Introduction
Completion of an introductory mathematics course is usually a
requirement in a wide range of postsecondary
programs in the sciences (such as chemistry and
computer science), social sciences (such as
psychology and economics), and professional programs
(such as business and engineering).
This situation presents special challenges for teaching
introductory and service courses, as these courses
attract a wide range of students. In a firstyear
calculus course, for instance, students may range
from mathematics majors to those seeking a single
mathematics elective. As a result, the background
and motivation of students taking the course vary
considerably, making it challenging for a course to
meet the needs of all students. On the one hand,
students with weak mathematical backgrounds might
not receive enough practice and become frustrated
with material seeming to pose an unrealistic
challenge to them, potentially even dropping the
course. On the other hand, stronger students might
not be pushed enough to realize their full
potential. The only feedback that top students would
receive is a perfect (or nearly perfect) score on
all their assignments and tests. Even though they
pay the same tuition as weaker students, and are in
a sense an equal partner in the process, they may
not receive much constructive feedback.
Larger institutions often attempt to solve this problem by
streaming students into a range of introductory
classes, although a wide range of abilities is
usually represented in each class nonetheless. This
approach is not practical at smaller institutions
because of limited enrollment.
The above problems are often compounded by the fact that,
according to surveys, about 85% of firstyear
students may feel some anxiety towards mathematics
(Perry, 2004). “Math anxiety” is a wellknown
fundamental impediment to learning in postsecondary
mathematics (Tobias, 1990).
Part of the solution to overcoming these challenges may lie
in the use of an online interactive and adaptive
learning environment that allows students to work on
a series of computer exercises. As they progress,
the software assesses each answer and provides the
student with instant feedback as well as setting the
difficulty of the next exercise based on the
response.
Adaptive technology is well established in testing. In fact,
most major computerbased standardized tests are
adaptive in nature. For instance, the Educational
Testing Service (ETS) offers computerbased adaptive
tests, such as GRE, GMAT and TOEFL. Private
certifications, such as those offered by Microsoft
and Oracle, also use these methods to deliver more
effective testing. The primary advantage of
computerized adaptive testing is that a much finer
distinction of ability can usually be made in a
shorter timeframe than with traditional testing. For
example, a twentyquestion True/False test would
traditionally provide only one of twentyone
possible grades from 0 to 20. However, in a
twentyquestion adaptive test the next question
asked could depend on all the previous answers, thus
doubling the number of possible distinct tests at
each step. Accordingly, if two students start with
the same initial question, they may each potentially
write one of 2^{19}=524,288 distinct tests.
In a nonTrue/False test, an even larger number of
distinct tests is possible if incorrect answers are
subclassified.
The focus of this article is on creating interactive and
adaptive assignments. Nonetheless, it is believed
that benefits similar to those of adaptive testing
can be derived from using adaptive technology as a
teaching tool incorporated into practice exercises
and assignments. The traditional approach to
adaptive testing has been to create a large bank of
multiplechoice questions as well as a method for
classifying the student's level, and adjusting
successive questions accordingly.
There are many wellknown pedagogical arguments
against using multiple choice questions, ranging
from not wanting students to work backwards from the
answers to wishing to avoid increased measurement
error due to random guessing that could make
branching decisions in adaptive testing problematic
(Bridgeman, 1992). Furthermore, producing large
banks of questions and solutions can be
resourceintensive, especially in the sciences,
where questions and solutions often require
nontextual information such as complex notations,
equations and diagrams. In quantitative science
courses, however, it is often possible to create
algorithmically generated questions. In calculus,
for example, a student could be presented with a
random polynomial to differentiate, while in
chemistry a student could be provided with a
chemical equation with random coefficients to
balance. In these cases, the student's answer could
be compared for correctness against a general
solution for that type of problem. A virtually
unlimited number of similar questions could be
produced from a single template. Note that, if the
underlying software is powerful enough, it is not
necessary to rely on multiple choice responses.
Having a virtually unlimited choice of possible questions
could make it possible to create adaptive
assignments. The following are examples of how an
assignment could be made adaptive:
·
Changing the parameters in a template to make
questions more challenging as a student improves.
For example, a general polynomial factoring question
could evolve from factoring quadratics to factoring
cubics.
·
In
a multistep question, the number of steps to assist
