can you think of any examples of public figures privately demeaning women, only to be shared in a public forum later?
Perhaps this fear of being found out as a woman-hater is not the ideal motivation for creating cultural values of respect and even admiration of women and men. As was mentioned above, most of the world historical leaders assumed that women were not as valuable as men. Women were treated as the totality of their reproductive role, as breeders of the species, rather than the valued human beings they are throughout the world today.
47
See Obermeyer, C.M. March 1999, Female Genital Surgeries: The Known and the Unknowable. Medical Anthropology Quaterly13, pages 79-106;p retrieved 5 December from
http://www.anthrosource.net/doi/abs/...q.19<IP_ADDRESS>
48
Read more:
http://www.smithsonianmag.com/histor...na-millennium-
49
See
http://www.unodc.org/unodc/en/data-a...me-Trends-and-
the-
Operations-of-Criminal-Justice-Systems.html
50
Retrieved 5 December, 2008 from
http://www.un.org/women/endviolence/docs/VAW.pdf
, Unite To End Violence Against Women, Feb. 2008
51
Retrieved 2016-11-19 from
http://www.bjs.gov/index.cfm?ty=tp&tid=317#terms_def
52
Retrieved 2016-11-19 from
http://www
.
bjs.ojp.usdoj.gov. Pages 5 and 8.
53
"
Reporting of Sexual Violence Incidents
".
National Institute of Justice
. Retrieved June 7, 2016.
54
American Medical Association (1995) Sexual Assault in America. AMA.


2.9: The Prime Number Theorem
We have already observed that there are 4 primes less than 10, 25 primes less than 100, and 168 primes less than 1000. And there are 78 498 primes less than 10 6 . So
40% of integers < 10 are prime;
25% of integers < 100 are prime;
16.8% of integers < 1000 are prime; and
7.8498% of integers < 10 6 are prime.
In other words, the fraction of integers which are prime numbers diminishes as we go up.
The first question to ask is whether prime numbers themselves “run out” at some stage, or whether they go on for ever. The answer is very like that for the counting numbers, or positive integers 1, 2, 3,4, 5,...:
the counting process certainly gets started (with 1); and no matter how far we go, we can always “add 1” to get a larger counting number.
Hence we conclude that the counting numbers “go on for ever”.
Problem 76
(a) (i) Start the process of generating prime numbers by choosing your favorite small prime number and call it p 1 .
(ii) Then define .
(b) (i) Since , n 1 must be divisible by some prime. Explain why p 1 is not a factor of n 1 . (What is the remainder when we divide n 1 by p 1 ?)
(ii) Let p 2 be the smallest prime factor of n 1 .
(iii) Define
(c) (i) Since n 2 > 1, n 2 must be divisible by some prime. Explain why p 1 and p 2 are not factors of n 2 . (What is the remainder when we divide n 1 by p 1 , or by p 2 ?)
(ii) Let p 3 be the smallest prime factor of n 2 .
(iii) Define
(d) Suppose we have constructed k distinct prime numbers p 1 , p 2 , p 3 , ... , p k . Explain how we can always construct a prime number p k+1 different from .
(e) Apply the above process with p 1 = 2 to find p 2 , p 3 , p 4 , p 5 .
Once we know that the prime numbers go on for ever, we would like to have a clearer idea as to the frequency with which prime numbers occur among the positive integers. We have already noted that
- there are 4 primes between 1 and 10,
- and again 4 primes between 10 and 20;
- but there is only 1 prime in the 90s;
- and then 4 primes between 100 and 110.
- And there are no primes at all between 200 and 210.
In other words, the distribution of prime numbers seems to be fairly chaotic. Our understanding of the full picture remains fragmentary, but we are about to see that the apparent chaos in the distribution of prime numbers conceals a remarkable pattern just below the surface.
The next item is only an experiment; but it is a very suggestive experiment. It is artificial, in that what you are invited to count has been carefully chosen to point you in the right direction. The resulting observation is generally referred to as the Prime Number Theorem. The result was conjectured by Legendre (1752–1833) and by Gauss (1777–1855) in the late 1790s – and was proved 100 years later (independently and almost simultaneously) in 1896 by the French mathematician Hadamard (1865–1963) and by the Belgian mathematician de la Vallée Poussin (1866–1962). You will need to access a list of prime numbers up to 5000 say.
Problem 77 Let denote the number of prime numbers : so , , , . You are invited to count the number of primes up to certain carefully chosen numbers, and then to study the results. The pattern you should notice works just as well for other numbers – but is considerably harder to spot.
The special values we choose for “ x ” are
the next integer above successive powers of the special number e ,
where e is an important constant in mathematics – an irrational number whose decimal begins 2.7182818 · · ·, and which has its own button on most calculators (see Problem 248 ).
(a) Complete the following table.
| n | e n | next integer N | π ( N ) |
|---|---|---|---|
| 1 | 2.718⋯ | 3 | 2 |
| 2 | 7.389⋯ | ||
| 3 | 20.08⋯ | ||
| 4 | 54.59⋯ | ||
| 5 | 148.41⋯ | ||
| 6 | 403.42⋯ | ||
| 7 | 1096.63⋯ | ||
| 8 | 2980.95⋯ | ||
| 9 | 8103.08⋯ | 1019 |
(b) Find an expression that seems to specify as a function of n . Hence conjecture an expression for in terms of x .
Durch planmassiges Tattonieren.
[Through systematic fumbling.]
Carl Friedrich Gauss (1777-1855),
when asked how he came to make so many profound discoveries in mathematics.


5.1: Importance of Early Childhood Mathematics and Developing Number Sense
Course Competency 5. Examine strategies that teach early math skills
Criteria 5.1. describe the continuum of number and counting development.
Introduction to Mathematics
In this chapter, you'll explore the importance of early math education, and best teaching practices in early childhood mathematics. You will be introduced to the joint position statement from the NCTM and NAEYC on early mathematics. You'll discover the mathematical strands and how we can support children develop skill and understanding of this content.
Math Learning Starts Early
“Mathematical knowledge begins during infancy and undergoes extensive development over the first 5 years of life. It is just as natural for young children to think mathematically as it is for them to use language, because “humans are born with a fundamental sense of quantity” (Geary, 1994, p. 1), as well as spatial sense, a propensity to search for patterns, and so forth” (Clements, Sarama, & DiBiase, 2004).
