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Mathematics Education in the Digital Era
Alison Clark-Wilson
Ornella Robutti
Nathalie Sinclair Editors
The Mathematics
Teacher in
the Digital Era
International Research on Professional
Learning and Practice
Second Edition
Volume 16
Series Editors
Dragana Martinovic, University of Windsor, Windsor, ON, Canada
Viktor Freiman, Faculté des sciences de l’éducation, Université de Moncton,
Moncton, NB, Canada
Editorial Board Members

Marcelo Borba, State University of São Paulo, São Paulo, Brazil
Rosa Maria Bottino, CNR – Istituto Tecnologie Didattiche, Genova, Italy
Paul Drijvers, Utrecht University, Utrecht, The Netherlands
Celia Hoyles, University of London, London, UK
Zekeriya Karadag, Giresun Üniversitesi, Giresun, Turkey
Stephen Lerman, London South Bank University, London, UK
Richard Lesh, Indiana University, Bloomington, USA
Allen Leung, Hong Kong Baptist University, Kowloon Tong, Hong Kong
Tom Lowrie, University of Canberra, Bruce, Australia
John Mason, The Open University, Buckinghamshire, UK
Sergey Pozdnyakov, Saint Petersburg Electrotechnical University,
Saint Petersburg, Russia
Ornella Robutti, Dipartimento di Matematica, Università di Torino, Torino, Italy
Anna Sfard, University of Haifa, Haifa, Israel
Bharath Sriraman, University of Montana, Missoula, USA
Eleonora Faggiano, University of Bari Aldo Moro, Bari, Italy
The Mathematics Education in the Digital Era (MEDE) series explores ways in
which digital technologies support mathematics teaching and the learning of Net
Gen’ers, paying attention also to educational debates. Each volume will address one
specific issue in mathematics education (e.g., visual mathematics and cyber-
learning; inclusive and community based e-learning; teaching in the digital era), in
an attempt to explore fundamental assumptions about teaching and learning
mathematics in the presence of digital technologies. This series aims to attract
diverse readers including researchers in mathematics education, mathematicians,
cognitive scientists and computer scientists, graduate students in education, policy-
makers, educational software developers, administrators and teacher-practitioners.
Among other things, the high-quality scientific work published in this series will
address questions related to the suitability of pedagogies and digital technologies
for new generations of mathematics students. The series will also provide readers
with deeper insight into how innovative teaching and assessment practices emerge,
make their way into the classroom, and shape the learning of young students who
have grown up with technology. The series will also look at how to bridge theory
and practice to enhance the different learning styles of today’s students and turn
their motivation and natural interest in technology into an additional support for
meaningful mathematics learning. The series provides the opportunity for the
dissemination of findings that address the effects of digital technologies on learning
outcomes and their integration into effective teaching practices; the potential of
mathematics educational software for the transformation of instruction and
curricula; and the power of the e-learning of mathematics, as inclusive and
community-based, yet personalized and hands-on.
Submit your proposal: Please contact the Series Editors, Dragana Martinovic
(dragana@uwindsor.ca) and Viktor Freiman (viktor.freiman@umoncton.ca) as well as
the Publishing Editor, Marianna Georgouli (marianna.georgouli@springernature.com).
Forthcoming volume:
• The Evolution of Research on Teaching Mathematics: A. Manizade, N. Buchholtz,
K. Beswick (Eds.)
Alison Clark-Wilson • Ornella Robutti
Nathalie Sinclair
Editors
The Mathematics Teacher

in the Digital Era
International Research on Professional
Learning and Practice
Second Edition
Editors
Alison Clark-Wilson Ornella Robutti
UCL Institute of Education Dipartimento di Matematica
University College London Università di Torino
London, UK Torino, Italy
Nathalie Sinclair
Faculty of Education
Simon Fraser University
Burnaby, BC, Canada
This work contains media enhancements, which are displayed with a “play” icon. Material in
the print book can be viewed on a mobile device by downloading the Springer Nature “More
Media” app available in the major app stores. The media enhancements in the online version
of the work can be accessed directly by authorized users.
ISSN 2211-8136 ISSN 2211-8144 (electronic)

ISBN 978-3-031-05253-8 ISBN 978-3-031-05254-5 (eBook)
https://doi.org/10.1007/978-3-031-05254-5
1st edition: © Springer Science+Business Media Dordrecht 2014

2nd edition: © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature
Switzerland AG 2022
This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether
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Introduction
The eight intervening years between this second edition of The Mathematics Teacher
in the Digital Era and the first edition have seen increased attention on the role of
the teacher within technology-enhanced educational contexts, leading to a more
developed understanding of the components of related teacher education pro-
grammes and initiatives for both pre- and in-service teachers. The shock to the edu-
cation system caused by the global coronavirus pandemic simultaneously highlighted
the key role that teachers and lecturers play in the nurturing of generations of learn-
ers, alongside increased global attention to the role that (educational) technology
plays as a mediator of teaching and learning. Studies that have taken place during
the pandemic have provided insights into how teachers’ practices have had to
evolve, whilst also highlighting theoretical and methodological gaps in our under-
standing of the relatively new phenomena of “hybrid”, “at distance” or “remote”
teaching in school and university settings (Bretscher et al., 2021; Clark-Wilson
et al., 2021; Crisan et al., 2021; Drijvers et al., 2021; Maciejewski, 2021).
As we reflect on the academic impacts of the first edition of the book, the chap-
ters within have offered theoretical constructs and methodological approaches,
which have provided other researchers in the field with research tools that are con-
tinuing to advance our collective understandings of the field. In this second edition,
we invited all of the authors who had contributed to the first edition to submit new
research that evidenced advances in their experiences, knowledge and practices. We
also invited new authors, whose research had emerged in the intervening years, to
offer new critical perspectives that broaden the international commentary, with con-
tributions from Argentina, Australia, Canada, France, Germany, Hong Kong,
Iceland, Italy, Mexico, Turkey and the United Kingdom.
vii
viii Introduction
A Journey Through the Text
The evolution of the research on technology in mathematics education has enabled

a more nuanced understanding of the teacher’s perspective to take account of their
trajectories of development from pre-service contexts through to in-service prac-
tices over time. Hence, we have chosen to loosely organise the text body in accor-
dance with teachers’ trajectories of experience with technology use. These
experiences concern those within: university undergraduate courses as learners of
mathematics; university-based pre-service teacher education courses; university-
based teacher education courses and research projects with in-service teachers as
participants.
We begin with chapters by Thurm, Ebers and Barzel, and Bozkurt and Koyunkaya
that address more practical considerations regarding the provision of support and
training for both in-service and pre-service teachers of mathematics.
The growth of large-scale, online professional development initiatives aimed at
teachers has resulted in new research that seeks to develop theoretical understand-
ing of the design and impact of such initiatives alongside the development of appro-
priate methodologies to inform both aspects. The chapter by Thurm, Ebers and
Barzel addresses aspects of the design of professional development for mathematics
teachers in Germany with a particular focus on the role of the professional develop-
ment facilitators within a regional professional development programme for 30 par-
ticipants who are all such facilitators. The programme was conducted online (due to
the Covid-19 pandemic) and Thurm and colleagues’ findings focus on the impact of
a module of the programme that supported participants’ understanding (and use) of
video-based case studies of mathematics teaching that embed multi-representational
technology. They use Prediger, Roesken-Winter and Leuders’ Three-Tetrahedron
Model as a framework to highlight the complexities of PD design that has a class-
room level, teacher PD level and facilitators’ PD level (Prediger et al., 2019). Their
findings, which highlight aspects of facilitators’ noticing, emphasise the need for
carefully structured prompts to support the analysis of video-based activities that
serve the dual needs of the facilitators and the teachers with whom they are working.
A pre-service teacher education context in Turkey is the subject of the qualitative
action research reported by Bozkurt and Koyunkaya in which they study the impacts
of a redesigned practicum course informed by the Instrumental Orchestration model
(Drijvers et al., 2010; Trouche, 2004). The course design emphasises the pre-service
mathematics teachers’ (PSTs, n=4) developing use of a dynamic mathematics soft-
ware (GeoGebra) from the university setting (through micro teaching to their peers)
as their practices move to school classrooms. Their study adopts a cyclical research
method that draws on data from the PSTs’ lesson plans, supported by analyses of
their teaching and associated interviews. The research findings offer insights into
how the PSTs initially overlooked the exploitation modes for the technology in their
planning but became more systematic in their approach through both the processes
of micro teaching and during the practicum itself. Given that many pre-service pro-
grammes stop short of requiring PSTs to apply their learning about mathematical
Introduction ix
technologies within authentic teaching situations, this chapter provides valuable