the learners could decrease as they improve.
·
The
size of the pool of templates could increase drawing
material randomly from a larger portion of the
course. The rate at which new questions are
introduced and reviewed could be adapted to an
individual's learning curve using artificial
intelligence algorithms.
Technology
An opensource (under a
GNU Public License) adaptive learning platform
called
Xero is being developed and is expected to be
available in the fall of 2007. It supports the
aforementioned adaptive models, using Bayesian
learning and neural network algorithms to aid with
parameter selection.
Currently there are no robust opensource software packages
for adaptive mathematics. While it is possible to
create simple interactive parameterized questions in
a number of platforms, such as WebCT/Blackboard,
there are two generalpurpose opensource
interactive environments:
WeBWorK (see Hauk and Segalla, 2005) and
WIMS (Xiao, 2001; Khaznadar, 2005) which can
create fairly complex mathematics questions. WIMS is
the more powerful of the two environments, and Xero
is derived from WIMS. WIMS has been
reviewed in MERLOT's Learning Materials
Collection, and several individual WIMS activities
have also been individually reviewed. With a few
changes, WIMS materials can be imported into Xero.
Xero provides much of the functionality of a course
management system: announcements, file management,
chat rooms, message boards, and functionality for
online tests, quizzes and assignments. However,
the main focus of Xero is providing a facility to
create adaptive templatebased exercises and an
environment for adaptive online assignments. The
Xero server interfaces with many wellknown
opensource mathematics software packages. For
example, the server uses, among others, computer
algebra systems such as Maxima (comparable to Maple)
and PariGP for symbolic calculations, the TeX
mathematical typesetting system for displaying
mathematics, and Gnuplot to create 2D/3D plots.
For example, a random function could be generated (say a*x^{b}*e^{cx+d}+
same function. A single
general function with enough terms could represent
virtually any function thstudent could be asked
to differentiate in an introductory calculus course.
For example, in the above function, letting c=d=0
and b=2 would provide a general quadratic function,
while a=1, b=f=g=0 would produce a general
exponential function with linear power.
Graphical
questions could also be parameterized. For example,
the exercise depicted in Figure 1 is a Xero exercise
that has been converted from the
Graphical Derivatives WIMS exercise reviewed by
MERLOT. In this exercise, a random function is
drawn, along with a number of functions related to
its derivative. Students are asked to identify the
relationship between the functions. Each time
students answer the question, they are presented
with the correct associations. This exercise can be
very effective in developing the intuition of
calculus students in service courses, especially
when they practice with a large number of these
examples. What follows is an example drawn from a
mathematics service course for computer science
students.
Example of Challenges in Teaching a Mathematics Service
Course for Computer Science Students
Computer Science students are often required to take several
mathematics courses, such as calculus and discrete
mathematics (which usually includes counting
techniques). The question, “How many ways can the
letters of ABRACADABRA be rearranged?” is a standard
discrete mathematics example that serves as a
counting model. The importance of this example was
emphasized in lectures, as it emphasizes valuable
counting techniques, and several questions with
different words to rearrange or objects in place of
letters were assigned as homework. Then, the
question “How many ways can 11 identical jelly beans
(5 Red, 2 Black, 2 Purple, 1 Orange, 1 Green), be
arranged in a row?” was placed on a test. Only about
20% of the students were able to answer it
correctly, even though it is identical to the above
question, with letters replaced with colored jelly
beans. In general, it appears that students in the
course did not internalize the above counting model.
This indicates that it may be necessary to assign more
questions of each type. There are, however, a large
number of counting techniques that must be learned
in a discrete mathematics course, and it is not
possible to provide more than a few questions of
each type on a typical assignment due to limited
marking resources as well as concerns about
assigning too much work to students in a service
course. Experience suggests that the reaction of
students can vary depending on whether the course is
for mathematics majors or for other students. In
courses taught to mathematics majors, when workload
or difficulty increases, students often exert more
effort and grades remain mostly unchanged. In
service mathematics courses, however, greatly
increasing workload or difficulty tends to result in
a combination of poor retention and decreased
grades. The anxiety that many students have towards
mathematics may be affecting their motivation. To
attempt to solve this problem, computerassisted
assignments were explored.
Implementation of Online Assignments
Online assignments were implemented in two service courses: a
firstyear applied calculus course with 20 students,
which was taught completely online and restricted to
nonmathematics majors; and a traditional
