
06454570
j
2015d.00589
Beckmann, Sybilla
Izs\'ak, Andrew
Two perspectives on proportional relationships: extending complementary origins of multiplication in terms of quantities.
J. Res. Math. Educ. 46, No. 1, 1738 (2015).
2015
National Council of Teachers of Mathematics (NCTM), Reston, VA
EN
F80
F70
G40
I20
mathematical concepts
multiplication
division
arithmetic
computation
geometric similarity
quantities
linear functions
slope
ratio
proportion
representation
http://www.nctm.org/Publications/journalforresearchinmathematicseducation/2015/Vol46/Issue1/TwoPerspectivesonProportionalRelationships_ExtendingComplementaryOriginsofMultiplicationinTermsofQuantities/
Summary: In this article, we present a mathematical analysis that distinguishes two distinct quantitative perspectives on ratios and proportional relationships: variable number of fixed quantities and fixed numbers of variable parts. This parallels the distinction between measurement and partitive meanings for division and between two meanings for multiplication  one rooted in counting equalsized groups, the other in scaling the size of the groups. We argue that (a) the distinction in perspectives is independent from other distinctions in the literature on proportional relationships, including the within measure space versus between measure space ratio distinction; (b) the psychological roots for multiplication suggest the accessibility of the two perspectives to learners; and (c) the fixed numbers of variable parts perspective, though largely overlooked in past research, may provide an important foundation for central topics that build on proportional relationships. We also suggest directions for future empirical research.