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«1. Introduction No more marking – online assignments are here! Can it really be that simple? We are three university level math instructors that ...»

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On Online Assignments in a Calculus Class

1. Introduction

No more marking – online assignments are here! Can it really be that simple? We are three

university level math instructors that describe our experience creating and managing online

assignments. Our goal is to elaborate on various issues related to this learning tool and to convince

our reader that online assignments have their place in post-secondary pedagogy.

We use the term online assignment to denote the set of questions and problems that are posted, submitted, graded, and recorded electronically through a course learning management system (LMS) of choice. Over the last decade, the use of online assignments in teaching postsecondary courses has been extensively researched in the areas of physics and chemistry (Carrier & Jones 2004; Riffell & Sibley 2004). Recently, research papers about the use of online assignments in mathematics have started to emerge (Krbavac 2006; Ledder 2009; Segalla & Safer 2006; York, Hodge & Richardson 2008). In general, online assignments have been seen as part of a move towards innovative, technologically improved and enhanced teaching and learning at many universities and colleges (Carrier & Jones 2004).

A common theme in research on the role of technology in classrooms is a challenge that twentyfirst century instructors face: How can technology be used in teaching so that students benefit the most? Or as Guess (2007) puts it,“The question, then, is: How can educators adapt their teaching methods to emerging technologies? And should they?” We note that this challenge about the place of technology in teaching and learning is not new. In 1972, Lacey wrote about using films in a classroom and was faced with the same dilemma on the use of technology for the purpose of teaching and learning. The aim of this article is to support the view that educators should adapt their teaching methods to emerging technologies only if that change would benefit student learning and would be feasible for instructors and institutions. We intend to demonstrate that many of the challenges posed by using technology in teaching and learning can be overcome with careful planning.

In this paper, we describe our experience creating and utilizing online assignments for several calculus classes at Simon Fraser University (SFU). We present our findings on the suitability of available software by considering the needs and perspectives of instructors, students, and administrators. We provide a list of questions to guide an instructor in choosing online assignment problems and a list of benefits and challenges associated with this endeavor. In an attempt to contribute to the discussion about the role of technology in classrooms, we propose a balanced use of both paper and online assignments in math classes. In support of our findings, we offer survey data from our science and engineering calculus courses as well as from social science calculus courses over the span of five years. We discuss recent developments and future possibilities for utilizing online problems, and conclude with a summary of the most important answers to our selfimposed questions about this emerging teaching tool.

2. Traditional Settings In the past two decades, students in first and second year math courses at SFU have been asked to submit weekly assignments that consisted of 20-25 problems for a total of 10 assignments per semester. Teaching Assistants (TAs, also known as tutors) marked these assignments. Due to large class sizes --- typically between 150 and 500 students --- and the limited number of TA hours for assignment-marking purposes, only one or two questions per assignment were marked completely.

Hence, students received feedback on a small fraction of their homework. To encourage assignment submission, up to five marks out of ten were awarded just for completing the assignment. The remaining marks were used for grading the chosen questions. Since the combined homework marks contributed 10% of the overall course mark, up to 5% of the total course grade was given as an incentive for nothing more than assignment submission.

The cost for the Department of Mathematics at SFU with this type of marking was significant. For example, in a typical fall semester there are about 2,000 SFU students enrolled in first year calculus courses. TAs are usually paid for one minute per question per assignment paper they mark. This minute not only includes actually grading the problem, but also the time allowed for collecting assignments, recording marks, and returning assignments. Just marking one question on all of the ten assignments requires about 333 TA hours per semester.

The above arrangement led to a situation in which students practically received no feedback on their assignments, instructors were not assured that students practiced concepts and developed skills sufficiently, and the costs of marking those assignments seemed inefficiently spent. These facts were the main reasons that the Department of Mathematics at SFU decided to introduce online assignments for its science/engineering calculus courses in summer semester 2003.

SFU subsequently adopted online assignments for the social science calculus course sequence in fall semester 2004. Indeed, similarly motivated moves to online assignments have been documented in the past decade, especially for physics courses (Kashy et al. 1993; Thoennessen & Harrison 1996; Bonham, Beichner & Deardorff 2003; Demirci 2007), and, recently, for mathematics courses (Seppälä, Caprotti & Xambó 2006; Segall & Safer, 2006; York, Hodge & Richardson 2008; Ledder 2009).

3. Software in Support of Online Assignments There are numerous educational software packages that support online assignments. It has become a standard for publishing companies to accompany their calculus and other math and science textbooks with software that supports online assignments. For example, software known as PHGradeAssist accompanies the standard textbook Calculus Early Transcendentals Version (Edwards & Penney, 2002). Another example is the software package known as WebAssign, originally developed by North Carolina State University, that accompanies the popular textbook Calculus: Early Transcendentals (Stewart 2002). There are also commercial companies that offer to create questions as part of their courseware packages or to utilize existing question banks for online assignments such as Lyryx Learning based in Calgary, AB, and Maplesoft from Waterloo, ON. In addition, many commercial learning management systems like WebCT come with online quiz creation capabilities. Beyond commercial software options, there are also open source packages that support online assignments. Two examples of open source software are LON-CAPA developed by Michigan State University, and WebWorK, which originated at the University of Rochester Department of Mathematics.

