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Sunday, November 22, 2009

Why are fractions so difficult to learn

As many adults know, learning the various fraction operations can be difficult for many people. It's not the concept of fraction that is difficult - it is the addition, multiplication, subtraction, simpifying, etc. - various operations that you do with fractions. And the simple reason why learning the various fraction operations proves difficult is the way they are typically taught in school books.

Just look at the amount of rules there are to learn about fractions:
1. Fraction addition - same denominators
Add the numerators, and use the same denominator

2. Fraction addition - different denominators
First find a common denominator by taking the least common multiple of the denominators. Then convert all the addends to have this common denominator. Then add using the rule above.

3. Finding equivalent fractions
Multiply both the numerator and denominator with a same number

4. Mixed number to a fraction
Multiply the whole number part by the denominator and add the numerator to get the numerator. Use the same denominator as in the fractional part of the mixed number.

5. (Improper) fraction to a mixed number
Divide the numerator by the denominator to get the whole number part. The remainder will be the numerator of the fractional part. Denominator is the same.

6. Simplifying fractions
Find the (greatest) common divisor of the numerator and denominator, and divide both by it.

7. Fraction multiplication
Multiply the numerators, and the denominators.

8. Fraction division
Find the reciprocal of the divisor, and multiply by it. 

If students simply try to memorize these without knowing where they came from, they will probably seem like a jungle of seemingly meaningless rules. By meaningless I mean that the rule does not seem to connect with anything about the operation - it is just like a play where in each case you multiply or divide or add or do various things with the numerators and denominators and that then should give you the answer.

Fraction math can then become blind following of the rules, tossing the numbers here and there, calculating this and that - and getting answers of which the kids have no idea if they are reasonable or not. And of course, it is quite easy to forget these rules, or remember them wrong - especially after 5-10 years.

manipulatives and use of pictures help understand fraction operations

Instead of merely presenting a rule, as many schoolbooks do, a better way is to teach children to visualize fractions, and perform some simple operations with these visual images or pictures, without knowingly applying any given 'rule'.

If a person is able to visualize fractions in his mind, he becomes more concrete - not just a number on top of other number without meaning. Then that person can estimate the answer before calculating, and evalute the reasonableness of the final answer, and perform many of the simplest operations in his head.

Of course textbooks DO show fractions with pictures, and they DO show one or two examples of how a certain rule connects with a picture. But that is not enough! A better way is to make students do lots of problems with fraction manipulatives - and DRAW fraction pictures for problems. That way they will form a mental visual model and can think through the pictures for simple problems.

If you think through pictures, you will easily see the need for multiplying or dividing both the numerator and denominator by the same number. But before voicing that rule, it is better that students get lots of 'hands-on' experience with fraction pictures they can draw themselves. They can even have fun splitting the pieces further, or conversely merging pieces together. They may find the rule, or you may tell them about it - and it will make sense. If they later forget the rule, they can always think back to splitting pieces, and re-discover it.

Another example is the lesson about teaching addition of unlike fractions . One can show how the individual fractions need to be 'split' into further pieces so that they are all same kind of pieces. One doesn't need to discuss "least common denominator" at this point. The teacher can simply use pictures or manipulatives. Then, the students will do the same with manipulatives, or by drawing pictures. After a while, some students might discover the 'rule' as to what kind of pieces the fractions need split. And in any case, they will certainly remember it better when they have been able to verify it themselves with numerous examples.

I'm not saying that the rules are not needed - because they are. You can't get through algebra without knowing the rules for fraction operations. But if 10 years from now the student maybe has forgotten algebra and the fraction rules, hopefully she will have retained the simple fraction pictures and is able to "do math" with the pictures in her mind, and not consider fractions as something she just "cannot do".

1.  Think back to when you started to learn how to add and subtract with fractions. How was it presented to you?

2.  Look at the list of  “rules” for operations involving fractions. Which one was (is) difficult for you and why?

3.   This article proposes the use of manipulatives (fraction circles) as a solution. Do you agree or disagree and why? How does your personal experience influence your opinion?

1 comment: said...

Fractions are definitely tricky at times. That's why I wrote three how-to articles dealing with fractions. Check out all of my how-to articles. The link to my articles is at: