Active learning in redesigning mathematics courses for engineering students

Abstract

Prepare, Participate, Practice": active learning in designing basic maths courses for engineering students at TU Delft works! The PRoject Innovation Mathematics Education (PRIME) at Delft University of Technology (TU Delft) is all about redesigning mathematics courses for engineers. This paper describes the process of developing, implementing, evaluating and implementing again of three basic courses at TU Delft using a blended learning approach developed by a growing team of teachers from the mathematics department. Our findings suggest that the approach taken enhances students' learning performance in maths education. The main results show that students have a more active learning experience compared to the traditional setup of these courses, leading to more engagement, more interaction and better results. An important role is played by meaningful examples taken from the engineering faculty where the students are studying, showing students from that faculty what role the mathematics play in their field of interest. This is also used to develop their skills in mathematical modelling.

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Delft University of Technology
Active learning in redesigning mathematics courses for engineering students
Cabo, Annoesjka; Klaassen, Renate
Publication date
2018
Document Version
Final published version
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The 14th International CDIO Conference
Citation (APA)
Cabo, A., & Klaassen, R. (2018). Active learning in redesigning mathematics courses for engineering
students. In C. Bean, J. Bennedsen, K. Edström, R. Hugo, J. Röslof, R. Songer, & T. Yamamoto (Eds.), The
14th International CDIO Conference: Proceedings – Full Papers (pp. 704-715). CDIO.
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ACTIVE LEARNING IN REDESIGNING MATHEMATICS COURSES
FOR ENGINEERING STUDENTS
Annoesjka Cabo
Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer
Science (EEMCS), Delft Institute of Applied Mathematics (DIAM)
and
4TU Centre for Engineering Education
The Netherlands
Renate Klaassen
4TU Centre for Engineering Education – Delft University of Technology
ABSTRACT
“Prepare, Participate, Practice”: active learning in designing basic maths courses for
engineering students at TU Delft works! The PRoject Innovation Mathematics Education
(PRIME) at Delft University of Technology (TU Delft) is all about redesigning mathematics
courses for engineers. This paper describes the process of developing, implementing,
evaluating and implementing again of three basic courses at TU Delft using a blended
learning approach developed by a growing team of teachers from the mathematics
department. Our findings suggest that the approach taken enhances students’ learning
performance in maths education. The main results show that students have a more active
learning experience compared to the traditional setup of these courses, leading to more
engagement, more interaction and better results. An important role is played by meaningful
examples taken from the engineering faculty where the students are studying, showing
students from that faculty what role the mathematics play in their field of interest. This is also
used to develop their skills in mathematical modelling.
KEYWORDS
Engineering education, blended learning, mathematics, team-based development, active
learning, CDIO- Standards: 1, 2, 8, 9, 10, 11, 12.
INTRODUCTION
In this paper we consider interfaculty education: mathematics for non-mathematics students
at TU Delft. Students need to have a sound mathematical background to pursue their studies
and in their future careers. Pinxten (2017) shows that students need 6 to 8 hours of
mathematics training in secondary education each week and a sufficient to very good grade
at the final exam to have a chance of success in studying Engineering. Continuation of
Proceedings of the 14th International CDIO Conference, Kanazawa Institute of Technology,
Kanazawa, Japan, June 28 – July 2, 2018.

