Long division? Ugh!

Claire Kenyon asks the following question on Lance Fortnow's blog:

Yesterday my sixth-grade son, doing his math homework, asked Is there a faster way to divide a number by another other than with long division? I had no good answer to that. Any suggestions for an alternative to tedious long divisions?

Here's a good alternative: not doing them.

A few years ago, I taught my then-six-year-old to do division, essentially by binary search. If you want to distribute 4832 cookies among 23 children, look at the sequence 23, 46, 92, 184, 368, 736, 1472, 2944, 5888, to figure out that each one will get at least 128, and you'll be left with 4832-2944 = 1888 cookies; of course, you can now give each one 64 more, using up 1472 and leaving you with 416, etc. (Of course, you will do this with powers of 10 if you are less of a sadist than I am). A year later, he probably forgot all about the mechanics of doing division, but still carried with him the ideas of division as distribution, perhaps the underlying "recursion", etc. Which is just as well.

Tony Ralston (an ex-president of the ACM!) has a rather interesting take on the theme of paper-and-pencil arithmetic, where he advocates mental arithmetic and basic number sense.

Two excerpts (PPA = paper-and-pencil arithmetic):

So is a facility with PPA necessary - or even desirable - for later study of mathematics? If a student arrives in secondary school, say in a first algebra course, unable to do PPA, is that student ipso facto disadvantaged compared to students with PPA skills? I take it that it is not PPA skill itself which critics of calculator use consider important. After all, there is little secondary school or other mathematics which requires much computation per se. Rather it must be the ancillary benefits of developing PPA skill which are considered important. These are usually subsumed under the rubric of numeracy or number sense and include, in addition to the obvious ones of knowing the addition and multiplication tables, such things as knowing which arithmetic operation to use when, having a good sense of number size and knowing strategies to check the answers to arithmetic operations. These are all important. Is there anything about them which a calculator-based curriculum could not instill? I think the answer is, no...

and on long division:

I should say a word about division. Although it is over 15 years now since the Cockcroft Report [1982] recommended that long division no longer be taught in British schools, this recommendation been, at most, spottily implemented. The California Board of Education standards [California, 1998, p43] would require students to master long division. This is mind-boggling. The only excuse can be that those who promulgated the California standards believe that long division is good for the soul. Not only does being able to do long division have no practical value whatever but, in addition, the time required to teach this algorithm to students is far, far in excess of any benefit which might accrue from learning it. Of course, students must learn what division is, when to apply it, what remainders are and how to do simple division problems mentally. But teaching long division is pertinent to none of these aims; it is as nonsensical as teaching the square root algorithm which was staple fare until recent times. I cannot help but believe that those who favor teaching long division in elementary school (and these include some research mathematicians [Klein, 1998]) are in the grip of some fantasy about what is important and useful in school mathematics⁶

Three comments on "selling science"

Lance Fortnow's blog entry Selling Theory raises several interesting issues, and I have three things to say about these.

Regarding the comment:

But [NYT's] Tuesday section Science Times has moved over the last couple of years from a general covering of science to a focus on medicine, environment and astronomy. Not just computer science but physics and chemistry get far less coverage than they once did.

One very important reason for this, in my opinion -- and I've often expressed this opinion in the past -- is that the latter fields (medicine, environment, astronomy) all have big goals that can be concretely stated so that an eighth grader can relate to them. "Cure cancer" or "300mpg cars", or "Flight to Mars" are all crisply stated, easy for a young mind to latch on to, and are more or less legitimate ambitions of the fields. Theoretical computer science, even Computer Science, lacks such ambitions. "Computers you could talk to" is about the best I can come up with that the median-quality high-schooler could understand. Perhaps something about doing functional genomics might appeal to smart undergraduates, but that's probably perceived more as a career in biology than in CS. Notice that physics, chemistry, etc. all have the same problem --- you could appeal to an eighth grader's imagination by talking about the expanding universe (or multiverse), but it's not tangible enough.

Lance writes

Computer science is a victim of its own success, by making computers powerful, easy to use and well-connected, we have turned computers into a commodity like cars with the mystique of computation and complexity lost on the users.

I don't think this is necessarily bad --- there are more users of automobiles than there are engineers working on building better ones, and even fewer scientists (or even "theoretical engineers") who think about the underlying laws of physics.

Finally, getting back to "selling theoretical computer science", I think we have a real problem. It's not so much that The New York Times doesn't write about what happens in TCS and why it is important. How many computer scientists that are not theorists appreciate the importance of theory? We need to do a much better job of convincing our colleagues in our departments, our graduate and undergraduate students that theoretical CS is not an arcane area of investigation, but a lively, vibrant area that, on the one hand, has deep philosophical and mathematical ramifications, and on the other hand, has real connections to the world around us. I think we simply cannot afford to have undergrad CS majors walk away with the feeling that this an anachronism in the day and age of the Internet (which they might if all they study are context-free languages and all-pairs-shortest-paths in the abstract form). The rest of CS has, in my opinion, done a better job in keeping up with their own growth. We don't seem to be offering any hint to the smart undergraduate that TCS is a constantly evolving discipline (just in the last ten years, we have had game theory, quantum computation, massive data set computations, etc. blossom from marginal curiosities into full-fledged multiple STOC/FOCS sessions). Jon Kleinberg's Information Networks, Michael Mitzenmacher's Algorithms at the end of the wire, the CMU course Algorithms in the real world, are all excellent examples of what we can do to improve the perception of TCS within computer science departments.