Arithmetic word problems describing discrete quantities: E.E.G evidence for the construction of a situation model

What we count dictates how we count: A tale of two encodings

We argue that what we count has a crucial impact on how we count, to the extent that even adults may have difficulty using elementary mathematical notions in concrete situations. Specifically, we investigate how the use of certain types of quantities (durations, heights, number of floors) may emphasize the ordinality of the numbers featured in a problem, whereas other quantities (collections, weights, prices) may emphasize the cardinality of the depicted numerical situations. We suggest that this distinction leads to the construction of one of two possible encodings, either a cardinal or an ordinal representation. This difference should, in turn, constrain the way we approach problems, influencing our mathematical reasoning in multiple activities. This hypothesis is tested in six experiments (N = 916), using different versions of multiple-strategy arithmetic word problems. We show that the distinction between cardinal and ordinal quantities predicts problem sorting (Experiment 1), perception of similarity between problems (Experiment 2), direct problem comparison (Experiment 3), choice of a solving algorithm (Experiment 4), problem solvability estimation (Experiment 5) and solution validity assessment (Experiment 6). The results provide converging clues shedding light into the fundamental importance of the cardinal versus ordinal distinction on adults' reasoning about numerical situations. Overall, we report multiple evidence that general, non-mathematical knowledge associated with the use of different quantities shapes adults' encoding, recoding and solving of mathematical word problems. The implications regarding mathematical cognition and theories of arithmetic problem solving are discussed.

The interaction of linguistic and arithmetic factors affects adult performance on arithmetic word problems

Performance on word problems is influenced by linguistic and arithmetic factors, and by their interaction. To study these factors and interactions, we manipulated linguistic and arithmetic factors independently in a within-participant design that included complexity parameters (a) in the domain of arithmetic: carry/borrow (no-carry/borrow vs. carry/borrow), operation (addition vs. subtraction), (b) in the domain of linguistics: nominalization (nominalized vs. verbalized form), and (c) linking the two domains: lexical consistency (linguistic predicate locally consistent vs. inconsistent with mathematical operation). Response times of 25 students solving 320 one-step word problems were measured. All four factors showed a main effect on response times, and interactions between linguistic and arithmetic factors affected response times. These interactions were observed when the linguistic and arithmetic factors were conceptually linked. Our results highlight that not only the linguistic and arithmetic complexities of an item contribute to the difficulty of a word problem, but linguistic and arithmetic factors interact. We discuss the theoretical implications for the numerical and the linguistic domain as well as the possible impact of domain-general characteristics, such as working memory limitations as a potential reason for the observed interactions between numerical and linguistic attributes.