Re: banning calculators

[John Clement <clement@HAL-PC.ORG>]

I think that the routine of drilling long arithmetic is part of the back to
basics movement which advocates automaticity in calculations.  Unfortunately
the part of the brain that does arithmetic is not the same part that does
algebra, so this sort of nonsense is not really a lead in to algebra.  In my
personal experience, math was hateful until I got to algebra.  Indeed I do
not remember doing a lot of practice with 5 digit divisors, but I had no
trouble with algebra.  Perhaps I blocked out that sort of painful memory.
My problem is very simple.  I remember many math facts by visualizing them,
and I have extreme difficulty remembering them by rote.  This problem
extends also to names and dates.        However this did not prevent me from
getting advanced degrees.  Because I am forced to rely on visualization, my
ability to estimate is very good.

I suspect that the ability to do rote calculations probably immediately
decreases as soon as a calculator is allowed, so that the overdrilled
abilities are probably useless.  The process of estimating and doing simple
in the head calculations probably does not fade, as it relies on a process
rather than just drilled facts.

I would go a big step further than just learning to estimate and do simple
mental calculations.  Students need to learn that answers should be
consistent with reality.  The only way to do that is to have them measure
things, do calculations, and the check the results by more measurements.
This provides them with a physical check that has a much larger impact.
Essentially this is an adaptation of McDermott's explore, confront, resolve
method of teaching concepts.  While the method actually goes farther back,
she is justly famous for exploiting it.  Essentially students should be
learning other subjects using the learning cycle which has been well
exploited by Anton Lawson, and a number of physics reformers.

Another point is that subjects like trig could benefit greatly by
integrating pictures, graphs, descriptions, and equations in the same
problem.  Again this idea which goes back a ways in the education literature
has been exploited by Hestenes and other in their curricula.  Trig problems
should be accompanied by both a reasonably accurate drawing of the angles,
and by measurement of the sides of the triangles as well as by algebraic
solutions.  My daughter has done about 1 graph for every 100 equation
problems and zero drawings were required.  I would teach trig initially by
having them do the drawings, make measurements, and then figure out things
using ratios.  Then after the idea that the ratio of the sides is always the
same for the same angle, I would introduce names for the ratios.  Again math
does it backwards.  Lawson cites evidence that shows that concrete thinkers
learn much better if definitions come after concept introduction.  Formal
thinkers do not have as much difficulty with the wrong order of
presentation.

Geometry has the same problem as trig.  Students come out of geometry unable
to look at a drawing and figure out the necessary relationships.  They can
often cite geometry rules, but they have no ability to apply them.  A good
example is that a significant number of students will not be able to
calculate how far a father has to walk when pushing a playground
merry-go-round.  They have the diameter of 2 meters, but then they calculate
the walk all the way around as 8 meters or use pi r^2.  I make them measure
the drawing in the book and estimate it first.  They often come up with a
little more than 3 times the diameter, but then are unable to say exactly
what the ratio should be.  Sometimes they will hit on pi, but usually I have
to ask if there is a number from math that might be helpful!!!

Drawing to scale and measuring should also have the side benefit of helping
students understand ratio and proportion.  Since 75% of my seniors come into
my regular HS physics class unable to understand that a pouring question
involves a ratio, this problem is severe.

John M. Clement
Houston, TX

> I'm not sure that if one KNOWS how to do long division that
> continually practising 8 digits divided by 5 digits makes much
> sense.  I have a particular complaint regarding what I have
> observed in the curriculum my grandson uses (8th grade -
> preAlgebra).  They continually practice long division problems
> over and over using divsion problems that are apparently
> carefully selected to be difficult to do.  I can take square
> roots and do long division by hand, but I wouldn't do it unless
> forced to do it.  The suggestion about estimating seems like a
> good idea.  The problem I see with my grandson is that numbers
> don't seem to really mean anything to him.  He can manipulate and
> get the right answers but
> has no FEEL for what numbers mean.  I still see this in some of
> my calc physics students who can manipulate their TI 83 to do
> everything but sing, but crank out answers that should be
> obviously ridiculous without blinking!
> James Mackey
>
> Justin Parke wrote:
>
> > This reminds me of a conversation I recently had with a retired
> engineer who went to talk to the head of elementary mathematics
> for the county in which he lives.  He wanted to know if long
> division is still taught in the elementary schools and how long
> was typically spent on it.  (i.e. do they do division of a 5
> digit number by a 4 digit number or only 3 by 2, for example.)
> His thought was to drastically decrease the amount of time spent
> practicing an algorithm and use that time to practice estimating
> what the result of the division *should* be and then confirming
> that with the calculator.  This is similar to the idea that
> students can do integrals while having little idea of what they mean.
> > What are your thoughts on cutting instruction on long division
> in favor of estimation/prediction?
> > Justin