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Thursday, October 19, 2023

Homeschooling

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Homeschooling
A person educating children at home

Homeschooling or home schooling, also known as home education or elective home education (EHE), is the education of school-aged children at home or a variety of places other than a school. Usually conducted by a parent, tutor, or online teacher, many homeschool families use less formal, more personalized and individualized methods of learning that are not always found in schools. The actual practice of homeschooling can vary. The spectrum ranges from highly structured forms based on traditional school lessons to more open, free forms such as unschooling, which is a lesson- and curriculum-free implementation of homeschooling. Some families who initially attended a school go through a deschool phase to break away from school habits and prepare for homeschooling. While "homeschooling" is the term commonly used in North America, "home education" is primarily used in Europe and many Commonwealth countries. Homeschooling should not be confused with distance education, which generally refers to the arrangement where the student is educated by and conforms to the requirements of an online school, rather than being educated independently and unrestrictedly by their parents or by themselves.

Before the introduction of compulsory school attendance laws, most childhood education was done by families and local communities. By the early 19th century, attending a school became the most common means of education in the developed world. In the mid to late 20th century, more people began questioning the efficiency and sustainability of school learning, which again led to an increase in the number of homeschoolers, especially in the Americas and some European countries. Today, homeschooling is a relatively widespread form of education and a legal alternative to public and private schools in many countries, which many people believe is due to the rise of the Internet, which enables people to obtain information very quickly. There are also nations in which homeschooling is regulated or illegal. During the COVID-19 pandemic, many students from all over the world had to study from home due to the danger posed by the virus. However, this was mostly implemented in the form of distance education rather than traditional homeschooling.

There are many different reasons for homeschooling, ranging from personal interests to dissatisfaction with the public school system. Some parents see better educational opportunities for their child in homeschooling, for example because they know their child more accurately than a teacher and can concentrate fully on educating usually one to a few persons and therefore can respond more precisely to their individual strengths and weaknesses, or because they think that they can better prepare their children for life outside of school. Some children can also learn better at home, for example, because they are not held back, disturbed or distracted from school matters, do not feel underchallenged or overwhelmed with certain topics, find that certain temperaments are encouraged in school, while others are inhibited, do not cope well with the often predetermined structure or are bullied there. Homeschooling is also an option for families living in remote rural areas, those temporarily abroad, those who travel frequently and therefore face the physical impossibility or difficulty of getting their children into school, and those who want to spend more and better time with their children. Health reasons and special needs can also play a role in why children cannot attend a school regularly and are at least partially homeschooled.

Critics of homeschooling argue that children may lack adequate socialization and therefore have poorer social skills. Some are also concerned that parents may be unqualified to guide and advise their children in life skills. Critics also say that a child might not encounter people of other cultures, worldviews, and socioeconomic groups if they are not enrolled in a school. Therefore, these critics believe that homeschooling cannot guarantee a comprehensive and neutral education if educational standards are not prescribed. Homeschooled children sometimes score higher on standardized tests and their parents reported via survey that their children have equally or better developed social skills and participate more in cultural and family activities on average than public school students. In addition, studies suggest that homeschoolers are generally more likely to have higher self-esteem, deeper friendships, and better relationships with adults, and are less susceptible to peer pressure.

Terminology

While "homeschooling" is the term commonly used in the United States and other nations in North America, "home education" is primarily used in the United Kingdom, elsewhere in Europe and many Commonwealth countries. Some believe that homeschooling has become more attractive and popular than ever before since the days of quick information retrieval on the Internet.

History

Frontispiece to Fireside Education, Samuel Griswold (Goodrich)

Early history

For most of history and in different cultures, homeschooling was a common practice by family members and local communities. Enlisting professional tutors was an option available only to the wealthy. Homeschooling declined in the 19th and 20th centuries with the enactment of compulsory school attendance laws. However, it continued to be practised in isolated communities. Homeschooling began a resurgence in the 1960s and 1970s with educational reformists dissatisfied with industrialized education.

Public schools

The earliest public schools in modern Western culture were established during the Reformation with the encouragement of Martin Luther in the German states of Gotha and Thuringia in 1524 and 1527. From the 1500s to 1800s the literacy rate increased until a majority of adults were literate, but development of the literacy rate occurred before the implementation of compulsory attendance and universal education.

Home education and apprenticeship continued to remain the main form of education until the 1830s. However, in the 18th century, the majority of people in Europe lacked formal education. Since the early 19th century, formal classroom schooling became the most common means of schooling throughout the developed countries.

Colonial North America

In 1647, New England provided compulsory elementary education. Regional differences in schooling existed in colonial America. In the south, farms and plantations were so widely dispersed that community schools such as those in the more compact settlements of the north were impossible. In the middle colonies, the educational situation varied when comparing New York with New England.

Most Native American tribal cultures traditionally used homeschooling and apprenticeship to pass knowledge to children. Parents were supported by extended relatives and tribal leaders in the education of their children. The Native Americans vigorously resisted compulsory education in the United States.

United States in 1960s

In the 1960s, Rousas John Rushdoony began to advocate homeschooling, which he saw as a way to combat the secular nature of the public school system in the United States. He vigorously attacked progressive school reformers such as Horace Mann and John Dewey, and argued for the dismantling of the state's influence in education in three works: Intellectual Schizophrenia, The Messianic Character of American Education, and The Philosophy of the Christian Curriculum. Rushdoony was frequently called as an expert witness by the Home School Legal Defense Association (HSLDA) in court cases. He frequently advocated the use of private schools.

Raymond and Dorothy Moore

During this time, American educational professionals Raymond and Dorothy Moore began to research the academic validity of the rapidly growing Early Childhood Education movement. This research included independent studies by other researchers and a review of over 8,000 studies bearing on early childhood education and the physical and mental development of children.

They asserted that formal schooling before ages 8–12 not only lacked the anticipated effectiveness but also harmed children. The Moores published their view that formal schooling was damaging young children academically, socially, mentally, and even physiologically. The Moores presented evidence that childhood problems such as juvenile delinquency, nearsightedness, increased enrollment of students in special education classes and behavioral problems were the results of increasingly earlier enrollment of students. The Moores cited studies demonstrating that orphans who were given surrogate mothers were measurably more intelligent, with superior long-term effects – even though the mothers were "mentally retarded teenagers" – and that illiterate tribal mothers in Africa produced children who were socially and emotionally more advanced than typical western children, "by western standards of measurement".

Their primary assertion was that the bonds and emotional development made at home with parents during these years produced critical long-term results that were cut short by enrollment in schools, and could neither be replaced nor corrected in an institutional setting afterwards. Recognizing a necessity for early out-of-home care for some children, particularly special needs and impoverished children and children from exceptionally inferior homes, they maintained that the vast majority of children were far better situated at home, even with mediocre parents, than with the most gifted and motivated teachers in a school setting. They described the difference as follows: "This is like saying, if you can help a child by taking him off the cold street and housing him in a warm tent, then warm tents should be provided for all children – when obviously most children already have even more secure housing."

The Moores embraced homeschooling after the publication of their first work, Better Late Than Early, in 1975, and became important homeschool advocates and consultants with the publication of books such as Home Grown Kids (1981), and Homeschool Burnout.

Simultaneously, other authors published books questioning the premises and efficacy of compulsory schooling, including Deschooling Society by Ivan Illich in 1970 and No More Public School by Harold Bennet in 1972.

