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Unschooling
With variations known as "natural learning", "autonomous learning", "self-directed learning", "life learning", "radical unschooling" and probably many others. I'll talk a bit more extensively about this as it's the style most different from school, so most people who haven't done it find it difficult to imagine how it works.
Unschooling is a philosophy of learning based on the belief that children learn best when they are learning what they are interested in, when they are interested, in their own way, and for their own reasons. The parent supports and encourages learning, rather than directing it. This partly involves honouring the child's passions, but also setting up the learning environment in such a way that the child will be exposed to things they might not have known about, so they will develop new interests and gain a well-rounded education.
In particular, the unschooling approach is philosophically opposed to any use of force, coercion or extrinsic reward systems, which are all seen as damaging to intrinsic motivation. Intrinsic motivation is regarded as the key to meaningful, permanent learning.
Parents with a strong philosophical commitment to unschooling usually take the stance that if a particular skill or piece of knowledge is truly necessary, the child will sooner or later come up against a need for it, and will be motivated to learn it because they need it. Therefore, so long as they know how to learn, curriculum plans are redundant. These families focus on providing a rich learning environment for their children, and trust that their children will learn what they need to know. Other unschoolers keep a close eye on what their children are learning naturally, and compare this to what they would like them to be learning, and find creative ways to "fill the gaps" without coercion. (These are often parents who have gravitated to unschooling because they have found it to be how their child learns best, but who had no prior commitment to the philosophy.)
Unschooling is a good approach for children who have never been to school, as they can simply continue learning all the time, through play and exploration, as they did when they were preschoolers. There is no transition, just a gradual morphing into a more mature learning programme. Unschooling also suits children who are very self-motivated and independent, who have strong ideas about what they want to learn. It can sometimes be helpful for children who have had bad experiences in school, who have lost their love of learning, and who are apathetic or downright resistant to any formal tuition, although a lengthy "decompression" period may be necessary until the child regains their intrinsic motivation. Children with very high energy levels (who are sometimes misdiagnosed as having ADHD in schools) are free to pursue lots of active learning, and will be much happier without a requirement to sit still on demand for lengthy periods. Some parents find their "hyperactive" or "oppositional" child is suddenly "cured" when they begin unschooling.
People not familiar with unschooling often wonder how the children will learn anything. Basically they learn in all the ways people do when they aren't at school. This includes preschoolers, adults, and children doing extra-curricular activities and learning things for pleasure. Experimenting, playing, asking questions and conversational learning, reading books, attending clubs or formal classes, finding tutors, study buddies or mentors, making things, practising hobbies, using computers, learning from real life experiences, visiting art galleries, museums, and so on, exploring the natural world, travelling ... the possibilities are endless! Unschooling students do sometimes choose to use textbooks and workbooks too, when it suits their learning goals. Most unschooling parents are happy to use formal curriculum when their child expresses a desire to learn something specific; they regard formal instruction as fine if the child has requested it, but not if it is imposed upon the child.
There are a few myths about unschooling:
Myth #1: Unschooling is not academically challenging and lacks rigour.
Not so! Unschooling may be more or less rigorous than other styles of homeschooling; it depends on the child. Bright students who are unschooling often move at a faster pace, cover a broader curriculum, and choose more challenging materials than they would have had access to in school.
Myth #2: Unschooled children are undisciplined and won't learn how to knuckle down and do unpleasant jobs that need to be done.
This is also false. Life presents many opportunities to learn discipline. Doing household chores, behaving properly, caring for pets, getting on with difficult people, and so on are all ways to learn discipline. It isn't necessary to take the joy out of learning to develop discipline.
Myth #3: Unschooling parents are lazy - they don't have to do anything.
Definitely false! Unschooling parents have to create a rich learning environment, set a good example of lifelong learning, understand and support and mentor their children, help them find resources, allow time for in-depth conversations, read to their children and play with them, teach them life skills and things they want to learn, take them to interesting places, expand their horizons, and so on. If you have naturally compliant children, getting them to plough through a couple of hours of workbook pages may be much easier!
Myth #4: Unschooling can't be mixed with other styles of homeschooling.
Not true. Many homeschoolers study some subjects formally and use a natural learning approach for others. Some families find unschooling suits their younger children, but their older children want a more formal curriculum. Others may find their younger children like a structured programme, and as the children get older they are more able to study independently and become more self-directed. These are all options.
Myth #5: Unschoolers don't go to university.
False. Many do go to university, and do very well, as they are used to self-directed learning.
Myth #6: Unschoolers won't get real jobs.
This may be partly true. If your definition of a "real job" is something that earns money but has no personal meaning, people who have been unschooled often don't want such a job. Unschoolers usually try to find rewarding work they love. Unschooling graduates are more likely than most people to start their own businesses or develop careers in creative industries, or find a way to turn their hobby into employment.