diligent study time in mathematics is a necessity for any engineering student to obtain their
bachelor degree and achieve academic success.
Mathematics at TU Delft is taught within the engineering faculties before or parallel to the
disciplinary courses in the engineering programmes. It allows the students, or so it is
hypothesized, that students use the mathematical theory and apply it in their disciplinary
engineering assignments. Despite the high expectations, the transfer of theory to practical
application in the disciplinary field is limited, as shown by student evaluations, performance
on exam questions and lecturer reports. From studies in childhood mathematics learning it is
known the more concrete object and materials are used to learn mathematics the more
difficult the transfer becomes of the mathematics to other disciplinary or isomorphic
assignments. Abstract mathematics allows for better transfer and better ability to understand
relational structures, allowing for math skills transfer to alternative math topics. (Kaminsky &
Sloutsky, 2012). Concrete objects increases the salience of superficial aspect and divert the
attention from the relational structures to be learned. The more complex the problems
become the more susceptible to diverted attention the learner is.
Finally, student engagement and intrinsic motivation are stimulated by establishing more
autonomous learning, a feeling of competency (self-efficacy) and relatedness to other
students who may struggle with the same materials (Deci & Ryan (2002) Bandura, (1997),
Artino (2012)). The present situations allow for little to no autonomy as the programme is
fixed and a schedule to be met. Once the students are behind there is little time or possibility
to catch up, bearing on the feelings of competencies. Frequent testing overburdens the
students and possibly makes them loose their intrinsic motivation and engagement with the
mathematics material.
To solve the issues mentioned above, a new teaching approach was developed in “PRIME” .
In this paper the following questions are researched upon: Does the new teaching method
activate/engage students (more), does it improve transfer, does it improve passing rates?
First we start by describing the project. Next the development of the new approach and the
didactical concept chosen are reflected upon. Then the implementation of the concept is
reported, followed by the consequences and improvements implemented after the first
operation of the courses. Data analysis of the results over the past two years are presented
and finally some suggestions for future research are discussed.
THE PROJECT: PRIME
In 2014, PRIME (PRoject Innovation Mathematics Education) was initiated in order to
conceive a different approach to the math courses for engineering students.
The organization of PRIME
The initial project team consisted of a group of six dedicated lecturers from Delft Institute of
Applied Mathematics (DIAM), an e-learning developer, an educational advisor and a project
leader. The project was supported by the Executive Board of the university. A large steering
group was assigned to the project to keep informed about the progress: it consisted of the
Vice-President of Education (Executive Board), the director of Student Affairs, the dean of
the faculty of Applied Physics, the dean of the faculty of Electrical Engineering, Mathematics
and Computer Science (EEMCS), the director of education of the faculty of Aerospace
Proceedings of the 14th International CDIO Conference, Kanazawa Institute of Technology,
Kanazawa, Japan, June 28 – July 2, 2018.

Engineering, the director of education of the faculty of EEMCS, the chair of the Mathematics
department, a student from the Mathematics student association. After two years of running
the project, the team has expanded into a team of a senior project leader, an assistant
project leader, 12 instructors, an e-learning developer, an educational advisor and four
student assistants. The steering group has remained the same, except for the student
member, who has been replaced by two students: one from Civil Engineering and one from
Aerospace Engineering. The steering committee gathers once every three to four months
with the project management team.
The goals of PRIME
Three goals were formulated:
1. Academic success: to improve study results
2. Transfer: to improve the connection between mathematics and engineering
3. Engagement: to increase students active participation in class and motivation for the
topic
In the following subsections each of the measures taken to address these goals is described
briefly.
Academic success
Once the student is motivated for mathematics, the next important challenge is to activate
him: active learning enhances retention and improves understanding of subsequent subjects
in the student’s learning path (Veenstra-van Dijk, 2000). Moreover, it is well known that
mathematics needs practice, in order to acquire the skill of interacting in a mathematical way
with their disciplinary field of study, needed to learn new concepts.(Kirschner et al., 2006).
Academic success is described as the measures teachers realise to sustain students’ time
on task. Engagement described below is the flip of the coin, the extent in which students are
engaged and motivated to realise the time on task.
Transfer
Showing the use of mathematics in the field of interest of the student is believed to enhance
motivation for learning (Chickering et al., 1987). With the help of lecturers and students from
the receiving faculties, contextual examples from the specific fields are worked out, to
illustrate the use of mathematics in the field of interest. Finding examples that are interesting,
not too hard to explain for the mathematicians, not too hard to understand for the students
turned out to be a challenge. A new smaller project carried out by Cabo & Makaveev (2018)
has resulted in a new method to investigate the use of mathematics in specific engineering
courses. The lessons learned from this project will be implemented in PRIME shortly. They
involve also incorporating projects in a later stage of the courses to apply their knowledge in
practice, an important feature of engineering education (Edström, 2008; Kamp, 2016).
Engagement
Engagement can be defined as the extent to which students actively participate in learning
activities (online presences, watching videos and doing assignments) and face to face
contact meetings (coming to class, being prepared, making use of the materials to digest the
learning materials). It equally includes the stimulation of student motivation by relating
Proceedings of the 14th International CDIO Conference, Kanazawa Institute of Technology,
Kanazawa, Japan, June 28 – July 2, 2018.

