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Flexibility in performing mental calculations has become an important topic for research in mathematics education, resulting in a surge of research done in this field over the last two decades. While contemporary research shows the same basic idea of mental flexibility, different operational definitions and approaches have led to results that are not directly comparable. The study reported here focuses on cognitive flexibility in arithmetic reasoning in second and fourth grade German and American students, where flexibility is defined more intuitively by referring to students’ knowledge and use of number patterns and relationships. Students were encouraged to sort 12 two-digit addition and subtraction problems reflecting a variety of number patterns into “easy” and “hard” groups and explain their reasoning. Qualitative results revealed a variety of reasoning patterns and profiles of flexibility and rigidity. Quantitative results indicated no country differences in cognitive flexibility and no grade differences for American students. However, German fourth graders exhibited significantly more flexible reasoning than German second graders.

Profiles of Cognitive Flexibility in Arithmetic Reasoning: A Cross-Country Comparison of German and American Elementary Students