John Holt

In 1976, educator John Holt published Instead of Education: Ways to Help People Do Things Better. In its conclusion, he called for a "Children's Underground Railroad" to help children escape compulsory schooling. In response, Holt was contacted by families from around the U.S. to tell him that they were educating their children at home. In 1977, after corresponding with a number of these families, Holt began producing the magazine Growing Without Schooling (GWS), a newsletter dedicated to home education. Holt was nicknamed the "father of homeschooling." Holt later wrote a book about homeschooling, Teach Your Own, in 1981.

In 1980, Holt said,

I want to make it clear that I don't see homeschooling as some kind of answer to badness of schools. I think that the home is the proper base for the exploration of the world which we call learning or education. The home would be the best base no matter how good the schools were.

One common theme in the homeschool philosophies of both Holt and that of the Moores is that home education should not attempt to bring the school to construct into the home, or a view of education as an academic preliminary to life. They viewed home education as a natural, experiential aspect of life that occurs as the members of the family are involved with one another in daily living.

Homeschooling can be used as a form of supplemental education and as a way of helping children learn under specific circumstances. The term may also refer to instruction in the home under the supervision of correspondence schools or umbrella schools. Some jurisdictions require adherence to an approved curriculum.

United States in 1970s

In the 1970s, a modern homeschooling movement began when American educator and author John Holt questioned the efficiency of schools and the sustainability of school learning, arguing that schools focus on strictly doing "skill drill" instead of other methods of learning. The influence of Raymond Moore is sometimes also held responsible for this movement on the religious right. A curriculum-free philosophy of homeschooling called "unschooling" also emerged around this time, although it would take a few more decades for this form of education to become popular. The term was coined in 1977 by Holt's GWS. The term emphasizes the more spontaneous, less structured learning environment in which a child's interests drive his pursuit of knowledge. Some parents provide a liberal arts education using the trivium and quadrivium as the main models.

COVID-19 pandemic

Filipino homeschooling students - blended (printed-digital modular) distance learning with self-learning materials during the 2020 COVID-19 pandemic in San Miguel, Bulacan

The COVID-19 pandemic led to school closures around the world, which is why over 300 million students had to study from home. Since the material to be learned was mainly outsourced to home and specified and checked by virtual schools, it can be said that this was mostly implemented in the form of distance education rather than traditional homeschooling in which parents educate their child independent from school. Because the transition to homeschooling often happened overnight without any possibilities of preparation for parents, teachers and children, this caused economic, educational, political and psychological distress.

Motivations

When homeschooling is a choice, families have different reasons for choosing it. This cake diagram shows the motivations regarded as most important for homeschooling in the United States as of 2007.

There are a multitude of sometimes complex reasons why parents and children choose to homeschool, some of which overlap with those for unschooling and may be very different depending on the country and (current) situation of parents and children.

Parents commonly cite two main motivations for homeschooling their children: dissatisfaction with the local schools and the interest in increased involvement with their children's learning and development. Parental dissatisfaction with available schools typically includes concerns about the school environment, the quality of academic instruction, the curriculum, bullying, racism and lack of faith in the school's ability to cater to their children's special needs. Some parents homeschool in order to have greater control over what and how their children are taught, to cater more adequately to an individual child's aptitudes and abilities, to provide instruction from a specific religious or and moral position, and to take advantage of the efficiency of one-to-one instruction and thus allow the child to spend more time on childhood activities, socializing, and non-academic learning.

Some African-American families choose to homeschool as a way of increasing their children's understanding of African-American history – such as the Jim Crow laws that resulted in African Americans being prevented from reading and writing – and to limit the harm caused by the unintentional and sometimes subtle systemic racism that affects most American schools.

Some parents have objections to the secular nature of public schools and homeschool in order to give their children a religious education. Use of a religious curriculum is common among these families.

Some parents are of the opinion that certain temperaments are promoted in school, while others are inhibited which may also be a reason to homeschool their children.

Another argument for homeschooling children may be the protection against physical and emotional violence, bullying, exclusion, drugs, stress, sexualization, social pressures, excessive performance thoughts, socialization groups or role models with negative impact and degrading treatment in school.

Some children may also prefer to or can learn more efficiently at home, for example, because they are not distracted or slowed down by school matters and can, for example, spend several hours dealing with the same topic undisturbed. There are studies that show that homeschooled children are more likely to graduate and perform better at university.

Homeschooling may also be a factor in the choice of parenting style. Homeschooling can be a matter of consistency for families living in isolated rural locations, for those temporarily abroad, and for those who travel frequently. Many young athletes, actors, and musicians are taught at home to accommodate their training and practice schedules more conveniently. Homeschooling can be about mentorship and apprenticeship, in which a tutor or teacher is with the child for many years and becomes more intimately acquainted with the child. Many parents also homeschool their children and return their child into the school system later on, for example because they think that their child is too young or not yet ready to start school.

Some children also have health issues and therefore cannot attend a school regularly and are at least partially homeschooled or take distance education instead.

COVID-19 has reinforced some parent's minds about homeschooling. The fact that parents realized remote learning was possible thanks to new technologies means that they have additional options to consider should their child face problems of any kind at school.

Teaching methods, forms and philosophies

Homeschooling is usually conducted by a parent, tutor, or an online teacher, but the concrete practice can be very different. The spectrum ranges from highly structured forms based on traditional school lessons to more open, free forms like unschooling, which is a curriculum-free implementation of homeschooling that involves teaching children based on their interests.

Many homeschool families use a wide variety of methods and materials and less formal educational methods, which represent a variety of educational philosophies and paradigms. Some of the methods or learning environments used include classical education (including Trivium, Quadrivium), Charlotte Mason education, Montessori method, theory of multiple intelligences, unschooling, Waldorf education, school-at-home (curriculum choices from both secular and religious publishers), A Thomas Jefferson Education, unit studies, curriculum made up from private or small publishers, apprenticeship, hands-on-learning, distance learning (both online and correspondence), dual enrollment in local schools or colleges, and curriculum provided by local schools and many others. Some of these approaches are used in private and public schools. Educational research and studies support the use of some of these methods. Unschooling, natural learning, Charlotte Mason Education, Montessori, Waldorf, apprenticeship, hands-on-learning, unit studies are supported to varying degrees by research by constructivist learning theories and situated cognition theories. Elements of these theories may be found in the other methods as well.

A student's education may be customized to support his or her learning level, style, and interests. It is not uncommon for a student to experience more than one approach as the family discovers what works best for their student. Many families use an eclectic approach, picking and choosing from various suppliers. For sources of curricula and books, a study found that 78 percent utilized "a public library"; 77 percent used "a homeschooling catalogue, publisher, or individual specialist"; 68 percent used "retail bookstore or another store"; 60 percent used "an education publisher that was not affiliated with homeschooling." "Approximately half" used curriculum from "a homeschooling organization", 37 percent from a "church, synagogue or other religious institution" and 23 percent from "their local public school or district." In 2003, 41 percent utilized some sort of distance learning, approximately 20 percent by "television, video or radio"; 19 percent via "The Internet, e-mail, or the World Wide Web"; and 15 percent taking a "correspondence course by mail designed specifically for homeschoolers."

Individual governmental units, e.g. states and local districts, vary in official curriculum and attendance requirements.