Emergent mathematics is the earliest phase of development of mathematical and spatial concepts. Emergent mathematics encompasses the skills and attitudes that a child develops in relation to math concepts throughout the early childhood period. Emergent mathematics and those foundational math skills are not optional. They are necessary to the success of each and every student.
It is widely accepted that literacy learning begins the day a child is born. Reading to infants and toddlers and preschoolers is an early predictor of positive literacy success. Mathematical understanding can be regarded in the same way. During the first months of life, a child begins to construct the foundations for mathematical concepts. For example, before a child can count, he/she can construct ideas about mathematics. They recognize “more,” “less,” and basic equality.
Read the articles, “Early Math Skills Predict Later Academic Success” by Nancy Christensen. You will notice that early childhood mathematics is critical for the future success of children (Duncan, 2007). As you read this article, reflect on what you have learned, what difference it makes to you as a future teacher and the children you will teach, and what you can do with this information.
Best Teaching Practices
As a teacher of mathematics, please remember these fundamental beliefs.
- Thinking about the problem, not just the answer, is what is most important. The teacher should be the facilitator, not the one with all the answers.
- Process is more important than product. Mathematics is not just “facts” to be memorized. The concepts children learn are of upmost importance; and then the memorization will come. Children must have experiences through play and using manipulatives and “reinvent” the concepts of mathematics in their own minds.
The National Council of Teachers of Mathematics (2014), in its publication Principles to Action, represents a unified vision of what is needed for teaching all students.
The Eight Mathematics Teaching Practices can be found in the Executive Summary.
- Establish mathematics goals to focus learning.
- Implement tasks that promote reasoning and problem-solving.
- Use and connect mathematical representations.
- Facilitate meaningful mathematical discourse.
- Pose purposeful questions.
- Build procedural fluency from conceptual understanding.
- Support productive struggle in learning mathematics.
- Elicit and use evidence of student thinking.
Read more about the Mathematics Teaching Practice in the Executive Summary.
Treat Children as Mathematicians
- Mathematicians often work for an extended time on a single problem. Allow your students ample time to work on a problem.
- Mathematicians collaborate with their colleagues and study the work of others. Allow your students to collaborate, argue, consult, defend, ask, explain, and pose questions to others.
- Mathematicians must prove for themselves that their solution is correct. Children must also prove to themselves that they have the correct solution. Give students time to think and discuss. Don’t be too quick to tell a student their answer is correct or incorrect; instead ask the student to explain their answer.
- The problems mathematicians work on are complex. Good problems ask students to find innovative solutions without a time limit. Problems should spark children’s thinking, rather than promote rote memorization.
- Mathematicians get satisfaction from the problem-solving process and take pride in their solutions. Children will understand the concepts and procedures more if they are allowed to use their own thinking processes. This also allows children to make connections to prior knowledge and real-life experiences.
- Mathematicians use unsuccessful attempts as stepping-stones to solutions. We need to emphasize mathematical thinking rather than just getting the correct answer.
NCTM/NAEYC Position Statement on Early Start in Learning Mathematics
The National Council of Teachers of Mathematics (NCTM) and the National Association for the Education of Young Children (NAEYC) published a Joint Position Statement (2010) affirming high-quality, challenging, and accessible mathematics education for 3- to 6-year-old children is a vital foundation for future mathematics learning. The ten research-based, essential recommendations to guide your classroom practice are:
- Enhance children’s natural interest in using mathematics to make sense of their world.
- Build on children’s background experience and knowledge.
- Base instruction on knowledge of children’s cognitive, linguistic, physical, and social-emotional development.
- Strengthen children’s problem-solving and reasoning processes as well as representing, communicating, and connecting mathematical ideas.
- Ensure a coherent curriculum that is compatible with sequences of important mathematical ideas.
- Provide for children’s deep and sustained interactions with mathematical ideas.
- Integrate mathematics with other activities and other activities with mathematics.
- Provide ample time, materials, and teacher support for children to engage in play where they explore mathematical ideas.
- Use a range of experiences and teaching strategies to introduce mathematical concepts, methods, and language.
- Support children’s learning by thoughtful and continuous assessment.
Allow children to work for long periods of time on one problem. Good problems can have multiple solutions and different ways to arrive at those solutions; therefore, allow children to use their own methods for solving a given problem. Children’s excitement about mathematics comes from their own thinking abilities; encourage social interaction that promotes children to act as young mathematicians by requiring them to prove their answer and all the steps they took to attain the answer. Additionally, allow children to be wrong many times before being right. It is important to encourage children to see “wrong” answers as a natural part of mathematical processes.
Mathematical Strands
Mathematical strands are the different domains or areas of mathematics knowledge. The Wisconsin Model Early Learning Standards has divided this knowledge into 6 areas. You will see the areas reflected in the standards below.
B. EL. 1 Demonstrates an understanding of numbers and counting.
B. EL. 2 Understands number operations and relationships.
B. EL. 3 Explores, recognizes, and describes, shapes and spatial relationships.
B. EL. 4 Uses the attributes of objects for comparison and patterning.
B. EL. 5 Understands the concept of measurement.
B. EL. 6 Collects, describes, and records information using all senses.
(Please note B.EL.1 and B.EL.2 refer to skills and knowledge in number sense. Some states keep those two components together, but Wisconsin has organized this knowledge into 2 separate standards.)
You will explore each of the strands throughout the chapter, starting with Number Sense below.
Number Sense
The earliest mathematical skills children develop deal with number recognition (identifying and naming numbers) and counting. Both are part of the foundational mathematic skill of number sense. (Refers to B. EL. 1 Demonstrates an understanding of numbers and counting.)
Number sense is defined as a “good intuition about numbers and their relations. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms” (Howden, 1989). When students consider the context of a problem, look at the numbers in a problem, and make a decision about which strategy would be most efficient in each particular problem, they have number sense. Number sense is the ability to think flexibly between a variety of strategies in context.
Building on subitizing, composing and decomposing numbers, using ten-frames, and the hundreds chart will help develop number sense in your students. When students have number sense, they gain computational fluency , are flexible in their thinking, and are able to choose the most efficient strategy to solve a problem. (Refers to B. EL. 2 Understands number operations and relationships.)
Number sense is a foundational idea in mathematics; it encourages students to think flexibly and promotes confidence with numbers. According to Marilyn Burns (2007), “students come to understand that numbers are meaningful and outcomes are sensible and expected.” And “just as our understanding of phonemic awareness has revolutionized the teaching of beginning reading, the influence of number sense on early math development and more complex mathematical thinking carries implications for instruction” (Gersten & Chard, 1999).
