Fillyjonk spoke to some of this above. I'm still at work and have a meeting in a couple of minutes, but I wanted to say that I have a young adult friend with no Thai language background who has been living in Thailand this year in homestay / au pair - plus school situation. She began dreaming in Thai after about 4 months and feels that's the point at which she had become quite fluent. Now seven months in she feels entirely comfortable in Thai. Unlike some immigrants, she dropped all her native language comforts when she arrived. She didn't return to family or other English speakers every time she entered her home. She didn't read English books or watch English TV or hang out with English friends. And (she thinks this is crucial) because she was living with a family of young children she was able to practice Thai in a low-stakes way by engaging in play with children who had simple vocabularies. She felt free to try and fail and try and fail and try and fail, with simple vocabulary in simple grammatical constructs.
Mountain mama to three great kids and one great grown-up
I think there is also a difference in both how subject matter is presented to young children, and the way they use their brains compared to adults. Adults, by training or development or whatever, are more, hmmm.... analytical is the wrong word because I think kids are too in their way.... more linear.... more judgmental though I don't necessarily mean negatively judgmental.
The methods for young children are introduction to a concept, practice and play. You just don't get that much past the early elementary ages. I think that this method can integrate ideas far better than the way we are taught in, say, high school, or in adult education. Visiting foreign countries and completely immersing oneself I think can bring adults back to that same space... so I don't think it is gone, just that we are trained out of it to some degree. To what extent, I don't think we can until we encounter a large population people who have never had exposure to the same education, yet can understand much of the same things. So, with my limited understanding of the issue, my first impression is that the "window" is an artificial construct.
Yes, there is something to be said for immersion and being exposed to something everyday, but I think the integration of language into human life cannot be matched by mathematics, except for those few who see the world in those terms. And those people are not at that point because someone taught them calculus, they learned calculus to better describe the relationships they are seeing.
I really have no idea what I'm talking about.... I merely giving my impressions.....
Give me a few minutes while I caffeinate.
At first glance it seems so but we are talking about second languages here. The integration of of second languages require quite an energy, intentional planning and implementation and in some cases extreme measures. I think we are just not as used to taking such measures for mathematics.
What would be the purpose of learning this second language? Is there a need for fluency, or enough of an understanding to operate within a certain context? Why am I learning a second language?
Maybe I'm misunderstanding this whole line of thinking..... I'm not seeing it as a perfect parallel. Perhaps because I think that a lot of mathematics is purely for the sake of mathematics, and any application comes later. So, in that sense, I think it is more like music... satisfying some indefinable human need that does not, however, serve an immediately apparent purpose. I can see the second language as analogous to those maths that we use purposefully to describe things and events... cosmology and astronomy and engineering.
Give me a few minutes while I caffeinate.
I don't agree that with language and music the hard work on basics (eg. grammar and spelling, music theory and scales) is easily deferred down the road while one builds a natural intuitive love of the subject, whereas with mathematics the hard work on basics must be done at the outset. I think that we assume that about mathematics because that's the story we all bought when we were learning math in school. Curriculum writers pretty much all buy it too. But I don't think it's true.
My dd is finishing up some geometry in an 8th grade textbook and the curriculum has been looking at rotations as they apply to tesselations. It seems about what I'd expect from an 8th grade curriculum. But last term I watched a 5-year-old explore rotations in art class:
He spent a couple of hours creating this mandala, and then a lot of time afterwards inspecting it from different angles to decide which motif was the "upside down of that one", and delightedly commenting that "lots of curved 'besides' make an upside-down, but it's backwards!" What he meant was that printing the small black circular motif several times with small amounts of rotation each time eventually takes you a diametrically opposite image, but the diametric image isn't a mirror-image. This is pretty sophisticated stuff and it doesn't require any knowledge of angles or cartesian graphs. Why do reflections and rotations only come into the curriculum at a 5th or 8th grade level when they are this cool and this accessible to 5-year-olds? Because culturally we believe that 5-year-olds should learn to count and add and multiply before they're ready for the fun stuff.
About incremental, daily exposure being important. I think there is a role for it. I think it's particularly useful when you want to be able to move beyond intellectual tasks to a sort of second-nature internalization, a mindless "automaticity" of the learned skill. Daily repetition allows learning to become deeply and intuitively ingrained. That's helpful with complex physical tasks like learning vibrato on the violin or cursive handwriting or cartwheels in gymnastics. It's helpful when you want to memorize a piece of music to the extent that it will play itself inside your head without conscious thought. It's helpful if you want to develop speed with multiplication facts to the point that you have essentially instant recall. If mindless mastery isn't the goal, shorter spurts of immersion when motivation is burning bright are probably a more natural and effective way to do the same learning.
