Doing Math vs. Understanding the Math

This is a topic I struggle with. Without that understanding, I'll be teaching the same students the same concepts if they leave for a few weeks or months, and I do not like to duplicate effort. I had one student that had to leave for about a month. When he returned, he had lost all of the progress we made, because he forgot the procedures and never learned the conceptual understanding. 

Teaching concepts is messier than teaching procedures, but I agree that it's really important--especially for retention. It's also critical for the kind of word problems that give you more (or less) than two numbers and a key word (There are 3 gizmoes on the shelf and 7 gizmoes on the floor, how many gizmoes are there altogether?).

The thing that I like here, though, is the dual focus. It's not only important to understand the idea, it's also important that once you understand it, you learn how to use the idea efficiently. If you understand the idea, you can make a good stab at figuring out a similar problem later, but you don't want to have to reinvent the wheel every single time.

The problem then is finding the balance. How much time do I devote to understanding, and how much to practice? I suspect that a few practice problems thoughtfully done are worth much more than a lot of problems done mindlessly. Research on memory indicates that new kinds of problems should be mixed in with familiar ones, and spread out over several days (or weeks), which makes sense if you're trying to build retention, fluency, and understanding. Unfortunately, I suspect that to do this properly, you have to pretty much make your own materials or infringe on copyright laws...

I am glad that this has come up because it is something that I struggle with daily. I lean towards building confidence in doing the problem then I infuse lessons that promote understanding. It is often tedious because students want to have mastery of a topic after only one lesson and without much practice. I do give a lot of practice problems to strengthen the foundation but I vary the type of problems and make sure to include word problems.

I also agree that without understanding, students are more prone to forgetting what they have learned but with all of the outside factors that my students deal with, its just something I have to work around.

Brooke, thank you for the link, I will be using this site.

- Alfons Prince

Back when I was a middle school classroom teacher, I worked almost exclusively with students with disabilities. I found that teaching fractions was incredibly hard, and did quite a bit of research on the subject. In doing so, I found the "big picture" model, which I have very successfully used.

The basis of the model, the big picture, is that fractions, division, decimals and percents are all the same (which helped me understand why students tend to struggle with all 4). For instance:

Fraction: 1/2

Division: 1 divided by 2

Decimal: .5

Percent: .5 = 50%

I have found that, without exception, when students understand this "big picture," it opens the way for understanding.

I also use pattern blocks to teach for understanding. The best resource I have found is here: http://www.curriculumsupport.education.nsw.gov.au/secondary/mathematics/years7_10/teaching/frac.htm