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Mathematical Structures and Modeling 2016. N. 1(37). PP. 80-86
UDC 519.237.5,378
to predict students’ success in the next class, we need to go beyond (reliable) grades from the previous class: an empirical study
A.M. SiEMSSEN
Student, e-mail: [email protected]
O. KoshElEva
Ph.D. (Phys.-Math.), Associate Professor, e-mail: [email protected] University of Texas at El Paso, El Paso, TX 79968, USA
Abstract. In a two-class sequence, it is important to be able to make sure that students graduating from the first class can succeed in the second one. If the cut-off for success in the first class is set too low, many ill-prepared students are allowed to take the second class and are thus doomed to fail it. If this cut-off is set too high, medium-prepared students who could potentially succeed in the next class waste time by unnecessarily repeating the first class. From this viewpoint, it is desirable to be able to predict the student’s success in the second class based on this student’s (reliable) grades in the first class. On the example of a two-class introductory computer science sequence, we show that in some situations, a reliable prediction is not possible. Namely, to get a good prediction, in addition to (reliable) grades for the exams, grades that reflect the students’ ability to solve simple problems, we also need to take into account less reliable (and more cheating-prone) grades on take-home assignments such as labs, grades that reflect the students’ ability to solve complex problems.
KEywords: linear regression, predicting student success, grading.
1. Introduction
NEEd to prEdict studEnts’ success. For many classes, the main objective is to prepare a student for the next class. Usually, if a student has a passing grade in the previous class, this student is eligible to sign up for the next class. Otherwise, if the student’s grade in the first class is unsatisfactory, the student has to repeat this class - or, if the student has already tried that several times and failed, the student is dropped from the program.
In many cases, this procedure works well. Usually, when a student shows excellent or very good knowledge of the material from the first class, this clearly indicates that the student is ready for the following class. On the other hand, if the student’s results in the first class are bad, this student is clearly not ready for the second class.
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However, in borderline cases, it is often not easy to predict the student’s success: sometimes, a student with a barely passing grade in the first class turns out to be not really ready for the following class. In view of such situations, it is desirable to predict the student’s success in the following class as accurately as possible.
How student’s success is predicted now. At present, the student’s success is predicted based on the student’s grade in the previous class.
In each class, there are usually several occasions on which the student’s knowledge is gauged: exams, quizzes, labs, projects, homework assignments, etc. The grade g for the class is usually a weighted combination of grades g1,... , gn for all these instruments for gauging the student’s level of knowledge:
n
g = wo + Wi ■ gi,
i=1
with appropriate weights wi.
How can we improve the current way of predicting the student’s success.
The weights assigned to different instruments are usually selected based on the instructor’s subjective understanding of the importance of different topics.
Because of this subjectivity, these weights are not always a very accurate representation of the material’s importance:
• the material that the instructor believes to be very important for the following class may be, in practice, not that important, and
• vice versa, the material that the instructor believes to be not very critical may turn out to be more important than the instructor thinks.
Thus, to get a more accurate prediction of student’s success, it is desirable to replace the subjective weights with the more objective weights - weights for which the corresponding weighted combination is the best fit for the grade in the next class. In other words, once we know, for sufficiently many students k = 1,... ,K, their grades g(k) for different assignments i from the first class and their grades s(k) for the second class, we can use, e.g., the Least Squares techniques to find the values wi of the weights which provide the best fit for the following equalities:
n
s(k) w ao + ^ a ■ g(k).
i= 1
This idea was proposed and analyzed in [2].
What we do in this paper. Our original idea was to simply apply the above general technique to a specific example of first two classes from the Computer Science introductory sequence. We expected a routine application, but what we found out was rather unexpected: that to adequately predict the student’s success in the following class, we need to go beyond reliable grades. This surprising result is presented in this paper.
82 A.M. Siemssen, О. Kosheleva. To Predict Students’ Success in the Next...
2. Case Study: Description
General description of the situation: CS1 followed by CS2. For Computer Science students in the University of Texas at El Paso Computer Science program, the first computer science class “Introduction to Computer Science” (CS1) is a prerequisite for the next class “Elementary Data Structures and Algorithms” (CS2). This sequence is in line with the 2013 Computer Science Curriculum [1] approved by the Association for Computing Machinery (ACM), the main Computer Science organization.