I think that developmental critical periods are unlikely to exist for learning tasks that were not crucial to human survival 20,000 years ago. In other words, I find it hard to believe that something that wasn't selected for on an evolutionary time scale will have hard-wired itself into our neurology. It's only in the last few (couple?) hundred years that reading has become an important aspect of survival fitness for our species. So I doubt reading resides in a part of the brain that is specifically devoted to just that task and which, if not given appropriate input during a critical period, will wither away. On the other hand, vision is something that is developed this way: twenty thousand years ago if you couldn't see well you were likelier that your tribe-mates to become food for lions. And so this is hard-wired in: the visual cortex at the back of the brain atrophies or gets devoted to other tasks if it doesn't receive "education" in the form of visual input from the eyes. I believe that language centres in the brain are very analogous: children who miss the critical period for language development can learn to vocalize speech sounds intelligibly and can accumulate some vocabulary, but they never develop syntax. The ability to understand and create syntax seems to be subject to critical-period considerations.
I choose to start teaching music to very young children. Why? Not because I believe there's a critical period, but because I know that young children's expectations are easily moulded into realistic proportions. They are less anxious about producing good short- and medium-term results when compared with external reference points. I have a lovely little five-year-old violin student who has just finished a year of lessons. A month ago she could still not quite play her very first piece, Twinkle Twinkle Little Star. She kept happily plugging away with her mom practicing for 15 minutes every day that whole time. Any adult who took almost a year of lessons and couldn't yet play a single simple tune would either have given up or become so anxious about the lack of progress that the anxiety impeded their learning. But for this kid violin was just a neat thing to work on every day with her mom's help, repeating the same simple little components, gradually, glacially stringing them together. Then about a month ago, the components skills gelled together, and since then she's been learning a new piece every week -- things have just taken off in a huge way! I have no doubt her visible progress will continue on this much faster trajectory now, because all the work she did for the first 11 months has built within her a physical and musical intuitiveness with the instrument. An adult who got through the same apparent learning in a week would not build that intuitiveness, and eventually, usually after 4 years or so, that would catch up with them making further progress very difficult. An adult who spent a year on the beginner stuff probably would progress beautifully from then on ... but it's impossible to find adults who will willingly endure all that time with so little success to show for it.
So in the case of music and math and second language learning, I believe that success comes from the process-oriented learning attitude of the young child, not any critical learning period. The product-orientation of we older folk creates expectations for ourselves that get in the way of thorough enjoyable learning that firmly roots the basics. That's not surprising: as more abstract thinkers we can project goals and plans and look down the road and compare reality with our hopes and dreams -- and we get frustrated with the pace, feel unsuccessful and maybe give up. That can make it look like there's a bit of developmental critical period -- but it's really more to do with attitude and expectations "ability to learn" per se.
Mountain mama to three great kids and one great grown-up
As for playing for with higher maths before learning the basics, in the form of mandala art for example or any other way, sure that can and should be encouraged/done. It is part of the process of learning math.
I totally agree with the last two paragraphs of Miranda's post. Too bad I can't quote
As for the purpose of learning, in general, I see my role as helping my children lay strong foundations so as to open up as many possibilities as possible for their later potential choices. And to do this while preserving their love of discovery and exploration. This also includes facilitating their development as self-directed learners who enjoy not just the end result but also the process of inquiry and acquisition of knowledge itself. I don't believe these things are mutually exclusive.
My DH wishes he studied engineering but he got intimidated by the math. He drowned in calculus basically and fell behind quickly. This altered his path and pushed him to a different direction. I think STEM careers are easier to pursue if you feel like you have the competence to withstand the process. I want to keep that possibility open for my kids, should they chose it. I want to help them remove as many stumbling blocks as possible.
I bring up learning second languages as a parallel but of course it is not the same. Not being able to do mathematics shuts one off from many fantastic, easily accessible career choices. I guess, among many other things, I also see education as a utility to a better/easier life/career choices.
@ Emaye I think the relevance of the critical period is that some people would argue that the reason to teach music, languages and potentially math early is that at certain points in childhood your brain is wired to be especially receptive to certain kinds of learning. That is one of the main reasons people (not necessarily you Emaye) might argue that kids need to learn stuff young-because if they don't, it will be actually impossible later. Am I correct in saying that you'd see it as circumstances which mean its important to learn this stuff young? What I'd say is that circumstances are very individual. For example, for a teenager or someone in their early twenties, prekids, to decide to pick up math or a language really doesn't seem to me to be a big deal. They might, possibly, be looking at deferring uni entrance by a few years. If they have work responsibilities then they might have to prioritise but honestly, I can study 24 hours a week, homeschool three kids, do a bunch of voluntary work do my urban homesteading thing and do the other things I do and still fritter away time on here. Its doable. Its hard but its not unpleasant. If you don't have time management at the start my god you will get it quite fast. I can tell you from repeated personal experience that it is not impossible to learn this stuff later in life.
