9 Fractions – exposition and students’ work In the textbook fractions are in focus for sub-chapter 1.2: Calculating with
9.1 Factorisation and simplification
The technique and formal writing of expansion is exemplified in the text- book. Simplification of fractions is written in two different ways. First, it is explained as dividing the numerator and denominator by the same num- ber:15 15: 5 3
25 25: 5 . Then it is said that this operation is mostly written: 5
15 25 5 3 5 5 3 5
, and the rule is given: “Before simplifying a fraction, one has to factorise numerator and denominator. Then numerator and denomi- nator must be divided by the same factors” (p. 26). In addition, it is em-
phasised that expansion and simplification do not alter the value of the fraction.
In the classroom, expansion was introduced by the example: 3 2 6 4 2 8
, and simplification was shown to be the opposite operation 6 : 2 3
8 : 2 4. In addition, this was written in MathType on the computer by cancelling the initial numerator and denominator, with the remaining factors written as superscripts: 63
84 3 4
. The focus was to find the factor in common for the numbers, 6 and 8, and perform mental factorisation. Most students fol- lowed this pattern of writing, shown in MathType.
Solving the first task, only two of 18 students, prime factorised the numbers 10 2 5 2
15 3 5 3
before cancelling equal factors. Five students made
the division visible; dividing both numerator and denominator by 5. The others followed the pattern the teacher had shown.
One frequent tendency, when the numbers involved were larger num- bers, was that the fractions were not reduced to their simplest form. One example is: 426
63090 6 90
. In this task 25 % of the students observed to solve the task, solved it this way. This tendency is in accordance with Brown and Quinn’s report (2006), and is even more frequent, when the results of addition/subtraction tasks should be simplified. Another tendency, report- ed by Shaw and colleagues (1982), but not so frequent, is to avoid simpli- fication of result fractions when numerator and denominator are equal. There is evidence of this in some students’ solutions.
In the classroom it was observed several times that students struggled to find common factors although the numbers were not that large. One such example is the episode with Glenn and Sven. They were observed discussing how it could be possible to go from 56
21 to 8
3. (They had found the correct answer in the textbook). They were challenged by the teacher to find the factors in each of the numbers 56 and 21.
T: 56 and 21 has one factor that goes into both numerator and denominator. If you write 21 as factors, what will it be?
Sven: Factors? Glenn: 7 times 3.
T: 7 times 3, yes, correct. And then you have to check. Either must 7 go into 56 or 3, this in order to simplify.
Sven: That… (he stops there) T: 56. Goes 3 or 7 into 56?
Glenn: 7 goes. T: Yes 7. Glenn: 7 times 8. T: 7 times 8, yes. Glenn: Yes.
T: Yes, and then you can simplify by seven, you see? Glenn: Should we just divide by 7?
T: Yes, when you simplify, you must divide by the common factor.
Here Sven seemed to be surprised by the word factors. Although the teacher had stressed the notion of factors and terms earlier, it seemed to mean nothing to Sven. However, when the teacher prompted them, Glenn seemed to grasp what factorisation is about, and he factorised both num- bers. However, he asked: “should we just divide by 7?” From this, it seemed that for him the problem was not only the numbers, he seemed to see no connection between simplification and division.
Some of the other students struggling with this task, found that they could use the calculator in the program TI-interactive.
Norwegian students are familiar with the use of hand-held calculators from the early grades (see section 2.3), and it was observed several times that they struggled to perform correct multiplication or division on num- bers without that tool. One example of this is from a lesson in November. Ronny is observed when he is going to divide 5 by negative 1:
Ronny: Let me see. How am I dividing 5 by minus 1 on this cell-phone? (He is struggling with his cell-phone. He has no calculator and the students were asked not to use the computer).
Glenn: Oh, 5 divided by minus 1? Five divided by… (Ronny waits and calls to Tore)
Ronny: Tore, what is the result of 5 divided by minus 1? Tore: That’s minus 1.
Ronny seems not to be satisfied by the answer. It is a session of group work and he walks around asking others the same question.
