Discusses students' inability to make the connection between manipulative materials and pencil-and-paper calculations in mathematics instruction. Outlines the development of mathematical ideas through the concrete, representational, and abstract phases of instruction. An annotated bibliography listing teacher resources for representational-level…

This paper discusses ways to use worked examples in teaching mathematics. It is argued that neither investigative teaching such as discovery learning nor lecturing and starting from the abstract are helpful as they are based on emotive associations with general labels rather than precise details of pedagogic strategies. (KHR)

The natural environment rings with mathematical sounds, as can be seen in the logarithmic spirals of elephant tusks, the hexagonal shapes of snowflakes and honeycomb cells, the rhythmic spouting of gysers, and in the circular growth of microscopic sea plants. Photographs reveal the mathematical relationships of familiar objects. (Author)

A classroom activity is described which encourages students to determine the use of an artifact from the pre-Columbian Iraquois. Primary emphasis is on creative and productive thinking. (JB)

Creativity can be distinguished from intelligence, but there is no consensus on how the recognized stages of the creative act can be taught. The steps to rational thinking can and should be intentionally taught and rationality may prepare a base for unexpected creative insights. (DB)

Aims to acquire better understanding of how informal learning experiences may interface with classroom science instruction by investigating one aspect of science learning, that of scientific reasoning ability. Investigates possible differences in students' scientific reasoning abilities relative to their informal learning environments, classroom…

Evaluates the thinking of third grade students in relation to an instructional program in probability which was informed by a research-based framework that included a description of students' probabilistic thinking. Reveals that overcoming misconceptions in sample space, applying both part-part and part-whole reasoning, and using invented language…

Abstract algebra courses tend to take one of two pedagogical routes: from examples of mathematics structures through definitions to general theorems, or directly from definitions to general theorems. The former route seems to be based on the implicit pedagogical intention that students will use their understanding of particular examples of an…

One of the missions of education is to prepare children for complex tasks that occur in their cultural environment. By means of abstracting, the effects of this complexity can be reduced. Recent research and theoretical development show us that young children already seem to be able to think abstractly. The acknowledgement of this potential in…

Author attempted to outline a concept of logic that involves standards of correct reasoning as well as some idea of how these might be carried out in a classroom. (Author/RK)

In teaching symbolic logic to students not majoring in mathematics, one can appeal to simple arithmetic examples, set theory examples, and the use of conjunction and disjunction as aids in presenting the conditional. (SD)

Examines the differences between concrete and abstract concepts and their implications for instructional design and teaching. How specific concepts are stored in and retrieved from memory is described, analogies are discussed as an instructional tool in abstract concept learning, and a possible instructional strategy for teaching abstract concepts…

Eighty-seven fifth graders in Israel were randomly assigned to four different filmmaking courses, covering photography only, scenario design and photography, editing and photography, or all three activities. Filmmaking had a significant effect on the cultivation of eight mental skills. The editing activity cultivated four logical inference skills.…

This research study deals with the modes of representation that ninth-graders choose in order to communicate their problem solving paths and justifications, and the relation between these modes of representations and achievement level. The findings are based on analysis of 350 answers to problems that demanded communication of reasoning,…

This paper uses data from two mathematics lessons to explore the nature of progressive discourse and examine critical features of teacher actions that contribute to mathematics classrooms functioning as communities of inquiry. Features found to promote progressive discourse include a focus on the conceptual elements of the curriculum and the use…