secondyear discrete mathematics course with 44
students, which was taught primarily to computer
science majors.
In
the calculus course, all assignments and tests, as
well as the final exam were online and used the Xero
technology mentioned above. In the discrete
mathematics course, online and paper assignments
were alternated such that the second and fourth of
five assignments were online. The material covered
on the assignments had significant overlap, since
some topics, such as techniques of proof, make poor
online assignments as their grading needs to be
subject to human judgment, while many quantitative
topics require repeated practice and so make better
online assignments. An anonymous paper survey was
administered in class following each online
assignment in the discrete mathematics class.
Surveys were not conducted in the calculus course,
due to logistical and other challenges involved in
creating an anonymous online survey. Hence, the
results that follow are primarily from the discrete
mathematics class, but are, nonetheless, consistent
with the results from the calculus class.
In the first discrete mathematics assignment, questions were
grouped in three groups of ten under the topic
headings: logic, sets and functions, and sequences
and series. Each time a set of questions was
attempted, a randomized series of questions were
generated. Students were required to answer the
questions in order. Each time a question was
answered, a solution or answer was immediately
displayed before the next question was attempted
(see Figure 2). After each set of 10 questions was
completed, students would be able to reattempt the
set as many times as they wished with new randomized
versions in order to improve their grade. In order
to provide more motivation to practice, the best
grade was the one that was kept. The second
assignment contained two sets of questions, one with
15 questions from combinatorial counting and the
other with 10 questions from probability.
There
was one major difference between the assignments.
The first assignment consisted mainly of questions
for which there were no examples in the lectures or
the textbook. The assignment was designed to be a
learning tool for students to learn new material.
However, in the second assignment a different
approach was attempted. Questions were chosen to
reinforce concepts taught in the lectures and book.
In each case, students had two weeks to complete the
assignment, the same as for a paper assignment. A
typical paper assignment had 15 questions, some with
multiple parts. Accordingly, a paper assignment
might effectively have the equivalent of 2530
online questions.
Below is an example of possible questions that can be
generated from a template for a counting problem:
How
many ways can 10 {distinct  identical}
{books  containers  jars} be placed {in  on} 5
{labeled  unlabeled} {boxes  shelves} if the order
that the objects are placed {in the boxes  on the
shelf} {matters  does not matter} such that any
{box  shelf} can have {any number of  at most 4
 at least 2} {books  containers  jars}.
In this case, integers in bold were randomly selected, and in
each presentation only one word/phrase from every
bracketed set was selected. For each attempt at a
set of counting problems, 15 templates were selected
from 33, and presented in a random order. All
templates could produce a variety of different
possible questions. For students, this created the
impression of there being an almost unlimited number
of different possible questions. A common comment on
student surveys, as expressed by one student
(student's emphasis) was a follows: “I don't think
I EVER got the same question twice.”
As online assignment questions for each student were randomly
generated, students were allowed to discuss possible
solutions to problems with each other. To further
encourage students, a message board was set up that
allowed for communication using mathematical
symbols, and bonus marks (ranging from 1 to 4
percent of the assignment value) were awarded to the
most active participants. Online office hours were
also held regularly using a mathematical whiteboard
program called enVision (Pollanen, 2006; Hooper,
Pollanen & Teismann, 2006), in which students were
allowed to participate anonymously.
Learning Outcomes
Traditional paper assignments usually serve a dual role: as a
learning tool, and as an assessment tool. By
allowing an unlimited number of attempts for the
online assignments, the emphasis is shifted towards
their serving as a learning tool. The primary method
of determining the effectiveness of the approach was
student surveys. One was conducted after each of the
online assignments in the discrete mathematics
course.
Survey Results:
In Table 1 (below), we summarize the quantitative results
from each of the surveys. The data on the amount of
time spent and number of questions answered is from
student estimates. The average number of questions
answered on assignment #2, as estimated from the
group of students participating in the survey, is
265. This is relatively close to the 252 questions
per student for the entire class, as estimated from
Webserver logs. It is also interesting to consider
the assignment averages. The assignment average for
all assignments in the previous year was 72.1%,
while the averages on the first and second online
assignments were respectively 78.9 and 72.0%.
Table 1: Summary of quantitative results from
surveys conducted after assignments #2 and #4

