In this section I will present some research on the concept of fraction as a rational number. In the next sections research on students work with frac- tions will be presented. Then research on simplification of algebraic ra- tional expressions will be reported.
Fractions are often experienced by students as one of the stumbling blocks in mathematics. The concept of fraction is complex, and according to Kieren it includes five different sub-constructs which represent five dif- ferent “rational number thinking patterns” (Kieren, 1980, p. 134). These sub-constructs are: part-whole, quotients, measures, ratio and operators. To form the full rational number concept, the learner must control these different constructs. However, too often students have learned algorithms and have not been offered opportunities to go in depth into the concept of rational numbers. A well-known example of this is Benny (Erlwanger, 1973) who is enjoying his work with fractions, but, is using different rules for different calculations with fractions.
Kieren (1980) emphasises that the different thinking patterns are not mathematically independent, nor are they psychologically independent. They are closely related, but represent different ways of thinking. Accord- ing to Kieren (ibid), the part-whole sub-construct is historically the main basis for students’ development of the fraction concept. This sub-
construct is according to Marshall (1993) related to situations in which continuous or discrete quantities are divided into equal parts. The fraction
a
b is then seen as a comparison between the part and the whole. The nu- merator represents the actual number of units in the part, and the denomi- nator represents the number of units in which the whole is partitioned. In part-whole situations a b . Although, the units are made smaller or big- ger, the part-whole relation is conserved. This unitising gives basis for the understanding of equivalent fractions (Lamon, 1999).
The quotient sub-construct is closely related to the part-whole sub- construct in that it depends on partitioning. According to Marshall (1993) the difference is that in part-whole situations a and b in the fraction a
b yield the same object or thing. In quotient situations a and b represent dif- ferent things or objects. The situation represents fair sharing/division of a elements into b groups. One example is “How would you share three piz- zas among four friends? How much would each person receive? (ibid, p. 275).” Lamon (1999) comments that in such a context the fraction 3
4 has multiple meanings. It stands for the division 3 divided by 4, and is at the
same time the result of that division, the quotient. However, it might also be interpreted as a ratio, pizza per person. In a quotient situation there are no constraints regarding the size of the fraction.
According to Marshall (1993) the ratio sub-construct is representing a situation in which two quantities are related to each other, not represent- ing a partition of one object. Some number of one object is compared to a number of a second object. One example of such a situation is: “In her recipe, Susie adds 1 cup of sugar for every 3 cups of water. How much sugar should she add for 6 cups of water?” Lamon (1999) asserts that the situation describes ordered pairs and that it implies a proportion. In the fraction a
b, a change in a will cause a predictable change in b.
For students this means that they have to grasp that the relationship be- tween the two quantities is constant, and that a change in one of them im- plies a change in the second. Lamon emphasises that although students seem to have grasped the concept of equivalent fractions, it does not mean that they have a valid conception of the invariance property of ratios.
Marshall (1993) describes the measure situation to be a situation in which a fraction 1
b “is used repeatedly to determine a distance”. Accord- ing to her, the most frequent representation is the number line, alternative- ly the ruler. Lamon (1999) claims that: “It is unlikely that any other frac- tion interpretation can come close to the power of the number line for building number sense” (p. 228). In addition, Lamon asserts that students are scaffolded by fractions as measurement to develop understanding for operations on fractions. She refers to her studies which show that students who started learning fractions through the measurement sub-construct of rational numbers, had a good conception of the relative sizes of fractions and were more likely to discover errors when operating on fractions, than students who had started out from other sub-constructs.
The last subconstruct, fraction as operator, is linked to the function concept by Marshall (1993). The operator serves as a function machine, mapping some set or region onto another set or region. For students it is understood as shrinking and enlarging, contracting and expanding or mul- tiplying and dividing (Lamon, 1999). Lamon (ibid) uses the example of the operator 2
3. This operator instructs one to take 2
3 of something, which means to divide by 3 and multiply by 2, or it might be regarded as one single operation. The result will be the same no matter in which order the calculations are made.
