ACTIMATH: A REMEDIAL COURSE IN MATHEMATICS ADAPTED TO
THE NEEDS OF THE STUDENT AND THEIR CHOSEN FIELD OF
STUDY
T. Neijens1, K. D’haeseleer1,
T. Stevens1, D. Coppens1, B. D’haenens1, L. Van Loon2, G. De Samblanx2,
G. Clarebout3, C. Biront4, E. Van Hoof5,6, L. Gielen7, A.Vermeyen1
1
KAHO Sint-Lieven (BELGIUM)
Lessius Mechelen (BELGIUM)
3
K.U.Leuven (BELGIUM)
4
HUBrussel (BELGIUM)
5
KHLim (BELGIUM)
6
LESEC (BELGIUM) (Leuven Engineering and Science Education Center)
7
KHLeuven (BELGIUM)
tim.neijens@kahosl.be, katrien.dhaeseleer@kahosl.be, tony.stevens@kahosl.be,
dimitri.coppens@kahosl.be, bart.dhaenens@kahosl.be, leon.vanloon@mechelen.lessius.eu,
gorik.desamblanx@mechelen.lessius.be, geraldine.clarebout@ped.kuleuven.be,
christian.biront@hubrussel.be, etienne.vanhoof@khlim.be, lut.gielen@khleuven.be,
annemie.vermeyen@kahosl.be
2
Abstract
The project ActiMath1 wants to close the gap between secondary education and higher education
concerning mathematics. Many starting college students fail their mathematics courses because their
foundation in mathematics is (too) weak. ActiMath’s goal is to make an interactive and adaptable
remedial course for the student, usable for different lecturers and bachelor/master courses.
Keywords: Maths, remedial, VLE, education, adaptive.
1
INTRODUCTION2
Many courses in colleges and universities have a need for mathematical concepts in one form or
another. Especially when mathematics is in a solely supporting role, problems occur. All studies
expect some basic foundations. In order to ensure that starting level, many colleges and universities
organize remedial math courses.
The need for such courses is well known. Bad results in these mathematically oriented courses often
lead to a delay in graduation, or even causes the student to drop out of college or university entirely.
Lecturers are often frustrated by having to explain simple mathematical concepts. This loss in time
leads to lower standards in the course.
For Flanders, most of these remedial courses are organized in a traditional way: theory lectures
followed by exercise session. This leaves little to no room for differentiation. Some participating
students do not need the extra lessons (although they are participating), while others have trouble
keeping up with the speed of the lectures. Summer courses tend to differ from college to college in
content and level, even if the field of study is the same. There is little to no cooperation between
lecturers.
While lecturers have the impression that these summer courses actually help the students to solidify
their mathematical knowledge, no research has been done to check this assumption.
1
This project was funded by the Association K.U.Leuven
2
These statements are based on a survey conducted on the lecturers of participating institutions.
2
ARTICLE SUMMARY IN BULLETS
Introduction
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Gap in math knowledge between secondary and higher education
Bad math results
No differentiation in content or learning style
Little cooperation between different institutions
Objectives
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Development of an adaptive remedial course
Stimulate cooperation between different institutions
Improve initial math knowledge and improve results
Approach
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Preparation
o Study current summer courses in content
o Study current summer courses in didactical approach
Technology and platform
o Toledo/BlackBoard as platform
 Common platform
 Testing and assessment
 Adaptive learning paths
 Multimedia
o Technology
 Automatic assessment
 Geogebra
 LaTeX
 STACK
Educational principles
o Scenarios
 Self-study, face-to-face instruction, cooperative learning
 Sample scenario
o Differentiation
 Learning style
 Entrance level
 Taxonomy (difficulty levels)
Evaluation
o Student evaluation
 Formative through entrance test and exercises
 Summative through final test
o Project evaluation
 Improvement of learning results
 Good student and lecturer experiences
Test runs
o September 2011 (partial), one year follow-up
o September 2012 (full)
Further research
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Determining an adequate entrance and final test
Preferred study method (grouped by subject)
Useability for other courses
Uniformization in course contents and initial expectations
Improving ActiMath through test results
OBJECTIVES
The main goal is to develop a remedial course for all participating colleges and universities
(Association K.U.Leuven). This remedial course needs to be adaptive. The contents and level have to
be catered to the needs of an individual student, taking into account the chosen field of study and their
personal knowledge of mathematics. For best results, the course should be useable in various
scenarios, catered to the learning style of the student ([3] Kolb (1984)), and to the teaching style of the
lecturer ([2] Ernest (1989)).
A second objective is to stimulate cooperation and sharing of information, material and experiences
between the lecturers of different colleges/universities, in order to heighten the effectiveness of
teaching in all participating institutions. This will lead to more uniform course descriptions and
expectations. This uniformity is, in turn, an asset for an association of universities and colleges.
Ultimately, we want to improve the foundation of mathematics and to close the gap between
secondary education and higher education. As a result, the failure rate in mathematically oriented
courses should be reduced in the starting semester/year. This should and will be tested in two testing
runs of the course.
4
4.1
APPROACH
Preparation
Through a survey, we studied the way the summer courses were given and the general opinion of the
lecturers. Summer courses for math are almost always taught from the same principle: theory first,
then exercises. Differentiation is minimal and only possible during the exercises.
In these courses, theory was provided through a syllabus and traditional lectures. Afterwards, or in
some cases integrated into the theory lectures, exercises are provided. Students solve these
exercises individually, while being assisted by the students sitting next to them and by the lecturer and
his teaching assistents should there be any questions.
As a second step we collected (again through surveys) the current course materials and tried to find
common principles and content, grouped by the chosen field of study. Discussing these stimulates
cooperation and consultation between the different lecturers. The differences between similar courses
