I learned maths AND was able to solve "story problems" in elementary school.
Looking back, the maths taught in elementary school is very basic, too. Just multiplication and division of integers, not even fractions yet.
To be honest, I don't care that much anymore if somebody doesn't want to learn maths. It's just their loss. But maths was one of the few things I enjoyed at school, so I absolutely see no point in removing it from the curriculum.
There are some school forms (like Montessori I think, quite popular here in Germany), where pupils can apparently choose for themselves what they want to learn at any particular moment. Not sure how exactly it works, but might be a good idea.
I am not saying that everybody is like me - but I think one example in this case is enough to show that the suggested way of getting rid of maths education is not the right thing for everybody. That's actually also maths: one counter-example is enough to disprove an assumption.
It seems especially dangerous to me as anti-maths types might dominate in politics and therefore might find the idea especially appealing.
Maybe it then boils down to the bigger question if talented people should be dragged down to the average levels of achievement.
In Montessori, there is some emphasis on choosing what you're interested in, but I think the more important part (when it comes to math) is that it's typically taught using strong visuals, manipulation of number blocks/cubes, etc. Instead of focusing on rote memorization of algorithms, the focus is on patterns and sensory experience. Kids "get" patterns; what they struggle with is algorithms applied to symbols.
I went through "normal" school and was always way ahead in math. My Montessori-educated sister, 3 years younger than me and not particularly mathematically inclined, could intelligently converse with me about my algebra homework when I was in sixth grade and she was in third. She wasn't as familiar with the symbols as I was, but she recognized many of the patterns (like difference of squares) in physical form.
Sure, kids don't like math. I myself didn't like math when I was a kid. Why? Well, because math takes work. Sometimes lots of work.
I believe a solution to this has to do with how math is taught, as opposed to just abolishing the subject altogether. I don't hate math any longer - I can sometimes see the beauty in it, and I sure as hell know that it comes in handy, especially when working with computers.
But even if it isn't taught properly - there is something to be said for doing math at an elementary level. Working on math problems tend to give you a certain kind of mental tenacity, that would come in handy later. The question, really, is how to make math seem fun/relevant - especially when it gets more abstract.
I think that intrinsic motivation is key, without it, it will be impossible to really learn. If you don't have intrinsic motivation to learn something, you can always go fotr the monkey approach, where you ape after your instructor to finish a task, or solve a problem. That is by no means "learning" though.
If I remember, since geometry is such a large topic, pre-calculus also teaches the parts of geometry necessary for traditional calculus study. That way, the geometry course can be made more simple.
Precalculus is the usual current designation in the United States for a secondary school course that includes functions and trigonometry, and perhaps also matrix algebra.
Many remedial courses in math in colleges will have the title "college algebra" while others will have the title "precalculus," with much the same content.
This story is dead on. I know this from personal experience and from teaching my own children. I could not add until I was 10. It was painful. I still remember it vividly. Suddenly everything clicked, and I rocketed through all the material I was behind on.
I have six children, five of which are old enough to be in school. We have been home schooling them, I believe I have some creds on this.
Two of them (both girls) took right to math early on. No problem there. Two of them (both boys) couldn't be bothered, no matter what method we tried.
One of the girls was doing math before she could read, because she was a late reader. Couldn't read until she was 9. But suddenly she got it, and took off, and is reading voraciously today. Her younger sister was reading at 4.
Both boys are now doing pre-algebra. One is 13, the other 11.
We decided not to sweat it when thy were way behind. We just kept getting them to do what they could. If they weren't getting some concept, we went slower, or went to other topics they could understand. Eventually everything started clicking.
Cookie-cutter education could never do this. I agree that national education standards that fit all children into specific requirements by age, are foolish and harmful. As guidelines they are fine, but if the government is behind them they are never just guidelines. They won't fix the problem of bad and lazy teachers (in fact, they make that problem worse), and they don't help good teachers teach better.
It sounds like you are suggesting teaching children only the things they are interested in and ready for at the time.
Compared to teaching on arbitrary set of topics at arbitrary times, you should have big gains in learning efficiency and enjoyment, since it is inefficient to spend time attempting to convey a concept the child is not ready for or is uninterested. The child will be spending a greater fraction of the time learning.