Assignment
#1 (Paper) 
Assignment
#2 (Online) 
Assignment
#4 (Online) 
About the
Same 
Yes 
No 
Average Time Spent 
7.7 hours 
6.8 hours 
9.8 hours 



Average Number of Questions Answered 
30 
265 
210 



Which of the first two assignments was more difficult? 
13%
(3/23) 
74% (17/23) 

13% (3/23) 


Which of the first two assignments was a better learning
tool? 
17%
(4/23) 
74% (17/23) 

9% (2/23) 


Which of the first two assignments did you enjoy
more? 
14%
(3/22) 
82% (18/22) 

5% (1/22) 


Should there be more online assignments? (Asked
in the first survey) 




95% (20/21) 
5% (1/21) 
Which online assignment was more difficult? 

17%
(4/23) 
70% (16/23) 
13% (3/23) 


Was the second online assignment an effective learning
tool? 




100% (23/23) 
0% (0/23) 
The surveys also allowed students to provide
qualitative feedback, the results of which support
the quantitative results. It would appear that
online assignments were well liked, as backed up by
qualitative comments from the survey:
“I
love online assignments.”
“I
really enjoyed this format.”
Although students thought they were difficult:
“Crazy hard questions... very good way of doing an
assignment.”
“It
was time consuming but I didn't mind.”
“Assignment #2 was challenging but helpful to see
the right answers right there.”
An important aspect of this seems to be that the solutions
were presented immediately, which aided in learning:
“Got immediate feedback, which corrected some of my
misunderstandings.”
“It told me right away if I was wrong and gave me the right
answers. Then I could figure out where I went wrong
and not make the same mistake next time.”
“In a lot of cases something which I thought I knew 100%
ended up being partly wrong. Thus I caught it.”
“I
was able to improve right away rather than wait a
couple of weeks.”
Overall, most students indicated that they learned more from
the online format:
“All assignments should be online because we learn more and
the tests could be made a bit more difficult to
compensate.”
“I
found I took more away from [the online
assignment].”
“It
really challenged what I thought I knew well.”
It appears from student comments that students were very
motivated to work on online assignments, compared to
regular paper assignments. This will now be analyzed
from the standpoint of Keller's ARCS Model of
Motivation.
ARCS Model of Motivation
During the 1980s Keller (see Driscoll, 1993 for a review) in
a series of papers (Keller, 1983; Keller, 1984)
developed the influential ARCS Model of Motivation
for instructional design. ARCS stands for Attention,
Relevance, Confidence, and Satisfaction, and
represents a set of conditions which must be met for
a learner to be motivated to learn.
Attention:
In order to motivate students to learn, you must
gain and keep the learner's attention.
As demonstrated above, most students enjoyed the online
format more than paper assignments, which is an
important factor in keeping their attention. Keller
suggests that varying the presentation and providing
visual stimuli is important as well, something that
students mentioned:
“I appreciated the visual aspect of the assignment, the
chance to see the solutions. The slight variations
of questions on each turn means that you have to
understand how the question is done. Plus, I found
the online aspect quite enjoyable.”
A large number of students indicated that the assignment was
entertaining or had “gamelike” qualities:
“In
an odd way it was more fun and entertaining.”
“Fun, learned immediately if I was doing things wrong. This
made it like a challenging game.”
A considerable amount of research (see Randel, Wetzel, and
Whitehill, 1992) has been conducted regarding the
positive impact that games can have on learning.
While they may be effective in maintaining
attention, they can have significant drawbacks:
complex educational games are often expensive to
develop and can be time consuming to use. It still
remains difficult at this time to see how they can
replace a substantial portion of any advanced
mathematics course, where multiple and diverse
topics can be covered by an instructor in any week.
It is interesting to see, though, that these online
assignments can act as valuable supplements to
traditional teaching and learning approaches by
changing the way that traditional assignment
questions are delivered to students; it appears
possible to maintain a high density of learning with
low development costs, and yet foster the motivation
associated with an atmosphere of gamelike learning.
Relevance:
In order to motivate students, it is important to
make the material relevant to the learner.
The number of topics that are required to be incorporated
into mathematics service courses, and the wide range
of student backgrounds often make it challenging to
incorporate many applications in these introductory
courses. Furthermore, students often find
applications frustrating until they have mastered
the basic underlying skills.
It is therefore necessary to consider the relevance of