What I really agree with is the idea that we can't necessarily assume kids will gain exposure to any given idea. And I also strongly feel personally (I've said this so much I'm sure I'm boring people now!) that we cannot take it for granted that our girls will be exposed to the same ideas as our boys. I am inclined to feel that you are right, having math/science knowledge opens a lot of doors. More important to me, actually, even is that it allows them to critically interact with the world-a kid who understands number won't be overwhelmed and switch off at a barrage of statistics but hopefully say "ok so lets look up the original study. Now was the sample size big enough? Was a double blind used? " . I strongly want that for my kids, enough that I very strongly suggest to them that learning arithmatic and being comfortable with it is a very good idea indeed. I dunno, more than anything to me its about equipping them for their life now really though-they see the newspapers, they talk with people, understanding how number work are useful to them now. With the possible exception of logarithms, I think almost everything you need in higher math would be stuff accessible through practical, living, math at a younger age. What helps with higher math more than anything is a sense of number and I don't think you can get that except by playing and experiementing with numbers.
@ Miranda really, really interesting to read yous post . Its very interesting to hear it regarding music also. Ok this is kind of OT but...perhaps its interesting, I don't know. I actually have come across maybe three adults who have learnt to some degree with Suzuki teachers as adults (I play viola, or violin where required, but I'm a proper amateur!). One in particular (ironically, in the context of this thread, a mathematician who teaches at the university) learnt with his children from when they were tiny, alongside them, and when they quickly overtook him he carried on with the same teacher, trying to approach it in a kid way and not become focused on the result. He had a particular interest as a teacher himself in doing this. He was highly process orientated and always stood out for this.
Also, I've been fairly active in my local folk music scene and there are a lot of adults there who are largely self-taught violinists who are very much playing in a process orientated way. Oftentimes someone will turn up with a violin (or ditto, another instrument) and everyone who already plays will sort them out so they can play at least a note or two in the right place. The Welsh folk music scene right now is very much about having fun and keeping the tradition alive so although technically we are performing in that we're playing in pubs, there is always room for learners, instrument switchers and so on. These guys tend not to have ever had lessons and there is a culture of avoiding formal instruction, but whats interesting to me is that they seem not to be significantly held back by things I'd see as absolutely un-negotiable, like a reasonable bow hold or not supporting the neck with your left hand. They go so far, realise that its not going to work for them to hold the ffidl (violin) like that and then work out another solution. The measure of their success is entirely in the music, whether their strategies for playing allow them to play at the level they are at. Its pretty interesting to me. (to me, who worked diligently through the Suzuki books and the UK grades this feels delightfully subversive )
I think this mostly comes from a study that says there is a critical period for language acquisition. If a child is raised by wolves, not exposed to any language in their early years, he won't be able to learn more than a few simple words. People extrapolated that this meant there was one for everything else like learning to read or learning a second language. But those are secondary types of language learning not the most fundamental language itself that has a critical period. But pronouncing certain sounds can't be learned as adults because of physical changes, I guess to the larynx, with age...
I don't know about math, though. Brains do better with abstract thinking after a certain age (10+?). Earlier years in school seem to just focus on mastering/memorizing arithmetic so when they are older they can get on with the more abstract stuff. My son comes across a ton of math concepts playing computer games. He was surprised to come across x and y axis on a tv show (Numb3rs) because he uses them in Minecraft. He's been doing simple algebra since he was 5. But he really doesn't have his "math facts" down at all. He frequently asks me to add things. I'm not sure how much of that is his wanting to confirm his own answer (or get a specific number instead of his ballpark estimation.) I don't think his lack of math skills (arithmetic) has slowed him down with learning concepts.
The whole critical period thing, not sure where it came from. My argument for starting math in childhood comes for the reasons Miranda decides to teach music in childhood, plus others, mainly -- they have a lot more time, less responsibility, and less distractions. Also see the walking across America example...
I think it came from a natural progression of the suggestion of wanting to start a child early on regular math practice in order for the task of higher math to be less daunting down the road. This naturally suggested the controversy of critical windows.
I think for adults, the problem comes because we see the whole road, and it can be discouraging unless we have early success to keep us motivated (and can be discouraging still). It was pointed out that many-most children focus on what's happening now. They are not seeing the long road to proficiency, they are seeing the next song--or in the case of Miranda's little student just the notes themselves. Adults need progress they need results. We embark on our journies with a purpose. Hanging out around A, B and C and grooving on our alphabetness doesn't cut it, for the most part, as it does with most kids. We have a goal in mind--X, Y, and Z, and we can be completely turned off a particular path because we dread the length of time and hard work to get us all the way to our chosen end.
Give me a few minutes while I caffeinate.
Young kids are process oriented and at some point seem to become product oriented. Not sure if that's natural maturity or a reaction to teaching methods. Possibly it's a combination. Ideally one should still enjoy the process as an adult...