As another homeschooler, there is a fine line between gauging interest and imparting self-discipline. Of course, such "hard work" looks radically different at home than it does in the classroom. It's important to teach your children that there are times when they need to focus on things that don't necessarily interest them, but, again, how this is carried out in the home would likely be more positive than a classroom setting.
That said, I agree with the parent, and the original blog post. Our older son picked up reading and writing quickly (without much prompting from us, but his younger brother is less interested in those things, but is showing greater creative ability, so we're running with that.
> It's important to teach your children that there are times when they need to focus on things that don't necessarily interest them
Why do you need to teach them that? life does that when you pursue something you are interested in. Whenever I want to achieve something there are things I don't want to do that are necessary to do what I want.
I completely agree with this. Just cultivate an attitude of ambition in them. The environment will show them what skills are necessary to achieve their ambitions. I don't believe you should teach anything without demonstrating it's use; the environment does this more effectively than any teacher can.
But if a child is never exposed to work (that is, focusing on things that don't necessarily interest them rather than simply floating between things they find fun), he may not be equipped to make the proper decisions later in life.
Don't misinterpret what I'm saying--I'm not suggesting that children are simply forced to learn things, especially without any sort of context. I am suggesting that sometimes children needed to be nudged to get over the "I don't want to think hard about this" hump. We've seen this in our own kids. Our older son didn't particular like to put forth the effort to read himself, but we asked him to do it just a bit each day, and quickly he realized this opened up a whole world of stuff that he could do on his own.
There are actually schools (http://en.wikipedia.org/wiki/Sudbury_school) that attempt to teach K-12 this way. It would be interesting to read a study of how Sudbury alumni perform after they graduate, but I can't find one right now.
Just adding my own experience to this: Until the age of 15, maths was considered a real "problem subject" for me. At every parent-teacher conference it was generally accepted that I would never be good at it (and I had no problem with that at all - I found the whole subject very boring and difficult).
Then one day, at age 15, a classmate showed me how to do quadratic equations. I have no idea why, but that day a really bright light suddenly switched on for me. I quickly caught up to, and then passed the abilities of everyone else in the class, and then started teaching myself calculus in my spare time. I went on to get a PhD in Pure Mathematics.
I came out of Algebra I with a C or a D and didn't feel like I understood any of it. Geometry came next and it was fun, I really liked proofs and somehow by the time I got to Algebra II, everything made sense.
From that point forward, however, math just seemed like rote memorization without any meaning or elegance. I shied away from it for years, only just returning to school to learn more math recently, in my 30s. I still feel like there's something fundamentally lacking in the way the whole math curriculum is set up, as I didn't see much that I really enjoyed in the period between 9th grade Geometry and Calculus II. I understand the need to build a solid foundation before learning more advanced material, but it seems very one-sided when there is such a breadth of interesting material out there.
I'm curious - before math clicked for you, were you good at other subjects and generally one of the brightest in your class? IE - did your teachers think you were dumb overall, or just that you were a smart kid who for some reason didn't get math?
I was always good in the sciences, especially physics, and also in music. My teachers seemed to think I was smart, but they definitely thought I would never get maths.
I was very different. I learned basic addition as a toddler, and then demanded that my uncle teach me whatever was "next". Bless his heart, he made me a basic math game on his commodore 64 with flashing lights, progressively harder levels, time limits and hints. Through his patient instruction and a lot of free time on that game, I learned to multiply before entering elementary school and passed a college calculus class before entering high school.
Then, somehow after going to high school I got really into sports and less nerdy subjects and never progressed much further in terms of math than where I was at the age of 14 or 15. I regret it and OpenCourseWare beckons.
I'm still in college, and I won't have kids for some time, but I am terrified by current state of education, especially elementary, in my country (Serbia).
Could you tell me how you have made your kids socially adapted, because that is the main benefit (IMHO) of schools.
cited in this submitted blog post gets things right is that children's readiness to learn this or that is surely influenced by early childhood experiences, and the role of an education (parent at home or teacher in school) is to serve up those experiences.
My wife the music major who is a piano teacher by occupation used to play counting and adding games with dice with our oldest son when he was little. I the Chinese major who was then working mostly as a translator and language teacher began guiding him through the excellent Miquon Math materials
as he reached typical first-grade age. My son took to math an something fun, not tedious, and eventually at fourth-grade age enrolled in the EPGY math distance learning courses.