(by similar we mean preceding the same course trajectory in college/university) are small. But these
differences actually allow to enrich the summer course by adding new exercises and viewpoints.
4.2
Technology and platform
4.2.1 Platform
ActiMath should be able to be used throughout the whole Association K.U.Leuven; one of the
requirements of the project is the use of a VLE (Virtual Learning Environment) (Toledo, based on
BlackBoard). Using BlackBoard to offer the course materials ensures that these materials will be
maintained, even if the initial project ends.
The use of such a common platform ensures that content is shared between all participating colleges
and universities in a structured way, since all lecturers have access to BlackBoard. This in turn eases
cooperation, even if the physical institutions are located in other regions.
Exercises can be monitored through BlackBoard. Preferably we will use a mathematics assessment
package (for example Maple T.A. or STACK) to give differentiated feedback (specific mathematical
errors are detected and commented). The choice will depend on financial and technological
(compatibility) constraints. An external site may be needed to do this, since it allows more possibilities
and is less rigid than a VLE.
The adaptive learning paths in Blackboard are an asset we will use fully in ActiMath. Adaptive learning
paths ensure a differentiation on the level of content and difficulty level (see also section 4.3
Educational Principles). From an entrance quiz and questionnaire about their school past we can
determine the entrance level and the subjects that have to be taken. Further progress and exercises
determine the actual path.
BlackBoard is very useful to provide multimedial content. Movies and applets can easily be embedded
in items, providing an additional differentiation, this time taking into account the different learning
styles as stated by [3] Kolb (1984).
Construction of the content on the site reflects the differentiation goal by giving the student the choice
of what he wants to do first. For each module or concept, he has a choice of learning possibility:
learning theory, doing exercises, trying examples, watching a web lecture (Fig. 1: Learning
possibilities). An exercise system will be implemented that gives differentiated feedback. Computer
usage (whether it is in self-study or more traditional learning) should be stimulated, as suggested by
[6] Lou et al. (2001) and [5] Sangwin and Pointon (2004). Mathematics, especially as a supporting
subject, is mostly done by computers or calculators. Easing the student into this technology should be
part of their mathematical education.
Fig. 1: Learning possibilities
An external site provides additional flexibility. While the main portion of ActiMath will be provided on
BlackBoard, the use of an external site should be considered, but only to meet the need for very
specific demands (automatic assessment for exercises). However, maintenance of an external site is
more difficult than that of a VLE. Usually, more technical knowledge (programming, database
structure, html, …) will be needed, increasing costs.
4.2.2 Technology
Automatic assessment is preferred, since the course will be available online. It provides an instant,
personalized and objective feedback. If possible, a system with a mathematics engine behind it
(STACK, Maple T.A.) is preferred.
Various applets help students to experiment with some concepts (geometry, functions). These applets
will be constructed with Geogebra (www.geogebra.org) since this program is freely available, easy to
use and easy to embed.
Theory shall be provided in pdf files, typesetted by LaTeX. LaTeX is ideal for typing mathematics.
Exercises can be constructed using STACK (stack.sourceforge.org), which is open-source software.
STACK provides differentiated feedback and is connected to an algebra package (Maxima) capable of
doing calculations in the background. Currently, it is unable to connect to the gradebook of
BlackBoard, so it is not useful for evaluation.
4.3
Educational principles
4.3.1 Scenarios
To determine the best mix and learning scenarios (self-study, contact education, cooperative
learning), we base ourselves on the previous remedial courses given by lecturers in the Association
K.U.Leuven and their experiences.
ActiMath should be usable in all scenarios. Different lecturers have different teaching styles, each with
its own merits. A lecturer should use the teaching style with which he or she is most comfortable, in
order to improve effectiveness ([2] Ernest (1989)). As has been pointed out in numerous studies,
students have their own learning style ([3] Kolb (1984)) in which they feel most comfortable.
Self-study is the more natural approach to ActiMath. The course materials are available online and are
complete. However, self-study still needs guidance from a lecturer and possibly from other students.
This way we can ensure useful feedback and correct interpretation of the topics. This can be
accomplished by adding a forum or by organizing one or two days of response college in which
students can ask questions and lecturers can give overviews.
Classical contact education is possible through ActiMath. Course materials are online and should be
used by the lecturers, referring to exercises an applets online. ActiMath fulfills a supporting role in this
scenario, since it will be mostly used for exercise and reference.
Where possible, group exercises should be constructed to stimulate cooperative learning. This is the
most effective way of learning, although it is less efficient (more time, more organizing), see also [4]
Slavin et al. (2009). Cooperative learning stimulates social contact and peer education, which in turn
trains cooperational skills. This scenario is most useful where mathematics has to be applied to real
life situations. The usability of mathematics motivates.
Where possible, a mix of these scenarios is preferred, and ideally this should be determined by
student and lecturers, depending on their personal learning and teaching styles.
The following figure (Fig. 2: Schematic overview of stakeholders) describes the interaction between
ActiMath and the different stakeholders.
Fig. 2: Schematic overview of stakeholders
4.3.2 Differentiation
Differentiation is partly achieved when giving the student the choice of learning scenario and speed.
This choice should reflect the students’ learning style. According to [3] Kolb (1984), there are 4 main
learning styles, as seen in Fig. 3: Learning styles (Kolb).