Number sense is a skill that allows students to work with numbers fluently and efficiently. This includes understanding skills such as quantities, concepts of more and less, symbols represent quantities (8 means the same thing as eight), and making number comparisons (12 is greater than 9, and three is half of six). Students must
- Understand numbers, ways of representing numbers, relationships among numbers, and the number system,
- Understand the meanings of operations and how they related to one another, and
Compute fluently and make reasonable estimates (NCTM, 2000).
Teachers must give students many opportunities to develop number sense through activities with physical objects, such as counters, blocks, or small toys. Most children need that concrete experience of physically manipulating groups of objects. After these essential experiences, a teacher can move to more abstract materials such as dot cards and ten frames. These materials would help to develop the more complex aspects of Number Sense represented in Wisconsin Model Early Learning Standard PERFORMANCE STANDARD: B.EL. 2 Understands number operations and relationshi ps.
The Development of Counting Skills/Number Sense
See the chart below to see how counting skills develop in young children.
Figure 9.5: Image by Ian Joslin is licensed by CC-BY-4.0
The information below is from the California Preschool Standards and is again provided as a reference to the age you may see these Number Sense skills appear. However, remember the Wisconsin Model Early Learning Standards include the skills for Number Sense in order of development but do not assign a specific age to any skills as all children develop at their own rate and time.
Examples in 1.0 would correlate to WMELS B.EL. 1 Demonstrates understanding of numbers and counting.
|
At around 48 months of age |
At around 60 months of age |
Supporting the Development of Number Sense Through Songs and Fingerplays
We know signing songs and participating in fingerplays is a great way to develop language skills in young children, but it is also a way to develop important concepts relating to number sense. See the Erikson Institute Early Math Collaborative for more information on Songs and Fingerplays with preschoolers to develop their number sense. Also, see their other resources to learn more about number sense and counting .
Giving children multiple opportunities to experience early number concepts and number sense is critical for early childhood teachers.
Children must experience mathematics through play and manipulatives as they make sense of numbers in their own minds.
Math Milestones
Children grow faster in the first two years of life; more than at any other period of development. During the first two years, the brain grows from 25% of its adult weight to 75% of its final adult weight. Additionally, gross motor skills develop; they use large muscle groups to crawl, sit, stand, and lift things. Fine motor skills develop more slowly: stacking blocks, stringing beads, zipping zippers, and buttoning buttons.
Children are learning about the world around them through experiencing it through sight, smell, sound, touch, and taste. They are also learning by thinking about things, forming hypotheses, testing them, and problem solving. This is Piaget’s sensorimotor stage of development. This stage is from birth to approximately 2 years of age, and is a period of rapid cognitive growth and development. Object permanence is the main development during the sensorimotor stage and is vital to mathematics because it is the understanding that an object continues to exist even if it is not seen. According to Geist, “object permanence is the first step toward representation. When a child knows something exists even when he can’t see it, it means he has a mental concept of the object in the absence of its presence. This is one of the bases upon which more advanced mathematics will be built. Numbers are representative of objects and quantities even if we do not see them” (Geist, 2009).
At a very early age, babies use mathematics concepts to make sense of their world. For example, they can signal “more” when they want more food. In addition, they know the difference between familiar and unfamiliar adults which is “sorting and classifying.” They have “spatial reasoning” as they play with boxes of different sizes, and they use “patterns” as they say words and phrases.
As early as 6 months old, children begin to develop an understanding of “more.” They ask for “more milk” or “more food.” Babies from 0-12 months begin to predict the sequence of events, such as running water means bath time, they begin to classify things into certain categories, such as toys that make noise and toys that don’t, and they begin to understand relative size, such as parents are big and babies are small (Morin, 2014).
At 12 to 18 months old, children start to recognize patterns and understand shapes as they play with objects. They begin to sort familiar objects by characteristics, and they enjoy filling and emptying containers. Many children at this age can complete simple puzzles when the puzzle pieces show the whole object.
Although each child progresses at different levels, generally children of 18 months to 24 months old can show “one” and “two” on their fingers. They know some number words. Although they do not understand quantity, they do begin to understand “take one” or “give me one.” Children can match same-sized shapes. Puzzles and toys of squares, triangles, and circles help children develop spatial reasoning.
At 24 to 30 months old, children are learning important mathematical concepts and skills through their play. They recognize patterns and begin to use logical reasoning to solve everyday problems. They can sort shapes and stack blocks by size, and be able to distinguish between “big and small” and “fast and slow.” Children will be able to say some numbers, although they may skip some numbers in the sequence. For example, they may say, “one, two, three, five, ten…” By the age of 3, children should be able to count one to ten. Children at this stage of development generally understand addition and subtraction with the numbers one and two.
At approximately three years old, a child may attend a preschool. But what does mathematics teaching and learning look like in a preschool classroom? All children have the potential to learn mathematics that is challenging and at a high-level (Sarama & Clements, 2009). There is a wide range of development between ages three and five. There will be children who are “almost potty trained” as well as children who have begun learning to read. “Sit-and-listen” approach is only effective if it is highly engaging and exciting, for example, a story being read to the class. Therefore, teaching should consist almost entirely of “active learning.” Teaching should emphasize student-directed/teacher-facilitated activities. Worksheets are inappropriate at this level.
At 3-4 years old, children can count up to 30; and be able to count backwards from ten. Additionally, they can use ordinal numbers “first,” “second,” and “third.” At this age, children will be able to use superlatives such as “big, bigger, biggest.” At age 4, children will understand the concept of addition as putting together and subtraction as taking from when objects are in front of them. They also recognize shapes, begin to sort things by color, shape, size, or purpose, understand that numerals stand for numbers, and have the spatial awareness to complete puzzles (Morin, 2014).
At age four, children begin to have one-to-one correspondence in which numbers correspond to specific quantities. For example, a child would point to the number of blocks on the table and say, “1, 2, 3, 4, 5. I have five blocks.”
At age five, children can identify the larger of two numbers and recognize numerals up to 20. They can copy or draw shapes, and understand basic time concepts such as days of the week. By age six, children can add and subtract within ten mentally and solve simple word problems.
Read more about Mathematical Milestones: Children’s Development of Mathematical Concepts: Ages 0-4 (Infants, Toddlers, & Preschoolers).