insights on the design decisions taken by the teacher educators to develop such an
approach.
The majority of the remaining chapters in the book report studies that involve
in-service teachers as participants within a range of research settings, each with a
different focus. We order these chapters according to teachers’ trajectories of devel-
opment with novel to them technologies. We adopt this phrase from Ng and Leung
(Chap. 10) as it better reflects our experience and expectation that it is not possible
for all teachers to be cognisant of all available (and educationally relevant) tech-
nologies at any point in time, irrespective of how mature the wider community
considers these technologies to be.
The study by Bakos explores how a novel multi-touch tablet technology,
TouchTimes, is used by two primary teachers in British Columbia, Canada, through
a lens that considers the teacher, the tool and the mathematical concept as an ensem-
ble. Rooted in the instrumental approach, and in particular Haspekian’s elaboration
of double instrumental genesis (2011, 2014), Bakos uses her case studies to reveal
three new orchestration types alongside sharing insights on how the agency exerted
by the tool extends our existing understandings of the nature of multiplication, and
the role of haptic devices within young children’s development.
Ng, Liang and Leung’s study also focuses on a more novel technology, 3D pens,
which enable 3-dimensional models to be drawn as physical objects. The 3D pen
warms and extrudes a plastic filament to produce a model that then hardens as it
cools. Ng, Liang and Leung’s method adopts the use of video-aided reflection with
a group of four in-service secondary school teachers in Hong Kong to support their
realisations of the affordances of such technologies as a potential teaching tool. In
their findings, Ng, Liang and Leung provide evidence for how the videos operate as
a boundary object between the teachers and researchers in the study (Robutti
et al., 2019).
Although the concept of silent animated films to show mathematical concepts
dates back to the early twentieth century and was further developed in the 1950s by
Nicolet, the design-based research developed by Kristinsdóttir examines aspects of
their design and use in her case study in an upper secondary mathematics classroom
in Iceland. Kristinsdóttir describes silent videos as short (< 2 min) videos that do not
pose a mathematical problem to be solved but rather invite the viewer to wonder, to
experience dynamically changing mathematical objects such that they might dis-
cover something new or consolidate previous thoughts about the mathematics
shown in the video. Each associated silent video task invites students to work in
pairs to prepare and record a voice-over for the video clip, which is then shared with
the class during a whole-class discussion that is led by the teacher. Framed by a lens
that focuses on the formative assessment dimension of such discussions,
Kristinsdóttir adapted Schoenfeld’s Teaching for Robust Understanding framework
(2018) to identify opportunities and challenges associated with such discussions.
McAlindon, Ball and Chang’s study also explores an innovative technology-
enhanced pedagogic approach, the flipped classroom, through a case study involving
an experienced teacher in an Australian secondary school. Defining the flipped
x Introduction
classroom as one in which the activities that would normally be conducted in the
classroom are flipped with those that would normally be conducted as homework,
they explore their case study teacher’s experiences and perceptions of a first imple-
mentation for the teaching of linear equations. This exploratory study, which involves
the teacher making qualitative comparisons with a parallel class that she taught using
her traditional approach, concludes positive outcomes such as improved student
engagement and improved formative assessment practices. Although the design pro-
cess for the teacher requires new technology skills and is time consuming, the authors
offer some guidelines to inform professional development initiatives that have the
goal to support mathematics teachers’ flipped classroom pedagogies.
Gueudet, Besnier, Bueno-Ravel and Poisard extend earlier research that featured
in the first edition of the book, which shone a theoretical lens on teachers’ classroom
practices at the kindergarten level from a Documentational Approach to Didactics
perspective (Gueudet et al., 2014). In the intervening years, evolutions of this theory
and its associated research methods have enabled the authors to consider a kinder-
garten teacher’s development as evidenced by both one of her documents (a micro
view) and the encompassing resource system (a macro view). The authors conclude
that both the micro and macro views are necessary to fully appreciate a teacher’s
design capacity within the context of long-term professional development concern-
ing digital technologies for education.
Staying in France, Abboud-Blanchard and Vanderbrouck report findings from a
study in France that explores the implementation of tablet computers in the French
primary school setting. Although tablets are no longer widely considered a new
technology, the authors’ contribution extends ideas reported in the first edition of
the book, which concludes three axes (cognitive, pragmatic and temporal) through
which to consider teachers’ adoption of new technologies within their mathematics
classrooms (Abboud-Blanchard, 2014). Abboud-Blanchard and Vanderbrouck
introduce the additional constructs of tensions and proximities, which they argue
align more specifically to classroom uses of tablet computers. In their chapter, the
authors articulate how these two new constructs evolve from Activity Theory, and
elaborations of Vygotsky’s and Valsiner’s respective Zone Theories.
Sandoval and Trigueros’ chapter is also situated in a primary school setting, this
time in Mexico. They offer new perspectives on the teaching of mathematics in
primary schools, with an emphasis on how two teachers integrate digital technolo-
gies to particularly meet the needs of learners from challenging socio-economic
contexts. In common with their contribution to the first edition of the book (Trigueros
et al., 2014), they adopt an enactivist approach to characterise teachers’ actions and
the resulting student activities that reveal high levels of participation in immersive
environments for learners who are commonly disenfranchised by education systems.
We move from primary school contexts to the secondary phase in the next two
chapters, which both follow teachers over a period of time with the aim to identify
aspects of their evolving practices. The first, by Simsek, Bretscher, Clark-Wilson
and Hoyles, is situated in England and focuses on three in-service teachers’ evolv-
ing use of a dynamic mathematical technology (Cornerstone Maths) for the teach-
ing of geometric similarity to 11–14 year olds over a period of months. The chapter
Introduction xi
extends the understanding of Ruthven and colleagues’ notion of curriculum script,

which is one of the five Structuring Features of Classroom Practice that was
described and critiqued in the first edition of the book (Ruthven, 2014). Simsek and
colleagues’ chapter contributes a case example of such a curriculum script for the
teaching of a specific mathematics topic, highlighting aspects of more productive
teaching practices which are often difficult to notice.
Villareal’s chapter, in which she describes research in Argentina, follows a sec-
ondary school mathematics teacher from her pre-service teacher education pro-
gramme into her role as a novice in-service teacher. The research dually categorises
the teacher’s evolving relationships with technology, which adopts Goos’ taxonomy
of sophistication (master, servant, partner and extension of self (Goos, 2000), along-
side Ruthven’s five Structuring Features of Classroom Practice (Ruthven et al.,
2009). These two frameworks offer an interesting and novel perspective for cate-
gorising the evolution of teachers’ classroom practices that have implications for
the design of teacher education programmes and initiatives.
A university in Canada is the setting for the research reported by Buteau, Muller,
Santacruz Rodriguez, Mgombelo, Sacristan and Gueudet, which expands research
understanding on the long-term development for a faculty-wide integration of pro-
gramming technologies within undergraduate-level courses for both mathematics
students and future mathematics teachers. Situated in the same context as the earlier
study by Buteau and Muller (2014), the instrumental orchestration framework is
used to examine the 20-year trajectory of this integration from the perspective of the
faculty members. The authors’ analysis of the course instructors’ and selected stu-
dents’ schemes concludes an orchestration and genesis alignment model that high-
lights the complexities of the instructor’s role as both policy maker and teacher with
responsibility for orchestrating the students’ instrumental geneses.
The Covid-19 pandemic provides the context for the research study that features in
the chapter by Sánchez Aguilar, Esparza Puga and Lezama. Set in South America, the
authors conducted a survey (n = 179) across five Latin American Countries (Argentina,
Chile, Colombia, Mexico and Uruguay) that aimed to elicit teachers’ perceptions of
the abrupt integration of digital technologies into their practices, triggered by wide-
spread and mandatory school closures in the first six months of 2020. This was framed
within a methodology that aims to capture the lived experience of teachers by giving
them a voice to express the obstacles that they faced. The study captures the broad
range of technologies in play, extending beyond solely mathematical technologies
(i.e., calculators, dynamic geometry software or spreadsheets) to include more general
technologies such as videoconferencing software and learning management plat-
forms. The findings revealed six categories of obstacles that capture both what they
did and how they felt as they worked to overcome the challenges that they faced.
The penultimate two chapters of the book offer theoretical contributions.
In the first edition of the book, the chapter by Arzarello, Robutti, Sabena, Cusi,
Garuti, Malara and Martignone introduced a new theoretical model, Meta-Didactical
Transposition (MDT), which was developed to respond to the need to consider the
complexity of teacher education with respect to the institutions in which teaching
operates, alongside the relationships that teachers must have with these institutions
xii Introduction
(Arzarello et al., 2014). The original MDT model (now referred to as MDT.1), an
extension of Chevallard’s Anthropological Theory of Didactics (1985, 1992, 1999),
describes the evolution of teachers’ education over time by analysing the different
variables involved: components that change from external to internal (internalisa-
tion); brokers who support teachers interacting with them; and dialectic interactions
between the community of teachers and researchers. The chapter by Cusi, Robutti,
Panero, Taranto and Aldon presents an evolution of MDT, namely, Meta-Didactical
Transposition.2 (MDT.2), which offers a deeper insight into the process of inter-
nalisation that captures the way in which the actors within the teachers education
programme develop shared praxeologies over time through the introduction of the
external (and, in some cases digital) components.
The final chapter, by Sinclair, Haspekian, Robutti and Clark-Wilson, charts the
development of theories that frame research on teaching mathematics with technol-
ogy from both a historical perspective and an epistemological one. Building directly
on Ken Ruthven’s chapter in the first edition of this book, it aims to highlight the
evolution of the relevant theories since 2014 and highlights trends in the ways that
these have been operationalised in recent studies. Furthermore, the authors seek to
make explicit the philosophical roots of the commonly adopted theories to provoke
the reader to consider what each might reveal—or conceal—concerning aspects of
teaching mathematics with digital technologies.
Alison Clark-Wilson
Ornella Robutti
Nathalie Sinclair
References
Abboud-Blanchard, M. (2014). Teachers and technologies: Shared constraints, common responses.

