数学是一门锻炼人思维的学科.阿诺德·罗斯(1906～2002)教授从1957年开始成立罗斯数学夏令营,其主要理念是"简单问题,深入思考(Think deeply about simple things)",该课程延续至今,并整理出"罗斯问题集",旨在激发学员对数学学习的兴趣,培养逻辑思维能力.

以一道高考解析几何题为楔子,引领学生学会发现特殊中的一般性,在类比、变式探究中关注运算的算理,看清问题本质,实现深度学习,进而发展逻辑推理、数学运算等核心素养.

经历探究、实验验证后,发现一道含参不等式试题及其解答均存在逻辑漏洞,笔者尝试在分析的基础上作代数论证,并对问题进行修正,通过对问题本质的探寻,实现学生逻辑思维素养的提升.

坚持素养为先,由学生自主创造"主题"并推动探究利于培养"四基",发展"四能".学生在对"这样的函数有多少个"的探究中,将函数、集合、分类计数原理、导数、不等式等知识有机结合,整体把握知识构架,实现深度学习,思维提升至"拓展抽象结构"的高阶水平.

从确定题眼入手,在"将面积之比为定值化归为直线过定点"这一核心思想的引领下,让学生更多关注简化运算的算理,依靠逻辑推理能力的支撑,逐步看清问题本质、突破难点,能"见"椭圆而"思"圆锥曲线,实现深度探究,完成从特殊到一般的数学抽象,"知"数学思想的精妙,达到分析、评价、创造的高阶思维水平.

培养学生的数学核心素养,仅关注事实性结论或解题技巧是远远不够的,如果学生能学会对问题的深度思考,便可经历从猜测到验证、从局部到整体、从现象到本质的思维过程,使碎片化的知识更加系统化.如何见微知著,形成逻辑严密的整体性思维,进而实现深度思考值得研究.本文以"数量积几何意义"的探究经历示例说明.

This paper employs the new quasi-Newton equation proposed by Zhang in 2001 with the tensor method,which is a Huang's quasi-Newton.With the combination of the dual principle and the structure principle of the secant method, Chen's algorithm is generalized to the whole Broyden's and two corresponding algorithms are presented in the first part.It is proved in the second part that the algorithms,invariant under an orthogonal matrix transformation on variables, possess local and super-linear convergence.Numerical experiments show that our algorithms are superior to Chen's.In addition,improvements of the algorithms are also given to enable them to be immune of the influence of the initial point and each iteration point,which effectively expand the application scope of such algorithms.

It is known that the quasi-Newton methods are one of the most effective techniques for the solution of unconstrained optimization. Under modification of conventional quasi-Newton equation to make full use of known information like the objective function value, this paper presents three BFGS type algorithms, which have almost the same efficiencies as that of Zhang([6]) and We(1[7]). These modified quasi-Newton methods are globally convergent and possesses superlinear convergence.