Informal learning

As a subset of homeschooling, informal learning happens outside of the classroom but has no traditional boundaries of education. Informal learning is an everyday form of learning through participation and creation, in contrast with the traditional view of teacher-centered learning. The term is often combined with non-formal learning and self-directed learning. Informal learning differs from traditional learning since there are no expected objectives or outcomes. From the learner's standpoint, the knowledge that they receive is not intentional. Topics such as planting a garden, baking a cake or even talking to a technician at work about the installation of new software can be considered informal learning. The individual is completing a task with different intentions but ends up learning skills in the process. Children watching their tomato plants grow will not generate questions about photosynthesis but they will learn that their plants are growing with water and sunlight. This leads them to have a base understanding of complex scientific concepts without any background studying. The recent trend of homeschooling becoming less stigmatized has been in connection with the traditional waning of the idea that the state needs to be in primary and ultimate control over the education and upbringing of all children to create future adult citizens. This breeds an ever-growing importance on the ideas and concepts that children learn outside of the traditional classroom setting, including informal learning.

Depending on the part of the world, informal learning can take on many different identities and has differing cultural importances. Many ways of organizing homeschooling draw on apprenticeship qualities and on non-western cultures. In some South American indigenous cultures, such as the Chillihuani community in Peru, children learn irrigation and farming technique through play, advancing them not only in their own village and society but also in their knowledge of realistic techniques that they will need to survive. In Western culture, children use informal learning in two main ways. The first as talked about is through hands-on experience with new material. The second is asking questions to someone who has more experience than they have (i.e. parents, elders). Children's inquisitive nature is their way of cementing the ideas they have learned through exposure to informal learning. It is a more casual way of learning than traditional learning and serves the purpose of taking in information any which way they can.

Structured versus unstructured

All other approaches to homeschooling are subsumed under two basic categories: structured and unstructured homeschooling. Structured homeschooling includes any method or style of home education that follows a basic curriculum with articulated goals and outcomes. This style attempts to imitate the structure of the traditional school setting while personalizing the curriculum. Unstructured homeschooling is any form of home education where parents do not construct a curriculum at all. Unschooling, as it is known, attempts to teach through the child's daily experiences and focuses more on self-directed learning by the child, free of textbooks, teachers, and any formal assessment of success or failure.

Unit studies

In a unit study approach, multiple subjects such as math, science, history, art, and geography, are studied in relation to a single topic. Unit studies are useful for teaching multiple grades simultaneously as the difficulty level can be adjusted for each student. An extended form of unit studies, Integrated Thematic Instruction utilizes one central theme integrated throughout the curriculum so that students finish a school year with a deep understanding of a certain broad subject or idea.

All-in-one curricula

All-in-one homeschooling curricula (variously known as school-at-home, the traditional approach, or school-in-a-box) are instructional methods of teaching in which the curriculum and homework of the student are similar or identical to those used in a public or private school. Purchased as a grade-level package or separately by subject, the package may contain all of the needed books, materials, tests, answer keys, and extensive teacher guides. These materials cover the same subject areas as public schools, allowing for an easy transition into the school system. These are among the most expensive options for homeschooling, but they require minimal preparation and are easy to use. Some localities provide the same materials used at local schools to homeschoolers. The purchase of a complete curriculum and their teaching/grading service from an accredited distance learning curriculum provider may allow students to obtain an accredited high school diploma.

Unschooling and natural learning

Natural learning refers to a type of learning-on-demand where children pursue knowledge based on their interests and parents take an active part in facilitating activities and experiences conducive to learning but do not rely heavily on textbooks or spend much time "teaching", looking instead for "learning moments" throughout their daily activities. Parents see their role as that of affirming through positive feedback and modeling the necessary skills, and the child's role as being responsible for asking and learning.

The term unschooling as coined by John Holt describes an approach in which parents do not authoritatively direct the child's education, but interact with the child following the child's own interests, leaving them free to explore and learn as their interests lead. "Unschooling" does not indicate that the child is not being educated, but that the child is not being "schooled", or educated in a rigid school-type manner. Holt asserted that children learn through the experiences of life, and he encouraged parents to live their lives with their child. Also known as interest-led or child-led learning, unschooling attempts to follow opportunities as they arise in real life. Children at school learn from 1 teacher and 2 auxiliary teachers in a classroom of approximately 30. Kids have the opportunity of dedicated education at home with a ratio of 1 to 1. An unschooled child may utilize texts or classroom instruction, but these are not considered central to education. Holt asserted that there is no specific body of knowledge that is, or should be, required of a child.

Both unschooling and natural learning advocates believe that children learn best by doing; a child may learn reading to further an interest about history or other cultures, or math skills by operating a small business or sharing in family finances. They may learn animal husbandry keeping dairy goats or meat rabbits, botany tending a kitchen garden, chemistry to understand the operation of firearms or the internal combustion engine, or politics and local history by following a zoning or historical-status dispute. While any type of homeschoolers may also use these methods, the unschooled child initiates these learning activities. The natural learner participates with parents and others in learning together.

Another prominent proponent of unschooling is John Taylor Gatto, author of Dumbing Us Down, The Exhausted School, A Different Kind of Teacher, and Weapons of Mass Instruction. Gatto argues that public education is the primary tool of "state-controlled consciousness" and serves as a prime illustration of the total institution — a social system which impels obedience to the state and quells free-thinking or dissent.

Autonomous learning

Autonomous learning is a school of education which sees learners as individuals who can and should be autonomous; i.e., be responsible for their own learning climate.

Autonomous education helps students develop their self-consciousness, vision, practicality, and freedom of discussion. These attributes serve to aid the student in his/her independent learning. However, a student must not start their autonomous learning completely on their own. It is said that first having interaction with someone who has more knowledge in a subject will speed up the student's learning and allow them to learn more independently.

Some degree of autonomous learning is popular with those who home educate their children. In true autonomous learning, the child usually gets to decide what projects they wish to tackle or what interests to pursue. In-home education, this can be instead of or in addition to regular subjects like doing math or English.

According to Home Education UK, the autonomous education philosophy emerged from the epistemology of Karl Popper in The Myth of the Framework: In Defence of Science and Rationality, which is developed in the debates, which seek to rebut the neo-Marxist social philosophy of convergence proposed by the Frankfurt School (e.g. Theodor W. Adorno, Jürgen Habermas, Max Horkheimer).

Hybrid homeschooling

Hybrid homeschooling or flex-school is a form of homeschooling in which children split their time between homeschool and a more traditional schooling environment like a school. The number of students who participated in hybrid homeschooling increased during the COVID-19 pandemic.

A commonly cited reason for choosing this model is that parents are not sure whether they can provide their children a comprehensive and neutral education at home or cannot devote themselves to homeschooling full-time due to time constraints or excessive stress. Some families also want their children to socialize with other children and find that schools are better suited for this purpose because social exchange does not only take place occasionally, but is an everyday experience there.

Homeschool cooperatives

A homeschool cooperative is a cooperative of families who homeschool their children. It provides an opportunity for children to learn from other parents who are more specialized in certain areas or subjects. Co-ops also provide social interaction. They may take lessons together or go on field trips. Some co-ops also offer events such as prom and graduation for homeschoolers.

