I was ambivalent about going to Las Vegas for the regional conference of the National Council of Teachers of Mathematics because my time is pretty limited. I’m really glad I went.
Some thoughts on fractions:
What IS a fraction? Partitioning. A part of a whole. More than that, each part is equal.
Are fractions numbers? Any mathematician would say yes, of course. They aren’t integers but they’re numbers.
First, kids learn that when you multiply two numbers you get a larger number. BUT …. when you multiply with fractions, you get a smaller number.
When you divide one integer into another, the answer is SMALLER than the number that was divided. However, when you divide an integer by a fraction, you get a LARGER number. When you multiply two fractions, you get a smaller number. When you divide one fraction by another, you get a larger number.
I never thought twice about it, dividing fractions, you invert the second fraction and multiply. Why?
If you invert the first fraction, you get the wrong answer.
Second, many kids don’t realize at first that you can have fractions on the number line.
Third, while HIGHER numbers go after one another on the number line, like 8 goes after 6, LOWER fractions go after one another on the number line, with 1/8 going before 1/6.
WHY? Well, as an adult, it might seem obvious, if you have more pieces, say 8 instead of 6, the size of each piece will be smaller.
One teacher said she just explains it to her students as “All the rules are backwards for fractions.” But …. why?
I learned math the old-fashioned way, where we were taught algorithms like invert and multiply and they worked fine for me. I went to some sessions using manipulatives and when the question was “What is 1/2 of 1/2?” of course you get a fourth by taking 1/2 of 1/2 – but why are you physically dividing your group of tiles (or whatever) to multiply?
When you divide fractions using manipulatives, if you divide 1/2 by 1/8 you get 4. WHY? One presenter explained it very well. She said when we divide we are basically asking the question
How many groups of size X go into Y?
So, if we divide 56 by 7 we are saying how many groups of size 7 go into 56? The answer is 8.
How many groups of size 1/8 go into 1/2 , well, that’s easy, 4.
In short, I had a fun time just thinking about math and how to teach it.
I also learned about a lot of resources for teaching mathematics. Mathsnacks.com has animations and videos (free). Casio has some cool calculators that can do simulations, graphs and even plug into the USB port on your computer for you to copy data for your students to analyze. There is also a Casio program you can download that emulates the calculator on your computer (so you can show the screen while teaching with a laptop).
Now, I’m excited to get home both because I have a lot of ideas for the statistics class I will be teaching next month and also for the game that we are developing using fractions.
I didn’t meet a single person who wasn’t enthusiastic about everything they learned here. So, one final take-away message – the regional conferences are where you get a lot of bang for your buck. I plan to put more regional conferences on my schedule for next year and no national/ international ones. I think you gain just as much for far less money and time away from the office. (Of course, this depends on your interest, but I’m very interested in applied research and best practices.)
Along those lines, I’ll be back in Las Vegas in a few weeks at the Western Users of SAS Software conference, talking about factor analysis, categorical data analysis and exploratory data analysis. Should be fun.