He enjoyed those, and spurted ahead to finishing the EPGY algebra course by the end of his fifth grade school year. By then he had also tested into the University of Minnesota Talented Youth Mathematics Program (UMTYMP),
and upper-division study of math as a dual-enrollment student at our state university. He is awaiting the last bit of college admission and financial aid news before deciding where to enroll for a double major in math and computer science.
To sum up, I'm not sure that homeschooling parents, or any parents, have to decide "not to sweat it" if a child hasn't learned much math yet. Maybe it's time to do activities that are supportive for learning math.
Our second son's strongest personal interest is literature--he loves to participate in National Novel Writing Month
but he applies his highly visual brain (very different from his older brother's "natural" way of approaching math) to learning math, and he is right on track to learn algebra at eighth-grade age, and rather more thoroughly than most students in Minnesota do. My two younger children (one boy, one girl) are also enjoying the Miquon Math and the Singapore Primary Mathematics materials that their older siblings used before them, well known for encouraging children to "use their noggins," as one homeschooling parent described them.
P.S. I grew to recover my childhood liking for math so much while observing my oldest son's study of math that I am now an elementary math teacher by occupation.
I agree with you, even though math was never a problem for me.
Another factor that could play into this is the learned frustration that comes with the increasing force of the teaching attempts. Some people might be bad at math when they're young, but then because it gets pushed on them so much they then acquire a personal dislike for the subject, so even if later they might be in a spot where they could learn the subject easily, they'll still have in their head that they suck at math and don't like it. It's a really hard thing to unteach.
Bear with me here, I have to lay down some foundation to get to my point.
Bias has a specialized definition in machine learning; in a (really thin) nutshell, it represents the set of concepts a given technique can represent. If your banana vs. ball classifier can only represent "roundness" and "yellowness", the bias will prevent it from seeing a yellow cube as anything but a banana. (If you really dig in, I find this concept is really a superset of the traditional meaning of "bias" and find it a lot more edifying, but that's beyond this post.)
A learning agent, even a human, can only learn what their biases permit. Much of intellectually growing up is the process of learning better biases, for instance putting away "magic" and replacing it with "chemistry". If you've ever felt your brain stretching as you learn something (a foreign language, Haskell, etc), that's your biases growing. Characterizing growing biases mathematically has been a great challenge for machine learning, but it clearly happens to humans.
Alright, getting up to the payload here: I have actually had similar concerns about our educational patterns before because you can't effectively teach something until you have the necessary biases in place. Unfortunately, you can't stop learning, you are always learning. So what happens when you "teach" something to students that lack the ability to apply the correct biases? You get the square yellow cube effect above... one way or another it will fit the biases you have (give or take growth, which takes time). The result is semantic gibberish. Now, by itself it is inevitable that you will go thorough quite a lot of gibberish as you grow up (see also "Kids Say the Darndest Things"), but why are we using precious school time to do that for math?
Furthermore, it is transparently obvious that things can be taught too soon. My 18-month-old is frolicking around my feet now, and he can't add. There's not much I can do to correct that right now. At some point he will, but this suffices to prove the point that there is a "too soon". So, when is it no longer "too soon"? Are we really sure that our traditional answer is anything more than traditional, that it has any actual truth?
Teaching things early is not harmless, either. Think about it; how many adults have downright childish issues with math? Childish misunderstandings, childish opinions, childish beliefs? Coincidence? Bad motivation? Or a schooling system that jammed it into their brains before they were ready. And those that don't have childish problems... is it merely because we were ready soon enough, rather that necessarily any actual unique skill? Some people never recover.
It is at least a question worth research and thought before we knee-jerk an answer of "if it was good enough for my grandpappy it's good enough for you".
(Getting a little more controversial, this is why I don't support really early sex education (i.e., elementary school). You will accomplish nothing except hilarious misunderstandings on a very important topic. This can have real, negative consequences, and the intentions count for nothing.)
Think about it; how many adults have downright childish issues with math? Childish misunderstandings, childish opinions, childish beliefs? Coincidence? Bad motivation?
