"Bear in mind that the wonderful things you learn in your schools are the work of many generations, produced by enthusiastic effort and infinite labor in every country of the world. All this is put into your hands as your inheritance in order that you may receive it, honor it, add to it, and one day faithfully hand it to your children. Thus do we mortals achieve immortality in the permanent things which we create in common." - Albert Einstein

Thursday, October 2, 2014

Understanding Fractions

In a previous post, Learning in Steps, a study, "Sources of Individual Differences in Children's Understanding of Fractions", which showed that number knowledge is a good predictor of future performance with fractions, has been highlighted. Number knowledge essentially encompasses arithmetic skills such as adding and subtracting. To appreciate further this correlation, it is useful to look at what skills are indeed required to understand fractions.

Fractions are currently being introduced to my son who is in third grade. The initial skills covered in this introduction include recognizing equivalent fractions as well as comparing two fractions and deciding which one is bigger. To help the student acquire these skills, the following models, for example, are used:

Equivalent Fractions

 4/8

2/4 

1/2

With the models above, a child can hopefully see that the fractions 4/8, 2/4 and 1/2 are equal. The models above, in fact, can be used to compare fractions as well. In deciding which one is bigger, 1/5 or 1/4, the following figures can be employed:


1/5

1/4

Again, in this case, the above figures should help a child visualize which fraction is bigger. It is straightforward to see that 1/4 is bigger than 1/5. Here is another example, when comparing 3/8 and 2/5, the following drawings can be utilized:

3/8

2/5

In this instance, it should be obvious that accuracy is required in the drawing. 2/5 is only slightly bigger than 3/8. 

The models above suggest prerequisite skills. In addition to drawing accurately, a child must know how to divide. The following fractional wall from NRICH illustrates how a rectangle can be divided into so many ways:


The white one (top rectangle) is divided into 24 equal parts. Each division here is therefore 1/24. The red rectangle is divided into 12 parts so each part is 1/12. The green one is partitioned into 8, so each section is 1/8. The pink rectangle is divided into 6 so each cut is 1/6. The green one shows what 1/4 looks like, brown is 1/3 and the purple one is a whole. Going from top to bottom, right across the middle, one can therefore see that 12/24, 6/12, 3/6, 2/4, and 1/2 are all the same.

The relevance of division in understanding fractions can not be overstated. Terezinha Nunes and coworkers at Oxford University have voiced this important point in a research brief published by the Teaching & Learning Research Programme:


To help us see the relevance of a child's arithmetic knowledge to understanding fraction, here is an example. A child can relate fractions to insights they may have drawn from experience as illustrated in an activity provided by NRICH:


Before child 9 joins the group, the 2 people at table 1 each gets 1/2 of the chocolate bar, the 3 people at table 2 each gets 2/3, and the 3 people at table 3 each gets a whole bar. Thus, from these initial numbers, for child 9 to get the biggest piece, he or she must choose table 3. With an additional person on table 3, each one now gets 3/4 of a chocolate bar. 

The scenario described above also helps illustrate that a fraction is composed of two parts. A fraction has a numerator and denominator:

And with the specific example of chocolate bars and children, the fraction is:

With this in mind, it can help a child see how the value of a fraction depends on the numerator and denominator. The larger the numerator becomes (the more chocolate bars), the larger the fraction each child may receive. The larger the denominator (the more children to share with), the smaller the fraction each child may get.

Division is quite different multiplication. Multiplication can be regarded by a child as nothing but repetitive addition. For example 5 times 3 can be taken as adding 3 + 3 + 3 + 3 + 3. Jumping from addition to multiplication is therefore natural. Division, on the other hand, maybe a little bit more complicated. However, it can be associated with the experience of sharing. And to be precise, "fair sharing". Thus, when using models or drawings, the divisions need to be accurate, a skill that a child in the primary years may still be developing. As seen in the previous example of comparing 2/5 versus 3/8, an accurate drawing is required for a child to arrive at the correct answer. 

Models do look good especially when these help a child visualize mathematical concepts. Take, for instance, another example below which takes the child to the next step, multiplying fractions. The models below help visualize the product, 7/8 times 4/7. In this figure a rectangle is divided into 56 squares (with 7 columns and 8 rows):
A child colors 7 columns but only up to the eight rows (This is what 7 times 8 implies). Doing so, the number of blue squares is then 28. (This is 7 times 4), and since the total number of sections above is 56. The answer to 7/8 times 4/7 is 28/56. 28 of the 56 squares are colored blue. One can go further by rearranging the blue squares in the following fashion (7/7 times 4/8) and see that the answer is in fact 1/2 (which is equivalent to 28/56).
Unfortunately, similar to the specific instance of comparing 2/5 and 3/8, one can see that accurate drawings are required. The above figure, for instance, is not really accurately drawn (the fifth row are in fact smaller rectangles and not squares).

Models are effective when these help children see what various mathematical operations do. In fact, individuals who understand fractions can easily appreciate the usefulness of these drawings. One needs to be careful, however, and be aware that sometimes models may be acquiring more from a child.

Forty years ago, I was taught that one way to compare fractions is to use multiplication. Using the specific example of 2/5 versus 3/8, the following is done:


Yes, there are no drawings and it does look dry, but as long as I multiply the numbers correctly, the answer is reliable. But even in this example, it shows likewise why understanding fractions depends a lot on a child's number knowledge or arithmetic.





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