Innoculating against innumeracy

For years parents have been told how important it is to read to their young children. Today I read that it may be just as important to do household math with young children. An article in the Wall Street Journal reports that

Math skill at kindergarten entry is an even stronger predictor of later school achievement than reading skills or the ability to pay attention, according to a 2007 study in the journal Developmental Psychology.

My first thought was surprise. How could math be even more important than reading? My next thought was that now conscientious parents will feel pressured to improve their children’s math skills prior to age 5.

When interests and abilities diverge

When I was growing up, I thought that sitting at a desk working with numbers was about the most boring job I could think of. I had very little idea what an actuary like my father actually did (I still have only hazy notions of how his workday was spent), but I knew it involved lots of numbers.

It’s not that I was bad at math. On the contrary, it came easily to me (except for one unit in third grade when we had to learn base 8), and I found it very boring. As a senior in high school I did my calculus homework to relax from more challenging subjects like literary analysis and chemistry. I enjoyed competing in Math League, but I had no interest in studying advanced math topics on my own in order to do better at the meets.

What I liked was writing. I had always been good at it, at least according to my teachers. (My mother also thought I wrote well, but I discounted her opinion as lacking objectivity.) I had always loved to read, and I longed to be able to write stories that other people would enjoy reading.

Blogging a thousand

I haven’t been keeping count, but according to WordPress, this is my one thousandth blog post. (It doesn’t seem like I’ve written that many, but I know I can go look at the whole list of them if I want to be convinced.) There’s nothing all that special about it being the thousandth post, but as long as it is, I thought I might as well write about the number 1000.

Have you wondered why we use both M and K to represent 1000?

M comes from the Latin word mille, meaning one thousand. We use it in words such as millennium and millipede.
K comes from the Greek word khilioi, which also means one thousand. We use it in words such as kilogram and kilometer.

Speaking of millipedes, does a millipede really have a thousand legs?

I’ve seen a number of different estimates, but all of them agree that no species of millipede has 1000 legs. One rare species does have up to 750 legs, but most have far fewer.

Do you know…

Whose face appeared on the thousand dollar bill? Why was it removed from circulation?

Who said “A picture is worth a thousand words“?

I originally planned to write a post with 1000 words, but I had trouble finding enough interesting trivia about 1000. So instead this is 1000 characters (not counting spaces).

It adds up to one smart dog

This article about a mathematically gifted Labrador retriever is a perfect example of the kind of undiscovered animal abilities Temple Grandin writes about in Animals in Translation (see my post from a few days ago). Grandin would not likely credit Beau with being able to calculate square roots or do algebra, but she would agree that he has an amazing ability.

Grandin told about a horse, Clever Hans, who could tap a hoof the right number of times to answer math questions. His owner was convinced his horse could count, since he wasn’t signalling Hans when he had reached the right number. A psychologist finally was able to show that the owner (or anyone else asking Hans questions) really was signalling Hans, they just were doing it without being aware of doing so. Whatever it was they were doing, another person could not detect, but Hans could. When the questioner was put out of Hans’ sight, or the questioner didn’t know the answer himself, Hans could not give the correct answer.

Beau’s owner and others who are convinced the dog is a math genius evidently haven’t read Grandin’s book. The article offers, as evidence that Beau is really doing math and not just watching for signals, that he can answer questions even when his owner is out of sight. But there is no indication that they have tested Beau with questions when the questioner is out of sight, or the questioner does not know the answer. It does say that Madsen (his owner) will ask him a question so complicated that you (the observer) are still trying to figure it out when Beau gives the answer. But Madsen presumably does have the answer figured out, and Beau is watching him very intently.

Grandin’s conclusion is that animals such as Hans and Beau are very intelligent – but it’s not the kind of intelligence that does math calculations. As she points out, no one knows how to train an animal to do what Hans did, and Beau is doing (though Madsen believes he taught math to Beau). These animals taught themselves to observe something so hard to detect that people have no idea how they do it. Other animals have taught themselves to predict when their owners are going to have a stroke, and no one knows how they do that, either. (They were trained to respond to seizures, but on their own they went beyond that to react before the seizure starts, something that humans do not know how to do.)

Abundantly weird

This is the post I was going to write this morning, when I got sidetracked looking at Doodle 4 Google. I finally did complete the search I meant to start, finding out about “abundant” numbers.

Occasionally when I help Al with his math homework (usually for his ELP class, though occasionally for his regular math class also), he refers to a term I’m not familiar with. Most of the time I know the concept, it’s just the terminology that has changed over the decades since I was in school.

