The NumWorks graphing calculator [LWN.net]

The NumWorks graphing calculator

Posted Oct 12, 2017 15:08 UTC (Thu) by Wol (subscriber, #4433)
In reply to: The NumWorks graphing calculator by gutschke
Parent article: The NumWorks graphing calculator

One only has to ask why do modern students make blunders with orders of magnitude. Someone who's learnt maths the hard way (ie long multiplication etc), on being presented with a sum like "3 times thirty", will *know* the answer is approximately 100.

Somebody who's been taught to just enter it into a calculator will quite happily accept an answer that is approximately 10, or approximately 1000.

I regularly do a "sloppy calculation" in my head when using a calculator, precisely to pick up "finger trouble" errors, and it regularly saves my bacon ... :-)

Cheers,
Wol

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The NumWorks graphing calculator

Posted Oct 12, 2017 16:34 UTC (Thu) by gutschke (subscriber, #27910) [Link]

I think I was in sixth grade, when I memorized log10(2) and log10(3). Turns out, that means with a second or two to think about it, I know the logarithms of all the numbers from 1 through 10. Although, admittedly, log10(5) and log10(7) are trickier and I'd have to interpolate.

There are an amazing number of problems that can be approximated, if you know these logarithms. You can usually get the order of magnitude and sometimes one or two significant digits. That's absolutely good enough to eyeball the results for plausibility.

Also, like most of you, I know the powers of two up to at least 16 -- and probably higher, if I give it a little thought. Turns out, a lot of problems that look difficult in base 10 look really easy in base 2.

Math is all about reducing a complicated problem to a different problem that is easy. And if you memorize a small number of constants from different problem domains, then your toolbox grows exponentially more powerful. Logarithms are particularly good for this, but some of the trigonometric functions are pretty useful, too.

I distinctly remember moments in university (back, when I still needed to do lots of math), when I would arrive at an approximate answer within seconds, whereas my peers took about one minute to get the exact answer from their calculators. Turns out, my answers were always within less than 10% of theirs, and often better -- which frequently is good enough.

The NumWorks graphing calculator

Posted Oct 14, 2017 7:22 UTC (Sat) by micka (subscriber, #38720) [Link]

I can testify that children taught operations by hand will absolutely have as much problems with decimal orders of magnitude as those taught using calculators. Order of magnitude (well... Sense based validation of results) is a different and mostly independent, unrelated, teaching. They can be quite conflated.
Take for example decimal comma positioning in manual multiplication. It will be placed by a calculus on operand comma position, not on estimated expected order of magnitude of the result. And learners will stick with it.