Homeschoolers are beginning to utilize Web 2.0 as a way to simulate homeschool cooperatives online. With social networks, homeschoolers can chat, discuss threads in forums, share information and tips, and even participate in online classes via learning management systems similar to those used by colleges.

Research

Test results

According to the Home School Legal Defense Association (HSLDA) in 2004, "Many studies over the last few years have established the academic excellence of homeschooled children." Home Schooling Achievement, a compilation of studies published by the HSLDA, supported the academic integrity of homeschooling. This booklet summarized a 1997 study by Ray and the 1999 Rudner study. The Rudner study noted two limitations of its own research: it is not necessarily representative of all homeschoolers and it is not a comparison with other schooling methods. Among the homeschooled students who took the tests, the average homeschooled student outperformed his public school peers by 30 to 37 percentile points across all subjects. The study also indicates that public school performance gaps between minorities and genders were virtually non-existent among the homeschooled students who took the tests.

A survey of 11,739 homeschooled students conducted in 2008 found that, on average, the homeschooled students scored 37 percentile points above public school students on standardized achievement tests. This is consistent with the 1999 Rudner study. However, Rudner said that these same students in public school may have scored just as well because of the dedicated parents they had. The Ray study also found that homeschooled students who had a certified teacher as a parent scored one percentile lower than homeschooled students who did not have a certified teacher as a parent. Studies have shown homeschooled students score higher on standardized tests than traditionally schooled youth Another nationwide descriptive study conducted by Ray contained students ranging from ages 5–18 and he found that homeschoolers scored in at least the 80th percentile on their tests.

In 2011, a quasi-experimental study was conducted that included homeschooled and traditional public students between the ages of 5 and 10. It was discovered that the majority of the homeschooled children achieved higher standardized scores compared to their counterparts. However, Martin-Chang also found that unschooling children ages 5–10 scored significantly below traditionally educated children, while academically oriented homeschooled children scored from one half grade level above to 4.5 grade levels above traditionally schooled children on standardized tests (n=37 homeschooled children matched with children from the same socioeconomic and educational background).

Studies have also examined the impact of homeschooling on students' GPAs. Cogan (2010) found that homeschooled students had higher high school GPAs (3.74) and transfer GPAs (3.65) than conventional students. Snyder (2013) provided corroborating evidence that homeschoolers were outperforming their peers in the areas of standardized tests and overall GPAs. Looking beyond high school, a study by the 1990 National Home Education Research Institute (as cited by Wichers, 2001) found that at least 33% of homeschooled students attended a four-year college, and 17% attended a two-year college. This same study examined the students after one year, finding that 17% pursued higher education.

On average, studies suggest homeschoolers score at or above the national average on standardized tests. Homeschool students have been accepted into many Ivy League universities. However, The Coalition for Responsible Homeschooling notes that "Our knowledge of homeschooling’s effect on academic achievement is limited by the fact that many of the studies that have been conducted on homeschoolers suffer from methodological problems which make their findings inconclusive."

Outcomes

Homeschooled children may receive more individualized attention than students enrolled in traditional public schools. A 2011 study suggests that a structured environment could play a key role in homeschooler academic achievement. This means that parents were highly involved in their child's education and they were creating clear educational goals. In addition, these students were being offered organized lesson plans which are either self-made or purchased.

Homeschooled youth are less likely to use and abuse illicit substances and are more likely to disapprove of using alcohol and marijuana. There are also studies according to which homeschooled children are less likely to be sexually abused than children in public schools.

In the 1970s, Raymond and Dorothy Moore conducted four federally funded analyses of more than 8,000 early childhood studies, from which they published their original findings in Better Late Than Early, 1975. This was followed by School Can Wait, a repackaging of these same findings designed specifically for educational professionals. They concluded that "where possible, children should be withheld from formal schooling until at least ages eight to ten." Their reason was that children "are not mature enough for formal school programs until their senses, coordination, neurological development and cognition are ready". They concluded that the outcome of forcing children into formal schooling is a sequence of "1) uncertainty as the child leaves the family nest early for a less secure environment, 2) puzzlement at the new pressures and restrictions of the classroom, 3) frustration because unready learning tools – senses, cognition, brain hemispheres, coordination – cannot handle the regimentation of formal lessons and the pressures they bring, 4) hyperactivity growing out of nerves and jitter, from frustration, 5) failure which quite naturally flows from the four experiences above, and 6) delinquency which is failure's twin and apparently for the same reason." According to the Moores, "early formal schooling is burning out our children. Teachers who attempt to cope with these youngsters also are burning out." Aside from academic performance, they think early formal schooling also destroys "positive sociability", encourages peer dependence, and discourages self-worth, optimism, respect for parents, and trust in peers. They believe this situation is particularly acute for boys because of their delay in maturity. The Moores cited a Smithsonian Report on the development of genius, indicating a requirement for "1) much time spent with warm, responsive parents and other adults, 2) very little time spent with peers, and 3) a great deal of free exploration under parental guidance." Their analysis suggested that children need "more of home and less of formal school", "more free exploration with... parents, and fewer limits of classroom and books", and "more old fashioned chores – children working with parents – and less attention to rivalry sports and amusements."

A study conducted by Ray in 2010 indicates that the higher the level of parents' income, the more likely the homeschooled child is able to achieve academic success.

Higher education admittance procedures were altered due to COVID-19 for the traditionally schooled student.

The ACT and SAT became test optional, yet the homeschooled applicant is required to submit college entrance exams.

Some homeschoolers averaged higher scores on these college entrance tests in South Carolina. Other scores (1999 data) showed mixed results, for example showing higher levels for homeschoolers in English (homeschooled 23.4 vs national average 20.5) and reading (homeschooled 24.4 vs national average 21.4) on the ACT, but mixed scores in math (homeschooled 20.4 vs national average 20.7 on the ACT as opposed homeschooled 535 vs national average 511 on the 1999 SAT math).

Some advocates of homeschooling and educational choice counter with an input-output theory, pointing out that home educators expend only an average of $500–600 a year on each student (not counting the cost of the parents' time), in comparison to $9,000–10,000 (including the cost of staff time) for each public school student in the United States, which suggests home-educated students would be especially dominant on tests if afforded access to an equal commitment of tax-funded educational resources.

Many teachers and school districts oppose the idea of homeschooling. However, research has shown that homeschooled children often excel in many areas of academic endeavour. According to a study done on the homeschool movement, homeschoolers often achieve academic success and admission into elite universities. According to the National Home Education Research Institute president, Brian Ray, socialization is not a problem for homeschooling children, many of whom are involved in community sports, volunteer activities, book groups, or homeschool co-ops.

In the UK, the government has noted that no figures are available on educational attainment for children educated at home: "This means no assessment can be made of the impact on educational attainment of being home schooled". There is some evidence from 2009 that home-educated children in the UK are more likely to be NEET, Not in Employment, Education or Training, at age 16 to 18.

Socialization

Using the Piers-Harris Children's Self-Concept Scale, John Taylor later found that, "while half of the conventionally schooled children scored at or below the 50th percentile (in self-concept), only 10.3% of the home-schooling children did so." He further stated that "the self-concept of home-schooling children is significantly higher statistically than that of children attending conventional school. This has implications in the areas of academic achievement and socialization which have been found to parallel self-concept. Regarding socialization, Taylor's results would mean that very few home-schooling children are socially deprived. He states that critics who speak out against homeschooling on the basis of social deprivation are actually addressing an area which favours homeschoolers.