In A. Clark-Wilson, O. Robutti, & N. Sinclair (Eds.), The mathematics teacher in the digital
era: An international perspective on technology focused professional development
(pp. 297–317). Springer.
Arzarello, F., Robutti, O., Sabena, C., Cusi, A., Garuti, R., Malara, N., & Martignone, F. (2014).
Meta-didactical transposition: A theoretical model for teacher education programmes. In
A. Clark-Wilson, O. Robutti, & N. Sinclair (Eds.), The mathematics teacher in the digital era:
An international perspective on technology focused professional development (pp. 347–372).
Springer.
Bretscher, N., Geraniou, E., Clark-Wilson, A., & Crisan, C. (2021, November). Learning from the
pandemic: Capitalising on opportunities and overcoming challenges for mathematics teaching
and learning practices with and through technology (Part 3). Paper presented at the British
Society for Research into Learning Mathematics.
Buteau, C., & Muller, E. (2014). Teaching roles in a technology intensive core undergraduate
mathematics course. In A. Clark-Wilson, O. Robutti, & N. Sinclair (Eds.), The mathematics
teacher in the digital era: An international perspective on technology focused professional
development (pp. 163–188). Springer.
Chevallard, Y. (1985). La transposition didactique. La Pensée Sauvage.
Chevallard, Y. (1992). Concepts fondamentaux de la didactique: perspectives apportées par une
approche anthropologique. Recherches en Didactique des Mathématiques, 12(1), 73–112.
Introduction xiii
Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didac-

tique. Recherches en Didactique des Mathématiques, 19(2), 221–266.
Clark-Wilson, A., Bretscher, N., Crisan, C., Geraniou, E., Gono, E., Neate, A., & Shore, C. (2021).
Learning from the pandemic: Capitalising on opportunities and overcoming challenges for
mathematics teaching and learning practices with and through technology (Part 2). Paper pre-
sented at the British Society for Research into Learning Mathematics June 2021.
Crisan, C., Bretscher, N., Clark-Wilson, A., & Geraniou, E. (2021). Learning from the pandemic:
Capitalising on opportunities and overcoming challenges for mathematics teaching and learn-
ing practices with and through technology. Paper presented at the British Society for Research
into Learning Mathematics March 2021.
Drijvers, P., Doorman, M., Boon, P., Reed, H., & Gravemeijer, K. (2010). The teacher and the tool:
Instrumental orchestrations in the technology-rich mathematics classroom. Educational
Studies in Mathematics, 1–22. https://doi.org/10.1007/s10649-010-9254-5
Drijvers, P., Thurm, D., Vandervieren, E., Klinger, M., Moons, F., van der Ree, H., … Doorman,
M. (2021). Distance mathematics teaching in Flanders, Germany, and the Netherlands during
COVID-19 lockdown. Educational Studies in Mathematics, 108(1), 35–64. https://doi.
org/10.1007/s10649-021-10094-5
Gueudet, G., Bueno-Ravel, L., & Poisard, C. (2014). Teaching mathematics with technology at the
Kindergarten level: Resources and Orchestrations. In A. Clark-Wilson, O. Robutti, & N. Sinclair
(Eds.), The mathematics teacher in the digital era (pp. 213–240). Springer.
Haspekian, M. (2011). The co-construction of a mathematical and a didactical instrument. In
M. Pytlak, T. Rowland, & E. Swoboda (Eds.), Proceedings of the seventh congress of the euro-
pean society for research in mathematics education (pp. 2298–3007). University of Rzeszów.
Haspekian, M. (2014). Teachers’ instrumental geneses when integrating spreadsheet software. In
A. Clark-Wilson, O. Robutti, & N. Sinclair (Eds.), The mathematics teacher in the digital era:
An international perspective on technology focused professional development (pp. 241–276).
Springer.
Maciejewski, W. (2021). Teaching math in real time. Educational Studies in Mathematics, 108(1),
143–159. https://doi.org/10.1007/s10649-021-10090-9
Prediger, S., Roesken-Winter, B., & Leuders, T. (2019). Which research can support PD facilita-
tors? Strategies for content-related PD research in the Three-Tetrahedron Model. Journal of
Mathematics Teacher Education, 22(4), 407–425. https://doi.org/10.1007/s10857-019-09434-3
Robutti, O., Aldon, G., Cusi, A., Olsher, S., Panero, M., Cooper, J., … Prodromou, T. (2019).
Boundary objects in mathematics education and their role across communities of teachers and
researchers in interaction. In G. M. Lloyd & O. Chapman (Eds.), (Vol. International handbook
of mathematics teacher education: Volume 3, pp. 211–240). Brill.
Ruthven, K. (2014). Frameworks for analysing the expertise that underpins successful integration
of digital technologies into everyday teaching practice. In A. Clark-Wilson, O. Robutti, &
N. Sinclair (Eds.), The mathematics teacher in the digital era: An international perspective on
technology focused professional development (pp. 373–394). Springer.
Ruthven, K., Deaney, R., & Hennessy, S. (2009). Using graphing software to teach about algebraic
forms: A study of technology-supported practice in secondary-school mathematics. Educational
Studies in Mathematics, 71(3), 279–297.
Schoenfeld, A. H. (2018). Video analyses for research and professional development: The teaching
for robust understanding (TRU) framework. ZDM, 50(3), 491–506. https://doi.org/10.1007/
s11858-017-0908-y
Trigueros, M., Lozano, M.-D., & Sandoval, I. (2014). Integrating technology in the primary school
mathematics classroom: The role of the teacher. In A. Clark-Wilson, O. Robutti, & N. Sinclair
(Eds.), The mathematics teacher in the digital era: An international perspective on technology
focused professional development (pp. 111–138). Springer.
Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized
learning environments: Guiding students’ command process through instrumental orchestra-
tions. International Journal of Computers for Mathematical Learning, 9, 281–307.
Contents
Professional Development for Teaching Mathematics

with Technology: Fostering Teacher and Facilitator Noticing �� 1
Daniel Thurm, Patrick Ebers, and Bärbel Barzel

Using the Instrumental Orchestration Model for Planning
and Teaching Technology-Based Mathematical Tasks
as Part of a Restructured Practicum Course�� 31
Gülay Bozkurt and Melike Yiğit Koyunkaya
An Ensemble Approach to Studying the Teaching
of Multiplication Using TouchTimes�� 65
Sandy Bakos
Using First- and Second-Order Models to Characterise
In-Service Teachers’ Video-Aided Reflection on Teaching
and Learning with 3D Pens �� 95
Oi-Lam Ng, Biyao Liang, and Allen Leung
Opportunities and Challenges That Silent Video Tasks
Bring to the Mathematics Classroom�� 119
Bjarnheiður Kristinsdóttir
Teaching Linear Equations with Technology:
A Flipped Perspective�� 149
Andrew McAlindon, Lynda Ball, and Shanton Chang

