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

Abstract

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.

Leveraging cognitive flexibility through flexible categorization training

“…When an attempt to solve a mathematical problem is not successful, it is essential for students to learn to alternate between different strategies. This entails the ability to reinterpret the problem; students have to switch between different strategies, which involves reinterpreting the problem in order to find alternative solutions (Gros et al, 2019;2021;Sander & Richard, 2005;Vicente et al, 2007).…”

“…To investigate the categories devised by the participants, we adopted a similarity analysis approach akin to the one used by Gros et al, (2021). Each participant's categorization was encoded using a co-occurrence matrix, which captured the frequency at which two problem statements were jointly classified under the same category.…”

A Cognitive Load Theory Approach to Defining and Measuring Task Complexity Through Element Interactivity

“…The effect of expertise levels on element interactivity and extraneous cognitive load can be seen when considering how learners semantically code problems, an example of strategy use. As indicated by Gros et al (2020Gros et al ( , 2021, learners with varying expertise levels may semantically code the same problem differently. For instance, novices might encode irrelevant features embedded in the problem statement, which can obstruct learning and problem-solving by imposing an extraneous cognitive load.…”

Using Primary Knowledge in Unpopular Statistics Exercises

“…When the presentation is not congruent with the knowledge needed to solve the problem, then the individual must recode the problem. Arithmetic problems are automatically coded on the basis of non-mathematical knowledge from everyday life (Gros et al, 2021). Familiarity can therefore be a hindrance rather than a help in problem-solving tasks.…”