Right here. I taught myself programming and I got pointers (references, same difference) within the first week I started reading about php. I enjoy learning new concepts except for math. I really want to learn math but I am so afraid of just opening a math book and not understanding anything in it. (Posttraumathematic stress disorder?)
I would try to find a book that balances managable and brain stretchy. _Calculus made easy_ was good for me at a certain level of understanding (yes algebra and geometry, no calculus). Try things at a library, and get recommendations, until you find one that turns you on. There's no penalty for trying a book and aborting after 1 page ;)
Are you in school? Get an independent study going with a professor on a topic she likes and you wouldn't mind learning. Emphasize that you want to get your training wheels for autodidactic math education.
May be you've seen these HN threads about self-learning maths already... if not, hopefully you'll find these textbooks useful. Here's a little HN link dump, in the order of time at which the discussions took place (from the oldest to the most recent):
One caveat is that many of these textbooks do not come with a solutions manual. So, in terms of practice, I think that's where the dearth of uploaded lecture materials could come in. Sometimes, there might be no e-learning/opencourseware material for the topic you're interested in. In such cases, I found that if you look through universities' Websites closely enough, professors upload a lot of course materials on their personal pages. You can do this by finding a random university's math dept. page, then searching from course/faculty listings.
Last of all, I'm pretty certain you've heard of this site already, as it's being circulated quite a lot lately, but many have professed that videos from khanacademy.org to be quite helpful when grasping basic mathematical concepts.
But I must acknowledge that some searches for submissions of the original article that I just tried on Google failed, although SearchYC works with enough keywords.
I don't think this sinks to the level of blogspam. The OP made a summary of the original and added his own comments. I looked at the first few entries on the home page and most of them seemed to be in this format. There is original content here and the guy writes well. To call it blogspam is (in my opinion) wrong.
Note: Wikipedia defines blogspam as "the post of a blogger who creates no-value added posts to submit them to other sites".
No math until after elementary school? what?
I remember doing multiplication timed tests in 2nd grade, waiting until 6th grade would seem to be catering to the lowest common denominator and doing a great disservice to many bright students.
Completely False. I started to learn mathematics starting in preschool and when I arrived in the US I was shocked by what passed as mathematics education in the US. The stuff American students where doing in 6th grade I had already done in 3rd grade. It might be that starting later is a good idea but the exact opposite was true for me. If I had been forced to learn mathematics the way American students learn it, well then I would have been bored out of my mind and would have given up learning it.
The American education system is constantly changing. I don't even know how they teach math to kids these days but I would probably think it was completely foolish. Apparently they changed the way they teach spelling and vocabulary and I think the new method is complete nonsense.
But the point of the article is that the current mantra seems to be that if 'kids aren't good at math in grade X' then 'teach even more math at grade X-1', which has diminishing returns when you start getting down to kindergarten and such.
> R–recitation. He wrote that by "recitation" he meant, "speaking the English language." He did "not mean giving back, verbatim, the words of the teacher or the textbook." The children would be asked to talk about topics that interested them–experiences they had had, movies they had seen, or anything that would lead to genuine, lively communication and discussion. This, he thought, would help them develop the capacity to reason and communicate logically.
He means "rhetoric." That's what the second "R" of the "Three Rs" (a.k.a. the Trivium) is supposed to mean.
This occurred to me when I took my first college psych course, and we were talking about "operational stages." The whole lecture was about how kids can't work with disconnected mental abstractions until they reach a certain level of cognitive development (the Formal Operational Stage.) It was also enforced that moving between stages of cognitive development just takes time, not more or better prerequisite education. In that light, it's kind of obvious that young children shouldn't be being taught math, any more than they should be being taught first-order logic.
This is silly. He's on his way to make a good point, but fails when his main points become 1) no math 2) focus on stories and language early on.
The good point he could have arrived at is to put less emphasis on things a child is not immediately acquiring. It would be like trying to force-feed mashed potatoes down a drinking straw. Perhaps sometimes it's math. Perhaps sometimes it's music.
Instead, he just says, stories and language are the important part! Leave math for later!
What a buffoon. He started with a bigger picture that ended with a close-minded detail.
To the original author, remind yourself that "not everyone is you." Kids can grow up to be scientists, artists, doctors, lawyers, even gasp mathematicians! And we all take different paths, even at a young age.
