How Old Is the Shepherd? An Essay about Mathematics Education.

U.S. students go through school with serious misconceptions about mathematics. According to Ms. Merseth, parents, the popular media, and the schools themselves reinforce these mistaken notions. EACH WEEKDAY during the school year, some 25 million children study mathematics in U. S. schools under the careful supervision of classroom teachers, using textbooks that publishers have spent millions of dollars to develop and print. With this impressive investment of human and monetary capital, why is it that our students perform as if their curriculum and instruction were on the cutting edge of mediocrity? Why is it that, on virtually every international comparison of teaching and learning of mathematics, the performance of children in the U.S. ranks near the bottom? In order to focus discussion, consider the following nonsensical problem: There are 125 sheep and 5 dogs in a flock. How old is the shepherd? Researchers report that three out of four schoolchildren will produce a numerical answer to this problem. [1] A transcript of a child solving this problem aloud reveals the kind of misinformed conception of mathematics that many children hold: 125 + 5 = 130 . this is too big, and 125 - 5 = 120 is still too big . while 125 / 5 = 25. That works! I think the shepherd is 25 years old. In this child's world, mathematics is seen as a set of rules -- a collection of procedures, actually -- that must first be memorized and then correctly applied to produce the answer. Examining the transcript, we see that this student is not without reasoning ability. Indeed, the accurate deduction about the appropriateness of the shepherd's age shows some sense-making. In spite of this reasoning, however, the child apparently feels compelled to produce a numerical answer. How frequently teachers see this eagerness to begin writing and manipulating numbers before the situation is fully understood! America has produced a generation of students who engage in problem solving without regard for common sense or the context of the problem. Why is this so? Three factors must bear the blame: societal beliefs about mathematics, the typical curriculum in use in our schools, and the preparation of our teachers. SOCIETAL BELIEFS A number of widely held beliefs about mathematics adversely affect the ability of children and adults to experience mathematics in productive, meaningful ways. These beliefs profoundly influence the way in which mathematics is taught, studied, and understood. Many individuals believe that mathematics is a largely rule-oriented body of knowledge that is acquired through the memorization of discrete number facts and algorithmic rules. For example, in a survey of eighth- and 12th-grade students, researchers found that 40% of students at both grade levels agreed with the statement that mathematics is a set of rules, while 50% of the eighth-graders and 25% of the 12th-graders stated that mathematics involves mostly memorizing. Lending further credence to this rule-bound conception of the field, eight out of 10 eighth-graders and two out of three 12th-graders held the opinion that there is always a rule to follow in solving a mathematics problem. [2] This belief in the sanctity of rules is clearly illustrated by the children's response to the shepherd problem. But in the real world of mathematics, nothing could be further from the truth. The work of mathematicians and of those individuals who use mathematics to design such technologies as weather satellites, the Patriot missile system, or the paths that our intercontinental telephone calls follow is not governed by simple rules and formulas. Instead, mathematicians participate in a problem-solving process that is interactive and often quite fluid. Some individuals have described this process as a "zigzag path" from conjectures to explorations of the conjectures through refutations and back to reformulated conjectures. …