This episode illustrates how simple calculations could be experienced as difficult. It also illustrates how dependent the students are on the tech- nological tools. It might be that the minus sign caused him and the others to wonder (see section 8.3.4). However, at this level this is expected to be known. All students used their cell-phones before they became used to the calculator in the IT-program, and it might be that the habit of relying on these tools prevented them from developing a good number sense
(McIntosh, et al., 1992).
When binomials were included in the fractions, all students seemed to be unsure about what to do, and how to simplify. The exposition part in the textbook introducing this, is extremely short. It is stated that the same rules are valid for algebraic fractions as for fractions with only numbers, and when simplifying fractions with binomials, one has to factorise by
setting common factors outside a bracket. Afterwards one can simplify by dividing numerator and denominator by the same expression. It is said to be the opposite of expanding the bracket.
The example task: 3 6
6 9
a a
was applied in the classroom. The immediate
suggestion was to cancel the a’s, the literal symbols, reported to be one strategy in Guzmán, Martínez and Kieran’s study (2011).
Students also offered other not efficient suggestions, and after a while the teacher applied the number example 25 5
5
in order to create mean- ing; to help students realise that the operations they suggested, did not work.
It seemed as if the students accepted her argument, but when asked why their suggestions will not work, the students only pointed to the fact that the answer would be wrong, or that there is a minus sign in the nu- merator. The teacher clearly waited for the mathematical notions of ‘terms’ and ‘factors’. After many hints and a time of silence Jarl answers ‘factors’ to the question of what makes a product; an example of a funnel pattern (Voigt, 1995). The teacher had to bring in the notion of ‘term’ her- self, and to explain what it means.
By gesturing, the teacher made circles around each binomial on the whiteboard. She emphasised that both binomials had to be dealt with as entities, and from this she deduced that the terms within the binomials cannot be cancelled; one has to factorise.
Few students had suggestions to what could be done. One was to change the operation sign within the binomial, another to factorise only the first term. The impression from the classroom was that the students really struggled, and the teacher let them take their time.
Her next move was to write the expression 3(a asking what could 2) be done. Now many students responded, and they told her to execute the distributed multiplication; 3(a2) 3 a . The teacher directed the at-6 tention to the latter expression and asked them to compare it with the ini- tial task, before emphasising that the operation executed in the factorisa- tion is division.
Neither was to factorise the denominator 6a easy. To set the factor 9 3 outside a bracket, was agreed upon, but some responses involved sub- traction (Storer, 1956). When the factorisation was completed, the simpli- fication seemed to be a simple matter.
In the example task presenting division of fractions with binomials:
3 4
: 1 3 3
literal symbols after having flipped the last fraction and transformed it into the operation of multiplication 3 3 3
1 4
a a
. Tore seemed to have grasped the notions, factors and terms emphasised by the teacher earlier, and he applies them in his argumentation.
The teacher wrote both binomials in brackets, and emphasised that the whole brackets constitute factors, and had to be handled as such. Some students suggested to carry out the multiplication. As a response, the teacher asked for possibilities to simplify. Again there were suggestions to subtract. The process of factorising the binomials took a rather long time. The problem was to see brackets as factors, as entities.
There were only few tasks offered on the work plan to practice factori- sation and simplification of fractions with binomials; one task in which it was asked for simplification only, one multiplication task, and one divi- sion task. In addition, there was one task, where the result had to be fac- torised in order to be simplified.
It was clear that many students were unsure when solving the first as- signed task 4 8
5 10 a a
. It was the task with most frequent questions from stu- dents. Some just needed the teacher’s confirmation on what they were do- ing. Others were stuck and needed help to factorise and/or to be told once again what factorisation of binomials was about. For many the problem was to find the common factor in the two terms in the numerator. They divided by 2 instead of by 4 in the numerator, and could not cancel the brackets. Others had problems seeing what should be left in the bracket, they did not perform division.
