I learned a valuable lesson in math, recently.
My oldest is just now hitting the pre-algebra, algebra-ish side of math. For me, this is like putting me back at square 1 with a 4.5 year old child and saying, "Ok, now you need to teach this kid to read!" It's nothing I can't overcome, but there's an awful lot to learn. Three kids later, I feel like I could teach basic reading to a classroom of kindergartners blindfolded and with one hand tied behind my back. No sweat! That is not, however, the way I'm feeling about algebra...yet. I'll get there, but it's a big learning curve, and mistakes are being made.
Our biggest mistake happened around 6th grade math, when the book launched into a new way of multiplying and reducing fractions. We were happily plugging along, multiplying our fractions across the top, and then along the bottom, and lastly, reducing, when WHAMO - reverse that. The book introduced cross simplifying, and then multiplication. What's the difference, you ask? My daughter sobbed and begged, "Please, please, PLEASE let me just multiply across and then reduce, the way I always do it." For the sake of returning peace to our household, I agreed. Bad move.
Now, in pre-algebra, we get these problems that are ungodly. When you multiply them out without simplifying first, you end up with an even ungodlier answer. And then when you have to finally reduce, it gets real ugly. Calculators come out. The student's eyes glaze over.
One day, I realized it: simplification. The key is in simplifying things as much as possible first. That lesson we skipped over back in 6th grade was a critically important foundational concept. It is the Key to the Algebraic Kingdom. In arithmetic, you multiply things out and usually make answers bigger. In algebra, you simplify things and make them as small as you can make them. You are constantly pulling numbers apart into smaller pieces, and getting rid of the ones that can cancel. It's a very different mindset from arithmetic.
I'm on my second round of teaching multiplication of fractions, this time to my 11 year old. I decided to spend an extra week on it and teach cross simplification right from the start.