50 comments
[ 2.6 ms ] story [ 111 ms ] threadLooking back, the maths taught in elementary school is very basic, too. Just multiplication and division of integers, not even fractions yet.
To be honest, I don't care that much anymore if somebody doesn't want to learn maths. It's just their loss. But maths was one of the few things I enjoyed at school, so I absolutely see no point in removing it from the curriculum.
There are some school forms (like Montessori I think, quite popular here in Germany), where pupils can apparently choose for themselves what they want to learn at any particular moment. Not sure how exactly it works, but might be a good idea.
It seems especially dangerous to me as anti-maths types might dominate in politics and therefore might find the idea especially appealing.
Maybe it then boils down to the bigger question if talented people should be dragged down to the average levels of achievement.
I went through "normal" school and was always way ahead in math. My Montessori-educated sister, 3 years younger than me and not particularly mathematically inclined, could intelligently converse with me about my algebra homework when I was in sixth grade and she was in third. She wasn't as familiar with the symbols as I was, but she recognized many of the patterns (like difference of squares) in physical form.
The Montessori math approach is outlined at http://www.infomontessori.com/mathematics/introduction.htm
Sure, kids don't like math. I myself didn't like math when I was a kid. Why? Well, because math takes work. Sometimes lots of work.
I believe a solution to this has to do with how math is taught, as opposed to just abolishing the subject altogether. I don't hate math any longer - I can sometimes see the beauty in it, and I sure as hell know that it comes in handy, especially when working with computers.
But even if it isn't taught properly - there is something to be said for doing math at an elementary level. Working on math problems tend to give you a certain kind of mental tenacity, that would come in handy later. The question, really, is how to make math seem fun/relevant - especially when it gets more abstract.
Steven Strogatz has some cool ideas: http://opinionator.blogs.nytimes.com/category/steven-strogat...
Many remedial courses in math in colleges will have the title "college algebra" while others will have the title "precalculus," with much the same content.
We had already had two classes in algebra, one in geometry, and one in functions, statistics, and trigonometry.
I have six children, five of which are old enough to be in school. We have been home schooling them, I believe I have some creds on this.
Two of them (both girls) took right to math early on. No problem there. Two of them (both boys) couldn't be bothered, no matter what method we tried.
One of the girls was doing math before she could read, because she was a late reader. Couldn't read until she was 9. But suddenly she got it, and took off, and is reading voraciously today. Her younger sister was reading at 4.
Both boys are now doing pre-algebra. One is 13, the other 11.
We decided not to sweat it when thy were way behind. We just kept getting them to do what they could. If they weren't getting some concept, we went slower, or went to other topics they could understand. Eventually everything started clicking.
Cookie-cutter education could never do this. I agree that national education standards that fit all children into specific requirements by age, are foolish and harmful. As guidelines they are fine, but if the government is behind them they are never just guidelines. They won't fix the problem of bad and lazy teachers (in fact, they make that problem worse), and they don't help good teachers teach better.
Compared to teaching on arbitrary set of topics at arbitrary times, you should have big gains in learning efficiency and enjoyment, since it is inefficient to spend time attempting to convey a concept the child is not ready for or is uninterested. The child will be spending a greater fraction of the time learning.
Sounds sensible to me.
That said, I agree with the parent, and the original blog post. Our older son picked up reading and writing quickly (without much prompting from us, but his younger brother is less interested in those things, but is showing greater creative ability, so we're running with that.
Why do you need to teach them that? life does that when you pursue something you are interested in. Whenever I want to achieve something there are things I don't want to do that are necessary to do what I want.
Don't misinterpret what I'm saying--I'm not suggesting that children are simply forced to learn things, especially without any sort of context. I am suggesting that sometimes children needed to be nudged to get over the "I don't want to think hard about this" hump. We've seen this in our own kids. Our older son didn't particular like to put forth the effort to read himself, but we asked him to do it just a bit each day, and quickly he realized this opened up a whole world of stuff that he could do on his own.
Then one day, at age 15, a classmate showed me how to do quadratic equations. I have no idea why, but that day a really bright light suddenly switched on for me. I quickly caught up to, and then passed the abilities of everyone else in the class, and then started teaching myself calculus in my spare time. I went on to get a PhD in Pure Mathematics.
