Innoculating against innumeracy

August 29, 2012

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.

Read the rest of this entry »

When interests and abilities diverge

January 21, 2012

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.

Read the rest of this entry »

Blogging a thousand

September 5, 2011

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

August 18, 2011

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

May 4, 2011

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

April 10, 2011

I like reading about science. For months now I’ve been checking in at 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, 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

March 22, 2011

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.