Some students cancelled the literal symbols before they added the re- maining numbers, as was also suggested by some students in the plenary session. The structure of the task was such that they then came to the same solution as was written in the textbook. Ronny and some peers were ob- served solving the task in this way: 4 8 4
5 10 a a a 8 5 a 12 4 15 5 10 , arguing that
they had reached the correct solution. Ronny speaks for them. He argues that there is no need for factorisation:
Ronny: Why should we have to factorise this here, it’s just 5 times a plus 10. We can do it like this? (He points to his solution)
T: What do you mean by doing it this way? Ronny: But the answer is correct.
T: You can’t do like this. Ronny: But why not?
she applied in the plenary session: 25 5 5
. She repeats that to cancel the 5’s is not what is to be done.)
Ronny: Yes, but that’s why there was a minus?
(The teacher then changes the minus in the example to a plus sign). T: I can do like this.
Ronny: Okay.
T: Okay? (She waits for Ronny’s response. He watches the white board). Ronny: But I got the correct result.
The teacher emphasised that it was just by coincidence that his result and the result in the textbook were identical. She applied the number example once again and told him and his peers to factorise, pointing to the fact that the whole numerator is one factor. Ronny does not seem to agree, but the teacher tells him and the others “to do it the hard way; by factorisation”. From the example above it is clear that these students did not see the necessity of factorisation in order to simplify; they did not see the binomi- al as an object (Sfard, 1991). It also illustrates the view they had on the ‘correct answer’. Since their solution was in line with the answer in the textbook, they did not question their solution method. Ronny argues as if he does not regard the process as important.
Eli was observed when she worked on the same task. She had factor- ised the fraction, however, not correctly. Her last move is then to cancel term by term within the bracket: 1( a )2
5( a 2 1 5 ) .
Hall (2004) experienced this phenomenon in his study. He commented that he as a teacher would not have been aware of such a reasoning if he had not interviewed the students. Here it is visible because of the cancel- ling markers. It might though be that also other students were reasoning in the same way as Eli, although seemingly cancelling the whole bracket.
When asked what she would have done if the expression in the numer- ator was 3a + 2, she indicated that a request for simplification is that one of the brackets could be completely cancelled.
In the two other assigned tasks, the main problem was the factorisation or to see the need for factorisation. Not many students, 6 and 11, respec- tively (dataset 18 students) solved the tasks. Four of them did not succeed in either of them. Also for these two tasks it was possible to come to the same result as was written as correct in the textbook; performing partly cancelling. The tasks were told to be worked out at home.
In the class Ane and ine solving one of the tasks had written: 2 6 3 2( 3) 15 : 5 15 5 3 a a a a . Ane asked:
Ane: Now we have done this. What are we then going to… Ine: Eh… Is it correct?
T: It is perfect.
Ine: Can we… What are we going to do now? T: Yes what should we do?
Ine: Can we remove this and this? (She highlights a+3 in the numerator and in the denominator).
T: Yes, can’t we?
Ane: And the others like that?
T: Yes, and then you simplify there, yes.
It seems as if they are not sure it is correct, and will not cancel any factor before they have asked the teacher; they want her confirmation. The stu- dents tell that they like this type of task.
In students’ work plan there were no more opportunities to practice factorisation of binomials. It is clear from the observation that these tasks did not serve their purpose. All assigned tasks were of such a structure that the result would be the same either they were factorised, or if the stu- dents started their solutions by cancelling the literal symbols. Actually, students could work out the tasks without having grasped the important difference between factors and terms and carry on with their mathematical work without their erroneous concept images being challenged. Former studies have though showed that it is likely that some students would sim- plify in the same way as these students did (Guzmán, et al., 2011; Storer, 1956).
In another assigned task with another structure the result had to be simplified in order to reach the solution written in the textbook.
Ronny was observed when recognising that his answer, 3 3
6
a , differed
from that in the textbook. He was claiming that he had no clue about how to reach the desired result. He turns to his classmate Jarl:
Ronny: But, you can’t simplify 3a, there is no a underneath. Jarl: What do you want to simplify?
Ronny: I don’t know, but …yes, look here … No, I can’t simplify. Jarl: You have to factorise.
Ronny: Yes, I have to factorise. Jarl: It will be 3 and a -1
Ronny was stuck about how to come to the answer in the textbook. He thought at first that it was impossible to simplify the fraction since the lit- eral symbol was present only in the numerator, this was his only strategy. The result 1
2
a in the textbook, made him ask Jarl for help.
