Analysis of Slow Learners in the Calculation of Symbolic Fractions and Development of an Online System to Support them

慢学习者不容易学习计算符号分数，即使他们有足够的技能来计算数值分数。怀疑的原因是，符号计算依赖于几个数学重写规则，这些规则通常隐藏在数字计算的黑匣子中，并且缓慢的学习者不能正确地使用这些规则。他们可以根据他们从经验中学到的一些模式自动计算分数，而不需要考虑计算步骤的机制。我们一直在开发一个基于Web的在线练习系统，以支持这些缓慢的学习者在符号计算。该系统对学生输入的符号表达式进行分析，将计算错误分为粗心错误、致命错误和其他错误。致命错误主要是由错误的重写规则引起的，这些规则看起来与任何数学重写规则相似，但不等于任何数学重写规则，并表明学生没有根据数学规则进行计算。该系统有两个专门的网页：逐步指导页面和解释页面。说明页面根据选定的重写规则向学生显示计算的一个步骤。通过使用这个页面，学生习惯于在他/她的符号计算中使用重写规则。当学生在练习页的计算中犯了致命的错误时，解释页就会自动显示出来。通过对实验结果的分析，说明了该方法和系统的有效性，并给出了一个数值反例来说明该方法和系统的不正确性。

Slow learners do not learn the calculation of symbolic fractions easily even if they have enough skills to calculate numerical fractions. The suspected reason is that symbolic calculations depend on several mathematical rewriting rules, which are usually hidden in black boxes at numerical calculations, and the slow learners cannot use the rules properly. They can calculate numerical fractions automatically according to a few patterns they have learnt from experience and without thinking of the mechanism of the calculating steps.We have been developing a Web-based online exercise system to support such slow learners in symbolic calculation. The system analyzes the input symbolic expressions sent by the students, categorizes the calculating errors into careless mistakes, fatal errors, and other errors. The fatal errors are mostly caused by incorrect rewriting rules that look similar but are not equal to any mathematical rewriting rules, and show that the student do not calculate upon mathematical rules. The system has two special Web pages: step-by-step instruction page and the explanation page. The instruction page shows one step of the calculation to the student according to the selected rewriting rule. By using this page, the student is accustomed to use rewriting rules in his/her symbolic calculation. The explanation page shows itself when the student makes a fatal error on his/her calculation in the exercise page. It explains that the exact calculating step is lead by an incorrect rule, and shows a numerical counter example to show the incorrectness.Slow learners who used the system showed better recognition than other students in the post-test, which demonstrates the effectiveness of the method and the system developed in this project.