Infants and Toddlers
Mathematics is all around us! And we use mathematical language all the time, even without realizing it. For example, when we sort laundry into colors and whites, we are sorting and classifying. Keeping score at a game is number and operations. Spatial relationships occur when we give someone directions, and when we make comparisons such as big and little, that is measurement. Everyday activities include patterns and order. Teachers must make mathematics visible to children through math talks (Greenberg, 2012).
Infants and toddlers understand number and operations as we talk with them about numbers, quantity, and one-to-one correspondence. According to Greenberg (2012), we can say the following to young children:
- You have two eyes, and so does your bear. Let’s count: 1-2.
- I have more crackers than you do. See, I have 1, 2, 3, and you have 1, 2.  Now I have the same as you!
- That’s the third time you have said mama. You’ve said mama three times!
Infants and toddlers understand geometry by recognizing and naming shapes. According to Greenberg (2012), we can talk with our children about understanding shapes and spatial relationships:
- Look, Samantha went under the stool and Hartley is on top.

- Some of the cookies are square and some are round.
Algebraic thinking begins as children see patterns, relationships, and change as they recognize relationships that make up a pattern or create repetitions. For example, you can say to infants and toddlers:
- Grandpa has stripes on his shirt – red, white, blue, red, white, blue.
- Let’s clap to the beat of this song.
- I put the blocks in the bucket; you dump them out. I put the blocks back in the bucket; you dump them out.
Data analysis begins as infants and toddlers gather, sort, classify, and analyze information to make sense of their world. Teachers can talk with children about data by saying the following:
- Let’s put the big lid on the big bowl.
- You always smile when mommy sings.
Adapted from Greenberg, 2012
Math talks begin as soon as a child is born. During bath time, diaper time, meal time, or on walks or shopping trips is the perfect time to talk about shapes and sizes, talk about patterns, and describe things that are different. Mathematics comprehension is more than just numbers. Math is also about sorting things into categories such as big, bigger, and biggest; solving problems using patterns; solving spatial problems; and using logic.
Read more about Math Talks with infants and toddlers in this article, More, All Gone, Empty, Full: Math Talk Every Day in Every Way .
Kindergarten and First Grade
As children enter kindergarten and first grade, there are many changes in the physical, cognitive, and social-emotional development. Children in kindergarten can sit and attend to a formal lesson for approximately 10 minutes. At this level, the focus of mathematics instruction should be through experiences. By first grade, children can sit for a 15-20 minute formal lesson and although they are developing selective attention (the ability to decide what to attend to and what to ignore), the focus should still be on mathematics teaching through experiences.
Children in kindergarten and first grade are developing their gross and fine motor skills. According to Thelen and Smith, the development of coordination comes from physical play such as running and jumping and skipping (Thelen & Smith, 1996). Fine motor skills are developed as children write numbers and letters and use a pencil.
Kindergarteners can add by counting on their fingers, and start with six on the other hand; they can copy or draw symmetrical shapes, and can follow multi-step directions using first and next. In first grade, students can predict what comes next in a pattern, know the difference between two- and three-dimensional shapes, count to 100 by ones, twos, fives, and tens, and can write and recognize numbers, and do basic addition and subtraction up to 20 (Morin, 2014).
The Value of Play
Play does not guarantee mathematical development, but it offers rich possibilities. As the teacher, it is up to you to observe and develop a classroom when children have many opportunities to discover mathematical concepts through play. Douglas Clements says, “Anyone who is pushing arithmetic onto preschoolers is wrong. Do not hurry children.” (Clements, 2001, pg. 270). Mathematics should be about joy and challenges, not pressure and compulsion.
For example, children begin to recognize mathematical patterns by listening to music. They naturally make relationships between different-size and different-shaped blocks by sorting. According to Clements (2001), “Good early mathematics is broader and deeper than mere practice in counting and adding. It includes debating which child is bigger and drawing maps to ‘treasure’ buried outside. Quality mathematics instruction includes providing loads of unit blocks, along with loads of time to use them; asking children to get just enough pencils for everyone in the group; and challenging children to estimate and check how many steps are required to walk to the playground.”
Classrooms that offer children opportunities to engage in play-based learning activities help children grow not only academically. “Play is a particular attitude or approach to materials, behaviors, and ideas and not the materials or activities or ideas themselves; play is a special mode of thinking and doing” (McLane, 2003, p. 11). As discussed in an earlier chapter, there is a strong connection between early mathematics learning and later success. One determining factor of later achievement is quality early mathematical experiences.
Play is critical to a young child’s learning. Mathematics activities must be developmentally appropriate by “integrating logical thinking, decision making comparison, and the making of relationships throughout the child’s day into everything they do” (Geist, 2009). This is the goal of emergent mathematics – to immerse children in mathematics, number, and number concepts. Mathematics at this level must be taught through experiences. Engaging young children in play-based learning, teachers can include the following:
- Rhythm and music allows children to recognize mathematical patterns,
- Blocks and shapes gives children opportunities to make relationships between them, such as same and different,
- Everyday activities such as counting help children develop number concepts,
- Manipulatives such as stringing beads, multicolored balls, blocks, etc. encourage children in imaginative play,
- Everyday routines and common activities such as snack time or distributing plates and napkins, and
- Math games encourage children to make mathematical relationships.
Adapted from Geist, 2009
Read the article, “Mathematics in the Preschool” by Douglas H. Clements for more information about quality early mathematical experiences.
Watch the video, “ The Decline of Play ,” by Peter Gray.
Play is how children learn! Mathematics games, fantasy play, puzzles, manipulatives, and games allow children to learn and grow.
As you read the vignettes below think about how number sense is being developed and fostered.


1.1: Anatomy and Physiology of the Heart
This segment is a prerequisite lecture in preparation for the EKG interpretation primer. A strong foundation in the anatomy and physiology of the heart is essential in understanding and interpreting EKGs. This segment will review that anatomy and physiology.
The above is a visual reminder of the structures of the heart. Of note, the tricuspid valve is also known as the right atrioventricular valve. The mitral valve is also known as the bicuspid valve or the left atrioventricular valve. Both terminology is correct. The tricuspid valve is tri cuspid because it has 3 flaps that seal together to form the valve, anchored along the wall of the adjacent ventricle wall. The bi cuspid valve only has 2 flaps.
The pulmonary and aortic valves are called semilunar valves. These valves are funnel-shaped and, while morphologically different from the cuspid valves, fulfill a similar function (preventing blood backflow). When you auscultate heart sounds with a stethoscope, the sounds you are hearing are the valves opening and closing.
