从经验逼近到算法建构：二分法的初高中衔接

From Empirical Approximation to Algorithmic Construction: Bridging the Gap between Junior and Senior High School Education on the Bisection Method

摘要: 二分法并非高中全新的知识。人教版九年级上册“二次函数与一元二次方程”中，学生已借助图像通过试值不断缩小区间得到近似根，这是二分法的朴素经验雏形。然而，部分高中课堂忽视这一前置经验，直接讲授零点存在定理与标准化迭代流程，学生只能机械记忆步骤。基于此，文章对比初高中相关内容的课标定位与教材编排，梳理出适用对象、理论依据、操作流程、终止判定、思维层级五个维度的认知差异，以弗赖登塔尔数学化理论与奥苏贝尔先行组织者理论为支撑，构建“唤醒水平数学化–问题链驱动垂直数学化–对比辨析厘清学段差异”三维衔接策略，并给出适配高一学情的教学片段。

Abstract: The bisection method is not entirely new knowledge in high school education. In the chapter “Quadratic Functions and Quadratic Equations” of the People’s Education Press Grade 9 Volume 1 textbook, students have already learned to approximate roots by iteratively narrowing intervals through trial values using graphical representations; this constitutes the rudimentary empirical foundation of the bisection method. However, some high school classrooms overlook this prior knowledge, instead directly introducing the Zero Point Existence Theorem and standardized iterative procedures, which forces students to mechanically memorize the procedural steps. By comparing the curriculum standard positioning and textbook organization of related content between junior and senior high school levels, this paper identifies cognitive gaps across five dimensions: objects of application, theoretical basis, procedural workflow, termination criteria, and cognitive hierarchy. Grounded in Freudenthal’s theory of mathematicalization and Ausubel’s theory of advance organizers, this study develops a three-dimensional bridging strategy: “activating horizontal mathematicalization - problem-chain-driven vertical mathematicalization - comparative analysis to clarify inter-level differences”, and provides instructional vignettes tailored to the learning characteristics of first-year high school students.