assignments from a shortterm point of view. Does
the online assignment help a student meet their
immediate objective of doing well in the course?
This is not an easy question to answer as – although
decades of research have been conducted – the role
of assignments and homework on achievement still
remains unclear (Trautwein and Köller, 2003).
In order to increase immediate relevance, the concept of an
“open test” was attempted for the second online
discrete mathematics assignment: all the questions
had the same wording and form as questions that
students had practiced with. This made the learning
objective quite transparent. It was announced when
the assignment was assigned that the next test would
consist of tons changed.
Although the problems on the first section of the
test were substantially more difficult than the test
problems from the previous year, students performed
much better. On the second part of the test,
students performed about the same as students in the
previous year.
Even without an open test, it can be argued that online
assignments are better for preparing students for a
test. In a textbook, questions are already
categorized by subject and technique, so approaches
to questions can often be guessed at based on the
section of the textbook they are found in.
Furthermore, the questions in each section are often
arranged in order of difficulty or type. However, on
a test, any particular question may come from one of
many sections, and the difficulty cannot always be
guessed in advance; an online assignment can be much
closer to a test situation.
Confidence:
To motivate students, they must gain confidence from
the activity.
One concern with the online assignments was that, as
questions were substantially more difficult than for
regular assignments, most students did poorly on
their first attempt. See Table 2 below for the
average scores for the first five attempts. These
scores are typical of what we saw in the other
online assignments in both the calculus and discrete
mathematics courses.
Table 2: Average score on each of the first five
attempts of the second online assignment
Attempt 
1 
2 
3 
4 
5 
Average Score 
18.93% 
33.06% 
40.80% 
46.24% 
48.31% 
This gave rise to the concern that the assignment
would negatively affect student confidence. To
address this concern, on the second survey a
question was asked regarding the effect of
assignments on confidence. Of the 23 students who
answered the survey, 22 indicated that the
assignment had a net positive effect on their level
of confidence. Only one student indicated that it
had a negative effect on confidence. Some comments
from students were as follows:
“At first it seemed really tough, but I took my time and got
help (which is something I don't normally do) and it
worked out fine. I think overall it had a positive
effect.”
“When I did badly the first time it motivated me to do better
and try harder the next time.”
“[It built] confidence over time.”
An important factor in managing confidence is communication.
Many students mentioned that the message board was
very useful, and reports regarding the class average
on the assignment were helpful:
“Definitely motivated me, especially when I heard that I
wasn't the only one struggling.”
Many students also mentioned that despite initially
performing poorly, the fact that they saw their
score improving provided them with an effective
measure of how much they were learning, and
motivated them further. It is interesting to note
that students using a significantly above average
number of attempts were generally weaker students in
the course, however, relative to the other students,
they typically ended up doing much better on the
online assignment than on the paper assignments
Satisfaction:
To motivate students, the learner must be
satisfied with the outcome.
One way for students to be satisfied is if they feel that
they have ownership over the learning process. This
may take the form of reaching their desired score:
“It
caused me to keep working until I got a mark that I
was happy with.”
“The low grade the first few times makes me work
harder to get a better one.”
Some feel that they gain a sense of independence:
“We
can realize our mistakes and [correct] our errors
without [the professor’s] help.”
Others believe that they gain a deeper
understanding:
“I can try different questions of the same type. It helps me
to understand what the question is about... like the
essentials of a problem.”
Discussion & Future Research
The concept of “unlimited attempts” for online assignments
and quizzes is not new. This functionality exists in
course management systems like WebCT and interactive
mathematics platforms such as WeBWorK, and is
usually accomplished in one of two ways. One way is
to allow students to repeat an entire assignment
(which is usually multiple choice with limited
randomization) after viewing their score, without
telling them which questions they answered
incorrectly. Another method (Hauk and Segalla, 2005)
is to allow students to repeat, without seeing the