When discussing the different sub-constructs and approaches in teach- ing fractions, Lamon concludes that the different interpretations do not give the same access to a deep conception of fractions or rational num- bers. She emphasises as does Kieren (1980) that the sub-constructs are intertwined and one construct can enhance another if the teacher is aware of the different interpretations.
Recent research has shown that there is a strong predictive relation be- tween fraction knowledge and later mathematical achievement (Bailey, Hoard, Nugent, & Geary, 2012; Siegler, Fazio, Bailey, & Zhou, 2013). Siegler et al. (ibid) applied two national longitudinal data sets; the ‘British Cohort Study’ from 1980 and 1986 and the ‘Panel Study of Income Dy- namics - Child development Supplement’ (PSID-CDS) from 1997 and 2002. In the first study 3677 children born in the United Kingdom during one week in 1970 formed the dataset. The latter dataset included 599 stu- dents from USA. The students in both datasets had been tested twice; first as ten years old and then as 15-17 years old. Analysis of the data showed that knowledge of fractions among students in high school correlated strongly with overall mathematics achievement, also with algebra. Knowledge of fraction and their overall mathematics achievement corre- lated stronger than algebra with overall mathematics achievement.
Across the dataset from both countries, elementary school children’s fraction knowledge and whole number division predict their mathematics performances in high school. This was the result even after controlling for IQ, reading abilities, working memory, family income and education, and knowledge about whole numbers. Bailey et al.’s (2012) study confirmed this result.
Wu (2001) suggests that fractions could be the gateway to algebra and the concept of variable, since the rules for operating on fractions can be generalised. He also claims that for students when meeting fractions, they are introduced for the first time to the world of abstractions (Wu, 2009). Both Wu (ibid) and Siegler et al. (Siegler, Thompson, & Schneider, 2011) propose that fractions should be dealt with as numbers located on the number line. This location on the number line is the property fractions have common with whole numbers. One advantage of applying the num- ber line according to Wu (2009), is that it is possible to present both prop- er and improper fractions in conceptually similar ways, and that equiva- lent fractions have the same location on the line. He proposes that defining “a fraction as a point on the number line is a refinement of, not a radical deviation from, the usual concept of a fraction as a ‘part of a whole”’ (ibid, p. 12). Siegler et al. (2011) proposed an integrated theory of numer- ical development which seems to be in accordance with the ideas in the work of Wu (2009). The main point is that “numerical development in-
volves coming to understand that all real numbers have magnitudes that can be ordered and assigned specific locations on number lines” (Siegler et al., 2011, p. 274). The researchers propose that knowledge of the mag- nitude of fractions would help students in algebra to consider if solution numbers are reasonable.
Although there has been a large amount of research reports on algebra and students’ problems in the topic, only a few have focused on the con- nection between students’ algebraic reasoning and their knowledge about fractions. However, researchers have claimed that proficiency in dealing with fractions is an important prerequisite in algebra learning e.g. (Brown & Quinn, 2007a, 2007b; Kilpatrick & Izsak, 2008; NMAP, 2008; Wu, 2001). Also researchers have reported that one problem in algebra learn- ing is that students have problems with fractions (Hoffer, Venkataraman, Hedberg, & Shagle, 2007).
4.6.1 Students’ problems with ordinary fractions
There is a long history of research on students’ work with fractions and on their conceptions of rational numbers (Eichelmann, Narciss, Schnaubert, & Melis, 2012). In upper secondary school it is expected that students should have control over and master operations on fractions. However, studies show that also students at higher levels struggle (Brown & Quinn, 2006, 2007a, 2007b). In upper secondary school when working with alge- bra, students are not asked to transform mixed numbers into improper fractions or opposite. Fractions larger than 1 mostly appear as improper fractions, however, the students have to solve tasks in which there are both whole numbers and fractions. It is also expected that they can trans- form fractions into whole numbers. This is not obvious for all students.
