This study introduces an effective intuitive rule focusing on "tending to the midpoint" in individuals' mathematical thinking. The rule concerns the intuitive tendency towards choosing the midpoint of a change, as a salient point among all the points. To test the proposed rule, 120 students in grade 9 were interviewed. We analyzed the…

We explore conditions for productive synthesis between formal reasoning and intuitive representations through analysis of college students' understanding of the limit concept in the definition of the derivative. In particular, we compare and contrast cognitive processes that accompany different manifestations of persistence of intuitions and tacit…

Students react similarly to a wide variety of conceptually unrelated situations. Describes a rule that is manifested when two systems are equal with respect to certain quantity A, but differ in another quantity B. Indicates that in such situations students often argue that the same amount of A implies the same amount of B. (Contains 30…

Analyzes the relationship between intuitions and structural schemata. Indicates that intuitions are generally based on structural schemata and the transition from schemata to intuitions is achieved by a particular process of compression. (Contains 38 references.) (Author/ASK)

Students' belief that a larger number has more factors is outlined as a particular example of 'the more of A, the more of B' intuitive rule. Discusses the robustness of this belief by demonstrating students' tendency to perceive conflicting evidence as an exception to the rule. Considers some pedagogical approaches. (Contains 12 references.)…

Divided into two parts, this article analyzes why some pupils feel reserve about instantaneous velocities and instantaneous flows. The second part relates reactions of pupils facing a problem that implicates the instantaneous rate of change. Describes some characteristics of this problem that enables the authors to explain its instructional…