The proposed rigorous mathematical thinking paradigm is based on two theories of learning: Vygotsky's sociocultural theory, with particular emphasis on his concept of psychological tools as mediators of cognitive processes, and Feuerstein's theory of Mediated Learning Experience. We examine the role of higher-order mental processes and psychological tools in rigorous mathematical thinking and analyze empirical data on cognitive and academic performance outcomes of the cognitively based program aimed at producing conceptual change in the underachieving students' comprehension of the mathematics concept of function.
The proposed rigorous mathematical thinking (RMT) paradigm consists of two major components. The first in the RMT approach (Kinard, 2001; Kinard & Falik, 1999) drives concept development and applications in mathematics, science, and technology education. The second component is an instructional practice that implements RMT principles in classroom instruction. The RMT theory and its instructional pedagogy are based on two theories of learning: Vygotsky's socio-cultural (Vygotsky and his early followers used the term cultural historical) theory, with particular emphasis on his concept of psychological tools as mediators of cognitive processes, and Feuerstein's (1990) theory of Mediated Learning Experience.
In this article we will examine the role of higher-order mental processes and psychological tools in rigorous mathematical thinking that promotes students' conceptual change in classroom learning situations. In addition, we will describe and discuss empirical data on cognitive and academic performance outcomes of the cognitively based program aimed at producing conceptual change in the students' comprehension of the mathematical concept of function.
Need for Rigorous Thought in Mathematics
There is deep concern in the U.S. that American mathematics and science education is falling well behind that of other industrialized societies. This is manifested in poor performance and low academic achievement in mathematics and science for the vast majority of America's students, generally compared with students in other Western and industrialized Asian countries (Schmidt, 1998; Hoff, 2000; Eisenkraft, 2001; Peak, 1996; TIMSS Policy Forum, 1997; Office of Educational Research and Improvement, U.S. Department of Education, 1997; Schmidt & Valverde, 1997; Schmidt, McNight, & Raizen, 1996). Mathematics and science education are seen as cornerstones of adequate functioning in a technological society. The lack of rigorous thinking and problem solving skills in students, particularly with reference to the content of instruction, is a frequently identified concern (Kinard & Falik, 1999). Simply learning calculations and mechanical processes, without understanding and manipulating the deeper structures of thinking, is clearly not sufficient for competence.
Pointing out what appears to be a lack of focus on rigorous thinking during mathematics instruction for young children, Bybee and Sund (1982, p. 255) state that teachers "have children adding and substracting before they even know the meaning of number." In a paper given at the 89th Annual Meeting of the National Council of Teachers of Mathematics (NCTM), Ball (2002) emphasizes the need for student engagement in rigorous mathematics concept development beginning at an early age. Ball also expresses the urgency for quality teacher professional development that will facilitate the nurturing of students through a rigorous mathematics-learning environment.
The need for rigorous thinking is clearly revealed in a study by Stigler and Hiebert (1997a, 1997b) of eighth-grade mathematics lessons in Germany, Japan, and the United States as part of the Third International Mathematics and Science Study (TIMSS). TIMSS data show that U.S. eighth grade students scored below their peers from 27 nations in mathematics and below their peers from 16 nations in science (Peak, 1996; TIMSS Policy Forum, 1997; Office of Educational Research and Improvement, U. …