Re: [Phys-L] just for fun

[John Denker <jsd@av8n.com>]

On 12/19/2013 09:07 AM, Bill Nettles wrote:

> I think calculators are being used to enable "math" teachers who
> don't want the hard work of forcing students to work hard.

There are lots of reasons to de-emphasize calculators, but IMHO 
"forcing students to work hard" is very far down on the list.

If the idea is to have people work hard, they don't need to go
to school at all.  They can get become migrant farm workers, or
catalog-fulfillment warehouse workers.  That allows them to work
super-hard day all and just barely survive with the help of food
stamps ... until somebody cuts off the food stamps.

One reason for going to school is to get a skilled job.  When
comparing skilled workers one to another, the more skilled ones
do not work any harder;  they just *accomplish more* per unit 
of work.  A skilled carpenter uses sharp tools.  Dull tools
result in more work and worse results.

Please do not allow students to blur the distinction between
amount of work and actual accomplishment.

My job is to help people to learn the easy way of getting good 
results ... to learn to use sharp tools.

> allow calculators when the football and basketball teams quit running
> wind sprints

That analogy is interesting, but tricky.  The ball players
can readily see that sprinting on practice days correlates
with not sucking on game days.

The connection between doing without calculators and achieving
something worthwhile is not quite so self-evident.  It requires
careful explaining ... and IMHO saying that it makes things 
harder is diametrically the wrong explanation.

Within limits, I want to minimize the amount of telling students
what they can't do, and maximize the amount of telling them
what they *can* do.  As applied to today's example, telling
students they *can* calculate stuff without depending on a
calculator is a positive thing.  It is empowering.

As others have mentioned:  A student should not need a calculator 
to divide 100 by 400.  Doing it in your head is quicker and easier 
... and (!) has a better chance of providing some insight into the
nature of the task.

As another example in the same direction, but somewhat further
down the road:  I see no sense in using a calculator to find 
the square root of 50.  There are three ways of doing it in 
your head at a level appropriate to the algebra-based intro
physics course.  Each one is accurate to better than 0.02 percent
... and teaches something interesting about the task in general.
And that's not to mention innumerable fancier schemes.....


Calculators are an unhappy medium.  I own several of the things, 
but I can't remember where they are, because it's been years 
since I used any of them.  I could install a scientific calculator 
app on my phone, but I've never been tempted to do so.

a) I can do simple problems by hand in less time than it would
 take to pick up a calculator, even if it were sitting right on
 top of the desk.

b) If the task is one level more complicated than that, I use a 
 spreadsheet.  The overwhelming deciding factor here is that with 
 a hand calculator, it is next to impossible to check the work
 or document the work.  In contrast, I can read the spreadsheet 
 formulas to check them for correctness.  I can put text into
 adjacent cells to explain and document what's going on.  I can 
 get colleagues to review the thing for correctness.

 Keep in mind that "check the work" is the foundation-stone of
 critical thinking.

c) If the task is even more complicated than that, I will switch
 from a functional programming language (spreadsheet) to an
 imperative programming language (probably C++).

==========

> I favor ditching calculators until absolutely needed for precise 
> numerical work.

Let's keep in mind that complete reliance on a calculator is 
*not* necessarily more accurate than doing key steps by hand.
As a classic example, suppose we wanted to know the square root
of 50, accurate to 25 decimal places.  Most hand calculators 
can't do that.  Ditto for most spreadsheet apps.  In contrast,
the calculation can be done with a modest amount of effort using 
pencil and paper ... especially if a calculator or spreadsheet 
is used to check some of the intermediate steps.

The example of using a 15-digit calculator to support a 25-digit
hand calculation makes it clear that the calculator is not the
enemy.  The objective is not to minimize the calculator; the
objective is to maximize the understanding ... the understanding
of the fundamental axioms and algorithms.

Another way to get high accuracy is to use a modern imperative 
programming language.  Virtually all of them have bignum packages 
nowadays.
   http://en.wikipedia.org/wiki/Arbitrary-precision_arithmetic

So, once again the basic calculator is either overkill or not
enough kill.

A partial counterargument arises in the narrow range of cases
where there is a calculator (but not a spreadsheet) built into 
your phone.  Generally speaking, a calculator you have with
you beats the spreadsheet app you don't have with you.  Still,
in the real world, the chance of being asked to do a serious
task with a calculator but without a spreadsheet is just about
nil.

================

To summarize the main point again: It makes sense to non-calculator
algorithms, but please don't do it to make things harder.  Do it 
to make things easier -- easier in the long run -- by encouraging 
students to develop and use appropriate, sensible computing skills 
and thinking skills.

As an additional point: figuring stuff out without a calculator 
can be fun, if you make a game of it.  Understanding stuff is 
more fun than mindless number-crunching.  Like most things, 
it's more fun after you build up some skill.

====

A very incomplete but still useful collection of suggestions
to help with doing math by hand can be found at
  http://www.av8n.com/physics/math-hints.htm