Half the students in the data-set of 18 students had solved this task. Three of them did not factorise the numerator, but two of them, Rakel and Tord, correctly divided all terms by 3. It might be that they know why this is functioning. They found at least a way to reach to the expected result.
Ruth in her solution seemed to perform some acrobatics in order to come to the desired result:
2 3 2 4 (3 ) 2 4 (6 2 ) 4 6 2 2 3 6 3 4 6 2 12 12 a a a a a a a a 6 a 1 122 1 2 a
This episode and Ronny’s solution are examples of how the answers in the textbook influenced students’ actions. Ronny asked for help and his peer instructed him. Ruth just searched for a way to adjust the result she had. The conclusion in Guzman et al.’s study (2011) about rational frac- tions and the use of CAS, was that when the result using CAS differed from students’ solutions, they had the opportunity to try to find another way to a correct result. The answers in the textbook can challenge stu- dents in the same way. It is, however, of crucial importance that the stu- dents are not left alone (ibid). They need the guidance of a teacher or peers who are more capable. In addition, they must be committed to learn (Entwistle & Peterson, 2004; Mellin-Olsen, 1981; Novak, 2002), not just to find the correct answer.
In September the students should solve a test task, dividing and simpli- fying rational fractions. The table below shows the test result.
Table 9-1: September test task 4c. Solutions (%). Dataset 27 students
3 15 2 10 : 6 8 a a % Alternative solutions Correct 19 Not finished 3 (A) 3 15 2 10 3 15 8 (24 120) :12 2 10 : 6 8 6 2 10 (12 60) :12 5 a a a a a a a a Partly can- celling 60 (B) 3 15 2 10 3 15 8 24 : 6 8 6 2 10 a a a a 12 a12012 12 12 a 6012 2 10 12 2 1 5 6 (C) 3 15 2: 10 8(3 15) 24 6 8 6(2 10) a a a a 2 a 1202 12 a 60 4 Other solutions 7 No response 11
All 24 students who solved the task, had flipped the last fraction, which means they know the algorithm for fraction division. Three students did not solve this task. Only five (19 %), had correctly factorised the binomi- als and simplified the rational expression. The others, except for two, had
multiplied the factors before trying to cancel/simplify. When simplifying rational expressions with binomials, it is crucial that students have a sense of the structure (Hoch & Dreyfus, 2004; Linchevski & Livneh, 1999), that they see the binomial as an object (Sfard & Linchevski, 1994b), and have strategic competence (Kilpatrick et al., 2001).
One student, (A) after multiplying the factors, he correctly divided all the terms by 12. He was not very successful, however, since he did not factorise the binomials. During seatwork, he was seen doing the same.
The solutions B and C were the most frequent solutions (60 %). In both of them, the students cancelled the literal symbols. Then the solu- tions differed. In solution B, the students divided all terms by 12. (This student has written the divisors as superscripts). In solution C the first two terms in both numerator and denominator were divided by 12, the two last terms were divided by 60. In B the remaining factor 1, was taken into ac- count. In C there should be a remaining factor 1 in each term in the de- nominator, these, however, just disappeared. Storer (1956) asserted that this type of error was probably caused by “striking without trace”. This was reported to be a recurring student error when working with numerical fractions (Eichelmann, et al., 2012). It therefore might be that they con- sider the simplification as an act of pure cancelling (Storer, 1956), forget- ting that it is division.
When comparing the solutions on the test with the solutions to the similar textbook tasks, only three students had correct solutions both in the textbook tasks and on the test. Four students solved the tasks correctly in the classroom, however, did not show competence in factorising bino- mials on the test, Ine and Ane who said they enjoyed this kind of task were in this group. Tore who in the classroom showed that he differentiat- ed between factors and terms, had not factorised any task. The same was the case with Ronny.
Selma solved the test task correctly, but solved just one of the text- book tasks related to factorising binomials. She was observed to struggle and asked the teacher for help. Here it seems to be that she has grasped