In 2003, the National Home Education Research Institute conducted a survey of 7,300 U.S. adults who had been homeschooled (5,000 for more than seven years). Their findings included:

  • Homeschool graduates are active and involved in their communities. 71% participate in an ongoing community service activity, like coaching a sports team, volunteering at a school, or working with a church or neighborhood association, compared with 37% of U.S. adults of similar ages from a traditional education background.
  • Homeschool graduates are more involved in civic affairs and vote in much higher percentages than their peers. 76% of those surveyed between the ages of 18 and 24 voted within the last five years, compared with only 29% of the corresponding U.S. populace. The numbers are even greater in older age groups, with voting levels not falling below 95%, compared with a high of 53% for the corresponding U.S. populace.
  • 58.9% report that they are "very happy" with life, compared with 27.6% for the general U.S. population. 73.2% find life "exciting", compared with 47.3%

Richard G. Medlin, Ph.D.'s research found that homeschooled children have better social skills than children attending traditional schools.

Legality and prevalence

Criticism

Resistance to homeschooling comes from some organizations of teachers and school districts. The National Education Association, a United States teachers' union and professional association, has asserted that teachers should be licensed and that state-approved curricula should be used.

Critics argue that homeschooled children can be indoctrinated if educational standards are not prescribed and if there is no regular monitoring by controlling authorities. There is also concern that homeschooling reduces a child's exposure to mandated reporters such as teachers, making children more susceptible to sustained, unreported abuse. The vast majority of homeschoolers are doing so for religious reasons. They are primarily concerned with producing "virtuous" adults, rather than "educated" adults. While there is not enough data to determine exact rates of abuse in homeschooling there is, however, strong evidence to suggest that parents who engage in maltreatment and educational neglect are more likely to use homeschooling as a guise.

Elizabeth Bartholet, a Harvard professor of law and faculty director of the Law School's Child Advocacy Program, recommended a ban on home education in 2019, calling it a risky practice.

Political scientist Rob Reich speculated in The Civic Perils of Homeschooling (2002) that homeschooling could threaten to "insulate students from exposure to diverse ideas and people."

Gallup polls of American voters have shown a significant change in attitude in the last 20 years, from 73% opposed to homeschooling in 1985 to 54% opposed in 2001. In 1988, when asked whether parents should have a right to choose homeschooling, 53 percent thought that they should, as revealed by another poll.

Bioaerosol

From Wikipedia, the free encyclopedia

Bioaerosols (short for biological aerosols) are a subcategory of particles released from terrestrial and marine ecosystems into the atmosphere. They consist of both living and non-living components, such as fungi, pollen, bacteria and viruses. Common sources of bioaerosols include soil, water, and sewage.

Bioaerosols are typically introduced into the air via wind turbulence over a surface. Once in the atmosphere, they can be transported locally or globally: common wind patterns/strengths are responsible for local dispersal, while tropical storms and dust plumes can move bioaerosols between continents. Over ocean surfaces, bioaerosols are generated via sea spray and bubbles

Bioaerosols can transmit microbial pathogens, endotoxins, and allergens to which humans are sensitive. A well-known case was the meningococcal meningitis outbreak in sub-Saharan Africa, which was linked to dust storms during dry seasons. Other outbreaks linked to dust events including Mycoplasma pneumonia and tuberculosis.

Another instance was an increase in human respiratory problems in the Caribbean that may have been caused by traces of heavy metals, microorganism bioaerosols, and pesticides transported via dust clouds passing over the Atlantic Ocean.

Common bioaerosol isolated from indoor environments

Background

Charles Darwin was the first to observe the transport of dust particles but Louis Pasteur was the first to research microbes and their activity within the air. Prior to Pasteur’s work, laboratory cultures were used to grow and isolate different bioaerosols.

Since not all microbes can be cultured, many were undetected before the development of DNA-based tools. Pasteur also developed experimental procedures for sampling bioaerosols and showed that more microbial activity occurred at lower altitudes and decreased at higher altitudes.

Types of bioaerosols

Bioaerosols include fungi, bacteria, viruses, and pollen. Their concentrations are greatest in the planetary boundary layer (PBL) and decrease with altitude. Survival rate of bioaerosols depends on a number of biotic and abiotic factors which include climatic conditions, ultraviolet (UV) light, temperature and humidity, as well as resources present within dust or clouds.

Bioaerosols found over marine environments primarily consist of bacteria, while those found over terrestrial environments are rich in bacteria, fungi and pollen. The dominance of particular bacteria and their nutrient sources are subject to change according to time and location.

Bioaerosols can range in size from 10 nanometer virus particles to 100 micrometers pollen grains. Pollen grains are the largest bioaerosols and are less likely to remain suspended in the air over a long period of time due to their weight.

Consequently, pollen particle concentration decreases more rapidly with height than smaller bioaerosols such as bacteria, fungi and possibly viruses, which may be able to survive in the upper troposphere. At present, there is little research on the specific altitude tolerance of different bioaerosols. However, scientists believe that atmospheric turbulence impacts where different bioaerosols may be found.

Fungi

Fungal cells usually die when they travel through the atmosphere due to the desiccating effects of higher altitudes. However, some particularly resilient fungal bioaerosols have been shown to survive in atmospheric transport despite exposure to severe UV light conditions. Although bioaerosol levels of fungal spores increase in higher humidity conditions, they can also be active in low humidity conditions and in most temperature ranges. Certain fungal bioaerosols even increase at relatively low levels of humidity.

Bacteria

Unlike other bioaerosols, bacteria are able to complete full reproductive cycles within the days or weeks that they survive in the atmosphere, making them a major component of the air biota ecosystem. These reproductive cycles support a currently unproven theory that bacteria bioaerosols form communities in an atmospheric ecosystem. The survival of bacteria depends on water droplets from fog and clouds that provide bacteria with nutrients and protection from UV light. The four known bacterial groupings that are abundant in aeromicrobial environments around the world include Bacillota, Actinomycetota, Pseudomonadota, and Bacteroidota.

Viruses

The air transports viruses and other pathogens. Since viruses are smaller than other bioaerosols, they have the potential to travel further distances. In one simulation, a virus and a fungal spore were simultaneously released from the top of a building; the spore traveled only 150 meters while the virus traveled almost 200,000 horizontal kilometers.

In one study, aerosols (<5 μm) containing SARS-CoV-1 and SARS-CoV-2 were generated by an atomizer and fed into a Goldberg drum to create an aerosolized environment. The inoculum yielded cycle thresholds between 20 and 22, similar to those observed in human upper and lower respiratory tract samples. SARS-CoV-2 remained viable in aerosols for 3 hours, with a decrease in infection titre similar to SARS-CoV-1. The half-life of both viruses in aerosols was 1.1 to 1.2 hours on average. The results suggest that the transmission of both viruses by aerosols is plausible, as they can remain viable and infectious in suspended aerosols for hours and on surfaces for up to days.

Pollen

Despite being larger and heavier than other bioaerosols, some studies show that pollen can be transported thousands of kilometers. They are a major source of wind-dispersed allergens, coming particularly from seasonal releases from grasses and trees. Tracking distance, transport, resources, and deposition of pollen to terrestrial and marine environments are useful for interpreting pollen records.