I do think I just wasn't ready for a long time.
From that point forward, however, math just seemed like rote memorization without any meaning or elegance. I shied away from it for years, only just returning to school to learn more math recently, in my 30s. I still feel like there's something fundamentally lacking in the way the whole math curriculum is set up, as I didn't see much that I really enjoyed in the period between 9th grade Geometry and Calculus II. I understand the need to build a solid foundation before learning more advanced material, but it seems very one-sided when there is such a breadth of interesting material out there.
Then, somehow after going to high school I got really into sports and less nerdy subjects and never progressed much further in terms of math than where I was at the age of 14 or 15. I regret it and OpenCourseWare beckons.
Could you tell me how you have made your kids socially adapted, because that is the main benefit (IMHO) of schools.
http://www.psychologytoday.com/blog/freedom-learn/201003/whe...
cited in this submitted blog post gets things right is that children's readiness to learn this or that is surely influenced by early childhood experiences, and the role of an education (parent at home or teacher in school) is to serve up those experiences.
My wife the music major who is a piano teacher by occupation used to play counting and adding games with dice with our oldest son when he was little. I the Chinese major who was then working mostly as a translator and language teacher began guiding him through the excellent Miquon Math materials
http://www.keypress.com/x6252.xml
as he reached typical first-grade age. My son took to math an something fun, not tedious, and eventually at fourth-grade age enrolled in the EPGY math distance learning courses.
http://epgy.stanford.edu/courses/math/elem.html
He enjoyed those, and spurted ahead to finishing the EPGY algebra course by the end of his fifth grade school year. By then he had also tested into the University of Minnesota Talented Youth Mathematics Program (UMTYMP),
http://mathcep.umn.edu/umtymp/
his first taste of somewhat conventional classroom instruction in any subject.
Our son of two non-mathy parents has gone on to do a lot of summer programs in math
http://www.dcu.ie/ctyi/files/Overseas%20Summer%20Brochure.do...
http://cty.jhu.edu/summer/catalogs/os/osmath.html#game
http://www.mathpath.org/newspapers/Colorado.Springs.Gazette....
http://www.math.ohio-state.edu/ross/
and upper-division study of math as a dual-enrollment student at our state university. He is awaiting the last bit of college admission and financial aid news before deciding where to enroll for a double major in math and computer science.
To sum up, I'm not sure that homeschooling parents, or any parents, have to decide "not to sweat it" if a child hasn't learned much math yet. Maybe it's time to do activities that are supportive for learning math.
Our second son's strongest personal interest is literature--he loves to participate in National Novel Writing Month
http://www.nanowrimo.org/
but he applies his highly visual brain (very different from his older brother's "natural" way of approaching math) to learning math, and he is right on track to learn algebra at eighth-grade age, and rather more thoroughly than most students in Minnesota do. My two younger children (one boy, one girl) are also enjoying the Miquon Math and the Singapore Primary Mathematics materials that their older siblings used before them, well known for encouraging children to "use their noggins," as one homeschooling parent described them.
P.S. I grew to recover my childhood liking for math so much while observing my oldest son's study of math that I am now an elementary math teacher by occupation.
http://www.ecae.net/category/saturday-school/
Another factor that could play into this is the learned frustration that comes with the increasing force of the teaching attempts. Some people might be bad at math when they're young, but then because it gets pushed on them so much they then acquire a personal dislike for the subject, so even if later they might be in a spot where they could learn the subject easily, they'll still have in their head that they suck at math and don't like it. It's a really hard thing to unteach.
Bias has a specialized definition in machine learning; in a (really thin) nutshell, it represents the set of concepts a given technique can represent. If your banana vs. ball classifier can only represent "roundness" and "yellowness", the bias will prevent it from seeing a yellow cube as anything but a banana. (If you really dig in, I find this concept is really a superset of the traditional meaning of "bias" and find it a lot more edifying, but that's beyond this post.)
A learning agent, even a human, can only learn what their biases permit. Much of intellectually growing up is the process of learning better biases, for instance putting away "magic" and replacing it with "chemistry". If you've ever felt your brain stretching as you learn something (a foreign language, Haskell, etc), that's your biases growing. Characterizing growing biases mathematically has been a great challenge for machine learning, but it clearly happens to humans.