Coronary Arteries
The coronary arteries provide the heart with its own dedicated blood supply. While the left ventricles do pump oxygen-rich blood to the rest of the body, the entire heart does still require their own blood supply. The coronary arteries are the first arteries that branch off the aorta.
For the time being, just be aware that there are coronary arteries, their job is to supply blood to the heart tissue, and make a mental note of where they are located. They will be revisited in a later segment.
What is Electricity and Polarization?
It is first necessary to understand what electricity is. Electricity is defined as the presence and the flow of electrical charge. All forms of electricity (including the kind that is used to power devices like lights in a home) involves a flow of positively and/or negatively charged atoms from one point to another. Atoms that have a non-zero charge are called ions.
To understand what an ion is, it is helpful to understand atomic theory as it relates to the position of electrons. An atom contains a nucleus (center) made of protons and neutrons. Protons are positively charged particles, neutrons are particles that have mass but are not charged (neutral = neutron). Electrons orbit around the nucleus of the atom, much like a planet orbits around a sun. While protons and neutrons are fairly stable in their position in the atom, electrons are constantly in motion around the outside of the atom. We can predict where they are at a given time using Bohr’s theory, and use that prediction to determine how that atom is electrically charged.
In short: Each electron inhabits an area called an “energy level.” Energy levels can be compared to levels in an apartment. Each level of the apartment complex has a set number of rooms, and those rooms must be filled in order to start filling the next level. The first level can contain up to 2 electrons. The second level up to 8, the third up to 18, and so on. One electron can only occupy one space, so an electron in the first energy level only counts for the first energy level; it does not get counted toward the number needed for the second energy level. We will use a sodium atom as an example:
A sodium atom contains 11 protons, 11 neutrons, and 11 electrons. According to Bohr’s theory, an atom wants to have all of its energy levels completely full, and can accept electrons from other atoms or give them away so that its top-most energy level is completely full. Sodium has 2 completely full levels - the first level with 2 electrons and the second level with 8 electrons, so a total of 10 of the atom’s 11 electrons have filled energy levels such that they are full. Sodium has a total of 11 electrons, so the 11th will start filling the third energy level.
Each atom wants to have completely filled energy levels with no extra and no vacancies. The third energy level contains 18 slots, and only 1 is being occupied by electrons in the sodium atom. This electron can be “given away” and used by another atom that has an almost-filled energy level (for example, hydrogen has only 1 electron and needs 1 more to fill its first energy level). Since it is much easier for the atom to give one electron away than it is to find 17 electrons to fill its third energy level, the electron is usually given away to another atom.
Atoms normally have 0 net charge, which means the number of protons and electrons are the same. However once an electron is either given away or accepted by another atom, the balance of 11 protons to 11 electrons has changed. Now the atom only has 10 electrons. The number of protons has not changed, so that means there are 11 positive charges and only 10 negative, leading to a net positive charge of 1.
A net positive charge is written as either 1+ or just +, so a sodium ion can be written as Na+. If we examined another atom that accepts 2 electrons to fill its outer-most energy level, like sulfur, would be written as 2-, meaning it has a negative charge of 2 (because it accepted 2 electrons instead of 1). Ions that have a positive charge like sodium are called cations; ions with a negative charge like sulfur are called anions.
Polarization and Ion Shifts in Cardiac Muscle
When ions flow from one point to another point, it transfers energy in that direction. That transfer of energy is measurable and is referred to as electricity. Polarization refers to a concentration of positive or negative charges that are higher than other surrounding areas.
Both muscle cells and the interstitial fluid surrounding muscle cells contain cations and anions. At rest, a muscle cell is typically in a polarized state at about -75 mV. That means: there are anions whose collective charge is -75 mV within the cell, where the area outside the cell is closer to neutral or is somewhat positively charged.
Muscle cells utilize several mechanics to control the flow of cations and anions into and out of the cell which causes an electrical gradient (that is, a gradient in which there is a greater charge on one side of the cell than the other side) to form. The cell membrane contains several types of “gates” that the ions can enter and exit the cell through:
Each type of channel has its own unique purpose and function in polarizing or depolarizing the cell.
Depolarization
Depolarization is the opposite process of polarization. While polarization creates a gradient in which one area has a higher concentration of cations or anions than the other, depolarization reverses that imbalance. In muscle cells, depolarization also activates other mechanics within the cell that cause the muscle cell to contract (described later).
Depolarization occurs as a result of an action potential. An action potential is the term for a sudden change in polarization in part of the polarized muscle cell, which then spreads to the other parts of the cell. The end of an action potential is the maximum depolarization, or the furthest the cell can get from its polarized state.
In skeletal muscle cells, that outside action is a neurotransmitter opening ligand-gated channels, which increase the voltage trapped within the cell by pumping more ions into the cell. When the cell reaches a certain voltage, voltage-gated channels open in the cell membrane, allowing other ions to flow in and out, depolarizing the cell by allowing ions to flow in more quickly, reaching a near-equilibrium in charge between the outside of the cell and inside. The voltage-gated channels only stay open for a short period of time; when they close, the cell begins to repolarize, using an active transport channel like the sodium-potassium pump to move positively charged ions back out of the cell to allow the cell to return to its baseline polarized state.
Cardiac muscle do not have ligand-gated channels, as they do not respond to neurotransmitters. Instead, cardiac muscle uses hyperpolarization activated cyclic nucleotide gated (HCN) channels, which are unique to cardiac muscle. They open in response to hyperpolarization, a state in which there is a greater-than-expected charge within the cell. In this case, it responds to negative hyperpolarization, or hyperpolarization that is measured as less than -75 mV (remember -80, for example, is a number less than -75 even though it represents a greater negative charge). They are poorly selective, which means they allow numerous types of ions in and are not very specific for which ones can flood in. Their poor ion selectivity means the depolarization process starts very quickly, as many types of cations are allowed to flood into the cell all at the same time.
Another specific type of channel that is understood to be cardiac-specific is called the fast Na+ channel. This is one type of voltage-gated channel that is selective for sodium and allows very rapid flow of sodium into the cell at a very specific voltage that occurs in the early to mid-phase of depolarization. The fast Na+ channel opens in response to a small amount of positive electrical stimulation but closes again when it exceeds a certain amount of positive electrical stimulation, well before action potential occurs. This allows rapid increase in depolarization, but slows the depolarization process immediately before action potential, allowing adjacent cardiac muscle cells to contract in conjunction. Closing the sodium channel before too much sodium flows in also prevents cell lysis: water tends to follow sodium, and too much sodium can pull too much water into the cell, stretching it and causing it to burst.