Eichelmann et al. (ibid) refer to (Shaw, Standiford, Klein, & Tatsuoka, 1982) who found that some junior high school students did not simplify fractions with equal numerator and denominator; they were not trans- formed into the number 1. This was seen at the end of adding fractions, and according to the author the reason might be that the students simply forgot to do it. It was also found that some regard fractions of type 1
a to be equal to a. No reason is given for this. The students were not asked to explain their reasoning.
Padberg (1986) found in his study of 861 students in grade 7 that one of the main problems for students was to work out tasks in which there were both fractions and whole numbers. The types of such tasks are:
, , , , : , :
a a a a a a
n n n n n n
b b b b b b
. There were two main categories of
n
n, or to treat the whole number as if it were a numerator with the same denominator as the closest fraction. The same was found in a study among 9 and 10 graders (Brown & Quinn, 2006). Padberg (1986) claimed that the students just followed rules; formal or invented. In many of these tasks it should have been simple for students to check if their results were ap- propriate, but they did not.
When adding and subtracting fractions with both equal and unequal denominators, most students succeeded in Padberg’s study (1986). In Brown and Quinn’s study (2006) of 9 and 10 graders, however, 48% did not find the correct sum of 5 3
12 8 In both studies the most common error was to treat the numerators and the denominators as distinct whole num- bers and follow the ‘rule’ a c a c
b d b d
. This ‘rule’ was explicitly ex- pressed by some students in Padberg’s study (1986). It is seen as a logical extension of the natural number system. Padberg found that this phenom- enon increased if the addition task followed after a multiplication task, which means that some students generalised the structure in the algorithm of fraction multiplication to fraction addition.
Quinn and Brown asked the students in grade 9 and 10 to find the product of 1 and 1
2 4. Only 42 % found the correct product. 26 % misap- plied the algorithm for multiplying fractions. Some of them (nearly 6% of all in the study) added the numerators and multiplied the denominators. Others (the same amount) had found the least common denominator be- fore executing the multiplication. The main error found by Wittmann (2013) when testing 428 German students in grade 6 and 7, was of this type: a c a c
b b b
. It was more likely that the tasks were correctly solved when the denominators were unequal. The same was found in the earlier study worked out by Padberg (1986). He reported that this error was more likely to occur if addition/subtraction tasks were solved before the multi- plication tasks. Padberg also found that some students solved multiplica- tion tasks in this way: a c a c
b b b b
. They multiplied the numerator and added the denominators. One reason, he suggested, might be that the stu- dents interpreted b multiplied by b to be equal to 2b. When the denomina- tors were unequal, some students tended to expand one or both to reach to
equal denominators. After this expansion some followed the erroneous path as with the equal denominators (Padberg, 1986; Wittmann, 2013).
In Padberg’s study students were asked to respond to the question: “Why do we multiply fractions in exactly this way?” None of the students in the study provided any answer. It seems to be that the rule was just memorised.
In division tasks Padberg (ibid) found that the same structure was used as for multiplication: a c: a c:
b b b . In his study 20 % did this at least once, while 4 % did it in all division tasks with equal denominators. One of the tasks was 4 2:
5 5. The researcher had expected that the students could solve this problem intuitively, but that was not the case. They tried to rely on rules. But it was obvious from the way the rule for fraction division was applied, that the rule gave no meaning for many of the students. Some students inverted both fractions, and some inverted the first one. Also for division; none of 681 students could give any reason for the rule and only few could formulate the rule in words. From the article, however, it has not been stated if the students have been taught the reason for the rules.