Alright, getting up to the payload here: I have actually had similar concerns about our educational patterns before because you can't effectively teach something until you have the necessary biases in place. Unfortunately, you can't stop learning, you are always learning. So what happens when you "teach" something to students that lack the ability to apply the correct biases? You get the square yellow cube effect above... one way or another it will fit the biases you have (give or take growth, which takes time). The result is semantic gibberish. Now, by itself it is inevitable that you will go thorough quite a lot of gibberish as you grow up (see also "Kids Say the Darndest Things"), but why are we using precious school time to do that for math?
Furthermore, it is transparently obvious that things can be taught too soon. My 18-month-old is frolicking around my feet now, and he can't add. There's not much I can do to correct that right now. At some point he will, but this suffices to prove the point that there is a "too soon". So, when is it no longer "too soon"? Are we really sure that our traditional answer is anything more than traditional, that it has any actual truth?
Teaching things early is not harmless, either. Think about it; how many adults have downright childish issues with math? Childish misunderstandings, childish opinions, childish beliefs? Coincidence? Bad motivation? Or a schooling system that jammed it into their brains before they were ready. And those that don't have childish problems... is it merely because we were ready soon enough, rather that necessarily any actual unique skill? Some people never recover.
It is at least a question worth research and thought before we knee-jerk an answer of "if it was good enough for my grandpappy it's good enough for you".
(Getting a little more controversial, this is why I don't support really early sex education (i.e., elementary school). You will accomplish nothing except hilarious misunderstandings on a very important topic. This can have real, negative consequences, and the intentions count for nothing.)
Right here. I taught myself programming and I got pointers (references, same difference) within the first week I started reading about php. I enjoy learning new concepts except for math. I really want to learn math but I am so afraid of just opening a math book and not understanding anything in it. (Posttraumathematic stress disorder?)
- http://news.ycombinator.com/item?id=108723
- http://news.ycombinator.com/item?id=201913
- http://news.ycombinator.com/item?id=458926
- http://news.ycombinator.com/item?id=755043
- http://news.ycombinator.com/item?id=1058359
- http://news.ycombinator.com/item?id=1193352
One caveat is that many of these textbooks do not come with a solutions manual. So, in terms of practice, I think that's where the dearth of uploaded lecture materials could come in. Sometimes, there might be no e-learning/opencourseware material for the topic you're interested in. In such cases, I found that if you look through universities' Websites closely enough, professors upload a lot of course materials on their personal pages. You can do this by finding a random university's math dept. page, then searching from course/faculty listings.
Last of all, I'm pretty certain you've heard of this site already, as it's being circulated quite a lot lately, but many have professed that videos from khanacademy.org to be quite helpful when grasping basic mathematical concepts.
http://news.ycombinator.com/item?id=1211198
But I must acknowledge that some searches for submissions of the original article that I just tried on Google failed, although SearchYC works with enough keywords.
Note: Wikipedia defines blogspam as "the post of a blogger who creates no-value added posts to submit them to other sites".
But the point of the article is that the current mantra seems to be that if 'kids aren't good at math in grade X' then 'teach even more math at grade X-1', which has diminishing returns when you start getting down to kindergarten and such.
> R–recitation. He wrote that by "recitation" he meant, "speaking the English language." He did "not mean giving back, verbatim, the words of the teacher or the textbook." The children would be asked to talk about topics that interested them–experiences they had had, movies they had seen, or anything that would lead to genuine, lively communication and discussion. This, he thought, would help them develop the capacity to reason and communicate logically.
He means "rhetoric." That's what the second "R" of the "Three Rs" (a.k.a. the Trivium) is supposed to mean.
The good point he could have arrived at is to put less emphasis on things a child is not immediately acquiring. It would be like trying to force-feed mashed potatoes down a drinking straw. Perhaps sometimes it's math. Perhaps sometimes it's music.
Instead, he just says, stories and language are the important part! Leave math for later!
What a buffoon. He started with a bigger picture that ended with a close-minded detail.
To the original author, remind yourself that "not everyone is you." Kids can grow up to be scientists, artists, doctors, lawyers, even gasp mathematicians! And we all take different paths, even at a young age.