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Hung Hsi Wu has written a document for the NMAP about fractions. This document "contains a detailed description of the most essential concepts and skills together with comments about the pitfalls in teaching them. What may distinguish this report from others of a similar nature is the careful attention given to the logical underpinning and inter-connections among these concepts and skills."
Basically, this is a document about what you need to know about fractions and how they work. This is how they should be understood. It is NOT a teacher's manual, and certainly not a student's textbook, but it is the mathematical basis of one, and it's a darn good start.
It is dense. Like a good math paper, there are no extra words to confuse. There is exactly as much precision and description but no more. That makes it slow reading if you are unfamiliar with the material.
Read it for yourself. Do you know everything it says? If not, does it help you identify your own lapses in understanding fractions? Does it help you to explain these concepts to your own youngster?
Over the next few days, I'll write more posts about this document, expanding on some of his writing. Perhaps we can make a KTM parent manual from this document with enough feedback and examples.
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4 comments:
Perhaps we can make a KTM parent manual from this document with enough feedback and examples.
That would be great.
I like what you've said about Wu's prose. I find reading Wu difficult in exactly the way I find reading a math textbook difficult.
What a great document, thanks for the link, Allison.
I also like the idea of a parent manual.
Allison asks, about Wu's paper on teaching fractions (which seems to be a draft chapter for a book on mathematical pedagogy), "Does it help you to explain these concepts to your own youngster?"
I just finished reading his draft chapter on whole numbers and am about halfway through the one on fractions, and definitely it helps to explain fractions as well as to diagnose strengths and weaknesses of curricula in how they explain fractions. (For example, I think Singapore is pretty solidly in line with what is is proposing, with the exception of not being as explicit as he advocates; but if you Wu-out on Singapore-educated kids, there should not be a very big conceptual hurdle.)
I am very motivated to alter how I am homeschooling my kids, based on what I have read so far. (And we aren't doing a bad job; my background is in mathematics and I have naturally tended to incorporate explanations in the style that he suggests -- see the paper on whole numbers and how he explains the long-division algorithm by expanding the dividend, emphasizing distribution and, well, doing it by hand; the paper is "Chapter 1: Whole Numbers (Draft) (July 15, 2000; REVISED September 1, 2002)", to be found at http://math.berkeley.edu/~wu/ )
Anyway, I look forward to using his philosophy to re-do fractions from the beginning through all 4 operations. I think that if you have a good number-line based approach like Wu advocates, there is no point in doing two operations one year, and getting to division of fractions in 6th grade or whatever the typical sequence is. It looks to me like the best thing is to have kids draw out many number lines, subdivide ruler-and-compass style, and reckon example after example of adding, subtracting, multiplying,and dividing fractions.
"Wu-out"
I like that.
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