Tensions and Proximities in Teaching and Learning Activities:
A Case Study of a Teacher’s Implementation of Tablet-Based Lessons�� 181
Maha Abboud and Fabrice Vandebrouck
Digital Resources in Kindergarten Teachers’ Documents
and Resource Systems: A Case Study in France �� 211
Ghislaine Gueudet, Sylvaine Besnier, Laetitia Bueno-Ravel,
and Caroline Poisard
xv
xvi Contents
Analysis of Primary School Teachers’ Roles in the Dynamics

of Mathematics Lessons That Integrate Technology
Resources in Challenging Socio-economic Contexts�� 235
Ivonne Sandoval and María Trigueros
Characterising Features of Secondary Teachers’
Curriculum Scripts for Geometric Similarity
with Dynamic Mathematical Technology�� 263
Ali Simsek, Nicola Bretscher, Alison Clark-Wilson, and Celia Hoyles
Instrumental Orchestration of the Use of Programming
Technology for Authentic Mathematics Investigation Projects �� 289
Chantal Buteau, Eric Muller, Joyce Mgombelo,
Marisol Santacruz Rodriguez, Ana Isabel Sacristán,
and Ghislaine Gueudet
Researching Professional Trajectories Regarding
the Integration of Digital Technologies: The Case of Vera,
a Novice Mathematics Teacher �� 323
Mónica E. Villarreal and Cristina B. Esteley

The Abrupt Transition to Online Mathematics Teaching
Due to the COVID-19 Pandemic: Listening to Latin American
Teachers’ Voices �� 347
Mario Sánchez Aguilar, Danelly Susana Esparza Puga,
and Javier Lezama
Meta-Didactical Transposition.2: The Evolution of a Framework
to Analyse Teachers’ Collaborative Work with Researchers
in Technological Settings �� 365
Annalisa Cusi, Ornella Robutti, Monica Panero, Eugenia Taranto,
and Gilles Aldon
Revisiting Theories That Frame Research
on Teaching Mathematics with Digital Technology�� 391
Nathalie Sinclair, Mariam Haspekian, Ornella Robutti,
and Alison Clark-Wilson
Index�� 419
Another random document with
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of intent listening. He was frontier born and bred. He had been in a
deadly conflict between white men and savages, and he had not
forgotten. Horses never forget that which has terrified them in their
youth. Something too faint for ears not sharpened by agony was on
the air. Would those whom he loved hear it in time?
“Noni! You precious! You are trying to tell us something! What is it,
my pretty beastie? Oh! you brave, beautiful fellow! I cannot
remember when you weren’t always with us. You must not, must not
die now!”
Poor Noni! It was useless. They would not be warned. He could
serve them no further. His dumb, silent life had been one long
example of duty faithfully performed. His virtues had been many, his
faults few. He had been well loved and he had loved much. He had
done what he could and he would sleep now. His glazing eyes fixed
on the face of his little mistress, but she could not see his parting
gaze because of the tears which filled her own.
Just then, while the deeply moved old Captain stooped to raise
Carlota to her feet, the sanctity of the forest death-chamber was
invaded.
“Sw-w-ish!”
Something whizzed swift and sharp between soldier and child,
grazing his shoulder and her blond curls, and buried itself in the
herbage a dozen feet away.
It was the arrow of an Apache!
CHAPTER XIII
BY THE CAPTAIN’S ORDERS
Toward sunset of that same day the little squad of cavalry was
slowly crossing the plain. It would not reach camp that night and
was watchful, though not expectant, of assault.
“I reckon the redskins had all they wanted for this particular time!”
exclaimed the lad who had longed for battle, till he had seen Benoni
die, and afterward had dreaded it. However, the whizzing arrow had
as promptly banished the dread, for it had barely escaped the breast
of little Carlota. From then he had fought like the born hero he was,
and his Captain was now regarding him with a grave smile of
approval.
“Yes, but not for long. There was sin in their eyes. We shall have
work and plenty of it.” And, after a moment, the officer spoke to the
girl who rode before him on his horse: “I’ve a word or two to say to
you, Carlota. I’d better say them now, before—”
“Oh! Señor Captain! will they come again?” she cried, in terror.
“Where can we go? It is so dreadful!”
“I go wherever my duty calls. What to do with you is the question.
At present, I can neither take nor send you to the fort. It’s too far. A
little way to the north of us is the railroad. One of its stations should
be in our direct line of march, and if we reach it, if all goes well, I
will leave you children there.”
“Shall we see any more Apaches?”
“Humph! You’re not so fond of Indians as you were, eh?”
“I never saw that kind before. Many, many have been to Refugio,
but they’ve always been good.”
“We soldiers believe that the good Indians are all dead.”
“Already, Señor Captain Gray Moustache?”
“Well, there hasn’t been a cataclysm to swallow them, as I know.
There, don’t stare; but if ever you come in the neighborhood of a
dictionary look that long word out. ‘Gray Moustache’ will bother you
no more with a humor you don’t understand.”
“I understand—some, dear Señor. And I didn’t mean that name for
harm. I always do name people something like them till I know their
regular one.”
“Indeed? Wish to be properly introduced, do you? Well, my name is
Sherman. But I like my nickname and, please, don’t look so like a
scared kitten. It’s never so bad but it might be worse. The old
Padres named that spot we’ve left behind us the ‘Spring of
Happiness’ in the ‘Mountains of Flowers.’ We found it so, too.”
“Why—where—Benoni died?” she asked, reproachfully.
“Exactly. Where we didn’t die. Where several of the tricky skunks
who would have killed us in ambush were not permitted. The
greatest regret I have is that, though he is past feeling wounded
pride, we were obliged to leave your noble horse in such vile
company. There were a half-dozen dead Apaches in the glen when
we left it, and a half-dozen white men so much the safer.”
When they had again proceeded in silence for some distance,
Carlota asked:
“Don’t Apaches go to railroad stations? and do they keep children at
them?”
“Under the circumstances, yes, to the last question; and to the first
—there’s little danger. There are too many trains passing. If only,
you midget, you were safely at home!”
“I will be, some day, after we find our father.”
“After all your experience, haven’t you dropped that crazy notion
yet? You are a child of ordinary common sense, I hope, Carlota.”
“Yes. That’s why I don’t see what else there is to do but to go on.”
“If a person does wrong I never heard that it was wise to keep on
doing it!” said the worried Captain, testily.
“Have I done that?” asked the child, really astonished.
“I call it wrong to make trouble and anxiety for a great number of
people, as your running away from home must have done. Even for
me, of whom you never heard before.”
Carlota wriggled herself aside.
“I will get right down, Señor Captain.”
“You’ll do nothing of the kind. Remember what I’ve said, and I’ll
scold no more. I will leave you two at the station. I will have
telegrams sent east and have an advertisement put in the leading
newspapers of the country. If the news comes to your father, as it
probably will, and certainly should, your troubles will be over. He’ll
attend to the rest. This is what I mean by your causing me trouble. I
shall do all this, not because you had any right to put it upon me,
but simply for humanity’s sake. Now, next time you are tempted to
act foolishly, stop and think if you’re going to worry anybody else
with your silliness. That’s all.”
It was the sure end of the severest lecture Carlota had ever
received, and the worst of it was that she felt she deserved it. She
could only say that she was truly sorry and resolve to “do a lot of
thinking next time before she did any acting.” Then she added, as a
bitter memory was stirred:
“I begin to understand that if we hadn’t run away from Refugio our
Benoni wouldn’t have died.”
“Not so fast, little girl, toward that conclusion. The issues of life and
death belong to God. We have no concern with them. Our business
is to do right, as nearly as we can, and—now let’s try a canter!”
At that moment a trooper rode up and saluted.
The Captain gave permission to speak and with a surprised
attention, listened to the other’s few words. Carlota tried not to hear
that which was not intended for her and was sadly startled when her
“Gray Moustache” gave her a hasty kiss on the tip of her nose and
said:
“That’s for good-by, my child. I learn that to take you to the station
myself would carry me far out of my way, for my first duty now is at
camp. I will write some directions on a leaf from my notebook and
enclose some money with it. You must give it to the station-master,
the telegraph operator, and he will attend to the matter. Good-by,
and a prompt reunion with your father!”
To lose this soldier, whom she had regarded as her own especial
friend, seemed a terrible misfortune, and her eyes filled as she felt
herself set upon the ground, while, with his squad about him, the
Captain loped away. Then she saw that Carlos was beside her and,
also, that two troopers had been assigned to their escort. However,
the faces of these men had neither the sternness nor the quizzical
pleasantry of their commander’s. They were the faces of those
detailed to perform a troublesome duty while their own desires were
elsewhere.
“Bouton, you take the girl and I will the boy.”
“All right.”
One horseman caught up Carlos and one Carlota, and, without
another word, rode off like mad across the plain. They handled the
twins very much as they would have handled bags of meal, and they
took a direction at right angles from that followed by their
commanding officer.
Carlos’s temper flamed and he opened his lips to remonstrate
against such contemptuous treatment, but remembered the Apaches
just in time to restrain his hot speech. It wouldn’t do to anger his
guardians then and there, and he did not know that they would not
have disobeyed the Captain’s orders to vent their own spite. Thus
they traveled for what seemed hours. Then they came, all in the
starlight, to a strange place where were two shining things laid flat
along the ground and the light of a lamp showed through the
window of a solitary shanty.
The cavalryman who carried Carlota dismounted and struck his saber
against the cabin door. After a brief delay this was opened by a
rough looking man, who held a candle above his head and was
speechless from astonishment.
The troopers saluted and said:
“By Captain Sherman’s orders, these children are to be left here until
further notice.”
The twins were promptly deposited upon the ground, where they
clasped hands and tried to realize this new thing that had befallen
them. Before they could do so, their military escort had again
saluted and disappeared in the distance, leaving them to make the
best they could of their forlorn situation.
CHAPTER XIV
A FIRST RIDE ON THE RAILWAY
“Come in by!” at last said the man in the doorway.

