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ICME-13 Monographs

Jason Silverman
Veronica Hoyos Editors

Distance Learning,
E-Learning and
Blended Learning
in Mathematics
Education
International Trends in Research
and Development
ICME-13 Monographs

Series editor
Gabriele Kaiser, Faculty of Education, Didactics of Mathematics, Universität
Hamburg, Hamburg, Germany
Each volume in the series presents state-of-the art research on a particular topic in
mathematics education and reflects the international debate as broadly as possible,
while also incorporating insights into lesser-known areas of the discussion. Each
volume is based on the discussions and presentations during the ICME-13 Congress
and includes the best papers from one of the ICME-13 Topical Study Groups or
Discussion Groups.

More information about this series at http://www.springer.com/series/15585

Jason Silverman Veronica Hoyos
•

Editors

Distance Learning,
E-Learning and Blended
Learning in Mathematics
Education
International Trends in Research
and Development

123
Editors
Jason Silverman Veronica Hoyos
School of Education National Pedagogical University
Drexel University Mexico City, Distrito Federal
Philadelphia, PA Mexico
USA

ISSN 2520-8322 ISSN 2520-8330 (electronic)

ICME-13 Monographs
ISBN 978-3-319-90789-5 ISBN 978-3-319-90790-1 (eBook)
https://doi.org/10.1007/978-3-319-90790-1
Library of Congress Control Number: 2018939465

© Springer International Publishing AG, part of Springer Nature 2018

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are exempt from
the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained herein or
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictional claims in published maps and institutional affiliations.

Printed on acid-free paper

This Springer imprint is published by the registered company Springer International Publishing AG
part of Springer Nature
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Contents

1 Research on Technologically Mediated Mathematics Learning

at a Distance: An Overview and Introduction . . . . . . . . . . . . . . . . . 1
Jason Silverman and Veronica Hoyos

Part I E-Learning and Blended Learning of Mathematics

2 The Blended Learning Concept e:t:p:M@Math: Practical
Insights and Research Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Fabian Mundt and Mutfried Hartmann
3 Challenges and Opportunities in Distance and Hybrid
Environments for Technology-Mediated Mathematics Teaching
and Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Veronica Hoyos, Maria E. Navarro, Victor J. Raggi
and Guadalupe Rodriguez

Part II Online Environments and Tutoring Systems for Leveling

College Students’ Mathematics Learning
4 Computer Assisted Math Instruction: A Case Study for
MyMathLab Learning System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Adam Chekour
5 Lessons Learned from a Calculus E-Learning System for
First-Year University Students with Diverse Mathematics
Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Lixing Liang, Karfu Yeung, Rachel Ka Wai Lui,
William Man Yin Cheung and Kwok Fai Lam

v
vi Contents

6 A Customized Learning Environment and Individual Learning

in Mathematical Preparation Courses . . . . . . . . . . . . . . . . . . . . . . . 93
Karin Landenfeld, Martin Göbbels, Antonia Hintze and Jonas Priebe

Part III Innovations on E-Math Learning and Teaching

7 Scripting Collaboration for Competence-Based Mathematics
Learning: A Case Study on Argumentation . . . . . . . . . . . . . . . . . . 115
Giovannina Albano and Umberto Dello Iacono
8 Effective Use of Math E-Learning with Questions
Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Yasuyuki Nakamura, Kentaro Yoshitomi, Mitsuru Kawazoe,
Tetsuo Fukui, Shizuka Shirai, Takahiro Nakahara, Katsuya Kato
and Tetsuya Taniguchi
9 Designing Interactive Technology to Scaffold Generative
Pedagogical Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Anthony Matranga, Jason Silverman, Valerie Klein
and Wesley Shumar
10 Supporting Teachers in Developing Their RiTPACK
Through Using Video Cases in an Online Course . . . . . . . . . . . . . . 165
Cosette Crisan

Part IV MOOC and Rich Media Platform for Mathematics Teacher

Education
11 Design and Impact of MOOCs for Mathematics Teachers . . . . . . . 185
Tamar Avineri, Hollylynne S. Lee, Dung Tran, Jennifer N. Lovett
and Theresa Gibson
12 Describing Curricular Materials for Mathematics Teacher
Education in an Online, Rich Media Platform . . . . . . . . . . . . . . . . 201
Daniel Chazan, Patricio Herbst, Dana Grosser-Clarkson,
Elizabeth Fleming, Janet Walkoe and Emina Alibegović
Chapter 1
Research on Technologically Mediated
Mathematics Learning at a Distance:
An Overview and Introduction

Jason Silverman and Veronica Hoyos

Abstract In this chapter, we provide an overview and introduction to this

monograph, which reports on the work of an international group of scholars that
joined together at the 13th International Congress on Mathematics Education to
share and build on current and emerging research in distance learning, e-learning
and blended learning in mathematics. We share work that emerged from Topic
Study Group 44: Distance learning, e-learning, blended learning, including
research and development in the use of digital teaching and learning platforms,
usage of this technology to scaffold mathematics instruction and tutoring, novel
interfaces for communicating and analyzing student thinking, and specialized
mathematics teacher education platforms.

Keywords Research on electronic and distance learning
Teaching and learn-

ing platforms Scaffolding mathematics instruction

This book emerged from the Topic Study Group 44 at the 13th International
Congress on Mathematics Education, ICME13, held in Hamburg, Germany on
2016, from July 24th to 31th, where an international group of scholars joined
together to share and build on current and emerging research in distance learning,
e-learning and blended learning. Specifically, in TSG44 we sought to push on the
boundaries of what was known on distance education, e-learning and blended
learning and teaching of mathematics through an examination and discussion of
recent research and development through these modalities and the common factors
that cut across them.

J. Silverman (&)
Drexel University, Philadelphia, USA
e-mail: silverman@drexel.edu
V. Hoyos
National Pedagogical University, Mexico City, Mexico
e-mail: vhoyosa@upn.mx

© Springer International Publishing AG, part of Springer Nature 2018 1

J. Silverman and V. Hoyos (eds.), Distance Learning, E-Learning and Blended
Learning in Mathematics Education, ICME-13 Monographs,
https://doi.org/10.1007/978-3-319-90790-1_1
2 J. Silverman and V. Hoyos

The papers published in this monograph are revisions and extensions of the
original papers presented during the TSG 44 sessions and reflect additional work
carried out by all participant authors after the conference ended. The monograph is
organized in four parts: The first part presents two chapters that focus on the
incorporation of new technologies into mathematics classrooms through the con-
struction or use of digital teaching and learning platforms (see chapters by Mundt &
Hartman, and Hoyos et al., in this book). The second part presents a wide range of
perspectives on the study and implementation of different tutoring systems and/or
computer assisted math instruction (see Chaps. 4–6 in this book, correspondingly
authored by Chekour; Liang et al.; and Landenfeld et al.). The third part presents
four new innovations in mathematics learning and/or mathematics teacher educa-
tion that involve the development of novel interfaces’ for communicating mathe-
matical ideas and analysing student thinking and student work (see Chaps. 7–10,
authored by Albano & Dello-Iacono; Nakamura et al.; Matranga, Silverman, Klein
& Shumar; and Crisan). Finally, the fourth part presents latest work on the con-
struction and implementation of new MOOCs and rich media platforms accom-
plished to carry out specialized mathematics teacher education (see chapters
authored by Avineri et al.; and Chazan et al., in this book).

1.1 Overview of Parts and Chapters

1.1.1 Part I: E-Learning and Blended Learning

of Mathematics

Chapter 2 introduces the reader in the construction of the e:t:p:M® platform,

developed by Mundt and Hartman from 2012 to 2016, at the University of
Education Karlsruhe (Germany). The authors worked to improve the quality of
higher education instruction through a platform that integrates digital and
internet-based technologies into regular (brick-and-mortar) classes. Specifically, the
authors looked at the articulation of online material and other technology to enable
a variety of options to be implemented in a blended-course format through the e:t:p:
M® approach. Additionally, teachers can modify the content and also student
activities can be tracked and reported for a continuous evaluation and improvement.
The potential benefits of this platform are clear as students can find materials for
specific content as well as document occasions of interactions with the materials,
colleagues, a mentor and/or the teacher; and teachers can create his/her materials
specific to their individual needs. In addition to sharing the e:t:p:M® development
and model, Mundt and Hartman’s chapter provides additional data and analysis
regarding the usage of the platform, for example measuring student accesses to the
platform as well as understanding relationships between interactions with the
content and the archived online lessons.
1 Research on Technologically Mediated Mathematics Learning … 3

Chapter 3, by Hoyos et al., from the National Pedagogical University in Mexico

City, addresses opportunities and challenges posed by the teaching and learning of
mathematics through digital platforms. Specifically, the chapter focuses on the
design and implementation of several different mathematics learning environments
that provided new teaching and learning opportunities for students in hybrid
environments. In this work, the authors establish a relationship between student
mathematical attainments and digital tools functionalities, by means of the elabo-
ration of teaching cycles that have influenced the design of the activity and stu-
dents’ learning improvement.
In the chapter by Hoyos et al., the authors documented usage of digital platform
tools for the administration, storage and deployment of resources and for facilitating
interactions within the resources stored in the platform. One important considera-
tion in this chapter is the teaching challenges of attending to the promotion of
reflection processes during the resolution of math problems in an online environ-
ment. This paper documents the challenges when teaching and learning of math-
ematics were completely online and mediated by technology.

