id: 06455369
dt: j
an: 2015e.00458
au: Paoletti, Teo J.
ti: Are all infinities created equal?
so: Math. Teach. (Reston) 107, No. 2, 98-103 (2013).
py: 2013
pu: National Council of Teachers of Mathematics (NCTM), Reston, VA
la: EN
cc: E60 A30
ut: mathematical concepts; infinity; cardinality; set theory
ci:
li: http://www.nctm.org/publications/article.aspx?id=39216
ab: From the text: Can one infinity be more than another infinity? Ask students
this question, and many will be puzzled; others will insist that
“infinity is infinity.” The question seems to pique their interest
and provides an opportunity to present the beautifully simple but
counterintuitive proofs concerning the size of infinity first
constructed by Georg Cantor. Such a question may not appear to fit
within certain specific content, but the concept of infinity emerges
within many contexts throughout the secondary school curriculum, from
first-year algebra to BC calculus. Two other topics in the secondary
school curriculum closely related to infinity are one-to-one and onto
functions, which have clear connections to Cantor’s work. A function
is “one-to-one” if each output value is the image of a unique
input; a function is “onto” if each element in the range is an
output for some input value from the domain. Too often, students get
little exposure to one-to-one and onto functions. The significance of
infinity is supported by NCTM’s standards [{\it National Council of
Teachers of Mathematics}, Principles and standards for school
mathematics. Reston, VA: NCTM (2000)], which underscore the importance
of developing a deep understanding of very large numbers as well as
comparing and contrasting the properties of different number systems at
the secondary school level. Further, these standards emphasize the
importance of students analyzing and evaluating the mathematical
thinking, arguments, and proofs of others. A historical exploration of
Cantor’s proofs concerning infinite sets is one way in which teachers
can develop students’ understanding of different sets of numbers
while allowing students to evaluate and critique the mathematical
arguments of others.
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