The children silently accepted the invitation and entered a bare little
room, to be instantly startled by a yell of dismay from their new
host:
“Injuns! Injuns! Apaches! The Saints!”
“We are not Indians!” retorted Carlos, indignantly.
“Och! is it so?” gasped the man, much relieved. “But it’s wearin’ their
clo’es ye are. Did they catch ye and rig yez like that? And did thim
soldiers catch ye back out of ’em?”
Despite his roughness, the man had a simple and timid speech, and
the boy replied by a brief account of their adventures, ending with
the question:
“What are you going to do with us now?”
“As sure as me name’s Dennis Fogarty, that thing I don’t know.”
“Do you live here?”
“Live, is it? Faith, nobody lives nowhere in this haythenish land.”
As no reply followed this assertion, Dennis strengthened it by
adding:
“It’s livin’ I’ve not done since I left Connemara, an’ that’s the truth
I’m tellin’ ye. This station of ‘Leopard’ has give up. The folks what
run it has moved on a peg, to the station beyant. ’Tis with them I
stays when I stays anywhere. But it’s mostly workin’ the track all
alone I am, barrin’ Mike Grady and a hansel o’ ‘Greasers’, to help. It’s
lonesome as sin, so it is. Moreover—”
Of his own accord, there seemed no prospect of Dennis’s pausing, so
Carlota interrupted:
“The Captain said we were to wait at the station till we were called
for by our father. Is this all the place there is?”
“Stay here, is it? All by your lone?”
“I—don’t—know. Is it the railroad? Are those shiny things outside the
rails? Do the trains come to-night?”
“Sure, yourself is a girrul, now ain’t ye? By the questions ye put all
to once. Let’s take ’em in turn an’ find out.”
“THE CAPTAIN SAID WE WERE TO WAIT”
The face of the lonely track-walker now shone with delight at his
company and, as soon as he had heard Carlota speak, he recognized
that she belonged to a higher social class than his own. He pulled
his forelock when he addressed her and his manner was both
respectful and protecting.
“Is it the railroad? ’Tis that same. Are the rails the rails? They be. Do
the trains come to-night? Why not? Same’s every other. But it’s not
for a train I do be waitin’. As I was a-sayin’, the folks that lives here
doesn’t. They’ve moved down to the next station on ‘Tuttle,’ more by
token there’s more good wather for fillin’ the ingines an’ drinkin’,
whilst here the bottom’s dropped out. Was a well sunk, one o’ thim
artaysians, same’s all down the track, yet somethin’ or ’nother
pumps it out down below, as fast as it fills an’ faster. So, says the
Superintendent, says he, to Misther Burnham, says he: ‘Take the
traps of your misthress an’ childer’ an’ go there,’ says he. So they
went, a week maybe; an’ ’twas I just stepped back here, the night,
to look for me shovel that’s lost. I’m abidin’ a hand-car will be along,
by and by; an’ I’m thinkin’ how Mike’ll be starin’ at a couple o’
Injuns, for passengers on it.”
“But we must stay here, right here. The Captain talked a deal about
‘Obedience,’ and his orders were: ‘Stay.’ So we must,” said Carlota,
firmly.
“But, sister, there is nothing to eat and nowhere to sleep!” protested
Carlos, who had not himself been “lectured.”
“There is the floor. And we can find something to eat—to-morrow.
We need only just sleep, till then.”
“An’ it’s not here ye’ll try it!” returned Dennis, indignantly. “What’s a
Captain more nor a man?”
“He’s a soldier. A gentleman. He knows.”
“He knows nought. He isn’t the ‘Super’, is he? He hasn’t the charge
o’ the railroad same as of soldiers an’ Injuns.”
The discussion was closed by a rumbling sound, which grew steadily
louder, and made Carlota clasp her brother’s hand. He, also, looked
toward Dennis with a new fear, almost as great as her own, till he
saw the trackman’s face beaming with satisfaction.
“There she comes, alanna! An’ I’m mindin’ the face o’ Mike, me
friend, when I trots the pair of ye out an’ he sees ye, first off. Here’s
me shovel—that’s right; an’ me pick, that I’ll maybe need an’ maybe
no. Ye see, it’s up an’ down the road I goes an’ it’s botherin’ to know
where’s handiest to be leavin’ me tools. Howiver—step out lively
now, whilst I shuts to the door.”
“Who is ‘she’? And I must not go, nor must Carlos!” declared Carlota,
with decision.
“Sharrucks! Out with ye, ye purty, silly girleen! Stay here, is it,
alone? To be scalped by the murderin’ Injuns? An’ is Dennis me
name? Out with ye. Be a duck of a colleen an’ bother no more. It’s
the car, I tell yez, an’ she’s all but come to the door. I’m thinkin’ it
may be the worse for us all if we hinder her, for ’tis a single track an’
big Number Nine’s about due. If we’re to meet up with a bed this
blessed night, it’s be ready we must when Mike slows her up.”
Carlota shrieked and darted back toward the shanty, for at that
moment there came into sight a low, rapidly moving thing which
seemed to threaten them with instant destruction. But Dennis
slipped his strong arm around her, saying:
“Sure, little misthress, ’tis nought but a hand-car will take us to
supper an’ bed. Fear nought, fear nought.”
“Is it far to this ‘Tuttle’ place?” asked Carlos, now more excited than
alarmed.
“A matter of a dozen mile, belike. Hi! Here she be! Hello, Mike!”
“Hello, Dennis! Who’s them?”
“Injuns! Didn’t ye hear them Apaches was makin’ fresh trouble the
now?”
“Quit foolin’, man! If ’twas redskins ye’d got, ’tisn’t Dennis Fogarty’d
stand there, grinnin’ that gait. On with ye and your company,
wherever they come from. I’m long behind an’ Number Nine
overdue,” returned the other, not easily duped.
Then Carlota was swung to the little platform car, her brother beside
her, and the men were working the handles which propelled it over
the rails. For a moment or two, the children felt as if they were
being hurled to ruin; then the sensation of flying exhilarated them,
and they cautiously looked about them.
“Isn’t—it—wonderful! Will it—run off—the rails?” whispered Carlota.
But nobody heard her; though a little later Carlos put down his head
and halloed in her ear:
“I should like to ride like this—forever!”
She did not answer, save by an ecstatic squeeze of his hand. Then
all at once, she caught sight of a dark object lying on the ground,
which they whizzed past and left behind. After another short
distance, a similar, uncanny, motionless body; and her heart sank
drearily. These were the carcasses of dead cattle which had been too
ignorant to get out of the way, had been slain by passing trains. In
fresh terror she hid her face on her brother’s breast and waited for
her own direful end. Then on and on, till it seemed she could not
endure the agony of suspense, and thought:
“I’d rather be killed quick—quick—than wait—this way!”
But, at last, with a little jerk the car came to a standstill and Dennis
cheerily called out:
“Misthress Burnham! Hello, Jack! Go tell your mother, lad, to come
see the fine present I’m after bringin’ her. A pair of young papooses,
to be raised along with yez all!”
CHAPTER XV
GETTING ACQUAINTED
As the trackmen and their charges stepped into the little circle of
light made by one kerosene lamp nobody spoke. The Indian-like
attire of the young Manuels deceived the Burnham household, till
the mother’s gaze rested upon Carlota’s face. Then she
comprehended that here was a child of white parentage, nor of a
class common to the plains, and exclaimed:
“You poor little creature! Come in, come in!”
As the little girl looked up she, also, realized that here was
somebody different from old Marta and the lively Anita. The woman’s
face was thin and worn but showed refinement, as did the
modulations of her quiet voice.
“She speaks like our father,” thought the wanderer and, impulsively,
flung her arms about the stranger’s neck.
The embrace was cordially returned.
“My child, you have found friends. Wherever you come from you are
safe with us. Come into the house, all of you. Come.”
“That settles it!” commented a boyish voice, and somebody laughed.
“What Ma says, goes. But where in the name of—”
“Jack, my son!”
“Yes, ma’am,” with a mock humility.
“Take care of your tongue.”
“Ma, that’s too big a contract, without your help. But, who are you,
anyway?” he demanded, turning to Carlos, who had become adept
in telling his story; and who had now scarcely finished it when there
came a rumble and jar which startled the Manuels, and sent all the
listeners to their feet.
“The express!”
Another moment and all had hurried to the outside platform, to
watch the incoming train, Carlota and her brother with the rest.
These two were greatly excited and, as before, the girl was terrified.
Perceiving this, Mrs. Burnham drew the child to herself, saying:
“Don’t tremble so, my dear. It cannot hurt you, and, if you had lived
long in this desolate region you would welcome every train that
arrives as a blessed link between you and civilization.”
So Carlota tried to conquer her fear and stand quiet, while the great
“Overland” with its dazzling headlight rolled up to that tiny station on
the plain. Yet, even when it stopped and the passengers began to