1.1.2 Part II: Online Environments and Tutoring Systems

for Leveling College Students’ Mathematics

In Chap. 4, A. Chekour, from the University of Cincinnati-Blue Ash College in

USA, describes an effort to utilize technology to support more effective develop-
mental mathematics learning and teaching. The chapter compares the academic
performance of students enrolled in developmental mathematics sections that utilize
computer-assisted instruction with those using traditional instruction. Results show
the potential of the computer-assisted instruction, both in aggregate and for both
males and females separately. The chapter also discusses the challenges and
opportunities for incorporating computer assisted instruction into university math-
ematics classes.
Chapter 5 by Liang et al., from the University of Hong Kong in Hong Kong,
presents an evaluation of the different user behaviours on an e-learning platform for
students with different levels of calculus knowledge. Having collected data from a
sample of 225 students who have used the platform, which includes both video
lessons and assessments, the authors focus on student interaction with a supple-
mental (i.e. not required) e-learning system. Results highlight relationships between
activity in the system and student performance on standard examinations. For
example, they document that while students with the necessity and urgency to catch
up (i.e. with less prior knowledge on calculus) tend to be more active on the
e-learning platform in general, many of them tend to ignore the importance the
quizzes, which are designed to provide practice and support the development of
fluency with the contents at hand.
4 J. Silverman and V. Hoyos

In Chap. 6, Landenfeld and her colleagues from the Hamburg University of

Applied Sciences in Germany discuss the online learning environment via MINT,
which was designed to provide differentiated mathematics support to undergraduate
science and engineering students. Assessment results allow for the system to rec-
ommend differentiated “paths,” that can include selections of video tutorials,
learning activities and exercises and the “Personal Online Desk” provide a view for
the student and others to view progress. In addition to sharing details of the system,
this chapter also shares results of analysis of student engagement with the
environment.

1.1.3 Part III: Innovations on E-Math Learning

and Teaching

Chapter 7, by Albano & Dello-Iacono, from the University of Salerno in Italy,

presents an innovative approach to competence-based mathematics learning,
through the use of a digital platform named DIST-M (Digital Interactive
Storytelling in Mathematics). DIST-M allows learners to define a model where the
roles of participants and the sequence of activities promote cognitive,
socio-cognitive and metacognitive processes. In this platform, students are engaged
in activities within a storytelling experience. The authors used both experiential and
discursive approaches to mathematics learning, integrating individual and social
tasks, defined by external scripts. The development DIST-M was based on the
assumption that such environment can be arranged in a way that a good exploitation
of platform tools and a well-structured collaboration among peers can act as an
expert support to students in achieving their learning goal. The environment also
supports the exploration of specific mathematics content—representation and
management of graphics and descriptive statistics, in the case of this paper—in
spatial activities that the authors have designed around thematic contexts, such as
the discovery of a new planet.
In Chap. 8, Nakamura et al., working in three different universities from Japan
(the Nagoya University, the Mukogawa Women’s University and the Nihon
University) and including the participation of the Sangensha LLC., and the
Cybernet Systems Co., Ltd., address two challenges that instructors encounter when
implementing the e-learning systems that are prominent in Japan: entry of mathe-
matical symbols and equations and the development of content for specific courses
and content. The authors share details of two input interfaces that integrate with
commonly used e-learning systems and allow students to input mathematical
symbols and equations using both computer and mobile devices, importantly, to
address the unique challenges of mathematics e-learning using mobile devices
(tablets and mobile phones). These input interfaces are one component of the
MeLQS e-learning, question specification allows for questions to be cross-platform,
1 Research on Technologically Mediated Mathematics Learning … 5

and users from many different e-learning systems collaborate and share questions
and tasks, thereby making the use of e-learning systems more generative.
Matranga et al. (Chap. 9), from the California State University San Marcos and
the Drexel University in USA, describe an online environment designed to support
the emergence of a set of professional practices within a group of mathematics
teachers. In the chapter, the authors share the design and features of the environ-
ment, highlighting the design process through which it was thought to meet specific
use cases identified by teachers. The chapter addresses the challenge of scaling
teacher professional development through using technology to emulate a boundary
encounter between a group of teachers and an existing community of educators with
productive pedagogical practices. Their findings show the promise of this approach,
specifically noting the emergence of productive pedagogical practices normative in
the target community.
Finally in this part, Crisan (Chap. 10), from the UCL Institute of Education,
University College London in UK, discusses her work supporting teachers as they
explore how digital technology supports students’ understanding and learning of
mathematics. Video cases that depict actual student engagement with specific
mathematics tasks, including audio and video of students synchronized with
recordings of their actual work, were specifically developed for this project and
participants engagement with these cases—and the student thinking that are
depicted in the cases—was analysed. Crisan reports that persistent engagement with
these video cases and the other supports provided in the online context show
promise for scaffolding teachers as they analyze student work and develop peda-
gogical solutions based on this analysis. Using a modified version of the
Technological Pedagogical Content Knowledge (TPACK) framework, Research-
informed TPACK (RiTPACK), Crisan presents additional evidence of teacher
development resulting from their engagement with the video cases and online
course.

1.1.4 Part IV: MOOC and Rich Media Platform

for Mathematics Teacher Education

Chapter 11 by Avineri et al., from three universities in USA (the North Carolina
School of Science and Mathematics, the North Carolina State University, and the
Middle Tennessee State University) and including the Victoria University in
Melbourne (Australia), specify design principles for the implementation of MOOCs
for professional development of mathematics teachers, based on recent research on
this topic. The chapter documents the design efficiency and discusses specific
impacts that participants report on changes into their teaching practices.
Specifically, some participants addressed changes to their approach to teaching
(e.g., increased focus on concepts as opposed to algorithms), others described how
their participation supported their refined attention to and understanding of their
6 J. Silverman and V. Hoyos

students’ thinking and their own personal improvement in knowledge of mathe-

matics. According to Avineri and colleagues, the research-based design principles
that guided the creation of the MOOC-ED courses have afforded educators’ choice
in professional learning, complemented with relevant, job-embedded activities,
access to the perspectives of experts, teachers, and students, and a network of
educators learning together around a common content area.
Chapter 12, by Chazan et al., from three institutions in USA (the University of
Maryland, the University of Michigan, and the Rowland Hall School at Salt Lake
City in Utah), describes how the LessonSketch platform has been used to implement
a larger project between math teacher educators. In particular, these authors use
Grossman’s pedagogies of practice to explore how teacher educators are repre-
senting practice, decomposing it, and providing opportunities for their students to
approximate practice through the curricular artefacts that they are creating. Chazan
et al., describe a practice-based approach to helping teachers explore the content of
mathematics teacher education, and report the novel ways in which a certain online
environment (LessonSketch in this case) supports new opportunities for teacher
candidates to practice the work of teaching. These authors note that professional
development experiences created with these platforms not only have pedagogical
characteristics and support learning about teaching, but also have curricular char-
acteristics that help shape what it is that teacher candidates should learn.

1.1.5 Purpose of This Monograph

This book addresses issues of collaboration, equity, access and curriculum in the
context of learning and teaching mathematics. For example, Mundt and Hartman
focus on the population of students entering Universities (Chap. 2) and propose an
online platform such as e:t:p:M® to address the challenges brought forth through
significant increases in undergraduate populations and associated challenges in
instruction and supervision. This is a consistent role posited by authors in this text
utilizing existing course management systems and tools, such as Moodle and
Blackboard Learn, as well as other custom designed platforms. While the vast
majority of online platforms offer similar features, such as organization of the
content, and integration of external software including e-mail, and discussions, the
authors noted that e:t:p:M® approach innovates because it could establish or
monitor a relationship between the usage of different mobile technology resources
with the blended courses it promoted. These results and others presented throughout
this volume confirm the existence of new teaching and learning opportunities when
working with students in hybrid environments.
With regards to wholly online mathematics learning and teaching, authors in this
volume reported the existence of challenges related with the promotion of reflection
processes when teachers or students solve mathematics complex tasks while par-
ticipating in a course at a distance (see Hoyos et al., and Matranga et al., chapters).
Using different contexts and approaches, the authors suggest that effective digital
1 Research on Technologically Mediated Mathematics Learning … 7

collaboration requires attention to individual’s (teacher or student) activity and

