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Re: 4 function calculators  and, why Arithmetic?
Posted:
Feb 9, 2013 5:46 PM


On Feb 9, 2013, at 3:56 PM, Jack Rotman wrote:
> [...]
> However, a more basic question must be raised  WHY do we see a > need to teach arithmetic? > Once upon a time, arithmetic was an advanced topic; pupils would > learn about number systems, properties, and the reasoning behind > operations. I've read people state that arithmetic was more of a > university topic than a grammar school (K8) topic. Learning > fractions, for example, involves many layers of knowledge  if we > want people to understand (and remember); given two months, students > could learn this type of arithmetic with fractions ... and another > month for decimals. > > Long before calculators became readily available, we had reduced > 'arithmetic' down to the procedures commonly used in a given culture > for computation. The backbone of arithmetic  number systems and > properties  was avoided as being either too difficult or > unnecessary. Correct computation is usually the most common > assessment. > > If we see 'arithmetic' as being the stuff commonly found in > arithmetic and prealgebra courses & books, then we have little > rationale for 'teaching' it. > I fully agree. > Correct computation is needed in very few fields and in few academic > courses  if we mean without technology. Our client disciplines > and occupations are much more interested in problem solving and > thinking (quantitative reasoning sounds good to them). > > I believe that arithmetic (described above) has no place in a modern > math curriculum. Arithmetic is a major problem for adults in our > country, especially those from low power/low economic backgrounds; > we have blocked many a student from pursuit of their dreams just > because they could not perform addition of numeric fractions and > similar 'skills'. Our treatment for this lack is a course which > induces students to compute correct answers within our course > without any longterm benefit (understanding and reasoning). >
While, of course, arithmetic can be a nice place where to develop some serious thinking which, moreover, carries over to algebra and beyond:
(1) For instance, seeing 345.67 as
3?10^+2 + 4?10^+1 + 5?10^0 + 6?10^1 +7?102
where, initially, + is to be read as "and"as in 3 apples + 4 bananasand ? is to be seen as just a separator is rather nice when dealing with
3x^+2 + 4x^+1 + 5x^0 + 5x^1 + 7x^2
In fact, algebra is "easier" than algebra because there is no carryover.
(2) Similarly, for instance,
5/7 = 0.71 + [...] 5/7 = 0.7142 + [...] 5/7 = 0. + [...] 5/7 = 0.714 + [...] 5/7 = 0.7 + [...]
where [...] is a precursor of Landau's "little o", o[h^n] read as "something too little to matter in the current situation". For more along such lines see
<http://www.freemathtexts.org/Standalones/RDA/index.php>
(3) In fact, since decimal numbers are essentially (Laurent) polynomials, this paves the way to the use of (Laurent) polynomial approximations as an alternative to limits (due to Lagrange) in the development of the differential calculus.
(4) In a different direction, one can develop "multidimensional arithmetic", aka fruit salad arithmetic, which, these days of "carts" and "price lists" could probably interest students which wouldn't feel dissed but would not feel lost. For more on this, see
<http://www.freemathtexts.org/References/AmatycReview/MathUndergroundPDF/17LanguageAnalysisLinAlgebra.pdf >
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