Collection

The main tools used to collect bioaerosols are collection plates, electrostatic collectors, mass spectrometers, and impactors, other methods are used but are more experimental in nature. Polycarbonate (PC) filters have had the most accurate bacterial sampling success when compared to other PC filter options.

Single-stage impactors

To collect bioaerosols falling within a specific size range, impactors can be stacked to capture the variation of particulate matter (PM). For example, a PM10 filter lets smaller sizes pass through. This is similar to the size of a human hair. Particulates are deposited onto the slides, agar plates, or tape at the base of the impactor. The Hirst spore trap samples at 10 liters/minute (LPM) and has a wind vane to always sample in the direction of wind flow. Collected particles are impacted onto a vertical glass slide greased with petroleum.

Variations such as the 7-day recording volumetric spore trap have been designed for continuous sampling using a slowly rotating drum that deposits impacted material onto a coated plastic tape. The airborne bacteria sampler can sample at rates up to 700 LPM, allowing for large samples to be collected in a short sampling time. Biological material is impacted and deposited onto an agar lined Petri dish, allowing cultures to develop.

Cascade impactors

Similar to single-stage impactors in collection methods, cascade impactors have multiple size cuts (PM10, PM2.5), allowing for bioaerosols to separate according to size. Separating biological material by aerodynamic diameter is useful due to size ranges being dominated by specific types of organisms (bacteria exist range from 1–20 micrometers and pollen from 10–100 micrometers). The Andersen line of cascade impactors are most widely used to test air particles.

Cyclones

A cyclone sampler consists of a circular chamber with the aerosol stream entering through one or more tangential nozzles. Like an impactor, a cyclone sampler depends upon the inertia of the particle to cause it to deposit on the sampler wall as the air stream curves around inside the chamber. Also like an impactor, the collection efficiency depends upon the flow rate. Cyclones are less prone to particle bounce than impactors and can collect larger quantities of material. They also may provide a more gentle collection than impactors, which can improve the recovery of viable microorganisms. However, cyclones tend to have collection efficiency curves that are less sharp than impactors, and it is simpler to design a compact cascade impactor compared to a cascade of cyclone samplers.

Impingers

Instead of collecting onto a greased substrate or agar plate, impingers have been developed to impact bioaerosols into liquids, such as deionized water or phosphate buffer solution. Collection efficiencies of impingers are shown by Ehrlich et al. (1966) to be generally higher than similar single stage impactor designs. Commercially available impingers include the AGI-30 (Ace Glass Inc.) and Biosampler (SKC, Inc).

Electrostatic precipitators

Electrostatic precipitators, ESPs, have recently gained renewed interest for bioaerosol sampling due to their highly efficient particle removal efficiencies and gentler sampling method as compared with impinging. ESPs charge and remove incoming aerosol particles from an air stream by employing a non-uniform electrostatic field between two electrodes, and a high field strength. This creates a region of high density ions, a corona discharge, which charges incoming aerosol droplets, and the electric field deposits the charges particles onto a collection surface.

Since biological particles are typically analysed using liquid-based assays (PCR, immunoassays, viability assay) it is preferable to sample directly into a liquid volume for downstream analysis. For example, Pardon et al. show sampling of aerosols down to a microfluidic air-liquid interface, and Ladhani et al., show sampling of airborne Influenza down to a small liquid droplet. The use of low-volume liquids is ideal for minimising sample dilution, and has the potential to be couple to lab-on-chip technologies for rapid point-of-care analysis.

Filters

Filters are often used to collect bioaerosols because of their simplicity and low cost. Filter collection is especially useful for personal bioaerosol sampling since they are light and unobtrusive. Filters can be preceded by a size-selective inlet, such as a cyclone or impactor, to remove larger particles and provide size-classification of the bioaerosol particles. Aerosol filters are often described using the term "pore size" or "equivalent pore diameter". Note that the filter pore size does NOT indicate the minimum particle size that will be collected by the filter; in fact, aerosol filters generally will collect particles much smaller than the nominal pore size.

Transport mechanisms

Ejection of bioaerosols into the atmosphere

Bioaerosols are typically introduced into the air via wind turbulence over a surface. Once airborne they typically remain in the planetary boundary layer (PBL), but in some cases reach the upper troposphere and stratosphere. Once in the atmosphere, they can be transported locally or globally: common wind patterns/strengths are responsible for local dispersal, while tropical storms and dust plumes can move bioaerosols between continents. Over ocean surfaces, bioaerosols are generated via sea spray and bubbles.

Small scale transport via clouds

Knowledge of bioaerosols has shaped our understanding of microorganisms and the differentiation between microbes, including airborne pathogens. In the 1970s, a breakthrough occurred in atmospheric physics and microbiology when ice nucleating bacteria were identified.

The highest concentration of bioaerosols is near the Earth’s surface in the PBL. Here wind turbulence causes vertical mixing, bringing particles from the ground into the atmosphere. Bioaerosols introduced to the atmosphere can form clouds, which are then blown to other geographic locations and precipitate out as rain, hail, or snow. Increased levels of bioaerosols have been observed in rain forests during and after rain events. Bacteria and phytoplankton from marine environments have been linked to cloud formation.

However, for this same reason, bioaerosols cannot be transported long distances in the PBL since the clouds will eventually precipitate them out. Furthermore, it would take additional turbulence or convection at the upper limits of the PBL to inject bioaerosols into the troposphere where they may transported larger distances as part of tropospheric flow. This limits the concentration of bioaerosols at these altitudes.

Cloud droplets, ice crystals, and precipitation use bioaerosols as a nucleus where water or crystals can form or hold onto their surface. These interactions show that air particles can change the hydrological cycle, weather conditions, and weathering around the world. Those changes can lead to effects such as desertification which is magnified by climate shifts. Bioaerosols also intermix when pristine air and smog meet, changing visibility and/or air quality.

Large scale transport via dust plumes

Satellite images show that storms over Australian, African, and Asian deserts create dust plumes which can carry dust to altitudes of over 5 kilometers above the Earth's surface. This mechanism transports the material thousands of kilometers away, even moving it between continents. Multiple studies have supported the theory that bioaerosols can be carried along with dust. One study concluded that a type of airborne bacteria present in a particular desert dust was found at a site 1,000 kilometers downwind.

Possible global scale highways for bioaerosols in dust include:

  • Storms over Northern Africa picking up dust, which can then be blown across the Atlantic to the Americas, or north to Europe. For transatlantic transport, there is a seasonal shift in the destination of the dust: North America during the summer, and South America during the winter.
  • Dust from the Gobi and Taklamakan deserts is transported to North America, mainly during the Northern Hemisphere spring.
  • Dust from Australia is carried out into the Pacific Ocean, with the possibility of being deposited in New Zealand.

Community dispersal

Bioaerosol transport and distribution is not consistent around the globe. While bioaerosols may travel thousands of kilometers before deposition, their ultimate distance of travel and direction is dependent on meteorological, physical, and chemical factors. The branch of biology that studies the dispersal of these particles is called Aerobiology. One study generated an airborne bacteria/fungi map of the United States from observational measurements, resulting community profiles of these bioaerosols were connected to soil pH, mean annual precipitation, net primary productivity, and mean annual temperature, among other factors.