Accommodating: Uses trial and error rather than thought and reflection. Good at adapting to
changing circumstances; solves problems in an intuitive, trial-and-error manner, such as
discovery learning. This student will do exercises first and tries to learn from them.

Diverging: Emphasizes the innovative and imaginative approach to doing things. Views
concrete situations from many perspectives and adapts by observation rather than by action.
Likes such activities as cooperative groups and brainstorming. This student will ask for
opinions of other students, lecturers and watches movies before forming his or her own
approach and ideas.

Assimilating: Pulls a number of different observations and thoughts into an integrated whole.
Likes to reason inductively and create models and theories. Likes to design projects and
experiments. This student tries to think and generalize concepts. He will build his own strategy
from the offered materials while looking at the theory.

Converging: Emphasizes the practical application of ideas and solving problems. Likes
decision-making, problem-solving, and the practicable application of ideas. Prefers technical
problems. From theory to exercises, this student tries to apply what he has learned.
Fig. 3: Learning styles (Kolb)
While a learning style defines how you start to tackle a learning, this actually is a cycle. By giving a
student access to all materials without imposing an order, he can choose where his cycle starts.
We can also differentiate in the contents the students have to learn (depending on choice of study and
starting level in mathematics, measured by an entrance test).
ActiMath can be perceived as a collection of modules. Some of these form the required or expected
starting level for a student. These modules will be tested in the entrance test. From the test, a few of
these modules can be set as ‘known’, while others should be trained. The student will be tested on
those modules that should be trained in a final test. He will, however, have access to all the modules
that are connected to his chosen field of study, if he needs reference or decides to train those modules
as well. A schematic overview is given in Fig. 4: Entrance test
Fig. 4: Entrance test
The course materials, and exercises in particular, will be grouped in a few difficulty levels, inspired by
Taxonomy of Bloom (1956) [1], which was later updated for mathematics by [5] Sangwin and Pointon
(2004). First, we will focus on basic math competences (basic calculation), since many lecturers
experience difficulties concerning even the simplest of calculations. Reviewing student exams, the real
math problems seem to originate from this lack of basic competences. They can construct equations
fairly easy, but fail at solving them correctly.
The difficulty levels for content (on which we focus) are:

Basic knowledge: reproduction of formulas, definitions, algorithms, and use of those
algorithms (differentiating a polynomial, solving a quadratic equation).

