MEDICIONES CON LA TÉCNICA P

The major interpretations students have in relation to the use of letters identified in this study confirm the findings of previous studies are: (1) each letter has a unique value or variables cannot take on multiple values; (2) letters stand for objects rather than numbers; (3) letters have no meaning; (4) a letter standing alone equals 1; and (5) letters are sequential and represent numerical position in the alphabet. The misinterpretations of the use of literal symbols as variables in algebra are briefly discussed with reference to the results inTable 5.

4.2.1.1 Different letters have different values

Analyses of the algebraic diagnostic test items and interviews revealed that students misinterpreted that every different letter must stand for a different number. They believed that different letters within an equation cannot take on the same numerical value consistent with findings of Stephens (2005), thinking that letters always have

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one specific value. For example, some students believe that when the literal symbol changes, then the value that it represents also changes. They are unwilling to accept that different symbols could stand for the same value. This view, however, is applicable to natural numbers, where each number has a unique symbolic representation and where different symbols stand for different numbers. This belief was clearly exhibited when students were asked when is ′ ? (Item 10). Many responded incorrectly. In fact, 24 out of 78 students answered never by selecting response ‘B’ (refer to Table 5). The results indicated that about 31% of students believed that each letter has a unique value. When Sam (S3) was interviewed and asked to explain his rationale, he made the claim that b and z cannot be the same answer in the following excerpt: (Interviewer: Teacher (T))

T: When is ? What you you think? S3: I think they are never the same.

T: Why do you think that?

S3: No, … It’s not that, (pause) ‘cause b can’t be equal to z. T: Why?

S3: Because they’re different letters.

It was clear that Sam (S3) (one of the 24 students) experienced difficulties in understanding and using variables as general number misinterpreting that every different letter must stand for a different number.

4.2.1.2 Letters represent labels for objects rather than numbers

The label or object misconception has been researched heavily since Kuchemann in the 1970s. For example, often students will interpret 3a as 3 apples instead of 3 times the number of apples. They perceived letters as representing objects rather then numbers. A very small proportion of the students, only 2 out of 78 students (from Item 5 in Table 5), 6 out of 78 (from Item 11) and about 8% of the students (Item 12) seemed to have this misconception. Further evidence from student’s (Pete: S4) interview confirms this misconception. Pete was asked what answer he selected for this question (Item 5) “Plums cost 8 cents each and bananas cost 5 cents each. If ‘p’

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stands for the number of plums and ‘b’ stands for ‘the number of bananas’ bought, what does 8p + 5b stand for?”

T: What answer did you select?

S4: I chose C. [The answer for C is 8 plums and 5 bananas]. T: Why did you choose that?

S4: Well, it’s just that it would have cents on it if it was the cost … it would equal so many cents. Yeah, I still think it’s 8 plums and 5 bananas, ... ‘p’ is for plums and ‘b’ is for bananas.

By selecting response ‘C’ for Item 5, Pete apparently perceived letters as labels for objects. Very often during the course, a very small proportion of the students make this error. They see a letter as standing for an object, or acting as a label, in which the letter, rather than clearly being a placeholder for a number, is regarded as being an object (Kuchemann, 1981). In this case, the letter is manipulated without being evaluated. When students view variables as objects, they are often used as labels for something. Variables are often used in geometry for points and lines. Moreover, variables are used in arithmetic to refer to units, such as ‘m’ for meters and ‘c’ for cents. When students enter algebra, they are suddenly referring to ‘m’ as the number of meters and ‘c’ as the number of cents.

4.2.1.3 Letters have no meaning

Students simply believe letters have no meaning in the realm of numeracy and that they belong to the literal realm (Perso, 1991). Sometimes, they would simply disregard the letters and solve the problem without taking them into account. Surprisingly, there were 17 out of 78 (students) or approximately 22% of the students (all from the low-achieving class) believed that letters have no meaning in mathematics by selecting ‘nothing’ (response ‘D’) to test item (item 11) ‘In the expression 5, ′ stands for’ (refer to Table 5).

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4.2.1.4 A letter standing alone equals 1

Mathematically speaking, the variable does not have a specific sign of its own. The values of a variable are determined when specific numbers are substituted for the literal symbol. Variables can stand for either positive or negative numbers (Vlassis, 2004). For example, the variable which is represented by the variable ‘x’ can stand for positive and also for negative numbers: ‘ ’ can stand for positive or negative

numbers as well (this happens because– 5 5 . When students are introduced to negative numbers they learn that the presence of the negative sign means ‘negative value’. Therefore, students would tend to interpret ‘x’ to stand for positive number and ‘ ’ to stand for negative numbers.

Besides, students also saw a one-to-one correspondence between letters and numbers resulting in the misconception that x standing alone must be equal to 1. Results from the test (item 11) indicated that 26 out of 78 students who appeared to have this misconception. Approximately 33% of the students responded by selecting ‘1’ (i.e., response ‘A’) to the question what does ′ stands for in the expression 5? (refer to Table 5).

4.2.1.5 Letters are sequential and represent numerical position in the

alphabet

One of the problems that some students have in what Warren (2003) discovered was the assigning of numerical values to letters according to their rank in the alphabet. When students do this, they will often assume that the variable ‘a’ is equal to 1, with the variable ‘b’ being equal to 2, and so on. Findings of prior studies on students’ difficulties with the use of the literal symbols in algebra are consistent with this view. There is evidence that some students associate literal symbols with natural numbers, in the sense that they respond as if there is a correspondence between the linear ordering of the alphabet and that of the natural number system. By examining the transcript of interview made with respect to this question, it appeared that the student used the letters as labels. For example Ramiah (S8) assigned a numerical value 5 to

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the literal symbol ‘ ’ to the following question (Item 1) “If 7, then ?” She failed to realise that the meaning or value of a variable is not dependent on the letter used as evident in the following excerpt:

T: If 7, then what? S8: I think it’s 12.

T: Why?

S8: Well, if 7, a could equal 3, b could equal 4 and c could equal 5, in a sequence.

T: Why do you think it would be a sequence, because they are alphabetical? S8: Yeah, that’s it, 3 4 5 12.

This discourse suggests that Ramiah (S8) assigned a numerical value 5 to the letter to get the answer 12. She is one of the four (about 5% in total) of the students that associated the letter ‘ ’ with the natural number 5 by selecting response ‘B’ for Item 1 (Table 5).

In sum, results showed that students had difficulty with letter usage in algebra and persistently misinterpreted letters as (1) representing different values, that is, different letters have different values; (2) abbreviated words or labels for objects; (3) having no meaning or ignoring letters; (4) standing alone equals 1; and (5) sequential and representing numerical position in the alphabet. Next, category 2 misconceptions related to the manipulation of operator symbols in algebra are discussed.