Division seems to be as difficult for students to master as subtraction. I have students who struggle with it if its a division problem. If I change it into a multiplication problem, they struggle less but its still a challenge.
According to two different articles I read, division is introduced to young children by using the idea of sharing, such as there are 12 pieces of candy shared between 3 children. How many pieces will they each get.
The second stage of teaching division has students looking at division as grouping rather than sharing. It also incorporates relating it to division. Add into the mix the fact that the division algorithm makes less sense to most students than any other operation because its a left to right rather than the usual right to left process.
Most of the students who are successful using the division algorithm have memorized the steps without understanding the concepts behind each step. I am the first to admit, I am one of those. I was told how to do it, I followed the steps and got the answer, What more did I need to know?
The students who struggle are the ones who do not know their multiplication tables well and who missed the step of placing a zero in when they cannot divide the number. These same students struggle when division is taken from using straight numbers such as 28/7 to dividing polynomials such as x^2 + 5x + 6/x+2. The algorithm is the same but it uses variables.
In my opinion, division is really just a way of finding the second factor of a number when you know one factor. Yes, sort of the reverse of multiplication. When my students struggle with 28/7, I'll often ask them "7 times what gives you 28? because it is sometimes easier for them.
It has been suggested students should not even attempt the division algorithm until they have their multiplication facts down cold, otherwise it is difficult for them to learn. Its also suggested students have division down as both sharing and grouping because they need both interpretations. With grouping if you have a problem like 900/30 it could be interpreted as 900 items split into 30 groups or 900 items split into groups with 30 items.
Add into this the idea of remainders. I don't think the elementary teachers or books have taken time to explain remainders properly. My students see it as remainders, not as so many items left over which are not enough to make a group itself.
I'd love to hear from you all on ideas you have to help make division easier for high school students who struggle. Hope you are having a good break.
Then comes the question of when will I ever use polynomial division? It is used in most crypto and error correcting algorithms which are used in most modern digital devices.