Maths: from intuitive to conceptual problem-solving

​Teaching elementary students how to go beyond intuition and use conceptual problem-solving helps improve their performance in arithmetics, new study finds.

When trying to solve arithmetic word problems, elementary students often use informal strategies allowing them to simulate mentally the action described in the text.


“Luc is playing with his 22 marbles at recess. During the recess, he loses 4 marbles. How many marbles does Luc have now?”

To solve this arithmetic word problem, students are thus likely to count down four times from 22 to the eventual solution, 18. However, as noted by the authors of a recent paper, this intuitive strategy is only effective here because the problem to be solved is a “low-cost mental simulation problem”: the mimicking action (counting down four times to represent a 4-marble loss) is easy to perform and not cognitively taxing. The situation would probably be different with the following problem: 

“Luc is playing with his 22 marbles at recess. During the recess, he loses 18 marbles. How many marbles does Luc have now?”

Here, we have a high-cost mental simulation problem where a mimicking action is too cognitively demanding to be performed successfully by young learners. Having to count down 18 times, they are likely to forget the starting number (22) and/or to lose track of the substractions. A much better strategy would be to count up from 18 to 22, which means reformulating the original substraction (22 - 18 = ?) into a much easier to solve indirect addition (18 + ? = 22).

Interestingly, the same elementary students who will likely fail to solve 22 - 18 = ?  would easily solve 18 + ? = 22, which means that what they are lacking in not so much mathematical content knowledge, but rather the ability to articulate problems in efficient ways, including by conceptualizing them in counter-intuitive ways (turning a substraction into an addition).

Developing such skills is precisely the objective of the Arithmetic Comprehension in Elementary School (ACE) program. To test its effectiveness, researchers tested at the end of first grade 103 French students from 5 different classes whose teachers had received a ACE training to a control group of 105 students from 5 traditional classes. During the year, ACE teachers used “semantic recoding exercises” such as the one described above. After presenting high-cost mental simulations problems (e.g., 22-18 = ?), the teacher would represent the mimicking action on the whiteboard, provide the solution, then ask students whether they could think of an easier strategy to solve the problem. If necessary, the teacher would show how substracting 18 from 22 is formally identical (although not intuitively similar) to measuring the distance between 18 and 22, then ask students which strategy they prefer and why. 

During the experiments, students were asked to solve 12 arithmetic word problems--both low- and high-cost mental simulation ones. As expected, but quite impressively, ACE students did much better than control students and manages to solve 50.6% (versus 29.8%) of the unseen complex problems.

Reference: Gvozdic and Sander, “Learning to be an opportunistic word problem solver: going beyond informal solving strategies”, ZDM Mathematics Education, December 2019.