### Ugh! Such a proud display of sheer irrationality

13Feb07

What **is** this … person… going on about? She decries teaching some mathematical reasoning (“cluster methods”) in elementary school and wants to emphasize rote algorithm drill-and-kill work instead? She blames 18-year-olds not being able to handle logical reasoning on them not having memorized traditional arithmetic algorithms? She equates “maths” with “rote arithmetic”? It annoys me that she plays to the parents’ fear and ignorance against progress in teaching kids to think.

What method do you really think would help kids to later be able to prove theorems in real analysis, operate group theory or grok category theory, “cluster” reasoning or repetitive algorithm crankery?

Argh!

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I agree with this lady to that extent that if my kids were told to do multiplication and division in this way, that I would be sure to teach them standard way.

I’m a bit torn between the two extremes. There’s certainly no point in turning kids into calculators, because electronic calculators do a much better job of that. But that doesn’t mean that no algorithms should be taught.

Ideally I’d like to see cluster-like reasoning be applied to derive an algorithm, be it ‘the’ standard method or something more along the lines of that other series of books in the video (the title has escaped me). This matches the general approach in math, where a problem is first explored and generalized, and then solved systematically for that whole class of problems.

N.B. it’s interesting how Mrs.McDermott deliberately slows down the division algorithm about 11 minutes into the video, maybe to cover up the similarity to the ‘standard’ method?

I can only imagine trying to take an advanced math course and being reduced to the “cluster” method in order to do simple multiplication and division. You can’t do higher math until you master the basics.

I agree that many kids will have trouble with the “rote algorithms.” I was a particularly gifted child and was able to reason through them and even break them up if I so chose. It allowed me to do math in my head, similarly to the cluster method. Of course, enough of my classmates had trouble with simple multiplication, but I don’t think it would have done them any good to not learn the correct algorithm.

Also, if a child can’t learn the algorithm, do you really think they have a future in group theory? I don’t. Most things in life require simple math skills, and being able to do calculations in the most efficient manner is, in my opinion, the highest priority.

I agree with the lady as well. The cluster method seems more like a search method which is hardly a proper algorithm to teach kids how to do mathematics

How do you prove a theorem? You search for theorems close to it that you know, and connect the pieces!

I thought she presented an excellent case against those books. It didn’t seem liked she decried teaching other methods. She decried teaching them to the exclusion of the standard method. I enjoyed seeing the other methods, incidentally. I liked the division algorithm and multiplication algorithm that looked like the standard algorithm only with a nicer handling of place values. Seeing that substituted for the standard method didn’t seem that bad, but those books looked awful.

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.”

I grew up during the American “New Math” wave of the 1960s. My elementary school switched text book series every year from 1st through 6th grade. I knew simple set theory (Venn diagrams, for example) very early, but didn’t really understand long division until my 6th grade teacher finally got it through my head. As it turned out, I loved and excelled at math after that and majored in it in college. This was before Computer Science was common outside of the major research schools, or I probably would have chosen CS.

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.

As a person who teaches math at the university level, I find the methods troubling because the arithmetic doesn’t progress to the automatic level. It’s hard to do algebra if you have to think hard about integer multiplication. It’s hard to do calculus if you are still thinking hard about algebra. And so on. Calculators seem to make that worse because using the calculator becomes the automatic response. You’ll see people reach for a calculator to figure out things like 30*7, which at the very least is extremely inefficient.

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.

The advantage of the traditional algorithms is that they are efficient and well-known. The disadvantage is that they provide very little insight, and–beyond a few digits–they’re too slow and complicated to use mentally. At that point, you’re driven to either paper or a calculator.

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.

Almost never the truth lies with only one side. I personally had issues with math division and multiplication because I often became emotional thinking and blocking myself into a fear like “my colleagues will laugh at me because I don’t know this” or “I wish the teacher went away from my desk so he doesn’t see that I’ll make a mistake”. Agreed, that is a psychological issue, but almost always I used clustering in my head to get over the blocking point and gain confidence.

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.

My perspective on this as a mathematician is that numbers are overemphasized in elementary and high-school mathematics courses. What really matters about numbers are properties like distributivity, and the standard algorithm is a good way to show that, but shouldn’t be drilled. Numbers are indeed very important kinds of structures, but they’re not the only interesting elementary structures in mathematics by far. There’s a great deal of other mathematics which would be equally useful later on in life regardless of field. Very elementary graph or group theory, for instance. There are also plenty of simple mathematical elements of computer science which one could teach – basic things about parity or sorting networks which have a rather visual nature to them and are easy platforms for involving oneself in some logic.

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?”.

Agreed.

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.

“Cluster methods” are unmatched for speed and “mental math”, and I would suggest to the video author that there is some value to their teaching–there is nothing more embarrassing than being off by an order of magnitude in a critical calculation and lacking the mental ability to see it quickly and easily. However, without the “standard algorithm” as a basis for solving complex problems with exact precision, all the speed and “guesstimating” in the world can’t serve as an alternative in important situations, and I’d say the author has that exactly right.

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.

CraigH said, “But as the video author has observed, higher mathematics relies on the accuracy of standard algorithms…”

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.