specific supports to accomplish an epistemological change required in order to
engage productively and solve such mentioned tasks. These supports can be
included in the computational device, learning environment or otherwise be pro-
vided by tutorial intervention.
A second broad theme in this volume is the construction and evaluation of
mathematics tutoring systems for supporting college students’ persistence and
success. Such tutoring systems are essential, both given the growth in undergrad-
uate students and continued issues regarding entering freshman’s preparation for
college level mathematics. While there are various commercial tutoring environ-
ments available, the authors in this volume (Chekour; Liang, et al.; and Landenfeld
et al.) notice the benefit of custom designed environments to address specific local
constraints and share information about their systems as well as suggestions to
improve students’ use of an e-learning platform.
The third part of this volume addresses a third theme: innovation in e-learning.
In this part, the authors discuss new approaches to mathematics learning and
mathematics teacher collaboration through the use of Web platforms and commu-
nication tools. Albano & Dello-Iacono introduce a general methodology to support
an e-learning-based approach to competence-based mathematics learning. These
authors designed and implemented certain computer-supported collaboration scripts
aimed to foster students’ shift from investigating, reasoning and communicating
what they have found. Nakamura et al.’s chapter described and displayed a series of
interfaces designed to minimize the challenges of mathematical symbols and syntax
in e-learning environments. In Matranga et al. chapter, the authors documented that
a specifically designed online collaborative environment had the potential to scaf-
fold teachers’ legitimate participation in reform-type conversations and activities
that were not common for these individuals without the online supports. Finally,
Crisan’s chapter provides another example on the use of varied multimedia for
teacher development resulting from their engagement with video cases and specific
online course.
The fourth part of the book addresses a final theme: the use of online rich media
platform for teacher education, including the development and implementation of
both visualizations of teaching and specially constructed MOOCs for mathematics
teacher education. Two of these applications are discussed in Chaps. 11 and 12, and
share theoretical and empirical evidence regarding both the effectiveness of the
specific design and medium as well as emerging advancements in this area. As an
example, Chazan et al. (Chap. 11) use the mathematics education literature on
curriculum to suggest that the curriculum creation process that is underway in
teacher education, when it happens online, is influenced by the digital nature of
technological artifacts.
This book is a scholarly collaboration on the part of professors, developers and
researchers in the broad fields of technologically-enhanced mathematics education
and serves as an effort to disseminate significant contributions and share interna-
tional perspectives on this important and timely area. The book provides an over-
view of the current state-of-the-art research and shares and discusses emerging
8 J. Silverman and V. Hoyos

work, including trends, ideas, methodologies, and results and represents a special
call to continue research and development and to grow a canon of research foun-
dations for distance learning, e-learning and blended learning in mathematics
education.
 Part I
E-Learning and Blended
Learning of Mathematics
Chapter 2
The Blended Learning Concept
e:t:p:M@Math: Practical Insights
and Research Findings

Fabian Mundt and Mutfried Hartmann

Abstract The chapter outlines the key ideas of the blended learning concept
e:t:p:M® and its further development in the field of Higher Mathematical Education.
e:t:p:M@Math aims to integrate digital technologies and face-to-face interactions to
simultaneously allow personalized and high-quality learning. Both practical
teaching experiences as well as research findings will be discussed. One focus is on
the description of the self-developed and designed e-Learning environment, its
possibilities and further development. Another focus is on the reflection of the
practical implementation into everyday teaching, especially the integration with
face-to-face seminars. In addition, first research insights will be presented and
explained.

Keywords Blended learning
E-learning
Distance learning
Learning analytics

2.1 Introduction: Challenging Trends in Higher

Education

In the winter term 2014/2015, the Federal Bureau for Statistics of Germany counted
2.7 million university students—a milestone in the history of the Federal Republic
of Germany (SB, 2015). Given that only ten years ago there were far less than 2
million students (Bildungsbericht, 2014), the magnitude of this increase becomes
even more significant. Many universities have adopted “bulk-instruction” with
heterogeneous student groups and an unfavorable student-to-instructor ratio
(Himpsl, 2014). In particular, high-demand introductory courses suffer under these
problematic circumstances. Therefore, the quality of education is lacking and the
need for reforms is apparent (Asdonk, Kuhnen, & Bornkessel, 2013).

F. Mundt (&)
M. Hartmann

University of Education Karlsruhe, Bismarckstraße 10, 76133 Karlsruhe, Germany
e-mail: fabian.mundt@ph-karlsruhe.de
M. Hartmann
e-mail: mutfried.hartmann@ph-kalrsruhe.de

© Springer International Publishing AG, part of Springer Nature 2018 11

J. Silverman and V. Hoyos (eds.), Distance Learning, E-Learning and Blended
Learning in Mathematics Education, ICME-13 Monographs,
https://doi.org/10.1007/978-3-319-90790-1_2
12 F. Mundt and M. Hartmann

Since this situation is unlikely to change in the foreseeable future—neither

nationally nor internationally (Dräger, Friedrich, & Müller-Eiselt, 2014; Maslen,
2012)—innovative teaching and learning concepts are necessary. In contrast to
widely-discussed MOOCs, one very promising approach involves integrating reg-
ular class sessions with the opportunities of digital (Internet-) technologies (see
Carr, 2012). One specific model that specifically aims at the integration of both
class sessions and digital content is e:t:p:M®.

2.2 The Blended Learning Concept e:t:p:M®

e:t:p:M®1 was developed as an introductory course in education in the winter term

2012 at the University of Education in Karlsruhe by Timo Hoyer and Fabian
Mundt. Detailed information about the project and its theoretical framework can be
found in Hoyer and Mundt (2014, 2016). The acronym, which indicates the indi-
vidual parts of the project, are described in depth below.

2.2.1 “e” for E-Learning

The core of the e-learning content consists of 11 studio recorded online lessons that
have been post-produced according to a creative framework. The lessons are all
between 20 and 30 min long and are comprised of a speaker as well as info boards,
images, animations and quotations. Additionally, the lessons are structured through
so called “Fähnchen“ (thematic headlines). The students can access the content via
an especially for the e:t:p:M® project developed responsive web-app (Fig. 2.1).2
Personal annotations can be added to every “Fähnchen” and then downloaded as
a PDF-file (Fig. 2.2). Furthermore, the web-app grants access to additional mate-
rials (texts, exercises etc.) and does not only contain general information about the
class but also an extensive FAQ-area and the possibility to get in touch with the
lecturers directly. The web-app also provides the user with a differentiated module
for analysis that enables the teacher to track the students’ interactions.3

1
Project website: http://etpm.ph-karlsruhe.de/demo/ [13.12.2016].
2
The web-app was developed with the open-source frameworks Laravel, Vue.js, Semantic UI and
Video.js.
3
As a tool for analyzing the interaction, an adjusted version of the open analytics platform “Piwik”
is used. All collected data is anonymized. The tracking function can be deactivated from inside the
web-app, which is highlighted for the users.
2 The Blended Learning Concept e:t:p:M@Math: Practical Insights … 13

Fig. 2.1 The responsive web-app (original version)

Fig. 2.2 The annotation function of the web-app

2.2.2 “t” for Text and Theory Based

Alongside every online lesson, the students are provided with a text that deepens
the content (primary as well as secondary literature). In addition to suggested
approaches to the text, the file contains questions that will be dealt with during the
attended seminar. All texts are formatted uniformly and have been edited for the use
in a seminar.

2.2.3 “p” for Practice-Oriented (and Attendance-Oriented)

The e-learning content of e:t:p:M® aims at a high personalization of the learning

content as well as its integration into the seminars. The latter are comprised of
information sessions (lecturers), FAQ-sessions (lecturers) and weekly mentoring
sessions (student mentors) (Fig. 2.3).
14 F. Mundt and M. Hartmann

Fig. 2.3 The e:t:p:M® concept

2.2.4 “M” for Mentoring

Especially in the beginning of studies at university, the support and care for
beginners is of high importance. In addition to subject-specific competences, the
students need to acquire a sense to navigate the foreign academic world. In e:t:p:M®
the class is separated into smaller groups who will be mentored by a tandem of
older students during the semester. The mentors are trained in a specifically
designed workshop and receive a certificate after completion.
The program received an award for extraordinary teaching methods in 2013 and
was evaluated positively several times.4

2.3 Using e:t:p:M® in an Introductory Course

in Mathematics

Based on the previously explained challenges for teaching at university and the very
positive feedback towards the project e:t:p:M®, the concept is being adapted for
other subjects outside the realm of pedagogy. At the moment, the authors work on
applying the program to an “Introduction in Mathematics” course, which started in
the winter term 2015 (Mundt & Hartmann, 2015). The current evolution of the
concept is presented below. Since the contents are more historical and theory
oriented, the application of e:t:p:M@Math requires adjustments. The online lessons
and web-app, in particular, are being revised extensively to meet the requirements
of mathematical learning.
The adaptation is informed and guided by “design-based research methodology”
(Wang & Hannafin, 2005). Specifically, it is situated in a real educational context
(mathematics introduction) and is focusing on the design and testing of significant
interventions (e:t:p:M@Math concept) (see Anderson & Shattuck, 2012). As part of
the design process, we refer to contemporary findings in the field of Higher

4
http://etpm-dev.ph-karlsruhe.de/etpm-evaluation/ [13.10.2015].
Another random document with
no related content on Scribd:
landscape. A three hours’ ride brought us to the Rabna Gja, the
eastern boundary of Thingvalla, and, winding up its rugged face, we
took our last look over the lovely plain beneath us, and then
manfully set across the same kind of arid lava plateau as that which
we had already traversed before arriving at the Almanna Gja.
Letters from High Latitude, being some account of a
voyage in the schooner yacht Foam in 1856 (London,
1859).

