The ‘fraction units’ framework
The following framework of levels of using fractions quantitatively is based on the analysis of thousands of students’ responses to questions designed to elicit their understanding of fractions as well as a synthesis of the research literature.
Emergent partitioning involves breaking things into parts and allocating the pieces. No attention is given to the specific size of the pieces. At this level, when a student uses the term half it generally means a piece, which may or may not be one of two equal pieces.
Halving to form two equal pieces is an early fractioning process. The term equal is emphasised here to draw attention to the need to be aware of the basis of determining equality. At Level 1, finding half way is typically used to halve. That is, the basis of determining half of a rectangular piece of paper relies on length rather than area. In a similar way, repeated halving with respect to length can form quarters or eighths.
2 Partitioning continuous quantities into specified numbers of equal parts is very difficult for those partitions not based on repeated halving (i.e. other than halves, quarters, eighths, sixteenths, etc). Instead of partitioning to create odd numbers of parts such as fifths, verifying partitions is recommended. Verifying partitioned fractions helps to establish the relationship between the part and the whole and links to Levels 3 and 4 in measurement.
Show me one-half
The methods used to form halves, quarters or eighths (such as stacking and repeated halving) cannot be readily applied to finding one-third.
One-quarter but not one-third
Constructing thirds and fifths by partitioning a continuous quantity is difficult. Although fifths are introduced in some syllabus documents before thirds, partitioning to create thirds is clearly easier than partitioning to create fifths. The emphasis at Level 2 is not on the student being able to partition into fifths and thirds but rather being able to verify that particular partitions represent fifths and thirds. Students can be provided with strips of paper partitioned as follows and asked to determine the indicated partitions as fractions of the whole.
Students at Level 2 verify continuous (and discrete) linear arrangements have been partitioned into thirds or fifths by checking the equality and number of parts forming the whole. Similarly, students could be given a short strip of paper (e.g. 1/3 of a longer strip of paper) and asked what fraction of the longer un-partitioned strip they have. This form of learning opportunity provides a way of moving beyond a focus solely on partitioning to teaching strategies that support students’ development of measurement iterating operations.
When iterating a fraction part such as one-third beyond the whole, the student re-forms the equal whole. Some students consistently regard an improper fraction produced via iteration of a unit fraction as a new whole (Tzur, 1999). That is, they think of 1/4 iterated five times as 5/5 and each part is considered as being transformed into 1/5. This belief could be attributed to students failing to reform the iterated four-fourths into the equivalent unit whole.
Five-quarters recast as five-fifths
Even when successfully creating seven-fifths of a drawing of a chocolate bar, some students view the resulting pieces as not being fifths but rather sevenths — “they turned into seven pieces instead of five pieces” (Hackenberg, 2007, p. 33). This reorganisation of iterated fraction units, recognising when the whole has been formed, is necessary to make the transition from an additive iteration of units to a multiplicative association between parts of an equal-whole.
At level 4 the student can coordinate composition of partitioning. For example, given one-half and asked to create one-sixth of a whole, the student finds one-third of one-half. This requires coordinating units at three levels to move between equivalent fraction forms.
Coordinating units at three levels with proper fractions
Moving between equivalent fraction forms can also include improper fractions (Hackenberg, 2007). For example, conceiving of 4/3 as an improper fraction means conceiving of it as a unit of 4 units, any of which can be iterated 3 times to produce another unit (the whole), a three-levels-of-units structure. One level is 4/3 as a unit, another level is the whole and the final level of units is one-third. Dealing with equivalent proper fractions also requires operating across three levels of units.
The following drawing shows the use of units at three levels: the four-thirds, the one-whole and the thirds.
Drawing four-thirds and showing coordination of units at three levels
Coordinating units at three levels is also used with composition of partitioning in the student’s answer to the following problem.
Sue and Michele share a two-finger Kit Kat. Just after Sue breaks her Kit Kat finger in half Alice comes along. How can Alice be given a fair share of the two Kit Kat fingers? Explain your answer.
Coordinating composition of partitioning and units at 3 levels
The explanation shows an understanding of the multiplicative relationship between the fractional value 1/3, the number of pieces and the whole. Further, each of the halves of the broken Kit Kat finger is divided into thirds to produce sixths (composition of partitioning).The coordination of units at three levels is evident in the drawings (halves, sixths and thirds of the equal wholes).
Unit coordination at different levels can also be seen in working with equivalent fractions displayed in regional models using the same equal whole.
Equivalent regional models
At this level, the student identifies the need to have equal wholes to compare fractional parts. They can also use fractions as numbers (unit-less quantities), i.e. 1/3 > 1/4. For example, in determining the relative size of fractions such as 1/3 and 1/6, care is taken in representing the two fractions with equal wholes (unlike the following response).
Comparing the size of 1/3 and 1/6 without referencing equal wholes.
In this student’s response, no attempt has been made to use equal wholes when comparing 1/3 and 1/6. It is also clear that for this student the fraction notation does not link to regional models of fractions. At this level the student is aware of the need for the fixed unit whole to compare quantity fractions. Coordinating units linked with the idea of a universal equal whole, is also important in addressing the distinct problem of fractions having multiple representations of the one quantity (1/3 = 2/6 = 3/9) within the same representational system.
As Lamon (1999, p. 22) has suggested, the hardest part for some students is understanding that “what looks like the same amount might actually be represented by different numbers.” The notational equivalence of fractions is implicitly dependent on the existence of a universal one, a whole that is always the same size.