step out of the curious carriages, that they might stretch their
stiffened limbs in a momentary walk, she shivered before the
monstrous thing as before a Juggernaut which must crush her if she
moved.
Then she heard greetings exchanged between those who had
arrived and the station-master’s family. The boy, Jack, was hail-
fellow-well-met with strange men in blue-jean suits, much begrimed
by soot and oil. He even caught a flaming light from the hand of one
and went bobbing about beside the cars, looking at the wheels and
tapping them with a little hammer, as if he were in charge of the
whole affair.
The trainmen jested with him, asking: “Is she all right, lad?” “When
are you going to join the crew, Jack?” and so on. Dennis was here,
there, and everywhere; and Carlota was sure that, at all times, he
was rehearsing her own story till, presently, she found herself
surrounded by staring strangers in a most unpleasant way.
Tuttle was a water-station and the trains delayed there longer than
at most other points. Over in front, where the engine puffed and
breathed like a living monster, some men had dropped a big, canvas
pipe from a huge, high tank, over whose sides the water was
splashing wastefully. The little girl’s thoughts flew to her mother’s
garden and the care with which each drop was there hoarded and
expended. Then she heard, as in a dream, all the staring people
talking, as if she were deaf and could not be offended.
“Apaches.” “Fight in the mountains.” “Escaped with their bare lives!”
“Wonder if we’ll be attacked!” “Left here by the cavalry. Will be
shipped east to their friends.” “Captives all their lives.” “Father an
Indian chief.”
Her head was dizzy. She could but dimly feel that these remarks
concerned her brother and herself; that they were as untrue as
possible; and that she had no strength left to correct them. Then
she saw another woman’s face bending close to her own. One of the
many faces which had come down from the car, as the water was
coming down from the tank yonder. Like the water, the faces were
wasting themselves in vain. She wished they could be stopped.
Especially, she wished this last woman would go away. She was old
and she spoke in a shrill, cracked voice.
“Indian captives, are they? How interesting! I’ve crossed the
continent a dozen times before, yet these are the first amusing
Indian relics I’ve ever seen. Apaches, eh? Decidedly thrilling. I wish
—”
“A-l-l A-b-oar-d!”
Slowly, the great pipe swung back to its own place on the tank. The
blue-jean figures, with their flaring torches, climbed into the already
moving train. The curious passengers hurried to their “sections,” to
dream of hold-ups and an Indian outbreak. Once more the heavy,
jarring rumble filled the whole earth; then gradually—swiftly—
completely passed away. Upon that platform in the wilderness there
was once more left but a handful of people to face the night alone.
Carlota’s tired, excited brain was full of visions; and Carlos clasped
his hands in a momentary despair for that far off House of Refuge,
whose safety he had so unwisely left. Alas! the world was not that
always brilliant, sunshiny place he once had fancied it, and a sob
rose in his throat.
“Come on! my White-Around-the-Gills-Young-Brave!” cried Jack,
bringing his hand down with a ringing slap on Carlos’ shoulder.
“Take care!”
“That’s what I’m doing. It’s getting near midnight and I expect you’ll
have to share my lofty chamber. So, march, propel, come along,
vamos! For an Indian, you’re the slowest—”
The word died in its utterance. A blow, as well directed as it was
unexpected, settled upon Jack’s wide mouth with a force that sent
him staggering backward.
Carlota instantly rallied from her half-swoon of fatigue, and
screamed:
“Carlos! Carlos! My brother! Boy—boy—go away!”
She would have rushed between the combatants had not Mrs.
Burnham, though herself vastly astonished, restrained her; while
Dennis flung himself into the business, hot foot.
“Go it, gossoons! Faith, the little one’s the better one! Och! Jack—
that’s a silly blow! The little one! THE LITTLE ONE!”
So crying, he hopped and pranced about the little platform, in high
glee, and, presently, found it too small to accommodate his rising
spirits.
“Here’s to ye, Mike, me friend! Sure it’s the nate night for wrastlin’.
So it is, so ’tis. Now, isn’t that the purty sight? Eh? an’ ye would,
would ye? Come on then! I’m for ye!”
When Mr. Burnham emerged from the tiny office, wherein he
arranged his business concerning the passing trains, he found the
lads in a fierce scuffle at the very threshold while, on the ground
outside, rolled Dennis and Mike in a frenzy of contest, yet that was,
moreover, a perfectly friendly and familiar one.
In the “ould counthry,” both trackmen had been famous wrestlers
and had won prizes at their parish festivals. Therefore, in this new
land, they lost no chance to keep themselves in practice, and now
stood up to shake hands with the best of good nature.
“Faith! That was a fine one, Mickey, me boy. Thanks to ye!” cried
Dennis, the victor.
“The same to yourself, Mr. Fogarty. If there’s one thing out of Ireland
I likes more nor another, ’tis a good wrastle with a neighbor,
betimes.”
“Yes, I know, I know. Clears a man’s head betther nor a Sunday o’
sleep. Let’s turn in now, Mr. Grady, an’ leave the misthress in peace.”
So, with their arms about each other’s shoulders, in a fashion
beautiful to see, the late belligerents departed toward a small
outbuilding where they slept and, in an incredibly short time, were
oblivious to all the world.
Then directed Mr. Burnham:
“Follow their example, lads. Go to bed, and no more nonsense.”
For an instant, Jack and his guest regarded each other, then both
sheepishly laughed—which astonished the wide-eyed Carlota even
more than their brief fight had done.
“That’s all right, little girl. They’ll be good friends now. Boys often
begin their acquaintance by a ‘scrap,’ test one another that way, so
to speak,” explained the station-master.
“I never saw him do such a dreadful thing before. Never—never—
never! I—I—it’s a mis’able world! And I wish—I—was in—my own—
Refugio!” wailed Carlota.
In her heart, Letitia Burnham echoed that wish, but aloud she
cheerfully said:
“It’s a pretty good world, after all, my child. But come, you poor,
tired dearie. I’m going to put you to sleep in my very own bed; and
to show you the prettiest sight in the wide west—I think—nay, I am
sure!”
Wondering what this might be and how anything very lovely could
be in that dreary place, Carlota sleepily followed her new friend to
the inner room.
CHAPTER XVI
THE NEXT MORNING
When Carlos awoke he saw Jack standing in the middle of the small
room where they had slept, trying to put on his own kid jacket and
this absurd attempt of an overgrown youth to squeeze himself into a
garment several sizes too small was so absurd that the watcher
giggled.
“Hello, you! Laughing at me, are you? I’d like to know why?”
“You can’t get into that. Besides, I want it myself.”
“Can’t have it. It’s become the property of Burnham and Co.”
“Is it time to get up?”
“Past time. Too late for breakfast.”
Jack said this so gravely that Carlos was disturbed, though he
lingered to stretch himself thoroughly and to look curiously around
the chamber. Save the shack at which he had been dropped on the
evening before it was the barest place he had ever seen. Even a
sheep-herder’s hut had more of convenience about it, but he had
gone to bed in the dark without observing this. After the manner of
lads, the pair had long lain awake exchanging confidences, until Mr.
Burnham’s voice from somewhere had warned them that there must
be silence.
“Oh! dear! I wish morning hadn’t come so soon!”
“Mid-day, you mean. Near dinner-time. Where’d you get this jacket,
anyway? I’d like to shoot a red man and steal one for myself.”
“Why don’t you, then?”
Jack ceased struggling with the garment and whistled.
“Humph! You’re ‘sassy,’ too. But come on. Get up. Here. The mother
has been in and says you are to put on these.”
Carlos sat up and stared at the outfit which the facetious Jack held
toward him. It consisted of a blue-jean costume, similar to that worn
by the trainmen, save that this was clean, if faded, and was one Jack
had outgrown.
“Nonsense!”
“Her name is Letitia, if it’s Mrs. Burnham you’re mentioning.”
Carlos sprang up and put on his stockings and moccasins, but failed
to find his leggins.
“I want to dress. What have you done with my clothes?”
“I told you the truth. My mother has them. Wait. She shall speak for
herself. Mother!”
“Yes, Jack!”
“Isn’t this boy to put on my old clothes? He says he won’t.”
“Dear lady! I never!” he cried, indignantly, to the person without,
who answered, promptly:
“For the present, Carlos, it will be better if you wear the things I left
for you. After breakfast I will explain. I’m going to cook it now, so
please don’t delay,” returned the voice from without.
“There, Jack Burnham! You’re an untruthful boy!” said Carlos,
indignantly.
“Hold on! That’s serious!”
“Breakfast isn’t over.”
“It is. Dennis and Mike had theirs two hours ago. I saw them go by
