When masters of abstraction run into a concrete wall: Experts failing arithmetic word problems

Abstract

Can our knowledge about apples, cars, or smurfs hinder our ability to solve mathematical problems involving these entities? We argue that such daily-life knowledge interferes with arithmetic word problem solving, to the extent that experts can be led to failure in problems involving trivial mathematical notions. We created problems evoking different aspects of our non-mathematical, general knowledge. They were solvable by one single subtraction involving small quantities, such as 14-2 = 12. A first experiment studied how university-educated adults dealt with seemingly simple arithmetic problems evoking knowledge that was either congruent or incongruent with the problems' solving procedure. Results showed that in the latter case, the proportion of participants incorrectly deeming the problems "unsolvable" increased significantly, as did response times for correct answers. A second experiment showed that expert mathematicians were also subject to this bias. These results demonstrate that irrelevant non-mathematical knowledge interferes with the identification of basic, single-step solutions to arithmetic word problems, even among experts who have supposedly mastered abstract, context-independent reasoning.

Should Mathematics Education Be About Building Concrete Walls?

“…We mostly disagree that these conclusions -which indeed contain some elements of insights -can be drawn from the study performed in (Gros et al, 2019) and shall explain why with reference to the broader context of the study. First, we shall briefly summarise its content.…”

“…Success with regards only to a single correct expected solution of a mathematics WP should therefore not be conflated with the concept of validity that is inherent to mathematical reasoning. A recent study (Gros, Sander, & Thibaut, 2019) addressed this question and concluded that:…”

“…The study found that in both cohorts a substantial proportion (larger in normal adults than in expert adults) of participants would incorrectly tag a problem as being unsolvable when according to the authors it was solvable. It was also found that worded questions involving the estimation of the size of sets -termed cardinal problems in (Gros et al, 2019) -were much more prone to being incorrectly classified as unsolvable compared to so-called ordinal problems involving quantities such as lengths or durations. Furthermore, cardinal problems would require a larger amount of time for a correct answer to be given by the subjects.…”

“…Since word problems contain redundant, non-mathematical information, we can suggest that the process of extracting the mathematically relevant content from a worded problem constitutes an operation of reduction. Under this definition, two differently worded problems are equivalent (or isomorphic to use the term from (Gros et al, 2019)) if their reduced (mathematical) form is the same. The claim, debatable in itself, that failing to solve a worded problem is equivalent to failing to solve the reduced mathematical problem can then only be made if the reduction of a worded problem into a mathematical one is unique.…”

“…This may lead to non-unique reductions of certain worded problems. We will illustrate this point with one of the examples given in Table 2 of (Gros et al, 2019):…”