This week, though, there was a concept that was new to me. Abundant numbers? Well, of course numbers are abundant – they’re infinite. But the paper explained what the term means: an abundant number is one that is less than the sum of its “proper divisors” (factors of the number other than the number itself).

The homework assignment didn’t mention “deficient numbers” but they are the opposite – where the sum of those proper divisors is less than the number itself. And then there are “perfect numbers” where the sum of the proper divisors is equal to the number itself.

The concept may have been new to me (or maybe it was mentioned in a long-forgotten math lesson but not required to learn for a test), but it’s been around since the time of the ancient Greeks. The terms abundant, deficient, and perfect in reference to numbers were coined by Nicomachus.

What I really wanted to know, though, was what significance there was to identifying numbers as abundant (or deficient or perfect). I went through several pages of hits on Google before I found my answer in this quote by Martin Gardner:

One would be hard put to find a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery–and more totally useless–than the perfect numbers.

On the other hand, on the same page (which contains a wonderful collection of quotes about education, books, mathematics, and more), I found this statement by Nicolai Lobachevsky:

There is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world.

Who knows? Maybe by the time Al someday has kids and they come to him for help with their math homework, someone will have found a practical use for the concept of abundant numbers.

But in the meantime, I found another concept: weird numbers. These are abundant numbers that are not also semiperfect, meaning that they can not be expressed as the sum of some of their proper divisors. They probably aren’t very useful either. But I like the name.

Reading science for the fun of it

I like reading about science. For months now I’ve been checking in at futurity.org every few days to see if they have any new and interesting articles. Some days there are new ones, but few really catch my interest. The biggest thing I’ve learned from reading articles there is how narrow the scope is of much scientific research.

In order to show that A causes B, you have to limit the effects of C, D, E, F, and G, or at least control for them in analyzing your data. That means that you are often studying a very small part of a very big picture. Put together all the scientific research being done around the world, and it starts adding up, which is why we see such incredible advances in certain fields. But the results of any individual research project can seem pretty underwhelming.

Today I came across Science 2.0, which covers a wide variety of scientific fields, and has contributors who are good at writing, not just at science. They may not have news quite as up-to-date as futurity.org, but the articles are a whole lot more interesting. (Obviously that’s just my opinion, but then this is my blog – who else’s opinion would you expect it to be?)

I happened to encounter Science 2.0 by way of The Daytime Astronomer, written by Alex “Sandy” Antunes. The particular article which I stumbled on (thanks to Thirty Three Things, a regular feature at the First Thoughts blog) was Which Science Kills More People? OK, so it’s not exactly a serious study of mortality rates, but I was glad to see that, despite the title, it was not an anti-science screed blaming chemicals for everything that’s wrong with modern life. (You do realize, don’t you, that you can die from an excess of dihydrogen monoxide?)

Most of the articles I read, in my brief excursion at Science 2.0 this evening, are more serious in nature, but they are also well-written and therefore enjoyable to read. I look forward to reading more of them, either when I have time to spare, or when I need a good topic for my own blog post.

Language without abstraction

I’ve been working on ideas for a speech I’m giving Saturday (for a Toastmasters contest), and one idea (which I’m thinking now won’t really work but I haven’t figured out what to do instead) had to do with the importance of words vs numbers. (Think of King Azaz and the Mathemagician in The Phantom Tollbooth.) I remembered having read about a language that has few if any numbers, where people manage with just words like “few” and “more.”

Looking for more information, I came across this fascinating article about the Pirahã, a tribe in northwestern Brazil. Don Everett, a linguist who first went there as a missionary with the Summer Institute of Linguistics, probably knows as much of their language as anyone else outside the tribe. Their language not only lacks numbers, Everett says, but any kind of abstractions. They have no interest in the distant past or future, or in anything that they cannot experience directly.

Imagine the challenges that presents to someone trying to share the gospel of Jesus Christ. I have read a number of accounts of the difficulties Bible translators have with languages that don’t have words that are key to understanding Bible stories and concepts. But I never heard before of a language without any abstract words.

Some linguists think that Everett is wrong. His claims fly in the face of much of what is widely accepted in academic circles regarding language and linguistics, particularly the theories advanced by Noam Chomsky. Everett himself was once an enthusiastic disciple of Chomsky, until he realized that the Pirahã language simply didn’t fit the theories.

