Here is what I have found out about the different styles based on research. So far, we have largely used a virtual school that is really mostly school at home along with some classical and Charlotte Mason techniques for History and a smattering really of most of the other styles here and there for enrichment. I find the problem of "school at home" being limiting and wearing on both educator and student to be somewhat true, but I also find the stated advantages to these styles to also be accurate. For more info about what we do in our home (and especially the unit studies we have done) you'll need to check out my website (listed on my profile).
School At Home
This style is really like having school, but in your own house. The parent/learning coach teaches using a combination of didactic and hands on approaches with the support of textbooks and workbooks purchased through a publishing house, home school consortium or the local bookstore.
The nice thing about this style of home schooling is that the parents know exactly what to teach and when, as long as they follow the guidelines of the curricula purchased. Lessons, practice and how much should be completed of each subject each day of schooling is laid out within the materials. This can be a great confidence builder for a year or two for parents and kids just starting out even if they would prefer a freer style down the road.
It does tend to be the more expensive of the styles and it can be difficult to motivate students if the materials do not fit their particular learning style. It also tends to be one of the styles that results in the earliest "burn-out".
This style could not be more opposite from the "school at home" style. Also known as "interest led", "natural" or "child-led" education, this style does not bear any resemblance to what many would view as a traditional education. In unschooling, children play games and are exposed to experiences that will encourage their learning. These experiences are mostly directed by the child's interests. Many people new to the concept wonder about how these children learn very much this way. However, the proponents of the style say when we wait and give them time and when they are ready they will ask, experiment and try to find out (this idea is largely backed by personal experiences and the fact that kids taught this way often do extremely well in their chosen fields).
According to my sources, the challenge with Unschooling comes when it is time for standardized testing because these children are not use to such formal assessments. Adjusting to College or integrating into a standard classroom can also be challenging for the same reason. These children really do learn "how to learn" though as usually parent and child are researching answers to the the child's questions together. These children are also likely to become experts in their area of passion and although they may have "holes" in areas of less interest, they will pick up "basics" (such as basic mathematics) through necessity as they arrive at problems requiring these skills to be solved. For more information about this style check out www.unschooling.com or http://naturalchild.org.
Charlotte Mason was an educator herself and outlined a philosophy centered around educating the child through exposure to enriched and "living" materials and observation. The idea is that kids will learn by interacting with the world in "real life" situations and through their exposure to "living books". Living books contain rich text we would consider great literature and that has stood the test of time (classics). Textbooks would usually not be considered "living".
Charlotte Mason kids learn science through observing nature on nature walks and experimentation. They learn history by reading about it through "living books", visiting history museums where they can view historical artifacts and by keeping a time line to help in visualizing the passage of time. Likewise, they learn art by being surrounded by masterworks and then trying their hand at it, music by exposure to classic works and practice.
This website, Charlotte Mason Method Homeschool, has a discussion Forum, links to resources, articles and online books about Charlotte Mason-styled education and learning, including an article all about "getting started". For even more information, Katherine Levison is one of the leading authors regarding this style.
I did not find a lot the specified specific benefits or drawbacks as clearly as were outlined for the other two styles already listed.
The Classical Approach
The Classical Approach to an at home education is one that has been around since the middle ages and in actuality is probably the most truly traditional of all the approaches (more so than sending a child to the "traditional" mortar and brick school in your neighborhood today). This approach uses the Trivium (or five tools for learning), including reason, record, research, relate and rhetoric to gain deeper and deeper learning for the student. "The Well Trained Mind" by Susan Wise Bauer and Jessie Wise is mentioned frequently in relation to this educational style. For more information check out the online magazine "Classical Homeschooling".
Without realizing it, we have used a classical approach history record book in our History Curriculum already. The classical approach uses a lot of "copy work", oral narration and recitation of items children are to have memorized. This "copy work" can become tedious for both learner and teacher, but it does offer the practice children need for memorization. From a distance, it may look a lot like the school at home approach and indeed, pre-made and organized curricula can be purchased specifically for this style sort of combining the two approaches. There are also mortar and brick schools that are considered "Classical Schools" that use this approach.