FOOTNOTE:
[4] The plain of Thingvalla is in a great measure clothed with
birch brushwood.
 LAND’S END AND LOGAN ROCK
(ENGLAND)
JOHN AYRTON PARIS

“The sunbeams tremble, and the purple light

Illumes the dark Bolerium;—seat of storms,
High are his granite rocks; his frowning brow
Hangs o’er the smiling ocean. In his caves,
Where sleep the haggard spirits of the storm,
Wild dreary are the schistose rocks around,
Encircled by the waves, where to the breeze
The haggard cormorant shrieks; and far beyond
Are seen the cloud-like islands, grey in mists.”

Sir H. Davy.
In an excursion to the Land’s End the traveller will meet with several
intermediate objects well worthy his attention, more worthy,
perhaps, than the celebrated promontory itself, as being monuments
of the highest antiquity in the kingdom. They consist of Druidical
circles, cairns, or circular heaps of stones, cromlechs, crosses,
military entrenchments, and the obscure remains of castles.
Having arrived at the celebrated promontory, we descend a rapid
slope, which brings us to a bold group of rocks, composing the
western extremity of our island. Some years ago a military officer
who visited this spot, was rash enough to descend on horseback;
the horse soon became unruly, plunged, reared, and, fearful to
relate, fell backwards over the precipice, and rolling from rock to
rock was dashed to atoms before it reached the sea. The rider was
for some time unable to disengage himself, but at length by a
desperate effort he threw himself off, and was happily caught by
some fragments of rock, at the very brink of the precipice, where he
remained in a state of insensibility until assistance could be afforded
him! The awful spot is marked by the figure of a horseshoe, traced
on the turf with a deep incision, which is cleared out from time to
time, in order to preserve it as a monument of rashness which could
alone be equalled by the good fortune with which it was attended.
Why any promontory in an island should be exclusively denominated
the Land’s End, it is difficult to understand; yet so powerful is the
charm of a name, that many persons have visited it on no other
account; the intelligent tourist, however, will receive a much more
substantial gratification from his visit; the great geological interest of
the spot will afford him an ample source of entertainment and
instruction, while the magnificence of its convulsed scenery, the
ceaseless roar, and deep intonation of the ocean, and the wild
shrieks of the cormorant, all combine to awaken the blended
sensations of awe and admiration.
The cliff which bounds this extremity is rather abrupt than elevated,
not being more than sixty feet above the level of the sea. It is
composed entirely of granite, the forms of which present a very
extraordinary appearance, assuming in some places the resemblance
of shafts that had been regularly cut with the chisel; in others,
regular equidistant fissures divide the rock into horizontal masses,
and give it the character of basaltic columns; in other places, again,
the impetuous waves of the ocean have opened, for their retreat,
gigantic arches, through which the angry billows roll and bellow with
tremendous fury.
Several of these rocks from their grotesque forms have acquired
whimsical appellations, as that of the Armed Knight, the Irish Lady,
etc. An inclining rock on the side of a craggy headland, south of the
Land’s End, has obtained the name of Dr. Johnson’s Head, and
visitors after having heard the appellation seldom fail to
acknowledge that it bears some resemblance to the physiognomy of
that extraordinary man.
On the north, this rocky scene is terminated by a promontory 229
feet above the level of the sea, called Cape Cornwall, between which
and the Land’s End, the coast retires, and forms Whitesand Bay; a
name which it derives from the peculiar whiteness of the sand, and
amongst which the naturalist will find several rare microscopic shells.
There are, besides, some historical recollections which invest this
spot with interest. It was in this bay that Stephen landed on his first
arrival in England; as did King John, on his return from Ireland; and
Perkin Warbeck, in the prosecution of those claims to the Crown to
which some late writers have been disposed to consider that he was
entitled, as the real son of Edward the Fourth. In the rocks near the
southern termination of Whitesand Bay may be seen the junction of
the granite and slate; large veins of the former may also be
observed to traverse the latter in all directions.
We now return to the Land’s End,—from which we should proceed to
visit a promontory called “Castle Treryn,” where is situated the
celebrated “Logan Stone.” If we pursue our route along the cliffs, it
will be found to be several miles southeast of the Land’s End,
although by taking the direct and usual road across the country, it is
not more than two miles distant; but the geologist must walk, or ride
along the coast on horseback, and we can assure him that he will be
amply recompensed for his trouble.
From the Cape on which the signal station is situated, the rock
scenery is particularly magnificent, exhibiting an admirable specimen
of the manner, and forms, into which granite disintegrates. About
forty yards from this Cape is the promontory called Tol-Pedn-
Penwith, which in the Cornish language signifies the holed headland
in Penwith. The name is derived from a singular chasm, known by
the appellation of the Funnel Rock; it is a vast perpendicular
excavation in the granite, resembling in figure an inverted cone, and
has been evidently produced by the gradual decomposition of one of
those vertical veins with which this part of the coast is so frequently
intersected. By a circuitous route you may descend to the bottom of
the cavern, into which the sea flows at high water. Here the Cornish
chough (Corvus Graculus) has built its nest for several years, a bird
which is very common about the rocky parts of this coast, and may
be distinguished by its red legs and bill, and the violaceous
blackness of its feathers. This promontory forms the western
extremity of the Mount’s Bay. The antiquary will discover in this spot,
the vestiges of one of the ancient “Cliff Castles,” which were little
else than stone walls, stretching across necks of land from cliff to
cliff. The only geological phenomenon worthy of particular notice is a
large and beautiful contemporaneous vein of red granite containing
schorl; is one foot in width, and may be seen for about forty feet in
length.
Continuing our route around the coast we at length arrive at Castle
Treryn. Its name is derived from the supposition of its having been
the site of an ancient British fortress, of which there are still some
obscure traces, although the wild and rugged appearance of the
rocks indicate nothing like art.

ROCKING STONES, LAND’S END, CORNWALL.

The foundation of the whole is a stupendous group of granite rocks,
which rise in pyramidal clusters to a prodigious altitude, and
overhang the sea. On one of those pyramids is situated the
celebrated “Logan Stone,” which is an immense block of granite
weighing about sixty tons. The surface in contact with the under
rock is of very small extent, and the whole mass is so nicely
balanced, that, notwithstanding its magnitude, the strength of a
single man applied to its under edge is sufficient to change its centre
of gravity, and though at first in a degree scarcely perceptible, yet
the repetition of such impulses, at each return of the stone,
produces at length a very sensible oscillation! As soon as the
astonishment which this phenomenon excites has in some measure
subsided, the stranger anxiously inquires how, and whence the stone
originated—was it elevated by human means, or was it produced by
the agency of natural causes? Those who are in the habit of viewing
mountain masses with geological eyes, will readily discover that the
only chisel ever employed has been the tooth of time—the only artist
engaged, the elements. Granite usually disintegrates into rhomboidal
and tabular masses, which by the farther operation of air and
moisture gradually lose their solid angles, and approach the
spheroidal form. De Luc observed, in the giant mountains of Silesia,
spheroids of this description so piled upon each other as to resemble
Dutch cheeses; and appearances, no less illustrative of the
phenomenon, may be seen from the signal station to which we have
just alluded. The fact of the upper part of the cliff being more
exposed to atmospheric agency, than the parts beneath, will
sufficiently explain why these rounded masses so frequently rest on
blocks which still preserve the tabular form; and since such
spheroidal blocks must obviously rest in that position in which their
lesser axes are perpendicular to the horizon, it is equally evident that
whenever an adequate force is applied they must vibrate on their
point of support.
Although we are thus led to deny the Druidical origin of this stone,
for which so many zealous antiquaries have contended, still we by
no means intend to deny that the Druids employed it as an engine of
superstition; it is indeed very probable that, having observed so
uncommon a property, they dexterously contrived to make it answer
the purposes of an ordeal, and by regarding it as the touchstone of
truth, acquitted or condemned the accused by its motions. Mason
poetically alludes to this supposed property in the following lines:

“Behold yon huge

And unknown sphere of living adamant,
Which, pois’d by magic, rests its central weight
On yonder pointed rock: firm as it seems,
Such is its strange, and virtuous property,
It moves obsequious to the gentlest touch
Of him whose heart is pure, but to a traitor,
Tho’ e’en a giant’s prowess nerv’d his arm,
It stands as fix’d as—Snowdon.”

