As teachers begin to implement mathematics curricula that capitalize fully on computing technology and that are focused on concepts and applications instead of on execution of by-hand symbolic manipulation routines, their well-established routines of thinking about mathematics and its teaching no longer apply in seamless fashion. This case study,…

Explores what kinds of calculus-related insights seem to typify calculus-related reasoning. Introduces "qualitative calculus" in which learning is focused on synthesis. Discusses the resemblance and difference between traditional calculus and qualitative calculus, advantages of learning qualitative calculus, and how understanding qualitative…

One of the most common misconceptions about probability is the belief that successive outcomes of a random process are not independent. This belief has been dubbed the "gambler's fallacy". The belief that non-normative expectations such as the gambler's fallacy are widely held has inspired probability and statistics instruction that attempts to…

Discussed are possible reasons behind the inconsistencies in the learning of calculus. Implicated are students' beliefs, mathematical paradigms including concept image and concept definition, language use, and curriculum sequencing. (KR)

Higher order thinking skills are inevitably developed or exercised relative to some discipline. The discipline may be formal or informal, may or may not be represented in a school curriculum, or relate to a wide variety of domains. Moreover, the development or exercise of thinking skills may take place at differing levels of generality. This paper…