Things have changed since I first started teaching. One of the big changes is the desire to have students understand the concepts behind the math. When I was in school, you didn't learn about concepts, you learned to follow a prescribed process.
You didn't vary from that process. It was accepted you didn't really need to know what you were doing, you just needed to know enough to follow the steps to obtain the correct answer.
Over the years, I've discovered my students often did not understand or know the steps needed to solve a problem.
For instance, why do we change a division problem with fractions into a multiplication with its reciprocal? After quite a lot of looking I finally found an explanation which makes sense. When you divide one fraction by another such as 3/4 / 2/3 you multiply the top and bottom by 3/2 to get one in the denominator.
The explanation is so clear and easy to see but its not as clear when you try to draw a picture to show the same thing. I honestly tried to figure it out myself.
I drew the first block divided into quarters. I divided the second block into thirds. I had to divide each quarter into 3 more so the whole block consisted of 12 segments. I then divided the 12 segments by 3 so I knew each 3rd equaled four segments from the first block.
So 3/4 equals 9 segments while 2/3 equals 8 segments. 3/4 / 2/3 is 9/8 or 1 1/8. I am not going to tell you how many times I watched the Learn Zillions video before I finally got the hang of it. I think I watched it close to 15 times.
Yes I'm stretching my mind learning this but its cool. We all need to learn something new every day. Let me know what you think. I look forward to hearing from you.