Nearly every national study of student achievement and every major research project has brought us to the conclusion that students do not have a good understanding of fractions. The rules and ideas around fraction computation are meaningless without a better understanding of fractions than that which students have been developing up to this time. It does not seem that traditional textbooks are improving much in this regard, either. It is due to the need to develop fraction concepts that I have devoted an entire chapter to concepts and have kept fraction computation as a separate issue. You will find that many of the teachers you are teaching have poor fraction concepts themselves. When you finish this chapter you will hear many of them admit that they never understood fractions before.
Fractional parts or fair shares of a whole are used as the building block of nearly every idea in this chapter. The discussion begins with an investigation of sharing activities, primarily based on the research of Susan Lamon. This provides not only an intuitive basis for children's understanding of fractional parts but also a connection to division as an interpretation of fractions (a/b = a ÷ b).
The main idea is to get children to recognize and be able to construct parts called fourths, or sixths, or thirds. This idea can be generalized from the very beginning. I believe it is a myth that halves and thirds are easier concepts than sixths or eighths. An iteration approach, or counting fractional parts, leads to a useful interpretation of numerator and denominator and helps with other ideas as well. When working with a set model, you will find confusion between the number of counters in the set and the name of the set. For example, with 12 counters in a whole, the sets called fourths have three pieces and the sets called thirds have four pieces. I prefer to confront this confusion head on and not avoid it.
Numerator and Denominator Meanings
Counting fractional parts gives rise to the conventions of fraction symbolism. I have found the formulations in the book to be initially difficult for teachers but worth the effort to develop. That is, the denominator names the kind or denomination of a fractional part. It is also a divisor of the whole. The numerator is a counter or enumerator of the fractional parts named by the denominator. It is therefore also a multiplier.
Students need some form of practice activities that will help them develop a real feel for these numerator and denominator meanings. If you distinguish between the part, the whole, and the fraction that names the part/whole relationship, three related activities result: Using any model, provide two of the three components (part, whole, fraction) and have students find the third. These exercises prove to be an excellent means of developing the meanings of top and bottom numbers (I generally avoid the words numerator and denominator). Parts-and-whole activities are challenging for teachers as well as students.
In Chapter 18, when the issue of percent problems is explored, there is an overt connection between these parts-and-whole exercises and the three typical percent problems. This is a good example of important and useful mathematical connections.
Fraction Number Sense
Interestingly, good definitions and the parts-and-whole exercises do little to develop an intuitive or comfortable feel for fractions. At the most basic end of the spectrum of ideas is the concept that the more equal-sized parts there are, the smaller those parts will be. While children understand this in terms of sharing a pizza, it is difficult for them to connect this inverse relationship to symbolic fractions. The idea must be addressed periodically whenever the opportunity presents itself.
For whole numbers, the use of 5 and 10 as anchors was important and was extended to any numbers ending in 5 or
0. For fractions, the important anchors are 0, 1/2 and 1 whole. To know to which of these benchmarks a fraction is closest or if it is more or less than 1 or 1/2 are very valuable ideas. There is an obvious extension to numbers greater than one. This relatively basic idea should be developed not just with symbolic fractions but with models as well. To see a shaded portion of a non-partitioned region, for example, it is important to be able to decide if more or less than 1/2 is shaded or is it a fraction closer to a whole such as 7/8 or 9/10.
Comparison of two fractions is the most sophisticated idea in this number-sense category. Depending on the particular fractions involved, nearly every idea about fractions can be brought to bear on this issue. I like to explore these ideas prior to equivalent fraction concepts because there is less of a temptation to employ a common-denominator algorithm to make all comparisons. Algorithms avoid the more global issue of number sense with fractions. Computational estimation with fractions, at least for addition and subtraction, relies almost entirely on fraction concepts rather than on any understanding of the algorithms. Children (and your teachers) can and should be involved with fraction estimation as they are learning about fraction ideas and long before they are asked to worry about computational procedures. However, these same types of comparisons should also be conducted when children are aware of equivalent fractions and should include fractions not in lowest terms.
Fraction equivalence is an area where the distinction between procedural knowledge and conceptual knowledge is very clear and evident. However, teachers need to see that teaching either the concept or the procedure does not directly help with the other. I try to make a distinction between these two ideas and to provide examples of instructional activities for each. You will note that the algorithm suggested is to multiply the top and bottom number by the same quantity as opposed to multiplication of the fraction by a fractional form of unity. The latter requires the multiplication algorithm and an appreciation of an algebraic argument.
It is also important to note that all of the preceding activities in the section on number sense avoid the possibility of two fractions being equal and do not involve fractions not in lowest terms. Teachers need to realize that these same activities should be done with fractions not in lowest terms. The comparison activities especially should allow for a pair of fractions to be equal.