abstract materials to their disciplinary field of study The extent to which students are
engaging in higher education is supposed to strengthen the learning outcomes. (Trowler,
2010, HEA report)
The courses innovated in PRIME
To start with three basic maths courses were considered for innovation: Calculus 1 and 2,
Linear Algebra (all first year courses) and Probability & Statistics (first or second year course).
Since the context examples are tailored to each individual program, the courses are not
exact copies of each other. However the content is mostly exchangeable, only the pace of
each course may differ. Bachelor programs with courses in PRIME are Aerospace
Engineering, Computer Science, Electrical Engineering, Civil Engineering and Mechanical
Engineering. In the near future courses like Differential Equations and Calculus 3 in certain
programs will also be innovated. A typical course consists of nine weeks of two or three two-
hour lectures, resulting in 18 to 27 contact hours.
BLENDED LEARNING CYCLE: “PREPARE, PARTICIPATE, PRACTISE”
A number of educational principles have been included to achieve the innovation and goals
of the project. Active participation in teaching sessions (Freeman et al, 2014), conceptual
understanding in the face to face contact (Rittle-Johnson et al,, 2015), adequate performance
feedback (Hattie 2007, Boud & Falchnikov, 2006) and a carefully balanced format of
contextual examples (Cabo & Makaveev, forthcoming) using contextual problems, with a
sufficient level of generalisation, to motivate the importance of maths in other fields of study
and equally support transfer. In other words: the students should prepare themselves before
coming to class, should participate actively by joining in-class-activities and after the face-
to-face session students should practise to process the new knowledge. A blended
approach was felt to best meet the requirements (Bonk et al., 2006; Szeto, 2014 in this
context, due to the workload of teachers, increasing student numbers, the stimulation of
autonomous learning, competency building and time on task. A video has been recorded,
available at the TU Delft website (2017), which stimulates students to study differently.
Figure 1: Blended learning cycle used in PRIME
Proceedings of the 14th International CDIO Conference, Kanazawa Institute of Technology,
Kanazawa, Japan, June 28 – July 2, 2018.

Practise
At home (or wherever the students want), a set of computer aided exercises can be done: an
online platform offers two or three types of exercises: basic, intermediate (with an optional
help function to guide the student through the exercises) and an assignment, to be handed in
online. At the moment a platform offered by an editor is being used, however the project
management is currently looking for an (open source) alternative.
BLENDED LEARNING: MATERIAL DEVELOPED (hybrid flipped classroom)
This model of blended learning is established as a sort of hybrid flipped classroom, as shown
in the sequence prepare, participate and practice, the flipped model of autonomous learning
and reflection and discussion in class to further explore the learning materials is not enough.
The practice step consolidation of the learned materials is essential to bring the math skills to
the next level of learning.
For each course new material has been developed by the project team. During frequent
meetings (once every one or two weeks), first a lesson plan was designed, with all the
learning outcomes listed. Then consensus had to be reached on which learning outcome
could go into the pre-lecture video, and how the others would be covered in the slides.
Exercises had to be chosen, contextual examples had to be collected from the faculties and
implemented into the course. An overview of the course, linking the separate subjects was
constructed, and included in the collaborative learning environment, showing students how
the subjects connect.
Evaluation and evolution of all the learning material is constantly being done: lecturers send
their comments to a special mailbox created for this. If possible changes are implemented
immediately by the student assistants. More drastic improvements are collected and stored.
During meetings where new courses are being prepared for their pilots, every remark on the
content, video, course structure, exercises and quizzes is taken into account, discussed,
reviewed and altered if necessary.
Using the Collaborative learning environment
All the material for the course is presented to the students in a well-organized page on
Brightspace, the collaborative learning environment in use at TU Delft since September 2017.
The lectures are structured by week and represent the blended learning cycle.
Proceedings of the 14th International CDIO Conference, Kanazawa Institute of Technology,
Kanazawa, Japan, June 28 – July 2, 2018.

