Problems were selected as part of the evolution of the class to challenge the students, to reinforce the search for patterns, and to evoke questions and problem-posing from the students. Through interactions with each other and by writing their thought processes as they solved problems, the students were actively engaged in doing mathematics rather than watching mathematics being done. Our study of Flatland (Abbott, 1884/1994) afforded opportunities for the students to examine their relationships with mathematics and to recognize their own mathematical power.

This is a study of the relationships that developed among the students in a small class of Mathematics for Critical Thinking. The organization of the class was based on the social constructivism of Vygotsky. There was an attempt to create an atmosphere that was sensitive and nurturing in which students could trust their classmates to be supportive of their efforts to solve mathematics problems. We worked together to establish the sociomathematical norms that would enhance the development of mathematical power within the students and promote the evolution of problem-solving skills.

There are strong implications that mathematics can be made more accessible to more students by restructuring the mathematics curricula to accommodate the different ways that students approach the construction of their mathematical knowledge. My findings call for college and university mathematics departments to re-examine the traditional lecture method for disseminating mathematics to large numbers of students in a lecture hall and consider more opportunities for students to engage in mathematics at a personal level. This could revolutionize mathematics education in ways that would empower all students.

The findings indicate that the relationships within the classroom enhanced the development of viable relationships with mathematics. There were also indications that the fact that the students were all females may have contributed to the workable relationships with mathematics.

Using the different ways of knowing presented by Belenky, et al. (1986) as a guide, it was determined that the students became less dependent on received knowledge and more competent and confident in their procedural and constructed areas of knowledge. The challenges of this class offered the students opportunities to engage in mathematics in a personal way that was empowering. They all left the class with new and positive attitudes towards mathematics and their abilities to actively engage in challenges and rewards that relationships with mathematics afford.