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In this paper we examine the possibility of differentiating between two types of nonexamples. The first type, intuitive nonexamples, consists of nonexamples which are intuitively accepted as such. That is, children immediately identify them as nonexamples. The second type, non-intuitive nonexamples, consists of nonexamples that bear a significant similarity to valid examples of the concept, and consequently are more often mistakenly identified as examples. We describe and discuss these notions and present a study regarding kindergarten children's grasp of nonexamples of triangles.

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Engaging students with multiple solution problems is considered good practice. Solutions to problems consist of the outcomes of the problem as well as the methods employed to reach these outcomes. In this study we analyze the results obtained from two groups of kindergarten children who engaged in one task, the Create an Equal Number Task. This task had five possible outcomes and five different methods which may be employed in reaching these outcomes. Children, whose teachers had attended the program Starting Right: Mathematics in Kindergartens, found more outcomes and employed more methods than children whose teachers did not attend this program. Results suggest that the habit of mind of searching for more than one outcome and employing more than one method may be promoted in kindergarten.

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Initial investigations suggest that students' intuitive decisions concerning the equivalency of two given infinite sets are largely determined by the way these sets are represented. So far the effects of two types of representations were investigated: the numerical-horizontal and the numerical-vertical representations. Our study was mainly aimed at determining which representations (numerical-horizontal, numerical-vertical, numerical-explicit or geometric) yielded higher percentages of one-one correspondence reactions. For these purposes, 189 middle class 10th to 12th graders were asked to react to 14 problems dealing with comparing infinite sets. The problems presented different representations of the same infinite sets. It was found that one-one correspondence justifications were mainly elicited by numerical-explicit and by geometric representations. The discussion suggests ways of adjusting these findings to two different approaches to teaching: analogy and conflict.

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