Figure 2. Example of a lesson on Brightspace
Overview
A graph representing an overview of the subjects presented in the course, shows students
the connection between different subjects covered
Figure 3. Overview of a course Calculus 1 (Civil Engineering)
Sub-parts of the Course design
The course consists of online exercises to practice the conceptual understanding of the
subject together with the book exercises. The exercises provide feedback and allow
repetition as much as needed by the students. 110 videos have been recorded covering an
introductory subjects, half of them are used as a type of homologation in which students
secondary education knowledge is upskilled (TU Delft, 2018). A slide pack is the
framework/benchmark for all the lecturers and students involved. It includes definitions,
theorems, contextual examples, interactive quiz questions, workflow of a lecture,
accomplished learning goals after having done all the lectures activities and homework.
Finally, there are interactive quiz questions including questions on conceptual understanding.
Depending on the results of the quiz, the lecturer can decide to further elaborate on the
subject and stimulates active participation of students in class. It is reported by lecturers and
students that the quizzes stimulate active participation of students in class.
Mathematical modelling
One of the learning goals of the newly designed courses in Calculus was to teach the
students the mathematical modelling cycle: this is the most important application of their
mathematical knowledge in practice.
Proceedings of the 14th International CDIO Conference, Kanazawa Institute of Technology,
Kanazawa, Japan, June 28 – July 2, 2018.

Figure 4. Mathematical model cycle
EVALUATION
The evaluation was focused on whether the new teaching method activated/engaged
students (more), is transfer improved and are the passing rates improving? The data are as
much as possible triangulated and emerge from data at the programme level, the lecturers
and the student evaluation. At this point we were not yet able to formulate any research
hypothesis.
Program directors and academic success
The program directors of the Bachelor curricula involved are pleased with the innovation: the
activity of the students has increased, and – after an initial dip in the results- the study
success rate has increased (Table 1) They appreciate the fact that mathematical modelling is
now part of the learning outcomes, and they hear from lecturers of their own faculty that they
feel more comfortable about the expected level of mathematical background of the students
of their own classes.
Table 1. Passing rates
2013/14
2014/15
2015/16
2016/17
LinearAlg AE
61%
72%
52%
75%
Prob&Stat AE
54%
19%
56%
67%
Prob&Stat EE
67%
79%
54%
70%
Calculus 1 CE
73%
68%
64%
68%
before PRIME
during PRIME
Proceedings of the 14th International CDIO Conference, Kanazawa Institute of Technology,
Kanazawa, Japan, June 28 – July 2, 2018.

Lecturers role in the hybrid flipped model
The impact on lecturers involved in blended learning has been investigated in different ways.
After the first pilot a survey was distributed among the nine lecturers who taught the course.
(Vos, 2016). In subsequent courses, for each course three meetings were held to discuss the
content and impact of the course: one before the course started, one in the middle of the
course (week 4 or 5 of the quarter) and one at the end, after the course had finished, but
before the exam was taken. The instructors commented on the use of the pre-lecture videos
and how to deal with the fact that students don’t watch them: 50 % of the instructors tend to
repeat the material of the videos, 50 % does not, or in a concealed way. The teachers are
positive about the interactive quizzes although some of them (40%) thinks it takes too much
of their instruction time. Using the slide pack is for 40 % of the teachers a burden: they are
used to teach the course in their own way. The other 60% however think it is helpful to
reduce preparation time. All teachers have seen that the students are more active during the
classes, and the attendance is higher than it was before the blended teaching. Working in a
team to develop and discuss course content was appreciated by 70% of the lecturers.
Observing each other’s classes was viewed as a relevant and stimulating experience,
helping to improve the quality of teaching. The support from the project lead was considered
sufficient (70%), could have been more (30%). The cultural change needed in the teaching
staff turns out to be a tough process. It takes more time to get the teachers along than it
takes to convince the students.
Hence the activation of students and the stimulation of time on task, may not have reached
its optimal balance yet.
Students Engagement
Apart from the official quality cycle (Evasys) - a survey that students fill out after having done
the exam (average response rate 30%) - the project management implemented the so-called
ContinueStartStop Survey. In this questionnaire the students are asked to write down what
they would like the lecturer to continue doing start doing, stop doing or. The survey is given
to the students during class, the response rate is quite high (70 - 90%). The general remarks
collected from this survey are grouped and the ones that appear the most are commented on
by the responsible lecturer together with the project management. These comments are
posted on Brightspace. The comments that relate to individual teachers are sent to the
teachers, and they discuss them in class.
In the second quarter of the academic year 2017-2018 a lunch meeting with students from
Civil Engineering, with part of the project team and the responsible lecturer was organized to
discuss the outcomes of the survey. The use of contextual examples was highly appreciated
there. The students confirmed that they liked the way of teaching and the videos, but also
gave some useful feedback on individual teachers and explanation of the online exercises.
Proceedings of the 14th International CDIO Conference, Kanazawa Institute of Technology,
Kanazawa, Japan, June 28 – July 2, 2018.

