What Kinds of Numbers Do Students Assign to Literal Symbols? Aspects of the Transition from Arithmetic to Algebra

Who can escape the natural number bias in rational number tasks? A study involving students and experts

“…The results from this experiment are in line with previous studies that have shown a natural number bias in students when judging the effects of arithmetic operations (Van Hoof et al ., ) and when interpreting literal symbols (Christou & Vosniadou, ; Van Dooren et al ., ). Thus, the most likely explanation for our finding is that the eighth‐grade students made use of a heuristic about the effects of operations (‘multiplication makes bigger, division makes smaller’) and/or applied a substitution strategy (substitute the variable x by concrete numbers) to solve the inequality problems.…”

“…For example, one could present the same algebraic expression problems without explicitly indicating the domain of the variable x . This mode of presentation would allow testing whether experts – just like school students and educated adults (Christou & Vosniadou, ; Van Dooren et al ., ; Van Hoof et al ., ) – show the tendency to spontaneously associate x with natural numbers. Further studies could make use of such a manipulation to investigate how explicit cues in rational number tasks influence the occurrence of the natural number bias.…”

“…When interpreting such symbols, students spontaneously tend to think only of natural numbers as standing for these symbols (e.g., that x can be 1, 2, 3, etc. ), even if they can stand for any real number (e.g., x can also be 1/2 or the square root of 2; Christou & Vosniadou, ; Van Dooren, Christou, & Vamvakoussi, ; Van Hoof et al ., ). In a series of experiments, Christou and Vosniadou () asked students in grades eight and nine to indicate possible values of an algebraic expression.…”

“…), even if they can stand for any real number (e.g., x can also be 1/2 or the square root of 2; Christou & Vosniadou, ; Van Dooren, Christou, & Vamvakoussi, ; Van Hoof et al ., ). In a series of experiments, Christou and Vosniadou () asked students in grades eight and nine to indicate possible values of an algebraic expression. They found that the large majority of students chose answers that can occur if the symbols stand for natural numbers only.…”

‘Negative of my money, positive of her money’: secondary students’ ways of relating equations to a debt context

“…In a series of experiments, Christou and Vosniadou [30] found that secondary students aged 12.5-14.5 years had a 'natural number bias' such that they tended to interpret symbols for real variables as representing natural numbers. The authors identified this bias as an important impediment in students' transition from arithmetic to algebra.…”