the window. They stopped to look in and shake their fists at you,
‘friendly like.’ They’re good wrestlers, the pair of them. Between us
we’ll give you plenty of exercise. Hello! Who’s that wants to come in?
Why?”
“To see my new bruvver.”
“Who is that?” asked Carlos.
“That—is the future Governor of New Mexico. Or President of the
United States. Teddy Burnham—here you are!”
With that he admitted a dark-headed little four-year-old, very short
and fat, and whose brown eyes were the sharpest possible to a
childish face. This youngster planted himself firmly just within the
room and ordered:
“Boy, come here, Teddy wants to see you.”
The stranger laughingly obeyed.
“So you are Jack’s brother, eh?”
“Yep. I like the girl to you.”
“That’s good. Most people do. She’s much nicer than I am.”
“Yep,” agreed the child, unflatteringly. “Now talk Injun.”
“I don’t know how. Say Jack, I cannot wear these things!” holding
up the overalls and making a funny grimace that sent Teddy into a
paroxysm of laughter. The effect of his remark was so unexpected
that Carlos, also, laughed, and not to be outdone, Jack joined in the
mirth.
“Isn’t you d’eadful!” shrieked the baby “Governor,” and then they all
laughed again.
This served the purpose of putting everybody on good terms; for,
despite their confidences of the night, daylight had found the lads
secretly shy of one another. Now they coolly faced and scrutinized
each other. What Jack saw, we already know; save that Carlos, clad
in the rough clothing of a workman, had lost something of his
refinement of appearance, while he had gained in manliness. What
Carlos saw was an overgrown lad of fifteen, shock-headed, freckled,
with arms and legs two sizes too large for his width of chest. A face
that was brimful of fun and good nature, honest, gray eyes, and a
mouth wide and upturned at the corners over strong white teeth.
Yet, although he instantly liked him, Carlos had a feeling that one
could place little dependence upon Jack. As he afterwards expressed
it to Carlota: “A fellow anybody could make to do anything that was
pleasant but nothing that wasn’t.”
When they were dressed, Jack opened the door and pointed out the
bench beside the wall, where a tin basin and one coarse towel
served the needs of the entire family’s toilet.
“I see. Thank you. But isn’t there any stream near?”
“Stream? No. Not by a long shot. What you want a stream for?”
“Why, for my bath.”
“Are you so dirty? Anything the matter with you that you must wash
yourself?”
His face dripping from its plunge in the small basin, the guest looked
up, surprised.
“Nothing the matter, except that I haven’t dipped—all over—since I
left home and I’ve ridden miles and miles. Of course, I’m dirty. How
could I help being?”
Jack whistled.
“Whew! Is that the kind of a fellow you are? Well, then, the sooner
you get over such namby-pamby notions the better. This isn’t any
place for a ‘tenderfoot.’”
“I’m not a ‘tenderfoot’. I’m a born Westerner. But it’s neither decent
nor healthy not to keep your body clean,” retorted Carlos.
“It’ll be healthy for you not to put on any ‘frills’ here, my Young-
Fuss-and-Feathers! And there’s Ma calling us to breakfast. High time,
too. I’m hungry enough to eat hay.”
Carlos, also, was hungry; and anxious about Carlota, who had been
so tired on the previous night; so he hurried into the house. This
had but three rooms. The larger was the living room of the family
and the waiting place of what few passengers ever entered it. A
small desk, where Mr. Burnham kept his accounts, was in one corner,
and a table, covered by an oilcloth square, was in the middle.
A girl ran forward from behind this table and clasped Carlos in her
closest embrace. For a moment, he did not recognize her. The
golden curls which had always been simply brushed, then left to
nature’s will, were now put back in a rigid little braid that completely
altered the child’s appearance. Her picturesque garments had been
replaced by a blue print frock, ugly in shape and color; while the
clumsy skirt which draped her limbs gave them an awkwardness of
movement most unlike the hitherto graceful Carlota.
But it was she; and with her soft clinging arms about his neck and
her sunshiny smile greeting him, the lad realized that the world was
not yet empty of all he had held dear and familiar.
Jack looked on, amazed by this rapturous embrace between the
twins. In that household expressions of sentiment were rare, save
on the mother’s part toward little Teddy. She idolized him, and it was
he—sleeping rosy and tumbled in his “trundle,” whom she had
shown to Carlota, at bedtime, as “the prettiest sight in the world.”
But—kiss a girl? A boy—a sister? “Whe-e-ew!” said master Jack, and
whistled so loudly that the twins loosed their arms and looked at him
in surprise that was tinged with alarm. Their experience of yesterday
had left them both apprehensive of what might happen next.
“What’s the matter?” asked Carlos. Then, since no answer came, he
crossed to where Mrs. Burnham was dipping mush from a kettle and
gravely bowed over the hand she kindly extended.
“Good morning, Madam. I hope you have rested well.”
In her surprise, the poor lady nearly dropped the dish of hot
“suppawn.” She had already been touched and gladdened by the
earlier civilities of Carlota. They reminded her of a past that was
widely different from the present, and of a time when she, too, had
had time for the small amenities of life. But to have the lad, also,
remember to be courteous sent a faint flush to her cheek and a
grateful warmth to her heart.
“Thank you, yes. Fairly well. And you?”
Jack could no longer whistle. He had to sit down in order to properly
recognize this “airish” gentlewoman who had stepped into his
mother’s shoes, and he sat thus, staring, when Carlota discovered
him. She went directly to him and offered her hand in greeting,
saying:
“Good morning, Jack. Teddy has told me all about you and what a
splendid brother you are. I’m sorry I called you a bad boy, last night,
but I thought you were going to—to kill my brother.”
“Gophers! I guess he ain’t easy killed. I—” He hesitated and uneasily
glanced toward his mother. He had never felt so big and clumsy. He
thought the little girl wasn’t half as pretty in his sister’s clothes—My!
but the mother must have liked her to let her wear them!—as she
had been in that queer, Indian attire of her own. Wow! Wouldn’t that
plaguey breakfast ever be ready? In his most boyish manner he
demanded this and with so much more disrespect than usual that
Mrs. Burnham stared, then smiled to herself, as she quietly
answered:
“It is ready for those who, also, are ready. But you haven’t finished
your preparations, my son.”
“I’d like to know what you mean?”
She touched her head significantly and he understood. The family
comb had done faithful duty on Carlos’s curls, but Jack, wishing to
impress the “tenderfoot” by his own manly independence had, for
once, omitted that part of his toilet which Mrs. Burnham had,
hitherto, compelled. Affecting a rude disdain, he now slouched
forward to the table, but chanced to look at Carlota.
She still retained that wide, innocent outlook which commonly
belongs to earlier childhood, and her blue eyes regarded him with
open astonishment. He was a curiosity to her. She had never seen
anybody like him. He was on a par with the hand-car, the railway,
the Apaches—any and all of the novelties which confronted her in
this land of the strangers. All her life she had been accustomed to
the exaggerated courtesy of the Spanish dependents at Refugio and
to the exact politeness of her gentleman father, who believed in
example rather than precept. These little civilities were as natural as
the breath in her nostrils and, above all that, the name of “mother”
suggested a personality higher than mortal.
She exclaimed, in a low tone:
“He is saying that—to his—mother!”
“Yes. But, for his own sake and in justice, I will explain that he is
exhibiting himself in a new character,” observed that lady.
“Why?”
“Ah! why? Probably, my dear, because he is a—boy! A being whose
nature, as yet, is all at sixes and sevens; but who will arrive at a true
manhood, by and by, please God.”
As she spoke she smiled at her son and yet she sighed; and it was
due to Carlota that as he hastily left the room he returned an
answering smile. When, after a brief delay, he came back his head
had been deluged with soapsuds, which still trickled over his blue
jumper, and his shock of hair was plastered as smooth as if it had
been glued into place.
Then, while Mrs. Burnham bowed her head in the “silent grace” that
was a remnant of her former life, the sun stole through the window
of the cabin and touched their reverent forms with a glory all his
own. Even so the busy housemistress felt her heart brightened by
the presence of these young strangers and silently wondered:
“Are they ‘angels unawares’? I have a feeling that they will prove
such.”
CHAPTER XVII
THE BURNHAMS
When the simple breakfast was over, Mrs. Burnham bade the lads
remain indoors for a moment, saying:
