ONLINE MATHEMATICS EDUCATION

E-Math for First Year Engineering Students

Steen Markvorsen and Karsten Schmidt

DTU Mathematics, Technical University of Denmark, Copenhagen, Denmark

Keywords:

E-learning, Multifaceted E-approach, First year University Mathematics.

Abstract:

We consider the technology enhanced learning of ﬁrst year engineering mathematics and in particular the

application of E-learning objects and principles in the course Mathematics 1 which has a yearly intake of 750

students at the technical University of Denmark. We show that with non-linear multimedia technology and

e-learning principles it is possible to strengthen and enhance the students’ desire and ability to prepare for the

teaching and to read and enjoy the textual representations of the course materials.

1 INTRODUCTION

With an enrollment of 750 students once a year the

ﬁrst year course Mathematics 1 at the Technical Uni-

versity of Denmark is one of the largest courses at uni-

versity level in Denmark. Since its ﬁrst re-formation

10 years ago a number of interesting and valuable as-

sets concerning undergraduate math education in gen-

eral have emerged from that course. These assets

have now been transplanted into the next 2010 re-

formation, which is our main concern in this report.

We are convinced that physical attendance to lec-

tures as well as to exercise sessions are still of utmost

importance for most students, but we are also con-

vinced that the overall outcome of the teaching can

be improved considerably with the aid of e-learning

techniques.

The purpose of this paper is to show that it is pos-

sible to strengthen the students’ preparation for the

lectures and it is possible to strengthen their ability

to learn from the textual representations of the math-

ematics in question. This can be done by introducing

a wide spectrum of web-based learning objects and

activities which build directly upon the students’ own

up-to-date knowledge and use of modern multimedia

technology.

The strategy has two parts: First of all the individ-

ual students’ preparation can be freed from the classi-

cal linear string of text and progression, which has

been the classical way to learn mathematics. Sec-

ondly the e-learning technologies have the potential

to create new feed back to the mathematical text in

a way which eventually will facilitate the ﬁnal syl-

labus or curriculum as something which is actually

owned by the individual student, based upon his or

her own experiences, work, and progression through

the course. This speciﬁc and explicit ownership of the

resulting text makes it readily available also for later

courses.

Speciﬁcally we show that it is possible to design

web based learning which satisﬁes these goals via the

following three-step strategy:

• Present the main topics, concepts, and points in

many very different ways

• Provide help through links and video examples

• Replace traditional textbooks by easily accessible

and easy-to-search eNotes, which contain more

layers of understanding, and offer more and dif-

ferent ways of reading and study.

We will give a few glimpses from the present ar-

chitecture of the course and discuss some results and

challenges concerning such a modern large scale and

multifaceted teaching of ﬁrst year undergraduate cal-

culus and linear algebra. The curriculum material in

question is worldwide usually considered as standard

service material and thus taught in standard courses as

a much needed background for almost all engineering

disciplines. Correspondingly these courses are also

usually taught from standard textbooks and in a very

standard way.

Our main goal with this paper is to point out that

even with a fairly well established essentially invari-

ant curriculum we have by now a unique chance to

apply the computer driven technologies and internet

facilities to enhance the teaching of undergraduate

360

Markvorsen S. and Schmidt K..

ONLINE MATHEMATICS EDUCATION - E-Math for First Year Engineering Students.

DOI: 10.5220/0003923503600363

In Proceedings of the 4th International Conference on Computer Supported Education (CSEDU-2012), pages 360-363

ISBN: 978-989-8565-06-8

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

mathematics and to inspire the young students even

more into the learning of basic linear algebra and cal-

culus.

2 BACKGROUND

In 2007-2008 DTU Mathematics conducted a thor-

ough investigation of the study habits among the stu-

dents in the course Mathematics 1, see (Ratleff et al.,

2009). The focus of the evaluation was partly on the

students’ participation in the various learning activi-

ties (preparations for the lessons, participation in the

lectures and in the exercises, and concerning their

strategies for homework) and partly on their use of

learning resources (text books, demo worksheets in

Maple, and the students’ own notes, respectively. The

primary goal of the investigation was to ﬁnd out about

the role and importance of the traditional textbook

for the learning process. The result was meant to

support a decision concerning the pertinent question

whether the textbooks (which were from 1992-1993)

should be reprinted, new texts should be written, or

we should simply use different learning resources.

