Learning to reason spatially is increasingly recognized as an essential component of geometry education. Generally taken to be the "ability to represent, generate, transform, communicate, document, and reflect on visual information," "spatial reasoning" uses the spatial relationships between objects to form ideas. Spatial thinking takes a variety…

A geometry textbook or mathematics journal that prints all the work that mathematicians use as they generate proofs of mathematical results would be rare indeed. The false starts, the tentative conjectures, and the arguments that led nowhere--these are conveniently omitted; only the final successful product is presented to the world. To students…

Building coherence in the development of mathematical ideas across the grades is key to improving students' mathematical learning in the United States. Knowing the mathematical experiences, understanding, skills, and habits of mind that students bring to a grade level and what the expectations are for the following grades can help teachers bridge…

This article describes how the concept of "key idea" can be used in high school geometry to connect students' informal explorations with rigorous mathematical proof. (Contains 6 figures.)

An approach to teaching geometry is promoted that allows students to decide for themselves what they could prove from given information. Such an approach allows pupil involvement in the personal process of discovering mathematical ideas and formulating problems. It is noted these methods will not work for all. (MP)

A system for categorizing components of geometry class discussions is described. The unit of observation is the teacher question; responses are coded at four cognitive levels. The system is easier to use than many in the literature; use of it can help teachers develop questioning techniques. (SD)

A problem sometimes called Moser's circle problem where a circular region has to be partitioned with chords without any three chords intersecting at one point, is discussed. It is shown that Moser's circle problem makes the students to use a variety of mathematical tools to find correct solutions to problems and gives an opportunity to think about…

Presents an activity in which students develop their own theorem involving the relationship between the triangles determined by the squares constructed on the sides of any triangle. Provides a set of four reproducible worksheets, directions on their use, worksheet answers, and suggestions for follow-up activities. (MDH)

Presents the hierarchical structure of quadrilaterals as an illustration of learning a geometric concept by moving from the levels of visualization and analysis to the level of formal deduction. The development discusses the classification of quadrilaterals, the inheritance of properties within the hierarchy, connections between algebra and…

Describes one teacher's reflection concerning the quest to develop an understanding of school mathematics that promotes and sustains students' opportunities for exploration and conjecture. Recounts that a particular student's exploration of the features of parabolas eventually led to an understanding of the quadratic formula precisely because of…