How To Teach Elementary Math
2 Dec 2008
How to Teach Fractions
Ho Math and Chess Learning Centre
Canada, BC certified math teacher
Vancouver, BC, Canada
The traditional way of teaching fractions addition is to tell children that if the denominators of two fractions are different then they must be converted to the same denominator by using LCD. Sometimes two diagrams of pie charts are drawn to show the reason why. It teaches the concept by drawing diagrams to show the reason and then the procedure is taught to children on how to do it. Why most students still get confused by this way of teaching fractions?
I was at shock the other day when a grade-8 students did 2 fractions of multiplications by flipping the second fraction in his test and he got all fractions multiplications wrong. This is a student who actually understands concept and could do fractions operations well, I was really confused on what he did and he simply said to me that he forgot that multiplication does not need to flip the second fraction.
I have been giving a lot of thought and feel that the notation of fraction is a great invention since it actually defines the meaning of what is a rational number. With the above said, fraction is also the most confusing concept in elementary math and continues to be confused when going to rational equation in high school.
Have we taught fraction clearly to children? Can the concept of fraction be expressed much more clearly? My answer is that we have not been teaching children in a clearer way but thought it would be understood if we simply explain the meaning of fractions to them by drawing diagrams.
A number written in the form of p/q is not necessarily a fraction. Its meaning is not really clear until the question is presented. So this is the first misconception on children to think that just because a number is written in the form of p/q then it is a fraction. p/q can be used to solve ratio, rate, %. Proportion, probability etc. problems so a number written in the from of p/q is not necessarily always a fraction.
Should the denominators be always changed to the same LCD? Look at the following example, ½ plus 2/4. Its LCD is 4 but why not change ¼ to ½ and ½ plus ½ is 1. Here we did not use LCD, which is 4. So clearly there is something missing here that we continue to tell children to always change all different denominators to the same LCD. What is missing is the unit fraction concept. ½ has an unit fraction of ½. 2/4 has 2 of unit fraction ¼ so they can not be added together because they have different unit fractions. By reducing 2/4 to ½ then they have the same unit fractions.
One of 3/8 pie and 3 of 1/8 pies have different meanings even though the final quantity are the same. This can be understood by introducing the concept of unit fraction.
The most confusing about fraction is that the fraction notation itself has 2 operations hidden. They are division and multiplication. So 2/5 can be thought as 2 divided by 5 but it also can be thought as 2 times by 1/5. No wonder our children get confused.
Children learn order of operation, but does the order of operation apply to a simple fraction operation? The operation of fraction is also from top to bottom, not necessary from left to right, so how does the order of operation apply to fraction? I have never seen any math textbook address this problem. For example, 18/3 times 6, should the student do 18/3 first then times 6 or should student do 6 divided by 3 first then do 18/2? Many children get confused on this point. If you do order of operation then perhaps one would do 18/3 then times by 6 from left to right but it is much easier to reduce the “big” number by reducing so 6/3 should be done first. This kind of problem about fraction has never been clearly mentioned to children in math book and the complexity of having 2 operations born with fraction was never taught to children in a clear way.
The top to bottom operation of fractions is very unique which is very different from a normal operation of left to right and this was also not clearly taught to children.
Is it enough to just draw some pie charts to expect children to master fractions operations? Clearly from my explanation above, it is not enough. Inherently the invention of powerful fraction has created a very complexity of operations but the center of teaching seems to only concentrating on how to get children to understand why there are different denominators instead of also explaining the different of meanings of the notation of p/q it self and how it should be operated correctly.
It is not enough we just try to teach children on the concepts of how fractions are operated, it is equally important we teach children on how a fraction is different from other math operation with its hidden division and multiplication. Why p/q is not necessary a fraction and what is unit fraction. How a sign is placed on a fraction? Where a negative sign is placed on a fraction and where is its preferred place -- top, down, or in the front? To show its value we like to place a “–“ in front of a fraction but to calculate we like to place it on the top.
2Y means 2 times Y. The number 23 does not mean 2 times 3. So why does 2 2/3 mean 2 + 2/3? Have we explained to children all these different meanings? If we have not, then no wonder they get confused.
How is 3/2 related to how an expression can be expressed in remainder form?
Is it true that we can not do fraction division? If this is true then how come when children go to high school then they can do polynomial division and its divisor is not a whole number?
We are at fault by not showing that we can actually do fraction division by using the same concept of whole number division. For example, 2 divided by ½ and use the regular notation of whole number division that we can do 2 divided by ½ and we get quotient 4 since 4 times ½ we get 2 and the remainder is 0. Why we did not tell children that we can do it? Why we decide not to do it this way? This again is one of many reasons that children do not understand fractions is that we have failed to let them see the reason but simply tell them not do fraction division.
I feel there are more work we need to do on how we introduce fractions to children and let them understand its operations and see the result on what would happen if they do not follow the suggested way.