Unique Attributes of Cardiac Muscle
Cardiac muscle tissue has several unique attributes that allow it to function effectively specifically in the heart. The first unique attribute cardiac muscle has is its ability to excite and stimulate its own depolarization without outside intervention. Additionally, voltage that stimulates depolarization in one cell can spread to other adjacent cardiac muscle cells. Cardiac muscle can do this because it contains a physical conductor between its adjacent cells in a space called an intercalated disc:
The intercalated discs are special connections between the cardiac muscle tissue that allows it to have branch-like connections to multiple other muscle fibers. Within the intercalated discs are areas called gap junctions, which allow electrical currents to be transferred to other cells. One cardiac muscle cell can stimulate the voltage-gated channels in adjacent cells connected by intercalated discs, commencing the depolarization process of the other cells in synchrony with the originator. This allows multiple muscle fibers to go through depolarization and reach action potential together as a unit.
The branch-like pattern that almost weaves cardiac muscle cells together is important as well because it allows the muscle cells to contract together in a net-like pattern, rather than in a strictly linear pattern. This net-like network of muscle tissue is more effective at pumping and pushing blood.
Desmosomes are units that anchor the cardiac muscle tissue to one another. This prevents them from pulling apart and tearing at the junction site as they contract all at the same time.


6.12: Probability Rules (1 of 3)
Learning Objectives
- Reason from probability distributions, using probability rules, to answer probability questions.
In our previous discussions of probability, we focused on determining the probability of one event at a time. For example, we used two-way tables in Relationships in Categorical Data with Intro to Probability to find the probability that a randomly selected female student from a community college is a Health Science major.
Now we shift our focus to describing the probabilities of all possible outcomes instead of the probability of just one outcome.
We think of all possible outcomes as variable values. Each variable value has a probability. The variable values together with their probabilities are a probability distribution .
Example
Probability Distribution for Blood Type
Consider the variable blood type . This is a categorical variable with variable values A, B, AB, or O. Using relative frequencies from large samples of randomly chosen individuals, we can estimate the probability of choosing a person with a given blood type. Using relative frequencies, the Stanford University’s Blood Center (bloodcenter.stanford.edu) gives the probabilities of human blood types in the United States as follows:
| Blood Type | O | A | B | AB |
| Probability | 0.45 | 0.41 | 0.10 | 0.04 |
This table is an example of a probability distribution. Each variable value is assigned a probability.
Notice the following important fact about this probability distribution:
The sum of all of the probabilities is 1. This makes sense because we have listed all the outcomes. Since each probability is a relative frequency, these outcomes make up 100% of the observations.
We can use the probability distribution to answer probability questions:
- Question: People with blood type O can donate blood to people with any other blood type. For this reason, people with blood type O are called universal donors. What is the probability that a randomly selected person from the United States is a universal donor?
- Answer P (universal donor) = P (blood type O) = 0.45. There is a 45% chance that a randomly selected person in the United States is a universal donor.
Example
Probability Distribution for Boreal Owl Eggs
Boreal owls are common in Canada and Alaska. They are fairly small, averaging 10 inches in length and weighing from 4 to 6 oz. They often make their nests in woodpecker holes. The number of eggs in a boreal owl nest generally ranges from 4 to 6 eggs. Using relative frequencies from large field observations, we can estimate the probability of a nest containing a certain number of eggs.
The variable is Boreal owl eggs in a nest . This is a quantitative variable with values 0, 1, 2, 3, 4, 5, or 6 eggs. The probability distribution gives the probability that a nest will have from 0 to 6 eggs.
| Number of Eggs | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| Probability | 0.2 | 0.1 | 0.1 | 0.25 | 0.25 | 0.05 | 0.05 |
This table is also an example of a probability distribution. Each variable value is assigned a probability.
Note: The sum of all of the probabilities is 1 . This is always true for a probability distribution.
We can use the probability distribution to answer probability questions:
- Question: Which is more likely: (1) To find a boreal owl nest with 3 eggs, or (2) To find a boreal owl nest with 4 eggs.
- Answer Both of these events are equally likely. P (3 eggs) = P (4 eggs) = 0.25. There is a 25% chance that if you find a boreal owl nest, it will have 3 eggs. You are equally likely to find a boreal owl nest with 4 eggs.
Notice the following important facts about probability distributions:
- The outcomes are random events. When we randomly choose a person, we do not know their blood type. But there is a predictable pattern in the outcomes that is described by the relative frequencies. When we randomly select a boreal owl nest, we do not know how many eggs it will contain, but there is a predictable pattern in the outcomes that is described by the relative frequencies.
- All outcomes are assigned a probability.
- The probabilities are numbers between 0 and 1. This makes sense because each probability is a relative frequency.
- The sum of all of the probabilities is 1. This makes sense because we have listed all the outcomes. Since each probability is a relative frequency, these outcomes make up 100% of the observations.
Contributors and Attributions
- Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution


12.11.2: Thought Questions
Thought Questions
Why do you think the outer planets have such extensive systems of rings and moons, while the inner planets do not?
14.
Ganymede and Callisto were the first icy objects to be studied from a geological point of view. Summarize the main differences between their geology and that of the rocky terrestrial planets.
15.
Compare the properties of the volcanoes on Io with those of terrestrial volcanoes. Give at least two similarities and two differences.
16.
Would you expect to find more impact craters on Io or Callisto? Why?
17.
Why is it unlikely that humans will be traveling to Io? (Hint: Review the information about Jupiter’s magnetosphere in The Giant Planets.)
18.
Why do you suppose the rings of Saturn are made of bright particles, whereas the particles in the rings of Uranus and Neptune are black?
19.
Suppose you miraculously removed all of Saturn’s moons. What would happen to its rings?
20.
We have a lot of good images of the large moons of Jupiter and Saturn from the Galileo and Cassini spacecraft missions (check out NASA’s Planetary Photojournal site, at http://photojournal.jpl.nasa.gov , to see the variety). Now that the New Horizons mission has gone to Pluto, why don’t we have as many good images of all sides of Pluto and Charon?
21.
In the Star Wars movie Star Wars Episode VI: Return of the Jedi , a key battle takes place on the inhabited “forest moon” Endor, which supposedly orbits around a gas giant planet. From what you have learned about planets and moons of the solar system, why would this be an unusual situation?


