The NumWorks graphing calculator [LWN.net]

The NumWorks graphing calculator

Posted Sep 29, 2017 10:29 UTC (Fri) by gdt (subscriber, #6284)
In reply to: The NumWorks graphing calculator by anselm
Parent article: The NumWorks graphing calculator

The problem being that modern calculators are so good that allowing them in secondary school exams would turn them into simple copying from the screen to the exam paper. To see an example, enter a polynominal into Wolfram Alpha, it will graph it, show the roots, and even show line-by-line working for obtaining the roots using all the different methods. So the examination bodies have choosen to restrict calculators to have little capability beyond what is possible with pen, paper and tables of functions.

There is a deeper question here, which is why in the presence of such computing power we persist in teaching mathematics as if that tool is not available. A way which is now very different to the way professional mathematicians, statisticians and engineers approach the same tasks. I was watching my daughter be taught graph theory and surprised that it didn't end with an introduction to a library like Python's NetworkX, which would have considerably helped them with their task of computing a 50 vertice graph. Similarly that the statistics topic didn't introduce R. There are obvious answers of course, and they are so difficult to overcome that it's clear how we got to this point. It's less clear how to get out beyond this point and towards a better mathematics experience for students.

to post comments

The NumWorks graphing calculator

Posted Sep 29, 2017 12:18 UTC (Fri) by aleXXX (subscriber, #2742) [Link] (4 responses)

Learning and understanding how to compute the roots of an polynomial or how to solve a (small) linear equation system manually yourself is a good thing.
Without doing it manually on paper you don't understand how it works, and it trains your brain.

I know this may sound old...
I still think that one of the best lectures I had (20 years ago) was one where the professor explained the stuff by drawing it on a blackboard. IMO the best way to make somebody understand a problem is a person writing/drawing and explaining and pointing at the same time. Works without any computer.

The NumWorks graphing calculator

Posted Sep 29, 2017 15:07 UTC (Fri) by gutschke (subscriber, #27910) [Link] (3 responses)

That's why it often makes sense to administer those type of tests without allowing any calculators.

There are a good number of numerical problems that can be computed without the help of calculators.

And once the student has mastered the mechanics of math, you can let them take future tests "open book" and with a calculator.

Of course, those tests will be much more difficult

The NumWorks graphing calculator

Posted Oct 12, 2017 15:08 UTC (Thu) by Wol (subscriber, #4433) [Link] (2 responses)

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

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.

The NumWorks graphing calculator

Posted Sep 29, 2017 13:13 UTC (Fri) by mjthayer (guest, #39183) [Link] (1 responses)

I have been wondering about some sort of Raspberry Pi-based calculator - a Pi Zero with a suitable case and touch screen would be great for the purpose. If the SD card were easily accessible, examiners could provide their own ready prepared ones and would know exactly what students had. A little effort for the examiners of course, but why not.

The NumWorks graphing calculator

Posted Sep 29, 2017 14:15 UTC (Fri) by cpitrat (subscriber, #116459) [Link]

A constraint that calculators have is that you expect its 4 AA batteries to last longer than the calculator itself. Hard to reach with a raspberry-pi ...

The NumWorks graphing calculator

Posted Sep 29, 2017 14:30 UTC (Fri) by tialaramex (subscriber, #21167) [Link]

Aha! I think I can answer this one

OK, so, ultimately the goal is for the student to learn stuff. We have exercises, tests, coursework, all that stuff _only_ in service of this goal. At the top end, everybody involved is aware that's what is going on, and if you propose to cheat the tests rather than learn the material they will stare at you like you just said you've got an idea how to defeat the safety limits and cut your own arms off with the wood saw. We do not much care that you are able to actually add 5 + 7 = 12‡, our goal in asking that is only to verify that you properly understood what's going on there so that when we later assume that knowledge you're not left staring blankly into space.

‡ Although it will help in some jobs to not need a calculator or other machine to do this sort of simple arithmetic that's not why we're teaching it to children in 2017.

But at high school level this idea about cheating being _futile_ is not going to be something we can expect most of the students to understand. So rather than replace 100 hours of her classes with material about "This is why you probably ought to pay attention and not cheat in tests" we just go out of our way to not let your daughter cheat until much later, by which time hopefully she'll have figured this out for herself.

The NumWorks graphing calculator

Posted Sep 29, 2017 22:28 UTC (Fri) by anselm (subscriber, #2796) [Link]

The problem being that modern calculators are so good that allowing them in secondary school exams would turn them into simple copying from the screen to the exam paper.

I probably should have mentioned that that was in the 1980s. Calculators (programmable or not) used to be a lot more primitive then.