For some reason, much of the pre-made materials for this approach are religiously based, and I have had trouble finding materials for those persons who are NOT religious or even specifically Christian to use with this approach. I am told, more and more Classical approach secular groups are arising so this drawback (for some) will change over time. From a cursory search for materials, supplies and lesson plans for this approach, science education may be a weakness because it would appear not much experimentation occurs. However, the Classical approach does seem to have a reputation for being one of the best ways to make sure children get a very good background in History.
Virtual Homeschooling is basically like school at home but with an added element to the "purchased" curricula which is that there is access to an online community (and sometimes a real-world community as well) all using the same set of curricula and curricular standards. Different Virtual Schools might have differing philosophies, so it is possible to find those that use an approach from almost any of the styles listed. The idea is that all your materials, lessons and activities can be prescribed by the school, and organized and laid out for you (minimizing leg work for the parent). Families that use enrollment in a "Virtual School" may have access to trained teachers in person, online chat rooms and class sessions done in a "go to meeting" or "Skype" type of program. The online classes use computers and these programs to link students from across state, or even international boundaries with an expert in a particular subject. These online classes are in addition to what the parent teaches using the program's set lesson plans, guide books, and workbooks/practice materials and are not the only lessons children receive. Much of a virtual school curriculum can be done offline and without the use of a computer as well as online.
The advantage here is that in many virtual schools the school seeks accreditation with various states and legally (in the US) the child is considered to be enrolled in a private or charter school. This means parents using these programs have fewer concerns over meeting legal requirements as it becomes the school's responsibility to help achieve these set standards and obtain the appropriate sample work (or at least teach you exactly what to record, how to record it and what to collect).
The disadvantages are much like those for families using the "school at home style". These families have more freedom over what is covered educationally than do families with kids in mortar and brick settings, but they have even fewer freedoms to diverge from the prescribed texts and curricula than their "school at home", "Charlotte Mason", and "Classical" counterparts. This can be a costly way to do schooling as well.
www.k12.com can give you examples of ONE of the leading virtual schools available internationally.
A Unit Studies educational approach tries to create cross-curricular units in which one central subject is studied, but many (or all) of the core curricular subjects are addressed within the unit. For example, a unit study of Ancient Rome would include learning Roman Numerals, a few Latin basic root words (that are intended to help with both Language Arts skills and understanding Rome), art and architecture from Ancient Rome, and the Mythology of Ancient Rome (Literature being another category of Language Arts). Writing assignments would center around lessons done about Ancient Rome and for science one might choose to focus on the structural advantages of arch construction, gadgetry of warfare as an in for simple machines and/or volcanoes as it ties in to the eruption of Vesuvius and all that has been learned from digs at Pompeii and Herculaneum.
Most Unit studies try to incorporate hands on activities or some hands-on culminating project. Unit studies may be child-directed (chosen because of an interest the child has) or learning coach/teacher/parent directed (a topic is chosen because of a need the adult perceives in the education of the child in question). The advantage to learning this way, is that it is often a lot more memorable and "real world" than teaching each of the subjects separately. It also helps kids to see connections between subjects and therefore more value in subjects they might otherwise not find interesting.
The thing that is tough about this approach is balancing all the subjects and making sure you are meeting educational benchmarks in a fashion conducive to expectations in states that dictate standardized testing even in the earlier grades. It can be done, but requires a great deal of organization and research on the part of the educator to use Unit Studies alone.
Rudolf Steiner founded this educational style around 1919 when he was asked to establish a school for the workers at the Waldorf Astoria Cigarette Factory in Stuttgart, Germany. Since he believed the human was a mix of spirit, soul and body and that there was a three stage emergence of the full being that occured in three developmental stages (early childhood, middle childhood and adolescence) his school was profoundly different from most schools of the day, and since. When I visited Waldorf Schools in St. Louis as part of my teacher training I found them to be incredibly comfortable because the environment is totally natural. Furniture is not made of plastic and metal, but of wood. Instead of playdough, kids mold bee's wax (which is amazing for your skin btw - softer hands in minutes). Kids work in an organic garden as part of their introduction to life sciences, nature and ecology. As you enter the school, you remove your outdoor shoes and slip into slippers . . .
