Three methods for robust grading

We propose three strategies by which a professor of a university course can assign final letter grades taking into account the natural uncertainty in students’ individual assignment and final numerical grades. The first strategy formalizes a common technique that identifies large gaps in the final numerical grades. For the second and third strategies, we introduce the notion of a borderline student, that is, a student who is close to, but below, the breakpoint for the next highest letter grade. Using mixed-integer linear programming and a tailor-made branch-and-bound algorithm, we choose the letter-grade breakpoints to minimize the number of borderline students. In particular, the second strategy treats the uncertainty implicitly and minimizes the number of borderline students, while the third strategy uses a robust-optimization approach to minimize the maximum number of borderline students that could occur based on an explicit uncertainty set. We compare the three strategies on realistic instances and identify overall trends as well as some interesting exceptions. While no strategy appears best in all cases, each can be computed in a reasonable amount of time for a moderately sized course. Moreover, they collectively provide the professor important insight into how uncertainty affects the assignment of final letter grades.     

Design-based research principles for successful peer tutoring on social media

ABSTRACT

Social media platforms such as WhatsApp are increasingly used in formal education settings. However, there is little research to guide educators on how to set up effective peer tutoring groups on social media platforms, particularly between tutors and tutees who rarely meet face-to-face. In the context of a WhatsApp mathematics tutoring project, we present evidence-based principles to guide the establishment and operation of peer tutoring groups on a social media platform. The development of the principles followed a design-based research framework, based on theories about peer learning, participation barriers to social media use in a low socio-economic setting, and input from participants, who were high school students and university students in Cape Town, South Africa. The refined principles provide guidance for others in similar settings who aim to use tutoring groups on a social media platform to achieve education goals.     

Measuring pre-service teachers’ self-efficacy in tutoring children in primary mathematics: an instrument

This article reports on the use of Rasch modelling to develop and validate an instrument measuring self-efficacy in tutoring children in primary mathematics (SETcPM). In response to the literature on teacher efficacy, the 20-item instrument aims to inform teacher educators, and is designed for novice pre-service teachers (nPSTs) preparing to teach mathematics in primary school (grades 1–7, ages 6–13). To ensure that the tasks of teaching are imaginable for nPSTs, the instrument targets the core activity of teaching mathematics: helping a generic child with mathematics tasks. We propose that SETcPM is measurable for the intended population and represents a central part of self-efficacy in teaching mathematics (SETM). Understanding the initial SETcPM of novices and mapping its development over the course of their training programme can contribute to a better understanding of SETM, and allow teacher educators to tailor their support.     

Relationships Between Achievement and Selective Variables in a Chemistry Course for Nonmajors

The main purpose of this study was to investigate the relationships between students' approaches to studying, prior knowledge, logical thinking ability, and gender and their performance in a nonmajors' college freshman chemistry course. Subjects for this study were 220 students (128 females and 92 males) enrolled in the second semester of a freshman chemistry course for nonmajors at a private university in New York State. Instruments used in this study included seven subscales of the Approaches to Studying Inventory and the Test of Logical Thinking (TOLT). The students' grades on an hour-long exam early in the semester were used as measures of the students' prior knowledge, while the semester cumulative final examination scores were used as measures of achievement in chemistry. Students in this study had slightly higher scores on reproducing orientation than on meaning orientation, a pattern that confirms Entwistle and Ramsden's (1983) findings with a similar group of nonmajors. The results of a stepwise multiple regression showed that prior knowledge, TOLT scores, and meaning orientation accounted for 32% of the variance on the final examination scores.     

A Problem-Solving Based Mathematics Course and Elementary Teachers' Beliefs

This study examined the impact a problem-solving based mathematics content course for preservice elementary education teachers (PSTs) had on challenging the beliefs they held with respect to mathematics and themselves as doers of mathematics. Nine PSTs were interviewed to gain insight into changes that occurred to their belief systems and what aspect(s) of the course were instrumental in producing those changes. Surveys to measure how strongly PSTs subscribed to five mathematical beliefs were administered to 137 PSTs who were enrolled in the course. Significant positive changes (p < .01) were observed for four of the five beliefs. When changes were studied by achievement level, students with final grades of A or B showed statistically significant changes (p < .005) in three of the five beliefs. Students who were interviewed consistently reported increased confidence in their mathematical abilities as a result of the course.     

Towards a reference curve for the grades of each course

A grade-reference curve (GRC) can be constructed for any course based on the grades of a course in the last several years. Among other things, the reference curve of a course can be used to test for any abnormality in the current semester's grades of a course. It can be a very important document about the course that serves students, teachers and decision makers in the institution. A GRC of a course should be updated on a regular basis. In this article, the construction of such curve is outlined and illustrated by an example. The use of the constructed curve to detect abnormality via statistical analysis is also discussed and illustrated by an example.     

