Elementos que otorgan identidad a la exposición
EXPERTOS ENTREVISTADOS ALFONSO CASTRILLÓN
From early school years students are familiar with carrying out the com- putations from left to right, however, addition and multiplication are commutative operations, and in strings of numbers within those fields it is no problem to compute also from right to left. When introduced to the op- erations of subtraction and division this switch is no longer a possibility. And in problems including multiple operations, the students have to re- flect on which operations have to be carried out first.
Herscovics and Linchevski (1994) had experienced as Vlassis (2004) that students calculated from right to left. In Herscovics and Linchevski’s study 22 grade 7 students were interviewed about equation solving. All were from the same class in a school in Montreal and none of them had followed algebra courses. During the interviews they experienced that students when solving the equation 364 = 796 – n, first transformed the equation to be n – 796 = 364. They assumed the reason was that the stu- dents had not developed a relational conception of the equal sign, and commented that if the signs had been positive, it would not have mattered if the direction was changed.
10 The notion ‘detachment of a term from the indicated operation’ is taken from Linchevski
They also found ‘bracket reasoning’ (Vlassis, 2004) (see the section above). In Herscovics and Linchevski’s (1994) terminology this was called ‘detachment of a term from the indicated operation’.
Another problem occurred when students performed addition before multiplication. A large proportion, nearly 30 %, of the students failed in the equation 6 9 n 60 because they performed the addition before multiplication. However, in similar tasks in which there was a minus sign instead of the plus sign, the students performed the operations in the cor- rect order. This was explained to be caused by the students’ view of the minus sign as a splitting factor.
In a follow up study Linchevski and Livneh (1999) interviewed 53 students at the end of grade 6. In this study the tasks were designed with the same structure as in the equations, but now as pure numerical strings; no letters included. These students had been taught about order of opera- tions.
The authors found the same problems as in the work with equations. In the expression 5 + 6 × 10 = ?, the task was performed by adding 5 and 6 before the multiplication. When there appeared to be a minus sign such as in the task: 17 - 3 × 5=, this happened for only three students, the others correctly carried out the multiplication before the subtraction. This may be due to an interpretation of the minus sign as a splitting factor resulting in bracket reasoning (Vlassis, 2004) as seen above. However, they (Linchev- ski & Livneh, 1999) also found that when there were both division and multiplication in the expression ( 24 : 3 2 ), the multiplication was per- formed first, and in tasks with both addition and subtraction ( 27 5 3 ) the addition was performed first.
Linchevski and Livneh (ibid) interpret these errors to be caused by students generalising the rules for order of operations incorrectly. They had learned that the order of operations is given as: brackets first, then multiplication, division, and then addition and subtraction. The research- ers suggest that one reason could be that students interpret the rule to mean that addition and multiplication were on the same level, and there- fore it did not matter which was performed first, and that the same was the case with division and subtraction. One could choose what was most con- venient. Another suggestion was that the students had interpreted the rule to mean that addition took precedence over subtraction and multiplication over division.
However, the task with both subtraction and addition ( 27 5 3 ), is similar to those referred to earlier in which the phenomenon was called the ‘detachment of intended operation’. For Vlassis (2004) when she asked students about it, they explained that the minus sign was seen as a
splitting sign; as a bracket, and what comes after, has to be operated on first. It might be that students have the same view of the division sign.
Linchevski and Livneh (1999) do not consider such an interpretation here, but later on when reporting other problems in the same article not involving order of operations, they use the examples 24 : 3 2 and
27 5 3 to make visible what they call detachment of the term from its indicated operation. Some students had then even put brackets around 5+3 and 3∙2.
It might be that the authors do not see this bracket reasoning as a rea- son for the problem, but as a description of the phenomenon only.
Linchevski and Livneh (1999) asked the students in the interviews if there were other ways of solving the problems, showing them alternative student solutions. Many students changed their minds, also from correct to incorrect, some kept to their solution. Others accepted both solutions, and meant that one task could give other correct solutions; it depended on the rules.
Linchevski and Livneh (ibid) used the findings to create an interven- tion program “teaching arithmetic for algebraic purposes” (Linchevski & Livneh, 2007). The aim was to help students to develop structure sense. The results indicated that students who were assumed to have low scores in algebra, progressed in that they avoided making the mistakes reported in the other studies. It was also observed that the other students improved from the instruction program.
Another problem related to order of operations occurs when powers are included in strings of numbers or in algebraic expressions. In the ex- ample 2 3 2some students tend to multiply the base by two before execut-
ing the exponentiation.
In higher grades there have been studies about order of operations (Lee & Messner, 2000). College students were reported to discuss the meaning of 32; the negative number with an exponent without grouping.
Some argued that 32 9 rather than 32 9. The argument they gave
was that the minus sign should be considered before the exponent.
4.5 Powers – base and exponent
Powers are mentioned in relation to order of operation in the foregoing section. However, the concept of power and what it means for the stu- dents, has been in focus for some studies (Pitta-Pantazi, Christou, & Zachariades, 2007; Weber, 2002).
Traditionally students have been introduced to the definition of powers as repeated multiplication of the base. This definition is meaningful as long as the exponents are natural numbers, but can cause troubles when
the powers are more complicated with negative exponents or with frac- tions as exponents (Weber, 2002).
For this thesis, the basic definition holds for most occasions, however, the students are introduced to scientific notation in which they have to ex- pand their conceptions of exponents from natural numbers to include all numbers within the set of integers, ℤ.
In a study (Cangelosi, Madrid, Cooper, Olson, & Hartter, 2013) identi- fying persistent errors students make when working with powers, and teasing out the reason for those errors, the researchers tested 904 students at a university and then interviewed 18 of them.
Students were asked to decide if some powers would be positive or negative and also to simplify them. Two persistent errors were found. One was what they called the ‘sticky sign’. The expression was 932. When
calculating the power, 13 out of 18 treated the expression as if it was in an imaginary bracket, their conception was that the minus sign was ‘stuck’ to the number 9, whether there was a bracket or not. The authors claim this is caused by a lack of a conception of the ‘additive inverse’ or as Vlassis (2008) labelled it; the opposite number.
According to the authors, the other error was due to a lack of the con- cept of the multiplicative inverse. The power was 23and 10 out of 18
failed. Alternative solutions were 213, 23,
3 / 1 2
,
1 / 3 2 , and 3 2. Some stu- dents’ conception of powers with negative exponents was that something should be flipped. However, what to flip and why, was not clear to them. They talked about flipping; not to find the multiplicative inverse of 2³.The authors recommend that textbook authors and teachers should in- corporate the additive and multiplicative inverses, when introducing stu- dents to the general laws for addition and multiplication.
One error, somewhat related to the ‘sticky sign’, was that some stu- dents were reported to equal ab and 2 ( )ab (Seng, 2010). 2
Errors occurring when calculating with powers are often caused by in- terferences from other symbol systems according to MacGregor and Stacey (1997). One example is the tendency shown by some students to mix exponentiation and multiplication. MacGregor and Stacey (ibid) refer to some students’ conceptions of exponents as an instruction to multiply, which causes them to write the multiplication as a power with the multi- plicator as the exponent. This confusion between what was ‘met before’ and new learning is also reported by de Lima and Tall (2006a), to cause problems when working with powers. This might cause students to multi- ply the base by the exponent.