As a test case, we considered students who successfully took CS1 in Fall 2014 and then took CS2 in Spring 2015.
How the knowledge of CS1 students was gauged. The knowledge of CS1 students was gauged by three midterm exams, a final exam, and 13 labs. The overall grade is a weighted combination of the grades for the exams and of the grades for the labs. To pass the class, the students must gain at least 70 points out of 100.
Midterm exams and final exams are performed in-class. All exams are proc-tored, so we are confident that the results of each exam properly reflect the students’ knowledge. Since the exam time is limited, problems presented at an exam have to be reasonably simple, to enable students to successfully solve them during the time allocated for the exam.
In contrast, labs are performed by students on their own time. Usually, a lab is due a week after it is assigned. Each student’s work is supposed to reflect this student’s individual work. Students are prohibited from seeking help with working on the lab. However, while the students are not allowed to explicitly ask for specific help with the lab assignments, they are encouraged to ask instructors and teaching assistants (and more advanced students from the class) for general help with understanding the material and with solving similar problems. Since students get help (indirect but still help) while working on the labs, the grade on each lab does not necessarily adequately reflect the student’s ability to individually solve the corresponding problem: in our experience, there is no guarantee that without outside help students will be as successful in solving a similar problem.
Since we wanted the overall grade to reflect the student’s individual knowledge, we were hesitant to give much weight to the labs when computing the overall grade for CS1. As a result, each lab was worth only 2 points out of 100, so that overall, the grade for all 13 labs could contribute, at best, to 26 points out of 100. An additional reason not to assign too many points for the labs is that labs are not proctored, so assigning too many points for the labs would create a temptation for cheating. With the current 26 points assignment, even if a student cheats on all the labs, this student still needs to show reasonably good knowledge on other assignments to gain 70 points needed to pass the class.
How the knowledge of CS students was gauged. Two sections of CS2 were taught by two different instructors. While the two instructors agreed on what level of knowledge corresponds to passing the class, they used different weights to assign grades for individual assignments and different thresholds for passing. To
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compensate for this difference, we multiplied the grade of the first instructor by 1.07 and the grade of the second instructor by 1.03. This way, the passing grade of both instructors becomes equal to the same 70 points threshold as for CS1.
Resulting grades. The resulting grades - sorted in the decreasing order by the re-scaled CS2 grade - are presented in the next page’s table. In this table:
• 6j is the grade for the i-th midterm exam (out of 100),
• l is the grade for the labs (out of 26),
• в/ is the grade for the final exam of CS1, and
• s is the (re-scaled) grade for the second class (CS2), also calculated out of 100.
Comment, Please note that some exams and labs include extra point questions, so some students got more than 100 points.
3. Case Study: Analysis and Its Results
Least square regression. We used least squares to come up with a linear function that provides the best fit for the above data:
s = ao + a • в\ + Й2 • в2 + аз • вз + ai • l + a/ • в/.
As a result, we got the following values:
ao a1 a2 a3 ai a/
-62.02 0.05 0.01 0.20 4.36 0.14
We see that the coefficient at the lab grade l is much larger than the coefficient at the exam grades, which means that the grade on the labs is a much more important predictor of the success in the next class than the grade on the previous class’s exams.
To make a fair comparison, let us re-scale the lab grade l, i.e., replace the
original grade l whose maximal value is 26 with a re-scaled lab grade gi = — • l
26
whose range is from 0 to 100 (the same as for each of the exams). Then, the coefficients of the re-scaled linear regression
s = ao + ai • ei + a2 • в2 + аз • вз + aai • gi + a/ • в/
take the following values:
ao a1 a2 a3 ai a/
-62.02 0.05 0.01 0.20 1.13 0.14
84 A.M. Siemssen, О. Kosheleva. To Predict Students’ Success in the Next...
el e2 e3 t ef s
96 108 87 25 102 112
95 100 95 27 102 107
91 101 95 27 109 106
93 102 75 26 107 106
89 107 117 27 103 103
93 106 100 26 102 100
90 0 90 26 86 96
79 99 92 26 98 93
96 97 113 27 106 93
85 99 98 25 82 92
93 98 98 24 94 91
89 100 78 25 98 91
76 92 85 25 89 90
86 99 107 27 10 90
83 80 103 25 94 89
90 101 83 27 95 88
85 106 103 27 101 87
94 97 92 25 88 85
98 98 100 26 101 85
86 101 60 27 51 82
89 97 83 26 105 82
85 82 92 23 95 81
84 96 85 26 98 81
81 93 65 24 94 79
90 84 85 26 99 79
90 98 87 26 76 78
86 76 77 24 84 77
88 91 92 25 86 77
81 97 85 25 82 75
51 87 90 23 92 71
90 101 83 23 81 71
85 107 115 22 84 70
96 83 80 25 99 68
80 81 68 20 70 68
94 92 88 20 76 64
96 83 87 24 108 64
83 92 92 20 82 59
85 87 68 26 88 49
89 52 57 18 92 46
96 97 88 18 89 37
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The coefficient a'£ = 1.13 at the re-scaled lab grade g£ is more than 5 times larger than the largest coefficient a3 = 0.20 at the exam grade. In this sense, we can say that the lab grade is at least 5 times more important to predict the grade in the next class than the grades on the previous class’s exams.
Conclusion. At the end of CS1, we have:
• grades for the exams which provide a very reliable knowledge of the students’ ability - but only about the students’ ability to solve simple problems;
• grades for the labs, which gauge the students’ ability to solve more complex problems - but much less reliably.
It would have been nice to be able to predict a student’s success in the next class based on this student’s reliable grades (i.e., grades for the exams). Unfortunately, our analysis shows that this is not possible: to get a good prediction, we need to go beyond reliable grades and take into account lab grades - which gauge the student’s ability to solve complex problems but which are not as reliable as the exam grades.
Acknowledgments. The authors are thankful to the faculty of Computer Science department for providing their data, their advice, and their encouragement.
References
1. Computer Science Curricula 2013: Curriculum Guidelines for Undergraduate Degree Programs in Computer Science. ACM Press, New York, 2013.
2. Niwitpong S., Chiangpradit M., Kosheleva O. Towards optimal allocation of points to different assignments and tests // Journal of Uncertain Systems. 2010. Vol. 4, № 4. P. 291-295.
86 A.M. Siemssen, О. Kosheleva. To Predict Students’ Success in the Next...
для прогнозирования успеха студентов в изучении следующего предмета мы должны выйти за рамки (надёжных) оценок, полученных за предыдущий предмет: эмпирическое исследование
А.М. Симссен
студентка, e-mail: [email protected]
О. Кошелева
к.ф.-м.н., доцент, e-mail: [email protected]
Факультет педагогического образования, Техасский университет в Эль Пасо, США
Аннотация. Для двухуровневой системы учебных предметов важно убедиться, что студенты, успешно сдавшие первый учебный предмет, смогут достичь успеха во втором. Если границу успешного завершения первого предмета установить слишком низко, многие плохо подготовленные студенты будут допущены ко второму предмету и, следовательно, будут обречены его завалить. Если границу успешного завершения установить слишком высоко, студенты средней подготовки, которые потенциально могли бы успешно сдать следующий предмет, будут терять время на лишнее повторение первого предмета. С этой точки зрения, хотелось бы предсказывать успех студента при изучении второго предмета на основе его (надёжных) оценок за первый предмет. На примере двух уровней предмета «Вводный курс компьютерных наук» мы покажем, что в некоторых случаях надёжный прогноз невозможен. А именно, чтобы получить хороший прогноз, в дополнение к (надёжным) оценкам за экзамены — оценкам, которые отражают способность студентов к решению простых задач — мы также должны учитывать менее надёжные (и чаще получаемые обманом) оценки за домашнюю работу, например, за лабораторные задания — оценки, которые отражают способность студентов к решению сложных проблем.
Ключевые слова: линейная регрессия, прогнозирование успеха студента, оцени-
вание.
