History of the Infinitely Small and the Infinitely Large in Calculus, with Remarks for the Teacher

A categorization of the “whys” and “hows” of using history in mathematics education

“…For instance, that history may serve as a source of inspiration for a given designer of teaching materials, including textbooks, worksheets, etc. (e.g., Van Maanen, 1997;Gravemeijer & Doorman, 1999;Radford, 2000b;Kleiner, 2001;Streefland, 2003;Van Amerom, 2003;Fung, 2004;Bakker & Gravemeijer, 2006;Barbin, 2007). Such designers may not be interested in restricting themselves to using history as either a tool or a goal; possibly, they would want to do both.…”

What does it mean for a student to understand the first-year calculus? Perspectives of 24 experts

“…Debate over what students should know and be able to do upon completion of the first-year calculus has been disputatious for over a century (Bressoud, 1992;Douglas, 1986;Ford, 1910;Hobbs & Relf, 1997;Kaput, 1997;Kleiner, 2001;MacDuffee, 1947;Maher, 1991;Maltbie, 1900;Munroe, 1958;Osgood, 1907;Renz, 1986;Tucker, 1999;Woods, 1929). Inability to arrive at consensus on essential learning points has been reported across the mathematical community (Ganter, 2001;Tucker, 1999;Zorn, 1991) with some questioning whether consensus is possible or even appropriate and others asserting its necessity.…”

Envisioning the infinite by projecting finite properties

“…24 centuries later one of the hallmarks of higher mathematics is the routine discussion of actually infinite objects and their properties, but this state of affairs is only possible through changes in epistemological outlook and the development of a fair bit of mathematical machinery (Kleiner, 2001;Russell, 1914). Thus it is unsurprising that students experience persistent and stubborn difficulties when learning about the mathematically infinite, difficulties that mirror Aristotle's concerns (e.g.…”