Definitions and examples in elementary calculus: the case of monotonicity of functions

This paper is devoted to the investigation of students’ understanding and handling of examples in the framework of an example-based introductory mathematics undergraduate course. The plan of the course included a wide use of graphs in standard lectures, tutoring sessions as well as in examinations. This study deals with the notion of increasing function, which has been introduced by means of both the standard definition and a range of examples and non-examples, most often conveyed through graphs. We have analysed students’ interpretations of the notion of increasing function as they applied them in a set of written examination tests. The data gathered have been completed by a number of interviews of students whose answers were difficult to interpret. The outcomes underline the importance of linguistic and semiotic competence and suggest that the design of innovative teaching paths should take care of the linguistic and semiotic skills needed to handle the representations involved.     

Correcting grade deflation caused by multiple-choice scoring

In a study involving three sections of 181 pre-calculus students at a four-year college, partial credit scoring on multiple-choice questions was examined over an entire semester. Questions were constructed by experienced teachers to pose carefully designed, incorrect alternatives, many of which implicitly suggested varying degrees of partial knowledge or understanding. Computing students' course grades based solely on percentage correct scoring resulted in a course-wide GPA of 1.38. This seemed inconsistent with the fact that students were screened for entry into the course. To avert apparent grade deflation due to such right/wrong scoring, student scores were recomputed based on supplementing percentage correct scores with partial credit. This resulted in a course-wide increase in GPA to 1.92. Using that same letter grade distribution, and curving percentage correct averages to achieve the 1.92 GPA, it was found that 10% of the students were given different grades from those determined by the partial credit scoring. Regarding the efficacy of awarding partial credit points, it was found that partial credit by itself measured achievement of relevant mathematical skills that were missed by percentage correct scoring. Overall, grades determined by partial credit scoring seemed more reflective of both the quantity and quality of student knowledge than grades determined by either traditional percentage correct scoring or any corrective curving.     

Mathematical tasks,study approaches,and course grades in undergraduate mathematics: a year-by-year analysis

Students approach learning in different ways, depending on the experienced learning situation. A deep approach is geared toward long-term retention and conceptual change while a surface approach focuses on quickly acquiring knowledge for immediate use. These approaches ultimately affect the students’ academic outcomes. This study takes a cross-sectional look at the approaches to learning used by students from courses across all four years of undergraduate mathematics and analyses how these relate to the students’ grades. We find that deep learning correlates with grade in the first year and not in the upper years. Surficial learning has no correlation with grades in the first year and a strong negative correlation with grades in the upper years. Using Bloom's taxonomy, we argue that the nature of the tasks given to students is fundamentally different in lower and upper year courses. We find that first-year courses emphasize tasks that require only low-level cognitive processes. Upper year courses require higher level processes but, surprisingly, have a simultaneous greater emphasis on recall and understanding. These observations explain the differences in correlations between approaches to learning and course grades. We conclude with some concerns about the disconnect between first year and upper year mathematics courses and the effect this may have on students.     

关于高等数学课程分层次教学的实践与思考

高等数学在高校课程体系中具有十分重要的地位．在高等教育进入大众化教育阶段的背景下，本文介绍了我校高等数学课程分层次教学的改革实践与经验，阐述了该教学模式对提高学生的数学素质和对教学质量的作用，同时也指出了存在的主要问题．     

The impact of taking a college pre-calculus course on students’ college calculus performance

Poor performance on placement exams keeps many US students who pursue a STEM (science, technology, engineering, mathematics) career from enrolling directly in college calculus. Instead, they must take a pre-calculus course that aims to better prepare them for later calculus coursework. In the USA, enrollment in pre-calculus courses in two- and four-year colleges continues to grow, and these courses are well-populated with students who already took pre-calculus in high school. We examine student performance in college calculus, using regression discontinuity to estimate the effects of taking college pre-calculus or not, in a national US sample of 5507 students at 132 institutions. We find that students who take college pre-calculus do not earn higher calculus grades.     

Preservice Teacher Beliefs About Proofs

This article describes a study of backgrounds, beliefs, and attitudes of teachers about proofs. Thirty preservice elementary teachers enrolled in a mathematics content course and 21 secondary mathematics teachers in an abstract algebra course were surveyed. The study explored four issues: preservice teachers' experiences/exposure to proof, their beliefs about what constitutes a proof and the role of proof in mathematics, and their beliefs about when proof should be introduced in grades K-12. Results of the survey are described as a means for discussing the backgrounds and beliefs future teachers hold with regard to teaching proofs in their own classrooms. Finally, a short collection of sample explorations and questions, which could be used to encourage the thinking and writing of proofs in grades K-12, is provided. One of these questions was posed to 215 secondary students; examples of their reasoning and a discussion of the various techniques employed by the students are included.     

