Fractions

Fractions are among the most abstract concepts presented in elementary school math. We need to ensure that a sound conceptual base is established. This means taking time and providing numerous, concrete experiences for children to construct an understanding of these concepts before every formalizing them.

Here's an example of work adding fractions in a 5th grade classroom.

Students were asked to find the sum of the fractions 1/4 + 1/2 + 2/8. Using materials, they were able to find that these three fractions had a sum of 1 whole.

Students also noticed that they could trade in the fourth in for two eighths and the half in for four eighths, leaving them with 8/8, another form of 1 whole. Likewise, they could trade the eighths and half in for fourths, or trade the fourth and eighths in to make a half. Thus proving that it was equivalent to 2/2, 4/4, or 8/8... all equivalents of 1.

Without ever thinking about "common denominators" the students were dealing with the concept. 

In a more challenging problem, they were asked to think about the sum of 4/5, 7/8, and 2/3. Without worrying about the exact sum, they reasoned that 2 < sum < 3. From earlier explorations, they already know that when a fraction was missing a unit fraction, it was as close to 1 as possible. Thus, three fractions close to one added together, must get close to three. It would have to be close to two because each of the fractions was at least 1/2.

Once assembled, children were easily able to see that these three fractions had a sum of about 2 1/3.

Powerful reasoning, with no need for "common denominators."

Yes, they will be exposed to the algorithm for adding fractions, but they will first construct it through many more investigations such as these. In the last example above, the student noted, "I could find the answer by thinking about 3 - 1/3 - 1/5 - 1/8. That's like taking away about 2/3 from three."

Female Math Stereotypes

This article is more than a year old, nothing new, but here is some research to back up what many of us who teach math have long known. Teaching math is different than teaching other subjects. One has to really love it and have a deep understanding and appreciation for the subject in order to be effective in the classroom. 

Believing Stereotype Undermines Girls' Math Performance: Elementary School Women Teachers Transfer Their Fear of Doing Math to Girls, Study Finds

ScienceDaily (Jan. 26, 2010) — Female elementary school teachers who are anxious about math pass on to female students the stereotype that boys, not girls, are good at math. Girls who endorse this belief then do worse at math, research at the University of Chicago shows.
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Happy New Year

2011 is a prime number year.
What's more, it's the sum of eleven consecutive primes!
2011=157+163+167+173+179+181+191+193+197+199+211

Football Math

So I'm sitting here at Giants Stadium, waiting for my Redskins to take the field. Thinking about the numbers in football. Like all sports it's full of statistics.
Football scores are quite complex. Six points for touch down. One point for the extra point after a touch down. Or, you could run the ball in after a touchdown for two points. Then of course there is the three points for a field goal. Finally, the saftey for two points. 6, 3, 2, 2, 1.
So I'm thinking, what kinds of numbers can the final score be? Could it be prime? What about square or cubic? What would the score look like written in Binary?
Well, I suspected the Redskins would lose. They have a lot of players out with injuries in key places and the quarterback has been struggling. That, with the fact that the Giants have been doing very well lately.
So the final score was 7-31. Both prime numbers. How many different ways could these scores have been achieved using the posibilities given above?

Binary

So boil it all down to one and zero and you have the simplest possible system. Two digits. Yes or no.

The Indian scholar Pingala is credited with the first known use of a binary system. He used a system of dashes, a short dash and a long dash to represent 1 and 0. The Chinese used hexagrams to represent numbers. These are squares made of a stack of 6 lines that are either solid or broken in the middle to represent 1 and 0. Then finally in 1703, Leibnitz uses the 1 and 0 digits as we know them and formalized the binary system as a practical device for computation.

Thinking About Zero

Well, it's the Thanksgiving Break from school and I have time to ponder some of those many ideas that have been rattling around in the back of my brain. Lately I've been thinking a lot about zero. We've been talking about how powerful this digit is in my math classes. This idea of a place-holder is so powerful. It allows us to create numbers of unimaginable magnitude.

"How can something be nothing?" asked the ancient Greeks. But those brave Indian mathematicians about 2200-2500 years ago started using a symbol to represent this idea of nothing. Not just nothing in terms of counting, but nothing of a particular place value in a numeral! This was a huge improvement over the Babylonian's empty space between digits. By the 9th Century AD, the Indian mathematicians were making use of zero in calculations, using it like any other number.

Of course the Maya had them all beat, using a symbol for zero as a placeholder in their base 20 system. However, this system was a parallel evolution and didn't seem to influence the systems we use today.