Re: [unrev-II] Valid contributions to the debate? - Mental arithmetic, ballpark assessments, mental models

From: Henry van Eyken (
Date: Mon Feb 14 2000 - 19:52:54 PST

From: Henry van Eyken <>

Eric Armstrong wrote on Sun, Feb. 13:

> My major thought:
> I'm not sure that calculators "free" us.
> Someone ran a test with calculators rigged to produce obvious,
> unmissable errors -- orders of magnitude off -- and found that
> college students would rarely question the result. If they did, and
> the calculator produced the same result a 2nd time, they'd go
> with it!
> Like the atrophy of memory from lack of use, the ability to do abstract
> thinking (via symbol manipulation, of which arithmetic is a simple
> manifestation) is also withering away. (Unfortunately, the rote right-
> to-left addition mechanism taught in school is all but useless for
> mental addition. Working from left to right and using various
> algebraic manipulations works a lot better for mental math. (e.g.
> 22 + 97 = (20 + 2) + (100 - 3) = (20 + 100) + 2 - 3 = 120 - 1)
> "Mental math" should be taught in schools, if only as preliminary
> exercise for symbolic logic, algebra, and other symbol-manipulation
> practices.

Agreed. Some domains I am much conscious of where I do thinking are mental
arithmetic and three-dimensional visualizing. (Sorry for making this
first-person.) I tend to associate mental arithmetic with applying mental tricks
(or algorithms, routines really!) like mental multiplication such as 25 x 34
(hard) = 3400 / 4 (easy), etc. Just little algorithms to move givens and
processes into short-term memory. Even inaccurate mental manipulation ought to
verify results obtained with a calculator with a "ballpark" check.

Calculators, especially pocket computers like yesteryear's Casios and Sharps,
could well be excellent devices for efficiently providing insights into whether
outcomes of calculations are resasonable or not. I wrote an item about that in
Fleabyte, "The little Red Engine That Could've"
f Little Red Engine that could

I feel sad really that technology has moved so fast that a contemplative
examination of the benefits it may provide is destroyed in the turmoil of its
wake. I felt there just has not been enough time to contemplate the potential
value of the devices described (cheap, programmable computers that fit the
pocket) between the time these thinhgs came to market and were taken off the
market! Very sad, to my mind.

Those instruments could have been the portal to make "programming logic"
(looping and decision structures) part of the math curriculum. But multimedia
glitter took over ...

Another valuable domain of practicing thought, I believe, is three-dimensional
geometry. The subject is a bit of a mind-bender. It is very useful to have
spatial models o help vusualize, say a DNA structure, but as a chemistry teacher
I had to work with manipulatible models of such simple structures as a
tetrahedron. Once a student has taken solid geometry, he needs no visible models
to visualize rotation of simple 3-D objects.

I guess what I am saying here is that educators might spend more time on "types
of thinking" than on "factual content of thinking."


Fleabyte -- -- is
an evolving, experimental web-publication
devoted to public computency, which, like
common literacy, is regarded as essential
to an environmentally healthy, democratic

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