Actimath: a Remedial Course in Mathematics Adapted to the Needs

Actimath: a Remedial Course in Mathematics Adapted to the Needs
of the Student and Their Chosen Field of Study
T. Neijens, B. D’haenens, J. Van Caudenberg, A. Vermeyen
KAHO Sint-Lieven, Association K.U.Leuven (Belgium)
[email protected], [email protected], [email protected],
[email protected]
Abstract
1.1 Background
Many courses in higher education have a need for mathematical concepts. Whether or not
mathematics fulfills a supporting role, all studies expect some basic knowledge. Therefore many
lecturers organize remedial math courses.
For Belgium, most are organized in a classical way: theory lectures followed by exercises, leaving little
room for differentiation.
1.2 Objectives
The main goal is to develop a remedial and adaptive course for all participating colleges and
universities (Association K.U.Leuven). The contents and level are determined by the chosen field of
study and the individual mathematics level. As such, we improve their basic mathematical knowledge
and close the gap between secondary and higher education.
1.3 Approach
We use the VLE (Virtual Learning Environment) of the Association, Toledo (based on BlackBoard).
This way, all teachers and students have access to ActiMath.
From a test, we can determine the specific subjects that have to be taken. The learning scenario (selfstudy, contact education …) depends on the students and lecturers. Construction of the content on the
site reflects this by giving the student the choice what he wants to do first (learning theory, doing
exercises, watching a web lecture). An exercise system will be implemented giving differentiated
feedback.
1.4 First Results
In September 2011, we tested ActiMath for students starting their first bachelor year at KAHO Aalst
(construction, real estate, electro-mechanics), totaling 120 students. From a test, we gave them a
suggested path to follow in ActiMath, and we instructed them on how to use the system. Afterwards
they could try out ActiMath at home.
Responses were positive, especially for the interactive exercises with feedback. It should be noted
that most students preferred a mix between self-study and contact education.
We will continue to add new material to ActiMath. A larger test will be held in September 2012,
hopefully with similar positive results.
1. Introduction
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.
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.
2. 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)).
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.
3. Approach
3.1 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
(through a syllabus and traditional lecture), then exercises (individually, assisted by other students and
teachers). Differentiation is minimal and only possible during the exercises.
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.
3.2 Technology and 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.
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 3.3
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.
Construction of the content on the site reflects the differentiation goal by giving the student the choice
of what he wants to do first. This way, we take into account the different learning styles as stated by
[3] Kolb (1984). 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 selfstudy or more traditional learning) should be stimulated, as suggested by [5] Lou et al. (2001) and [4]
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
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). Theory shall
be provided in pdf-files, typesetted by LaTeX. LaTeX is designed for typing mathematics.
3.3 Educational principles
3.3.1 Scenarios
The following figure (Fig. 2: Schematic overview of stakeholders) describes the interaction between
ActiMath and the different stakeholders.
Fig. 2: Schematic overview of stakeholders
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
self-contained. 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. ActiMath fulfills a supporting role in this scenario, since it will be mostly used for
exercise and reference.
3.3.2 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).
Fig. 3: Learning styles (Kolb)
•
Accommodating: Uses trial and error rather than thought and reflection, does exercises first
and tries to learn from them.
•
Diverging: Emphasizes the innovative and imaginative approach to doing things, asks for
opinions before forming his/her own approach and ideas.
•
Assimilating: Pulls a number of different observations and thoughts into an integrated whole,
builds his own strategy from the offered materials while looking at the theory.
•
Converging: Emphasizes the practical application of ideas and solving problems. From theory
to exercises, this student tries to apply what he has learned.
While a learning style defines how you start to tackle a learning task, 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 also differentiate in the contents the students have to learn. ActiMath can be perceived as a
collection of modules, which 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 and afterwards tested. 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 [4] Sangwin and Pointon
(2004).
•
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.
3.4 Evaluation
Students will be evaluated throughout the course by doing exercises. 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 feedback provided afterwards. The entrance test should be viewed as
formative evaluation. A final test will evaluate the required and previously not 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 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.
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. First results
We tested ActiMath over the course of 4 days in September 2011 (19-22), the test group consisted of
116 students of the Professional Bachelor studies (Construction, Real Estate, Electro Mechanics) in
one institution (KAHO Sint-Lieven, Aalst). They were first given an entrance test to determine their
starting level and materials to be studied. Afterwards, use of ActiMath was explained. From there on,
students could use the platform to study the mathematics course. During those 4 days, they could use
the computers at the institution, and a teacher was present to solve mathematical and technical
problems. After this, they used their computers at home.
Two weeks later, we asked the participating students to fill out a questionnaire:
•
Usage: 55% of the respondents claimed they still used ActiMath frequently. 25% claimed that
they didn’t use ActiMath anymore, of which 63% stopped because they finished the course. All
in all, this is a good result. If asked for future use, slightly under 50% of the students claimed
they would use ActiMath after 3 months.
•
Reception: 75% of the students think ActiMath is pleasant to use. The mix in different learning
possibilities (theory, exercises, movies …) is ActiMath’s biggest asset. Most students (95%)
prefer a mix of traditional teaching (theory lectures followed by exercise sessions supervised
by a teacher) and self-study. They think ActiMath cannot replace remedial mathematics
courses. However, it is a very valuable addition.
•
Preferred learning materials: Exercises and theory are the most visited learning materials,
which corresponds to the answer that students still like traditional teaching. While movies are
frequently watched, it is always to support theory and exercises, and never as a standalone
means of learning.
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, and adapted where needed.
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] 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.
[5] 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.