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This is one of my favorite MMM posts so far:
Math as a Foreign Language
You learn [a foreign] language by listening to others speak, by looking words up in the dictionary, by learning the rules of grammar and then memorizing the “irregular” applications that break all those rules. Finally, you are able to communicate with others in this new language and the more often you do so, and the more challenging the conversations become, the more you grow in your ability to communicate.
You can always tell who had the benefit of a good foundation through formal instruction (grammar, syntax, pronunciation, vocabulary) and who just "picked it up" informally, on their own and without academic support. The latter tend to be more limited in the range and depth of communication in that foreign language. There is a limit to what they are able to accomplish with the "tools" in the "toolbox" so to speak. Those with the strong foundation move through their discipline with ease, with confidence and with mastery. I think the same applies to many other disciplines be it ballet, basketball, the violin or dare I say, mathematics?
Yes, learning math is much like learning a language. When you master the basics through diligence, perseverance, and the benefit of good instruction, one day without realizing it, you’ve opened the door to a whole new world.
Beautiful.
10 comments:
You don’t discover the language on your own. You don’t invent the language. Well actually, according to Krashen, you do. Perhaps the trend has changed from the 80s and 90s but there was a big movement to minimize the explicit teaching of grammar from textbooks. The student literally discovers the rules of grammar on his own. It takes a lot of work to find a textbook that lays it all out for you. Hmmm. I've got a "Whole language/fuzzy grammar" French text and a 1901 text. Time for another compare and contrast?
And the "Natural Approach" people like Stephen Krashen are working as hard as they can to eliminate formal grammar from the curriculum.
This too is now acceptable since grammar doesn't work for everyone (but evidentally it does work for the majors who will get double doses of it in upper divsion courses) Texts which explicitly teach grammar are hard and intimidating and what in the world would a non-grammary person do if it were not for the Natural Approach? See, MATH is like foreign language, you should learn it on the fly, piece meal. It "works" for some people.
Krashen: "In support of this position are studies showing that even advanced students with a great deal of interest and experience with grammar are able to access only a small amount of their grammatical knowledge when actually using language. Even when students are deliberately focused on form and taught rules carefully, the impact of grammar study is weak"
And you know whose fault it is? The university professors who only know linguistics and literature but nothing about pedagogy.
Sound familiar?
The student literally discovers the rules of grammar on his own. It takes a lot of work to find a textbook that lays it all out for you.
I've said this before, but for the benefit of those who weren't on KTM1 or didn't see it the first (or second!) time, I highly recommend the Warriner's English Grammar series for structured grammar study. I learned from those in at least the seventh, eighth, and ninth grades in two different schools (one public, one private) in the late 1970s, and they were fantastic. Head on over to Amazon and find yourself a used one!
I wasn't on KTM I.
Thanks for the recommendation.
I have a theory on constructivism.
With nearly any skill, there exists a small subset of the population which will quickly and easily at it with barely any effort exerted. We've all seen it - musicians who can memorize, repeat, and modify anything after a single listen, athletes who glide while the rest of us stagger, and, yes, those damned geniuses who busted the curve every time without ever picking up a textbook.
The constructivist looks at the natural learners, and thinks, 'He learned it so easily just by diving in, while everyone else struggled with formal learning. Therefore, we should skip the boring formal learning and just have everyone jump sraight in!'
There is even a certain plausibility to it - after all, if you're going to build a learning program, why not model it on the experiences of those who seemed to have succeeded at it. The problem is, if you ask a prodigy how he managed to learn so quickly, he probably couldn't explain it to you, and he certainly couldn't teach it to you. That's what a prodigy is - an exceedingly rare individual who hit the genetic lottery, and was born with a brain pre-configured for a particular skill. It's no more plausible to expect me to play chess like Kasparov or than it is to expect even a gifted ten year-old to suddenly 'get' algebra simply by exposure. If I want to be a grand master, I'm going to have to work my tail off for another decade or six, and Junior is going to have to practice those standard algorithms until he can do it in his sleep.
A less obvious point is that often, even prodigies benefit from formal study. Michael Jordan was not just a phenomenal athlete; he was also a pathologically competitive individual who worked harder than anybody else in the league. He treated practices like the regular season, the regular season like the playoffs, and the playoffs like World War III.
I think you have a good point, Independent George.
David Mulroy says that one reason for all the "immersion" classes is that adolescents and adults don't know enough formal grammar to learn foreign languages.
When I talked to Mitchell Dobbs, the famed middle school ELA teacher (fantastic writing instructor) he said that one of the things he would change, if he were to teach middle school again, is that he would give formal instruction in grammar.
He had a very nice way of putting it, something like, "Everyone tells you not to teach grammar, but kids will learn what you ask them to learn."
In other words, kids like to learn, and if you teach well they'll like learning grammar.
Christopher's teacher spent some time teaching sentence diagramming this year -- turns out she had to shoehorn it in. It's not in the curriculum.
Warriner's English Grammar
Yes, I managed to track them all down - I'll get the ISBN numbers posted if they aren't already up on ktm1.
A less obvious point is that often, even prodigies benefit from formal study. Michael Jordan was not just a phenomenal athlete; he was also a pathologically competitive individual who worked harder than anybody else in the league.
This is the lesson of the 10-year rule.
It takes 10 years for a genius to reach his peak, same way it takes 10 years for the rest of us.
Mozart wrote his first classics 10 years after beginning serious study at the age of 4.
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