I am a homeschooling parent who came to realize that my young children (in grades 1 and 3 this year) are into mathematics for its beauty. They are amazed at the fact that adding two whole numbers in any order yields the same result (commutativity), grouping when adding may be helpful (associativity), that for particular numbers n, x, and y, we have that n = x^2 = y^3: 1 = 1^2 = 1^3; 64 = 8^2 = 4^3, etc. They are mesmerized by the fact that 1 = 1, 3 = 1 + 2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, ...; special sequences of numbers such as (in this case) triangular numbers (which they know as "step shapes"). They don't take for granted that 1n = n for any integer (I don't think they are ready for real numbers in general). They can see that division is akin to factoring: 12/4 = 3 because 3 and 4 are factors of 12 (though they are not too up on the vocabulary yet)... They can see informally that a - b = - (b - a): 8 - 5 = 3 implies that 5 - 8 = -3.
I hope I am fortunate enough to see them through their entire grade school through high school learning.
My concern is: why do educators/teachers and parents a debate so much about basic skills versus critical reasoning as if these things removed from context of the ACTUAL beautiful ideas in mathematics will pull school children into understanding mathematics any more than knowing words alone is enough to make a great writer? Why do we either (a) think that its all basic skills, or (b) word problems about so-called everyday stuff that are more social studies than mathematics? How about focusing on mathematics as subject like we do Language Arts? We don't reduced Language Arts to writing letters and business contracts in which grammar and syntax are important? We want children to enjoy the beauty of language; shouldn't we do the same with mathematics?
Of course we want children to enjoy the beauty of language and, accordingly we might as well want them to enjoy the beauty of mathematics. However, we must not forget that both language and mathematics serve a very clear and particular purpose. In first place they are tools, and very useful ones indeed. Then, their usefullness must be emphasized and used as a motivation. The purely aesthetical "joy" can (or maybe not) come afterwards... After all, beauty is more subjective than applicability...
I think Cris Wellington was very lucky with the kids. Most children are unlikely to be able to get interested in the beauty of mathematics, all the more so in an ordinary classroom. Therefore, a pragmatic approach to the formation of curricula and methods is fully justified. Individual support for gifted children is needed. This is possible in small classes, with tutorial teaching.
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