In the previous post, I played with sieves some more. Sieves are a device for getting something calculated. At least since Eratosthenes, 2200 years ago, this simple and illustrative tool has been part of mathematics. Yet you don’t see sieves mentioned much in the K-12 math curriculum. I’m not suggesting there is any evil design here, nor do I suspect any anti-sieve bias – I think we are simply seeing the side effects of a pendulum swinging away from printed look-up tables of any sort.

If you look today for the kind of look-up tables that in the past were a mainstay of K-12 math textbooks: reference tables in the back of the book, you will see very little. In high school textbooks, you might still find small tables for sin, cosine and tangent, perhaps for logarithms. In earlier grades, you might find a list of prefixes for the metric system, like milli- and Mega- and nano-, with explanations how big or how little each is. You are unlikely to find, at any grade level, a simple multiplication table shown for reference purposes. Instead, the multiplication table is assumed to be committed to memory.

In contrast, textbooks half a century ago would pay great attention to look-up tables. Even if an extensive logarithm table wasn’t part of the reference section of the textbook, it would be published separately and used by students. Yet the textbook would have a section on how to use such tables. Look-up tables weren’t considered a crutch – they were considered the real thing. There would likely also have been tables for square roots, cube roots, reciprocals, and lots of other stuff.

It’s fairly easy to justify why all that stuff is gone or de-emphasized, and that is the availability of calculators and computers that can quickly do the calculations that once were the domain of those look-up tables. Why sift through pages of tables to find the cosine of 79 degrees if you can find out with a few keystrokes on your scientific calculator or spreadsheet? Those vestigial log and trig tables that I did see in the high school textbook appendices seem disconnected from anything students or teachers do.

Other look-up tables, not math related, are also their way out. Every year several phone books, white pages and yellow pages, are dumped on our doorstep and we turn right around and put them in the recycling bin. It’s been over a decade since I’ve looked up a number in a paper phone book. And when is the last time that you looked up a word in a paper dictionary or a paper encyclopedia? Or a quote in a book of quotes?

And yet it is clear that the issue is paper, not look-up. What is Google other than superbly organized look-up? Google Maps, Google Earth, Google News, iTunes music downloads: all look-up. The face of look-up has changed. Instead of a person flipping pages, look-up now happens behind the screen. Who knows or cares what’s happening behind that screen, it is somehow doing the look-up, not you.

For many centuries, there has been a trade-off in mathematics and engineering between calculation and look-up. More look-up meant you needed to do less calculation, and less look-up meant more calculation. The trade-off shifted back and forth dependent on the cost of producing tables, and the cost of computation. Publishing tables only took off when paper became relatively inexpensive. Napier, around 1614, published extensive logarithm tables, which took years of hard work to complete. Such tables saved generations of engineers time and errors in doing calculations. The earliest computing machinery, mechanical and later electronic, was very expensive and included almost no storage capacity whatever – just enough for the two numbers to be added and their sum. When addressable storage became feasible – still extremely expensive – we saw the rise of stored-program digital computers and the use of look-up tables became possible again, but now under the covers of the machine. With a modern calculator such as a TI-84, you’d have a hard time finding out to what extent the calculator is relying on table look-up internally for it to calculate any of its functions. Same with computers – not too many years ago, a three-dimensional role playing game might pre-compute the look of your avatar at a number of different angles, e.g. in 15 degree intervals, and only ever show your character at those angles. Trade-offs between the cost and speed of computing versus the cost and capacity of the various hierarchies of storage continue to shift back and forth with interesting consequences: the iPod only became feasible after massive storage capacity became small, cheap, fast, and yet would keep its content even with power turned off.

There is rich and interesting mathematics involved both in creating look-up tables, and in using them. This mathematics highlights the computational structure of numbers and operations. Much of it can be introduced in grade-appropriate forms throughout K-12. I intend to share my thinking about this in future posts.

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