Where have they gone? Calculators

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Modern 4-function calculator

Back in the good old forties and fifties (and no, I wasn't around, he asserted defensively), people could do mathematics in their heads.  My grandmother, for instance, knew her multiplication tables up to 16× for example 16×12 = 192, and so on.  In school, my generation learned up to the 12× tables, but now I believe the standard is to teach up to 10× in around Grade 3, and stop.  Multiplication is viewed by a large proportion of people as the worst thing in mathematics, and among some American citizens, "I can't do math!" actually means "I have trouble multiplying."  (They will deny this, but that is true.)  Some elementary school teachers, however, still teach multiplication tables beyond multiplying by 10, under the mistaken belief that they're teaching higher math.  It's certainly better than just teaching up to only the 10-times table, and getting kids onto calculators right away.  But it certainly isn't higher mathematics.

Scientific Calculator of 1980s vintage

Then, in the eighties, inexpensive hand calculators began to find their way into the classroom.  There was some resistance, because calculator use was considered to get in the way of learning multiplication.  In about half the cases, students who used calculators exclusively for multiplying didn't suffer much conceptual shortfalls.  To explain: in math, some ideas don't really get through into your brain (your cognitive structures, as we say) unless you have some special mathematical experiences.  It was long thought that having a lot of multiplication facts to draw on was important.  But some kids sailed through not having these.  (They still don't know what 9×7 is, but that doesn't get in their way.)  But other kids do seem to keep stumbling around forever, never being able to really connect with the math they're being taught, or until they practice doing mental math.  So we don't know whether it is a matter of confidence, or some cognitive principle.  (I am not an expert.)

Graphing Calculator

By the time the nineties arrived, there were not only those little arithmetical four-function calculators that you could get for free, but there were graphing calculators, that could get you quick approximate calculus results.  They could find the area under a parabola, etc.  They could actually sketch a parabola, or a sine curve.  Just at about this time, smart cell phones were becoming affordable to everyone, and, to add insult to injury, some of the better ones actually had built-in basic calculators!

From the point of view of math teachers, these sophisticated calculators were a hindrance to testing.  If you're trying to find out whether your students could use some basic, important methods to sketch a certain type of graph, letting them use a clever calculator to sketch the graph defeats the purpose.  At this point it is as well to dispel some confusions about this whole thing.  Some parents (and some silly teachers) ask the question: when will students need to sketch so many damn graphs?  The answer is: the curve-sketching we test them with tells us whether they have understood the connection between certain calculations they do, and their geometrical implications.  It is a low-level skill that paves the way for fluent use of calculus and algebra.  A truck driver need not know how to replace a brake pad.  But there are certain benefits to knowing those things.  A common Texas Instruments calculator that had some of these capabilities cost merely around $90 at about this time.

My own solution to this problem was to split the test into two parts, and have one part completely without any sort of electronic aids, and the other part with free use of basic calculators.  Other teachers have gone a different route: they allow calculators on every test, but make the tests computationally challenging.  Yet other teachers use other approaches; they all have their advantages and their disadvantages.

A few years later, calculators were invented that could actually do symbolic mathematics.  Symbolic mathematics is what you do when you're factoring a polynomial, or solving an equation exactly; for instance finding x if you're given that x³ + 35x = 12x².  (The answer is that x could be 5, 7, or 0.)  The techniques for programming calculators to solve problems like this, and far more sophisticated problems, was developed in the eighties and nineties first for large computers, and then for even sophisticated handheld calculators.  Some of these calculators (that cost around $200) are amazing, but few students know how to use them, and I had quite a difficult time trying to figure out how to test students in upper-level courses, when they insisted that they needed to use their nuclear-powered calculators on their tests.

Then, of course, the Iphone and Android phones made an appearance, and Google (who are the driving force behind the Android system) opened up their system to anyone who wished to create application programs to install on the phones.  People stopped calling them application programs, and called them Apps, and anyone with a phone could actually create them.  Many of them are free, but of course they (the apps) try to sell you all sorts of merchandise, which you could easily buy by accident, if you're not careful.  So now, phones were becoming able to perform the sorts of graphing calculations that expensive calculators could do in the nineties.  Furthermore, Casio began to release a truly inexpensive Scientific Calculator that could do most sorts of calculations, including trigonometric and logarithmic calculations, for just about $12.  The expensive Texas Instruments, and Casio programmable calculators could still do some calculations that involved matrix algebra and other fancy math that the inexpensive Casio could not quite handle, but the fact that most of what a Casio Scientific could do, a phone could also do, brings us to the next giant leap.

Several Internet sites, notably Google and Wolfram Mathematica, offered the service of doing any calculation on demand.  You could just go on the site and ask: Solve x^3 + 35x = 12x^2, and it would actually give you the answer!  Let's try this now:

On Google, too, the query Solve x^3 + 35x = 12x^2 takes you to a website or two where the problem is solved in great detail, which is certainly impressive.  (Other suggested websites are places where x^3, and 35x, and so on are present in various paragraphs, which is obviously useless.

In other words, if you have Internet access, it appears that you don't need to have a scientific calculator, most of the time.  I'm not going to chase down some of the chains of implications that that observation could lead to.  But there certainly are some interesting observations we can make.

Doing mathematics is a skill.  People who are good at it have learned procedures, and seen examples where various problems have been solved, so they're able to solve a variety of new mathematical problems by joining the dots.  A calculator does the same thing, and in addition is able to remember various mathematical bits of data: for instance, how to find  sin(17π) exactly.  This is all programmed permanently into the electronic chip that is the heart of the calculator, and it will never forget it.  So this could come in handy for someone who happens to need that information.

But now, even if these capabilities are not programmed into smart phones, you can access the Internet through them, and get the information via a search engine, or a mathematical website.  Communication is taking over some of the tasks for which we used computational technology.  It is a distant cousin of "It isn't what you know, but who you know!"  To be absolutely precise: it doesn't matter if you don't know the information, provided you can find it on the Web.

As you will have noticed, the development of new, sophisticated calculators has ground to a halt.  New fancy phones, however, and being put on sale every five minutes.  Most people use the phones for entertainment (e.g. texting a friend while they're bored riding an elevator--and the boredom tolerance of young people is pretty low).

So, the most fancy calculators available today are not much more capable than calculators of 20 years ago.  (There might be a few, but something tells me that if they're not being aggressively marketed, they probably don't exist, or are out of the reach of typical citizens.)  On the other hand, if you have a conventional computer, e.g. a Windows 7 or more recent type, it is possible to get software such as Wolfram Mathematica, or Maple, for around $700, which is very, very powerful.  I mean that they can answer questions that most of us don't even know how to ask.  For instance, Wolfram Mathematica attempts to answer almost any question, e.g. what is the latitude of Tegucigalpa.

This is, in principle, a distant cousin of Distributed Computing, where a problem is solved by a system of several interlinked computers, each working on an aspect of a computation, and requesting intermediate results from whichever computer has obtained it already.  As long as there are locations on the Internet (the Internet is basically a network of computers; everything you get from it is sitting on some hard drive somewhere) that are willing to share their information, you could ask for it, instead of owning a little piece of hardware that will figure it out as needed.

For the specialist, however, dedicated mathematical tools are still useful.  But there's no real need, presently, for having mobile devices that can compute for you; most of the time, you can do the calculations at your desk.  Field engineers, I suppose, could use a combination of mobile browsers and conventional pocket calculators for their needs, so we can expect that pocket calculators will continue to not be developed very aggressively in the foreseeable future.

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