Math and art

January 3, 2014

Most people don’t think of math and art as having much in common. Math follows strict rules; art is all about being creative. But math can produce some really cool images, such as those at Jos Leys’ Mathematical Imagery.

I found my way here from the Astronomy Picture of the Day from Monday, a “mathematical visualization of a generalization of a fractal into three dimensions” which looks more like an imaginative illustrator’s conception of an alien life form. I took a brief look at more information about Mandelbulb sets, and was reminded why I did not major in math. (I did very well in math up through calculus, I just found nothing in it that made me want to learn more.)

The art created by Jos Leys, however, is amazing, so I’m glad there are people who do study math and create the kind of software that can create such images. Some of these look like beautiful jewelry, others would make great posters, and some look like inspiration for some interesting characters in children’s picture books.

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 »

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.)

Learning math with a fifth grader

October 5, 2010

My younger son takes some ELP classes at school. (I think it stands for extended learning program but I’m not sure.) He’s done different subjects, including reading, writing, and science, but the one he has done most is math. He regularly brings home logic puzzles and brainteasers of various sorts. When he gets stuck, I help out, and sometimes the problems take some serious thinking even for me.

Recently he has been bringing home word problems, which come with an explanation of strategies on how to solve the problems more easily. I know a lot of people dread word problems, and while I’ve never had much difficulty doing them, I do find it hard to explain how to go about figuring them out.

When he brought home the “Think 1” set of problems, we went through them together, and I was impressed how easy the Think 1 technique made problems that might otherwise have seemed quite daunting. They were all along the lines of “Bill can paint a house in 1 hour and Bob can paint a house in 2 hours. If they work together, how long will it take them to paint 3 houses?”

I don’t know how I would have gone about doing it before, but the Think 1 strategy says to start by figuring out how much Bill and Bob will get done in 1 hour. It’s not hard to figure out that Bill will paint one house and Bob will paint half a house, so together they’ll paint one and a half houses in one hour. And from there it’s easy to see that in two hours they’ll finish the three houses, since two times one and a half is three.

Even if the numbers get a bit more complicated, the same method works. On one of the more advanced problems, we ended up with a fraction of an hour that wouldn’t even work out to an even number of minutes, but it did work out to an even number of minutes and seconds. I don’t have to try to set up the problem using algebra now, just use the Think 1 method.

Tonight I learned the 2-10 method with him. The idea is that a lot of times the word problems would be easy if they had nice numbers like 2 and 10, instead of fractions like 5/7 or big numbers like 12,000. So you restate the problem using 2 and 10 in place of those numbers, and figure out what you need to do – add, subtract, multiply, and/or divide. Then once you know what to do with the numbers 2 and 10, you substitute the original numbers back in and do the same arithmetic operations.

It does still require knowing how to deal with fractions and decimals, which I know I hadn’t learned as a fifth grader – and my son is a bit shaky on. So I helped him with that part. And there was one problem where I didn’t find the 2-10 method nearly as helpful as noticing that $1.28 and $1.92 were both multiples of $0.64. (Otherwise he would have had to divide 1.28 by 1.92, which he could have done, but why go to that trouble if you don’t have to?)

Anyway, if you or your child could use some help of that nature with word problems, you might want to check out Becoming a Problem Solving Genius by Edward Zaccaro. As the reader comments at point out, it works well for a variety of ages and skill levels, because the explanations are simple, using cartoon illustrations, and there are various levels of problem difficulty.

I wonder what I’ll learn next from Zaccaro.

Puppy math

October 15, 2008

Whether you like math or hate it, you ought to take a look at Jon Scieszka’s Math Curse. I don’t remember who gave it to us, but it is a fun book to read. The word problems are not your typical “if Train A leaves station X at 8:30…” For instance:

I have 1 white shirt, 3 blue shirts, 3 striped shirts,
and that 1 ugly plaid shirt my Uncle Zeno sent me.

1. How many shirts is that all together?
2. How many shirts would I have if I threw away that awful plaid shirt?
3. When will Uncle Zeno stop sending me such ugly shirts?

Another one details the stops the school bus makes and how many children get on at each stop. Then the question is… True or False: What is the bus driver’s name?

If you want a serious math book, don’t get this one. But if you like some humor along with a low key message that math really is all around us, check this out.

Now to the puppy part. In the spirit of Math Curse,

Kyra was 7 weeks old when we got her on September 6, and weighed 11 pounds.
Today she weighs 22 pounds.

1. How many weeks did it take her to double her weight?
2. When should we celebrate her birthday (in dog years)?
3. How long before she weighs too much for me to hold her on the bathroom scale?
4. If I feed her 5 scoops of food a day, why does she want to chew on my socks and my husband’s pay stub?

Here she is, all 22 pounds plus 1 rope bone. (Multiply by 4 paws, add 20 teeth, and how many pieces will she divide the insole from my shoe into?)

Birthday Math

January 14, 2008

My sister, who loves puzzles as much as I do (and who enjoys number puzzles more than I do, I think), emailed me birthday greetings two years ago with a description of a number puzzle she had come up with recently. She took the digits of the date of my birthday (day and month only), and found a way to put them in a mathematical expression to equal the age I was on that day.

My birthday is Jan. 14, represented in digits as 1/14. As a mathematical expression as written, that would be one fourteenth. But if you take out the “/” and use the same digits, in the same order, but are allowed to change the grouping of digits and use any valid operators, you can make 11 x 4, which was my age (44) two years ago.

Naturally I had to respond to the challenge to do likewise for the birthday she would be having that year.

On 11/5/2006 she would be 50. I had to use the year as well, and use an exponent. I’m not sure how to use a superscript here for the exponent, so I will use the notation ** to represent using one. (Thus 3**2 means 3 squared, or nine. And if you don’t remember, any number to the power of zero equals 1.)

( 11 x 5 ) + 2**0 + 0 – 6 = 50

So I thought I would come up with one for my birthday this year. Today is 1/14/2008, and I am 46 years old.

( 11 x 4 ) + 2 + 0 + ( 0 x 8 ) = 46

If you are so inclined, see if you can do one for your birthday.