The rocks are covered with a species of Byssus long and rough to
the touch, forming a kind of hoary beard; in many places they are
deeply furrowed, carrying with them a singular air of antiquity, which
combines with the whole of the romantic scenery to awaken in the
minds of the poet and enthusiast the recollection of the Druidical
ages. The botanist will observe the common Thrift—(Statice
Armeria) imparting a glowing tinge to the scanty vegetation of the
spot, and, by growing within the crevices of the rocks, affording a
very picturesque contrast to their massive fabric. Here, too, the
Daucus Maritimus, or wild carrot; Sedum Telephium, Saxifraga
Stellaris, and Asplenium Marinum, may be found in abundance.
The granite in this spot is extremely beautiful on account of its
porphyritic appearance; the crystals of feldspar are numerous and
distinct; in some places the rock is traversed by veins of red feldspar,
and of black tourmaline, or schorl, of which the crystalline forms of
the prisms, on account of their close aggregation, are very indistinct.
Here may also be observed a contemporaneous vein of schorl rock in
the granite, nearly two feet wide, highly inclined and very short, and
not having any distinct walls. On the western side of the Logan Rock
is a cavern, formed by the decomposition of a vein of granite, the
feldspar of which assumes a brilliant flesh-red and lilac colour; and,
where it is polished by the sea, exceeding even in beauty the
Serpentine caverns at the Lizard.
A Guide to the Mount’s Bay and the Land’s End (London,
2d Ed., 1824).
 MOUNT HEKLA[5]
(ICELAND)
SIR RICHARD F. BURTON
The Hekla of our ingenuous childhood, when we believed in the
“Seven Wonders of the World,” was a mighty cone, a “pillar of
heaven,” upon whose dreadful summit white, black, and sanguine
red lay in streaks and patches, with volumes of sooty smoke and
lurid flames, and a pitchy sky. The whole was somewhat like the
impossible illustrations of Vesuvian eruptions, in body-colours, plus
the ice proper to Iceland. The Hekla of reality, No. 5 in the island
scale, is a commonplace heap, half the height of Hermon, and a
mere pigmy compared with the Andine peaks, rising detached from
the plains, about three and a half miles in circumference, backed by
the snows of Tindafjall and Torfajökull, and supporting a sky-line
that varies greatly with the angle under which it is seen. Travellers
usually make it a three-horned Parnassus, with the central knob
highest—which is not really the case. From the south-west, it shows
now four, then five, distinct points; the north-western lip of the
northern crater, which hides the true apex; the south-western lip of
the same; the north-eastern lip of the southern crater, which
appears the culminating point, and the two eastern edges of the
southern bowls. A pair of white patches represents the “eternal
snows.” On the right of the picture is the steep, but utterly
unimportant Thrihyrningr, crowned with its benchmark; to the left,
the Skardsfjall, variegated green and black; and in the centre, the
Bjólfell, a western buttress of the main building, which becomes
alternately a saddleback, a dorsum, and an elephant’s head, trunk,
and shoulders.
We came upon the valley of the Western Rángá[6] at a rough point,
a gash in the hard yellow turf-clad clay, dotted with rough lava
blocks, and with masses of conglomerate, hollowed, turned, and
polished by water: the shape was a succession of S, and the left side
was the more tormented. Above the ford a dwarf cascade had been
formed by the lava of ’45, which caused the waters to boil, and
below the ford jumped a second, where the stream forks. We then
entered an Iceland “forest,” at least four feet high; the “chapparal”
was composed of red willow (Salix purpurea), of Grá-vidir, woolly-
leaved willow (Sulix lapponum), the “tree under which the devil
flayed the goats”—a diabolical difficulty, when the bush is a foot high
—and the awful and venerable birch, “la demoiselle des fôrets,”
which has so often “blushed with patrician blood.” About mid-
afternoon we reached Næfrholt (birch-bark hill), the “fashionable”
place for the ascent, and we at once inquired for the guide. Upon
the carpe diem principle, he had gone to Reykjavik with the view of
drinking his late gains; but we had time to organize another, and
even alpenstocks with rings and spikes are to be found at the farm-
house. Everything was painfully tourist.
In the evening we scaled the stiff slope of earth and Palagonite
which lies behind, or east of Næfrholt; this crupper of Bjólfell, the
Elephant Mountain, gives perhaps harder work than any part of
Hekla on the normal line of ascent. From the summit we looked
down upon a dwarf basin, with a lakelet of fresh water, which had a
slightly (carbonic) acid taste, and which must have contained lime,
as we found two kinds of shells, both uncommonly thin and fragile.
Three species of weeds floated off the clean sandstrips. Walking
northward to a deserted byre, we found the drain gushing under
ground from sand and rock, forming a distinct river-valley, and
eventually feeding the Western Rángá. This “Vatn” is not in the map;
though far from certain that it is not mentioned by Mackenzie, we
named it the “Unknown Lake.” Before night fell we received a
message that three English girls and their party proposed to join us.
This was a “scare,” but happily the Miss Hopes proved plucky as they
were young and pretty, and we rejoiced in offering this pleasant
affront of the feminine foot to that grim old solitaire, Father Hekla.
Before the sleep necessary to prepare for the next day’s work, I will
offer a few words concerning the “Etna of the North,” sparing the
reader, however, the mortification of a regular history. It was
apparently harmless, possibly dormant, till a. d. 1104, when
Sæmund, the “Paris clerk,” then forty-eight years old, threw in a
casket, and awoke the sleeping lion. Since that time fourteen regular
eruptions, without including partial outbreaks are recorded, giving an
average of about two per century. The last was in 1845. The air at
Reykjavik was flavoured, it is said, like a gun that wants washing;
and the sounds of a distant battle were conducted by the lava and
basaltic ground. The ashes extended to Scotland. When some
writers tell us that on this occasion Hekla lost 500 feet in height, “so
much of the summit having been blown away by the explosions,”
they forget or ignore the fact that the new crater opened laterally
and low down.
Like Etna, Vesuvius, and especially Stromboli, Hekla became mythical
in Middle-Age Europe, and gained wide repute as one of the gates of
“Hel-viti.” Witches’ Sabbaths were held there. The spirits of the
wicked, driven by those grotesque demons of Father Pinamonti
which would make the fortune of a zoological society, were seen
trooping into the infernal crater; and such facts as these do not
readily slip off the mind of man. The Danes still say “Begone to
Heckenfjæld!” the North Germans, “Go to Hackelberg!” and the
Scotch consign you to “John Hacklebirnie’s house.” Even Goldsmith
(Animated Nature, i. 48) had heard of the local creed, “The
inhabitants of Iceland believe the bellowings of Hekla are nothing
else but the cries of the damned, and that its eruptions are contrived
to increase their tortures.” Uno Van Troil (Letter I.) who in 1770,
together with those “inclyti Brittannici,” Baron Bank and Dr. Solander,
“gained the pleasure of being the first who ever reached the summit
of this celebrated volcano,” attributes the mountain’s virginity to the
superstitions of the people. He writes soberly about its marvels; and
he explains its high fame by its position, skirting the watery way to
and from Greenland and North America. His companions show less
modesty of imagination. We may concede that an unknown ascent
“required great circumspection”; and that in a high wind
ascensionists were obliged to lie down. But how explain the “dread
of being blown into the most dreadful precipices,” when the latter do
not exist? Moreover, we learn that to “accomplish this undertaking”
they had to travel from 300 to 360 miles over uninterrupted bursts
of lava, which is more than the maximum length of the island, from
northeast to southwest. As will be seen, modern travellers have
followed suit passing well.
The next morning (July 13) broke fair and calm, reminding me
“Del bel paese la dove il sì suona.”
The Miss Hopes were punctual to a minute—an excellent thing in
travelling womanhood. We rode up half-way somewhat surprised to
find so few parasitic craters; the only signs of independent eruption
on the western flank were the Raudkólar (red hills), as the people
call their lava hornitos and spiracles, which are little bigger than the
bottle-house cones of Leith.
At an impassable divide we left our poor nags to pass the dreary
time, without water or forage, and we followed the improvised
guide, who caused not a little amusement. His general port was that
of a bear that has lost its ragged staff.—I took away his alpenstock
for one of the girls—and he was plantigrade rather than
cremnobatic; he had stripped to his underalls, which were very
short, whilst his stockings were very long and the heraldic gloves
converted his hands to paws. The two little snow fonds (“steep
glassy slopes of hard snow”), were the easiest of walking. We had
nerved ourselves to “Break neck or limbs, be maimed or boiled
alive,” but we looked in vain for the “concealed abysses,” for the
“crevasses to be crossed,” and for places where a “slip would be to
roll to destruction.” We did not sight the “lava wall,” a capital
protection against giddiness. The snow was anything but slippery;
the surface was scattered with dust, and it bristled with a forest of
dwarf earth-pillars, where blown volcanic sand preserved the ice.
After a slow hour and a half, we reached the crater of ’45, which
opened at 9 a. m. on September 2, and discharged lava till the end of
November. It might be passed unobserved by the inexperienced
man. The only remnant is the upper lip prolonged to the right; the
dimensions may have been 120 by 150 yards, and the cleft shows a
projecting ice-ledge ready to fall. The feature is well-marked by the
new lava-field of which it is the source: the bristly “stone-river” is
already degrading to superficial dust. A little beyond this bowl the
ground smokes, discharging snow-steam made visible by the cold
air. Hence doubtless those sententious travellers “experienced at one
and the same time, a high degree of heat and cold.”
Fifteen minutes more led us to the First or Southern Crater, whose
Ol-bogi (elbow or rim) is one of the horns conspicuous from below.
It is a regular formation about 100 yards at the bottom each way,
with the right (east) side red and cindery, and the left yellow and
sulphury; mosses and a few flowerets grow on the lips; in the sole
rise jets of steam and a rock-rib bisects it diagonally from northeast
to southwest. We thought it the highest point of the volcano, but the
aneroid corrected our mistake.
From the First Crater we walked over the left or western dorsum,
over which one could drive a coach, and we congratulated one
another upon the exploit. Former travellers “balancing themselves
like rope-dancers, succeeded in passing along the ridge of slags
which was so narrow that there was scarcely room for their feet,”
the breadth being “not more than two feet, having a precipice on
each side several hundred feet in depth.” Charity suggests that the
feature has altered, but there was no eruption between 1766 and
1845; moreover, the lip would have diminished, not increased. And
one of the most modern visitors repeats the “very narrow ridge,”
with the classical but incorrect adjuncts of “Scylla here, Charybdis
there.” Scylla (say the crater slope) is disposed at an angle of 30°,
and Mr. Chapman coolly walked down this “vast” little hollow. I
descended Charybdis (the outer counterscarp) far enough to make
sure that it is equally easy.
Passing the “carriage road” (our own name), we crossed a névé
without any necessity for digging foot-holes. It lies where sulphur is
notably absent. The hot patches which account for the freedom from
snow, even so high above the congelation-line, are scattered about
the summit: in other parts the thermometer, placed in an eighteen-
inch hole, made earth colder than air. After a short climb we reached
the apex; the ruddy-walled northeastern lip of the Red Crater (No.
2): its lower or western rim forms two of the five summits seen from
the prairie, and hides the highest point. We thus ascertained that
Hekla is a linear volcano of two mouths, or three including that of
’45, and that it wants a true apical crater. But how reconcile the
accounts of travellers? Pliny Miles found one cone and three craters;
Madam Ida Pfeiffer, like Metcalfe, three cones and no crater.
On the summit the guides sang a song of triumph, whilst we drank
to the health of our charming companions and, despite the cold wind
which eventually drove us down, carefully studied the extensive
view. The glorious day was out of character with a scene niente che
Montagne, as the unhappy Venetians described the Morea; rain and
sleet and blinding snow would better have suited the picture, but
happily they were conspicuous by their absence. Inland, beyond a
steep snow-bed unpleasantly crevassed, lay a grim photograph all
black and white; Lángjökull looking down upon us with a grand and
freezing stare; the Hrafntinnu Valley marked by a dwarf cone, and
beyond where streams head, the gloomy regions stretching to the
Sprengisandur, dreary wastes of utter sterility, howling deserts of
dark ashes, wholly lacking water and vegetable life, and wanting the
gleam and the glow which light up the Arabian wild. Skaptár and
Oræfa were hidden from sight. Seawards, ranging from west to
south, the view, by contrast, was a picture of amenity and
civilization. Beyond castellated Hljódfell and conical Skjaldbreid
appeared the familiar forms of Esja, and the long lava projection of
the Gold Breast country, melting into the western main. Nearer
stretched the fair lowlands, once a broad deep bay, now traversed
by the network of Ölfusá, Thjórsá, and the Markarfljót; while the
sixfold bunch of the Westman Islands, mere stone lumps upon a
blue ground, seemingly floating far below the raised horizon, lay
crowned by summer sea. Eastward we distinctly traced the Fiskivötn.
Run the eye along the southern shore, and again the scene shifts.
Below the red hornitos of the slope rises the classical Three-horned,
not lofty, but remarkable for its trident top; Tindfjall (tooth-fell) with
its two horns or pyramids of ice, casting blue shadows upon the
untrodden snow; and the whole mighty mass known as the Eastern
Jökull Eyjafjall (island-fell), so called from the black button of rock
which crowns the long white dorsum; Kátlá (Költu-gjá), Merkrjökull,
and Godalands, all connected by ridges, and apparently neither lofty
nor impracticable.
Ultima Thule; or a Summer in Iceland (London and
Edinburgh, 1875).

