I remember growing up and being assigned the requisite 20 to 30 problems per night. The problems were never the ones with an answer in the back so you couldn't check your work.
I know the thought behind this is simply, if a student has trouble the more practice they get, the better they'll get. Unfortunately, this does not work very well.
If the student has no idea how to do the problem, they will probably not get the hang, no matter how many times they do the problem. If they can't check the answer, they have no idea if their final product is correct. So if they learn it wrong, they will continue doing it wrong.
In the past, I have left the teacher's edition open on a table so students could check their answers. In the upper level classes, students would often ask me questions if they did not do it correctly. The younger ones, often copy the answer without showing the work but I found some of the calculators out there online will list steps so students can see the process.
I like something said at the recent technology conference. It was suggested you assign way fewer problems but have students include a written explanation of what they did rather than just solve it. The explanation provides a form of assessment. It allows the student to share their understanding of the process.
I know, some people believe its enough for students to show they can solve a problem but I don't think so. I think its important a student is able to explain why they are doing a certain step. Can student explain why they want to isolate a variable. Why is it important?
Does a student understand when they must follow the reverse of the order of operations and when its not as important? Can they explain when you must do that and when you don't have to? Can they explain why they chose to solve a certain problem using proportions? Does it matter if you divide before you square something?
So many times math teachers expect students to do certain steps because it was the way they learned it. I read a book which made it clear when you are solving a problem such as 5(x + 2) = 20, you can divide both sides by 5 first rather than distributing it first. I'd never though about it because I was taught to distribute first before trying to solve.
I am starting to offer students the opportunity to answer fewer questions as long as they explain what they did and why. So far, no takers but I'm sure as time passes, I'll get them moving that way. I'd like to hear what everybody's thoughts on this.