Search

This paper further develops an instructional method called here "process-object linking and embedding". The idea is to link the familiar mathematical processes to objects in a familiar situation, then re-embed the new link through mathematical symbols into their mathematical construction. It makes use of children's extra-mathematical, ethnomathematical, or everyday knowledge to link situations to and "unpack" processes in the mathematics with which they are already confident. It is designed as a means of overcoming the problem of intuitive gaps recurring in children's mathematical development. This process is illustrated here in the case of two attempts to teach integers. It is based on the principle that intuitions arising outside mathematical experience can be imported into mathematics: strategies and concepts arising in the extra-mathematical situation can be modeled with the aid of tools and representational devices. The design of appropriate activity does need to address the whole social situation in which the child's intuitions arise. Here we find a method in which the intuition for "fairness" is carried into a classroom game in which integers are modeled particularly effectively. Contains 41 references. (PVD)

This paper argues that operations with negative numbers should be taught using a curriculum that is grounded in algebraic geometry. This position is supported by the results from a study that compared the conceptual understanding of grade 9 students who received the Algebra Project transition curriculum to a control group of grade 6 gifted students who received a traditional introductory algebra course. The overall scores on an open-ended examination showed that the Algebra Project students, who performed lower than the traditional students at the beginning of the year, had surpassed the traditional group by the end of the year. Further examination of the students' problem-solving strategies revealed that the Algebra Project students had developed an understanding of integer addition and subtraction, based on vector operations, while the traditional group of students still exhibited confusion from the use of the conventional sign rules. The study results show how all operations with integers can be made more intuitive to students by providing them with physical experiences that correspond to vector operations in space/time coordinates. These results not only reinforce the view that all students should have the opportunity to learn the important ideas of mathematics, but that all students need to learn the traditionally "higher-order mathematics" that provide geometrical grounding for abstract algebraic concepts. (Contains 19 references.) (Author)

Metaphorical reasoning explains how people can interpret abstract representations through a complex activity and then apply them to new problems. In particular, metaphors can facilitate both conceptual understanding and problem solving by: (1) intuitively justifying mathematical operations, (2) integrating mathematical knowledge, (3) enhancing the computational environment, and (4) improving recall. In this study audiotaped interviews of (n=12) novice middle school students and (n=5) expert master's graduates solving three tasks involving negative numbers were analyzed. Through a variety of spatial and quantitative metaphors, these students reasoned metaphorically, not only to understand and solve these problems, but also to evaluate and justify their solutions. Experts articulated more metaphors and reasoned with them selectively. In contrast, novices employed metaphorical reasoning less skillfully, but they used it more frequently. Appendices include: Arithmetic is Motion Along a Linear Path metaphor, Stock Market problem, Ordering problem, and Images of Arithmetic Expressions. Contains 85 references and 12 figures. (MKR)

An influential proposal about aquiring mathematical knowledge is that it entails linking instruction-based concepts to intuitions derived from informal activities. In the case of non-positive numbers, informal knowledge is unlikely to emanate from observing physical objects, because non-positive objects or sets of objects do not exist. However, it is hypothesized that such intuitions could derive from experience with actions that undo other actions, such as decrementing a collection or returning to a starting point. Game-like activities involving positive and non-positive actions are used in an exploratory study of 4- to 7-year-old children to examine this hypothesis. The results suggest that children do develop action-based intuitions of non-positive quantities prior to formal instruction, although this knowledge must be described as qualitative and precursory. Similar informal activities could be adapted for use in the early grades to promote an intuitive basis for formal concepts that will be presented later. Activities emphasizing ordinal concepts may be especially useful, as these appear less likely than cardinal concepts to be acquired through casual experience. (Author)

In finite probability theory the only probability-zero event is the impossible one, but in standard Kolmogorov probability theory probability-zero events occur all the time. Prominent logicians, probability experts and philosophers of probability, including Carnap, Kemeny, Shimony, Savage, De Finetti, Jeffrey, have successfully argued that a sound probability should be regular, that is, only the impossible event should have zero probability. This intuition is shared by physicists too. Totality is another desideratum which means that every event should be assigned a probability. Regularity and totality are achievable in rigorous mathematical terms even for infinite events via hyper-reals valued probabilities. While the mathematics of these theories is not objectionable, some philosophical arguments purport to show that infinitesimal probabilities are inherently problematic. In this paper we present a simpler and natural construction - based on Sergeyev's calculus with Grossone (in a formalism inspired by Lolli) enriched with infinitesimals - of a regular, total, finitely additive, uniformly distributed probability on infinite sets of positive integers. These probability spaces - which are inspired by and parallel the construction of classical probability - will be briefly studied. In this framework De Finetti fair lottery has the natural solution andWilliamson's objections against infinitesimal probabilities are mathematically refuted. Further, in contrast with the classical case, Grossone-like uniform distribution and binomial distribution are infinitely divisible. [ABSTRACT FROM AUTHOR]