Brown and Quinn (2006) asked students to reduce the fraction 24 36 to the lowest factors. The result was that 9 % could not demonstrate any method for reducing it, and 9 % did not reduce it to the lowest factors. 73% of the students reduced it correctly. These were 9th and 10th graders. Padberg (1986) reports that it was more likely that the students simplified fractions if simplification was the aim of the task. More often it happened that result or solution numbers were not reduced. The reason was suggest- ed to be that the students forgot to do it, because simplification was not in focus, and their feeling was that the task was done.
In their study, Brown and Quinn (2006) found that some (4 %) solved the addition task in this way: 5 3 5 3 4 5 7
12 8 12 8 4 12 12
. These stu-
dents expanded the fraction by addition, not seeing the fraction as a ratio; an indication that those students lack the conception of equivalent frac- tions.
When fraction tasks were formulated as word problems many students had problems. As with decimal numbers (Fischbein, Deri, Nello, & Ma- rino, 1985) students had problems choosing the correct operation (Brown & Quinn, 2006; Wartha, 2009).
In a longitudinal study in Germany Wartha (ibid) focused upon stu- dent’s understanding of fractions, and the development of their concep-
tions from grade 5 until the end of grade 7. A pencil and paper test was administered among 2000 students from these three school years, then 36 students were interviewed on the basis of their results on the tests. Wartha (2009) gave students word problems to solve. The students understood that the amounts should be smaller and many students chose division or subtraction, this happened both with unit fractions and other fractions. The researcher claims that the reason for this is the over generalisation of the properties of division and subtraction within the set of natural num- bers, where those operations ‘make smaller’. The same was found in word problems in Brown and Quinn’ study (2006).
Another frequent error found, was that some students avoided ordinary fractions and converted the fraction into decimal numbers, which made the solution process even more complicated.
Wartha (2009) also found that many students do not regard fractions as numbers, and this causes problems when trying to compare fractions. The study confirmed that although many students are able to apply the algorithmic rules for operations on fractions, it does not mean that they have a well-developed concept image of fractions as rational numbers. Wittmann (2013) when studying the consistency of error patterns in operations on fractions, found four distinct groups of students.
The first group consisted of students who solve all fractions tasks with one operation resulting in a correct result. Out of 428 students in seventh grade between 25 % and 33 % belonged to this group. Within this group there are two sub-groups. One contains the students who solve the prob- lems in a flexible way which means that they take the number into consid- eration, and execute mental computations when that is convenient. The other sub-group represents students who apply one approach, whether it is the most appropriate method or not.
The second main group represents students who consistently show the main error patterns. Wittmann (ibid) asserts that those students might have “internalized an incorrect procedure while performing automation exer- cises”. The students in these two first main groups represent students who are consistent in their work.
The third group consists of students who mainly proceed well, but oc- casionally make mistakes. The researchers assume that the reason for the inappropriate approaches might be due to careless mistakes caused by in- tuitive forms of error patterns, by numbers fitting into an “error pattern”, or caused by the sequencing of the tasks.
In group four are students who are not consistent at all. Wittmann sug- gests that the reason might be that the students’ choices of approaches are dominated by the given numbers.
The studies reported above indicate that although students meet frac- tions in early school years, some will still have problems and have a rather limited conception of rational numbers even when entering upper second- ary school. In upper secondary school, the algorithmic rules are just reca- pitulated, and students are expected to move on to more advanced mathe- matics. Wittmann’s study about consistency in carrying out fraction tasks, shows that at least in German seventh grade there is a group of students that is consistent in applying error patterns, and one group in which the students have no clear approach at all. Both groups need help and guid- ance in applying algorithms, but even more so to gain a concept image of rational numbers that can help them to see the reason behind the rules. 4.6.2 Students’ problems with algebraic fractions
There are many studies showing that students have problems simplifying algebraic fractions or strings of fractions. The older studies are focused on typical errors found in testing, not so much on students reasoning when solving the tasks. In more recent studies students have been interviewed on the basis of tasks, or they have been asked to reflect on the solution process. Some studies in this area will be referred to in this section.