FOOTNOTES:
[5] Heklu-fjall derives from Hekla (akin to Hökull, a priest’s cope),
meaning a cowled or hooded frock, knitted of various colours,
and applied to the “Vesuvius of the North” from its cap and body
vest of snow. Icelanders usually translate it a chasuble, because
its rounded black shoulders bear stripes of white, supposed to
resemble the cross carried to Calvary.
[6] Rángá (“wrong,” or crooked stream) is a name that frequently
occurs, and generally denotes either that the trend is opposed to
the general water-shed, or that an angle has been formed in the
bed by earthquakes or eruptions.
 VICTORIA FALLS
(AFRICA)
DAVID LIVINGSTONE
We proceeded next morning, 9th August, 1860, to see the Victoria
Falls. Mosi-oa-tunya is the Makololo name, and means smoke
sounding; Seongo or Chongwé, meaning the Rainbow, or the place
of the Rainbow, was the more ancient term they bore. We embarked
in canoes, belonging to Tuba Mokoro, “smasher of canoes,” an
ominous name; but he alone, it seems, knew the medicine which
insures one against shipwreck in the rapids above the Falls. For
some miles the river was smooth and tranquil, and we glided
pleasantly over water clear as crystal, and past lovely islands densely
covered with a tropical vegetation. Noticeable among the many trees
were the lofty Hyphæne and Borassus palms; the graceful wild date-
palm, with its fruit in golden clusters, and the umbrageous
mokononga, of cypress form, with its dark-green leaves and scarlet
fruit. Many flowers peeped out near the water’s edge, some entirely
new to us, and others, as the convolvulus, old acquaintances.
But our attention was quickly called from the charming islands to the
dangerous rapids, down which Tuba might unintentionally shoot us.
To confess the truth, the very ugly aspect of these roaring rapids
could scarcely fail to cause some uneasiness in the minds of new-
comers. It is only when the river is very low, as it was now, that any
one durst venture to the island to which we were bound. If one went
during the period of flood, and fortunately hit the island, he would
be obliged to remain there till the water subsided again, if he lived
so long. Both hippopotamus and elephants have been known to be
swept over the Falls, and of course smashed to pulp.
Before entering the race of waters, we were requested not to speak,
as our talking might diminish the virtue of the medicine; and no one
with such boiling, eddying rapids before his eyes, would think of
disobeying the orders of a “canoe-smasher.” It soon became evident
that there was sound sense in this request of Tuba’s, although the
reason assigned was not unlike that of the canoe-man from
Sesheke, who begged one of our party not to whistle because
whistling made the wind come. It was the duty of the man at the
bow to look out ahead for the proper course, and when he saw a
rock or snag to call out to the steersman. Tuba doubtless thought
that talking on board might divert the attention of his steersman, at
a time when the neglect of an order, or a slight mistake, would be
sure to spill us all into the chafing river. There were places where the
utmost exertions of both men had to be put forth in order to force
the canoe to the only safe part of the rapid, and to prevent it from
sweeping down broadside on, where in a twinkling we should have
found ourselves floundering among the plotuses and cormorants,
which were engaged in diving for their breakfast of small fish. At
times it seemed as if nothing could save us from dashing in our
headlong race against the rocks which, now that the river was low,
jutted out of the water; but just at the very nick of time, Tuba
passed the word to the steersman, and then with ready pole turned
the canoe a little aside and we glided swiftly past the threatened
danger. Never was canoe more admirably managed: once only did
the medicine seem to have lost something of its efficacy. We were
driving swiftly down, a black rock, over which the white foam flew,
lay directly in our path, the pole was planted against it as readily as
ever, but it slipped just as Tuba put forth his strength to turn the
bow off. We struck hard, and were half-full of water in a moment;
Tuba recovered himself as speedily, shoved off the bow, and shot the
canoe into a still shallow place, to bale out the water. Here we were
given to understand that it was not the medicine which was at fault;
that had lost none of its virtue; the accident was owing entirely to
Tuba having started without his breakfast. Need it be said we never
left Tuba go without that meal again?
We landed at the head of Garden Island, which is situated near the
middle of the river and on the lip of the Falls. On reaching that lip,
and peering over the giddy height, the wondrous and unique
character of the magnificent cascade at once burst upon us.
It is a rather hopeless task to endeavour to convey an idea of it in
words, since, as was remarked on the spot, an accomplished painter,
even by a number of views, could but impart a faint impression of
the glorious scene. The probable mode of its formation may perhaps
help to the conception of its peculiar shape. Niagara has been
formed by a wearing back of the rock over which the river falls; but
during the long course of ages, it has gradually receded, and left a
broad, deep, and pretty straight trough in front. It goes on wearing
back daily, and may yet discharge the lakes from which its river—the
St. Lawrence—flows. But the Victoria Falls have been formed by a
crack right across the river, in the hard, black, basaltic rock which
there formed the bed of the Zambesi. The lips of the crack are still
quite sharp, save about three feet of the edge over which the river
rolls. The walls go sheer down from the lips without any projecting
crag, or symptoms of stratification or dislocation. When the mighty
rift occurred, no change of level took place in the two parts of the
bed of the river thus rent asunder, consequently, in coming down the
river to Garden Island, the water suddenly disappears, and we see
the opposite side of the cleft, with grass and trees growing where
once the river ran, on the same level as that part of its bed on which
we sail. The first crack, is, in length, a few yards more than the
breadth of the Zambesi, which by measurement we found to be a
little over 1,860 yards, but this number we resolved to retain as
indicating the year in which the Fall was for the first time carefully
examined. The main stream here runs nearly north and south, and
the cleft across it is nearly east and west. The depth of the rift was
measured by lowering a line, to the end of which a few bullets and a
foot of white cotton cloth were tied. One of us lay with his head over
a projecting crag, and watched the descending calico, till, after his
companions had paid out 310 feet, the weight rested on a sloping
projection, probably fifty feet from the water below, the actual
bottom being still further down. The white cloth now appeared the
size of a crown-piece. On measuring the width of this deep cleft by
sextant, it was found at Garden Island, its narrowest part, to be
eighty yards, and at its broadest somewhat more. Into this chasm,
of twice the depth of Niagara-fall, the river, a full mile wide, rolls
with a deafening roar; and this is Mosi-oa-tunya, or the Victoria Falls.

