Unless you’re working with polynomials, you could work for years in higher mathematics without doing long division.

If you actually want to lay the groundwork for higher mathematics, you need to introduce the concepts of “structures”, “algorithms” and “proofs”. And this is something that the curricula discussed in the video are attempting to do: The first book’s cluster methods are an excellent lead-in to distribution laws, and the second book’s variety of multiplication techniques are an excellent introduction to algorithms.

Now, you could make a pretty powerful case that most students will never need to learn higher mathematics, and that it’s therefore safe to omit the groundwork needed for undergraduate mathematics. You could also make the argument that teaching advanced techniques is hopeless, because a significant minority of teachers (particularly in the US) can’t function at the level expected from college freshmen.

But average high-school graduates over the last half-century or so have proven unable to apply basic math to their lives, so I don’t think the standard curriculum should be entirely sacrosanct, particularly in the hands of competent teachers.

]]>Today’s US math students are often woefully unprepared. They may be able to use calculators, but they can neither realize nor test the accuracy of their answers. For those adept at “cluster methods” for rapid mental estimation and solving of simple problems, there is some advantage and convenience in light applications. But as the video author has observed, higher mathematics relies on the accuracy of standard algorithms, and there is now a dearth of competent students because of the failings of our newer curriculum.

]]>What annoyed me the most about this “Where’s the math” inititiative is that it plays on the general public’s misconceived preconception that mathematics is about being able to crunch numbers and produce exact results. This is the basic perspective on mathematics that the vast majority of non-mathematically educated adults have — I’d even venture most engineers think this.

It could be that the “new maths” movement has failed to make clear that this isn’t “new” mathematics at all — it’s the maths professional mathematicians have been doing for centuries, and “old maths” is just stuff that’s even older, some of it dating back to Antiquity.

In any case, despite the “how to use calculators” section, I support strongly the kind of program she decries simply because the cluster method is both more applicable in the world we live today — how many adults educated on algorithm-churning can quickly guesstimate the order of magnitude of 15 x 14? –, teaches how to explore & exploit *structures* (cluster methods are really exploiting the distributivity of multiplication) as well as how to find solutions to new problems by gradually approximating it with solutions to problems you know and/or combining solutions to known problems. That’s how most theorem-proving gets done in undergraduate mathematics: you know basic theorems and heuristically combine them to get new, interesting theorems.

What’s the advantage of the traditional algorithm? Well, it’s faster for a civil engineer in the field who needs to compute precise results to small arithmetic problems quickly and has no calculator or slide rule available.

If you feel proud and/or tranquil that your kid is being taught traditional maths at school, try and ask him to quickly produce a lower and an upper bound to the result of 375 x 412.

]]>What the students *really* need for later on in life is a firm grasp of logic. Mathematical proofs are a great way to teach that, but are completely de-emphasized. If you’re not proving theorems, you’re not doing mathematics. Instead, you have courses which would be more aptly named “Calculator Ownership”, or else, “How To Be A Calculator”. Neither one of these is what we really want to teach.

I also have the fear that the current system is churning out students the vast majority of whom have very little respect for mathematics, or even knowledge of what it’s about. This is frightening as the study of mathematics is essentially funded by the government, who are elected by these people. So far, we’ve been able to sneak by, but it doesn’t rest well with me. It would be far better to be inspiring and get across little of “practical value” than to be uninspiring, but produce students well versed in carrying out algorithms which their computers will do millions of times faster and more accurately than they will.

You never hear students in a music or art class asking “Why are we doing this?”.

]]>IMO, clustering is faster for mental calculus while traditional methods are useful to get fast and precise results on paper.

Even now, I sometimes find myself trying to calculate the change the teller is giving me via traditional methods in my head, to resort quickly to clustering and approximate and evaluate fast if I was not cheated (sometimes I even stop the calculus when I conclude I am in an acceptable ballpark).

In conclusion, I’d say that the best way, like in many cases, is to find balance.

]]>The nice thing about the cluster “algorithm” is that it gets you an approximation very quickly, which you can then refine. This is basically how I do “back of the envelope” calculations at work: Chunk the numbers up, and look for a fast strategy to get an answer within 10%. If it matters, I can fill in the last few digits with a bit more work.

Even when reading the news, this kind of fast calculation is invaluable: “If counter-insurgency warfare historically requires a 10:1 force advantage, and if there are roughly 125,000 US soldiers in Iraq, which has a population of 25 million people, we can only afford to radicalize 12,500 Iraqis (times two gives us 25,000, which is 1/1000th of 25 million), which is 1 in every 2000 Iraqis. We’re going to need more troops.” Notice how fast the chunked division can be.

In comparison, most adults learned (and use) the traditional multiplication algorithm. And, unsurprisingly, most of these people are very poor at mental calculation. So I’m not so sure we should treat the traditional math curriculum as sacrosanct–it produces a large number of people who can’t apply even the most basic math to their day-to-day lives.

]]>If you want to see real irrationality over math education, check out some of the letters in Notices in recent years, wherein eminent mathematicians derive the very best methods of math education from first principles, without need for statistics and other such nonsense.

]]>My perspective on this stuff is that the kids with good aptitude for math do well with the emphasis on mathematical reasoning instead of raw drill in arithmetic, except for some gaps like my trouble with long division. But the kids who aren’t gifted in whatever it is that makes math fun for me risk being shortchanged in basic skills that would be valuable for them. And when you think about how much math is now needed for other fields, its a shame to lock those folks out so early.

]]>This situation of trying to open up these fundamental mathematical operations reminds me of a quotation from Alfred Whitehead, which one could probably read to support either side of the situation.

“It is a profoundly erroneous truism repeated by all copybooks, and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of operations that we can perform without thinking about them. Operations of thought are like cavalry charges in a battle–they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”

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