Biogeochemical impacts

Bioaerosols impact a variety of biogeochemical systems on earth including, but not limited to atmospheric, terrestrial, and marine ecosystems. As long-standing as these relationships are, the topic of bioaerosols is not very well-known. Bioaerosols can affect organisms in a multitude of ways including influencing the health of living organisms through allergies, disorders, and disease. Additionally, the distribution of pollen and spore bioaerosols contribute to the genetic diversity of organisms across multiple habitats.

Cloud formation

A variety of bioaerosols may contribute to cloud condensation nuclei or cloud ice nuclei, possible bioaerosol components are living or dead cells, cell fragments, hyphae, pollen, or spores. Cloud formation and precipitation are key features of many hydrologic cycles to which ecosystems are tied. In addition, global cloud cover is a significant factor in the overall radiation budget and therefore, temperature of the Earth.

Bioaerosols make up a small fraction of the total cloud condensation nuclei in the atmosphere (between 0.001% and 0.01%) so their global impact (i.e. radiation budget) is questionable. However, there are specific cases where bioaerosols may form a significant fraction of the clouds in an area. These include:

  • Areas where there is cloud formation at temperatures over -15 °C since some bacteria have developed proteins which allow them to nucleate ice at higher temperatures.
  • Areas over vegetated regions or under remote conditions where the air is less impacted by anthropogenic activity.
  • Near surface air in remote marine regions like the Southern Ocean where sea spray may be more prevalent than dust transported from continents.

The collection of bioaerosol particles on a surface is called deposition. The removal of these particles from the atmosphere affects human health in regard to air quality and respiratory systems.

Alpine lakes in Spain

Alpine lakes located in the Central Pyrenees region of northeast Spain are unaffected by anthropogenic factors making these oligotrophic lakes ideal indicators for sediment input and environmental change. Dissolved organic matter and nutrients from dust transport can aid bacteria with growth and production in low nutrient waters. Within the collected samples of one study, a high diversity of airborne microorganisms were detected and had strong similarities to Mauritian soils despite Saharan dust storms occurring at the time of detection.

Affected ocean species

The types and sizes of bioaerosols vary in marine environments and occur largely because of wet-discharges caused by changes in osmotic pressure or surface tension. Some types of marine originated bioaerosols excrete dry-discharges of fungal spores that are transported by the wind.

One instance of impact on marine species was the 1983 die off of Caribbean sea fans and sea urchins that correlated with dust storms originating in Africa. This correlation was determined by the work of microbiologists and a Total Ozone Mapping Spectrometer, which identified bacteria, viral, and fungal bioaerosols in the dust clouds that were tracked over the Atlantic Ocean. Another instance in of this occurred in 1997 when El Niño possibly impacted seasonal trade wind patterns from Africa to Barbados, resulting in similar die offs. Modeling instances like these can contribute to more accurate predictions of future events.

Spread of diseases

The aerosolization of bacteria in dust contributes heavily to the transport of bacterial pathogens. A well-known case of disease outbreak by bioaerosol was the meningococcal meningitis outbreak in sub-Saharan Africa, which was linked to dust storms during dry seasons.

Other outbreaks have been reportedly linked to dust events including Mycoplasma pneumonia and tuberculosis. Another instance of bioaerosol-spread health issues was an increase in human respiratory problems for Caribbean-region residents that may have been caused by traces of heavy metals, microorganism bioaerosols, and pesticides transported via dust clouds passing over the Atlantic Ocean.

Common sources of bioaerosols include soil, water, and sewage. Bioaerosols can transmit microbial pathogens, endotoxins, and allergens and can excrete both endotoxins and exotoxins. Exotoxins can be particularly dangerous when transported through the air and distribute pathogens to which humans are sensitive. Cyanobacteria are particularly prolific in their pathogen distribution and are abundant in both terrestrial and aquatic environments.

Future research

The potential role of bioaerosols in climate change offers an abundance of research opportunities. Specific areas of study include monitoring bioaerosol impacts on different ecosystems and using meteorological data to forecast ecosystem changes. Determining global interactions is possible through methods like collecting air samples, DNA extraction from bioaerosols, and PCR amplification.

Developing more efficient modelling systems will reduce the spread of human disease and benefit economic and ecologic factors. An atmospheric modeling tool called the Atmospheric Dispersion Modelling System (ADMS 3) is currently in use for this purpose. The ADMS 3 uses computational fluid dynamics (CFD) to locate potential problem areas, minimizing the spread of harmful bioaerosol pathogens include tracking occurrences.

Agroecosystems have an array of potential future research avenues within bioaerosols. Identification of deteriorated soils may identify sources of plant or animal pathogens.

Elliptic geometry

From Wikipedia, the free encyclopedia

Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.

The appearance of this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally, including hyperbolic geometry.

Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interior angles of any triangle is always greater than 180°.

Definitions

In elliptic geometry, two lines perpendicular to a given line must intersect. In fact, the perpendiculars on one side all intersect at a single point called the absolute pole of that line. The perpendiculars on the other side also intersect at a point. However, unlike in spherical geometry, the poles on either side are the same. This is because there are no antipodal points in elliptic geometry. For example, this is achieved in the hyperspherical model (described below) by making the "points" in our geometry actually be pairs of opposite points on a sphere. The reason for doing this is that it allows elliptic geometry to satisfy the axiom that there is a unique line passing through any two points.

Every point corresponds to an absolute polar line of which it is the absolute pole. Any point on this polar line forms an absolute conjugate pair with the pole. Such a pair of points is orthogonal, and the distance between them is a quadrant.

The distance between a pair of points is proportional to the angle between their absolute polars.

As explained by H. S. M. Coxeter:

The name "elliptic" is possibly misleading. It does not imply any direct connection with the curve called an ellipse, but only a rather far-fetched analogy. A central conic is called an ellipse or a hyperbola according as it has no asymptote or two asymptotes. Analogously, a non-Euclidean plane is said to be elliptic or hyperbolic according as each of its lines contains no point at infinity or two points at infinity.

Two dimensions

Elliptic plane

The elliptic plane is the real projective plane provided with a metric. Kepler and Desargues used the gnomonic projection to relate a plane σ to points on a hemisphere tangent to it. With O the center of the hemisphere, a point P in σ determines a line OP intersecting the hemisphere, and any line L ⊂ σ determines a plane OL which intersects the hemisphere in half of a great circle. The hemisphere is bounded by a plane through O and parallel to σ. No ordinary line of σ corresponds to this plane; instead a line at infinity is appended to σ. As any line in this extension of σ corresponds to a plane through O, and since any pair of such planes intersects in a line through O, one can conclude that any pair of lines in the extension intersect: the point of intersection lies where the plane intersection meets σ or the line at infinity. Thus the axiom of projective geometry, requiring all pairs of lines in a plane to intersect, is confirmed.

Given P and Q in σ, the elliptic distance between them is the measure of the angle POQ, usually taken in radians. Arthur Cayley initiated the study of elliptic geometry when he wrote "On the definition of distance". This venture into abstraction in geometry was followed by Felix Klein and Bernhard Riemann leading to non-Euclidean geometry and Riemannian geometry.

