The Competition between Numbers and Structure: Why Expressions with Identical Algebraic Structures Trigger Different Interpretations
Linchevski, Liora, Livneh, Drora, Focus on Learning Problems in Mathematics
The common perception of school algebra as generalized arithmetic implies that algebraic structural rules are perceived as rules that draw their legitimization and meaning from rules that are valid in the world of numbers (Buxton, 1984; Davis, 1985; Smith, 1997). This widespread view has generated the search for a model that describes the relations between students' understanding of the number system and of the algebraic one (Collis, 1971; Lee and Wheeler, 1989; Linchevski and Herscovics, 1996b). In the context of the school curriculum, it has motivated a teaching approach that may be described as teaching arithmetic for "algebraic purposes" (Davis, 1985; Arcavi, 1994; English and Sharry, 1996; Milton, 1999). The proponents of this approach maintain that difficulties students experience with algebra originate in the lack of a suitable arithmetic foundation and claim that the failure of students to appreciate the rules of working with "letters" is largely due to their failure to understand the rules of workin g with numbers. They assume that understanding of the structural rules in arithmetic guarantees understanding of the corresponding parts in algebra (Kuchemann, 1981; Booth, 1984; Kieran, 1989; Lee and Wheeler, 1989; MacGregor, 1996; Milton, 1999).
This widespread view of school algebra has also motivated the ongoing research on students' "non-algebraic" behavior (structural "bugs") in numerical contexts. However, although in some cases there are grounds to suggest that certain "non-algebraic" behavior of students may be attributed to problems with arithmetic, by the same token it has also been suggested that the presence of numbers does not always make the task "easier" (Lins, 1990; Kieran and Sfard, 1999).
Non-algebraic behavior in a numerical context
The following describes one of many actual instances where students seem to be disregarding structural aspects of numerical expressions when solving numerical problems. These seventh-grade students (1) had been investigating in class, for quite some time, equivalent numerical expressions. They had been introduced to the conventions related to the order of operations, to the role of brackets and to the possibilities of "removing" and "inserting" brackets with and without changing the numerical expressions, such as the commutative, associative and distributive laws, and had manipulated numerical expressions using their newly acquired knowledge.
In one of the questions on a test, they were asked to evaluate the expression: 53-3x5+15. Rather than the expected answer of 53, Ron arrived at 23. His work showed the following calculations: 53-3x5+15=53-15+15=53-30=23. We assume that, in the first step, Ron followed the correct order of operations and multiplied 3 by 5. However, after inserting the result (15) back into the expression, he "detached" (Herscovics and Linchevski, 1994; Linchevski and Herscovics, 1996a; Linchevski and Livneh, 1999; Kirshner, 1989) the first 15 from the subtraction sign, put mental brackets around the two 15s and got 23 as an answer. Yet, in another exercise with the same algebraic structure: 46-8x3+7, Ron followed the correct order of operations and got the expected answer: 29. His written work showed: 46-8x3+7=46-24+7=22+7=29. Does Ron know the order of operations or not?
Other students in Ron's class also arrived at different interpretations of expressions with identical algebraic structures. For example, Rachel's answer for 160/(5x2), was l6; she multiplied 5 by 2 and then divided 160 by 10. Is she over-generalizing the order of operations, thinking that multiplication takes precedence over division? Yet, in another item on her exam paper, 48/8x4, Rachel did not start to evaluate the expression by first multiplying 8 by 4 and then dividing 48 by 32 (getting 3/2 as an answer), as she had done in the previous case. This time she correctly followed the order of operations; she divided 48 by 8 and then multiplied the obtained answer (6) by 4, and arrived at the correct answer (24). …