I've lost count of the number of times that Colin has posted an interesting article on mathematics education on Hacker News over a weekend. This submission here is particularly good. The problem indicated is quite stark, and very commonplace. Most pupils in a mathematics lesson in elementary school swiftly learn that "getting the right answer" is the point, and some check out and soon begin to doubt their own ability to REASON to the right answer.
The author of the submitted article writes, about a seventh grade class, "The basic premise of the activity is that students must sort cards including probability statements, terms such as unlikely and probable, pictorial representations, and fraction, decimal, and percent probabilities and place them on a number line based on their theoretical probability." As the author makes clear, the particular lesson arises from the new Common Core State Standards in mathematics,[1] which are only recently being implemented in most (not all) states of the United States, following a period of more than a decade of "reform math" curricula that ended up not working very well. I am favorably impressed that the lesson asked students to put their numerical estimates of probability on a number line--the real number line is a fundamental model of the real number system and its ordering that historically has been much too unfamiliar for American pupils.
The author continues by elaborating on his main point: "When did we brainwash kids into thinking that math was about getting an answer? My students truly believe for some reason that math is about combining whatever numbers you can in whatever method that seems about right to get one 'answer' and then call it a day." I like the author's discussion of that issue, but I think she misses one contributing causal factor--TEACHER education in the United States in elementary mathematics is so poor[2] that most teacher editions of mathematics textbooks at all levels differ from the student editions mostly just in having the answers included[3] and don't do anything to develop teacher readiness to respond to a different approach in a student's reasoning.
What I LOVE about the Singapore Primary Mathematics series,[4] which I have used for homeschooling all four of my children, is that the textbooks encourage children to come up with alternative ways to solve problems and to be able to explain their reasoning to other children. The teacher support materials for those textbooks are much richer in alternative representations of problems and discussions of possible student misconceptions than typical United States mathematical instruction materials before the Common Core. Similarly, the Miquon Math materials[5], which I have always used to start out my children in their mathematics instruction before starting the Singapore materials, take care to encourage children to play around with different approaches to a problem and to THINK why an answer might or might not be correct. (Those materials, both of them, are very powerful for introducing the number line model of the real number system to young learners, as well as introducing rationales as well as rote procedures for common computational algorithms. I highly recommend them to all my parent friends.)
I try to counteract the "what's the correct answer" habit in my own local mathematics classes (self-selected courses in prealgebra mathematics for elementary-age learners, using the Art of Problem Solving prealgebra textbook[6]). I happily encourage class discussion along the lines of "Here is a problem. [point to problem written on whiteboard] Does anyone have a solution? Can you show us on the whiteboard how you would solve this?" Sometimes I have two or three volunteer pupils working different solutions--which sometimes come out to different answers [smile]--at the same time. We DISCUSS what steps make mathematical sense according to the field properties of the real numbers and other rules we learn as axioms or theorems in the course, and we discuss ways to reality-check our answers for plausibility. We don't do any arithmetic with calculators in my math classes.
[3] When I last lived overseas, I had access to the textbook storage room of an expatriate school that used English-language textbooks from the United States, and I could borrow for long-term use surplus teacher editions of United States mathematics textbooks. They were mostly terrible, including no thoughtful discussion at all of possible student misconceptions about the lesson topics or of alternative lesson approaches--but they were all careful to show the teachers all the answers for the day's lesson in the margins next to the exercise questions.
My kids are now in middle school. The curriculum that they used in K-5 encouraged kids to come up with their own approaches. I'll admit that I was rather shocked, as it seemed quite messy and inefficient, but I discussed it with a neighbor who's a high school calculus teacher. He told me: "Be patient, just you wait, they'll surpass you in math." And he was right!
Also, our school kept several sets of the Singapore Math books on hand, and loaned them out to families who wanted to give it a try at home.
So I think the "correct answer" is not the problem, because that's just math. The "correct algorithm" is the problem. Or maybe, the algorithm is the answer. Oddly enough, the one course where the algorithm is indeed the answer -- high school geometry -- is the one that most people remember fondly.
