A Probe Into The Development Of Six - Year Students From Arithmetic Thinking To Algebraic Thinking

Arithmetic’s basic object is number and algebra’s basic objects are number and symbol. Studying from arithmetic to algebra is a leap from arithmetic thinking to algebraic thinking, which not only updates the content of knowledge but also transforms the cognitive structure. Due to the particularity of educational system in Shanghai, the sixth grade is the transition of elementary school to junior high school.Word problems’ teaching in sixth grade is not only the further development of higher grade in elementary school, but also the thinking base of algebraic abstract relations in seventh grade. Therefore, the author studied the transformation of arithmetic thinking to algebraic thinking through the word problem’s teaching.According to the data, the author mainly studied the development of arithmetic thinking to algebraic thinking from the following three aspects:(1) First of all,analyze the students’ errors, and then find out the cause of cognitive errors.(2)Analysis of students’ transition from arithmetic thinking to algebraic thinking in detail.(3) Analysis of teachers’ teaching strategies can help students have a smooth transition of thinking.During the transition of thinking, there are 13 kinds of common cognitive errors,among which there are several affections of arithmetic thinking as follows: persistent in arithmetic method; focus on the solution of equation and ignore the ultimate answer to problem; it is difficult to regard programmed operation as a part of the answer;students can’t understand the meaning of the equal sign in algebra; with arithmetic method to calculate the answer first, and then assigned to x; the negative affection of reverse thinking when analyzing the equivalent relations; have difficulty in understanding of the para-tactic sign; list the algebraic expression instead of equation.During the transition of thinking, there are several reasons of students relying on arithmetic thinking: the word problem’s structure is simple; students’ mathematical foundation is weak; students’ reverse thinking ability is stronger. When thinking between arithmetic and algebraic patterns, students often master the steps of algebraic solution but not learn the essence of algebraic thinking, such as the understanding of“1”. For some certain types of questions, their equivalent relationship is formula or line segment, students regards these formulas or line segment as the rudiment of the equation. Therefore, students from arithmetic thinking to the early stages of algebraicthinking are based on the formula and schema. On the question of choosing arithmetic method or algebraic method, students are unwilling to use algebraic method if they are familiar with arithmetic thinking; students are willing to use algebraic method when the problem’s equivalent relationship is fixed, such as formula or line segment.According to the literature and data, the author thinks that the following strategies can help students have a smooth transition of thinking:(1) Teachers should pay more attention to cultivate students’ ability to analyze the equivalent relationship.(2) Teachers should pay more attention to develop the students’ ability of problem representation.(3) Teaching students to solve problems both arithmetically and algebraically can guide students to discover the similarities and differences between arithmetic and algebraic approaches, so they can understand the power of a more general algebraic approach.