Just hearing the F word can cause kids (adults too) to freak out. And if you think about it, there are lots of reasons students feel flummoxed by fractions. For one thing, there's the misleading vocabulary, as when we reduce a fraction to lowest terms even though it doesn't involve a reduction in value. Or when we call a fraction "improper" just because its value is greater than one.

Then there are apparent inconsistencies between arithmetic with natural numbers and arithmetic with fractions. Multiplying 10 by 5, for example, increases the value from 10 to 50. But multiply 10 by 1/5, and you end up with only 2. Conversely, whereas dividing 10 by 2 yields a smaller number (5), 10 divided by 1/2 results in a larger number (20).

Yet as confusing as fraction arithmetic can be, a lot of this confusion can be prevented if students have a conceptual understanding of fractions before teachers target procedural understanding. In elementary school, students need to interact with concrete representations of fractions until they can see the effects of operations involving fractions. In other words, teachers need to develop students' understanding of fractions using manipulatives--actual and/or virtual (National Library of Virtual Manipulatives is an awesome--and free--site).

Unfortunately, for various reasons--such as lack of training and lack of time (gotta get through the curriculum before the test)--elementary teachers often move on to abstract/algorithmic treatment of fractions before students grasp them concretely/conceptually. They then cross their fingers hoping students will remember to flip and multiply, not add denominators, etc. on the test.

But even when students remember what to do on the test, many of them forget afterward. And though some students memorize procedures for good, they often don't understand or think about what they're doing. Instead, they just apply algorithms. Take, for example, problems where students have to place fractions such as 3/4, 1/2, 2/3, and 4/5 in order from least to greatest. Some students will convert to a common denominator, while others may go with decimals. But few of them--even high school students--will take a more analytical approach unless we model/facilitate this for them. Just asking, "So would you rather make three out of every four free throws or one out of every two or...?" (or you can go with other contexts such as money) can be enough for light bulbs to go on for many kids.

Asking students questions like this and providing them quality instruction and practice using manipulatives can help them nail one of the most dreaded math topics. And when they do, I often hear teachers use the F word: fantastic!