A process of students and their instructor developing a final closed-book mathematics exam

This article describes a study, from a Canadian technical institute's upgrading mathematics course, where students played a role in developing the final closed-book exam that they sat. The study involved a process where students developed practice exams and solutions keys, students sat each other's practice exams, students evaluated classmates' solutions to the practice exams, and finally the instructor used questions from the practice exams to develop the ‘live’ final exam. Phenomenography is used to analyse interview data and report students' experiences. Through the results, claims are made that students experienced deep approaches to learning and worked as partners in learning, teaching and assessment during the process of developing the final exam with their instructor.     

Likert-scale questionnaires as an educational tool in teaching discrete mathematics

In this paper, we report on the results of an experiment in teaching discrete mathematics to students majoring in business informatics. We supplemented our problem-based approach to teaching the course with a set of Likert-scale surveys or questionnaires that helped improve the students’ performance. On the one hand, these surveys gave us feedback and, on the other, encouraged the students to reflect on the subject-matter. The experiment was quite successful, as the grades obtained by the students on the exam were significantly higher than usual. Here, we describe the structure of the surveys and the method of evaluation of the experimental results.     

Preservice Teachers of High School Mathematics: Success,Failure, and Persistence in the Face of Mathematical Challenges

The goal of this study was to identify variables related to success and resilience in an undergraduate, high school mathematics teacher education program. Over a five‐year period, we tracked the academic performance and achievement motivation goals of multiple cohorts of students. Students who successfully completed their degrees had higher grade point average (GPAs) upon entering the program, earned higher grades in their first college mathematics course, and failed fewer courses than students who left the program or university. Learning and performance motivational goals did not predict success in the program. Performance goals decreased over time. Nearly half the successful students repeated one or more mathematics courses. Ten students completed their degrees, obtained a teaching license, and are teaching despite the need for multiple repetitions of the same mathematics courses. These persistent students did not differ from their peers in motivational goals. Our results suggest that although students with higher GPAs and initial mathematics grades were more likely to complete the program, students who experienced challenges in mathematics courses were able to succeed. We discuss the implications of these results for recruiting, advising, and retention of students in mathematics education programs.     

人口与基础教育发展趋势预测模型研究及应用

以西藏人口与基础教育发展为研究对象,应用灰色系统理论建立了一个新的出生人口预测模型,还分别建立了小学和初中招生数、分年级在校生规模、在校生总量的预测模型,并结合近几年西藏自治区教育统计结果,给出了应用实例.     

基础教育发展趋势预测模型研究及应用

以基础教育发展为研究对象 ,在应用新增人口的离散预测模型基础上 ,分别建立了小学和初中招生数、分年级在校生规模、在校生总量、专任教师需求和补充规模的预测模型 ,并结合近几年山东省教育统计结果 ,给出了计算实例 .     

新中考中的排课问题

为提高初中学业水平和综合素质教育,近几年有些地方中考进行改革,要求学生进行学科选择。新中考课程分为选修和必修两大类。必修课程为语文、数学、英语,选修课程为其他六门课程,从中选出三项。把最后总成绩作为中考录取的标准。跟传统排课不同,这里排课要求每人一张课表,问题变得复杂困难。本文以北京某初三课程为例,研究了新中考的排课问题。一般情况下约束和变量是上百万级的,无法求解。本文利用整数规划建模,然后把该问题转化成多阶段问题,每个阶段给出小问题的解,这样使得问题的求解变得可行。最终的排课结果,仅比预期增加三位老师就可以实现新中考的排课问题。本文的求解过程给新中考排课带来了新的启发。     

A metacognitive intervention in mathematics at university level

The paper reports the main results of an instructional study. The study was aimed at improving the performance in mathematics of a group of university students of biology who repeatedly failed the final examination of a compulsory course in mathematics. The main difficulties of these students seemed to be metacognitive and affective in nature. The training therefore worked on metacognitive and affective features: knowledge about cognition, monitoring, beliefs, emotions and attitudes. The intervention was successful: at the end of the course all students passed the examination that they had failed so often. The results also suggest that it may be possible (and necessary) to ‘teach learning to learn’ mathematics.     

A commutative l-monoid for classification with fuzzy attributes

In this paper the properties of an algebraic fuzzy structure are investigated in detail. The structure is suitable for modeling classifications through clusters composed of conventional sets and fuzzy attributes. We show that the structure is an integral commutative l-monoid. The expressive power of the structure is such that several situations can be viewed as classification problems, e.g., fuzzy assessment of students, user modeling for fuzzy hypermedia systems, spaces of the cognitive states of the user of a tutoring system, financial investments, medical diagnoses. The problem of getting the unknown classification beginning from the final classification is deeply investigated and it is shown that the problem is strictly related to the solution of an equation in the monoid. Thus it is possible to construct procedures of the type `what happens if' which permit to attain significant results both on the theoretical side and the applicative one. Finally, by means of this approach, both the absolute and the relative relevance of an attribute are defined and evaluated, given a universe of discourse and a set of classifications. Moreover, this couple of features allow to develop a sophisticated analysis of how a new attribute can be obtained beginning from a set of attributes.