Número de Articulos por Repositorio
CASUÍSTICA FRAUDE FALSO POSITIVO TOTAL
In this thesis I have used the notion literal symbol instead of the notion ‘variable’ (see the end of section 4.1). No example task, assigned task, or part of the exposition in the textbook offered opportunities to reason about the meaning or use of literal symbols in algebra. Some few tasks, howev- er, in the algebra test, were given in order to test if students accepted let-
ters as substitutes for general numbers, that different letters could take on the same value, and if students were able to differentiate between the val- ue or number of an object and the object itself (see section 4.1).
The results from the test tasks show that all but one student seemed to accept literal symbols representing number values. In three tasks asking students to pick out expressions representing a certain value of a general number, item 3, 40 % solved all three tasks correctly. One student did not respond, and four students regarded inequivalent expressions to be equiv- alent. The others picked out only one of two correct expressions, although in the text it was expressed that they should pick out “the or those expres- sions”. Taking into account that some students might have been satisfied by picking out the first correct expression in the list, it might be inferred that at least 20 % had some problems with equivalent expressions in addi- tion to the 16 % equalling inequivalent expressions. In item 4 in the alge- bra test; the ‘student-professor problem’ (Clement, Lockhead, & Monk, 1981; MacGregor & Stacey, 1997) was given:
In a school there are 10 students to each teacher. 18 Which of the expression(s) represent the correct relation?
L=number of teachers, E = number of students. Mark the, or those correct expres- sion(s).
10L=E 10E=L L=10E E=10L 10L+E 11E
76 % picked out two expressions (60 %, though, picked out expressions representing the opposite relation). This strengthens the interpretation that some of the students had problems which expressions were equivalent in item 3. In the spring though, the proportion solving all three tasks correct- ly in item 3, had risen to 67 %; a considerable improvement (see appendix 9.1, and the diagram in appendix 6.1)
Another task similar to a task in the CSMS project (Küchemann, 1981) asked students to consider if a statement is always true, never true or true for a special case. The result is shown in the table underneath.
Table 8-1: Algebra test, item 6 with results (%) and alternative answers Task 6a) Correct Alternatives No response x + y + z = x + p + z
This:
is always true is never true could be true, if ……...
48 Never true 48
4
48 % of the students regarded correctly the equality to be true if y was equal to p. 48 % of the students regarded the equality to never be true. This means that nearly half the students did not seem to regard the two letter symbols to be able to take on the same value. In the spring, nearly 70 % chose the correct alternative and gave a correct explanation, which is an indication of a positive development. They had then had the experi- ence of working with functions and had seen that two different letter sym- bols could take on the same value, which might be one reason for the bet- ter result.
An important ability according to Küchemann (ibid) is to distinguish between the objects themselves and the numbers/values of objects. One task similar to one in the CSMS study was chosen to check this:
Table 8-2: Algebra test item 7 with results (%) and alternative solutions Task 7 Correct Alternatives
One cake costs c kroner. One sandwich costs s kroner.
I buy 3 cakes and 4 sandwiches: What does the expression 3c + 4s mean?
40 Expression representing numbers of objects (20%)
No response or had stopped in the mid- dle of the solution process (36 %) Letters given specific values (4%)
In this item, 40 % or ten students gave an acceptable interpretation of the expression. Five students (20 %) interpreted clearly the expression to be the amount of sandwiches and cakes. They did not differentiate between the price of the cakes/sandwiches and the objects themselves (see section 4.1). The choice of the letters in the task makes it probably more likely to happen since students let c represent the object cake and s the object
sandwich19. One student answered by assigning values to the letters. In the
spring, the result was not better; then only nine students (38 %) stated cor- rectly that the expression was the total price.
The interviews confirmed the problems, but when given a similar ex- ample in which they had to convert text into mathematical notation, it seemed to be easier. However, when led back to the initial problem, it still was a source of difficulty to differentiate between the price (value) of the object and the object itself. In the lessons, students solved no similar tasks.
In the CSMS-project (Küchemann, 1981; Booth, 1984) and in the KIM-project (Brekke, 2005) it was reported that some students conjoined terms. This was rarely seen during observation, and only by two students on two occasions; an indication that all accepted the lack of closure.
The problem to differentiate between the number or value of objects and the objects themselves, might cause problems for some students in transforming word problems into algebraic equations; one student ex- pressed this explicitly when converting text into equations when working with straight lines. The problems of deciding if expressions are equivalent or not, might also create difficulties when solving equations, and when reducing expressions to simpler forms. It was though evident that a larger group of students succeeded in picking out equivalent expressions in the spring than in the autumn.
No example or assigned task was offered for students to discuss or compare expressions, or to interpret the letter symbols. The concept of equivalence was not mentioned in the textbook. This means that there was no new opportunity given (Hiebert & Grouws, 2007; Kilpatrick, et al., 2001) for the students to develop a proper concept image (Tall & Vinner, 1981) of the concept of equivalence, which is the basis for all manipula- tive work with algebraic expressions (Kieran, 2007).