1.14.49: Osmotic Coefficient
There is possible disadvantage in an approach using the mole fraction scale to express the composition of a solution. Granted
- that our interest is often in the properties of solutes in aqueous solutions,
- that the amount of solvent greatly exceeds the amount of solute in a solution, and
- that the sensitivity of equipment developed by chemists is sufficient to probe the properties of quite dilute solutions,
the mole fraction scale for the solvent is not the most convenient method for expressing the composition of a given solution [1-3]. Hence another equation relating \(\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) to the composition of a solution finds favor.
By definition, for a solution containing a single solute, chemical substance \(\mathrm{j}\) [4],
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
Or, in terms of the standard chemical potential for water at temperature \(\mathrm{T}\) and standard pressure \(\mathrm{p}^{0}\),
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}\left(\ell ; \mathrm{T} ; \mathrm{p}^{0}\right)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell ; \mathrm{T}) \, \mathrm{dp} \nonumber \]
\(\mathrm{M}_{1}\) is the molar mass of water; \(\phi\) is the practical osmotic coefficient which is characteristic of the solute, molality mj, temperature and pressure. By definition \(\phi\) is unity for ideal solutions at all temperatures and pressures.
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0 \text { at all T and } \mathrm{p} \nonumber \]
Further for ideal solutions, the partial differentials \((\partial \phi / \partial T)_{p}, \left(\partial^{2} \phi / \partial T^{2}\right)_{p} \text { and } (\partial \phi / \partial \mathrm{p})_{\mathrm{T}}\) are zero. For an ideal solution [5],
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
We rewrite equation (d) in the following form:
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})-\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
Hence with an increase in molality of solute in an ideal aqueous solution, the solvent is stabilized, being at a lower chemical potential than that for pure water. We contrast the chemical potentials of the solvent in real and ideal solutions using an excess chemical potential, \(\mu_{1}^{E}(a q ; T ; p)\);
\[\begin{aligned}
\mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}) &=\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{1}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}) \\
&=(1-\phi) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}
\end{aligned} \nonumber \]
The term \((1 - \phi)\) is often encountered because it expresses succinctly the impact of the solute on the properties of the solvent. At a given molality (and fixed temperature and pressure), \(\phi\) is characteristic of the solute.
In the case of a salt \(\mathrm{j}\) which on complete dissociation forms ν ions the analogue of equation (a) takes the following form.
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
Footnotes
[1] N. Bjerrum, Z. Electrochem., 1907, 24 ,259.
[2] G. N. Lewis and M. Randall, Thermodynamics, revised by K. S. Pitzer and L. Brewer, McGraw-Hill, New York, 2nd edn., 1961, chapter 22.
[3] Mole fractions of solvent \(\mathrm{x}_{1}\) for aqueous solutions having gradually increasing molality of solute \(\mathrm{m}_{j}.
| (A) | \(\mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}^{-1}=10^{-3}\); | \(\mathrm{x}_{1}=0.999982\) | \(x_{j}=1.8 \times 10^{-5}\) |
| (B) | \(\mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}=10^{-2}\); | \(\mathrm{x}_{1}=0.99982\) | \(x_{\mathrm{j}}=1.8 \times 10^{-4}\) |
| (C) | \(\mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}=10^{-1}\); | \(\mathrm{x}_{1}=0.9982\) | \(x_{j}=1.8 \times 10^{-3}\) |
| (D) | \(\mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}{ }^{-1}=0.5\); | \(\mathrm{x}_{1}=0.9911\) | \(\mathrm{x}_{\mathrm{j}}=8.9 \times 10^{-3}\) |
| (E) | \(\mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}=1.0\) | \(\mathrm{x}_{1}=0.9823\) | \(\mathrm{x}_{\mathrm{j}}=1.77 \times 10^{-2}\) |
[4]
\[\left[\mathrm{J} \mathrm{mol}^{-1}\right]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]-[1] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\mathrm{kg} \mathrm{mol}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \nonumber \]
The definitions of ideal solutions expressed in equations (i) and (ii) are not in conflict.
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right) \nonumber \]
Thus for an ideal solution these equations require that, \(v-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}=\ln \left(\mathrm{x}_{1}\right)\) But
\[\ln \left(\mathrm{x}_{1}\right)=\ln \left[\mathrm{M}_{1}^{-1} /\left(\mathrm{M}_{1}^{-1}+\mathrm{m}_{\mathrm{j}}\right)\right]=-\ln \left(1.0+\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right) \nonumber \]
Bearing in mind that \(\mathrm{M}_{1} = 0.018 \mathrm{~kg mol}^{-1}\), then for dilute solutions \(\ln \left(1.0+\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right)=\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\).


21.3: Ureters
The kidneys and ureters are completely retroperitoneal, and the bladder has a peritoneal covering only over the dome. As urine is formed, it drains into the calyces of the kidney, which merge to form the funnel-shaped renal pelvis in the hilum of each kidney. The renal pelvis narrows to become the ureter of each kidney. As urine passes through the ureter, it does not passively drain into the bladder but rather is propelled by waves of peristalsis. As the ureters enter the pelvis, they sweep laterally, hugging the pelvic walls. As they approach the bladder, they turn medially and pierce the bladder wall obliquely. This is important because it creates an one-way valve (a physiological sphincter rather than an anatomical sphincter ) that allows urine into the bladder but prevents reflux of urine from the bladder back into the ureter. Children born lacking this oblique course of the ureter through the bladder wall are susceptible to “vesicoureteral reflux,” which dramatically increases their risk of serious UTI. Pregnancy also increases the likelihood of reflux and UTI.
The ureters are approximately 30 cm long. The inner mucosa is lined with transitional epithelium (Figure \(\PageIndex{1}\)) and scattered goblet cells that secrete protective mucus. The muscular layer of the ureter consists of longitudinal and circular smooth muscles that create the peristaltic contractions to move the urine into the bladder without the aid of gravity.


5.4: What is the future of education?
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Introduction
Education is evolving so rapidly in the way of technology which makes it hard for the world to keep up. J. Johnson, C. Chapman and J. Dyer (2006) state, “The past three decades have seen the rapid development of new and emerging technologies revolutionizing the way we live, work, and learn” (J. Johnson, C. Chapman and J. Dyer, 2006, P.1). Education started off as a traditional classroom and is now integrating more toward technology based learning.