I once thought I would spend my life doing what Don Everett and his wife Keren headed to the Amazon to do. Keren still works at learning Pirahã, with the goal of translating Scripture into their language. Don now considers himself an atheist, and his interest in Pirahã is for the language itself. The couple separated years ago, after Don concluded that he found no more spiritual meaning in the Bible than the Pirahã do. (Don did succeed in translating some passages, but the stories elicited no interest among the people of this tribe.)

I have long thought that one strong bit of evidence for a spiritual dimension to life is that people of all times and cultures have spiritual experiences and beliefs (not all people, but some people in all cultures). That the Pirahã do not (if Don Everett’s understanding is correct) does not weaken my belief in spiritual realities. But it is strange.

News for number lovers

When I read Norman Juster’s The Phantom Tollbooth, I always find myself curiously annoyed that the princesses Rhyme and Reason (who need to be rescued from the Castle in the Air to bring sanity back to a land in chaos) determine that words and numbers are equally important. (I am relieved to learn from an interview with Juster about his famous book that he wasn’t trying to make any grand point regarding that conflict – like most of the book, it was just about having fun with language and ideas.)

I’ve always loved words (and obviously Juster has also). Numbers can occasionally be interesting, but mostly they’re just useful tools. Without numbers, I couldn’t be writing this blog post because there wouldn’t be any computers. Without the technology that numbers enable, we’d probably all be cave dwellers. But I could at least share stories around the fire. Without words, there would be no human society (at least not as I think of society), and no use for technology even if it existed.

Recently I took a quiz that is supposed to be able to give some indication whether a person has an autism spectrum disorder. While I don’t have autism, my score is closer to that of many autistic people than to the average non-autistic person. I’m not good at chitchat or social situations in general, I prefer a library to a party, and I notice patterns in things a lot (I’m not sure if I can quite say “all the time” but it seemed close enough).

But I cannot say I am exactly fascinated by numbers, as people with autism spectrum disorders often are. (My younger son, who is considered autistic, isn’t much of a numbers person either. I help him almost every week with his homework for his Extended Learning Program math class, and while the math is easy enough for me, I have trouble figuring out how to help him arrive at the answers himself.)

All that said, after reading an article about an important mathematical discovery, I decided I needed to post this if only because I figured it would interest my sister Margaret (in case she hasn’t already read this). For myself, I find myself reacting about as I would to the discovery of some new species of beetle – well, that’s nice, and I’m sure it’s important to some people, but I just can’t get too excited about it.

Frankly, I’d never even heard the term “partition numbers” before – or if I had, it had completely escaped my notice. Even after reading the article, I am uncertain what makes them so important. And I can’t muster the interest to figure out for myself why they are, though I’m more than willing to believe that this discovery about them is as breathtaking as the article says it is.

I did enjoy the article though – after all, it’s full of words.

Whether and how to change the flag

Even since I first heard of Puerto Rico when I was a child, I’ve heard arguments about whether or not it should become our 51st state. As with other complex and controversial topics, the arguments I hear or read often sound convincing – until someone else presents a contrary view.

On the whole, I tend to lean toward agreeing with the proponents of statehood, both for reasons of principle and pragmatism. As to principle, why should citizens of this country not have the same kind of voting rights and elected representatives at the federal level? The practical reasons have to do with the economic boost that statehood proponents believe would occur, as it has with other states that entered the Union.

Trying to predict economic outcomes, of course, is difficult at best. But I do think that statehood proponents have a point when they point out the flaws in the economic arguments of opponents to statehood. The latter group claim that since rates of poverty are so high in Puerto Rico, having Puerto Ricans pay federal income tax would generate little revenue, while more tax dollars from the existing fifty states would flow into Puerto Rico.

The question is whether the current state of the economy in Puerto Rico would persist. The opponents of statehood seem to assume that it would. The proponent of statehood point to studies that purport to show that the island’s economy would experience a significant boost. People who know far more about economics than I do can’t agree on the matter, so I’m not going to try to render an opinion. But I do know that the economy is so complex, influenced by so many interdependent factors, that you can’t change a few factors and expect the others not to change also.

The purpose of this post isn’t to argue for or against statehood, however. If the subject interests you, there are a variety of website that discuss the matter. The U.S. Council for Puerto Rico Statehood is – as the name says – for statehood. No Statehood for Puerto Rico and ProEnglish oppose it. This one gives a fairly balanced view, I think, of the issues from both perspectives.