Figure 5. Outcome of the ContinueStartStop Survey in the 1st semester of 2017-2018 with
684 respondents
Transfer
Most important reviews were on the contextual examples: some of them were too difficult to
understand for the students, some of them were too difficult to explain for the teachers, some
were not realistic enough. Also some videos had to be recorded anew because they had too
much content. Furthermore, a lot of the interactive questions were adapted because either
they did not connect well enough to the videos, or they did not test concepts well enough or
they took too much time to answer.
CONCLUSIONS
After three runs of the courses Calculus 1 for Civil Engineering and Probability and Statistics
for Electrical Engineering, they seem to have reached a steady state. The rest of the courses,
that have run two times, or only one time, need adjustments.
Working in large teams of teachers improves the quality of the courses and the consensus
on how to teach the course, this is noticed by the students. Blended learning is welcomed by
students, blended teaching is a challenge for some of the teachers. Finding suitable and
meaningful examples to illustrate the use of mathematics is an equally tough challenge. Help
from students from the relevant programs might turn out to be crucial to improve this. In one
instance (Cabo, 2018) this turned out to be the solution. On the other hand the use of this
kind of examples in the courses is really appreciated by the students.
Proceedings of the 14th International CDIO Conference, Kanazawa Institute of Technology,
Kanazawa, Japan, June 28 – July 2, 2018.

FUTURE DEVELOPMENTS AND RESEARCH
In the near future, the courses are being improved using student evaluations and teacher
experiences. It is interesting to find out if three runs with this intensity of adjustment and
evaluation is the standard to get to a steady state situation of a course. Additionally, many
research questions have emerged as a result from designing and re-designing these courses.
A lot of data is being collected from the students. Well-defined research questions should
guide the relevance of the learning analytics data gathered until now and from the next
academic year onward. In particular we will investigate how online individual learning paths
enhance student’s learning, and how active learning (time on task, engagement, motivation)
effects the understanding of the mathematics taught and how the mathematic and
disciplinary based assignments can be validated for conceptual understanding of the
discipline.
In the academic year 2017-2018 a pilot has been done at Civil Engineering by grouping
students having a similar, somewhat better, mathematical background from secondary
school: Did these students perform better in the mathematics course in higher education than
their less prepared counterparts? Did they appreciate the extra information and deepening of
the learning experience they were offered? Is it worthwhile expanding this experiment to
other faculties?
Teachers that are late adapters or have problems getting used to this PRIME approach will
be supported with extra training activities. This will contribute to lifelong learning and faculty
development on teaching mathematics in the PRIME model.
The lessons learned from the project investigating how to better implement the connection
between mathematics and aerospace engineering, will be incorporated in PRIME and further
expanded. What the most efficient way is to embed discipline based examples in
mathematics or computational learning is to be explored.
The ambition is to involve the multiple stakeholders in the data collection and analysis to
generate more evidence based support for the things that have intuitively been done until
now and extend this to a larger community within TU Delft and beyond.
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BIOGRAPHICAL INFORMATION
Annoesjka J. Cabo, Dr. is working as a statistician and as a lecturer of Mathematics. Since
August 2016 she is the director of studies of interfaculty education at the faculty of EEMCS at
the Delft University of Technology. She is the leader of PRIME: Project Innovation
Mathematics Education, a university wide initiative to innovate mathematics education for
engineering students. As such she is involved in developing learning material, researching
data from the project and coordinating a growing group of people involved in the process.
Renate Klaassen, Dr. is an educational consultant, working at the TU Delft Teaching and
Learning Services. She has been heavily involved in educational advising on the innovation
of the BSc in Aerospace Engineering, and various other curriculum reforms at TU Delft.
Currently, she is TU Delft Programme Co-ordinator and Researcher of the 4TU.Centre for
Engineering Education. Consultancy activities include assessment (policy, quality and
professionalization), internationalisation of university education and design education. Areas
of research interest pertain to content, language integrated learning in higher education,
Conceptual Understanding in Engineering Education and Interdisciplinary learning.
Corresponding author
Dr. Annoesjka J. Cabo
Delft University of Technology
Van Mourik Broekmanweg 6
2628 XE Delft
+31-6404 301 36
a.j.cabo@tudelft.nl
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