Most pupils in a mathematics lesson in elementary school swiftly learn that "getting the right answer" is the point, and some check out and soon begin to doubt their own ability to REASON to the right answer.
It's been some years since I was in K-12 (I'm in my 30s), but I remember the opposite being the case: almost all the points on homeworks and exams were typically awarded for "showing your work". A correct answer with no work shown got few to no points; meanwhile an incorrect answer could get close to full credit if it showed the correct reasoning but ended up with the wrong result just because of a trivial addition/etc. error.
People tended to criticize this from the opposite direction, arguing that "in the real world" getting the right answer is all that matters, and that giving kids near full credit for incorrect answers was coddling them.
Homeschooling as in not sending your kids to school and teaching them yourself at home? This is such an exotic concept to me as a european, would you mind telling my why you'd do that?
My personal reason for choosing homeschooling for my four children, after discussions with my wife about our childhood educations in two different countries, was to ensure flexibility so that we would never be slowing down our children's education. That has allowed them to learn more at younger ages than is typical in United States schools, even in a good school district like the one we live in. This school year, all three of my children who are still minors are actually enrolled in public school programs (one full-time at the local high school, and the younger two full-time in an online public school). Next year we will try something different again. Flexibility and the opportunity to mix and match learning resources with children's needs is why we homeschool mostly, relying on outside resources as needed.
My badly neglected personal website[1] (I've hardly updated any page on it for YEARS), provides some more details of my thinking about homeschooling, current to a decade ago and perhaps not fully representative of my motivations today.
I can't answer for the OP, but I was homeschooled and I was around a lot of people from other cultures that were also homeschooled.
In the case of the Americans, they were homeschooled primarily because the parents felt they could do a better job than the US school system, and because they wanted to use Christian-friendly learning materials. For social contact and 'real-life' social education, they'd put their children into a variety of (sports) clubs.
In the case of my parents, it was a practical necessity, as we lived in a developing country. The nearest international school was a long drive away, it was expensive, and reintegration into the Dutch school system would have been much more difficult.
Instead, their solution was the get a just-out-of-school teacher to have one or more exciting years volunteering and practicing their craft on me and my siblings, which my parents would augment with personal attention and long-distance help from retired teachers back in Holland. The teacher would use a Dutch 'long-distance learning' approach that was the defacto system for Dutch expats. For high-school material, this meant a book that explained, day by day, what should be read and what exercises should be done.
Now, I personally feel the results were so good, that I would do everything to make it possible to either home-school my children (if I have 'em), or to get them into some alternative school (montessori, etc.). Aside from some difficulties adjusting to Dutch society, my siblings and I did quite well, and I believe on the reasons we are all active learners and autodidacts is a direct result of not having been part of the normal system.
The primary argument I hear against home-schooling when normal schooling is possible is that it hampers a child's social development. I never bought that. Kids get a lot of that from playing out in the street, or being part of clubs. Furthermore, for a significant portion of kids I believe their 'normal' schooling can have a negative social impact, through various degrees of bullying, and the group pressure to believe that learning is stupid. I did not have a clue that voluntary study was stupid and nerdy until I returned to high school at age seventeen...
(of course, I also understand that for many people home-schooling isn't a practical option. I think that's a shame. And I also understand that some parents might not be any good at teaching, and that a mandatory educational system can be a benefit in that case.)
The most common reason around here are parents who don't want their kids religious beliefs distorted by the general public and secular school system. There are religious specific schools but they tend to be expensive and far apart. It is easier to just pull your kids from school and have a parent teach them, supplemented with clubs and special activities.
Another reason is parents might not want their kids to have to cope with the social pressures of the school resulting from both income disparity and bullying.
Some kids just don't do well in large group activities with generic and/or dumbed down lessons. 1:1 teaching and custom tailored educational experiences can really help some kids excel. When the household has enough money and a well enough educated parent, to support a parent staying home doing full time teaching, these kids get educational opportunities they wouldn't get in a large public school system.
Similarly, for whatever reason sometimes the family lives in a school district with either terrible schools or schools with such a cultural mis-match that it doesn't make sense to even bother sending your kids there.
