ABSTRACT. Conceptual analyses of Newton's use of the Fundamental Theorem of Calculus and of one 7th-grader's understanding of distance traveled while accelerating suggest that concepts of rate of change and infinitesimal change are central to understanding the Fundamental Theorem. Analyses of a teaching experiment with 19 senior and graduate mathematics students suggest that students' difficulties with the Theorem stem from impoverished concepts of rate of change and from poorlydeveloped and poorly coordinated images of functional covariation and multiplicatively-constructed quantities.John Dewey once said that theory is the most practical of all things (Dewey, 1929). Theory is the stuff by which we act with anticipation of our actions outcomes and it is the stuff by which we formulate problems and plan solutions to them. It is in this sense that I consider this theoretical investigation of students' calculus concepts to be a highly practical endeavor. Mine is a theoretical paper driven by practical problems. The theoretical side has to do with imagery and operations in the constitution of students' understanding of the Fundamental Theorem of Calculus; the practical side is motivated by our general lack of insight into the poor quality of calculus learning and teaching in the United States.A primary theme I will develop is that students' difficulties with the Fundamental Theorem of Calculus can be traced to impoverished images of rate. To develop this theme I will need to make several digressions -one to explicate my use of "image", one to explain what I mean by images of rate, and a third to clarify issues surrounding the Fundamental Theorem itself. IMAGERY AND OPERATIONSBy "image" I mean much more than a mental picture. Rather, I mean "image" as the kind of knowledge that enables one to walk into a room full of old friends and expect to know how events will unfold. An image is constituted by coordinated fragments of experience from kinesthesia, proprioception, smell, touch, taste, vision, or hearing. It seems essential also to include the possibility that images entail fragments of past affective experiences, such as fearing, enjoying, or puzzling, and fragments of past cognitive experiences, such as judging, deciding, inferring, or imagining. 1 Images are less well delineated than are schemes of actions or operations (Cobb and von Glasersfeld, 1983). They are more akin to

We conclude that age was the most important factor, followed by BMI, race and lifestyle factors in explaining steroid hormone variability.

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ABSTRACT. Six fifth-grade children participated in a four-day teaching experiment on complex additively-structured problems, which was followed by in-depth interviews of individual children. The teaching experiment was meant to investigate children's difficulties in holding in mind at once situations in which one or more items played multiple roles. Two important difficulties were identified: (1) distinguishing between "difference" as the result of subtracting and "difference" as the amount by which one quantity exceeded another; and (2) indirect evaluation of an additive comparison. Sources of these difficulties, along with pedago~cal and curricular recommendations for addressing them, are discussed.