Sunday, August 12, 2007

Fraction Friction

I was speaking with a friend that teaches adults preparing for their GED’s. They were describing the moans and groans heard as the teacher announced they would begin to review fractions. It reminded me how many of my middle grade students hate fractions and love decimals. There is something very reassuring about decimals to students in grades 6-8, and possibly even older.

I have a theory about why fractions are so hated among this crowd. These theories have no research to back them up, so I hope that readers out there will feel free to call me to the mat on this. First lets do a little comparison between fractions and decimals.

AdditionSubtractionMultiplicationDivisionComparison
Fractions
Decimals


So the ways in which decimals are better than fractions are the addition, subtraction, and comparison (by which I mean being able to compare the magnitude of numbers). I would say this is at the heart of my middle school student’s love of decimals. These are the operations which students feel most confident with. In fact, I would venture to say that students often don’t really feel the “know” a fraction. If they must give answer they like to give a decimal. Students love to give answers like 3.291098094. At this point the reason for this is almost beyond my grasp.

I will say of the three areas I think that comparison is the most important. I believe that this is because each student has a sort of internal measuring tape. They feel they know a number when they know where it goes on the measuring tape. It can be difficult for a student to place with a degree of certainty 4/13 on such a mental measuring tape.

How do we help students feel more comfortable with fractions? I propose the following steps:

➢ Emphasize fraction’s strengths. They solve division problems easily 3 divided by 23 is 3/23 for example. They can be multiplied easily. A good example is .125 * .75 is 3/32. Division too.
➢ Help student to understand the meaning of fraction in many different ways. As rates, as ratios, as the solution to division problems. As a number with a unit e.g., 3 eighths. Eighths can be thought of as a unit, like feet or inches. This is why we must have a common denominator to add.
➢ Help students find ways to estimate the size of fractions. This may help them place the answer on their mental number line. For example, 4/13 is a little smaller than 4/12 = 1/3 so 4/13 is just a little smaller than one-third.
➢ Help students understand through pictures and a variety of situations why multiplying fractions can lead to smaller numbers and division can lead to bigger ones. This should include a thorough discussion of the meanings of multiplication and division. Too often students just see these as ways to make numbers bigger and smaller.

It really is an interesting question. Why do students dislike fractions so? Fractions predate decimals by a good 600 years. I often wish I could watch a middle school math class prior to Simon Stevin to see what the kids were bitching about then.

3 comments:

Tony said...

To be fair, those ancient fractions you are talking about were unit fractions. They were easy to compare and a pain in the ass to work with. I have heard that European students used to the metric system barely learn fractions at all.

Finding the LCD or GCF amounts to an extra step, as far as I'm concerned. It's true that working with fractions will help understand later concepts like rational functions, but the kids don't have the kind of foresight to understand that. To them, it just feels like extra torturous, arbitrary work. Honestly, when in real life do you ever have to divide a fraction by a fraction?

I have long felt that we can hold off on most fraction lessons until much later, when students are mature enough to grasp the concepts.

Matt said...

From my perspective fractions are definitely part of the human condition. People will think in fractions and ratios. Proportional thinking is an integral part of understanding our world. I think your comment shows many of the same bias that I talk about in my post. If you were to hold off teaching student fractions until later then you are not letting your students take advantage of fractions for the things they are good at.

Student mistakes with decimals show that decimals aren't that much easier to understand, and things like 1/3=.3, .9999999=1. There are difficult concepts here as well. I don't know that decimal division is any more enlightening than fraction division.

Overall, I think proportional thinking is a worthwhile concept that needs fractions to aid in the understanding. Seems like a worthwhile reason to teach fractions. I just think they should be taught better.

Tony said...

I agree they should be taught better, and I agree that many of the teachers don't really understand the fractions they are teaching. I am biased toward metric decimal notation, and I freely admit it. Both systems have pros and cons, but both are necessary and enlightening.