More Truth, Less Inconvenience
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This post is a continuation of the other posts about the video on YouTube entitled “An Inconvenient Truth” with M.J McDermott (not to be confused with Al Gore’s film) which concerns the dismal state of American basic math education in public schools. You can view it here.
M.J. had two good premises, but her conclusion does not jibe. “Their methods suck.” (True.) “My method is better.” (True.) “Therefore mine is the one everyone should use.” (Nahhhhh.)
Why don’t you experiment a lot and discover what works best for you, and keep refining it? It can be so much more fun and rewarding to do that. Respect your mind, not the opinions and emotional responses that were put there by others in the past. Try this stuff out, then decide.
It’s important to mention that people who think it’s OK not to learn the basic arithmetical operations because “you can do it with a calculator” are just plain damn dumb. That’s like saying, “Hey, this ‘walking’ stuff sucks. It takes effort! Why do we need to learn to walk? That takes years! Let’s just give everyone a wheelchair!’
Not having the basics down cold, and being asked to educe basic mathematical algorithms for yourself, is like not knowing how to read, and having to derive the sense out of every word you read as you go along, without knowing what the letters mean. It would be amazingly cool, and I’m sure some autists have done something like it, but expecting every school child to do that is nothing short of educational child abuse. And then subjecting them to standardized testing? Well, that’s just medieval.
Learning the basic operations can be much easier than the ways generally taught in school. Learning them only takes years if you have ignorant educators, or good educators limited by ignorant administrators and politicians. The “No-Child-Left-Unstressed” act only exacerbates the situation. It’s great for teachers - bad teachers who don’t know how to teach and need someone looking over their shoulders to make sure they are “on track”. If a person can’t be trusted enough to do the job s/he was hired to do, why should s/he be doing it?
On the other hand, that law, born of ignorance and arrogance, limits and straight-jackets good teachers into using only the “standard” methods of doing things. “Standards” are nice and quaint. They are “minimums.” They do not inspire - and no valuable learning comes without inspiration. Inspired teachers are punished in today’s schools. Inspired kids drop out, hate school, and usually are more intelligent than the drones that the “No-Child-Left-Untested” law creates. Thank you, standards!
My point is, this way to multiply is better than the “standard” in probably every way, yet people keep clamoring for the “standard.”
Why is it that people see a change for the worse, and say “We need to change back to what wasn’t worse” instead of raising standards and look forward to a change for the better, which is easily at hand?
I’m afraid the “new” math suffers from lack of willingness to do work, and “old school” math suffers from lack of imagination.
How about using both effort and imagination?
If Newton and Euler stayed “old school” we wouldn’t have calculus.
Back to multiplication.
An algorithm is more like a recipe than a law. It’s only a word; don’t be intimidated by it. An algorithm is basically a set of well-defined instructions.
First of all, it’s silly to think that there is one best algorithm for all multiplication problems. There are lots of algorithms, and lots of problems. You don’t use the same kind of hammer for every kind of nail, do you?
Karl Menninger, in “Calculator’s Cunning” (a brilliant book, by the way) has dozens and dozens of ways to do multiplication. He suggests that people who can reckon really well need to have a lot of arrows in their quivers. That was in 1931, and the situation has changed a bit. Dozens of methods might be a bit much now, but we should all certainly have more than one tool for each operation.
Each time you learn a new method, it helps you understand other methods, as long as you learn a bit about why each method works.
Think of music. There are plenty of ways to play a song, but it’s the same song. Some of the “standards” are pretty hokey. Some of them are absolutely great. But if we insisted that we stick with what we did before, a lot of the music that you love would never have been made. (If you like Steve Miller, that might be a good thing.)
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Comment by Zza, Stanislav
Multiplication is just a convolution product on a vector of integers with a ‘carry’ operation tacked on. I can’t get excited about finding alternative ways to do it (which most parents won’t know). There doesn’t seem to be any compelling ‘efficiency’ reason to do that (remember this is a country that couldn’t convert to metric). Mental math is great for small numbers, but what if you want to multiply two 100-digit numbers together? If you know the usual algorithm, you can put the numbers in vectors, take Fourier Transforms, multiply, inverse transform and carry. If you understand how the usual algorith works it’s all perfectly transparent.
Comment by Brian
Stanislav,
Sounds like an interesting concept, although, as I am not a mathematician, just a recreational math hobbyist, I can’t say that it’s “all perfectly transparent” to me. Nor do I think it is to most other readers of the Chronicles.
Care to elaborate? I love to learn new stuff.
I don’t imagine that most of us want to multiply two 100-digit numbers often, but when we do, it’d be great to have a superior method.
In the meantime, I can’t see how it would effect that fact that if we were taking 28 students on a field-trip, and each had to pay 12 dollars, there’d be a better way to figure out the total owed than with mental math.
Constantly seeking more efficient methods, the above could even be done more easily than with the “mystery algorithm.” Readers can try that here.
You are right, mental math is great for small numbers. The “mystery algorithm”, with a small tweak, and pencil and paper (just to write the answer, not to do any “work”), is pretty easy to do with multiplications up to six digits by six digits long.
Even with numbers larger than that, it remains much easier than the “standard” algorithm.
But my point in the Chronicles isn’t that “my” way is the best. The point is that there generally is a better way to do anything. Apparently the method you suggest is much better for very large numbers, and I’d love to learn it.
I have to admit that I am not familiar with fourier transforms. They probably aren’t within the scope of this site, but I certainly would like to learn about them.
Let’s consider your points of:
I’d hope you will share with us a detailed description of the method you mentioned.
Thanks for writing in!
Brian