The assessment showed for example that the stu-

dents to a large degree change their study habits dur-

ing the ﬁrst year of study. They typically ﬁnd their

own strategy for survival as a way to get through and

pass the course. One of these strategies is expressed

in the title of the ensuing report: ”I read less, but un-

derstand more” – which is a direct citation from one

of the students who were interviewed.

For the present paper the following three speciﬁc

conclusions from the 2007-2008 assessment are of

particular importance:

1. The students’ preparation for the teaching was

generally decreasing throughout the year. After

the 3.rd week of the course 65% of the students

reported that they had generally prepared for the

classes; this number had decreased to 30 % after

week 19. The time used for the preparation had

likewise declined in the same period by more than

40 %.

2. The importance of the textbooks for and in the

learning process was also strongly declining dur-

ing the year of study. The students were asked

how much of their learning outcome could be as-

cribed to the individual activities and resources.

The average was 12 % for the text books after

week 3 in the course and then only 6% after week

19. Moreover it was clear from the responses that

the textbooks were mostly used for skimming the

text and for looking up formularies and examples.

The number of students who reported that they

had read contiguous text in the books dropped

from 47% after week 3 to only 17 % after week

19.

3. The importance of Maple as a facilitator for doing

e.g. the homework and exercises was, on the other

hand, increasing. In the above question concern-

ing the learning outcome the corresponding num-

ber for the effect of using the Maple demo work

sheets increased from 6 % in after week 3 to 12 %

after week 19.

The 2007-2008 assessment showed that the ex-

tensive use of Maple – in particular the Maple demo

work sheets – to some extent also substituted the clas-

sical use of the textbooks. There were both positive

and negative effects of this. Some students reported

that the more experimental work with Maple implied

a better understanding of the concepts and topics,

whereas others admitted that they often used Maple

as a black box solver without actually understanding

the mathematics behind. We refer to (Schmidt et al.,

2010) which contains a deeper analysis of the pitfalls

and potentials that are associated with the extensive

use of Maple in a ﬁrst year mathematics course like

Mathematics 1.

The decrease in the time for and quality of prepa-

ration for the teaching and the decline in the actual

classical intended use of the textbooks seems to be

quite correlated. One of the most important starting

points for the current development and re-thinking of

the course is our conviction that a carefully paced and

possibly individually enhanced preparation is a much

needed prerequisite for the students’ outcome from

lectures and exercises. Moreover we are convinced

that the ability and the desire to read and to cope with

the mathematical concepts in a textual representation

is of utmost importance for transfer – i.e the students’

ability to carry over and actually use the mathematics

in other disciplines and technology applications.

3 WEB-BASED LEARNING

OBJECTS

Based on the ﬁndings reported in the above section,

DTU Mathematics decided in 2010 to replace the

traditional textbooks and the corresponding printed

weekly menus by a Mathematics 1 portal.

As mentioned above we believe that physical at-

tendance, lectures and group exercises are crucial

hallmarks of effective teaching and learning, but we

also face the facts and challenges that the course in

question is mandatory for more than 750 ﬁrst year

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students, that it must give fundamental mathematical

training and services to 11 very different study lines

at DTU, and that each one of our students at the same

time is struggling to ﬁnd the working methods and

styles which are best suited for him or her at DTU.

Therefore we have introduced a wide range of web

based learning objects and activities into the teach-

ing agenda - to support and drive the good old clas-

sical curriculum but now also to open up and offer

plenty of windows and pathways into the core mate-

rial. Traditional teaching materials such as textbooks

and printed exercises and weekly sheets are replaced

by the internet portal where YouTube-based video in-

structions, e-Notes and electronic exercises suggest

and offer a much less linear form of learning than

before. In order to create both new solid challenges

for the strongest students and at the same time keep

the weaker students well motivated and engaged we

have found that the assets mentioned in the introduc-

tion (the three bullets) are of invaluable importance

for this enterprize.

In particular there are then obvious new chal-

lenges that the authors of the e-Notes (replacing the

traditional textbooks) are facing. The e-Notes must

contain several layers corresponding to different pur-

poses: They must display the concepts and the theory,

they must serve as the target for links from the online

exercises, they must provide inspirations for videos,

they must take advantage of and unfold parallel vi-

sually enhanced exercises, and ﬁnally they must pro-

vide several choices for printing ”your own book” -

depending on which parts of the content the reader

wants to emphasize. Similarly the e-Notes should of-

fer ﬂexible and different ways of reading: studying

contiguous text, skimming, ﬁnding theorems and for-

mulas, checking examples etc.

