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...stripped of her clothes and hacked to death with sharpened oyster shells (Osborne, 1992; Gibbon, 1975). It appears that her crime was teaching about the roundness of the earth at a time when Christian leaders wanted to revive the notion of a flat earth centered in a tabernacle shaped universe (Alic, 1998).
In 1299, the city of Florence banned the use of Hindu-Arabic numerals. It was thought that the new system facilitated dishonest dealings because it was easy to modify a zero to a nine or a six. It was also thought that the place value system allowed deceitful merchants to inflate values by adding a new number to the end of a row (Flegg, 1989). During the late Middle Ages many Europeans rejected the very notion of a zero, regarding it as a creation of Satan (Menninger, 1969).
We can also find more recent examples of resistance to changes in mathematics education. In Colonial America, reading and writing were seen as the chief purposes of education, and mathematics was often not part of the school curriculum. In 1900 parents in New York City complained "that their children were using methods that were different from those they themselves employed." (Kilpatrick, 1992, p. 17)
The mathematical knowledge needed by any particular culture is not static. If we assume that mathematics and mathematics education that met the needs of our parents are appropriate for the new century, we are deeply mistaken. As Steen (1990) states:
To develop effective new mathematics curricula, one must attempt to foresee the mathematical needs of tomorrow's students. It is the present and future practice of mathematics-at work, in science, in research-that should shape education in mathematics. To prepare effective mathematics curricular for the future, we must look for patterns in the mathematics of today to project, as best we can, just what is and what is not truly fundamental. (p. 2-3)
Many Americans are convinced that they can never learn mathematics. This persuasive attitude is an example of what psychologists call learned helplessness. McLeod & Ortega (1993) define learned helplessness in the mathematics education context as "a pattern of behavior whereby students attribute failure to lack of ability" (p. 28). These authors contrast learned helplessness with mastery orientation. In mastery orientation students have confidence in their ability to solve challenging problems. Learned helplessness is negatively related with persistence, while mastery orientation is positively connected with persistence.
McLeod & Ortega (1993) found a student's self-concept could be modified by social context. They describe how classroom conversation, such as a teacher's characterization of a problem as "easy" can profoundly demoralize students. The National Council of Teachers of Mathematics [NCTM] Assessment Standards for School Mathematics (1995) defines mathematical disposition as "interest in, and appreciation for, mathematics; a tendency to think and act in positive ways; includes confidence, curiosity, perseverance, flexibility, inventiveness, and reflectivity in doing mathematics (p. 88). The critics of the Standards dismiss this notion of disposition as nonsense and advocate a back-to-basics approach. In the words of Jennings (1996), "get a math book, make students practice problems, have them do simple addition, subtraction, and multiplication in their heads, give them standardized tests, and drop the group work." This back-to-basics orientation seems more rooted in nostalgia than actual research. McLeod and Ortega (1993) give us reason to hope that if we address the affective components of mathematics education, as suggested in the NCTM Standards, we can improve students' achievements.
The point is not to dichotomize between conceptual understanding of mathematics and basic skills. Instead, many researchers believe that computational skills can be achieved and applied to new mathematical situations even more powerfully, if those skills are put in a meaningful context. The goal ought to be for students to develop strong number sense and to acquire conceptual understanding of number meaning and number relations.
In spite of our belief that understanding and skills can and should develop together, we must make it clear that we assume the primary goal of mathematics instruction is conceptual understanding. But we must also make it clear that setting conceptual understanding as the primary goal does not mean ignoring computation skills. In fact, we have found that instruction for understanding can help students construct skills that...
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