What I found interesting this evening was a far easier question: How could we rearrange the stars on our flag to add in one more? Fifty-one is three times seventeen, but it would hardly work to have three rows of seventeen stars. You could split seventeen into eight and nine, and have six rows of eight and six row of nine, but then you wouldn’t have the nice symmetry of today’s flag, with longer rows of stars at both top and bottom.

Of course, if Puerto Rico became a state, might there be other territories desiring the same status? How would you make a flag with fifty-two stars, or fifty-three? Fortunately, a mathematician and a computer can offer practical solutions to these questions far more easily than economists and politicians can answer the thornier questions regarding statehood.

Check here for an interactive flag calculator that lets you see possible configurations for anywhere from one to one hundred stars – with three exceptions for which there are no valid patterns (at least not using the six most common star configurations). Many numbers offer two or more possible patterns (try clicking on the long, short, alternate, equal, wyoming, and oregon buttons when they are not grayed out).

Not a day over 30

[Published 1/E/7DA]

I am 30 years old today. And as my sister is reading this and will no doubt protest at this statement, let me point out that I did not say what numbering system I am using.

I first learned about alternatatives to the decimal system when I was in fourth grade. The class was divided into two groups for math, the “Houghton-Miflin” group, which used a blue textbook published by Houghton-Miflin, and the “Addison-Wesley” group, which used a green textbook published by Addison-Wesley. I was in the Houghton-Miflin group, which was for the more advanced students. While the teacher (whom I can’t remember at all, as math was the one subject where we left our regular classrooms and went to another teacher’s room) worked with the Addison-Wesley group, those of us in the Houghton-Miflin group worked largely on our own.

That was fine until we got to a chapter on base 8. I was good at math and had never been stuck on any concept before, but I simply could not make sense of this strange system. I don’t know if Houghton-Miflin didn’t do a good job of explaining it, or if I just had some kind of mental block. I simply could not make any of the problems come out with the answers given in the back of the book.

Fortunately a classmate (he had dark curly hair, but I have no idea what his name was, though I think it started with an “A”) came to my rescue. He showed me a way to do the problems that enabled me to get the right answer. It had something to do with subtracting or adding two at certain points, and it worked, as I now was able to get the answers in the back of the book. But I had absolutely no idea why it worked that way.

Some fifteen years later, I began studying computer science, because I wanted to become a computer programmer. One of the skills we had to master was converting numbers between our familiar decimal system, and the binary system used by computers. Because binary numbers are awkward for humans to read and write (the year 2010 in binary is 11111011010), programmers use octal (base 8) or hexadecimal (base 16) representations of these numbers.

If you’re not familiar with different numbering systems, I’m not sure whether I can explain it better than my Houghton-Miflin textbook did. Here’s a site which explains them, though as I already understand it I can’t say whether this is a good introductory explanation. In a nutshell, just as the decimal system has the “ones” place in the first position on the right (assuming only whole numbers with no decimal point), then the “tens” to the left of that, then the “hundreds” then “thousands” and so on, each “place” being ten times the one to its right because it is base ten, any other number system works the same way but multiplying by whatever base you are using.

So binary has the ones place, then the twos place, then the fours then the eights and so on. The number we call 14 (I like that number, as I tend to get presents on that day of the month at the beginning of each year) would be represented as 1110, because it is made up of one “eight” (2 x 2 x 2) plus one “four” (2 x 2) plus one two (and no ones). In octal, that same number would be 16, because there is one eight (one in the eights place) and six ones.

Of course it gets confusing trying to discuss these numbers using digits from the decimal system. The number that we call fourteen in decimal couldn’t be called fourteen in octal, but it wouldn’t make sense to call it sixteen either, even though we write it as 16. Because the “teen” comes from the word ten. So if we actually used octal in everyday life, perhaps 16 would be called “six-ayet” or something like that. And the number 60 (six times eight plus zero times one, or 48 in decimal notation), might be called “six-ets.”

Hexadecimal gets more confusing, because the ones place can have number greater than nine. After all, the next place over is the sixteens place, and what we call sixteen would be written as 10. So how do you write the numbers we know as ten through fifteen? I don’t know who came up with the solution, but it is to use the letters A through F. So twenty-six in decimal becomes 1A in hexadecimal: one sixteen and ten ones. Forty-seven becomes 2F: two sixteens and fifteen ones.

That (2F) was my age yesterday. But today, adding one more year, the F becomes a 0, carry one (just like adding 1 to 9 in the decimal system) to the sixteens place, and my age is 30. Of course, it doesn’t look as good when I write my age in octal: I am now 60 (six eights and zero ones). And forget about binary – I don’t want my age to be 110000!