Musical education begins with clapping games, singing and rhythm and then in about second or third grade kids are taught to play the recorder. Reading and writing are not a part of the curriculum until third grade, but plenty of games are played that help increase literary awareness so that when kids are taught to read and write in a more formal way they pick it up very easily and quickly. It is all about experiential learning and contact with the natural things of the world. Many schools emphasize an avoidance of early access to television, computer and other "screen" and electronic technologies as well as plastics and non-organic chemicals and products (note: the founding of this style predates the environmental movement). Kids learn by doing. For example, they make their own muesli (as a class) for morning snack, and pick fruits or veggies from the garden to make into an afternoon snack. In each snack preparation, measuring is taking place along with teamwork. After having a fairy tale read to them, the kids might recreate that fairy tale through dress up and dramatic presentation.
Even though this style was founded in mortar and brick schools many families are using the style in their home schools. Training is available (though expensive) as are plenty of pre-made supplies and curricula.
Founded by Educator and Physician Maria Montessori, much like an "unschooling" approach, these schools emphasize a child directed approach to education. However, in the Montessori school, there is a prescribed set of choices of specific games out and available that will steer the children in the direction of learning certain things.
Montessori trained teachers are trained to ask guiding questions that will help nurture and direct the child's thinking and questioning. If a child is floundering he or she is brought back into corrected thinking through their own reasoning, but with the help of thoughtful, well-planned questions.
The Montessori Foundation has a fairly extensive website with more information as well as resources for schools and information about training. They are in the process of building a section specifically designed for home school families attempting to use a Montessori approach. Montessori games and activities are often used in regular classrooms - especially in preschools and elementary mathematics so you may be more familiar with some of these than you think.
I actually disagree with this as a title for an Educational Method, as Howard Gardner set forth a hypothesis about brain development, learning and (mainly) the nature of intelligence, not a guideline or philosophy about how education should occur. However, many people do use it this way.
Howard Gardner hypothesized that there are seven different intelligences (and added an eighth later) and we all exhibit these intelligences in different degrees. People that claim to use this as a method are really saying, they accept the hypothesis that there are multiple intelligences and incorporate this belief into their teaching methods in a thoughtful way that puts the child's talents and strengths to use in their learning.
Although we know there are different learning styles there is no "Learning Styles" approach to education just as there really isn't truly a "Multiple Intelligences" approach to education. Rather, this approach simply represents a philosophy or belief that a child's natural strengths (or intelligence) should be valued and tapped into in order to help each individual learn whatever subject matter is at hand in much the same way teachers today try to make sure to address all four learning styles in the classroom (a clarification: learning styles are very different from intelligences and in fact, Gardner addresses learning styles along with intelligences in some of his writing). MI approach then becomes a bit of an eclectic mix of styles depending upon the individual who is receiving the education and their specific needs as the approach to teaching will consider the learning style AND combination of intelligences the child leans on most in the choice of method used for a specific individual.
An Eclectic homeschooler draws upon any of the homeschooling styles as that parent, learning coach or teacher deems most beneficial for their family situation and philosophy. This style usually means the parent is taking the best that each of the other styles has to offer and using it to teach in his or her home. The advantage of course, is the freedom to go with what works in each moment for each kid while still having a structure to help guide the parent through subjects he or she may not already know much about.
The disadvantage of course, is that it can be an overwhelming way to go about educating a child because of the sheer number of resources to weed through in order to find the best match for each child and subject (and its subsets of information and skills). Eclectic homeschooling can be done very inexpensively or it can become highly expensive depending on choices made in regard to purchased materials for curricula or activities.
What the proponents and opponents of a style have to say about it online and in books doesn't always match the reality for all. If your experience differs from what I have found in my research, PLEASE add another answer!!! I always love to learn more.