Comprehension: the meaning of certain symbols in certain formulas, why do formulas work?

Application: using the appropriate method for simple questions and problems.
Depending on the chosen course trajectory, not all of these levels are required for each student.
Sometimes only a basic knowledge of the concept is needed. This is the case where the concept
plays a purely supporting role. For example, when a student needs to calculate zeroes of a quadratic
polynomial, only knowledge of the formula is needed, not the way this formula was constructed.
The other three difficulty levels (analysis, synthesis, evaluation) are beyond the scope of a summer
course. These levels are trained and achieved in the courses in the regular curriculum.
4.4
Evaluation
Students will be evaluated throughout the course by doing exercises. If possible, these results can be
saved using the gradebook provided by BlackBoard. This evaluation is formative, we will allow the
student to take these exercises or tests multiple times, in order to learn from the mistakes they make
and the automated or personal feedback provided afterwards. The entrance test should be viewed as
a formative evaluation.
A final test will evaluate the required and previously not (adequately) known modules. While this
evaluation is summative, it cannot be used to withhold the student from his chosen field of study.
Instead, it should motivate him to review the course materials again.
The project itself will be evaluated through the results of the students on the final test (in comparison
with the entrance test) and with the general successes in their studies afterwards. While this is
subjective (not everybody will take this summer course), it gives a good indication to the merits (or
flaws) of ActiMath.
From questionnaires we can determine if ActiMath was easy and interesting to use. Opinions,
feedback and suggestions will be collected from students and lecturers. This in turn can be used to
improve ActiMath.
While ActiMath is running, the developers will also be in contact with the users. This way, technical
problems can be reported and solved, immediately or after the conclusion of the course.
4.5
Test runs
The first test run will start in September 2011 (year 2011-2012). The test group will consist of the
students of the Professional Bachelor studies (Construction, Real Estate, Electro Mechanics),
preferably across all participating institutions. We will determine through questionnaires how the
students and lecturers experienced ActiMath. We will follow the students in their first year to
determine their results, comparing them to the results of the students of year 2010-2011.
This test run will be held on the VLE Blackboard, using the adaptive learning paths implemented.
Evaluation (entrance test, exercises and final test) will be done using multiple choice questions.
Where needed, the course materials and method can be adapted to suggestions and results.
The second test run will start in September 2012 (year 2012-2013). This test group will consist of the
students of the Professional Bachelor studies, those of Industrial and Trade Engineering and those of
Trade Sciences across all participating institutions. Again through questionnaires and studying
students’ results, the effectiveness of ActiMath can be determined.
5
FURTHER RESEARCH
Determining a good entrance and final test is no easy task and warrants further research of each field
of study and their demands on entrance level.
Through questionnaires we can determine the preferred or most effective method of studying
(exercises or theory first, lots of examples or not, …) and adjust this course (and maybe others)
accordingly.
The use of an online and complete course could be implemented in other subjects (physics, chemistry,
biology) or maybe even languages.
Cooperation between lecturers can (and should) lead to uniformization in content and desired starting
level. Further research can determine which materials are absolutely necessary to ensure a good
starting level in mathematics.
After a first test run in September 2011, experiences will be collected through questionnaires and test
results in order to improve ActiMath. Comparing these results to the actual exam results hopefully
yields a positive and relevant correlation.
REFERENCES
[1]
Bloom, B.S.(ed.), 1956, Taxonomy of Educational Objectives, McKay New York.
[2]
Ernest, P., 1989, The Impact of Beliefs on the Teaching of Mathematics, Mathematics Teaching:
The state of the Art, Falmer Press London, p. 249-254.
[3]
Kolb, D.A., 1984, Experiential Learning: experience as the source of learning and development,
Prentice-Hall New Jersey.
[4]
Slavin, R.E., Groff, C., and Lake, C., 2009, Effective Programs in Middle and High School
Mathematics: A Best-Evidence Synthesis, Review of Educational Research, v. 79(2), p. 839-911.
[5]
Sangwin, C.J., and Pointon A., 2004, Assessing Mathematics Automatically Using Computer
Algebra and the Internet, Teaching Mathematics Applications, v. 23(1), p. 1-14.
[6]
Lou, Y., Abrami, P.C., and D’Appollonia, S., 2001, Small Group and Individual Learning with
Technology: A Meta-Analysis, Review of Educational Research, v. 71(3), p. 449-521.