Cantarella...how does homeschooling work? What do parents teach their kids and how do they know what to teach them? How are the kids assessed?
I can't answer for Cantarella, because every family is as individual as every child within the family is individual.
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So homeschooling for us worked very well, albeit many years ago as my youngest is now approaching 32 years. Our first two children attended school for 2 and 3 years and appalled by what they were receiving as "education" we withdrew them from the system.
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At the time and at the place where we lived homeschooling was illegal if you were not a teacher, or not under the supervision of a trained teacher. So we did it illegally. We had a visit from the police to whom we told our story and they went away and did not return.
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We collected materials from wherever we could find them - usually 2nd hand because they were the most didactic - bought, borrowed and donated. Our children progressed well, went on to become respectively a priest, an electrician and a nurse (now a mother). Our two youngest intend to homeschool their children, in fact as already doing so though none of their children is of school age yet.
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what to teach them? There are many educational services which can offer curriculums to suit you and your family's needs. In fact there are many threads here dealing with the question.
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How are the kids assessed? If my children wanted to enroll in a particular course I would send them with a sample of their work. Sometimes they were told that our homeschooled students are our best and would not even want to see the sample of their work. They did a matriculation course as in intro to university studies.
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We saved much time on daily schooling and also in years of schooling.