Critical reflective thinking in Euclidean geometry for grade nine mathematics students

The vision of the mathematics classroom that is presented in the National Council of Teachers of Mathematics' Curriculum and Evaluation Standards for School Mathematics (1989) has inspired many of us to want to change the way in which we teach. Indeed, the NCTM has said that many of the mathematics classrooms need to change as transmission models of teaching no longer fit what we know about the teaching and learning process. Learning is no longer viewed as a passive exercise and many have come to believe that students must think, be mentally active, generate meaning, and construct their own understanding. -- This however, is not a sudden realization as Dewey (1933) and Polya (1957), for example, have both recognized the power of reflection and the need for purposeful critical thinking. Polya (1987) noted that as teachers what we need to teach is purposeful thinking (Dossey, 1988). Polya (1987) also claims that such thinking can be identified with problem solving (Taback, 1992). John Dossey, the President of the NCTM, has stated that the knowledge, skills, attitudes and purposeful thinking discussed in Polya's publications of 1945,1954 and 1962 have provided the foundation for their work as a Council (Dossey, 1988). Thus, this change called for by the NCTM is not so much of a new initiative in theory as it is in practice. -- Likewise, this project is grounded in similar theory and signifies a shift away from the transmissive practices of the past. It signifies a shift towards a more transformative way of thinking and away from textbook learning, rote memorization and the mastery of traditional school subjects through traditional methodologies. This project is not a research paper but is instead an alternate unit of Euclidean Geometry which requires an alternate teaching style. It is something that other teachers can lay hands on and use in their classroom as a means of promoting critical reflection and higher order thinking skills. While this unit may represent a transition in approach it does reflect the changes called for - and the standards of - the NCTM (1989) and the Intermediate Mathematics Curriculum Guide (1995) Thus, it is the general purpose of this unit to implement those changes in an environment of cooperative learning. The unit will introduce students to the dynamics of group structure, encourage students to reflect and employ a problem centred discovery approach to mathematics. -- As Vermette (1994) states, "cooperative learning is a powerful and engaging strategy worthy of thoughtful implementation by all high school teachers" (p.38). Conard (1988) goes even further and suggests that cooperative learning methods are important not only for academic achievement but also for our survival as human beings in a complex world. Indeed, several cooperative learning models have been developed for use as teaching tools however, the model employed in this unit resembles what Sharan and Sharan (1990) describe as group investigation. -- The unit is based on critical reflection, problem solving and communication and thus literature reviews are provided for each. There is also a literature review on the process of change as this unit does represent a change for both student and teacher. Chapter Three, titled Methodology, provides a brief description and analysis of the unit whereas Chapter Four is the complete unit itself. The project then concludes with reflections on my experiences with the unit and suggestions for prospective users.