Comparison with Euclidean geometry

Comparison of elliptic, Euclidean and hyperbolic geometries in two dimensions

In Euclidean geometry, a figure can be scaled up or scaled down indefinitely, and the resulting figures are similar, i.e., they have the same angles and the same internal proportions. In elliptic geometry, this is not the case. For example, in the spherical model we can see that the distance between any two points must be strictly less than half the circumference of the sphere (because antipodal points are identified). A line segment therefore cannot be scaled up indefinitely. A geometer measuring the geometrical properties of the space he or she inhabits can detect, via measurements, that there is a certain distance scale that is a property of the space. On scales much smaller than this one, the space is approximately flat, geometry is approximately Euclidean, and figures can be scaled up and down while remaining approximately similar.

A great deal of Euclidean geometry carries over directly to elliptic geometry. For example, the first and fourth of Euclid's postulates, that there is a unique line between any two points and that all right angles are equal, hold in elliptic geometry. Postulate 3, that one can construct a circle with any given center and radius, fails if "any radius" is taken to mean "any real number", but holds if it is taken to mean "the length of any given line segment". Therefore any result in Euclidean geometry that follows from these three postulates will hold in elliptic geometry, such as proposition 1 from book I of the Elements, which states that given any line segment, an equilateral triangle can be constructed with the segment as its base.

Elliptic geometry is also like Euclidean geometry in that space is continuous, homogeneous, isotropic, and without boundaries. Isotropy is guaranteed by the fourth postulate, that all right angles are equal. For an example of homogeneity, note that Euclid's proposition I.1 implies that the same equilateral triangle can be constructed at any location, not just in locations that are special in some way. The lack of boundaries follows from the second postulate, extensibility of a line segment.

One way in which elliptic geometry differs from Euclidean geometry is that the sum of the interior angles of a triangle is greater than 180 degrees. In the spherical model, for example, a triangle can be constructed with vertices at the locations where the three positive Cartesian coordinate axes intersect the sphere, and all three of its internal angles are 90 degrees, summing to 270 degrees. For sufficiently small triangles, the excess over 180 degrees can be made arbitrarily small.

The Pythagorean theorem fails in elliptic geometry. In the 90°–90°–90° triangle described above, all three sides have the same length, and consequently do not satisfy . The Pythagorean result is recovered in the limit of small triangles.

The ratio of a circle's circumference to its area is smaller than in Euclidean geometry. In general, area and volume do not scale as the second and third powers of linear dimensions.

Elliptic space (the 3D case)

Note: This section uses the term "elliptic space" to refer specifically to 3-dimensional elliptic geometry. This is in contrast to the previous section, which was about 2-dimensional elliptic geometry. The quaternions are used to elucidate this space.

Elliptic space can be constructed in a way similar to the construction of three-dimensional vector space: with equivalence classes. One uses directed arcs on great circles of the sphere. As directed line segments are equipollent when they are parallel, of the same length, and similarly oriented, so directed arcs found on great circles are equipollent when they are of the same length, orientation, and great circle. These relations of equipollence produce 3D vector space and elliptic space, respectively.

Access to elliptic space structure is provided through the vector algebra of William Rowan Hamilton: he envisioned a sphere as a domain of square roots of minus one. Then Euler's formula (where r is on the sphere) represents the great circle in the plane containing 1 and r. Opposite points r and –r correspond to oppositely directed circles. An arc between θ and φ is equipollent with one between 0 and φ – θ. In elliptic space, arc length is less than π, so arcs may be parametrized with θ in [0, π) or (–π/2, π/2].

For It is said that the modulus or norm of z is one (Hamilton called it the tensor of z). But since r ranges over a sphere in 3-space, exp(θ r) ranges over a sphere in 4-space, now called the 3-sphere, as its surface has three dimensions. Hamilton called his algebra quaternions and it quickly became a useful and celebrated tool of mathematics. Its space of four dimensions is evolved in polar co-ordinates with t in the positive real numbers.

When doing trigonometry on Earth or the celestial sphere, the sides of the triangles are great circle arcs. The first success of quaternions was a rendering of spherical trigonometry to algebra. Hamilton called a quaternion of norm one a versor, and these are the points of elliptic space.

With r fixed, the versors

form an elliptic line. The distance from to 1 is a. For an arbitrary versor u, the distance will be that θ for which cos θ = (u + u)/2 since this is the formula for the scalar part of any quaternion.

An elliptic motion is described by the quaternion mapping

where u and v are fixed versors.

Distances between points are the same as between image points of an elliptic motion. In the case that u and v are quaternion conjugates of one another, the motion is a spatial rotation, and their vector part is the axis of rotation. In the case u = 1 the elliptic motion is called a right Clifford translation, or a parataxy. The case v = 1 corresponds to left Clifford translation.

Elliptic lines through versor u may be of the form

or for a fixed r.

They are the right and left Clifford translations of u along an elliptic line through 1. The elliptic space is formed from S3 by identifying antipodal points.

Elliptic space has special structures called Clifford parallels and Clifford surfaces.

The versor points of elliptic space are mapped by the Cayley transform to ℝ3 for an alternative representation of the space.

Higher-dimensional spaces

Hyperspherical model

The hyperspherical model is the generalization of the spherical model to higher dimensions. The points of n-dimensional elliptic space are the pairs of unit vectors (x, −x) in Rn+1, that is, pairs of antipodal points on the surface of the unit ball in (n + 1)-dimensional space (the n-dimensional hypersphere). Lines in this model are great circles, i.e., intersections of the hypersphere with flat hypersurfaces of dimension n passing through the origin.

Projective elliptic geometry

In the projective model of elliptic geometry, the points of n-dimensional real projective space are used as points of the model. This models an abstract elliptic geometry that is also known as projective geometry.

The points of n-dimensional projective space can be identified with lines through the origin in (n + 1)-dimensional space, and can be represented non-uniquely by nonzero vectors in Rn+1, with the understanding that u and λu, for any non-zero scalar λ, represent the same point. Distance is defined using the metric

that is, the distance between two points is the angle between their corresponding lines in Rn+1. The distance formula is homogeneous in each variable, with du, μv) = d(u, v) if λ and μ are non-zero scalars, so it does define a distance on the points of projective space.

A notable property of the projective elliptic geometry is that for even dimensions, such as the plane, the geometry is non-orientable. It erases the distinction between clockwise and counterclockwise rotation by identifying them.

Stereographic model

A model representing the same space as the hyperspherical model can be obtained by means of stereographic projection. Let En represent Rn ∪ {∞}, that is, n-dimensional real space extended by a single point at infinity. We may define a metric, the chordal metric, on En by

where u and v are any two vectors in Rn and is the usual Euclidean norm. We also define

The result is a metric space on En, which represents the distance along a chord of the corresponding points on the hyperspherical model, to which it maps bijectively by stereographic projection. We obtain a model of spherical geometry if we use the metric

Elliptic geometry is obtained from this by identifying the antipodal points u and u / ‖u2, and taking the distance from v to this pair to be the minimum of the distances from v to each of these two points.

Self-consistency

Because spherical elliptic geometry can be modeled as, for example, a spherical subspace of a Euclidean space, it follows that if Euclidean geometry is self-consistent, so is spherical elliptic geometry. Therefore it is not possible to prove the parallel postulate based on the other four postulates of Euclidean geometry.

Tarski proved that elementary Euclidean geometry is complete: there is an algorithm which, for every proposition, can show it to be either true or false. (This does not violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.) It therefore follows that elementary elliptic geometry is also self-consistent and complete.

Politics of Europe

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