FALLS OF THE ZAMBESI.

Looking from Garden Island, down to the bottom of the abyss,
nearly half a mile of water, which has fallen over that portion of the
Falls to our right, or west of our point of view, is seen collected in a
narrow channel twenty or thirty yards wide, and flowing at exactly
right angles to its previous course, to our left; while the other half,
or that which fell over the eastern portion of the Falls, is seen in the
left of the narrow channel below, coming towards our right. Both
waters unite midway, in a fearful boiling waterfall, and find an outlet
by a crack situated at right angles to the fissure of the Falls. This
outlet is about 1,170 yards from the western end of the chasm, and
some 600 from its eastern end; the whirlpool is at its
commencement. The Zambesi, now apparently not more than
twenty or thirty yards wide, rushes and surges south, through the
narrow escape-channel for 130 yards; then enters a second chasm
somewhat deeper, and nearly parallel with the first. Abandoning the
bottom of the eastern half of this second chasm to the growth of
large trees, it turns sharply off to the west, and forms a promontory,
with the escape-channel at its point, of 1,170 yards long, and 416
yards broad at the base. After reaching this base, the river runs
abruptly round the head of another promontory, and flows away to
the east, in a third chasm; then glides round a third promontory,
much narrower than the rest, and away back to the west, in a fourth
chasm; and we could see in the distance that it appeared to round
still another promontory, and bend once more in another chasm
toward the east. In this gigantic, zigzag, yet narrow trough, the
rocks are all so sharply cut and angular, that the idea at once arises
that the hard basaltic trap must have been riven into its present
shape by a force acting from beneath, and that this probably took
place when the ancient inland seas were cut off by similar fissures
nearer the ocean.
The land beyond, or on the south of the Falls, retains, as already
remarked, the same level as before the rent was made. It is as if the
trough below Niagara were bent right and left, several times before
it reached the railway bridge. The land in the supposed bends being
of the same height as that above the Fall, would give standing-
places, or points of view, of the same nature as that from the
railway bridge, but the nearest would be only eighty yards, instead
of two miles (the distance to the bridge) from the face of the
cascade. The tops of the promontories are in general flat, smooth,
and studded with trees. The first, with its base on the east, is at one
place so narrow, that it would be dangerous to walk to its extremity.
On the second, however, we found a broad rhinoceros path and a
hut; but, unless the builder were a hermit, with a pet rhinoceros, we
cannot conceive what beast or man ever went there for. On reaching
the apex of this second eastern promontory we saw the great river,
of a deep sea-green colour, now sorely compressed, gliding away, at
least 400 feet below us.
Garden Island, when the river is low, commands the best view of the
Great Fall chasm, as also of the promontory opposite, with its grove
of large evergreen trees, and brilliant rainbows of three-quarters of a
circle, two, three, and sometimes even four in number, resting on
the face of the vast perpendicular rock, down which tiny streams are
always running to be swept again back by the upward rushing
vapour. But as, at Niagara, one has to go over to the Canadian shore
to see the chief wonder—the Great Horseshoe Fall—so here we have
to cross over to Moselekatsé’s side to the promontory of evergreens,
for the best view of the principal Falls of Mosi-oa-tunya. Beginning,
therefore, at the base of this promontory, and facing the Cataract, at
the west end of the chasm, there is, first, a fall of thirty-six yards in
breadth, and of course, as they all are, upwards of 310 feet in
depth. Then Boaruka, a small island, intervenes, and next comes a
great fall, with a breadth of 573 yards; a projecting rock separates
this from a second grand fall of 325 yards broad; in all, upwards of
900 yards of perennial Falls. Further east stands Garden Island;
then, as the river was at its lowest, came a good deal of the bare
rock of its bed, with a score of narrow falls, which, at the time of
flood, constitute one enormous cascade of nearly another half-mile.
Near the east end of the chasm are two larger falls, but they are
nothing at low water compared to those between the islands.
The whole body of water rolls clear over, quite unbroken; but, after a
descent of ten or more feet, the entire mass suddenly becomes a
huge sheet of driven snow. Pieces of water leap off it in the form of
comets with tails streaming behind, till the whole snowy sheet
becomes myriads of rushing, leaping, aqueous comets. This
peculiarity was not observed by Charles Livingstone at Niagara, and
here it happens, possibly from the dryness of the atmosphere, or
whatever the cause may be which makes every drop of Zambesi
water appear to possess a sort of individuality. It runs off the ends of
the paddles, and glides in beads along the smooth surface, like
drops of quicksilver on a table. Here we see them in a
conglomeration, each with a train of pure white vapour, racing down
till lost in clouds of spray. A stone dropped in became less and less
to the eye, and at last disappeared in the dense mist below.
Charles Livingstone had seen Niagara, and gave Mosi-oa-tunya the
palm, though now at the end of a drought, and the river at its very
lowest. Many feel a disappointment on first seeing the great
American Falls, but Mosi-oa-tunya is so strange, it must ever cause
wonder. In the amount of water, Niagara probably excels, though not
during the months when the Zambesi is in flood. The vast body of
water, separating in the comet-like forms described, necessarily
encloses in its descent a large volume of air, which, forced into the
cleft, to an unknown depth, rebounds, and rushes up loaded with
vapour to form the three or even six columns, as if of steam, visible
at the Batoka village Moachemba, twenty-one miles distant. On
attaining a height of 200, or at most 300 feet from the level of the
river above the cascade, this vapour becomes condensed into a
perpetual shower of fine rain. Much of the spray, rising to the west
of Garden Island, falls on the grove of evergreen trees opposite; and
from their leaves, heavy drops are for ever falling, to form sundry
little rills, which, in running down the steep face of rock, are blown
off and turned back, or licked off their perpendicular bed, up into the
column from which they have just descended.
The morning sun gilds these columns of watery smoke with all the
glowing colours of double or treble rainbows. The evening sun, from
a hot yellow sky imparts a sulphureous hue, and gives one the
impression that the yawning gulf might resemble the mouth of the
bottomless pit. No bird sings and sings on the branches of the grove
of perpetual showers, or ever builds his nest there. We saw hornbills
and flocks of little black weavers flying across from the mainland to
the islands, and from the islands to the points of the promontories
and back again, but they uniformly shunned the region of perpetual
rain, occupied by the evergreen grove. The sunshine, elsewhere in
this land so overpowering, never penetrates the deep gloom of that
shade. In the presence of the strange Mosi-oa-tunya, we can
sympathize with those who, when the world was young, peopled
earth, air, and river, with beings not of mortal form. Sacred to what
deity would be this awful chasm and that dark grove, over which
hovers an ever-abiding “pillar of cloud”?
The ancient Batoka chieftains used Kazeruka, now Garden Island,
and Boaruka, the island further west, also on the lip of the Falls, as
sacred spots for worshipping the Deity. It is no wonder that under
the cloudy columns, and near the brilliant rainbows, with the
ceaseless roar of the cataract, with the perpetual flow, as if pouring
forth from the hand of the Almighty, their souls should be filled with
reverential awe.
The Zambesi and its Tributaries 1858-1864 (London,
1865).
 THE DRAGON-TREE OF OROTAVA[7]
(CANARY ISLANDS)
ALEXANDER VON HUMBOLDT
Orotava, the ancient Taoro of the Guanches, is situated on a very
steep declivity. The streets seem deserted; the houses are solidly
built, and of gloomy appearance. We passed along a lofty aqueduct,
lined with a great number of fine ferns; and visited several gardens,
in which the fruit trees of the north of Europe are mingled with
orange trees, pomegranates, and date trees. We were assured, that
these last were as little productive here as on the coast of Cumana.
Although we had been made acquainted, from the narratives of
many travellers, with the dragon-tree in M. Franqui’s garden, we
were not the less struck with its enormous size. We were told, that
the trunk of this tree, which is mentioned in several very ancient
documents as marking the boundaries of a field, was as gigantic in
the Fifteenth Century as it is in the present time. Its height appeared
to us to be about fifty or sixty feet; its circumference near the roots
is forty-five feet. We could not measure higher, but Sir George
Staunton found that, ten feet from the ground, the diameter of the
trunk is still twelve English feet; which corresponds perfectly with
the statement of Borda, who found its mean circumference thirty-
three feet, eight inches, French measure. The trunk is divided into a
great number of branches, which rise in the form of a candelabrum,
and are terminated by tufts of leaves, like the yucca which adorns
the valley of Mexico. This division gives it a very different
appearance from that of the palm-tree.
Among organic creations, this tree is undoubtedly, together with the
Adansonia or baobab of Senegal, one of the oldest inhabitants of our
globe. The baobabs are of still greater dimensions than the dragon-
tree of Orotava. There are some which near the root measure thirty-
four feet in diameter, though their total height is only from fifty to
sixty feet. But we should observe, that the Adansonia, like the
ochroma, and all the plants of the family of bombax, grow much
more rapidly than the dracæna, the vegetation of which is very slow.
That in M. Franqui’s garden still bears every year both flowers and
fruit. Its aspect forcibly exemplifies “that eternal youth of nature,”
which is an inexhaustible source of motion and of life.
The dracæna, which is seen only in cultivated spots in the Canary
Islands, at Madeira, and Porto Santo, presents a curious
phenomenon with respect to the emigration of plants. It has never
been found in a wild state on the continent of Africa. The East Indies
is its real country. How has this tree been transplanted to Teneriffe,
where it is by no means common? Does its existence prove, that, at
some very distant period, the Guanches had connexions with other
nations originally from Asia?[8]
 THE DRAGON TREE.
The age of trees is marked by their size, and the union of age with
the manifestation of constantly renewed vigour is a charm peculiar
to the vegetable kingdom. The gigantic Dragon-tree of Orotava (as
sacred in the eyes of the inhabitants of the Canaries as the olive-tree
in the Citadel of Athens, or the Elm of Ephesus), the diameter of
which I found, when I visited those islands, to be more than sixteen
feet, had the same colossal size when the French adventurers, the
Béthencourts, conquered these gardens of the Hesperides in the
beginning of the Fifteenth Century; yet it still flourishes, as if in
perpetual youth, bearing flowers and fruit. A tropical forest of
Hymenæas and Cæsalpinieæ may perhaps present to us a
monument of more than a thousand years’ standing.
This colossal dragon-tree, Dracæna draco, stands in one of the most
delightful spots in the world. In June, 1799, when we ascended the
Peak of Teneriffe, we measured the circumference of the tree and
found it nearly forty-eight English feet. Our measurement was taken
several feet above the root. Lower down, and nearer to the ground,
Le Dru made it nearly seventy-nine English feet. The height of the
tree is not much above sixty-nine English feet. According to tradition,
this tree was venerated by the Guanches (as was the ash-tree of
Ephesus by the Greeks, or as the Lydian plane-tree which Xerxes
decked with ornaments, and the sacred Banyan-tree of Ceylon), and
at the time of the first expedition of the Béthencourts in 1402, it was
already as thick and as hollow as it now is. Remembering that the
Dracæna grows extremely slowly, we are led to infer the high
antiquity of the tree of Orotava. Bertholet in his description of
Teneriffe, says: “En comparant les jeunes Dragonniers, voisins de
l’arbre gigantesque, les calcus qu’on fait sur l’ âge de ce dernier
effraient l’imagination.” (Nova Acta Acad. Leop. Carol. Naturæ
Curiosorum 1827, vol. xiii., p. 781.) The dragon-tree has been
cultivated in the Canaries, and in Madeira and Porto Santo, from the
earliest times; and an accurate observer, Leopold von Buch, has
even found it wild in Teneriffe, near Igueste....
The measurement of the dragon-tree of the Villa Franqui was made
on Borda’s first voyage with Pingré, in 1771; not in his second
voyage, in 1776, with Varela. It is affirmed that in the earlier times
of the Norman and Spanish conquests, in the Fifteenth Century,
Mass was said at a small altar erected in the hollow trunk of the
tree. Unfortunately, the dragon-tree of Orotava lost one side of its
top in the storm of the 21st of July, 1819.
Personal Narrative of Travels to the Equinoctial Regions of
America during the years 1799-1804 (London, 1825); and
Aspects of Nature (Philadelphia, 1849).