The following is a list of some of the basic features of existing software packages that support

online assignments:

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Question banks may be provided or purchased, as in the case with software that accompany textbooks, or shared, as is the case with all open source software. Almost all packages allow instructors to create their own questions. The questions appear in a variety of formats, including multiple-choice, true/false, fill-in-the-blank with a formula, numerical value, string answer, or open-ended. Communication tools usually include chat rooms, a discussion board, and an internal mailing system.

4. Questions to Guide an Instructor in Choosing Online Assignment Problems It is our experience that an instructor preparing a set of problems for an online assignment faces the same challenges regardless of whether the questions are created from scratch, modified from previous versions, or chosen from an open source or commercial question bank. We list below

some questions that arise when an instructor is creating an online assignment:

How can we choose online assignment questions that will best complement other • elements of the course such as lectures, readings, paper assignments, and exams?

In an attempt to make our students read the course textbook and go through their class notes on a regular basis, we created a series of true-false questions, matching questions, and fill-in-the-blank problems. These were constructed so that the online questions are concept driven and not dependent on any textbook. Furthermore, in an attempt to offer more guidance to our emerging problem solvers, we composed questions that required students to enter missing details of each step of the solutions.

What types of online questions are best suited for learning various mathematical concepts • and skills?

Different concepts may require different types of questions. For example, the problem to evaluate 1+1 would require a numerical response question. The problem to find the average of numbers 1 and a would require the formula response question. In this case, the system would accept correct answers written in different forms, like 1+ a a, 0.5 ⋅ (a + 1), + 0.5, and so on.

What types of online questions are the most appropriate for testing complex math ideas?

• We find that we often resort to combining several question modes to present a more € complex problem. A typical example in € year calculus courses is the problem of first € sketching a graph of a given function. Online problems of this type that we constructed are combinations of several multiple-choice, fill-in-the-blank, formula, and numerical response questions. After a sequence of correct answers, upon pressing the final click, the graph of the function in the question is displayed! We have also learned that feedback from students can be used to enhance the presentation and tease out the best format. In the system that we currently use, each online question is accompanied with a feature called Send Feedback. This feature has three settings: feedback to resource author, question about resource content, and question/comment/feedback on course policy. For example, students pointed out that the solution of a particular problem required several steps, but that they were asked to submit only the final answer. If the answer was wrong, then the students would not know where they went wrong and could not address their own problem solving errors. We responded by breaking down the problem to require step-bystep answers and give hints along the way based on common errors.

Can online questions be used to communicate mathematical ideas, i.e. be used to • introduce key concepts or applications that have not been seen in the lecture or the textbook?

The possibilities around creating an online assignment question are limited only by the degree of the author’s imagination and the intention of the instructor. Online assignment questions can contain pictures, applets, web links, animations, clips from movies, and so on. This flexibility allows the instructor to present material relevant to course topics that would otherwise be beyond students’ reach. For example, an online electronic medium is the right environment to introduce some of the concepts that play important roles in the contemporary mathematics, like a visual proof of a mathematical fact or some basics of experimental mathematics. Furthermore, we use online assignments to introduce our students to the correct way of presenting written mathematics. For example, if we ask a student to prove a certain property by using a particular theorem, we write the answer in the proper form, using the appropriate mathematical phraseology and notation. In the text of the proof we left blanks that the student needs to fill-in. Hence the student is put in a position of being an active participant in the development of a mathematical proof.

• To what degree should online assignments be used for drill exercises?

Online assignments are an ideal medium for repetitious exercises that are often a necessary step for a student to perfect a math skill or procedure. The challenge for an instructor is to resists the temptation to overuse this aspect of online assignments and make the task too monotonous for a student. Our experience with online drills is that less is more. However, when accompanied with a string of appropriate hints these exercises become an efficient and convenient tutorial tool. For example, consider the problem to evaluate 3 − 2 ⋅ (−1). There are three common erroneous answers: 0, 1, and -1. The first answer, 0, is obtained when students do not note the multiplication sign. Hence if a student enters “0” as their answer, the question “Did you do the multiplication?” would pop up. The answer “1” would be accompanied with the hint, “What can you tell about € the product of two negative numbers?” The answer “-1” would be matched with the hint, € “What is the order of operations?”

• Are the available online questions suitable for any teaching styles?

In our experience the answer to this question varies from instructor to instructor. The responses range from, “I’ve been successfully teaching in my own way for years” to “The online questions are too simple (or too complicated) to really contribute to student’s learning.” Although not everyone sees the value of using online tools for testing mathematical knowledge, we have experienced that some of our most skeptical colleagues started using online assignments in their classes after getting involved in testing and editing the existing problems or creating online problems on their own. One of the reasons that the pool of online assignment questions in our institution has been steadily growing is that instructors are creating new problems to materialize their own views and ideas about presenting certain topics.

• Are the available online questions suitable for any learning styles?

Students work on online assignments remotely, within the provided time frame, usually a few days. The fact that each student works on their own time, requires the student to be self-disciplined, organized, independent, and resourceful. The resources that a student is expected to use are the textbook, lecture notes, Discussion Board contributions, and the math help centre supporting the course. The feedback we received from students is that online assignments force them to read the textbook and lecture notes on a regular basis, see Figure 1, and helped them to learn the course material better, see Figure 2.

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