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[Carolyn once said that math was "a seamless whole" inside her head,...]
I don't know if this ties in with the idea of a seamless whole, but it has occurred to me that discrete skills are needed first before one can appreciate the connectedness of math. Without these concrete skills, math is more like a seamless black hole.
This became apparent to me again when teaching a group of seventh and eighth graders brought up on EM and currently using CMP who are a tabula rasa when it comes to the simplest bits of math knowledge. They can't do any operations with fractions (e.g. change mixed numbers to improper fractions let alone addition and division), can't divide decimals, don't have knowledge of even rudimentary geometry... One wonders what they have been doing for seven and eight years.
The seventh graders are currently in the CMP stretching and shrinking stage. Their homework consisted of finding the scale factor of two rectangles the width of which goes from 1.5 cm to 3 cm. So the idea was to divide 3 by 1.5 (they can't do it because they can't divide decimals). When I tried to show an alternative way of division using fractions to demonstrate the connectedness of math (seamless whole), I ran into trouble, too. They don't have the discrete skills of seeing 1.5 as 1 1/2, then changing this mixed number to 3/2 and dividing 3 by 3/2 (they absolutely can't divide fractions and moreover don't see 3 as 3/1. It would have been spectacular to make them experience with understanding that the more complicated decimal division problem 3/1.5 virtually solves itself when you divide the respective fractions (3 divided by 3/2). Invert and multiply but they have never heard of reciprocals and how they work. The 3 cancels and 2 is left standing without much ado!
So the upshot is: they use Connected Mathematics but can't see the connectedness of math because they don't have discrete skills (skills they could have learned through drill and kill but haven't). So to them, math is a seamless black hole from which not even light can escape.
This one's going in the Greatest Hits file. (on the sidebar)
wholes, not parts
top down teaching
whole math taught wholly
1 comment:
I just checked out my kid's CM book and the index page for "algoritm for addition" It says "you will devise an algorithm for addition, here are some samples to try, make sure it works for all case and mixed numbers". The homework is "use your method to add these fractions". Ditto for subtraction, multiplication and division. Not only is there no standard method, NO COMPLETE METHOD, NO MATTER HOW WACKY is described anywhere in the book. "Common denominator" isn't even in the book index. Yet a lady at CMP, and the BACK OF THE DON'T PANIC LETTER clearly state that students will learn how to use common denominators to add, multiply tops and bottoms, and invert and multiply to divide. How are they going to do this with a book that doesn't even contain ONE method to do any of these things? Your post gives the answer - THEY DON'T.
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