“I want to explain to you, Carlos and Carlota, why I had you make
this change of clothing. One reason is, you will feel better for putting
on fresh, even if plainer garments. I do not suppose you always
wore the one costume when you were at home, did you?”
Carlota laughed and replied:
“No, indeed! We had plenty of changes, though all were made the
same.”
“So I judged. Also that, probably, there are no other children in this
country attired in just that fashion.”
“My father thought it was the very best sort of dress for us,”
returned the little girl, earnestly.
“Dear child, I do not doubt it for a moment, and I wish nobody need
be more hampered by their clothes than you have been. I see, too,
that those simple skirts you have on now are a burden to you, but
you’ll soon get used to them. Maybe soon, also, you’ll return to your
own home and habits.”
As she said this she sighed and Teddy shrewdly remarked:
“She doesn’t not believe it, though. She allays bweaves herself that-
a-way when she doesn’t not sepect fings.”
They all laughed and the mother exclaimed:
“Why, Teddy! How observant! Yet small boys are not the truest
prophets. There are other reasons why it is better you should keep
your kid garments—”
“Kid, kidder, kiddest. Kid garments, garments of a kid. A pair of
kids,” mumbled Jack who had, by this time, quite forgotten the silent
rebuke of Carlota’s eyes. Carlos heard the monologue and was
inclined to resent it but, instead, found himself listening to Mrs.
Burnham.
“So I will wrap them carefully, and mark them with your names and
addresses. You should keep them with you. The blankets and the
other things which the troopers left with you at Leopard are here in
my room. They may be useful to you, and aren’t apt to wear out
soon. The blankets are the finest I’ve ever seen, though some of the
Indians who pass here, on the march, have those nearly like them.”
“They were gifts to our mother. They were woven by some Navajo
women to whom she’d been kind. She was always kind to
everybody. My father says that she nursed the sick, gave drink to the
thirsty, food to the hungry, and rest to the weary. Oh! dear lady, I
think you must be like her!” cried Carlota, impulsively.
Letitia Burnham’s eyes filled. She had already taken the motherless
wanderer into her inmost heart and had welcomed her as a gift from
God which she was thankful to retain, even for a little time. Yet she
was greatly concerned for her small guests, knowing how slight a
thing may turn the current of a life, and how doubtful it was that
news of their whereabouts would at once bring their absent father.
Before he had retired, Mr. Burnham had promptly acted upon the
suggestions in Captain Sherman’s brief note. But, would the
telegrams and advertisements reach the eyes for which they were
intended? And the Captain’s information had been very scant.
Adrian Manuel had gone “north”, but nobody knew where. The
children had never been told the name of their great aunt, Mrs.
Sinclair, nor even, until the Disbrows’ arrival at Refugio, of her
existence. Then Miguel had spoken of her as “a wicked old woman”,
and had honestly considered her such—simply because she gave
annoyance to his master. Neither child had mentioned the lawyers
save as “enemies”, and the further cross-examination which the
Captain had intended making in the presence of the station-master—
for their mutual benefit—had been forgotten in his hurried departure
elsewhere. After he had parted from his little charges he had
remembered this fact, but trusted to the station-agent’s intelligence
to learn what more there was to know.
Moreover, for some time past, Mrs. Burnham had lived in expectation
of a removal, and this fact, added to the foregoing, made the
children’s future a doubtful one.
Household duties were simple in that narrow cabin, and though
there was always sewing to be done, that could be taken out of
doors. So, as soon as the place was in order, Mrs. Burnham took
Carlota’s hand and said:
“Come, I have one other treasure to show you. Bare as this isolated
station may seem to you I have learned to love it. We have lived
here for some years and were only temporarily at Leopard. I didn’t
wish to go there and was glad to come back, because of—that!”
Carlota’s gaze followed the pointing finger. At some little distance
from the cluster of buildings was a small heap of stones. Around the
heap there had been set a slender fence of tule reeds, strung
together by strips of the same growth. A cactus, larger than ordinary
and loaded with brilliant flowers, stood at one end of the enclosure
while at the other a struggling tree made a bit of shade. A rude shed
had been fixed beyond the spot, and within this a bench, whereon
Mrs. Burnham and Carlota now sat down.
“What lovely blossoms! We have some cacti of that kind in our
mother’s garden.”
“This one blooms—in my child’s.”
The girl looked up in surprise, but instantly understood. She slipped
her hand into the mother’s and softly asked:
“Was it long ago?”
“Five years. She would have almost been a woman now and I often
think what she would have been to me as such. Then I look abroad
and am glad she is not here to suffer the intolerable loneliness of the
plains. The young are not fond of solitude. Her name was Mary.”
“Why, my own mother’s name! I’m glad of that. Maybe she wouldn’t
have suffered. Anyway, you know she doesn’t now. My father says
that though our mother was so very, very happy on this earth, she is
far happier now, with God.”
“Dear little comforter, so I try to believe of my own daughter,” said
the woman, laying her hand on Carlota’s head.
“Was that her cactus?”
“Yes. She set it here. Her father planted the tree, which a brakeman
brought her from a distant station. There were other things here,
too, but they are gone. At first, I felt too desolate to care for them,
and, when I had rallied so that I could, it was too late. Yet since it
was here that she had made her garden it was here I had her put to
rest.”
Carlota looked curiously at the stones. Away off in the distance, also,
she could see beside the track another of the dead cattle which had
so frightened her while on the hand-car. Mrs. Burnham noticed the
glance and answered it:
“We had to put the tule reeds as a precaution against the coyotes.”
The girl shivered and exclaimed:
“How dreadful! Yet the stones don’t make any difference. The dear
God knows about her, just the same. And—and—the cactus is very
beautiful.”
“Yes, dear, yes. ‘The cactus is very beautiful.’ There is no life so dark
or barren but may have its cactus bloom. Now since you have told
me all about yourself I’ll tell you what is needful you should know
about this Burnham family. It will do for the ‘story’ that Teddy is
always begging.”
She smiled upon the little man who now approached, with his fat
hands full of a rare yellow blossom which he offered to Carlota.
“Posies, girl, for you.”
“Thank you, niño. You’re a darling, darling baby! I do love flowers
better than—than almost anything, I guess.”
Teddy climbed up beside her and watched as, almost unconsciously,
she began to pull one of the strange blooms to pieces.
“Girl! No, no! You mustn’t bweak them. It might hurt them, my
muvver says.”
“I’m not hurting it, Teddy. See? Little by little, I take it apart, gently,
for I’m trying to find out its name. Though, maybe, Señora, you
know it?”
“No. I’ve not seen them often. But—at your age—do you understand
botany?”
“I don’t understand it—much. Only, most always, if I have a new
plant I can find its class and, often, its genus. Like this. Don’t you
know?”
“Once I knew. I was a teacher in my girlhood.” The sight of the child
analyzing the desert flower had carried the exile’s thoughts back
over many years, to a pleasant New England schoolroom and a class
of eager maidens who learned from her. Yet she promptly banished
her momentary regret, reflecting: “The cactus is very beautiful! And
my blessings do outnumber my deprivations.”
It was a wonderfully skillful young hand which dissected the
unknown flower and, when it lay with all its parts separated and
arranged, Mrs. Burnham’s interest was as great as Teddy’s. Eager to
see, he thrust his dark head between Carlota and her “subject” in a
way that hindered her study, so she left its finishing until another
time.
“I’ll put the rest of the blossoms in my box, Teddy, but I’m quite sure
it’s an orchid. I think it is a ‘Plantanthera.’ I do miss my father so

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New Literacies and Teacher Learning Professional

Primary Mathematics 3A Hoerst

Handbook of Research on Education Institutions, Skills,

Introduction to Crowd Management: Managing Crowds in

International Perspectives on English Teacher

Digital Lawyering : Technology and Legal Practice in

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The Teacher Educator s Handbook A narrative approach to

Editorial Board Members

The Mathematics Teacher

ISSN 2211-8136 ISSN 2211-8144 (electronic)

1st edition: © Springer Science+Business Media Dordrecht 2014

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