Nonstandard Student Conceptions About Infinitesimals

This is a case study of an undergraduate calculus student's nonstandard conceptions of the real number line. Interviews with the student reveal robust conceptions of the real number line that include infinitesimal and infinite quantities and distances. Similarities between these conceptions and those of G. W. Leibniz are discussed and illuminated by the formalization of infinitesimals in A. Robinson's nonstandard analysis. These similarities suggest that these student conceptions are not mere misconceptions, but are nonstandard conceptions, pieces of knowledge that could be built into a system of real numbers proven to be as mathematically consistent and powerful as the standard system. This provides a new perspective on students' “struggles” with the real numbers, and adds to the discussion about the relationship between student conceptions and historical conceptions by focusing on mechanisms for maintaining cognitive and mathematical consistency.

Making −x Meaningful

How do your algebra students read the symbol −x? Common responses are “negative x,” “minus x,” “the opposite of x,” or “the additive inverse of x.” The most common response is “negative x.” But are all these responses meaningful? Definitely not! In fact, the first two responses are very misleading, if not incorrect. In many classrooms, teachers are quite careful to name a real number less than zero (or to the left of zero on the real-number line) a negative number. Students quite easily grasp the meaning of the phrase negative number. But suddenly we bring out the expression − x and read it “negative x”! Trouble begins. Students immediately assume that this symbol stands for a number less than zero simply because its verbal name contains the word negative.

Axioms of symmetry: Throwing darts at the real number line

We will give a simple philosophical “proof” of the negation of Cantor's continuum hypothesis (CH). (A formal proof for or against CH from the axioms of ZFC is impossible; see Cohen [1].) We will assume the axioms of ZFC together with intuitively clear axioms which are based on some intuition of Stuart Davidson and an old theorem of Sierpiński and are justified by the symmetry in a thought experiment throwing darts at the real number line. We will in fact show why there must be an infinity of cardinalities between the integers and the reals. We will also show why Martin's Axiom must be false, and we will prove the extension of Fubini's Theorem for Lebesgue measure where joint measurability is not assumed. Following the philosophy—if you reject CH you are only two steps away from rejecting the axiom of choice (AC)—we will point out along the way some extensions of our intuition which contradict AC.