Saturday, October 6, 2007

Who Invented "Lattice Multiplication"?

I can't seem to get the issue of lattice multiplication off my mind. The negative reactions to this perfectly sound algorithm are not grounded in any reasonable arguments, as far as I can see. But I can't help but suspect that many people, both those who are vehemently opposed to teaching alternative approaches such as lattice multiplication, as well as those who are open to or neutral on this and similar ideas, believe that somehow some hippie math ed reformers at the University of Chicago Mathematics Project (UCSMP) dreamed it up one night and stuck it into EVERYDAY MATHEMATICS (unless other hippie math ed reformers at TERC beat them to it with their INVESTIGATIONS IN NUMBER, DATA & SPACE curriculum). Alternate theories, some involving Satan, Osama Bin Laden, or Josef Stalin have not been verified as of this writing.

However, looking at Frank Swetz's CAPITALISM & ARITHMETIC, I came upon the following earlier today. I hope this doesn't cause riots among some of the anti-reformers:

A ready variant of this method is the following gelosia or graticola technique, so named for its similarity to the contemporary lattice grillwork used over the windows of the high-born Italian ladies to protect them from public view. Following Byzantine custom, the wives and daughters of Venetian nobles were usually kept sequestered. Spying unobserved from such vantage point, these ladies often saw scenes that disturbed them or made them "jealous."

The computational technique employed is basically the same as that used in per quadrilatero; however, the cells are partitioned by diagonal lines so that when the product of a row entry and a column entry result in two digits the units digit is written below the diagonal line and the ten's digit above. In this manner, carrying becomes an integral part of the algorithm. When all partial products are obtained, the entries within diagonal columns are added as before and the resulting total product written along a side and base of the array.

This method is quite old. It probably originated in India, was known to be popular in Arab and Persian works, and was finally accepted into European arithmetics in the fourteenth century. Due to its organizational efficiency and the ease it provides in multipling any two multidigit numbers, it was quite popular as a computational scheme; however, it was difficult to print and read and thus fell out of favor. It is from the gelosia grid and principle that the computational device known as Napier's Bones (1617) evolved.

Horrible to discover that this "fuzzy" approach has been around for many, many centuries, was popular and fell into disfavor only because of the limitations of 15th century printing methods (Swetz's book is focused on arithmetic in 15th century Europe). But of course, if it has organizational efficiency and ease of use, it just might be an okay thing to teach to kids as ONE way to do multiplication.

I read only a couple of days ago on an anti-reform math list that "[s]ince your child is only in first grade one thing you really need to be aware of going forward is that it is unlikely you child will be taught the standard techniques (now elevated to "algorithms" ) for doing basic arithmetic operations in EM." This was quite a surprise to me, since Everyday Math most definitely presents the "standard techniques or algorithms" for addition, subtraction, and multiplication. Long division, a point of much contention in the Math Wars, is not presented in the EM curriculum, though during my work in Pontiac, MI, the teachers at all five of the elementary schools where I coached introduced it anyway. I suspect they are not isolated cases. But aside from division, there is no doubt that the claim by this parent about EM is completely false. Sounds great, though, when you want to scare the bejeezus out of folks, especially those inclined to believe anything they read or hear that puts unfamiliar or "new" mathematics teaching tools in a negative light.

In any event, it appears that we can't lynch the authors at UCSMP or TERC for lattice multiplication. I have no doubt, however, that some charge or other can be made to stick and that sooner or later those who dare to try to help kids think about and do mathematics more deeply and effectively will be punished for their temerity.

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