John Paul Cook , April Richardson , Zackery Reed , Elise Lockwood , O. Hudson Payne , Cory Wilson
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Students’ productive use of equivalence transformations
Equivalence transformations – that is, transformations that produce an object that is equivalent to the original – are a unifying conceptual thread in K-16 mathematics. Though researchers have already established that productive reasoning about equivalence transformations hinges on an awareness that the transformed objects are equivalent to the given object, research (a) has not yet explored the various ways in which students might attend to equivalence, and (b) has primarily examined equivalence transformations on only one type of object, leaving open the question of what commonalities might be present in students’ reasoning across transformations of multiple types of objects. In this study, we present our analysis of task-based clinical interviews with university students. This paper’s primary contribution to the literature involves the description and illustration of three common, unified ways in the students productively reasoned about the equivalence of the objects they produced with transformations. Our findings extend the theoretical scope of an existing equivalence framework and suggest that these ways of reasoning can inform efforts to help students overcome the widespread reports of difficulties they experience. We conclude with a discussion of the theoretical implications for research on equivalence transformations across K-16 mathematics.
期刊介绍:
The Journal of Mathematical Behavior solicits original research on the learning and teaching of mathematics. We are interested especially in basic research, research that aims to clarify, in detail and depth, how mathematical ideas develop in learners. Over three decades, our experience confirms a founding premise of this journal: that mathematical thinking, hence mathematics learning as a social enterprise, is special. It is special because mathematics is special, both logically and psychologically. Logically, through the way that mathematical ideas and methods have been built, refined and organized for centuries across a range of cultures; and psychologically, through the variety of ways people today, in many walks of life, make sense of mathematics, develop it, make it their own.