solution at each step, an individual question until
they are successful. This is the first study that
examines the impact of allowing students to repeat
highly randomized templatedriven assignments that
provide solutions or answers after every question.
The longterm goal of the Xero project is to create an
adaptive learning environment that can create an
individualized learning experience by fitting
assignments to personal learning curves and
motivational profiles. The method of allowing
students to repeat the assignment an unlimited
number of times can be viewed as a very elementary
form of adaptivity in that students requiring more
practice generally take more attempts than stronger
students.
A few of the question templates were designed to present
problems with multiple steps. These were
consistently the least favorite for students, though
few students gave reasons. Perhaps these sorts of
questions lose the “gamelike” qualities. This might
present a problem in creating adaptive assignments
using a multistep adaptive model, even though
pedagogically it seems like a good idea.
Another challenge in moving to fullyadaptive assignments
involves student perceptions regarding fairness. In
our assignments, students were made aware of the
method in which questions were generated. However,
how will a student react to having assignments with
different numbers of questions and questions of
different difficulties than other students that are
generated by complex artificial intelligence
algorithms, even when the goal is to bring them to
the same level as other students at the end of the
assignment? Even in this experiment, one student
commented that: “Some people seemed to have easier
questions. Probably just lucky, but slightly
unfair.” As a result of the large number of
questions answered by each student, from the law of
large numbers in statistics, it can be seen that the
expected deviations in grades as a result of
randomization should be insignificant. Perceptions
can, however, often be more important than reality.
One interesting perception that students had was about the
academic integrity of the assignment. Many students
commented that they thought it would be difficult to
cheat on the online assignments, as one student
wrote: “I also don't know how you could possibly
cheat? – Can't share answers with others.” Many
studies suggest that academic dishonesty is rampant
in universities (Lambert, Hogan and Barton, 2003).
However, most students seemed to think there was
less cheating on the online assignments. This
suggests that a lot of cheating on assignments may
be casual: it is easy to share answers on a paper
assignment, but much more difficult to have somebody
else complete an online assignment for you. Also,
using “open testing” might remove some of the
incentive to cheat. However, in discussions with
other faculty, it seems that many faculty distrust
the integrity of online assignments. This gap in
perceptions between students and faculty would be
worth considering further.
It is interesting to note that the student retention rate in
the discrete mathematic course improved from 81% to
91%, although it is difficult to draw a conclusion
from this. It is also noteworthy to consider that
about half of the students in the bottom 25% of the
class, the portion that might be considered “at
risk”, performed extremely well on the online
assignment by attempting it an above average number
of times. Perhaps these students have poor
mathematical skills and this assignment format
allowed them to get the practice they needed.
However, the other half of these atrisk students
did extremely poorly on the online assignment. From
logs it can be seen that they spent very little time
on the assignment, although some of them mentioned
to the instructor that they had worked very hard.
Perhaps these students do not realize how much
effort other students are spending on the
assignment, or are unable to accurately gauge their
own effort. It might be useful to provide each user
with a log of the amount of time they have spent on
the assignment, as well as an average for the class,
to see if this motivates students to work harder.
Summary and Conclusions
In this paper a model for online assignments in mathematics
service courses was presented. In the model students
are presented with highly randomized templatedriven
assignments as well as solutions or answers at each
step. The average difficulty level of the questions
is chosen to be much greater than that for a paper
assignment, but students are provided with an
unlimited number of attempts, as well as discussion
boards and incentives to discuss difficulties.
It is shown that students reacted positively to this model
and ended up answering 79 times more questions than
for a paper assignment of similar length. It is also
shown that this model is consistent with the ARCS
Model for Motivation, and results in a number of
positive learning outcomes.
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