Traditional Classrooms
Traditional classrooms have teachers set in their ways and who have a system. These teachers have their own way of thinking and like to teach things a certain way. The classroom is set a certain way, the students read from the book, and there is little or no technology. Perhaps this does not have to be a bad thing. There are a few subjects that are difficult to learn through the computer, like math for example. Patrick Suppes, a great educator, gives his expert opinion on traditional classrooms: “Hearing lectures and listening to someone else’s talk seem to be almost psychologically essential to learning complex subjects, at least as far as ordinary learners are concerned” (Suppes, 1969, par. 4). Also a traditional classroom seems to be more regulated. Some teachers are not willing to change the way they teach, perhaps some students do not want to change the way they are taught by learning new technology.
On the other hand, traditional classrooms often lack the technological aspect of education. Certain students need individual attention to learn or understand the material. In a traditional classroom, students who require more individual attention may be left behind. The opportunity for a customized learning experience is not as possible as it is in a technologically integrated classroom. The use of technology allows students to learn at a comfortable pace while integrating more collaborative learning tools, like group projects, that better simulate a real world environment. This classroom also frees the teacher to give more personalized attention to those who need it. “We need to continue moving teacher education away from the traditional sink-or-swim model of field experience…” (Zeichner, 2006, P.9)
Technology Integration
"Across the nation, on the Web and in the home, classrooms are evolving beyond the traditional learning environment with alternatives that are no longer bound by geography and customary modes of operation" (Coxx, 2004). This integration of technology makes it possible for teachers to grow and more importantly, possible for students to grow. What is technology integration? “Technology integration is using computers effectively and efficiently in the general content areas to allow students to learn how to apply computer skills in meaningful ways” (Dockstader, 1999, P.1). Incorporating technology into the curriculum does not mean just teaching the students how to use the programs, but how to apply these software to real-world situations. Technology integration is having instruction drive technology, not technology drive instruction. In other words, technology should be used as a tool for teaching, not the focus of the lesson. Holland (2005) explained how a third grade teacher needed to teach her students to distinguish between fact and opinion according to the core curriculum (Holland, 2005, par.4). She taught the concept using a digital camera and a simple processing activity. The students took pictures of them, printed them out, and wrote five facts and five opinions about the picture. Her students got so much out of the lesson, like using analytical skills to decipher the difference between what fact is and what opinion is. This is a necessary real work tool they were taught in an entertaining way.
Some people however, not just teachers, find it harder to adapt to the new technology. “For many adults, all this digital activity is a source of high anxiety” (Tapscott, 1999, P.7). For the younger generation who grew up with the technology, it is easier for them to make this transition. According to Holland’s article, some teachers that have been teaching in the traditional fashion for most of their career refuse to learn advanced technology. Not because these teachers do not want to prepare their students for the future, but because they find it difficult to incorporate technology into the curriculum. One teacher in Holland’s (2005) article writes, “I am not a computer person.” Then Holland goes on to say,”Well maybe you’re not, but face it-we live in a digital age-an age for which we must prepare our students. We weren’t all born knowing to double-click a mouse any more than we exited the womb reciting multiplication facts. We all learn in baby steps” (Holland, 2005, P.3). To some degree, people focus too much on the debate of how the student should be taught. The teachers are forgetting why they are here and are taking time away from the students learning by fighting the integration.
Author's Opinion
The integration of technology into the classroom and using it the fullest potential will have a positive impact of increasing the engagement of the students in our classroom. I remember trying to keep myself awake in class because the teacher did nothing to keep me engaged in the material. I will incorporate much technology into my classroom so that my students will be able to experience hands-on and will be inspired to collaborate with other classmates. I think Suppes (1969) expressed it best when he said, “I would claim that the wise use of technology and science, particularly in education, presents a major opportunity and challenge” (Suppes, 1969, par. 1). However, just because I feel so strongly towards the use of technology and think it is the future of education, does not mean there is not more to be done. Technology is growing so quickly, there is still much we do not know and bugs to be worked out.
Conclusion
Education has progressed immensely over the last fifteen years. It has gone from lectures and textbooks to digital cameras and computers in the classroom. Everybody is working together to make this transition from a traditional-centered classroom to a technology-integrated classroom. The future of education is the key to our children’s learning process and their opportunities ahead. “It is difficult to predict how education will change over the next decade let alone the next century, but there will most certainly be changes in pedagogies which more readily recognize the way young people learn with new technologies” (J. Johnson, C. Chapman and J. Dyer, 2006).
References
Coxx, A. (2004). Moving out of the traditional classroom. Retrieved November 16, 2010, from articles.cnn.com/2004-08-13/politics/b2s.overview_1_charter-school-public-schools-honor-roll?_s=PM:EDUCATION
Dockstader, J. (1999). Teachers of the 21st Century Know the What, Why, and How of Technology Integration. Retrieved January 29, 2009, from www.thejournal.com/the/printarticle/?id=14141
Holland, J. (2005). When Teachers Don't Get It: Myths, Misconceptions, and other Taradiddle. Retrieved February 1, 2009, from www.techlearning.com/story/showArticle.jhtml
Johnson, J., & Chapman, C., & Dyer, J. (2006). Pedagogy and Innovation in Education with Digital Technologies. Retrieved February 5, 2009, from www.formatex.org/micte2006/pdf/135-139.pdf
Suppes, P. (1969). Computer Technology and the Future of Education. Retrieved February 5, 2009, from suppes-corpus.stanford.edu/articles/comped/85-6.pdf
Tapscott, D. (1999). Educating the Net Generation. Educational Leadership. Retrieved February 1, 2009, Vol. 56, No. 5, pp. 6–11. from www.ascd.org/readingroom/edlead/abstracts/feb99.html
Zeichner, K. (2006). Reflections of a University-based teacher Educator on the Future of College-and University-based teacher Education. Retrieved February 5, 2009, jte.sagepub.com/cgi/content/abstract/57/3/326


12.2: Assignment- Practice with Excel
Assignment Objective
This assignment gives you practice using what you have learned about Excel without step-by-step directions.
Directions
Open the Excel Practice Application (.doc) document and use Excel to create a worksheet that meets the criteria listed. Try to complete the spreadsheet using what you remember. If you cannot remember a procedure, refer to the trainings provided previously.
CC licensed content, Original
- Introduction to Computer Applications and Concepts Excel Practice. Authored by : Jim Shannon. Provided by : Extended Learning Institute of Northern Virginia Community College. Located at : http://www.nvcc.edu/eli/index.html . License : CC BY: Attribution