FOOTNOTES:
[7] This famous tree was blown down by a storm in 1868. Its age
was estimated from five to six thousand years.—E. S.
[8] The form of the dragon-tree is exhibited in several species of
the genus Dracæna, at the Cape of Good Hope, in China, and in
New Zealand. But in New Zealand it is superseded by the form of
the yucca; for the Dracæna borealis of Aiton is a Convallaria, of
which it has all the appearance. The astringent juice, known in
commerce by the name of dragon’s blood, is, according to the
inquiries we made on the spot, the produce of several American
plants. At Laguna, toothpicks steeped in the juice of the dragon-
tree are made in the nunneries, and are much extolled as highly
useful for keeping the gums in a healthy state.
 MOUNT SHASTA
(UNITED STATES)
J. W. BODDAM-WHETHAM
Mount Shasta is the most striking feature of Northern California. Its
height is about 14,500 feet above the sea—very nearly the height of
Mount Blanc. Mount Blanc is broken into a succession of peaks, but
Shasta is one stupendous peak, set upon a broad base that sweeps
out far and wide. From the base the volcanic cone rises up in one
vast stretch of snow and lava. It is very precipitous to the north and
south, but east and west there are two slopes right up to the crater.
It is a matter of doubt whether Shasta is dead or only sleeping.
Vesuvius slept calmly for centuries, and then spread death and
desolation for miles around. The base of the mountain is
magnificently watered and wooded, and forms a splendid hunting-
ground. The woods are full of deer and bears; and now and then a
mountain-goat, an animal very like the chamois of the Alps, is seen
in the higher part of the mountains.
Well-provided with blankets and provisions, we started with a guide,
and a man to look after the horses, at a very early hour, and rode
through a beautiful forest of pines, silver firs, and cedars. Along the
banks of the streams were aspens, willows, and the trees known by
the name of the “Balm of Gilead,” whose vivid green leaves were
already changing to a rich orange or an apple-red—forming a
beautiful contrast of colours with the glazed green of the cedars and
the green-tinted white of the silver firs.
After an easy ascent to a height of about 8,000 feet, we reached the
limits of vegetation. Thence our upward path lay over snow, ice, and
lava—lonely, isolated barrenness on every side, relieved only by an
occasional solitary dwarf-pine, struggling to retain life amidst fierce
storms and heavy-weighing snow. Many of them were quite dead,
but embalmed by frost and snow in a never-decaying death.
With a few loads of this fuel we soon made a splendid fire, the
warmth of which was most welcome in the cold rarefied atmosphere.
Scarcely had we finished a capital supper ere night descended, and
great clouds and fitful fogs began to drift past. These in their turn
broke, and the moon threw a weird light over the forest below;
whilst above rose piles upon piles of pinkish lava and snow-fields,
reaching far up into the sky, whose magnificent blue grew more
sparkling and clear every moment.
Wrapping ourselves in our bundles of blankets, we crept as close as
possible to the huge fire, and before long my companions were fast
asleep and snoring. I could not sleep a wink, and mentally registered
a vow never again to camp out without a pillow. No one can tell till
he has tried it, the difference there is between going to sleep with a
pillow under the head and a stone or a pair of boots or saddle as its
resting-place.