Schools in the US are not all equal. Even a few miles apart, schools can be VERY different. And kids are all different too. We have the opportunity to be flexible about educating our children, which I think works out for the best for the majority of such special cases in the long run.
I have seen home schooled kids who have serious issues with:
1) Deadlines
2) Structured Schedules (like showing up for work on time).
3) Coping with social situations that are outside their comfort zone.
4) Separating from their parents.
5) Recognizing that not everyone learns the same way or at the same speed or is as smart as they are.
Sometimes they end up in trouble and unable to be self-sufficient in our society. Sometimes they find a niche that works for them and are very happy. Sometimes they have none of these issues and you'd never know they were home schooled.
The author of the submitted article writes, about a seventh grade class, "The basic premise of the activity is that students must sort cards including probability statements, terms such as unlikely and probable, pictorial representations, and fraction, decimal, and percent probabilities and place them on a number line based on their theoretical probability." As the author makes clear, the particular lesson arises from the new Common Core State Standards in mathematics,[1] which are only recently being implemented in most (not all) states of the United States, following a period of more than a decade of "reform math" curricula that ended up not working very well. I am favorably impressed that the lesson asked students to put their numerical estimates of probability on a number line--the real number line is a fundamental model of the real number system and its ordering that historically has been much too unfamiliar for American pupils.
The author continues by elaborating on his main point: "When did we brainwash kids into thinking that math was about getting an answer? My students truly believe for some reason that math is about combining whatever numbers you can in whatever method that seems about right to get one 'answer' and then call it a day." I like the author's discussion of that issue, but I think she misses one contributing causal factor--TEACHER education in the United States in elementary mathematics is so poor[2] that most teacher editions of mathematics textbooks at all levels differ from the student editions mostly just in having the answers included[3] and don't do anything to develop teacher readiness to respond to a different approach in a student's reasoning.
What I LOVE about the Singapore Primary Mathematics series,[4] which I have used for homeschooling all four of my children, is that the textbooks encourage children to come up with alternative ways to solve problems and to be able to explain their reasoning to other children. The teacher support materials for those textbooks are much richer in alternative representations of problems and discussions of possible student misconceptions than typical United States mathematical instruction materials before the Common Core. Similarly, the Miquon Math materials[5], which I have always used to start out my children in their mathematics instruction before starting the Singapore materials, take care to encourage children to play around with different approaches to a problem and to THINK why an answer might or might not be correct. (Those materials, both of them, are very powerful for introducing the number line model of the real number system to young learners, as well as introducing rationales as well as rote procedures for common computational algorithms. I highly recommend them to all my parent friends.)
I try to counteract the "what's the correct answer" habit in my own local mathematics classes (self-selected courses in prealgebra mathematics for elementary-age learners, using the Art of Problem Solving prealgebra textbook[6]). I happily encourage class discussion along the lines of "Here is a problem. [point to problem written on whiteboard] Does anyone have a solution? Can you show us on the whiteboard how you would solve this?" Sometimes I have two or three volunteer pupils working different solutions--which sometimes come out to different answers [smile]--at the same time. We DISCUSS what steps make mathematical sense according to the field properties of the real numbers and other rules we learn as axioms or theorems in the course, and we discuss ways to reality-check our answers for plausibility. We don't do any arithmetic with calculators in my math classes.
[1] http://www.corestandards.org/Math/
[2] http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf
http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...
http://www.nytimes.com/2013/12/18/opinion/q-a-with-liping-ma...
[3] When I last lived overseas, I had access to the textbook storage room of an expatriate school that used English-language textbooks from the United States, and I could borrow for long-term use surplus teacher editions of United States mathematics textbooks. They were mostly terrible, including no thoughtful discussion at all of possible student misconceptions about the lesson topics or of alternative lesson approaches--but they were all careful to show the teachers all the answers for the day's lesson in the margins next to the exercise questions.
[4] http://www.singaporemath.com/category_s/252.htm
[5] http://miquonmath.com/
[6] http://www.artofproblemsolving.com/Store/viewitem.php?item=p...