Originally our working title for the e-Notes was

transfer notes. We consider them as an integral –

yet compactly distilled – part of the whole enterprise.

They function as a constant reference for all the other

activities so that there is in principle a link to the rel-

evant section, example, or theorem from every other

activity in the course. The idea is thus twofold – to

keep the e-Notes as a text and to actively help lift-

ing this text into every corner of the course. Techno-

logically the text is constantly only one click away –

through the portal of the course. We have the hope –

to be seen and reported – that the students in this way

will be able to ’carry’ and ’transfer’ what they have

learnt much more readily into their next courses in the

engineering disciplines.

Figure 1: All lectures are available on video.

Figure 2: Look and listen to a careful pencast solution.

Figure 3: Prepare for the lecture – view the appetizer video!.

Figure 4: An exercise with clearly marked questions.

Figure 5: The exercise with 3 unfolded hints.

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3.1 Balanced and Useful Sample Codes

The Maple system is a well functioning example of a

merger between a calculator, pen and pencil, formu-

laries, and editor tool which is used throughout the

course for solving the math problems, for document-

ing homework, and for handling larger project exer-

cises. All the ﬁgures in the eNotes are produced from

simple Maple commands – so that the students can

do their own modiﬁcations of those ﬁgures directly in

their own work sheets and thereby enhance their study

and reading of the eNotes even further, see Figure 6.

Figure 6: A glimpse of a page from the eNotes.

4 EVALUATION AND RESULTS

As a contribution to the evaluation of the 2010 refor-

mation of the course Mathematics 1, DTU Mathemat-

ics has conducted a series of investigations in collab-

oration with LearningLab DTU, both before and after

the 2010 reformation. We display here a few selected

results, which contain information about the students’

preparation for the teaching and about their outcome

from the respective textual representation of the math-

ematical content and syllabus.

After week 8 in the study year 2009-2010 (i.e.

before the reformation of 2010) and correspondingly

after week 8 in 2010-2011 the students were asked

to prioritize the various teaching activities and state

from which activities they have learnt the most – in

their own opinion. The total score concerning prepa-

ration, lectures, and excercises in classes displayed as

in Table 1. The table shows that the active individ-

ual preparation before the teaching has been strength-

ened at the expense of the more passive listening to

lectures.

Correspondingly the students were asked to pri-

Table 1: Increasing preparation.

Year Preparation Lectures Exercises

2009 13.1 % 44.3 % 42.6 %

2010 21.6 % 37.0 % 41.5%

oritize the learning resources. The total scores con-

cerning textbooks/eNotes, Maple Demos, and the stu-

dents’ own notes from lectures are shown in Table 2.

The table shows that the students’ appreciation of the

new eNote representation of the mathematical con-

tent of the course has increased considerably at the

expense of the Maple Demos and their own notes.

Table 2: From textbooks to eNotes.

Year (e-)Text Maple Demos Own notes

2009 26.8% 38.0 % 35.2 %

2010 45.8 % 28.8 % 25.5 %

5 CONCLUSIONS

Taken as a whole the two tables 1 and 2 indicate that

the combined effects of a wide spectrum of web-based

learning objects has helped fulﬁl the overall aim: To

strengthen the students’ desire and ability to prepare

for the teaching and to read and enjoy the textual rep-

resentations of the course contents.

ACKNOWLEDGEMENTS

The investigation and the data that we mention in

section 4 have been designed and collected by P. M.

Hussmann (LearningLab DTU), P. Ratleff (Danish

School of Education), H. Rotz

en (DTU Informatics),

and K. Schmidt (DTU Mathematics).

REFERENCES

Ratleff, P., Schmidt, K., and Hussmann, P. M. (2009). Rea-

ding less and understanding more. In MONA, (2009)

3, pp. 21-40.

Schmidt, K., Rattleff, P., and Hussmann, P. M. (2010). The

impact of CAS use in introductory engineering math-

ematics. In Progress in Industrial Mathematics at

ECMI 2008. Mathematics in Industry, Volume 15, Part

2, (2010) 653-659. Springer Verlag.

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