Discusses several mathematics problems involving the generalization of rules from patterns. Cautions against using conjectures as proof that a generalization will always hold true. (MKR)

Discusses the process of generalization. Illustrates the process by generalizing the classic problem of how a farmer can get a fox, a goose, and a bag of corn across a river in a boat that is large enough only for him and one of the three items. (MDH)

Using a geoboard to find polygons with given perimeters presents a problem which refines skills and provides an opportunity to inquire and discover. (MK)

Describes a method of teaching the order of mathematical operations based upon the psychological theory of conceptual generalization. (MDH)

Research relevant to improving instruction for learning disabled students with mathematical problem solving deficits is reviewed. Using a cognitive-metacognitive framework, techniques for teaching both specific and comprehensive mathematical problem solving strategies as well as techniques to enhance strategy generalization are presented.…

The general combinatorial problem of counting the number of regions into which the interior of a circle is divided by a family of lines is considered. A general formula is developed and its use is illustrated in two situations. (PK)

Worksheets for duplication are provided for this activity designed for students in grades 6-10. The objective of the activity is to have students discover generalizations about the patterns formed. (MK)

Explores families of parabolas that result from the graphs of quadratic functions. Computer software allows students to quickly depict a series of graphs and make conjectures about emerging patterns. Discusses cases in different parameters in the quadratic are varied to produce different effects in which the parabolas. (MDH)

Suggests that problem-solving extensions are appropriate experiences for differentiating learning experiences for students with high abilities. The extensions fall into four major categories: pattern and generalization, new concepts and vocabulary, creativity, and making value judgments. (PK)

Presents methods for teaching two mathematical concepts that utilize visualization. The first illustrates a visual approach to developing the formula for the sum of the terms of an arithmetic sequence. The second develops the relationship between the slopes of perpendicular lines by performing a rotation of the coordinate axes and examining the…

A multisensory program using a counting technique was effective in teaching math skills to three elementary students with mild disabilities. Results showed significant gains in acquisition of target skills as well as maintenance and generalization to novel math problems. (Author/DB)

Ten remedial mathematics exercises are provided for children who have failed to integrate or apply their math skills. The exercises provide remediation through systematic experimentation, rather than abstract drills, by using number-configuration distinction with blocks, fractioned candy bars, decimal match sticks, graphed pictures, etc. (JDD)

Presents a method that connects the area formulas for triangles, rectangles, parallelograms, and trapezoids by focusing on the relationships between the bases and heights of each figure. Transformations allow figures to be reconceptualized to establish a general concept of area that can be applied to other figures. (MDH)

Investigates the question concerning the maximum number of lines of symmetry possessed by irregular polygons. Gives examples to illustrate and justify the generalization that the number of lines of symmetry equals the largest proper divisor of the number of sides. Suggests related classroom activities. (MDH)

Presents 5 activities for the K-1, 2-3, 4-5, 6-8 grade levels and for in the home in which students explore the concept of combinations. Each activity includes a lesson plan to investigate a combinatorics problem appropriate for that grade level. Provides reproducible worksheets. (MDH)